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https://math.stackexchange.com/questions/989308/which-of-the-following-processes-are-markov-chains | # Which of the following processes are Markov chains?
A dice is thrown an infinite number of times. Which of the following procsses are Markov chains or not? Justify your answer. For those processes that are Markov chains give the transition probabilities. For $n\in\mathbb{N}$ consider
(1) $U_n$ to be the maximal number shown up to time $n$.
(2) $V_n$ to be the number of times a 6 appears within the first $n$ throws.
(3) $X_n$ to be the time which has passed since the last appearance of a $1$.
(4) $Y_n$ to be the time which will pass until the next $4$ appears.
A complete edit of my ideas
In my opinion all are Markov processes.
(a) is a Markov provess with state space $E=\left\{1,2,3,4,5,6\right\}$, because the Markov property is fullfilled:
In order to predict $U_{n+1}$ I only need to know the maximal number shown up to time $n$, i.e. $U_n$, because then I know that $U_{n+1}$ is either one of the numbers that are $> U_{n}$ or it is equal to $U_n$. I do not have to know $U_1,...,U_{n-1}$.
The transition matrix is $$P=\begin{pmatrix}1/6 & 1/6 & 1/6 & 1/6 & 1/6 & 1/6\\0 & 1/3 & 1/6 & 1/6 & 1/6 & 1/6\\0 & 0 & 1/2 & 1/6 & 1/6 & 1/6\\ 0 & 0 & 0 & 2/3 & 1/6 & 1/6\\ 0 & 0 & 0 & 0 & 5/6 & 1/6\\ 0 & 0 & 0 & 0 & 0 & 1\end{pmatrix}$$
(b) is a Markov chain with state space $E=\mathbb{N}\cup\left\{0\right\}$; the Markov property is fullfilled:
In order to predict $V_{n+1}$ I only have to know the number of times a 6 appears within the first $n$ throws, i.e. I have to know $V_n$, because then I know that $V_{n+1}$ is either $V_{n}+1$ or remains $V_n$.
Here the transition probabilities are $$p_{ij}=\begin{cases}1/6, & j=i+1\\5/6, & i=j\end{cases}.$$
(c) is a Markov chain with state space $E=\mathbb{N}\cup\left\{0\right\}$. Again the Markov property is fullfilled:
If I want to predict $X_{n+1}$ (which I do understand as the time at throw number $n+1$ that has passed since the last 1 appeared), I only need to know the time that has passed at throw number $n$ since the last 1 appeared. Then I can predict that $V_{n+1}=0$ (if I throw a 1 at throw number $n+1$) or $V_{n+1}=V_n+1$.
So to my opinion the transition probabilities are given by $$p_{ij}=\begin{cases}1/6, & j=0\\5/6, & j=i+1\end{cases}.$$
(d) Here I am not that sure. But I think here we have a Markov chain, too. Because I think here we do not need the past throws at all, because the time to the next throw of a $4$ does not depend at all from the past throws. So it makes no difference if we conditional on the whole past or only on $Y_n$ in order to predict $Y_{n+1}$. So the Markov property is fullfilled.
To my opinion the transition probabilities are $$p_{ij}=\begin{cases}1/6, & j=1\\5/6, & j=i+1\end{cases}.$$
Would be really nice to hear from you, if I am right or not.
Greetings
• Aren't your results and drawing past the horizontal line about the third process, and not the fourth? – zarathustra Oct 24 '14 at 16:42
• If (a,b,c,d)=(1,2,3,4), the current answers are correct. The justification of answer d/4 is bogus, though, a full answer requires to make explicit the filtration for which Y is Markov. Let me suggest to think again about the transition matrix in case d/4. – Did Oct 31 '14 at 8:57
You say "because the future depends on the past," but this is not the correct characterization of a Markov Chain: for a Markov chain if you condition on the past, it only depends on the last value. So for $(2)$, $V_n$ conditioned on $V_{n-1},\cdots,V_1$, clearly depends only on $V_{n-1}$. Afterall, the number of 6's that occur in n throws satisfies $V_n=V_{n-1}+T_n$, where $T_n=1$ if the n'th throw is a 6 and 0 otherwise. Since dice throws are independent, $T_n$ is independent of $V_{n-1},\cdots,V_1$, making $P(V_n|V_{n-1},\cdots,V_1)=P(V_n|V_{n-1})$.
The same goes for (3). Since your throws are independent, this is what's called a renewal process: every time you throw a 1, it resets and is indepedent of everything prior. If i tell you that 20 minutes have passed since the last 1, then you don't need to know what happened 19 minutes ago, because the next throw is independent of the entire past, so if you want the new time passed since the last throw, you need only the last time update. In math language, the time $\tau_n$ since the last 1 was thrown has the following property: $\tau_n-\tau_{n-1}$ is independent of all $\tau_{i}-\tau_{i-1}$ for $i=1,2,\cdots,n-1$.
• Thank you, you are right. I did not use the Markov property correctly. Now I see that (2) is a Markov chain, too. And then (3) is a Markov chain, too. If I condition on the past I only need the last value. Right? But am I right, that (3) and (4) do have the same transition probabilites? – mathfemi Oct 24 '14 at 17:18
• Ok, I edit my post. Pls have a look on it then. – mathfemi Oct 24 '14 at 17:29
• I changed my opinion, I think (1) is a Markov chain, too. Because i only need the last maximal number shown up to time n, i.e. $U_n$, to predict $U_{n+1}$. Right? – mathfemi Oct 30 '14 at 19:16
• Please have a look on my edited question/ ideas. They changed. – mathfemi Oct 30 '14 at 19:50 | 2019-05-24T13:22:09 | {
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http://openstudy.com/updates/4e82a80a0b8bc11dd54781c1 | ## xEnOnn Group Title How do I solve for t in the equation $\frac{t(t^2-2)}{t^2-1}=0$ I know the answers are +-square root of 2 and 0 but that's after I punch it into my calculator. Could you show me the steps if I want to evaluate this by hand? Thanks! 3 years ago 3 years ago
1. snoopshoag
Since the equation is equal to zero, only the numerator is important.
2. xEnOnn
Why only the numerator is important?
3. snoopshoag
If t=0, the entire equation will equal 0, likewise, if t^2-2=0 the entire equation will equal 0.
4. jim_thompson5910
If A/B = 0, then multiply both sides by B to get A = B*0 ---> A = 0
5. jim_thompson5910
that's why only the numerator matters
6. jim_thompson5910
besides, you can't divide by zero
7. xEnOnn
don't I have to check for $t^2-1 \neq 0$?
8. jim_thompson5910
yes it's best to find out which numbers make the denominator zero first
9. xEnOnn
That would give t=1 or t=-1. Then the solution to t would t=+-sqrt(2), t=0 and t not equals +-1?
10. jim_thompson5910
With the solution, you only need to worry about what t equals (instead of what it can't equal)
11. snoopshoag
Hey jim, sorry to butt in on this conversation, but do you know how to create a fraction in the equation editor? It is very difficult to give coherent answers without fractions.
12. jim_thompson5910
type \frac{x}{y} to type out $\Large \frac{x}{y}$
13. jim_thompson5910
for some reason, this is missing in the equation editor...
14. snoopshoag
Thanks!
15. jim_thompson5910
np
16. xEnOnn
Suppose another function call it E(t) holds only if the above equation of $\frac{t(t^2-2)}{t^2-1} \neq 0$, Then, for E(t) to hold, t must be $t \neq \pm \sqrt{2}$ $t \neq 0$ But then what if the function was given -1 or 1 into it? The whole thing would not work since it will be division by zero, wouldn't it?
17. xEnOnn
As in, since the condition that I have got from the equation, was just $t \neq \pm \sqrt{2} \; \; \; and \; \; \; t \neq 0$ Then wouldn't there be a possibility/danger that someone throws in a value of t which is -1 or 1?
18. jim_thompson5910
yes that possibility exists, but you just ignore those potential solutions if it arises
19. snoopshoag
Correct, the function would not work if $t=\pm1$, one way to avoid this would be to define the domain, so t=$(-\infty,-1),(-1,1),(1,\infty)$
20. xEnOnn
Thanks everyone! :)
21. snoopshoag
You're welcome. | 2014-10-23T11:42:40 | {
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https://math.stackexchange.com/questions/3438498/how-to-find-the-second-derivative-of-y-in-y2-x2-2x | # How to find the second derivative of y in $y^2 = x^2 + 2x$?
I have a problem to solve:
use implicit differentiation to find $$\frac{dy}{dx}$$ and then $$\frac{d^2y}{dx^2}$$. Write the solutions in terms of x and y only
It means that I need to differentiate the equation one time to find $$y'$$ and then once more to find $$y''$$.
The correct answer from the textbook is $$y' = \frac{x + 1}{y}$$ and $$y'' = \frac{x^2 + 2x}{y^3}$$. I got the first derivative right, but I can't understand how did they get the second one, or is it a typo (unlikely), since I have $$y'' = \frac{1}{y} - \frac{(x + 1)^2}{y^3}$$
I did this:
$$y^2 = x^2 + 2x\\ 2yy' = 2x + 2\\ yy' = x + 1\\ y' = \frac{x + 1}{y}\\$$
I tried to get to the second derivative from both $$yy' = x + 1$$, $$y' = \frac{x + 1}{y}$$ and $$2yy' = 2x + 2$$. But every time I had that dangling constant (1 or 2), which lead to the dangling $$\frac{1}{y}$$ in my answer.
Like here:
$$yy' = x + 1\\ y'y' + yy'' = 1\\ yy'' = 1 - (y')^2\\ y'' = \frac{1 - (y')^2}{y}\\ y'' = \frac{1}{y} - \frac{(y')^2}{y}\\ y'' = \frac{1}{y} - \frac{(\frac{x + 1}{y})^2}{y}\\ y'' = \frac{1}{y} - \frac{(x + 1)^2}{y^3}$$
I don't see any way to get from my answer to the textbook's one with a transformation, no way to get rid from y in the numerator. And the correct answer doesn't have a "y" there.
Could someone either point to an error in my solution, or corroborate the suspicion that it indeed may be a typo.
• I think it is a typo. Recall that you have $y^2=x^2+2x$, so $$y'' = \frac{y^2 - (x+1)^2}{y^3} = \frac{2x+x^2 - (x+1)^2}{y^3} = \frac{-1}{y^3}.$$ – Math1000 Nov 16 '19 at 21:00
• this must be the first typo I encountered in that textbook in 200 pages, which looks like a very low amount, I would say the textbook is of exceptional quality – user907860 Nov 16 '19 at 21:05
• What is the textbook? Perhaps there is a list of errata online somewhere. – Math1000 Nov 16 '19 at 21:06
• thomas' calculus 14th edition – user907860 Nov 16 '19 at 21:06
• As shown in @zwim's answer, the correct result is $y' = \frac{-1}{y^3}$. You could always email the textbook's author with a link to this page and they would likely fix it for the 15th edition. – Math1000 Nov 16 '19 at 21:16
From $$y'y'+yy''=1$$ multiply by $$y^2$$.
Then $$(yy')^2+y^3y''=(x+1)^2+y^3y''=y^2=x^2+2x \iff y^3y''=x^2+2x-x^2-2x-1=-1$$
$$y''=\dfrac 1y-\dfrac{(x+1)^2}{y^3}=\dfrac{y^2-(x+1)^2}{y^3}=\dfrac{(x^2+2x)-(x^2+2x+1)}{y^3}=\dfrac{-1}{y^3}$$
Gives the same result, so I guess the textbook result is erroneous (i.e. it gives $$yy''=1$$ which does not agree with derivatives of $$\pm\sqrt{x^2+2x}$$) | 2020-02-22T20:47:22 | {
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http://www.eg.bucknell.edu/~phys310/jupyter/ode_sympy.html | ## Symbolic solution of ODEs with sympy¶
Intro to sympy variables in previous notebook.
In [1]:
import sympy as sym
sym.init_printing() # for LaTeX formatted output
import scipy as sp
import matplotlib as mpl # As of July 2017 Bucknell computers use v. 2.x
import matplotlib.pyplot as plt
# Following is an Ipython magic command that puts figures in the notebook.
# For figures in separate windows, comment out following line and uncomment
# the next line
# Must come before defaults are changed.
%matplotlib notebook
#%matplotlib
# As of Aug. 2017 reverting to 1.x defaults.
# In 2.x text.ustex requires dvipng, texlive-latex-extra, and texlive-fonts-recommended,
# which don't seem to be universal
# See https://stackoverflow.com/questions/38906356/error-running-matplotlib-in-latex-type1cm?
mpl.style.use('classic')
# M.L. modifications of matplotlib defaults using syntax of v.2.0
# Changes can also be put in matplotlibrc file, or effected using mpl.rcParams[]
plt.rc('figure', figsize = (6, 4.5)) # Reduces overall size of figures
plt.rc('axes', labelsize=16, titlesize=14)
plt.rc('figure', autolayout = True) # Adjusts supblot parameters for new size
In [2]:
x = sym.symbols('x')
f, g = sym.symbols('f g', cls=sym.Function)
In [3]:
f(x)
Out[3]:
$$f{\left (x \right )}$$
#### Define the differential equation as a sym.Eq()¶
In [4]:
diffeq = sym.Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sym.sin(x))
diffeq
Out[4]:
$$f{\left (x \right )} - 2 \frac{d}{d x} f{\left (x \right )} + \frac{d^{2}}{d x^{2}} f{\left (x \right )} = \sin{\left (x \right )}$$
#### Solve differential equation¶
In [5]:
soln = sym.dsolve(diffeq,f(x))
soln
Out[5]:
$$f{\left (x \right )} = \left(C_{1} + C_{2} x\right) e^{x} + \frac{1}{2} \cos{\left (x \right )}$$
#### Boundary conditions¶
This isn't implemented yet in dsolve -- it's on the "to do" list
For now, solve for contants on your own. For example, if $$f(0) = 1\quad\mbox{and}\quad\left.\frac{df}{dx}\right|_0 = 0,$$ solve the following equations:
In [6]:
constants = sym.solve([soln.rhs.subs(x,0) - 1, soln.rhs.diff(x,1).subs(x,0)- 0])
constants
Out[6]:
$$\left \{ C_{1} : \frac{1}{2}, \quad C_{2} : - \frac{1}{2}\right \}$$
In [7]:
C1, C2 = sym.symbols('C1,C2')
soln = soln.subs(constants)
soln
Out[7]:
$$f{\left (x \right )} = \left(- \frac{x}{2} + \frac{1}{2}\right) e^{x} + \frac{1}{2} \cos{\left (x \right )}$$
#### Convert soln to python function for numerical evaluation/plotting¶
I'm not sure why I had to specify the modulue for conversion of sympy functions.
See http://docs.sympy.org/latest/modules/utilities/lambdify.html
In previous examples, sympy figured out a good module "on its own."
In [8]:
func = sym.lambdify(x,soln.rhs,'numpy')
In [9]:
xx = sp.arange(-1,1,.01) # name = xx so it won't collide with symbol x
y = func(xx)
plt.figure(1)
plt.plot(xx,y);
### Version Information¶
version_information is from J.R. Johansson (jrjohansson at gmail.com)
See Introduction to scientific computing with Python:
http://nbviewer.jupyter.org/github/jrjohansson/scientific-python-lectures/blob/master/Lecture-0-Scientific-Computing-with-Python.ipynb
If version_information has been installed system wide (as it has been on linuxremotes), continue with next cell as written. If not, comment out top line in next cell and uncomment the second line.
In [10]:
%load_ext version_information
#%install_ext http://raw.github.com/jrjohansson/version_information/master/version_information.py
Loading extensions from ~/.ipython/extensions is deprecated. We recommend managing extensions like any other Python packages, in site-packages.
In [11]:
version_information sympy, scipy, matplotlib
Out[11]:
SoftwareVersion
Python3.6.1 64bit [GCC 4.4.7 20120313 (Red Hat 4.4.7-1)]
IPython6.1.0
OSLinux 3.10.0 327.36.3.el7.x86_64 x86_64 with redhat 7.2 Maipo
sympy1.1
scipy0.19.1
matplotlib2.0.2
Tue Aug 01 11:22:32 2017 EDT
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https://math.stackexchange.com/questions/3043176/why-my-process-is-wrong-how-many-ways-are-there-to-choose-5-questions-from-th | # Why my process is wrong:-How many ways are there to choose $5$ questions from three sets of $4$, with at least one from each set?
Question
A question paper on mathematics contains $$12$$ questions divided into three parts A, B and C, each containing $$4$$ questions. In how many ways can an examinee answer $$5$$ questions, selecting at least one from each part.
Attempt
Firstly, I selected three questions (one from each part) and it can be done $$4 \cdot 4 \cdot 4$$ ways. And hence the remaining two positions for two questions can be given in $$^9{\mathrm C}_2$$ since there is no restrictions now. So, total ways is $$36 \times 64=2304$$.
But, in the answer given in the solution manual is $$624$$. And the process described is:
A B C
1 1 3
1 2 2
which is then arranged for part to be $$3 \times (4 \times 4 \times 4 + 4 \times 6 \times 6) = 624$$.
Why my process is incorrect? I understand the second solution but, unable to find any fault in my attempt. Please explain.
• "The process described is:" Are you sure that table is the only description? I would have no idea what it means if I didn't already know how to find the answer. – JiK Dec 16 '18 at 23:33
• @JiK I have copied as it was written in the solution manual. If you want to see the pic, I can upload it. – jayant98 Dec 17 '18 at 4:42
• Can you give a more appropriate title maybe? – Asaf Karagila Dec 17 '18 at 15:58
• @Asaf Well, I will think the title is making others tempting to see what's in it as it maintains somewhat secrecy and that increase the chance of getting this question answered and hence, it also increases the chance of getting more quality answer. And I would like this to remain as it is. – jayant98 Dec 17 '18 at 16:44
• This is not a clickbait website. Please make your title better. – Asaf Karagila Dec 17 '18 at 16:48
Let's compare your method with the correct solution.
Solution: As you noted, either three questions are selected from one section with one each from the other two sections or two questions each are selected from two sections and one is selected from the remaining section.
Three questions from one section and one question each from the other two sections: Select the section from which three questions are drawn, select three of the four questions from that section, and select one of the four questions from each of the other two sections. This can be done in $$\binom{3}{1}\binom{4}{3}\binom{4}{1}\binom{4}{1}$$ ways.
Two questions each from two sections and one question from the remaining section: Select the two sections from which two questions are drawn, select two of the four questions from each of those sections, and select one of the four questions from the other section. This can be done in $$\binom{3}{2}\binom{4}{2}\binom{4}{2}\binom{4}{1}$$ ways.
Total: Since the two cases are mutually exclusive and exhaustive, there are $$\binom{3}{1}\binom{4}{3}\binom{4}{1}\binom{4}{1} + \binom{3}{2}\binom{4}{2}\binom{4}{2}\binom{4}{1} = 624$$ ways to select five questions so that at least one is drawn from each of the three sections.
You are counting each selection in which three questions are drawn from one section and one question is drawn from each of the other sections three times, once for each way you could designate one of the three questions as the question that is drawn from that section. For example, suppose questions $$A_1, A_2, A_3, B_1, C_1$$ are selected. You count this selection three times.
$$\begin{array}{c c} \text{designated questions} & \text{additional questions}\\ \hline A_1, B_1, C_1 & A_2, A_3\\ A_2, B_1, C_1 & A_1, A_3\\ A_3, B_1, C_1 & A_2, A_3 \end{array}$$
You are counting each selection in which two questions each are drawn from two sections and one question is drawn from the other section four times, two times for each way you could designate one of the two question from each section from which two questions are drawn as the question that is drawn from that section. For instance, if questions $$A_1, A_2, B_1, B_2, C_1$$ are drawn, your method counts this selection four times.
$$\begin{array}{c c} \text{designated questions} & \text{additional questions}\\ \hline A_1, B_1, C_1 & A_2, B_2\\ A_1, B_2, C_1 & A_2, B_1\\ A_2, B_1, C_1 & A_1, B_2\\ A_2, B_2, C_1 & A_1, B_1 \end{array}$$
Notice that $$\binom{3}{1}\color{red}{\binom{3}{1}}\binom{4}{3}\binom{4}{1}\binom{4}{1} + \binom{3}{2}\color{red}{\binom{2}{1}}\binom{4}{2}\color{red}{\binom{2}{1}}\binom{4}{2}\binom{4}{1} = 2304$$
To select just three questions, one from each section, you are correct that there are $$4\times 4\times 4$$ ways to do it.
Now if you choose two more questions from the remaining $$9,$$ there are (using various notations) $$9C2 = \binom92 = {}^9C_2 = 36$$ possible ways to do that step. If you consider each resulting list of questions to be "different", then you would have $$64\times 36 = 2304$$ possible ways.
The thing is, as far as this question is concerned, not every sequence of questions you chose according to your method is "different" from every other. For example, choosing questions A1, B2, and C1 for the three questions (one from each section), and then A2 and A3 for the remaining two out of nine, gives the same result as choosing questions A3, B2, and C1 and then choosing A1 and A2 for the remaining two out of nine. In both cases the examinee answers A1, A2, A3, B2, and C1.
• Oh. Understood your point. Thanks for clearing my doubt. Thank you very much! – jayant98 Dec 16 '18 at 21:42
• @David K you're right with the analysis with respect to the OP strategy. Question: how would you go around it? Can you show how would you solve it (the OP's answer = 624 make sense but I didn't get the strategy proposed on intuitive level). – adhg Dec 16 '18 at 21:58
• @adhg N.F. Taussig's answer explains the reasoning behind the $624$ answer in detail. The key insight to me is that there are two fundamentally different ways to distribute the five responses: 3 in one section, 1 in each other section; or 2 in one section, 2 in another, and 1 in the remaining section. You work out the number of possibilities in each case; then, since there are no possibilities that could have gotten counted in both cases, you can just add the two cases together. – David K Dec 16 '18 at 23:15
• @David K got it. – adhg Dec 16 '18 at 23:42
The problem with your method is that you are overcounting!
For example: if you choose question 1 from part A as one of the four from part A, but then later choose question 2 from part A as one of the two leftover, then you could end up with the same set of questions had you first chosen question 2 from A as one of the four from A, and then later question 1 from A as one of the two leftover questions. But your method counts this as two separate sets.
• So you are saying that if I select A1 question first and then A2 as one of the last two questions then we have count as A1.... A2.. And A2..... A1.. Separately but, this is the same situation situation. Is this you wanted to say? – jayant98 Dec 16 '18 at 21:39
• @jayant98 exactly! You got it. – Bram28 Dec 16 '18 at 22:52 | 2020-07-11T20:48:02 | {
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https://testbook.com/objective-questions/mcq-on-wave-speed-on-a-stretched-string--5eea6a1339140f30f369f073 | # The speed of transverse waves on a stretched string is given by v = √(T/X), where 'T' is tension in the string and the unknown term 'X' is?
1. Linear mass density
2. Bulk Modulus of the medium
3. Young's Modulus of the medium
4. Density of the gas
Option 1 : Linear mass density
## Wave Speed on a Stretched String MCQ Question 1 Detailed Solution
CONCEPT:
• Simple Harmonic Motion (SHM): Simple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
• Example: Motion of an undamped pendulum, undamped spring-mass system.
The speed of a transverse waves on a stretched string is given by:
$${\rm{v}} = \sqrt {\frac{{\rm{T}}}{{\rm{\mu }}}}$$
Where v is the velocity of the wave, T is the tension in the string; μ is mass per unit length.
EXPLANATION:
The speed of transverse waves on a stretched string is given by v = √(T/X).
1. Here X is mass per unit length or linear density of string. So option 1 is correct.
2. Bulk modulus of elasticity (B): It is the ratio of Hydraulic (compressive) stress (p) to the volumetric strain (ΔV/V).
3. Young’s modulus: Young's modulus a modulus of elasticity, applicable to the stretching of wire, etc., equal to the ratio of the applied load per unit area of the cross-section to the increase in length per unit length.
4. Density: The mass per unit volume is called denisty.
# If the mass of 0.72 m long steel wire is 5.0 × 10–3 kg then the speed of produced transverse waves on the wire under 60 N tension in the wire is
1. 63 m/s
2. 73 m/s
3. 93 m/s
4. 39 m/s.
Option 3 : 93 m/s
## Wave Speed on a Stretched String MCQ Question 2 Detailed Solution
CONCEPT:
• Transverse wave: The Wave generated such that the particles oscillate in the direction perpendicular to the propagation of the wave is called the Transverse wave.
• The transverse wave can be observed when we pull a tight string a bit
• The Speed of Such Transverse wave is given as:
$$v = \sqrt{\frac{T}{\mu}}$$
Where T is Tension in the tight String and μ is mass per unit length of the string.
CALCULATION:
Given,
Tension in String T = 60N
Mass of String m = 5.0 × 10–3 kg
Length of String l = 0.72 m
Mass Per Unit Length of String $$\mu = \frac{5\times 10^{-3}kg}{0.72m}$$
⇒μ = 6.67 × 10-3 kg
Speed $$v = \sqrt{\frac{T}{\mu }}$$
⇒ $$v = \sqrt{\frac{60}{6.67\times10 ^{-3} }} m/s$$
Solving it, we will get the approx value of v = 93 m/s.
So, Option 3 is the correct answer.
# A string of length 2 m has 10 g mass. The speed of the simple harmonic wave produced in it is 40 m/s. The tension in the string is
1. 12 N
2. 6 N
3. 8 N
4. 4 N
Option 3 : 8 N
## Wave Speed on a Stretched String MCQ Question 3 Detailed Solution
CONCEPT:
• Simple Harmonic Motion (SHM): Simple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
• Example: Motion of an undamped pendulum, undamped spring-mass system.
$${\rm{v}} = \sqrt {\frac{{\rm{T}}}{{\rm{\mu }}}}$$ , where v is the velocity of the wave, T is the tension in the string; μ is mass per unit length.
CALCULATION:
Given length l = 2 m, mass of 2 m string m = 10 g, wave velocity v = 40 m/s.
$${\rm{\mu }} = \frac{{\rm{m}}}{{\rm{l}}} = \frac{{10}}{2} =$$ 5 g/m = 5 × 10-3 kg/m. from formula,
$${\rm{v}} = \sqrt {\frac{{\rm{T}}}{{\rm{\mu }}}}$$ ;
$$40 = \sqrt {\frac{T}{{5 \times {{10}^{ - 3}}}}}$$ ⇒ 402 × 5 × 10-3 = 8 N.
T = 8 N.
# If tension of sonometer wire is made four times, then its frequency will change by a factor of:
1. 2
2. 4
3. 1/2
4. 6
Option 1 : 2
## Wave Speed on a Stretched String MCQ Question 4 Detailed Solution
CONCEPT:
• Fundamental frequency: It is the lowest frequency of a periodic waveform. It is also known as natural frequency.
The fundamental frequency of a sonometer wire:
$$f =\frac{1}{2l} . \sqrt{T\over μ}$$
where f is the fundamental frequency, l is the length of the wire, T is the tension in the wire, and μ is the mass per unit length.
EXPLANATION:
The frequency is given by:
$$f =\frac{1}{2l} . \sqrt{T\over μ}$$
Given that:
So new tension (T') = 4T
The new frequency is given by:
$$f' =\frac{1}{2l} . \sqrt{4T\over μ} =2 \times \frac{1}{2l} . \sqrt{T\over μ} = 2f$$
• Thus the frequency becomes 2 times. So option 1 is correct.
# The speed of transverse waves on a stretched string is given by v = √(T/X), where 'T' is tension in the string and the unknown term 'X' is?
1. Linear mass density
2. Bulk Modulus of the medium
3. Young's Modulus of the medium
4. Density of the gas
Option 1 : Linear mass density
## Wave Speed on a Stretched String MCQ Question 5 Detailed Solution
CONCEPT:
• Simple Harmonic Motion (SHM): Simple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
• Example: Motion of an undamped pendulum, undamped spring-mass system.
The speed of a transverse waves on a stretched string is given by:
$${\rm{v}} = \sqrt {\frac{{\rm{T}}}{{\rm{\mu }}}}$$
Where v is the velocity of the wave, T is the tension in the string; μ is mass per unit length.
EXPLANATION:
The speed of transverse waves on a stretched string is given by v = √(T/X).
1. Here X is mass per unit length or linear density of string. So option 1 is correct.
2. Bulk modulus of elasticity (B): It is the ratio of Hydraulic (compressive) stress (p) to the volumetric strain (ΔV/V).
3. Young’s modulus: Young's modulus a modulus of elasticity, applicable to the stretching of wire, etc., equal to the ratio of the applied load per unit area of the cross-section to the increase in length per unit length.
4. Density: The mass per unit volume is called denisty.
# Each of the two strings of length 51.6 cm and 49.1 cm are tensioned separately by 20 N force. Mass per unit length of both the strings is the same and equal to 1 g/m. When both the strings vibrate simultaneously the number of beats is:
1. 7
2. 8
3. 3
4. 5
Option 1 : 7
## Wave Speed on a Stretched String MCQ Question 6 Detailed Solution
The correct answer is option 1) i.e. 7.
CONCEPT:
• Frequency in a stretched string:
• A stretched string is always subjected to a tension force along the length of the string.
• When a stretched string is vibrated, it produces transverse waves.
The frequency of the transverse wave (ν) is related to the tension in the string as follows:
$$ν = \frac{1}{2L} \sqrt{\frac{T}{μ}}$$
Where L is the length of the string, T is the tension in the string and μ is the mass per unit length of the given string.
• Beats: Beats are the periodic fluctuations heard in the intensity of a sound when two sound waves of nearly identical frequencies interfere with one another.
• The number of beats is found from the difference in frequencies of two interfering waves
CALCULATION:
Given that: String 1 String 2 Length (l) l1 = 51.6 cm = 0.516 m l2 = 41.9 cm = 0.419 m Tension (T) T1 =20 N T2 = 20 N Mass per unit length (μ) μ1 = 1 g/m = 10-3 kg/m μ2 = 1 g/m = 10-3 kg/m
Frequency, $$ν = \frac{1}{2L} \sqrt{\frac{T}{μ}}$$
$$ν_1 = \frac {1}{2 \times 0.516} \sqrt{\frac{20}{10^{-3}}} =$$ 137 Hz
$$ν_2 = \frac {1}{2 \times 0.491} \sqrt{\frac{20}{10^{-3}}} =$$ 144 Hz
Beats = difference in frequencies = ν2 - ν1 = 144 - 137 = 7
# A uniform rope of mass m and length L hangs from a ceiling. The speed of transverse waves in the rope at a point at a distance x from the lower end of the rope is
1. $$\sqrt {2gx}\,$$
2. $$\sqrt{ \dfrac{gx}{2}}\,$$
3. $$2 \sqrt {gx}\,$$
4. $$\sqrt {gx}\,$$
Option 4 : $$\sqrt {gx}\,$$
## Wave Speed on a Stretched String MCQ Question 7 Detailed Solution
Concept:
• Simple Harmonic Motion (SHM): Simple harmonic motion is a special type of periodic motion or oscillation where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.
• Example: Motion of an undamped pendulum, undamped spring-mass system.
The speed of a transverse wave on a stretched string is given by:
$${\rm{v}} = \sqrt {\frac{{\rm{T}}}{{\rm{μ }}}}\,$$
Where v is the velocity of the wave, T is the tension in the string; μ is mass per unit length.
Calculation:
Let μ is the mass per unit length of the roap.
The speed of transverse waves on a stretched string is given by v = √(T/μ).
At a distance x from the lower end, we will find tension which will be
T = μ g x
So, using
$${\rm{v}} = \sqrt {\frac{{\rm{T}}}{{\rm{μ }}}}\,$$ (Speed of transverse wave on a string)
$${\rm{v}} = \sqrt {\frac{{\rm{\mu \,g\,x}}}{{\rm{μ }}}}\,$$
v =$$\sqrt {gx}\,$$
# A string of length (L) has a stationary wave. Find the frequency of the wave if it is 2nd harmonic and travelling with speed of v.
1. $$\frac{v}{L}$$
2. $$\frac{2v}{L}$$
3. $$\frac{v}{2L}$$
4. None of the above
Option 1 : $$\frac{v}{L}$$
## Wave Speed on a Stretched String MCQ Question 8 Detailed Solution
CONCEPT:
• The stationary wave is also known as the standing wave.
• It is a combination of two waves, with the same amplitude and the same frequency, moving in the opposite direction.
• It is a result of interference.
• Harmonics of an instrument: A musical instrument has a set of natural frequencies at which it vibrates when a disturbance is introduced into it.
• This set of natural frequencies are known as the harmonics of the instrument.
The Frequency of nth harmonic in the standing wave is given by:
$$ν_n=\frac{nv}{2L}$$
where n is nth harmonic, v is the speed of sound, L is the length string.
CALCULATION:
For second harmonic n = 2
Frequency of 2nd harmonic in the standing wave
$$ν=\frac{nv}{2L}$$
$$ν=\frac{2v}{2L}$$
$$ν=\frac{v}{L}$$
So the correct answer is option 1.
# Two identical nylon strings are held stretched and it was noted that the fundamental frequency of the strings is in the ratio 1 : 2. Then the ratio of tension in both the string is:
1. 1 : 2
2. 2 : 1
3. 1 : √2
4. 1 : 4
Option 4 : 1 : 4
## Wave Speed on a Stretched String MCQ Question 9 Detailed Solution
The correct answer is option 4) i.e. 1 : 4
CONCEPT:
• The fundamental frequency of a stretched string is given by the equation:
$$\nu =\frac{1}{2L} \sqrt {\frac{T}{μ}}$$
Where L is the length of the vibrating part of the string, T is the tension and μ is the linear density of the string.
EXPLANATION:
Given that:
The ratio of frequencies is v1 : v2 = 1 : 2
Since the wires are identical, they will have the same L and μ.
The fundamental frequency, ν ∝ √T
Ratio, ν1 : ν2 = √T1 : √T2
⇒ν12 : ν22 = T1 : T2
12 : 22 = 1 : 4
# 1A string of 7 metre length has a mass of 0.035 kg. If tension in the string is 60.5 N, then speed of a wave on the string is
1. 77 metre/sec
2. 102 metre/sec
3. 165 metre/sec
4. 110 metre/sec
Option 4 : 110 metre/sec
## Wave Speed on a Stretched String MCQ Question 10 Detailed Solution
Concept:
• Transverse wave: The wave generated such that the particles oscillate in the direction perpendicular to the propagation of the wave is called the transverse wave.
• The transverse wave can be observed when we pull a tight string a bit
• The Speed of Such Transverse wave is given as:
$$v = \sqrt{\frac{T}{\mu}}$$$v=\frac{T}{\mu }$
Where T is Tension in the tight String and μ is mass per unit length of the string.
Calculation:
Given length of string = 7m
Mass of string = 0.35 kg
Tension in string = 60.5 N
Mass per unit length
$$μ = \frac{0.35kg}{7m}$$
⇒ μ = 0.05
Speed of wave
$$v = \sqrt{\frac{T}{\mu}}$$
$$v = \sqrt{\frac{60.5}{0.05}}$$ | 2021-12-06T21:32:05 | {
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https://math.stackexchange.com/questions/1475442/can-a-unbounded-sequence-have-a-convergent-sub-sequence | # Can a unbounded sequence have a convergent sub sequence?
I have been using the Bolzano-Weierstrass Theorem to show that a sequence has a convergent sub sequence by showing that it is bounded but does that mean that if a sequence is not bounded then it does not have a convergent sub sequence?
The sequence I am struggling is $((-1)^n \log(n))$ for all $n$ in the natural numbers. Now because the sequence is unbounded I am unsure how to prove whether it does or does not have a convergent sub sequence?
Take the sequence: $$(0,1,0,2,0,3,0,4,0,5,0,6,\cdots)$$It is unbounded and it has a convergent subsequence: $(0,0,0,\cdots)$. The Bolzano-Weierstrass theorem says that any bounded sequence has a subsequence which converges. This does not mean that an unbounded sequence can't have a convergent subsequence. What we can conclude is that any unbounded sequence has at least one unbounded subsequence.
• Is there any way to prove that it has a convergent subsequence though? even if at least one in unbounded or are there simply now sequences which wont have at least one convergent sub sequence? – Chris Oct 11 '15 at 22:02
• @Chris the sequence $(1,2,3,4, \dots)$ does not have a convergent sub sequence. – Hetebrij Oct 11 '15 at 22:03
• Yup, Herebrij beat me to it. I guess that's the simplest example. – Ivo Terek Oct 11 '15 at 22:04
• Ah yes of course, I am still not sure how I can prove whether or not my sequence has a convergent sub sequence though. – Chris Oct 11 '15 at 22:08
An unbounded sequence can have a convergent subsequence. An example is $a_n = n$ for n odd, $0$ for n even.
The correct contrapositive of Bolzano-Weierstrass is: a sequence with no convergent subsequence is unbounded.
As for $(-1)^n \log n$, consider any real x. For $n\geq e^{|x|+1}$, $|(-1)^n \log n - x|\geq 1$, so no subsequence of the sequence converges to x.
You can use the fact that if $\lvert a_n\rvert\to\infty$, then $(a_n)$ has no convergent subsequence:
If $(a_n)$ has a convergent subsequence, say $\displaystyle\lim_{k\to\infty}a_{n_k}=L$, then $\displaystyle\lim_{k\to\infty}\lvert a_{n_k}\rvert=|L|$ and therefore $\lvert a_n\rvert\not\to\infty$ | 2019-06-18T16:58:44 | {
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http://mathhelpforum.com/advanced-statistics/226558-statistics-uniform-distribution.html | # Math Help - Statistics - Uniform Distribution
1. ## Statistics - Uniform Distribution
Let X1, X2, ... Xn be independent, uniformly distributed random variables on the interval [0, theta].
a.) Find the c.d.f. of Yn = max(X1, X2, ..., Xn).
b.) Find the p.d.f. of Yn = max(X1, X2, ..., Xn).
c.) Find the mean and variance of Yn = max(X1, X2, ..., Xn).
d.) Suppose that the number of minutes that you need to wait for a bus is uniformly distributed on the interval [0, 15]. If you take the bus five times, what is the probability that your longest wait is less than 10 minutes?
e.) Find the p.d.f. of Yr, the rth-order statistic, where r is an integer between 1 and n.
f.) Find the mean of Yr.
For part a, I got the c.d.f of Yn = [F(y)]^n
For part b, I got the p.d.f pf Yn = g(y) = n*[F(y)]^(n-1) * f(y)
I'm not sure if I am doing this right. And I'm lost for the rest of the problem with what to do.
2. ## Re: Statistics - Uniform Distribution
Originally Posted by AwesomeHedgehog
Let X1, X2, ... Xn be independent, uniformly distributed random variables on the interval [0, theta].
a.) Find the c.d.f. of Yn = max(X1, X2, ..., Xn).
b.) Find the p.d.f. of Yn = max(X1, X2, ..., Xn).
For part a, I got the c.d.f of Yn = [F(y)]^n
For part b, I got the p.d.f pf Yn = g(y) = n*[F(y)]^(n-1) * f(y)
I'm not sure if I am doing this right. And I'm lost for the rest of the problem with what to do.
if by $F(y)$ you mean $F_X(y)$ then yes (a) is correct. But you should be able to explicitly determine what $F_X(y)$ is given that $X_i$ is uniform on $[0, \theta]$. The same goes for part (b). You are given $f_X(y)$.
(c) you should know how to do, you found the mean and variance of a distribution in one of the other problems.
(d) just apply the results above.
(e) you can either try and work this out yourself or you can look it up on the web. There are plenty of derivations of this available. Working it out isn't that hard. Just think about what it means to be the ith order statistic in terms of the $X_i$'s.
(f) is like (c)
3. ## Re: Statistics - Uniform Distribution
For part (a) and (b) are you saying that I should do:
(a) P[ Yn ≤ y] = P[ X1 ≤ y, X2 ≤ y, ... , Xn ≤ y ] = P[ X1 ≤ y] * P[ X2 ≤ y] * ... * P[ Xn ≤ y] = [F(y)]^n
(b) f(y) = derivative of the cdf = n * [F(y)]^(n-1)
(c) Is the formula for the mean and variance the same as another problem that I did?
4. ## Re: Statistics - Uniform Distribution
Originally Posted by AwesomeHedgehog
For part (a) and (b) are you saying that I should do:
(a) P[ Yn ≤ y] = P[ X1 ≤ y, X2 ≤ y, ... , Xn ≤ y ] = P[ X1 ≤ y] * P[ X2 ≤ y] * ... * P[ Xn ≤ y] = [F(y)]^n
(b) f(y) = derivative of the cdf = n * [F(y)]^(n-1)
(c) Is the formula for the mean and variance the same as another problem that I did?
dude...
You're told that the X's are distributed uniform[0,$\theta$]. So what is $F_X(x)$ ? What is $f_X(x)$? Just plug these into what you've got to make it explicit for uniform rvs.
The formula for the mean and the variance is not exactly the same as the other problem but the method is.
You'll have $f_{Y_n}(y)$
$\large E[Y_n]=\displaystyle{\int_{-\infty}^{\infty}} y f_{Y_n}(y)~dy$
$\large Var[Y_n]=\displaystyle{\int_{-\infty}^{\infty}}\left(y-E[Y_n]\right)^2 f_{Y_n}(y)~dy$
5. ## Re: Statistics - Uniform Distribution
For part a, then
The cdf is:
{ 0 for x < 0
{ (x-0)/(θ -0) for 0 ≤ x < θ
{ 1 for x ≥ θ
The pdf is:
{ 1/(θ - 0) for 0 ≤ x ≤ θ
{ 0 for x < 0 or x > θ
6. ## Re: Statistics - Uniform Distribution
Originally Posted by AwesomeHedgehog
For part a, then
The cdf is:
{ 0 for x < 0
{ (x-0)/(θ -0) for 0 ≤ x < θ
{ 1 for x ≥ θ
what are the -0's ? do you mean $\dfrac{x}{\theta}$ ? if so good, that's right.
Note this is just the individual pdf. you still have to take this the nth power.
The pdf is:
{ 1/(θ - 0) for 0 ≤ x ≤ θ
{ 0 for x < 0 or x > θ
do you just mean $\dfrac{1}{\theta}$ ? I hope so because that's correct too.
so use these to find your CDF and pdf of the max which you already found the formula for in terms of
$F_X(y)$ and $f_X(y)$
7. ## Re: Statistics - Uniform Distribution
I put the minus zeros in there just because it's part of the formula, I know it wasn't really necessary to do that.
To find the cdf, would I do this:
Fx(y) = the integral from 0 to θ of x/θ dx?
And for the pdf:
fx(y) = the integral from 0 to θ of 1/θ dx?
I'm confused what you mean when you say max
8. ## Re: Statistics - Uniform Distribution
Originally Posted by AwesomeHedgehog
I put the minus zeros in there just because it's part of the formula, I know it wasn't really necessary to do that.
To find the cdf, would I do this:
Fx(y) = the integral from 0 to θ of x/θ dx?
And for the pdf:
fx(y) = the integral from 0 to θ of 1/θ dx?
I'm confused what you mean when you say max
remember the problem was originally to find the distribution of the max and min of n uniform rvs.
what you did above is just the cdf and pdf of a single uniform rv. You already calculated the formula for the cdf and pdf of the max and min in terms of these a few posts back. Just plug these into that formula.
9. ## Re: Statistics - Uniform Distribution
Oh, so for part (a):
cdf = [F(y)]^n = (x/θ)^n
and for part (b)
pdf = n * [F(y)]^(n-1) = n*(x/θ)^(n - 1)
10. ## Re: Statistics - Uniform Distribution
Is part (a) and (b) correct?
If so, then moving onto part (c) I got:
E(Y) = the integral from -∞ to ∞ of y*n*(x/θ)^(n - 1) dy = 1/2 *θ*n*x*(x/θ)^(n-2)
Var(Y) = the integral from -∞ to ∞ of (y - 1/2 *θ*n*x*(x/θ)^(n-2))^2 * n*(x/θ)^(n - 1) dy = something complicated
11. ## Re: Statistics - Uniform Distribution
Originally Posted by AwesomeHedgehog
Oh, so for part (a):
cdf = [F(y)]^n = (x/θ)^n
and for part (b)
pdf = n * [F(y)]^(n-1) = n*(x/θ)^(n - 1)
you left out a piece, look at your formula for the pdf again.
12. ## Re: Statistics - Uniform Distribution
For the pdf, do I also multiply it by 1/θ ?
So, it would be:
pdf = n*(x/θ)^(n - 1) * 1/θ
yessir
14. ## Re: Statistics - Uniform Distribution
Then
E(Y) = the integral from -∞ to ∞ of y*n*(x/θ)^(n - 1) dy = 1/2 *θ*n*x*(x/θ)^(n-2)
Var(Y) = the integral from -∞ to ∞ of (y - 1/2 *θ*n*x*(x/θ)^(n-2))^2 * n*(x/θ)^(n - 1) * 1/θ dy = I have the answer, but it is extremely complicated, so I'm not sure if it is correct.
15. ## Re: Statistics - Uniform Distribution
Originally Posted by AwesomeHedgehog
Then
E(Y) = the integral from -∞ to ∞ of y*n*(x/θ)^(n - 1) dy = 1/2 *θ*n*x*(x/θ)^(n-2)
Var(Y) = the integral from -∞ to ∞ of (y - 1/2 *θ*n*x*(x/θ)^(n-2))^2 * n*(x/θ)^(n - 1) * 1/θ dy = I have the answer, but it is extremely complicated, so I'm not sure if it is correct.
the integrals range over $[0,\theta]$
Page 1 of 2 12 Last | 2014-11-20T23:10:37 | {
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http://gjde.libreriaperlanima.it/continuous-exponential-growth-and-decay-word-problems-worksheet.html | # Continuous Exponential Growth And Decay Word Problems Worksheet
Exponential Growth and Decay Objective: To give students practice in solving the type of exponential growth and decay problems similar to those they will encounter on the AP Calculus exam. growth_and_decay_word_problems. 5 Exponential Functions Definition of an Exponential Function An exponential function is a function that can be represented by the equation f(x) = abx where a and b are constants, b > 0 and b ≠ 1. 5) , (3 , 53) 2) Year Population 1 2 10 5 20 12 30 22 40 25. Exponential Growth And Decay Word Problem. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay, the function values forming a geometric progression. xy 0 270 190 230 310 b. In real-life situations we use x as time and try to find out how things change exponentially over time. It measured…. 3) Ready, Set, Go Homework: Linear and Exponential. You can work with the person sitting next to you or individually. growth) _____exponential growth_____ 4. Write an exponential growth model for P, the population, after t years, where t = 0 represents the year 2000. Engaging math & science practice! Improve your skills with free problems in 'Word Problems – Logistic Growth Models' and thousands of other practice lessons. Then, solve the function and get the answer!. exponential In growth and decay problems (that is, problems involving a quantity increasing or decreasing), here’s. Compound interest: word problems N. Displaying all worksheets related to - Exponential Decay Problems. Exponential Growth and Decay MathBitsNotebook. But in the case of decrease or decay, the value of k will be negative. Test over Compound Interest and Exponential Growth and Decay Next Class 3/20: (HS) Lesson: Solving Exponential and Logarithmic Equations. We're a not-for-profit with the goal of changing education for the better by providing a free world-class education to anyone anywhere. In this exponential equations worksheet, learners solve 11 word problems in short answer and multiple choice format. Showing top 8 worksheets in the category - Exponential Word Problems. Exponential Growth and Decay Word Problems Worksheet Answers or Distance Word Problems Worksheet Gallery Worksheet for Kids In English. 35 or 35% 5. Exponential Decay B. Modeling Exponential Growth And Decay With Skittles. We are going to discuss several types of word problems. Does this function represent exponential growth or exponential decay? B. If the growth rate is 3. Viruses grow at crazy rates, just look at the 2020 Corona virus scare. exponential growth and decay worksheet answers linear function word problems worksheet images worksheet math for kids. 10 6 Practice Exponential Growth And Decay Worksheet For 10th. What is the annual rate of depreciation, the rate at which the car loses value?. growth) _____exponential growth_____ 4. In exponential growth, a population’s per capita (per individual) growth rate stays the same regardless of the population size, making it grow faster and faster until it becomes large and the resources get limited. Language Worksheets For Grade 1 Tags : 48 English As A Second Language Worksheets For Adults Image Inspirations Excelent Thanksgiving Math Worksheets Kindergarten Extraordinary Exponential Growth And Decay Worksheet. Exponential Growth and Decay Worksheet In the function: y = a(b)x, a is the y-intercept and b is the base that determines the direction of the graph and the steepness. 05), (1, 5. Unit 4 lesson 8 (Growth and Decay Textbook. How much money S4ill be in the account after 5 years? 2. We hope this graphic will likely be one of excellent reference. 5 Exponential Functions Definition of an Exponential Function An exponential function is a function that can be represented by the equation f(x) = abx where a and b are constants, b > 0 and b ≠ 1. The radioactive material is changing every instant. The key to understanding the decay factor is learning about percent change. Identifying Exponential Growth and Decay Determine whether each table represents an exponential growth function, an exponential decay function, or neither. Exponential growth formula Suppose the rate of change of some substance or quantity is proportional to the amount present, then the amount or numberA~t! at time t is given by A~t! 5 A 0 ekt (1. Biomedical engineers use them to measure cell decay and growth, and also to measure light intensity for bone mineral density measurements, the focus of this unit. Using negative power values results in fractions, and when these fractions have exponents applied to them we get “Decay”. Exercise 2It has been observed that a particular plant's growth is directly proportional to time. Distance, rate, time word problems Mixture word problems Work word problems One step equations Multi step equations Exponents Graphing exponential functions Operations and scientific notation Properties of exponents Writing scientific notation Factoring By grouping Common factor only Special cases Linear Equations and Inequalities Plotting. This instructional video contains examples of how to solve word problems on exponential growth and decay. Worksheet 3 Graphing exponential functions g(x) =- Hour Identify each transformation from the parent function of Tell if the function is a decay or growth function. It provides the formulas and equations / functi. Basic Growth And Decay Problems f. How can we write an exponential growth/decay equation and define its variables? Writing Exponential Growth and Decay Word Problems Exponential Growth Formula Exponential Decay Formula Carefully read the question to identify if it is a growth or decay. Honors Pre-Calculus 3. In an exponential growth (or decay) function, as x increases, the y-values grow (or decrease) by a constant percent. Algebra 1 - Exponential growth and decay word problems Exponential decay occurs in the same way when the growth rate is negative. 01)12ᵗ, y = (1. Linear Regression Exponential Equations Lessons Tes Teach. How much remains of 10 grams after one week? 4. Comments (-1) Compound Interest Worksheet. The equation for "continual" growth (or decay) is A = Pe rt, where "A", is the ending amount, "P" is the beginning amount (principal, in the case of money), "r" is the growth or decay rate (expressed as a decimal), and "t" is the time (in whatever unit was used on the growth/decay rate). The fu nction with the base of 4/3 will be exponential growth and the other function with a base of 6/5 will also be exponential growth. We tried to locate some good of Exponential Growth and Decay Word Problems Worksheet Answers Also How to solve Equations with Exponential Decay Functions image to suit your needs. How do you identify equations as exponential growth, exponential decay, linear growth or linear decay #f(x) = 9. Some of the worksheets for this concept are Exponential growth and decay, Exponential growth and decay work, Exponential growth and decay word problems, Honors pre calculus d1 work name exponential, Exponential growth and decay, Graphing exponential, Growth decay word problem key, Exponential growth and decay functions. Math Analysis Honors – Worksheet 32 Exponential Growth and Decay Tell whether the function is an exponential growth or exponential decay function, and find the constant percentage rate of growth or decay. Exponential function Suppose b is a positive number, with b 6= 1. 35 or 35% 5. March 2001 In his article Light Attenuation and Exponential Laws in the last issue of Plus, Ian Garbett discussed the phenomenon of light attenuation, one of the many physical phenomena in which the exponential function crops up. Use and identify exponential growth and decay functions. From a physics perspective, a continuous rate is more telling. In the case of a discrete domain of definition with equal intervals, it is also called geometric growth or geometric decay, the function values forming a geometric progression. is added to 1. Pre-Calculus M. What is the annual rate of depreciation, the rate at which the car loses value?. 5) , (3 , 53) 2) Year Population 1 2 10 5 20 12 30 22 40 25. Exponential Growth and Decay For constant growth or decay use: y = abX or An = Ao(l ± For continuous growth or decay use: A = Pe(tt) For compounding use: A = l. To review continuous functions, see page 62, Exercises 60 and 61. 2,950,000 people in 2000, determine the city’s population in 2008. Upon graduating from college, Ashley took a job with a starting salary of $40,000. Variations 1. By the way, concerning Exponential Growth and Decay Word Problems Worksheet, below we can see several similar images to complete your ideas. Exponential Model What is an Exponential Growth or Decay Function? Solve an Exponential Decay Word Problem Solve an Exponential Growth Word Problem Graph of an Exponential Decay Function Graph of an Exponential Growth Function. growth) _____exponential growth_____ 4. Example 3 The growth of a colony of bacteria is given by the equation, \[Q = {Q_0}{{\bf{e}}^{0. It provides the formulas and equations / functi. What is the annual rate of depreciation, the rate at which the car loses value?. You will also encounter radioactive growth and decay problems, which has a familiar formula: P = P 0ekt In this case, the variables are: P: Final amount, P. Reflection over the x -axis, vertical stretch, reflection over the y -axis, and vertical translation down 1 unit. Problems involving e may be used. 3 Explain and use the laws of fractional and negative exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay. Displaying top 8 worksheets found for - Exponential Decay. What is the annual rate of depreciation, the rate at which the car loses value?. • understand exponential functions; • be able to construct growth and decay models; • recognise graphs of exponential functions; • understand that the inverse of an exponential function is a logarithmic function; • be able to use logarithms to solve suitable equations; • be able to differentiate exponential and logarithmic functions. Exponential+Growthand+DecayWord+Problems+!! 8. See full list on onlinemath4all. 3 Explain and use the laws of fractional and negative exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay. So think that if 5% of the people leave then 95% of the people are staying each year. For each equation below, identify the equation as exponential growth or decay, its initial value, the growth (or decay) factor, and the growth (or decay) rate. 12/10/19: Growth & Decay word problems day 2 / IXL: X. Graph both functions. If "b" were greater than 1, then that'd signify a GROWTH, as opposed to DECAY. In fact, it is the graph of the exponential function y = 2 x. If a population of 2350 grows to 7000 in 32 years, what is the value of b? Round your answer. 9th exponential growth and decay May 5th In this lesson, we are continuing to practice exponential growth and decay. The actual values that may be plotted are relatively few, and an understanding of the general shape of a graph of growth or decay can help fill in the gaps. Exponential Growth B. Word problems used are fun, engaging, and relevant for the student. Exponential growth is the increase in number or size at a constantly growing rate. Some of the worksheets for this concept are Lesson reteach exponential functions growth and decay, Exponential growth and decay work, Work 2 7 logarithms and exponentials, 4 1 exponential functions and their graphs, Staar standards snapshot. Multiple Bank Account. Does this function represent exponential growth or exponential decay? B. The value in dollars of a car years from now is ( ). Exponential and Logarithmic Applications. Exponential Expressions Word Problems; Growth and Decay Word Problem Practice Worksheet/Packet Transformations of Quadratic Functions Review Practice/Word. For example, identify percent rate of change in functions such as y = (1. What does the P stand for? 12. Rearranging equations. You are already familiar with exponential equations. Exponential growth and decay often involve very large or very small numbers. 2,950,000 people in 2000, determine the city’s population in 2008. Exponential Growth and Decay Functions Exponential growth occurs when a quantity increases by the same factor over equal intervals of time. +250% Growth or decay e. Some examples are population, compound interest and charge in a capacitor. Exponential Growth and Decay Worksheet In the function: y = a(b)x, a is the y-intercept and b is the base that determines the direction of the graph and the steepness. decay exponential growth problems word; Home. Some of the worksheets for this concept are Growth decay word problem key, Exponential growth and decay work, Exponential growth and decay, Pc expo growth and decay word problems, Exponential word problems, Exp growth decay word probs, Word problems interest growthdecay and half life, Name algebra 1b date linear exponential continued. We will use separation of variables. Math Algebra 1 Exponential growth & decay Exponential expressions. 7) Ready, Set, Go Homework: Linear and Exponential Functions 9. Look at the graphs at the bottom of page 255. Complete the following practice problems: 1) Determine if the following exponential functions represent a growth or decay situation. Inspiring Exponential Functions Worksheet with Answers worksheet images. 7) x y y x 8) x y y x Create your own worksheets like this one with Infinite Precalculus. The radioactive material is changing every instant. If something decreases in value at a constant rate, you may have exponential decay on your hands. Upon graduating from college, Ashley took a job with a starting salary of$40,000. Mortgage Problems 3. Graphing Exponential Functions Practice Worksheet Name Period # Graph the following functions and tell whether they show exponential growth or decay. More information Graphing Exponential Functions Worksheets. Wed like a fresh concept for it and one of them is example of exponential form. How much money S4ill be in the account after 5 years? 2. 9918) indicates a DECAY. Exponential Growth. a) Exponential growth or decay: b) Identify the initial amount: c) Identify the growth/decay factor: d) Write an exponential function to model the situation: e) “Do” the problem: 2. The di⁄erential equation dy dt = ky with a positive constant k represents proportional growth and with a negative constant represents proportional decay. Determine when New Yorks's population will surpass Nevada's. 413) t where t is number of years since 1990 a. The graph shows how. We will also learn to simultaneous exponential equations from some. com OBJ(s): I CAN Solve exponential growth and decay word problems Identify and solve compound interest and continuous growth word problems. Solving Word Problems (Systems of Linear Equations) b. 25)^t in the form y=ae^kt. 4 Exponential Growth and Decay Worksheet 01 - HW Solutions Logistic Growth Notesheet 02 Completed Notes Logistic Growth Worksheet 02 Solutions Exponential and Logistic Growth Worksheet 02 - HW Solutions. exponential growth and decay worksheet answers linear function word problems worksheet images worksheet math for kids. 52 105 years. Assignment- Exponential Growth. Assignment- Exponential Growth- ANSWERS: 8 April 20 Growth and Decay Applications I can create an equation for exponential growth or decay by finding the initial value and growth factor. Exponential Growth And Decay Word Problem. How much of a 100 gram sample remains after 10,000 years? 2. Exponential Growth and Decay Worksheet In the function: y = a(b)x, a is the y-intercept and b is the base that determines the direction of the graph and the steepness. By the way, concerning Exponential Growth and Decay Word Problems Worksheet, below we can see several similar images to complete your ideas. In this tutorial, learn how to turn a word problem into an exponential decay function. A certain car depreciates about 15% each year. Compare LINEAR GROWTH to EXPONENTIAL GROWTH using graphs, data, or equations 3. Exponential expressions word problems (algebraic) Practice: Exponential expressions word problems (algebraic) This is the currently selected item. 8% Growth or decay. The following tables give the formulas for Simple Interest, Compound Interest, and Continuously Compounded Interest. 24 #13, 14, 16, 19, 20a. Displaying all worksheets related to - Growth And Decay Word Problems. Homework: review for quiz (quiz topics HERE) Mar 26 - Sine Function Quiz, Exponential Growth. In 1985, there were 285 cell phone subscribers in the small town of Centerville. Exponential Functions, Growth, and Decay Tell whether the function shows growth or decay. K G BM2a jd Yed Iw Gi Yteh D xI Knhfai Dnoi nt4em IA ElAg4eBbarea 2 l1 2. o y wAMldl k urMihg jhYt Xse FrqensPeur tvze hd 9. 5% per hour. Exponential Growth And Decay Word Problem - Displaying top 8 worksheets found for this concept. Let's do a couple of word problems dealing with exponential growth and decay. In the following problem, identify the value of the variables. Exponential Growth and Decay Worksheet In the function: y = a(b)x, a is the y-intercept and b is the base that determines the direction of the graph and the steepness. UNIT 5 WORKSHEET 6 Exponential Growth and Decay Graph each of the following functions. Therefore. Assume you invest $5,0Q0 in an account paying 8% interest compounded monthly. Displaying top 8 worksheets found for - Exponential Growth And Decay. Inverse, Exponential and Logarithmic Functions Students can learn the properties and rules of these functions and how to use them in real world applications through word problems such as those involving compound interest and exponential growth and decay that they will find on their homework. Exponential Growth And Decay Word Problem. Exponential Growth and Decay Word Problems Worksheet Answers or Distance Word Problems Worksheet Gallery Worksheet for Kids In English. exponential In growth and decay problems (that is, problems involving a quantity increasing or decreasing), here’s how to decide whether to choose a linear function or an exponential function. Graphing exponential functions. Sum of the angles in a triangle is 180 degree worksheet. 3)x d) y 20(0. Example: The value of a mortgage brokerage company has decreased from 7. It was from reliable on line source and that we love it. ©L 62J0 81v2u gK HumtGaT HSFoSfIt ew Za QrJe w PL YLICJ. The population of a town changes by an exponential growth factor, b, every 4 years. Exponential Decay Depreciation of value and radioactive decay are examples of exponential decay. U4D8_S Applications – Exponential Growth and Decay Problems. Exponential Growth/Decay. Write LINEAR and EXPONENTIAL equations/functions from data tables 4. What was its value in 2009? 2. Graded Classwork #1 DELAYED OPENING (EXCEPT FOR JUNIORS) 6-3-6-4 QUIZ CORRECTIONS DUE 6-5 and 6-6 Word Problems HW: Worksheet. Then graph. Exponential Growth and Decay Word Problems 1. What is the rate of growth or rate of decay? 3. If the city had. Homework And Decay Exponential Growth. It provides the formulas and equations / functi. Solve word problems involving exponential growth and exponential decay; Evaluate an exponential function at a given point (may be in Exponential Functions) Determine the equation of an exponential function given a point or two points (may be in Exponential Functions) Graph an exponential function (may be in Exponential Functions) State the. 5%!per!year. Determine when New Yorks's population will surpass Nevada's. What does the t stand for? 13. In fact, certain populations may decrease instead of increase and we could still use the general formula we used for growth. Exponential Growth an Decay Task Cards 16 task cards with or without QR codes 8 problems - Identify growth or decay function, initial value, rate of decay or growth 8 problems - Write a growth or decay function and evaluate for a given value Extras! - A full sheet recording sheet to use with the t.$\endgroup$– Matthew Leingang Jan 15 '15 at 17:02. !The!value!!!of!the!car!decreases!by!16%!each!year. Word problems too! Specifically on exponential growth and decay, and half-life. Some of the worksheets for this concept are Growth decay word problem key, Exponential growth and decay work, Exponential growth and decay, Pc expo growth and decay word problems, Exponential word problems, Exp growth decay word probs, Word problems interest growthdecay and half life, Name algebra 1b date linear exponential continued. Complete a total of 5 problems. What is the rate of growth or rate of decay? 3. It measured…. 3) A population of 800 beetles is growing each month at a rate of 5%. Such quantities give us an equation of the form dy dt = ky:. The graph of an exponential decay function falls from left to right. Upon graduating from college, Ashley took a job with a starting salary of$40,000. This algebra and precalculus video tutorial explains how to solve exponential growth and decay word problems. But here, if you realize not have tolerable time. This section covers: Introduction to Exponential Growth and Decay Solving Exponential Growth Problems Using Differential Equations Exponential Growth Word Problems We can use Calculus to measure Exponential Growth and Decay by using Differential Equations and Separation of Variables. What does the r stand for? Problem Solving: f(t) = - Decay. Start by browsing the selection below to get word problems, projects, and more. P is principle (starting amount). What is the expected population in 2018? 1. Exponential Growth and Decay Word Problems Show all work when using equations. Write an exponential equation, find the amount after the specified time. 20 Exponential Growth and Decay Worksheet Algebra 2 – exponential functions algebra worksheet by pecktabo math exponential functions 20 problems 4 determine whether it is an exponential function given an equation 2 determine whether it is linear or exponential given a table 3 evaluate given x value 4 match the function to the graph 2 graph the exponential function describe the domain and rang. Then use the model to answer the question. It is important to know the general nature and shape of exponential graphs. Students write and solve exponential growth and decay problems given a word problem. Exponential Decay B. Exponential and Logarithmic Applications Worksheet 1. v Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name_____ Exponential Functions Date_____ Period____. Lecture 5 : Exponential Growth and Decay Many quantities grow or decay at a rate proportional to their size. If the revenue is following an exponential pattern of. 5 D1 Worksheet Name_____ Exponential Growth and Decay Exponential Growth: Exponential Decay: Compound Interest: Compound Interest Continuously: 1. 1) For a period of time, an island's population grows at a rate proportional to its population. Eight random students will be selected to present their solution to the class. Inverse, Exponential and Logarithmic Functions Students can learn the properties and rules of these functions and how to use them in real world applications through word problems such as those involving compound interest and exponential growth and decay that they will find on their homework. Here it is. Then, solve the function and get the answer!. Graphing transformations of exponential functions. When an original amount is reduced by a consistent rate over a period of time, exponential decay is occurring. What does the t stand for? 13. Exponential Growth and Decay Worksheet 1. Practice: Exponential expressions word problems (algebraic) This is the currently selected item. EX #3:A slow economy caused a company’s annual revenues to drop from $530, 000 in 2008 to$386,000 in 2010. Exponential Growth and Decay Word Problems Worksheet Inspirational Worksheet Quadratic formula. But in the case of decrease or decay, the value of k will be negative. Radioactive Decay Problems 5. In section. !Find! the!valueof!theinvestment!after!30yr. In each of the following cases, write a formula for the quantity,Q grams, of air-freshener remaining t days after the start and sketch a graph of the function. Let’s solve an example of each of these problems. HOMEWORK: Yesterday's Worksheet #7-10 Worksheet "Exponential Equations Not Requiring Logarithms" Growth and Decay Word Problems. 8)x e) y 20(1 0. I am really bad with exponential growth and decay word problems. In this exponential equations worksheet, learners solve 11 word problems in short answer and multiple choice format. Worksheet Template Tips And Reviews. b) Find when the worth of the company will be 4. b > 1 for growth, 0 < b < 1 for decay. 02)ᵗ, y = (0. Exponential Growth and Decay Word Problems Worksheet Answers or Distance Word Problems Worksheet Gallery Worksheet for Kids In English. Exponential Word Problems: Compound Interest and Continuous Growth. Displaying all worksheets related to - Growth And Decay Word Problems. Exponential Growth and Decay For constant growth or decay use: y = abX or An = Ao(l ± For continuous growth or decay use: A = Pe(tt) For compounding use: A = l. The corresponding initial value problems lead to the solution having slightly different form, e. Derivatives of exponential functions involve the natural logarithm function, which itself is an important limit in Calculus, as well as the initial exponential function. State the domain and range. What are the equations for growth and decay How can we use the equations to solve growth and decay problems? Dec 147:47 PM Starter: What is exponential growth and decay? Video 7 Billion Dec 147:54 PM Exponential Growth and Decay The Exponential Function Decay 01 General Equation: y= abx Apr 139:38 AM Apr 139:43 AM Dec 15. Some of the worksheets below are Exponential Growth and Decay Worksheets, Solving exponential growth/decay problems with solutions, represent the given function as exponential growth or exponential decay, Word Problems, …. Classroom Task: X Marks the Spot – A Practice Understanding Task. Filename alg103kc_lesson17. Exponential Growth and Decay Word Problems Write an equation for each situation and answer the question. 532 # 1-17 odd In the next two sections, we examine how population growth. Exponential Growth And Decay Worksheet Pecktabo Math 2015 Answer Key. A new 2006 Honda Accord was valued at $25000. Exponential growth can be amazing! The idea: something always grows in relation to its current value, such as always doubling. Problem Solving Exponential Growth And Decay Holt - Displaying top 8 worksheets found for this concept. Growth Problems Answer Key Exponential Growth and Decay Word Problems Exponential Growth and Decay Word Problems. students in the class). To solve this differential equation, there are several techniques available to us. 5 Exponential Functions Definition of an Exponential Function An exponential function is a function that can be represented by the equation f(x) = abx where a and b are constants, b > 0 and b ≠ 1. Exponential function Suppose b is a positive number, with b 6= 1. Our digital library saves in combination countries, allowing you to get the most less latency time to download any of our books like this one. Exponential equations and logarithms are used to measure earthquakes and to predict how fast your bank account might grow. Mar 27 - Exponential Decay Mar 30 - Simple Interest Mar 31 - Compound Interest, Exponent Quiz Apr 1 - Review simple and compound interest, present. Exponential function Suppose b is a positive number, with b 6= 1. Exponential expressions word problems (algebraic) Practice: Exponential expressions word problems (algebraic) This is the currently selected item. Exponential equation: Note: "b", being less than 1 (. Helpful tips: At the top of each video, you can access that lesson's worksheet and quiz. The equation for "continual" growth (or decay) is A = Pe rt, where "A", is the ending amount, "P" is the beginning amount (principal, in the case of money), "r" is the growth or decay rate (expressed as a decimal), and "t" is the time (in whatever unit was used on the growth/decay rate). 97)ᵗ, y = (1. The general equation for exponential growth is y C(1 r)t. For example, taking b = 2, we have the exponential function f with base 2 x The graph of the exponential function 2x on the interval [ 5,5]. A new 2006 Honda Accord was valued at$25000. Growth Problems Answer Key Exponential Growth and Decay Word Problems Exponential Growth and Decay Word Problems. Find an exponential function f(t) = ke at that models this growth, and use it to predict the size of the population at 8:00 PM. Solving number problems - CCEA. If we start with only one bacteria which can double every hour, how many bacteria will we have by the end of one day?. 3 Explain and use the laws of fractional and negative exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay. Exponential Word Problems: Compound Interest and Continuous Growth. Exponential Growth & Decay Word Problems b) Use your equation to estimate the approximate cost per day in 2010. Algebra 1 - Exponential growth and decay word problems Exponential decay occurs in the same way when the growth rate is negative. Slope: 400 — 2 100 150 O A point: (0, 100) Equation of the line: or Y 0) 100= - 150x + 100 To find the exponential equation, use the general form y = ab Substitute (0, 100): Substitute (2, 400):. Question 1103993: Write the exponential function y=8(1. Interpret the different parts of the formula for exponential growth or decay, for instance, the growth rate, the initial amount, the final amount, the growth factor. Last summer, I attended several weeks of workshops and conferences. It provides the formulas and equations / functi. Graphing exponential functions. A new 2006 Honda Accord was valued at $25000. Write an exponential equation, find the amount after the specified time. Inverse, Exponential and Logarithmic Functions Students can learn the properties and rules of these functions and how to use them in real world applications through word problems such as those involving compound interest and exponential growth and decay that they will find on their homework. It is important to know the general nature and shape of exponential graphs. In this exponential equations worksheet, learners solve 11 word problems in short answer and multiple choice format. Linear Regression Exponential Equations Lessons Tes Teach. 25)^t in the form y=ae^kt. You should also try to understand how changing any one of these a ects the. For example a colony of bacteria may double every hour. 413) t where t is number of years since 1990 a. Determine the equation and represent the function that defines the cost of squid based on weight. Interpret and rewrite exponential growth and decay functions. -2- Worksheet by Kuta Software LLC 5) y x x y 6) y ( ) x x y Write an equation for each graph. Exponential Decay Formula: N t = N 0 * e-rt where: N t: The amount at time t N 0: The amount at time 0 r: Decay rate t: Time passed. students in the class). Math Algebra 1 Exponential growth & decay Exponential expressions. 4 Graph an exponential function of the form f(x) = ab^x. , like (2) including a combination of additive and multiplicative constants. 3)x d) y 20(0. Compare to the next, perspective picture. 122 8 nbsp 5 Review Worksheet doc Unit 4. Exponential growth is the increase in number or size at a constantly growing rate. We hope this graphic will likely be one of excellent reference. 9918) indicates a DECAY. Graph both functions. For all exponential decay functions;. Exponential Function Word Problems Exponential growth is modelled by y= y 0ekt There are four variables, the initial amount, y 0, the time t, the growth factor k, and the current amount y:You should be comfortable with nding any one of these four, given the other three. In each of the following cases, write a formula for the quantity,Q grams, of air-freshener remaining t days after the start and sketch a graph of the function. 3)x d) y 20(0. Displaying all worksheets related to - Growth And Decay Word Problems. If the growth rate is 3. Displaying all worksheets related to - Exponential Decay Problems. Some of the worksheets for this concept are Growth decay word problem key, Exponential growth and decay work, Exponential growth and decay, Pc expo growth and decay word problems, Exponential word problems, Exp growth decay word probs, Word problems interest growthdecay and half life, Name algebra 1b date linear exponential continued. Continuously compounded interest: word problems. exponential growth and decay worksheet answers linear function word problems worksheet images worksheet math for kids. Exponential Growth and Decay Worksheet In the function: y = a(b)x, a is the y-intercept and b is the base that determines the direction of the graph and the steepness. What is the annual rate of depreciation, the rate at which the car loses value?. What is your initial value? C. The Khan Academy is an organization on a mission. Exponential+Growthand+DecayWord+Problems+!! 8. If the size of the colony after t hours is given by y(t), then we know that dy=dt = 2y. 2 Microbial Growth Exponential Growth The Logistic Equation: Verhulst(1845) Growth under nutrient limitation 3 Continuous Culture Microbial Growth in the Chemostat Competition for Nutrient 4 Bacteriophage and Bacteria in Chemostat 5 Food Chain H. Also check out the following: - Exponential Growth and Decay Presentation which g. What are the equations for growth and decay How can we use the equations to solve growth and decay problems? Dec 147:47 PM Starter: What is exponential growth and decay? Video 7 Billion Dec 147:54 PM Exponential Growth and Decay The Exponential Function Decay 01 General Equation: y= abx Apr 139:38 AM Apr 139:43 AM Dec 15. Make sure you label all important points. 17) When will New York's population reach 15 million people? 18) Nevada's population in 1990 was 14. In the Exponential Growth and Decay Gizmo™, you can explore the effects of C and r in the function y = C(1 + r)t. Exponential Growth and Decay Word Problems Show all work when using equations. Solving problems with exponential growth. The opposite of “Exponential Growth”, is when we apply exponents to fractions which results in “Exponential Decay”. Exponential Functions and Equations Worksheet Exponential and Logarithmic Equations Worksheet Graphing Exponential Functions Worksheet Answers Exponential Growth and Decay Word Problems Worksheet Exponential Functions Growth and Decay Worksheet Answers. Vary the initial amount and the rate of growth or decay and investigate the changes to the graph. !The!value!!!of!the!car!decreases!by!16%!each!year. 5% annual decrease in population. I can use an exponential growth equation to solve applications. The number of subscribers increased by 75% per year after 1985. You have to be efficient with these! This will be the largest part of the test. Let's Practice:. Exponential Growth B. If the growth rate is 3. exponential growth and decay word problems is the PDF of the. The following tables give the formulas for Simple Interest, Compound Interest, and Continuously Compounded Interest. growth, radioactive decay, and temperature of heated objects. Problems involving e may be used. Exponential Growth and Decay Worksheet With Answers PDF. Exponential Growth an Decay Task Cards 16 task cards with or without QR codes 8 problems - Identify growth or decay function, initial value, rate of decay or growth 8 problems - Write a growth or decay function and evaluate for a given value Extras! - A full sheet recording sheet to use with the t. Modeling Exponential Growth And Decay With Skittles. solving exponential growth and decay problems, lesson 7 growth and decay mr g s algebra 2 cp site, exponential growth practice word problems, exponential growth and decay printable worksheets, ixl exponential growth and decay word problems algebra, exponential practice mini test 7 1 1 7 1 2 7 2, word problems interest growth decay and half life, exponential growth and decay jackson school. Problem solving introduction - CCEA. This post has been setting in my drafts for a VERY long time! Exponential Functions were Unit 3 in Algebra 2 this year. How do you identify equations as exponential growth, exponential decay, linear growth or linear decay #f(x) = 9. 7) Ready, Set, Go Homework: Linear and Exponential Functions 9. Does this function represent exponential growth or exponential decay? B. Complete the following practice problems: 1) Determine if the following exponential functions represent a growth or decay situation. Exponential Growth and Decay Word Problems Worksheet Answers or Distance Word Problems Worksheet Gallery Worksheet for Kids In English. Write an exponential decay function to represent this situation. for representing their answer could be written? Present their solutions in terms of exponential use showing that all different factorizations yielded 26 x 33. The opposite of “Exponential Growth”, is when we apply exponents to fractions which results in “Exponential Decay”. In each of the following cases, write a formula for the quantity,Q grams, of air-freshener remaining t days after the start and sketch a graph of the function. How much money S4ill be in the account after 5 years? 2. That's the difference between a positive and negative growth rate. Word problem with growth rates. Exponential Functions Word Problems WS Exponential Growth and Decay Worksheet Comments (-1) Compound Interest Notes. Some of the worksheets for this concept are Growth decay word problem key, Exponential growth and decay, Exponential growth and decay work, Exponential growth and decay word problems, Exp growth decay word probs, Exponential growth and decay, Name algebra 1b date linear. Choosing linear vs. 35 or 35% 5. Exponential Growth and Decay Word Problems Write an equation for each situation and answer the question. Word problems too! Specifically on exponential growth and decay, and half-life. Solving problems with exponential growth. Determine when New Yorks's population will surpass Nevada's. The exponential equation represents an exponential decay because the rate of decay is 0. Does this function represent exponential growth or exponential decay? B. Finance: Future. Apr 3, 2007 #1 1) "Solve the equation: The. Exponential Growth/Decay. decay exponential growth problems word; Home. If "b" were greater than 1, then that'd signify a GROWTH, as opposed to DECAY. What is the expected population in 2018? 1. Exponential Growth and Decay Word Problems Find a bank account balance if the account starts with$100, has an annual rate of 4%, and the money left in the account for 12 years. In this exponential equations worksheet, learners solve 11 word problems in short answer and multiple choice format. 97)ᵗ, y = (1. 8% Growth or decay. 24 #13, 14, 16, 19, 20a. 3) A population of 800 beetles is growing each month at a rate of 5%. For example a colony of bacteria may double every hour. a) How many will there be in 30 hours? b) Write a function, N, that gives the number of bacteria t hours from now. is added to 1. It occurs when the instantaneous exchange rate of an amount with respect to time is proportional to the amount itself. ChalkDoc lets algebra teachers make perfectly customized Linear Functions worksheets, activities, and assessments in 60 seconds. Exponential Growth And Decay. Of course, when Precalculus students are solving this problem, they have no idea about what a differential equation is, making the word relative seem superfluous to the problem. 4 Exponential Growth and Decay Notesheet 01 Completed Notes Growth and Decay Word Problems 01 Solutions 6. In the following problem, identify the value of the variables. We can find the continuous decay rate by converting the discrete growth into a continuous pattern: This helps me understand why the natural log is natural-- it's describing what nature is doing on an instant-by-instant basis. The equation for "continual" growth (or decay) is A = Pe rt, where "A", is the ending amount, "P" is the beginning amount (principal, in the case of money), "r" is the growth or decay rate (expressed as a decimal), and "t" is the time (in whatever unit was used on the growth/decay rate). Exponential growth and decay by percentage. It measured…. Then the exponential function with base b is the function f defined by f(x) = bx. This instructional video contains examples of how to solve word problems on exponential growth and decay. Exponential equations and logarithms are used to measure earthquakes and to predict how fast your bank account might grow. Does this function represent exponential growth or exponential decay? B. Give a formula for the exponential function f(x) has the values in the next table. Finance: Compound interest. v Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Name_____ Exponential Functions Date_____ Period____. What was its value in 2009? 2. Question 1103993: Write the exponential function y=8(1. Identifying Exponential Growth and Decay Determine whether each table represents an exponential growth function, an exponential decay function, or neither. Exponential Growth and Decay Exponential growth can be amazing! The idea: something always grows in relation to its current value, such as always doubling. notebook January 30, 2015 Writing Exponential Growth and Decay Functions in Real Life!!! Remember this from yesterday? A(t) = a(1 + r)t Sometimes you have to use given values to find r! 15. Exponential Growth And Decay. We hope this graphic will likely be one of excellent reference. Exponential Growth / Decay Graphing Writing the Equation Given Graph & Table : 2/14. Using negative power values results in fractions, and when these fractions have exponents applied to them we get “Decay”. The process i s repeated; each time the growth factor is multiplied to the preceding expression. , like (2) including a combination of additive and multiplicative constants. Linear Regression Exponential Equations Lessons Tes Teach. Basic Growth And Decay Problems f. Home > Math Worksheets > Algebra Worksheets > Exponential Growth and Decay There are some things in this world and universe that grow and dwindle at a ridiculously fast rate. 4 properties of exponential functions worksheet. The graph of an exponential decay function falls from left to right. In this exponential equations worksheet, learners solve 11 word problems in short answer and multiple choice format. a) Exponential growth or decay: b) Identify the initial amount: c) Identify the growth/decay factor: d) Write an exponential function to model the situation: e) “Do” the problem: 2. Growth And Decay Word Problems. In some applications, a quantity y demonstrates exponential growth or decay on a huge range. Does this function represent exponential growth or exponential decay? B. Of course, when Precalculus students are solving this problem, they have no idea about what a differential equation is, making the word relative seem superfluous to the problem. a) Write an equation that expresses the number of beetles at time x. Worksheet by Kuta Software LLC Kuta Software - Infinite Calculus Exponential Growth and Decay Name_____ Date_____ Period____ Solve each exponential growth/decay problem. Complete problems 1-8 on the Exponential Growth and Decay Word Problems worksheet. 3 Explain and use the laws of fractional and negative exponents, understand exponential functions, and use these functions in problems involving exponential growth and decay. Exponential Growth B. In each of the following cases, write a formula for the quantity,Q grams, of air-freshener remaining t days after the start and sketch a graph of the function. How much money S4ill be in the account after 5 years? 2. Typically, exponential growth (A = p(1+r)t) focuses on populations, financial investments, or biological. Here the same scale is used on both axes to emphasize. Each output should be a multiple of 2 with the exception of the last day when all students become infected (unless there are 16, 32, 64, etc. 02)ᵗ, y = (0. • understand exponential functions; • be able to construct growth and decay models; • recognise graphs of exponential functions; • understand that the inverse of an exponential function is a logarithmic function; • be able to use logarithms to solve suitable equations; • be able to differentiate exponential and logarithmic functions. We get a final graph of 11. is added to 1. It occurs when the instantaneous exchange rate of an amount with respect to time is proportional to the amount itself. To review continuous functions, see page 62, Exercises 60 and 61. Solve word problems involving exponential growth and exponential decay; Evaluate an exponential function at a given point (may be in Exponential Functions) Determine the equation of an exponential function given a point or two points (may be in Exponential Functions) Graph an exponential function (may be in Exponential Functions) State the. where $$k$$ is a constant called the growth/decay constant/rate. Assume you invest $5,0Q0 in an account paying 8% interest compounded monthly. You!buy!anew!car!for!$24,000. The following diagram shows the exponential growth and decay formula. 5% per hour. r is the interest rate (growh/decay rate) as a decimal. More information Graphing Exponential Functions Worksheets. Bookmark File PDF Exponential Growth And Decay Word Problems Worksheet Answers challenging the brain to think better and faster can be undergone by some ways. exponential growth and decay word problems worksheet answers - Learning regarding the specific value of money is among the primary training kids of today may learn. The worksheet contains the exact problems we cover in the video. She can expect a 6% raise in pay each year. 4 Exponential Growth and Decay Notesheet 01 Completed Notes Growth and Decay Word Problems 01 Solutions 6. Make sure you label all important points. Math homework help video on writing exponential growth models, predicting expected value, and using the change of base formula to determine after how many years a product will be worth a certain amount. Exponential Growth/Decay Worksheet Answer the following questions about the exponential decay problems. That's the difference between a positive and negative growth rate. To review one-to-one functions, see Lesson 2-1. Word problems used are fun, engaging, and relevant for the student. Does this function represent exponential growth or exponential decay? B. growth and decay (1) growth and decay notes (1) growth and decay word problems (1) headbands (2) inequalities (1) interactive notebook (1) intercepts (1) interest formula (1) interest rate foldable (1) interest rate interactive notebook (1) linear and exponential project (1) linear equations (7) linear equations in the real world (1) linear. The problem is "In year 2000, the population in a place was 100. Correct Answer is : y = C(1 + r ) t Q2 A businessman made a profit of $12,143 in 1990. Solve real-life problems involving exponential growth and decay. Some of the worksheets for this concept are Exponential growth and decay word problems, Exponential growth and decay, Exponential growth and decay work, Exp growth decay word probs, Growth decay word problem key, College algebra work 2 exponential growth. Answer: The exponential function is f(t) = 80 e. 5 12/11/19: Growth & Decay IXL examples 12/12/19: Study Guide for Exponential functions, Growth and Decay. Lesson 11-7 Homework Lesson 11-7 Writing Exponential Growth and Decay Functions Lesson 11-7 HW KEY Lesson 11-7 Writing Exponential Growth and Decay Functions NOTES VIDEOS: Exponential Growth and D…. Determine the equation and represent the function that defines the cost of squid based on weight. 2)ᵗ/10, and classify them as representing exponential growth or decay. 025)x f) y 40(1 0. If the revenue is following an exponential pattern of. 5% per hour. Radioactive Growth and Decay: Word problems involving exponential functions are not necessarily limited to bank accounts. Homework And Decay Exponential Growth. Some of the worksheets for this concept are Exponential growth and decay word problems, Exponential growth and decay, Exponential growth and decay work, Exp growth decay word probs, Growth decay word problem key, College algebra work 2 exponential growth. After 8 years what is the population of the town to nearest whole number? (For exponential decay word problems the decay factor is 1 – r, were r is the percent expressed as a decimal. Exponential Growth and Decay Word Problems 1. Start by browsing the selection below to get word problems, projects, and more. Having exponential decay, you may think, means "decaying REALLY fast". Let's Practice:. The value in dollars of a car years from now is ( ). The graph shows how. Problem Solving Exponential Growth And Decay Holt - Displaying top 8 worksheets found for this concept. From a physics perspective, a continuous rate is more telling. ANSWER KEY 1. When $$k > 0$$, we use the term exponential growth. U4D8_S Applications – Exponential Growth and Decay Problems. Exponential Growth and Decay Word Problems 1. Does this function represent exponential growth or exponential decay? B. k=_ If t is measured in years, indicate whether the exponential function is growing or decaying and find the annual and continuous growth/decay rates. Example 3 The growth of a colony of bacteria is given by the equation, \[Q = {Q_0}{{\bf{e}}^{0. EX #3:A slow economy caused a company’s annual revenues to drop from$530, 000 in 2008 to $386,000 in 2010. Exponential Growth And Decay Word Problem. How many years will it take for her to start earning$60,000 per year? 2. For example, taking b = 2, we have the exponential function f with base 2 x The graph of the exponential function 2x on the interval [ 5,5]. Exponential Decay. Does this function represent exponential growth or exponential decay? B. the graph of (x) = 2x, reflected across the x axis. Linear Regression Exponential Equations Lessons Tes Teach. Some of the worksheets displayed are Name algebra 1b date linear exponential continued, Exponential growth and decay word problems, Concept 17 write exponential equations, Exponential word problems, Exponential function word problems, Exponential equations not requiring logarithms, Solving exponential and logarithmic. We will use separation of variables. In this tutorial, learn how to turn a word problem into an exponential decay function. Some of the worksheets for this concept are Exponential growth and decay, Exponential growth and decay work, Exponential growth and decay word problems, College algebra work 2 exponential growth and decay, Honors pre calculus d1 work name exponential, Exponential population growth, Graphing. b) What would the population be in 2000 if the growth continues at the same rate. Exponential Decay B. Exponential Growth and Decay Worksheet With Answers PDF. Comments (-1) Compound Interest Worksheet. • understand exponential functions; • be able to construct growth and decay models; • recognise graphs of exponential functions; • understand that the inverse of an exponential function is a logarithmic function; • be able to use logarithms to solve suitable equations; • be able to differentiate exponential and logarithmic functions. In some applications, a quantity y demonstrates exponential growth or decay on a huge range. Return to Exercises. Displaying all worksheets related to - Growth And Decay Word Problems. The process i s repeated; each time the growth factor is multiplied to the preceding expression. Homework: worksheet. 11 Exponential and Logarithmic Functions Worksheet Concepts: • Rules of Exponents • Exponential Functions – Power Functions vs. From 2000 - 2010 a city had a 2. Exponential Growth and Decay Word Problems Show all work when using equations. Exponential growth can be modeled by the equation y = C (1 + r) t, where C is the initial amount, r is the growth rate and t is the time. Let's Practice:. Look at the graphs at the bottom of page 255. Math homework help video on writing exponential growth models, predicting expected value, and using the change of base formula to determine after how many years a product will be worth a certain amount. See full list on onlinemath4all. If it grows at a rate of 30% annually. The equation for "continual" growth (or decay) is A = Pe rt, where "A", is the ending amount, "P" is the beginning amount (principal, in the case of money), "r" is the growth or decay rate (expressed as a decimal), and "t" is the time (in whatever unit was used on the growth/decay rate). Differential Equations Worksheets This section contains all of the graphic previews for the Differential Equations Worksheets. Also check out the following: - Exponential Growth and Decay Presentation which g. Problem solving introduction - CCEA. Exponential Growth And Decay Word Problem. this is on of my questions: An air-freshener starts with 30 grams and evaporates. Word problems too! Specifically on exponential growth and decay, and half-life. Connections to Previous Learning: Students should be familiar with solving exponential growth and decay problems using the universal growth formula y = Cekt. Exponential Growth / Decay Graphing : 2/12: HW #23 Graphing Exponential Functions Must do at least 4 graphing problems. The following diagram shows the exponential growth and decay formula. Solving an exponential word problem. Exponential Growth And Decay. Some of the worksheets for this concept are Exponential growth and decay, Exponential growth and decay work, Exponential growth and decay word problems, Honors pre calculus d1 work name exponential, Exponential growth and decay, Graphing exponential, Growth decay word problem key, Exponential growth and decay functions. Mortgage Problems 3. Start by browsing the selection below to get word problems, projects, and more. If we start with only one bacteria which can double every hour, how many bacteria will we have by the end of one day?. Use EXPONENTIAL FUNCTIONS to find INPUT given OUTPUT or find OUTPUT. At the end of the lesson we will come together to produce a general idea. Math · Algebra 1 · Exponential growth & decay Practice: Exponential expressions word problems (numerical) Initial value & common ratio of exponential functions. c) Use this model to predict about when the population of Brownville will first r each 1,000,000. Analyzing the Game: Once all students have been infected, I will have them fill out the exponential_zombies_worksheet table using the values from the class. Word problem with growth rates. Comments (-1) Compound Interest Worksheet. Make sure you have memorized this equation, along with. ©L 62J0 81v2u gK HumtGaT HSFoSfIt ew Za QrJe w PL YLICJ. 959 years Ashley would be earning \$60,000/year. 3)x d) y 20(0. You have to be efficient with these! This will be the largest part of the test. Exponential word problems almost always work off the growth / decay formula, A = Pe rt, where "A" is the ending amount of whatever you're dealing with (money, bacteria growing in a petri dish, radioactive decay of an element highlighting your X-ray), "P" is the beginning amount of that same "whatever", "r" is the growth or decay rate, and "t" is time. a) How many will there be in 30 hours? b) Write a function, N, that gives the number of bacteria t hours from now. • If b >1, the function is an EXPONENTIAL GROWTH function, and the graph INCREASES from left to right • If 0 < b < 1, the function is an EXPONENTIAL DECAY function, and the graph DECREASES from left to right • Another way to identify the vertical intercept is to evaluate f(0). Experiencing, listening to the other experience, adventuring, studying, training, and more practical happenings may urge on you to improve. Please read the study guide: Exponential Functions before doing these questions. | 2020-10-28T09:03:34 | {
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http://zafarindustries.com/where-was-jmegb/618c52-extreme-value-theorem-open-interval | (a,b) as opposed to [a,b] maximum and an absolute minimum on the interval [0,3]. The absolute minimum is \answer {-27} and it occurs at x = \answer {1}. Inequalities and behaviour of f(x) as x →±∞ 17. 9. The extreme value theorem gives the existence of the extrema of a continuous function defined on a closed and bounded interval. In calculus, the extreme value theorem states that if a real-valued function f is continuous on the closed interval [a,b], then f must attain a maximum and a minimum, each at least once.That is, there exist numbers c and d in [a,b] such that: The function is a In this section we learn to reverse the chain rule by making a substitution. proved. We solve the equation f'(x) =0. function f(x) yields: The absolute maximum is \answer {2e^4} and it occurs at x = \answer {2}.The absolute minimum is \answer {-1/(2e)} and it occurs at x = \answer {-1/2}. Suppose that the values of the buyer for nitems are independent and supported on some interval [u min;ru min] for some u min>0 and r 1. Built at The Ohio State UniversityOSU with support from NSF Grant DUE-1245433, the Shuttleworth Foundation, the Department of Mathematics, and the Affordable Learning ExchangeALX. The next theorem is called Rolle’s Theorem and it guarantees the existence of an extreme value on the interior of a closed interval, under certain conditions. We learn to compute the derivative of an implicit function. If f'(c) is defined, then Derivatives of sums, products and composites 13. If a function $$f\left( x \right)$$ is continuous on a closed interval $$\left[ {a,b} \right],$$ then it attains the least upper and greatest lower bounds on this interval. three step process. This is used to show thing like: There is a way to set the price of an item so as to maximize profits. Solution: The function is a polynomial, so it is continuous, and the interval is closed, so by the Extreme Value Theorem, we know that this function has an absolute maximum and an absolute minimum on the interval . f'(x) =0. Terminology. For example, (0,1) means greater than 0 and less than 1.This means (0,1) = {x | 0 < x < 1}.. A closed interval is an interval which includes all its limit points, and is denoted with square brackets. interval. We determine differentiability at a point. interval around c (open interval around c means that the immediate values to the left and to the right of c are in that open interval) 6. 2. This is what is known as an existence theorem. © 2013–2021, The Ohio State University — Ximera team, 100 Math Tower, 231 West 18th Avenue, Columbus OH, 43210–1174. Thus f'(c) \geq 0. Notice how the minimum value came at "the bottom of a hill," and the maximum value came at an endpoint. This theorem is sometimes also called the Weierstrass extreme value theorem. THE EXTREME-VALUE THEOREM (EVT) 27 Interlude: open and closed sets We went about studying closed bounded intervals… and x = 2. In such a case, Theorem 1 guarantees that there will be both an absolute maximum and an absolute minimum. Extreme Value Theorem Theorem 1 below is called the Extreme Value theorem. Proof of the Extreme Value Theorem Theorem: If f is a continuous function defined on a closed interval [a;b], then the function attains its maximum value at some point c contained in the interval. Then there are numbers cand din the interval [a,b] such that f(c) = the absolute minimum and f(d) = the absolute maximum. But the difference quotient in the Proof: There will be two parts to this proof. Open interval. The Inverse Function Theorem (continuous version) 11. In this section we learn a theoretically important existence theorem called the For the extreme value theorem to apply, the function must be continuous over a closed, bounded interval. The absolute extremes occur at either the endpoints, x=\text {-}1, 6 or the critical The Mean Value Theorem states that the rate of change at some point in a domain is equal to the average rate of change of that domain. The absolute maximum is shown in red and the absolute minimumis in blue. In this section we compute derivatives involving. point. We solve the equation it follows that the image must also If the function f is continuous on the closed interval [a,b], then f has an absolute maximum value and an absolute minimum value on [a,b]. However, if that interval was an open interval of all real numbers, (0,0) would have been a local minimum. Extreme Value Theorem: If a function is continuous in a closed interval , with the maximum of at and the minimum of at then and are critical values of Proof: The proof follows from Fermat’s theorem and is left as an exercise for the student. Vanishing Derivative Theorem Assume f(x) is a continuous function defined on an open interval (a,b). These values are often called extreme values or extrema (plural form). The quintessential point is this: on a closed interval, the function will have both minima and maxima. Which of the following functions of x is guaranteed by the Extreme Value Theorem to have an absolute maximum on the interval [0, 4]? In this section we learn the second part of the fundamental theorem and we use it to at the critical number x = 0. right hand limit is negative (or zero) and so f'(c) \leq 0. The Extreme Value Theorem (EVT) does not apply because tan x is discontinuous on the given interval, specifically at x = π/2. Compute limits using algebraic techniques. Then, for all >0, there is an algorithm that computes a price vector whose revenue is at least a (1 )-fraction of the optimal revenue, and whose running time is polynomial in max ˆ We use the logarithm to compute the derivative of a function. at a Regular Point of a Surface. These values are often called extreme values or extrema (plural form). Extreme Value Theorem. The #1 tool for creating Demonstrations and anything technical. To find the relative extrema of a function, you first need to calculate the critical values of a function. The next step is to determine all critical points in the given interval and evaluate the function at these critical points and at the endpoints of the interval. is increasing or decreasing. of a function. The standard proof of the first proceeds by noting that is the continuous image of a compact set on the We find extremes of functions which model real world situations. and the denominator is negative. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value. When moving from the real line $${\displaystyle \mathbb {R} }$$ to metric spaces and general topological spaces, the appropriate generalization of a closed bounded interval is a compact set. When you do the problems, be sure to be aware of the difference between the two types of extrema! discontinuities. We compute Riemann Sums to approximate the area under a curve. A point is considered a minimum point if the value of the function at that point is less than the function values for all x-values in the interval. In this section we discover the relationship between the rates of change of two or The quintessential point is this: on a closed interval, the function will have both minima and maxima. It ... (-2, 2), an open interval, so there are no endpoints. This becomes 3x^2 -12x= 0 It states that if f is a continuous function on a closed interval [a, b], then the function f has both a minimum and a maximum on the interval. 3. To find the relative extrema of a function, you first need to calculate the critical values of a function. We find limits using numerical information. Solution: First, we find the critical numbers of f(x) in the interval [\text {-}1, 3]. For the extreme value theorem to apply, the function must be continuous over a closed, bounded interval. This video explains the Extreme Value Theorem and then works through an example of finding the Absolute Extreme on a Closed Interval. within a closed interval. the interval [\text {-}1,3] we see that f(x) has two critical numbers in the interval, namely x = 0 The Inverse Function Theorem (Differentiable version) 14. If has an extremum on an open interval , then the extremum occurs at a critical point. Determine all critical points in the given interval and evaluate the function at these critical points and at the endpoints of the interval 3. Extreme Value Theorem If is continuous on the closed interval , then there are points and in , such that is a global maximum and is a global minimum on . https://mathworld.wolfram.com/ExtremeValueTheorem.html. A local minimum value … Use the differentiation rules to compute derivatives. Establish that the function is continuous on the closed interval 2. max and the min occur in the interval, but it does not tell us how to find fundamental theorem of calculus. Solution: First, we find the critical numbers of f(x) in the interval [\text {-}1, 6]. Extreme Value Theorem: f is continuous on [a;b], then f has an absolute max at c and and absolute min at d, where c;d 2[a;b] 7. number in the interval and it occurs at x = -1/3. Using the Extreme Value Theorem 1. How would you like to proceed? Does the intermediate value theorem apply to functions that are continuous on the open intervals (a,b)? The idea that the point $$(0,0)$$ is the location of an extreme value for some interval is important, leading us to a definition. more related quantities. (If the max/min occurs in more than one place, list them in ascending order).The absolute maximum is \answer {4} and it occurs at x = \answer {-2} and x=\answer {1}. Chapter 4: Behavior of Functions, Extreme Values 5 so by the Extreme Value Theorem, we know that this function has an absolute is differentiable on the open interval , i.e., the derivative of exists at all points in the open interval .. Then, there exists in the open interval such that . The Extreme Value Theorem ... as x !1+ there is an open circle, so the lower bound of y = 1 is approached but not attained . An open interval does not include its endpoints, and is indicated with parentheses. If f'(c) is undefined then, x=c is a critical number for f(x). them. Closed Interval Method • Test for absolute extrema • Only use if f (x) is continuous on a closed interval [a, b]. maximum and a minimum on If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). From MathWorld--A Correspondingly, a metric space has the Heine–Borel property if every closed and bounded set is also compact. In order to utilize the Mean Value Theorem in examples, we need first to understand another called Rolle’s Theorem. integrals. Continuous, 3. The extreme value theorem states that if is a continuous real-valued function on the real-number interval defined by , then has maximum and minimum values on that interval, which are attained at specific points in the interval. https://mathworld.wolfram.com/ExtremeValueTheorem.html, Normal Curvature 2.3. We compute the derivative of a composition. • Extreme Value Theorem: • A function can have only one absolute maximum or minimum value (y-coordinate), but it can occur at more than one x-coordinate. The Extreme Value Theorem (EVT) says: If a function f is continuous on the closed interval a ≤ x ≤ b , then f has a global minimum and a global maximum on that interval. •Note: If the interval is open, then the endpoints are. interval , so it must Among all ellipses enclosing a fixed area there is one with a smallest perimeter. In this section we use the graph of a function to find limits. The main idea is finding the location of the absolute max and absolute min of a If we don’t have a closed interval and/or the function isn’t continuous on the interval then the function may or may not have absolute extrema. A local minimum value … 4 Extreme Value Theorem If f is continuous on a closed interval a b then f from MATH 150 at Simon Fraser University Play this game to review undefined. In this section we examine several properties of the indefinite integral. the point of tangency. Fermat’s Theorem Suppose is defined on the open interval . We solve the equation f'(x) =0. Let f be continuous on the closed interval [a,b]. This becomes x^3 -6x^2 + 8x = 0 and the solutions are x=0, x=2 and x=4 (verify). Extreme Value Theorem If a function f {\displaystyle f} is continuous on a closed interval [ a , b ] {\displaystyle [a,b]} then there exists both a maximum and minimum on the interval. The Extreme value theorem requires a closed interval. average value. fails to hold, then f(x) might fail to have either an absolute max or an absolute min In order for the extreme value theorem to be able to work, you do need to make sure that a function satisfies the requirements: 1. Finding the absolute extremes of a continuous function, f(x), on a closed interval [a,b] is a Portions of this entry contributed by John We learn how to find the derivative of a power function. The absolute extremes occur at either the endpoints, x=\text {-}1, 3 or the critical A set $${\displaystyle K}$$ is said to be compact if it has the following property: from every collection of open sets $${\displaystyle U_{\alpha }}$$ such that $${\textstyle \bigcup U_{\alpha }\supset K}$$, a finite subcollection $${\displaystyle U_{\alpha _{1}},\ldots ,U_{\alpha _{n}}}$$can be chosen such that $${\textstyle \bigcup _{i=1}^{n}U_{\alpha _{i}}\supset K}$$. The absolute maximum is \answer {3/4} and it occurs at x = \answer {2}.The absolute minimum is \answer {0} and it occurs at x = \answer {0}.Note that the critical number x= -2 is not in the interval [0, 4]. In this section, we use the derivative to determine intervals on which a given function need to solve (3x+1)e^{3x} = 0 (verify) and the only solution is x=\text {-}1/3 (verify). Remark: An absolute extremum of a function continuous on a closed interval must either be a relative extremum or a function value at an endpoint of the interval. Closed interval domain, … We have a couple of different scenarios for what that function might look like on that closed interval. The largest and smallest values from step two will be the maximum and minimum values, respectively The function Theorem 1. This theorem is called the Extreme Value Theorem. Depending on the setting, it might be needed to decide the existence of, and if they exist then compute, the largest and smallest (extreme) values of a given function. In order to use the Extreme Value Theorem we must have an interval that includes its endpoints, often called a closed interval, and the function must be continuous on that interval. For example, let’s say you had a number x, which lies somewhere between zero and 100: The open interval would be (0, 100). The derivative is 0 at x = 0 and it is undefined at x = -2 and x = 2. Plugging these special values into the original function f(x) yields: The absolute maximum is \answer {17} and it occurs at x = \answer {-2}.The absolute minimum is \answer {-15} and it occurs at x = \answer {2}. In this lesson we will use the tangent line to approximate the value of a function near The extreme value theorem gives the existence of the extrema of a continuous function defined on a closed and bounded interval. numbers x = 0, 4. tangent line problem. numbers x = 0,2. Noting that x = 4 is not in Plugging these values into the original function f(x) yields: The absolute maximum is \answer {2} and it occurs at x = \answer {0}.The absolute minimum is \answer {-2} and it occurs at x = \answer {2}. We don’t want to be trying to find something that may not exist. Plugging these special values into the original interval around c (open interval around c means that the immediate values to the left and to the right of c are in that open interval) 6. Depending on the setting, it might be needed to decide the existence of, and if they exist then compute, the largest and smallest (extreme) values of a given function. I know it's pretty vital for the theorem to be able to show the values of f(a) and f(b), but what if I have calculated the limits as x -> a (from the right hand side) and x -> b (from the left hand side)? If has an extremum Extreme Value Theorem If a function f {\displaystyle f} is continuous on a closed interval [ a , b ] {\displaystyle [a,b]} then there exists both a maximum and minimum on the interval. Related facts Applications. If the interval is open or the function has even one point of discontinuity, the function may not have an absolute maximum or absolute minimum over For example, consider the functions shown in … Fermat’s Theorem. from the definition of the derivative we have f'(c) = \lim _{h \to 0^-} \frac {f(c+h) -f(c)}{h} = \lim _{h \to 0^-} \frac {f(c+h) -f(c)}{h} The difference quotient in the left The closed interval—which includes the endpoints— would be [0, 100]. In this section we learn the Extreme Value Theorem and we find the extremes of a (The circle, in fact.) Are you sure you want to do this? This has two important corollaries: . the equation f'(x) =0 gives x=2 as the only critical number of the function. The Weierstrass Extreme Value Theorem. If has an absolute maximum or absolute minimum at a point in the interval, then is a critical number for . We learn the derivatives of many familiar functions. No maximum or minimum values are possible on the closed interval, as the function both increases and decreases without bound at x = π/2. It is also important to note that the theorem tells us that the values of a continuous function on a closed interval. History. Explore anything with the first computational knowledge engine. • Three steps/labels:. Thus f(c) \geq f(x) for all x in (a,b). This video explains the Extreme Value Theorem and then works through an example of finding the Absolute Extreme on a Closed Interval. Unlimited random practice problems and answers with built-in Step-by-step solutions. Which of the following functions of x is guaranteed by the Extreme Value Theorem to have an absolute maximum on the interval [0, 4]? If you update to the most recent version of this activity, then your current progress on this activity will be erased. Join the initiative for modernizing math education. There are a couple of key points to note about the statement of this theorem. This is a good thing of course. Solving Thus, a bound of infinity must be an open bound. So the extreme value theorem tells us, look, we've got some closed interval - I'm going to speak in generalities here - so let's say that's our X axis and let's say we have some function that's defined on a closed interval. Extreme Value Theorem If f is continuous on a closed interval [a,b], then f has both a maximum and minimum value. Proof of the Extreme Value Theorem Theorem: If f is a continuous function defined on a closed interval [a;b], then the function attains its maximum value at some point c contained in the interval. Incognito. Select the third example, showing the same piece of a parabola as the first example, only with an open interval. Differentiation 12. A continuous function ƒ (x) on the closed interval [a,b] showing the absolute max (red) and the absolute min (blue).. Example . Extreme Value Theorem If f is continuous on a closed interval [a,b], then f has both a maximum and minimum value. [a,b]. Try the following: The first graph shows a piece of a parabola on a closed interval. this critical number is in the interval (0,3). In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval [a, b], then it takes on any given value between f(a) and f(b) at some point within the interval.. know that the absolute extremes occur at either the endpoints, x=0 and x = 3, or the The Extreme Value Theorem, sometimes abbreviated EVT, says that a continuous function has a largest and smallest value on a closed interval. Hence f'(c) = 0 and the theorem is That makes sense. In this example, the domain is not a closed interval, and Theorem 1 doesn't apply. In this section we learn about the two types of curvature and determine the curvature interval , then has both a The first derivative can be used to find the relative minimum and relative maximum values of a function over an open interval. The Heine–Borel theorem asserts that a subset of the real line is compact if and only if it is both closed and bounded. In this section we prepare for the final exam. The Extreme Value Theorem ... as x !1+ there is an open circle, so the lower bound of y = 1 is approached but not attained . The Extreme Value Theorem guarantees both a maximum and minimum value for a function under certain conditions. First, since we have a closed interval (i.e. If f(x) exists for all values of x in the open interval (a,b) and f has a relative extremum at c, where a < c < b, then if f0(c) exists, then f0(c) = 0. In this section we compute limits using L’Hopital’s Rule which requires our If either of these conditions • Three steps In this section we use definite integrals to study rectilinear motion and compute As an instantaneous rate of change of two or more related quantities = -1/3, John and,... Minimum values of a function { 0 } and it is differentiable everywhere evaluate the is... Here is a way to set the price of an item so as to maximize.... Will solve the equation f ' ( c ) \geq f ( x ) in the interval a! 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extreme value theorem open interval 2021 | 2021-07-29T13:18:24 | {
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http://math.stackexchange.com/questions/106036/solving-a-recurrence-relation-equation-is-there-more-than-1-way-to-solve-this | # Solving a Recurrence Relation/Equation, is there more than 1 way to solve this?
1) Solve the recurrence relation
$$T(n)=\begin{cases} 2T(n-1)+1,&\text{if }n>1\\ 1,&\text{if }n=1\;. \end{cases}$$
2) Name a problem that also has such a recurrence relation.
The answer I got somewhere is here:
Here $T_0=0,T_n-2T_{n-1}=1$ for $n=1,2,\dots$
Multiply by $z^n$ and sum over $n\ge 1$, we get $(A(z)-T_0)-2zA(z)=\dfrac{z}{1-z}$
$\therefore\qquad A(z)=\dfrac{z}{(1-2z)(1-z)}=\dfrac1{1-2z}-\dfrac1{1-z}=\sum\limits_{n\ge 0}2^nz^n-\sum\limits_{n\ge 0}z^n$.
Thus $T_n=2^n-1$.
I'm still in the process of understanding how to solve recurrence relations.. and I'm seeing that there's multiple methods to solving recurrence relations in general. So my question is, is there multiple ways to solve this? If so, can someone state the answer? Also, how do I know what method to use when solving these recurrence relations? thanks
EDIT: did some reading, the first method i'm reading about is mathematical induction.. i'm getting the impression that i can prove that the equation is 2N-1.. so can I solve it this way too?
Also, for the 2nd question, I have Towers of Hanoi, are there any other examples someone can maybe list? thanks
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For example noticing that $T(n)=2T(n-1)+2-1\ \$ so that $T(n)+1=2(T(n-1)+1)\ \$. Use the change of variable $U(n)=T(n)+1\ \$ to conclude... (promised, I didn't see the answer at least consciously !). The change of variable trick is often useful for recurrences. – Raymond Manzoni Feb 5 '12 at 19:07
thanks, keeping it in mind as i go along reading right now – user1189352 Feb 5 '12 at 19:37
Yes, there are many ways to solve this recurrence. The method that you found uses generating functions and is towards the sophisticated end. Here are three others.
(1) Guess and prove. Calculate the first few terms of the sequence:
$$\begin{array}{r|rr}n&1&2&3&4&5&6\\ T(n)&1&3&7&15&31&63 \end{array}$$
The numbers in the bottom row should look familiar: they’re one less than consecutive powers of $2$. Thus, you might at this point guess that in general $T(n)=2^n-1$. Of course, now you have to prove your guess. This typically requires a proof by induction. Getting the induction off the ground (i.e., checking the base case) is easy: $T(1)=1=2^1-1$. Now you need to show that if $T(n)=2^n-1$ is true when $n=k$, it’s also true when $n=k+1$.
Assume as your induction hypothesis that $T(k)=2^k-1$ for some $k\ge 1$. By the definition of $T$ you know that $T(k+1)=2T(k)+1$. Your induction hypothesis tells you that $T(k)=2^k-1$, so $T(k+1)=2(2^k-1)-1=2\cdot2^k-2+1=2^{k+1}-1$, which is exactly what we wanted. It follows by induction that $T(n)=2^n-1$ for all $n\ge 1$.
Added: It isn’t always so easy to guess the right formula. As Gadi A reminds me, there is a wonderful tool available to help with this, the On-Line Encyclopedia of Integer Sequences, known as OEIS. If you search it for a sequence containing $\langle 1,3,7,15,31,63\rangle$, you’ll get 29 matches, of which the first, A000225, turns out to be the right one for this problem. From the COMMENTS section of the entry:
Also solutions (with minimum number of moves) for the problem of Benares Temple, i.e. three diamond needles with n discs ordered by decreasing size on the first needle to place in the same order on the third one, without ever moving more than one disc at a time and without ever placing one disc at the top of a smaller one.
(2) Unwind the recurrence. Imagine that $n$ is a fairly large number, and start calculating $T(n)$ in terms of earlier and earlier terms of the sequence until you spot a pattern:
\begin{align*} T(n)&=2T(n-1)+1\\ &=2\big(2T(n-2)+1\big)+1\\ &=2^2T(n-2)+2+1\\ &=2^2\big(2T(n-3)+1\big)+2+1\\ &=2^3T(n-3)+2^2+2+1\\ &\qquad\qquad\qquad\vdots\\ &=2^kT(n-k)+2^{k-1}+2^{k-2}+\dots+2+1\tag{1}\\ &=2^kT(n-k)+\sum_{i=0}^{k-1}2^k\;.\tag{2} \end{align*}
Now plug $k=n-1$ into $(2)$ to get \begin{align*}T(n)&=2^{n-1}T(1)+\sum_{i=0}^{n-2}2^i\\ &=2^{n-1}+\big(2^{n-1}-1\big)\\ &=2^n-1\;, \end{align*}
where I used the formula for the sum of a geometric series to evaluate the summation.
If you use this approach, you should then go on to prove the formula by induction, just as in the previous method, because the formula in $(1)$ was a guess $-$ a very solid guess, but still a guess requiring proof.
Another point to notice here is that the last step would have been very slightly easier if we’d backtracked the sequence by calculating $T(0)$ from $T(1)$: $T(0)=\frac12\big(T(1)-1\big)=0$. Then we could have substituted $k=n$ into $(2)$ to get the desired result directly.
(3) Turning the non-homogeneous recurrence into a homogeneous one by a change of variable. Let $S(n)=T(n)+d$ for some as yet unknown $d$. Then $T(n)=S(n)-d$, and the recurrence can be rewritten as
\begin{align*} S(n)-d&=\begin{cases}2\big(S(n-1)-d\big)+1,&\text{if }n>1\\ 1-d,&\text{if }n=1 \end{cases}\\\\ &=\begin{cases} 2S(n-1)-2d+1,&\text{if }n>1\\ 1-d,&\text{if }n=1\;. \end{cases} \end{align*}
Move the constants to the righthand side in the recurrence: $$S(n)=2S(n-1)-d+1\;.$$ Thus, if we set $d=1$, we get rid of the constant term: $$S(n)=2S(n-1)\;.$$ This is just exponential growth: $S(n)$ doubles with each increase of $1$ in $n$. And $$S(1)=T(1)+d=1+1=2=2^1\;,$$ so it’s very easy to see that $S(n)=2^n$ for every $n\ge 1$. Finally, $T(n)=S(n)-d=2^n-1$.
For your second question, I could cook up other examples, but the Tower of Hanoi problem is by far the best known satisfying this recurrence.
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wow thank you so much for the indepth answer. going to be studying this one for a while, i appreciate the extra examples to help me make sure i grasp this too. tyvm!!! – user1189352 Feb 5 '12 at 19:39
You can show your appreciation by upvoting ... +1 – ldog Feb 5 '12 at 19:49
A wonderful answer. It might be worthwhile to mention the OEIS as an effective "real life" tool for the "guess and prove" method. – Gadi A Feb 5 '12 at 19:50
@Gadi: Thanks; I can’t believe that I didn’t think of that, considering how often I’ve used it! – Brian M. Scott Feb 5 '12 at 20:05
"If you search it for a sequence containing $\{1,3,7,15,31,64\}$": I bet you meant $63$. – robjohn Feb 5 '12 at 21:29
show 5 more comments
Add $1$ to both sides of the recurrence to get $$T(n)+1 = 2(T(n-1)+1)\tag{1}$$ Thus, $T(n)+1$ simply doubles with each increase in $n$. Thus, $$T(n)+1=c\cdot2^n\tag{2}$$ Plugging $T(1)=1$ into $(2)$, we get that $c=1$. Therefore, we get the solution $$T(n)=2^n-1\tag{3}$$
I see that the method I am describing is simply a rewording of $(3)$ from Brian M. Scott's answer, but I will leave it as it might prove simpler to read before jumping into Brian's rather extensive answer.
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the simplicity is much much appreciated, as this is still all kinda complicated to me ;D – user1189352 Feb 5 '12 at 20:13
$$T(n)=2T(n-1)+1=2(2T(n-2)+1)+1=4T(n-2)+(1+2)$$ $$=8T(n-3)+(1+2+4)=16T(n-4)+(1+2+4+8)=\dots=$$ $$2^kT(n-k)+(1+2+\dots+2^{k-1})=\dots=1+2+\dots+2^{n-1}=2^n-1$$ | 2014-03-17T10:48:19 | {
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https://math.stackexchange.com/questions/1986529/power-set-of-x-is-a-ring-with-symmetric-difference-and-intersection/1988151 | # Power Set of $X$ is a Ring with Symmetric Difference, and Intersection
I'm studying for an abstract algebra exam and one of the review questions was this:
Let $X$ be a set, and $\mathcal P(X)$ be the power set of $X$. Consider the operations $\Delta$ = symmetric difference (a.k.a. "XOR"), and $\bigcap$ = intersection.
a) Does $\Delta$ and $\bigcap$ make $P(X)$ into a ring?
b) If so, is it a ring with unity?
c) Is the ring commutative?
d) Is it a field?
For parts a) and b), I think it does form a ring with unity, but I'm not quite sure how to get started on proving it.
For part c), it is a commutative ring since $\mathcal P(X)$ is closed under symmetric difference and intersection, right?
Not even sure how to get started on d).
I'd really like to understand this question fully, so any kind of input would be tremendously helpful. Thank you!
• Do you know what's a ring? A ring with unity, and so on? For instance, I can show you how to prove that $\cap$ is a commutative operation (which means $a \cap b = b \cap a$ for $a,b \in P(X)$). $\forall x, (x \in a \cap b \longleftrightarrow (x \in a$ and $x \in b) \longleftrightarrow (x \in b$ and $x \in a) \longleftrightarrow x \in b \cap a)$, so $a \cap b = b \cap a$. – nombre Oct 26 '16 at 19:47
• Well, if you know what a ring is, then you can confirm that a set with operations is a ring or not. Do you understand how to confirm that the ring axiom hold? It is a commutative ring since it's closed under... Well no, there are more axioms than just having a binary operation defined. And none of that directly proves the multiplication is commutative, either way. How about you systematically start to verify the axioms? It is all very mechanical. – rschwieb Oct 26 '16 at 20:36
I believe that the best way to study the properties of this structure is to note that $\mathcal P(X)$ with these operations is isomorphic to the set $\mathcal{F}$ of functions $X \to E$, where $E = \{ 0, 1\}$ is the field with two elements, and $\mathcal{F}$ is endowed with pointwise sum and product, that is, for $f, g \in \mathcal{F}$ we have $$(f + g)(x) = f(x) + g(x),\qquad (f \cdot g)(x) = f(x) \cdot g(x).$$ The isomorphism is given by $$X \supseteq A \mapsto \left(f : x \mapsto \begin{cases}1 & \text{if x \in A}\\0 &\text{if x \notin A}\end{cases}\right)$$
• For short, $A\mapsto\chi_A$, the characteristic function of $A$. – user26857 Oct 27 '16 at 20:44
• Wow thanks for this, it illuminated and made it a ton easier. I didn't realize this picture was much bettter. – Santropedro Mar 23 '17 at 2:44
• @Santropedro, you're welcome! – Andreas Caranti Mar 24 '17 at 8:57
Let symmetric difference be correspond to addition and intersection correspond to multiplication. Also let $\varnothing$ be empty set and correspond to $0$, as well as $E$ be universe of set and correspond to $1$. Then we can prove that symmetric difference and intersection of set satisfy each ring axiom.
1. Commutativity of addition: $$X\bigtriangleup Y=(X-Y)\cup (Y-X)=(Y-X)\cup (X-Y)=Y\bigtriangleup X$$
2. Additive identity: $$X\bigtriangleup \varnothing =(X-\varnothing )\cup (\varnothing -X)=X\cup \varnothing =X$$
3. Additive inverse: $$X\bigtriangleup X=(X-X)\cup (X-X)=\varnothing \cup \varnothing =\varnothing$$ So additive inverse of $X$ is $X$ itself.
4. Associativity of addition: By this post Prove that two sets are the same $$(X\bigtriangleup Y)\bigtriangleup Z=X\bigtriangleup (Y\bigtriangleup Z)$$
5. Commutativity and associativity of multiplication are due to fact that intersection is commutative and associative.
6. Multiplication identity is satisfy for $$X\cap E= X$$
7. Multiplication is distributive with respect to addition: \begin{align} (X\bigtriangleup Y)\cap Z&=((X-Y)\cup (Y-X))\cap Z \\ &=((X-Y)\cap Z)\cup ((Y-X)\cap Z) \\ &=(X\cap Z-Y\cap Z)\cup (Y\cap Z-X\cap Z) \\ &=(X\cap Z)\bigtriangleup (Y\cap Z) \end{align}
Hence set under symmetric difference and intersection is commutative ring with unity.
Since it is not true that for any set X, there is a set $Y$ that $X\cap Y=E$, set under symmetric difference and intersection is not field. | 2019-08-18T04:54:11 | {
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http://starke-automotive.com/weather-in-jfpov/5e5gcu.php?tag=63e9ef-do-all-functions-have-an-inverse | Imagine finding the inverse of a function … The inverse relation is then defined as the set consisting of all ordered pairs of the form (2,x). Explain your reasoning. There is one final topic that we need to address quickly before we leave this section. The inverse of a function has all the same points as the original function, except that the x's and y's have been reversed. Note that the statement does not assume continuity or differentiability or anything nice about the domain and range. Not all functions have inverses. Sin(210) = -1/2. so all this other information was just to set the basis for the answer YES there is an inverse for an ODD function but it doesnt always give the exact number you started with. Define and Graph an Inverse. Inverse Functions. Inverse of a Function: Inverse of a function f(x) is denoted by {eq}f^{-1}(x) {/eq}.. There is an interesting relationship between the graph of a function and the graph of its inverse. While it is not possible to find an inverse of most polynomial functions, some basic polynomials do have inverses. how do you solve for the inverse of a one-to-one function? Answer to Does a constant function have an inverse? as long as the graph of y = f(x) has, for each possible y value only one corresponding x value, and thus passes the horizontal line test.strictly monotone and continuous in the domain is correct In fact, the domain and range need not even be subsets of the reals. As we are sure you know, the trig functions are not one-to-one and in fact they are periodic (i.e. It is not true that a function can only intersect its inverse on the line y=x, and your example of f(x) = -x^3 demonstrates that. Thank you! Explain.. Combo: College Algebra with Student Solutions Manual (9th Edition) Edit edition. Does the function have an inverse function? both 3 and -3 map to 9 Hope this helps. Such functions are called invertible functions, and we use the notation $$f^{−1}(x)$$. Hello! Question 64635: Explain why an even function f does not have an inverse f-1 (f exponeant -1) Answer by venugopalramana(3286) (Show Source): You can put this solution on YOUR website! Definition of Inverse Function. If the function is linear, then yes, it should have an inverse that is also a function. Explain why an even function f does not have an inverse f-1 (f exponeant -1) F(X) IS EVEN FUNCTION IF Such functions are often defined through formulas, such as: A surjective function f from the real numbers to the real numbers possesses an inverse as long as it is one-to-one, i.e. Suppose we want to find the inverse of a function … Warning: $$f^{−1}(x)$$ is not the same as the reciprocal of the function $$f(x)$$. Before defining the inverse of a function we need to have the right mental image of function. There are many others, of course; these include functions that are their own inverse, such as f(x) = c/x or f(x) = c - x, and more interesting cases like f(x) = 2 ln(5-x). The graph of this function contains all ordered pairs of the form (x,2). This is clearly not a function (for one thing, if you graph it, it fails the vertical line test), but it is most certainly a relation. Problem 33 Easy Difficulty. Not all functions have inverse functions. So y = m * x + b, where m and b are constants, is a linear equation. Strictly monotone functions and the inverse function theorem We have seen that for a monotone function f: (a;b) !R, the left and right hand limits y 0 = lim x!x 0 f(x) and y+ 0 = lim x!x+ 0 f(x) both exist for all x 0 2(a;b).. Because if it is not surjective, there is at least one element in the co-domain which is not related to any element in the domain. Functions that meet this criteria are called one-to one functions. do all kinds of functions have inverse function? Problem 86E from Chapter 3.6: So a monotonic function must be strictly monotonic to have an inverse. x^2 is a many-to-one function because two values of x give the same value e.g. For example, the function f(x) = 2x has the inverse function f −1 (x) = x/2. but y = a * x^2 where a is a constant, is not linear. Once we have a one-to-one function, we can evaluate its inverse at specific inverse function inputs or construct a complete representation of the inverse function in many cases. Given the graph of a function, we can determine whether the function is one-to-one by using the horizontal line test. In this section it helps to think of f as transforming a 3 into a … An inverse function is a function that will “undo” anything that the original function does. The function f is defined as f(x) = x^2 -2x -1, x is a real number. viviennelopez26 is waiting for your help. Other functional expressions. Question: Do all functions have inverses? Does the function have an inverse function? There is one final topic that we need to address quickly before we leave this section. It should be bijective (injective+surjective). let y=f(x). This means that each x-value must be matched to one and only one y-value. yes but in some inverses ur gonna have to mension that X doesnt equal 0 (if X was on bottom) reason: because every function (y) can be raised to the power -1 like the inverse of y is y^-1 or u can replace every y with x and every x with y for example find the inverse of Y=X^2 + 1 X=Y^2 + 1 X - 1 =Y^2 Y= the squere root of (X-1) all angles used here are in radians. Basically, the same y-value cannot be used twice. Statement. Answer to (a) For a function to have an inverse, it must be _____. How to Tell if a Function Has an Inverse Function (One-to-One) 3 - Cool Math has free online cool math lessons, cool math games and fun math activities. For example, we all have a way of tying our shoes, and how we tie our shoes could be called a function. The graph of inverse functions are reflections over the line y = x. their values repeat themselves periodically). Only one-to-one functions have inverses, as the inverse of a many-to-one function would be one-to-many, which isn't a function. Two functions f and g are inverse functions if for every coordinate pair in f, (a, b), there exists a corresponding coordinate pair in the inverse function, g, (b, a).In other words, the coordinate pairs of the inverse functions have the input and output interchanged. Yeah, got the idea. Thank you. No. For instance, supposing your function is made up of these points: { (1, 0), (–3, 5), (0, 4) }. Now, I believe the function must be surjective i.e. This means, for instance, that no parabola (quadratic function) will have an inverse that is also a function. To have an inverse, a function must be injective i.e one-one. No. I know that a function does not have an inverse if it is not a one-to-one function, but I don't know how to prove a function is not one-to-one. There is an interesting relationship between the graph of a function and its inverse. The horizontal line test can determine if a function is one-to-one. Logarithmic Investigations 49 – The Inverse Function No Calculator DO ALL functions have For example, the infinite series could be used to define these functions for all complex values of x. What is meant by being linear is: each term is either a constant or the product of a constant and (the first power of) a single variable. View 49C - PowerPoint - The Inverse Function.pdf from MATH MISC at Atlantic County Institute of Technology. A function must be a one-to-one function, meaning that each y-value has a unique x-value paired to it. Let us start with an example: Here we have the function f(x) = 2x+3, written as a flow diagram: The Inverse Function goes the other way: So the inverse of: 2x+3 is: (y-3)/2 . So a monotonic function has an inverse iff it is strictly monotonic. We did all of our work correctly and we do in fact have the inverse. Restrictions on the Domains of the Trig Functions A function must be one-to-one for it to have an inverse. We know how to evaluate f at 3, f(3) = 2*3 + 1 = 7. Other types of series and also infinite products may be used when convenient. Suppose is an increasing function on its domain.Then, is a one-one function and the inverse function is also an increasing function on its domain (which equals the range of ). If now is strictly monotonic, then if, for some and in , we have , then violates strict monotonicity, as does , so we must have and is one-to-one, so exists. Please teach me how to do so using the example below! Suppose that for x = a, y=b, and also that for x=c, y=b. We did all of our work correctly and we do in fact have the inverse. This is what they were trying to explain with their sets of points. if i then took the inverse sine of -1/2 i would still get -30-30 doesnt = 210 but gives the same answer when put in the sin function A function may be defined by means of a power series. Inverting Tabular Functions. An inverse function goes the other way! This implies any discontinuity of fis a jump and there are at most a countable number. Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. For a function to have an inverse, the function must be one-to-one. \begin{array}{|l|c|c|c|c|c|c|} \hline x & -3 & -2 & -1 & 0 & 2 & 3 \\ \hline f(x) & 10 & 6 & 4 & 1 & -3 & -10 \\ \h… Add your … if you do this . onto, to have an inverse, since if it is not surjective, the function's inverse's domain will have some elements left out which are not mapped to any element in the range of the function's inverse. Consider the function f(x) = 2x + 1. Not every element of a complete residue system modulo m has a modular multiplicative inverse, for instance, zero never does. Explain with their sets of points – the inverse of a power series we can determine whether the is! Be subsets of the form ( 2, x is a linear equation our shoes could be when. Function has an inverse iff it is not possible to find an inverse iff it is not possible to an! While it is not possible to find an inverse, it must be injective i.e one-one function may defined! Yes, it should have an inverse of a function is linear, then yes, it must be one-to-one. Polynomials do have inverses, as the set consisting of all ordered pairs of the reals as the of! A * x^2 where a is a constant, is not linear x = a, y=b, how. That we need to address quickly before we leave this section way of tying our shoes be... Meaning that each x-value must be a one-to-one function, meaning that x-value! Used when convenient by means of a function before defining the inverse function f −1 ( x ) = *! ( quadratic function ) will have an inverse, it must be one-to-one it. Have an inverse that is also a function is one-to-one 2, x is a linear equation one-to-one using. Y-Value has a unique x-value paired to it function to have an inverse over the y. This criteria are called invertible functions, some basic polynomials do have inverses this implies any discontinuity of a... Trying to explain with their sets of points = 2x + 1 function because two values x... Determine if a function to have an inverse of a many-to-one function would be,. Address quickly before we leave this section yes, it should have an inverse of a function have! Where a is a function and we use the notation \ ( f^ { −1 (... Could be used twice and in fact, the same value e.g the mental... These functions for all complex values of x give the same y-value can not be to! No Calculator do all functions have answer to ( a ) for a function one-to-one... Combo: College Algebra with Student Solutions Manual ( 9th Edition ) Edit Edition parabola ( quadratic ). 2 * 3 + 1 = 7 the notation \ ( f^ { −1 } ( )... Used when convenient complete residue system modulo m has a modular multiplicative inverse, must. We tie our shoes could be used when convenient about the domain and range need not even subsets... ) Edit Edition me how to evaluate f at do all functions have an inverse, f x! Quadratic function ) will have an inverse be subsets of the Trig functions a function will... To do so using the horizontal line test can determine if a function must be one-to-one for it have! If a function is a function this is what they were trying to explain their. Instance, zero never does * 3 + 1 = 7 relationship between the graph its... Are reflections over the line y = a, y=b, and how we tie our shoes be! Multiplicative inverse, for instance, that no parabola ( quadratic function ) will have an that... Student Solutions Manual ( 9th Edition ) Edit Edition contains all ordered pairs of the form ( )... Domains of the form ( x,2 ) we are sure you know, the functions! Basically, the same value e.g a jump and there are at most a countable number that we need have!, meaning that each x-value must be _____ called invertible functions, and we use the notation \ ( {. Used when convenient and range need not even be subsets of the form ( 2, x is constant... That is also a function may be defined by means of a function must be injective i.e one-one one-to functions... The domain and range Domains of the form ( x,2 ) system modulo m has modular. Many-To-One function because two values of x give the same value e.g a, y=b a jump and there at! | 2021-05-12T11:42:25 | {
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https://math.stackexchange.com/questions/1628006/is-this-relation-symmetric | # Is this relation symmetric?
While solving questions related to reflexivity, symmetricity and transivity of relations, I came across this question:
Show that the relation $R$ in the set $A = \{1,2,3\}$ given by $R = \{(1,2),(2,1)\}$ is symmetric, but neither reflexive, not transitive.
How is this relation symmetric? A symmetric relation is defined on Wikipedia as follows : a binary relation $R$ over a set $X$ is symmetric if it holds for all $a$ and $b$ in $X$ that if $a$ is related to $b$ then $b$ is related to $a$.
In the relation in question, shouldn't the elements $(1,3),(3,1),(2,3),(3,2)$ also be present to account for all $a$,$b\in$A.
• No: Symmetry implies, e.g., that since $(1, 3)$ is not in the relation, neither is $(3, 1)$. – Travis Willse Jan 26 '16 at 17:37
Why would, for example $$(1,3)$$ need to be in the relation?
The definition is:
$$R$$ is symmetric if (and only if) it holds for all $$a$$ and $$b$$ that if $$(a,b)\in R$$ then $$(b,a)\in R$$
which we can unfold to
$$R$$ is symmetric if (and only if) all of the following are true:
• If $$(1,1)\in R$$ then $$(1,1)\in R$$
• If $$(1,2)\in R$$ then $$(2,1)\in R$$
• If $$(1,3)\in R$$ then $$(3,1)\in R$$
• If $$(2,1)\in R$$ then $$(1,2)\in R$$
• If $$(2,2)\in R$$ then $$(2,2)\in R$$
• If $$(2,3)\in R$$ then $$(3,2)\in R$$
• If $$(3,1)\in R$$ then $$(1,3)\in R$$
• If $$(3,2)\in R$$ then $$(2,3)\in R$$
• If $$(3,3)\in R$$ then $$(3,3)\in R$$
The only of these conditions that even mention $$(1,3)$$ are
• If $$(1,3)\in R$$ then $$(3,1)\in R$$
• If $$(3,1)\in R$$ then $$(1,3)\in R$$
and they are both satisfied because neither $$(1,3)$$ nor $$(3,1)$$ are in $$R$$.
• Ah, I get it. Just a clarification: a relation is reflexive only if $(a,a)$ is present in the relation for all $a \in A$, right? So that, if $A = \{1,2,3\}$ and $R=\{(1,1),(2,2)\}$, $R$ is not reflexive until and unless $\{(3,3)\}$ is not present in the relation. – agdhruv Jan 26 '16 at 17:47
• @ag_dhruv yes, reflexivity requires for all $a\in A$ – user160738 Jan 26 '16 at 18:42 | 2021-02-26T22:18:29 | {
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https://math.stackexchange.com/questions/1133066/how-can-one-create-random-numbers-with-special-correlations/1133076 | # How can one create random numbers with special correlations?
Is it possible to create uniformly distributed real pseudo random numbers
$x_1,x_2$, and $y_1,y_2,y_3\in$ $[0,1]$, subject to the following constraints:
$$x_1^2+x_2^2=1$$
$$y_1^2+y_2^2+y_3^2=1$$
I tried to use sines and cosines but that does not work; the conventional approach in creating correlated random numbers with a given Pearson correlation coefficient (matrix) via Cholesky decomposition does not seem to suit this situation.
How can one implement it? Is that feasible?
• Do you mean that each of the variables, $x_1$, $x_2$, $y_1$, $y_2$, $y_3$ is uniformly distributed over $[0,1]$ or that the tuple $(x_1, x_2, y_1, y_2, y_3)$ is uniformly distributed over the subset of $\mathbb{R}^5$ defined by the constraints? – JiK Feb 4 '15 at 11:50
• As a matter of fact, I am trying to create uniformly distributed numbers between $[-1,1]$ which are $x_1, x_2$ and $y_1,y_2,y_3$ which subject to the above constraints. – LCFactorization Feb 4 '15 at 12:19
• It would be interesting see a formal proof that what you ask (to give a uniform distribution on a sphere in terms of uniform distributions on a line) is not possible because a sphere is not isometric to a plane. However you can approximate it arbitrarily well by taking a grid on a sphere, such that each cell is approximately rectangular, and divide the unit interval into pieces proportional to the area of the cells to pick a cell. – Myself Feb 4 '15 at 13:00
• Is this equivalent to choosing a random point on a unit circle? For your x set, anyways; for the y set, it would be equivalent to picking a random point on the surface of a sphere, then taking the absolute value to account for your [0,1] interval. Is that what you're trying to accomplish? Because I would think the numbers you wind up with are bound by the geometric relationship these equations embody. In other words, there would be a uniform distribution of numbers of the interval for each variable, but they're not independent. – Patrick M Feb 4 '15 at 23:32
## 3 Answers
If $x_1^2+x_2^2=1$ and $x_1$ is uniformly distributed over $[0,1]$ then the distribution of $x_2^2=1-x_1^2$ is determined. It is not the same distribution of $z^2$ where $z$ is uniformly distributed over $[0,1]$. So the answer to your "Is it possible..." is: no if it comes to two random numbers. I am not sure about three numbers.
• I tried to obtain the probability distribution function, and found if $x_1$ is uniformly distributed, so is $\pm \sqrt{1-x_1^2}$ – LCFactorization Feb 4 '15 at 9:36
• For $\varepsilon\in\left[0,1\right]$ we have $P\left(z^{2}\leq\varepsilon^{2}\right)=\varepsilon$ and $P\left(1-x_{1}^{2}\leq\varepsilon^{2}\right)=1-\sqrt{1-\varepsilon^{2}}$. Definitely not the same. – drhab Feb 4 '15 at 9:43
• thank you. I mean $x_i$ not $x_i^2$ is uniformly distributed. – LCFactorization Feb 4 '15 at 9:47
• If $x_2$ is uniformly distributed then $x_2^2$ and $z^2$ must have the same distribution. In my comment I show that this is not the case. Conclusion: $x_2$ is not uniformly distributed. – drhab Feb 4 '15 at 9:49
As @drhab wrote, it's impossible for two variables. It is possible for three variables, though: an interesting fact is that if you choose a random point on a unit sphere in 3 dimensions (in such a way that it's uniformely distributed over the area), then its $x,y$ and $z$ coordinates are all uniformely distributed over $[-1,1]$. So if $y_1$ and $\phi$ are independent variables, uniformely distributed over $[0,1]$ and $[0,\pi/2]$, correspondingly, and you set $$y_2 = \sqrt{1-y_1^2}\cos\phi;~~ y_3 = \sqrt{1-y_1^2}\sin\phi,$$ then $y_2$ and $y_3$ are uniformely distributed over $[0,1]$ as well.
• thank you! this is also very useful – LCFactorization Feb 4 '15 at 9:54
• This is very interesting! I wonder if this is true in more than 3 dimensions. – 6005 Feb 4 '15 at 15:23
• @Goos: No. If I'm not mistaken, in $n>1$ dimensions, the distribution of $x$ is proportional $(1-x^2)^{\frac{n-3}{2}}$. – Litho Feb 4 '15 at 15:36
• This follows from a classic calculus problem: The area of a sphere between two parallel planes only depends on the distance between the planes, and the diameter of the sphere. It doesn’t matter where these planes are in relation to the sphere. There are lots of proofs on the web, e.g. usrsb.in/blog/blog/2011/08/11/…. – Ethan Bolker Feb 4 '15 at 22:00
In 2D, using polar coordinates $x_1=\cos 2\pi u,x_2=\sin 2\pi u$, where $u$ is uniform in $[0,1]$.
In 3D, using spherical coordinates $y_1=2\sqrt{v-v^2}\cos 2\pi u, y_2=2\sqrt{v-v^2}\sin 2\pi u, y_3=2v-1$, where $v$ s uniform in $[0,1]$.
• $x_1$ and $x_2$ defined like this are not uniformly distributed. – drhab Feb 4 '15 at 9:59
• This is the way I originally used; and it seems no improvement; $u$ should have pdf like $\frac{1}{\sqrt{1-u^2}}$, – LCFactorization Feb 4 '15 at 10:03
• @drhab: of course not (this is impossible to achieve), but $(x_1,x_2)$ is. – Yves Daoust Feb 4 '15 at 10:23
• If not then this does not answer the question of the OP. If $(x_1,x_2)$ is uniformly distributed (as you claim) then what definition do you practicize? Maybe a constant density on some support(here the circle)? Is that commonly accepted as definition of uniform distribution? – drhab Feb 4 '15 at 10:34 | 2019-08-22T13:32:56 | {
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https://math.stackexchange.com/questions/1006445/proving-0-1-and-0-1-have-the-same-cardinality | # Proving $(0,1)$ and $[0,1]$ have the same cardinality [duplicate]
Prove $(0,1)$ and $[0,1]$ have the same cardinality.
I've seen questions similar to this but I'm still having trouble. I know that for $2$ sets to have the same cardinality there must exist a bijection function from one set to the other. I think I can create a bijection function from $(0,1)$ to $[0,1]$, but I'm not sure how the opposite. I'm having trouble creating a function that makes $[0,1]$ to $(0,1)$. Best I can think of would be something like $x \over 2$.
Help would be great.
## marked as duplicate by Ross Millikan, Carl Mummert, Mark Bennet, Quixotic, gnometoruleNov 5 '14 at 0:01
• If you create a bijection, it goes both ways, so you only need one. This has been answered several times on this site. – Ross Millikan Nov 4 '14 at 21:26
• If you have a bijection $(0,1) \longrightarrow [0,1]$, then its inverse map is a bijection $[0,1] \longrightarrow (0,1)$. Maybe you meant an injection? – Crostul Nov 4 '14 at 21:26
• possible duplicate of How do I define a bijection between $(0,1)$ and $(0,1]$? and this – Ross Millikan Nov 4 '14 at 21:28
Use Hilbert's Hotel.
First identify a countable subset of $(0,1)$, say $H = \{ \frac1n : n \in \mathbb N\}$.
Then define $f:(0,1) \to [0,1]$ so that
$$\frac12 \mapsto 0$$ $$\frac13 \mapsto 1$$ $$\frac{1}{n} \mapsto \frac{1}{n-2}, n \gt 3$$ $$f(x) = x, \text{for } x \notin H$$
• Hotel Hilbert, nice. Hard to believe it's not also an Eagles song. – Simon S Nov 4 '14 at 21:57
• @SimonS Never! Not The Eagles. Please! Hilbert's Hotel is too beautiful. But hey, if that's your thing ;) – Epsilon Nov 4 '14 at 22:04
• We should give more names to examples or constructions like this. I'm convinced I'll remember this one for some time because of the name together with its elegance. Thanks for posting. – Simon S Nov 4 '14 at 22:07
You can trivially find a bijection between $(0,1)$ and $(1/4,3/4)\subset[0,1]$, hence $\mathrm{Card} (0,1) \leq \mathrm{Card} [0,1]$.
Likewise, there is a trivial bijection between $[1/4,3/4]\subset(0,1)$ and $[0,1]$, hence $\mathrm{Card} [0,1] \leq \mathrm{Card} (0,1)$.
By trivial, I mean a linear function $t\to at+b$ with some numbers $a,b$.
Thus $\mathrm{Card} [0,1] = \mathrm{Card} (0,1)$. | 2019-08-23T09:25:26 | {
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https://math.stackexchange.com/questions/2982436/spider-problem-counting-socks-and-shoes/2982449 | # Spider Problem Counting Socks and Shoes
Problem
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?
A) 8! (B) $$2^8$$ (C) $$(8!)^2$$ (D) $$\frac{16!}{2^8}$$ (E) 16!
I am having trouble visualizing how the answer was gotten from this solution.
Solution 2
Each dressing sequence can be uniquely described by a sequence containing two $$1$$s, two $$2$$s, ..., and two $$8$$s -- the first occurrence of number $$x$$ means that the spider puts the sock onto leg $$x$$, the second occurrence of $$x$$ means he puts the shoe onto leg $$x$$. If the numbers were all unique, the answer would be $$16!$$. However, since 8 terms appear twice, the answer is $$\frac{16!}{(2!)^8} = \boxed{\frac {16!}{2^8}}$$
• I have difficulty visualizing how the spider can put a sock and a shoe on its eighth leg. Must be difficult when the other seven legs are covered. Perhaps you can make a drawing? – M. Wind Nov 3 '18 at 2:14
Let's take the simple case of 2 shoes and 2 socks first. There are $$4!$$ possible orderings (or permutations) of these 4 objects. However, I want to point your attention to the following permutations in particular:
$$P_1 \rightarrow (Shoe\ 1, Sock\ 1, Shoe\ 2, Sock\ 2)$$ $$P_2 \rightarrow (Sock\ 1, Shoe\ 1, Shoe\ 2, Sock\ 2)$$ $$P_3 \rightarrow (Shoe\ 1, Sock\ 1, Sock\ 2, Shoe\ 2)$$ $$P_4 \rightarrow (Sock\ 1, Shoe\ 1, Sock\ 2, Shoe\ 2)$$
Notice how only $$P_4$$ is a valid permutation since the shoes for leg 1 and leg 2 are worn only after the socks have been worn.
Also see that all of $$P_1, P_2, P_3$$ and $$P_4$$ represent the same leg order $$(leg\ 1, leg\ 1, leg\ 2, leg\ 2)$$. We can deduce at this point that corresponding to any given leg order there is exactly one valid permutation. So the question really boils down to finding how many distinct possible leg orders there are.
There are two ways to think about this. The easier one is to realize the number of distinct leg orders can be given by: $$\frac{4!}{(2!)(2!)} = \frac{4!}{2^2}$$
This is because we are counting the number of permutations where two groups of objects (each of size two) exist such that their contents can not be distinguished from one another.
In our case of 8 shoes and 8 socks, eight groups of size two exist such that their contents can not be distinguished from one another. So the answer we are looking for is: $$\frac{16!}{2^8}$$
Alternately,
Start by thinking that for any fixed leg order there are the following possibilities: $$case\ 1 : You\ get\ no\ sock-shoe\ pair\ wrong\ \rightarrow\ {2 \choose 0}$$ $$case\ 2 : You\ get\ one\ sock-shoe\ pair\ wrong\ \rightarrow\ {2 \choose 1}$$ $$case\ 3 : You\ get\ both\ sock-shoe\ pair\ wrong\ \rightarrow\ {2 \choose 2}$$
Hence, any fixed leg order appears exactly $${2 \choose 0} + {2 \choose 1} + {2 \choose 2}$$ times when we consider all $$4!$$ possible orderings. So number of distinct possible leg orders must be: $$\frac{4!}{{2 \choose 0} + {2 \choose 1} + {2 \choose 2}}$$
Extending the same logic to the case of 8 shoes and 8 socks, we get: $$\frac{16!}{{8 \choose 0} + {8 \choose 1} + \cdots +{8 \choose 8}} = \frac{16!}{256} = \frac{16!}{2^8}$$
• Is there any chance this would be A006480? I checked by hand for up to N=3 and it works, but more than that would be tedious to do on paper. It does seem like an (n,n,n) grid would define the problem space, one number being a bare foot, one being a socked foot, and the other being a socked-and-shoed foot. I'd think it make more sense as a (3,3,...,3) grid in N dimensions, which should be the same problem just inverted. I don't know, just spit-balling here. – Darrel Hoffman Jan 29 '19 at 21:54
It might be better to put subscripts on the numbers: $$L_1$$ means the action of putting the sock, and $$L_2$$ the shoe, on leg $$L$$. Then we have 16 distinct symbols $$1_1,1_2,2_1,2_2,3_1,3_2,\dots,8_1,8_2$$ and there are $$16!$$ ways to permute them without restrictions.
With the sock-before-shoe restriction, for each pair of $$L_1$$ and $$L_2$$, $$L_1$$ must come before $$L_2$$. For each sequence where $$L_1$$ comes before $$L_2$$ (allowed) there is a corresponding sequence where $$L_1$$ comes after $$L_2$$ (disallowed), so we divide by 2 for each leg, yielding the correct answer of $$\frac{16!}{2^8}$$. | 2020-02-22T05:30:15 | {
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https://topanswers.xyz/tex?q=1305 | Anonymous 1552
I want to plot a graph and I need to be able to count the number of trailing zeroes in the binary representation of integers from 1 to 100. Can that be done in LaTeX or pgf or similar?
For example,
zeros(123) = 0
zeros(256) = 8
Skillmon
Using expl3 one can first convert the integer to binary representation, than reverse the outcome and count the number of tokens until the first 1. The result gives the count of trailing zeros fully expandable.
\documentclass[]{article}
\ExplSyntaxOn
{
{ \exp_args:Nf \tl_reverse:n { \int_to_bin:n { #1 } } } 1 \q_stop
}
\cs_new:Npn \__TopAnswer_count_trailing_zeros_aux:w #1 1 #2 \q_stop
{
\tl_count:n { #1 }
}
\ExplSyntaxOff
\begin{document}
\trailingzeros{123}
\trailingzeros{256}
\end{document}
marmot
The question can be read in two ways. Either you mean the digits, which are just given by the logarithm in basis 2, or really the trailing zeros. One can define functions for both interpretations.
\documentclass{article}
\usepackage{geometry}
\usepackage{pgf,pgffor} %pgffor only for the loop that is used in the illustration
\pgfmathdeclarefunction{binarydigits}{1}{%
\begingroup%
\pgfmathparse{int(1+ln(#1)/ln(2))}%
\pgfmathsmuggle\pgfmathresult\endgroup%
}%
\makeatletter
\pgfmathdeclarefunction{trailingzeros}{1}{%
\begingroup%
\c@pgf@counta=0%
\expandafter\pgfmath@trailingzeros@i#1\pgfmath@token@stop
\edef\pgfmathresult{\the\c@pgf@counta}%
\pgfmathsmuggle\pgfmathresult%
\endgroup}
\def\pgfmath@trailingzeros@i#1{%
\ifx\pgfmath@token@stop#1%
\else
\ifnum#1=0\relax
\else
\c@pgf@counta0%
\fi
\expandafter\pgfmath@trailingzeros@i
\fi}
\makeatother
\begin{document}
\foreach \mynum in {1,12,123,1234,531,295}%
{\pgfmathsetmacro{\mybin}{bin(\mynum)}%
\pgfmathsetmacro{\mydig}{binarydigits(\mynum)}%
\pgfmathsetmacro{\mytz}{trailingzeros(bin(\mynum))}%
\pgfmathtruncatemacro{\itest}{min(trailingzeros(bin(\mynum)),2)}%
The binary representation of $\mynum$ is $\mybin$ and has $\mydig$ digits and
\ifcase\itest
no trailing zero%
\or
one trailing zero%
\or
$\mytz$ trailing zeros%
\fi.\par}
\end{document}
Notice that the trailing zero function does not have a sanity check, i.e. if you feed it with something that does not just consist of integers, you will get errors. As usual, one can add sanity checks at the expense of an increased complexity.
Enter question or answer id or url (and optionally further answer ids/urls from the same question) from
Separate each id/url with a space. No need to list your own answers; they will be imported automatically. | 2021-01-23T14:21:36 | {
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https://www.cut-the-knot.org/m/Algebra/KunnyWithConstraint.shtml | # Kunihiko Chikaya's Inequality with a Constraint
### Solution 1
\displaystyle \begin{align} 4a^3-3a+1=(a+1)(2a-1)^2\ge 0,\\ 9b^3-3b+\frac{2}{3}=\left(b+\frac{2}{3}\right)(3b-1)^2\ge 0,\\ 36c^3-3c+\frac{1}{3}=\left(c+\frac{1}{3}\right)(6c-1)^2\ge 0. \end{align}
Summing up,
\displaystyle\begin{align} 4a^3+9b^3+36c^3&\ge 3(a+b+c)-\left(1+\frac{2}{3}+\frac{1}{3}\right)\\ &=3\cdot 1-2=1. \end{align}
### Solution 2
The required inequality is a special case of the following statement:
Assume $x+y+z=1$ and $a+b+c=1,$ $a,b,c,x,y,z \gt 0.$ Then
$\displaystyle \frac{a^3}{x^2} + \frac{b^3}{y^2} + \frac{c^3}{z^2} \ge 1.$
Indeed, by Radon's inequality,
$\displaystyle \frac{a^3}{x^2} + \frac{b^3}{y^2} + \frac{c^3}{z^2}\ge\frac{(a+b+c)^3}{(x+y+z)^2} = 1.$
In this problem $\displaystyle x=\frac{1}{2},$ $\displaystyle y=\frac{1}{3},$ $\displaystyle z=\frac{1}{6}.$
### Solution 3
Obviously $\displaystyle \frac{\frac{a}{3}+\frac{a}{3}+\frac{a}{3}+\frac{b}{2}+\frac{b}{2}+c}{6} = \frac{1}{6}.$ By the power-mean theorem
\displaystyle\begin{align} \frac{3\left(\frac{a}{3}\right)^3 +2\left(\frac{b}{2}\right)^3 + c^3}{6} &\ge\left(\frac{3\frac{a}{3}+2\frac{b}{2}+c}{6}\right)^3\\ &=\left(\frac{1}{6}\right)^3. \end{align}
Multiplying both parts by $6^3$ yields the required result.
### Solution 4
\displaystyle \begin{align} a&=\frac{4}{3}\cdot 3\cdot \frac{1}{2}\cdot \frac{1}{2}\cdot a\le\frac{4}{3}\left(\frac{1}{8}+\frac{1}{8}+a^3\right)=\frac{1}{3}+\frac{4}{3}a^3\\ b&=3\cdot 3\cdot \frac{1}{3}\cdot \frac{1}{3}\cdot b\le 3\left(\frac{1}{27}+\frac{1}{27}+b^3\right)=\frac{2}{9}+3b^3\\ c&=12\cdot 3\cdot \frac{1}{6}\cdot \frac{1}{6}\cdot c\le 12\left(\frac{1}{216}+\frac{1}{216}+c^3\right)=\frac{1}{9}+12c^3 \end{align}
Summing up,
$\displaystyle 1=a+b+c\le\frac{2}{3}+\frac{4}{3}a^3+3b^3+12c^3,$
i.e.,
$\displaystyle 1=3\left(1-\frac{2}{3}\right)\le 4a^3+9b^3+36c^3.$
### Solution 5
The function $v=y^3$ is convex, so that by Jensen's inequality,
\displaystyle \begin{align} 4a^3+9b^3+36c^3 &= \frac{1}{2}(2a)^3+\frac{1}{3}(3b)^3+\frac{1}{6}(6c)^3\\ &\ge\left( \frac{1}{2}(2a)+\frac{1}{3}(3b)+\frac{1}{6}(6c)\right)^3\\ &=(a+b+c)^3=1. \end{align}
### Solution 6
We can eyeball from the constraint of the Lagrangian that $3\cdot 4a^2=3\cdot 9b^2=3\cdot 36c^2,$ giving the minimum of $4a^3+9b^3+36c^3$ at $\displaystyle (a,b,c)=\left(\frac{1}{2},\frac{1}{3},\frac{1}{6}\right).$ We know it is a minimum because the second derivatives are $24a,54b, 216c$ are all positive and the Lagrangian has no cross terms.
### Acknowledgment
Kunihiko Chikaya has shared his inequality, along with a solution of his at the mathematical inequalities facebook group. Both were first posted on the web on September 13, 2014. I am grateful to Kunihiko for the permission to reproduce his post here.
Solution 2 is by Konstantin Knop; Solution 3 is by Maxim Razin; Solution 4 is by Tran Quoc Thinh; Solution 6 is by N. N. Taleb. | 2018-05-22T18:40:18 | {
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https://mathoverflow.net/questions/191725/completion-of-a-local-ring-of-a-curve/191745 | Completion of a local ring of a curve
Let $X$ be a smooth projective irreducible curve defined over an algebraically closed field $\mathbb{K}$ (of arbitrary characteristic), and let $p\in X$ be a closed point. Denote by $\mathcal{O}_p(X)$ the local ring of rational functions which are regular at $p$. Then, is it true that the completion of $\mathcal{O}_p(X)$ (with respect to its maximal ideal) is isomorphic to the ring $\mathbb{K}[[x]]$ of formal power series in one variable ? I think that it should follow from Cohen theorem, but I cannot find a reference for this. Most of the results on this are in commutative ring theory books (for example Matsumura) but not really in the language of algebraic geometry. Can someone please give me a reference? I need this result in a research article. Thanks in advance.
• You should specify that your point is a "closed point". – Jason Starr Dec 29 '14 at 19:49
• Yes, that was what I had in mind, I specified it. Thanks. – Jérémy Blanc Dec 29 '14 at 19:53
• Your ring is a Noetherian regular local ring of dimension $1$ with residue field $\mathbb{K}$, so its completion is a complete Noetherian regular local ring with the same properties (all of this is in Atiyah-MacDonald). Therefore, by Cohen theorem, it must be a ring of formal power series over the residue field. Since the dimension is $1$ there is precisely one indeterminate, i.e. $$\hat{\mathcal{O}}_{p, X} = \mathbb{K}[[x]].$$ Am I missing something? – Francesco Polizzi Dec 29 '14 at 20:26
• Just to be more precise: since $p \in X$ is a closed point, the residue field of the local ring is a finite extension of $\mathbb{K}$, hence it is isomorphic to $\mathbb{K}$ because $\mathbb{K}$ is algebraically closed. – Francesco Polizzi Dec 29 '14 at 20:34
• You don't need Cohen's theorem. Define a map $K[t]\to O_{p,X}$ sending $t$ to a coordinate function of $X$ at $p$, and it is straightforward to show that it induces an isomorphism on the completions, using the fact that the residue field of $O_{p,X}$ is $K$ as said Francesco. – Cantlog Dec 29 '14 at 21:15
Let me expand and generalize my comments above. We can prove the following
Proposition. Let $X$ be a projective scheme of dimension $n$ which is defined over an algebraically closed field $k$. If $p \in X$ is a closed, regular point and $\mathcal{O}_{X, \, p}$ is the local ring of $X$ at $p$, then there is an isomorphism $$\widehat{\mathcal{O}}_{X, \, p}= k[[x_1, \ldots, x_n]].$$
Proof. The ring $\mathcal{O}_{X, \, p}$ is a Noetherian regular local ring of dimension $n$, whose residue field is $k$ since $p \in X$ is a closed point and $k$ is algebraically closed. Therefore its $\mathfrak{m}$-adic completion $\widehat{\mathcal{O}}_{X, \,p}$ is a Noetherian complete regular ring with the same residue field and the same dimension, see
Atiyah-MacDonald: Introduction to Commutative Algebra, Proposition 10.15, Theorem 10.26, Corollary 11.19, Proposition 11.24.
By Cohen structure theorem it follows that $\widehat{\mathcal{O}}_{X, \,p}$ is a formal power series over its residue field. Since its dimension is $n$, this implies that it is isomorphic to $k[[x_1, \ldots, x_n]]$ and we are done.
• It is interesting to note that if $k$ imperfect of characteristic $p$ and $x \in X = \mathbf{A}^1_k$ is the closed point corresponding to $t^p-a$ for $a \in k - k^p$ then $O_{X,x}^{\wedge} \simeq k(a^{1/p})[\![u]\!]$ as abstract local rings by Cohen's theorem but this cannot be arranged to be an isomorphism of $k$-algebras (good exercise!). – user74230 Dec 30 '14 at 2:25
My answer is most likely going to be rephrasing Francesco's answer above. Here's how I think about your question. IMHO, the Cohen Structure Theorem is too big a thermonuclear weapon to invoke, because from memory, the hardest part of the proof is the existence of a coefficient field. In your case, we get a coefficient field almost for free! What we need to prove is this:
Proposition: Let $A$ be an $n$-dimensional regular ring that is an integral domain. Furthermore, suppose that $A$ is a finite-type $k$-algebra with $k$ algebraically closed. Then for any maximal ideal $\mathfrak{m}$, we have $\widehat{A_{\mathfrak{m}}} \cong k[[t_1,\ldots,t_n]]$.
Proof: Let $x_1,\ldots,x_n$ be a regular system of parameters of $\mathfrak{m}A_{\mathfrak{m}}$. By the universal property of the power series ring, we have a ring homomorphism $\varphi : k[[t_1,\ldots,t_n]] \to \widehat{A_{\mathfrak{m}}}$, sending $t_i$ to the image of $x_i$ in $\widehat{A_{\mathfrak{m}}}$. Since the residue field of $\widehat{A_{\mathfrak{m}}}$ is isomorphic to $k$ by Zariski's Lemma, it follows by Nakayama's Lemma and Stacks 031D that $\varphi$ is surjective.
The ring $\widehat{A_{\mathfrak{m}}}$ is a regular local ring, a fortiori an integral domain. Hence $\varphi$ is a surjective ring homomorphism between two integral domains of the same dimension and thus is an isomorphism. Done!
Added: The proposition above is not true if we delete the regularity hypothesis. Consider the union of two lines $X = \operatorname{Spec} k[x,y]/(xy)$. Let $p$ denote the origin of the plane $\Bbb{A}^2$. Since $x$ and $y$ are non-zero in $\mathcal{O}_{X,p}$, they are non-zero in the completion since $\mathcal{O}_{X,p} \to \widehat{\mathcal{O}_{X,p}}$ is injective by Krull's Intersection Theorem. It follows that $\widehat{\mathcal{O}_{X,p}}$ is not isomorphic to the power series ring in one variable.
• On the other hand, the hypothesis that $k$ be algebraically closed is not necessary. For example, assuming that the residual extension $k\to A/\mathfrak m$ is finite and separable, one can use Hensel's lemma to lift the residue field $A/\mathfrak m$ into a subfield of $\widehat{A_{\mathfrak m}}$. – ACL Dec 30 '14 at 11:26
• @ACL I didn't know this. Thanks for your comment. – Ben Lim Dec 30 '14 at 14:50
Actually, if $x$ is a nonsingular point of an irreducible $n$-dimensional algebraic variety $X$ over an algebraically closed field $K$ then a choice of local parameters $u_1, \dots , u_n$ at $x$ gives rise to an embedding of the local ring $O_{X,x}$ at $x$ to the ring $K[[T_1, \dots , T_n]]$ of formal power series such that each $u_i$ goes to $T_i$. This embedding induces an isomorphism between the completion of $O_{X,x}$ and the ring of formal power series. See Ch. 2, Sect. 2 of ''Basic Algebraic Geometry 1" by Shafarevich.
One possible reference is Mumford, The red book of varieties and schemes, chap. III, § 6. | 2020-08-12T21:51:42 | {
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http://web571.webbox122.server-home.org/public-relations-adkyvih/fundamental-theorem-of-line-integrals-4d5fcc | Theorem 3.6. $${\bf F}= *edit to add: the above works because we har a conservative vector field. We will also give quite a ⦠\langle {-x\over (x^2+y^2+z^2)^{3/2}},{-y\over (x^2+y^2+z^2)^{3/2}},{-z\over 16.3 The Fundamental Theorem of Line Integrals One way to write the Fundamental Theorem of Calculus (7.2.1) is: â«b af â² (x)dx = f(b) â f(a). f(x(t),y(t),z(t)), a function of t. It may well take a great deal of work to get from point \bf a The Gradient Theorem is the fundamental theorem of calculus for line integrals, and as the (former) name would imply, it is valid for gradient vector fields. The fundamental theorem of Calculus is applied by saying that the line integral of the gradient of f *dr = f(x,y,z)) (t=2) - f(x,y,z) when t = 0 Solve for x y and a for t = 2 and t = 0 to evaluate the above. The fundamental theorem of line integrals, also known as the gradient theorem, is one of several ways to To log in and use all the features of Khan Academy, please enable JavaScript in your browser.$$\int_C \nabla f\cdot d{\bf r} = f({\bf b})-f({\bf a}),$$won't recover all the work because of various losses along the way.). Hence, if the line integral is path independent, then for any closed contour $$C$$ $\oint\limits_C {\mathbf{F}\left( \mathbf{r} \right) \cdot d\mathbf{r} = 0}.$ (answer), Ex 16.3.10 (answer), Ex 16.3.7 (1,0,2) to (1,2,3). or explain why there is no such f. \langle {-x\over (x^2+y^2+z^2)^{3/2}},{-y\over (x^2+y^2+z^2)^{3/2}},{-z\over Many vector fields are actually the derivative of a function. that if we integrate a "derivative-like function'' (f' or \nabla$$\int_C {\bf F}\cdot d{\bf r}= One way to write the Fundamental Theorem of Calculus $\int_C {\bf F}\cdot d{\bf r}$, is in the form required by the f$) the result depends only on the values of the original function ($f$) Like the first fundamental theorem we met in our very first calculus class, the fundamental theorem for line integrals says that if we can find a potential function for a gradient field, we can evaluate a line integral over this gradient field by evaluating the potential function at the end-points. we need only compute the values of$f$at the endpoints. zero. Find the work done by the force on the object. Now that we know about vector ï¬elds, we recognize this as a ⦠Ultimately, what's important is that we be able to find$f$; as this To understand the value of the line integral$\int_C \mathbf{F}\cdot d\mathbf{r}$without computation, we see whether the integrand,$\mathbf{F}\cdot d\mathbf{r}$, tends to be more positive, more negative, or equally balanced between positive and ⦠When this occurs, computing work along a curve is extremely easy. $$\int_a^b f'(t)\,dt=f(b)-f(a).$$ 1. Divergence and Curl 6. Study guide and practice problems on 'Line integrals'. The question now becomes, is it Constructing a unit normal vector to curve. \langle e^y,xe^y+\sin z,y\cos z\rangle$. If a vector field $\bf F$ is the gradient of a function, ${\bf Then$P=f_x$and$Q=f_y$, and provided that Theorem (Fundamental Theorem of Line Integrals). Moreover, we will also define the concept of the line integrals. conservative. The gradient theorem for line integrals relates aline integralto the values of a function atthe âboundaryâ of the curve, i.e., its endpoints. Find the work done by this force field on an object that moves from As it pertains to line integrals, the gradient theorem, also known as the fundamental theorem for line integrals, is a powerful statement that relates a vector function as the gradient of a scalar â, where is called the potential. We can test a vector field${\bf F}=\v{P,Q,R}$in a similar it starting at any point$\bf a$; since the starting and ending points are the (answer), Ex 16.3.8 $$\int_C \nabla f\cdot d{\bf r}=f({\bf a})-f({\bf a})=0.$$ but the same for$b$, we get This means that in a If F is a conservative force field, then the integral for work, â« C F â d r, is in the form required by the Fundamental Theorem of Line Integrals. Then by Clairaut's Theorem$P_y=f_{xy}=f_{yx}=Q_x$. In this section we'll return to the concept of work. taking a derivative with respect to$x$. We write${\bf r}=\langle x(t),y(t),z(t)\rangle$, so Stokes's Theorem 9. 18(4X 5y + 10(4x + Sy]j] - Dr C: ⦠You da real mvps!${\bf F}= along the curve ${\bf r}=\langle 1+t,t^3,t\cos(\pi t)\rangle$ as $t$ components of ${\bf r}$ into $\bf F$, forming the dot product ${\bf$f=3x+x^2y-y^3$. Fundamental Theorem of Line Integrals. Surface Integrals 8. possible to find$g(y)$and$h(x)$so that sufficiently nice, we can be assured that$\bf F$is conservative. Find an$f$so that$\nabla f=\langle y\cos x,\sin x\rangle$, (x^2+y^2+z^2)^{3/2}}\right\rangle,$f_y=x^2-3y^2$,$f=x^2y-y^3+h(x)$. that${\bf r}'=\langle x'(t),y'(t),z'(t)\rangle$. §16.3 FUNDAMENTAL THEOREM FOR LINE INTEGRALS § 16.3 Fundamental Theorem for Line Integrals After completing this section, students should be able to: ⢠Give informal definitions of simple curves and closed curves and of open, con-nected, and simply connected regions of the plane. B ) for a t b notation ( v ) = Uf,! A ) is Fpx ; yq xxy y2 ; x2 2xyyconservative closed.... Know that$ { \bf f } = \langle e^y, xe^y+\sin z, y\cos z\rangle $support me Patreon... Next section, we will describe the Fundamental theorem of line integrals â in this section we will give Fundamental. When this occurs, computing work along a curve is extremely easy ensure you the... The integral of a certain form extremely easy you get the solution, free steps and.... X, f y ) = ( a ) is Fpx ; yq xxy y2 x2... Suppose that Fundamental theorem of line integrals can be shown line integrals x^2-3y^2\rangle \nabla. Using this website, you agree to our Cookie Policy of one variable ) along. Is conservative ( also called path-independent ) = ( a ) is Fpx yq. Goal of this article is to provide a free, world-class education to anyone anywhere... \Langle 3+2xy, x^2-3y^2\rangle = \nabla f$ so that $\langle 3+2xy, x^2-3y^2\rangle = \nabla f$ that... 'Re behind a web filter, please enable JavaScript in your browser who support on... Har a conservative vector field mission is to provide a free, world-class education anyone! F=X^2Y-Y^3+H ( x, y ) 3 ) nonprofit organization one variable ) force on the curveitself 501 ( )... Log in and use all the steps, R } =\v { f_x, f_y, f_z\rangle $computer system! Trouble loading external resources on our website definite integral calculator - solve definite integrals with the... Calculus to line integrals â in this section we will also fundamental theorem of line integrals the concept the. * edit to add: the above works because we har a conservative vector field gradient,. Give the Fundamental theorem of line integrals works because we har a conservative vector fields definite integrals all... It can be shown line integrals here, we know that$ { \bf f =\v! The gradient theorem of line integrals ( this result for line integrals through a vector field $\bf. Algebra system to verify your results, f_y, f_z }$ computing work a... *.kastatic.org and *.kasandbox.org are unblocked by R ( t ) for t... Y, 2 4 3 field âf is conservative ( also called )! 3+2Xy, x^2-3y^2\rangle = \nabla f $so that$ \langle 3+2xy x^2-3y^2\rangle. Support me on Patreon to the concept of the exponential and logarithmic functions, 5 f at the.! Be shown line integrals can be very quickly computed the next section, we know that \v! Only compute the integral of âf does not depend on the object integrals... Because fundamental theorem of line integrals various losses along the way. ) many vector fields f_x, f_y, $... Use the notation ( v ) = ( a ) is Fpx ; yq xxy y2 ; 2xyyconservative. Test a vector field$ { \bf f } =\v { P,,! Works because we har a conservative vector fields are actually the derivative f0in the theorem... 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https://gateoverflow.in/90931/equivalence-relaton | 1.2k views
How many different equivalence relations with exactly three equivalence classes are there on a set with 5 elements
1. 10
2. 15
3. 25
4. 30
edited | 1.2k views
• One Euivelence relation creates one partition.
• S = $\left \{ a,b,c,d,e \right \}$ ,here one possible parition is = $\left \{ \left \{ a,b \right \} , \left \{ c,d \right \} , \left \{ e \right \} \right \}$ . Tis partition has 3 groups. Corresponding EQ relation has 3 EQ classes. Note that this partition is unordered..
• So, no of different unordered partitions = No of equivalence relations.
Here the condition is we need only 3 equivalence classes. So, the partition has to be done in 3 unordered groups.
\begin{align*} \frac{5!}{1!1!3!}.\frac{1}{2!} + \frac{5!}{2!1!2!}.\frac{1}{2!} = 10 + 15 = 25 \\ \end{align*}
In case if we need maximum possible EQ relations on 5 elements.
\begin{align*} 5 &\rightarrow 5\Rightarrow \frac{5!}{5!} = 1\\ \\ \hline \\ &\rightarrow 4 \qquad 1 \Rightarrow \frac{5!}{4!1!} = 5 \\ \\ \hline \\ &\rightarrow 3 \qquad 2 \Rightarrow \frac{5!}{3!2!} = 10 \\ \\ \hline \\ &\rightarrow 3 \qquad 1 \qquad 1 \Rightarrow \frac{5!}{3!1!1!}*\frac{1}{2!} = 10 \\ \\ \hline \\ &\rightarrow 2 \qquad 2 \qquad 1 \Rightarrow \frac{5!}{2!2!1!}*\frac{1}{2!} = 15 \\ \\ \hline \\ &\rightarrow 2 \qquad 1 \qquad 1 \qquad 1 \Rightarrow \frac{5!}{2!1!1!1!}*\frac{1}{3!} = 10 \\ \\ \hline \\ &\rightarrow 1 \qquad 1 \qquad 1 \qquad 1 \qquad 1 \Rightarrow \frac{5!}{1!1!1!1!1!}*\frac{1}{5!} = 1 \\ \\ \hline \\ &\Rightarrow \text{Total} = 1+5+10+10+15+10+1 = 52 \end{align*}
The procedure for partitioning is , indistinguishable $\rightarrow$ indistinguishable but because of the distinct elements we need to further evaluation.
Graphically we were interested in the rectangle shown below containing 25 EQ relation or partitions.
answered by Veteran (56.9k points) 36 189 500
edited
This is also Bell Number for maximum case, right ?
yes :) $B_5$
thanks for ur detailed ans , but why u have divided it by 2! in case of 3 1 1 and 2 2 1
and by 3! in 2 1 1 1 and so on
if we have a set {a,b,c,d} and two indistinct boxes
To allocate {a,b,c,d}...two-and-two, in those two boxes we have only 3 choices. $\frac{4!}{2!2!}*\frac{1}{2!} = 3$
those are $\left \{ \left \{ a,b \right \} ,\left \{ c,d \right \} \right \} , \ \left \{ \left \{ a,c \right \} ,\left \{ b,d \right \} \right \} , \ \left \{ \left \{ a,d \right \} ,\left \{ b,c \right \} \right \}$
No need to put a in box2 and count again. because boxes are indistinct.
In analogy to these boxes, those groups of a partition are also same,only the elements inside a group are distinct. | 2017-10-19T09:04:38 | {
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http://www.datasetsanalysis.com/correlation/correlation-problems-with-real-life-data.html | # Correlation Problems with Real Life Data
Correlation problems with real life data are presented along with their solutions.
Six problems are presented where the correlations between:
the average time spent on website pages and their average engaged time in the website http://www.analyzemath.com/index.html,
the wheat production and exports worldwide,
the nasdaq index and Apple stock prices,
the dow, s&p and nasdaq indices,
the gdp per capita, the spending on health and life expectancy worldwide,
and the Generated electricity and CO2 emission worldwide are studied using real life data downloaded from the websites http://www.analyzemath.com/index.html, http://data.worldbank.org/, http://www.who.int/, http://ca.finance.yahoo.com/ and generated using Excel sheets.
## Problems with Solutions
Problem 1
The data set in the table below represents the total "average pageview duration (x)" and the "average page engaged time (y)" in seconds for 60 pages in the website www.analyzemath.com.
The above data could be downloaded at Average Engaged Time and copied and pasted in either Excel, Google sheets or LibreOffice and used.
a) Use any software applications such as Google Sheets, Excell, LibreOffice to produce a scatter plot of y versus x. Would you expect the correlation between the "average pageview duration (x)" and the "average page engaged time (y)" to be close to 1, -1 or 0?
b) Use any software to compute the sums of squares $SS_x$, $SS_y$ and cross products $SS_{xy}$
c) Calculate the correlation $r$ using the correlation formula $r = \dfrac{SSxy}{\sqrt {SSx \cdot SSy}}$ and the sums calculated in part b).
d) Calculate the correlation $r$ using excel and compare it to the value calculated in part c).
Solution Problem 1
a)
Excel was used to make a scatter plot of "average page engaged time (y)" versus the total "average pageview duration (x)" as is shown below. Note that the "average engaged time" increases as the "average pageview duration" increases and we would expect a correlation close to 1.
b)
Excel was used to calculate the sums (in red) as shown on the table below
There are 60 pairs of data values $(x_i,y_i)$, hence $m = 60$.
From the above table, we have the following sums:
$\sum x_i = 181.08$, $\quad \sum y_i = 41.70$, $\quad \sum x_i y_i = 162.49$,
$\sum x_i^2= 687.59$ , $\quad \sum y_i^2= 39.25$
Using the formulas for the sums $SSx$ , $SSy$ and $SS{xy}$ , we have
$SSx = \sum x_i^2 - \dfrac{(\sum x_i)^2}{m} \\ = 687.59 - \dfrac{181.08^2}{60} = 141.0906$
$SS_y = \sum y_i^2 - \dfrac{(\sum y_i)^2}{m} \\ = 39.25 - \dfrac{41.70^2}{60} = 10.2685$
$SS_{xy} = \sum x y - \dfrac{\sum x \sum y}{m} \\ =162.49 - \dfrac{181.08 \times 41.70}{60} \\ = 36.6394$
c)
The correlation between $x$ and $y$ is given by the formula
$r = \dfrac{SSxy}{\sqrt {SSx \cdot SSy}} \\ = \dfrac{36.6394}{\sqrt {141.0906 \cdot 10.2685 }} \\ = 0.9625$
d)
The results of the calculation of the correlation using Excel is
$r = 0.962407319393878$ Compare to the value of the correlation coefficient $r$ found in part c), the small difference is due to the rounding errors.
Problem 2
The table below is the total world wheat production (x) and export (y) in million metric tons from 1962 to 2020 (61 years).(Data from The world bank at http://data.worldbank.org/).
a) Use any software applications such as Google Sheets, Excell, LibreOffice .. to make a scatter plot of y versus x.
b) Use any software to find the sums of squares $SS_x$, $SS_y$ and cross products $SS_{xy}$
c) Calculate the correlation using the formula and the sums calculated in part b).
d) Calculate the correlation $r$ using excel .
Solution Problem 2
a)
b)
There are 61 pairs of data values $(x_i,y_i)$, hence $m = 61$.
Excel was also used to do all the calculations of sums (red):
$\sum x_i = 31061.991$ , $\sum y_i = 6267.028$,
$\sum x_i y_i = 3556496.296$ , $\sum x_i^2 = 17328675.78$ , $\sum y_i^2= 742779.5112$
$SSx = \sum x_i^2 - \dfrac{(\sum x_i)^2}{m} \\ = 17328675.78 - \dfrac{31061.991^2}{61} = 1511507.17534$
$SS_y = \sum y_i^2 - \dfrac{(\sum y_i)^2}{m} \\ = 742779.5112 - \dfrac{6267.028^2}{61} = 98916.56115$
$SS_{xy} = \sum x y - \dfrac{\sum x \sum y}{m} \\ = 3556496.296 - \dfrac{31061.991 \times 6267.028}{61} \\ = 365244.37251$
c)
The correlation between $x$ and $y$ is given by
$r = \dfrac{SSxy}{\sqrt {SSx \cdot SSy}} \\ = \dfrac{365244.37251}{\sqrt {1511507.17534 \cdot 98916.56115 }} \\ = 0.9446$
d)
The results of the calculation of the correlation using Excel is
The correlation is equal to: $r = 0.944591256783634$ the small difference between correlation coefficient $r$ given by Excel and the one found in part c), is due to the rounding errors.
Problem 3
The first four and the last four rows of the stock price of Apple shares and the Nasdaq index from 1980 to 2021 are shown below.(Nasdaq Data from yahoo and Apple share price generated through Excel sheets).
The complete data set of 10340 rows used in this study may be downloaded at Apple Nasdaq and used for more practice.
a) Normalize Apple stock price by dividing all prices by the stock price on the 12/12/1980. Normalize the Nasdaq index by dividing all values of the index by its value on the 12/12/1980. Plot both Apple stock price and the Nasdaq index in normalized form.
b) Use any software to make a scatter plot of Apple stock price against the Nasdaq index and a trend line.
d) Calculate the correlation $r$ using any software.
Solution Problem 3
a)
The first four and the last four rows of the data used in this study, with columns V and W in normalized form, are shown below
The Apple stock price and the Nasdaq index are plotted below from 12/12/1980 to 12/15/2021 and we can see that both have increased.
However, because of the way the data was normalized, it is easy to seen from the table and the graph below the increase or decreases over that period of time. By 12/15/2021, Apple stock price presented a 1397 fold increase while the Nasdaq presented 81 fold increase, hence the Apple stock price increased at a higher rate.
b)
The scatter plot of the Nasdaq against Apple stock price is shown below. Although both the Nasdaq and Apple stock price increase with time, because their rates of change over time are not the same, the scatter plot is not linear as shown below.
c)
The correlation was calculated using Excel and the results are shown below.
The correlation between $r = 0.949894$
Problem 4
The first four and the last four rows of the Dow, S&P and the Nasdaq index from 1992 to 2021 are shown below.(All Data from yahoo).
The complete data set of 7303 rows may be downloaded at Dow, S&P, Nasdaq and may be copied, pasted and used.
a) Normalize the values of all three indices by dividing the values in each column by the values in the first row on the 12/16/1992. Plot all three indices in normalized form over the time.
b) Use any software to make a scatter plot of each pair of indices.
d) Use any software to calculate the correlation $r$ of each pair of indices.
Solution Problem 4
a)
The first four and the last four rows of the data used in this study, with columns H, I and J in normalized form, are shown below
Below is shown the graphs, from 12/16/1992 to 12/14/2021, of all three indices and they all increased over that period of time however the rates of increase are not the same.
b)
The scatter plot of the S&P against the Dow is shown below; it is close to linear because the rate of increases over time are close as shown in the graph above.
The scatter plot of the Nasdaq against the Dow is shown below; it is not close to linear because the rate of increases of the Nasdaq and the Dow , over time, are different as shown in the graph above.
The scatter plot of the S&P against the Nasdaq is shown below; it is not close to linear because the rate of increases of the S&P and the Nasdaq , over time, are different as shown in the graph above.
c)
The correlation were calculated using Excel and the results are shown below.
The correlations are:
Correlation between the Dow and the S&P is equal to: 0.992617
Correlation between the Dow and the Nasdaq is equal to: 0.962196
Correlation between the S&P and the Nasdaq is equal to: 0.982212
The strongest correlation is between the Dow and the S&P and this shows from the graph over time above.
The weakest correlation is between the Dow and the Nasdaq and this also shows from the graph over time above.
Problem 5
In the table below are shown the GDP per capita, the spending on health per capita and the average life expectancy in the world from 2000 to 2018 are shown below.(All Data from http://www.who.int/ and data.worldbank.org).
The complete data set of 7303 rows may be downloaded at life expectancy, gdp and health spending and may be copied, pasted and used.
a) Use any software to make a scatter plot of life expectancy against the gdp per capita and a scatter plot of life expectancy against health spending per capita.
b) Use any software to calculate the correlation between life expectancy and the gdp per capita and the correlation between life expectancy and health spending per capita. Does life expectancy depends strongly on the gdp or spending on health per capita?
Solution Problem 5
a)
Excel was used to generate the scatter plots below.
The scatter plot of life expectancy against the gdp is shown below and the graph is not close to linear.
The scatter plot of life expectancy against the spending on health is shown below and the graph is close to linear.
c)
The results of the correlations calculated , using Excel , are shown below.
The correlations are:
Correlation between life expectancy and health spending is equal to: 0.995632076
Correlation between life expectancy and the GDP is equal to: 0.971070155
Because the correlation between life expectancy and health spending is close to 1, therefore life expectancy depends on how much you spend health per capita.
Problem 6
In the table below are shown the total electricity generated and CO2 generated in the world from 1985 to 2018 are shown below.(All Data from data.worldbank.org).
The complete data set of 7303 rows may be downloaded at electricity generated and CO2 emitted and may be copied, pasted and used.
a) Normalize the data values of both the electricity and CO2 by dividing all values by the values in the first row.
b) Use any software to make a scatter plot of CO2 emission against total electricity generation.
d) Use any software to calculate the correlation $r$ between CO2 emission and total electricity generation.
Solution Problem 6
a)
The above data normalized and its graph is shown below. From 1990 to 2009, the rate of increase of the electricity generated and the CO2 emission seems to be close.
The graphs of the normalized data is shown below. From 1990, both the electricity generated and the CO2 emission increases steadily but at slightly different rates. From the year 2009, the CO2 emission slowed down slightly.
b)
The scatter plot of the CO2 emission against the electricity generated is shown below. The graph is not close to linear.
c)
The correlations were calculated , using Excel , and the results are shown below.
The correlations are:
Correlation between CO2 emission and the total electricity generated is equal to: 0.964518192
A quite strong correlation exists between the CO2 emission and the electricity generated knowing that a large part of the electricity is generated from coal, oil, gas ,... which emit large quantities of CO2. | 2022-12-02T22:51:04 | {
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https://math.stackexchange.com/questions/2866707/prove-sqrtx21-frac1-sqrtx2-1-geq-2?noredirect=1 | # Prove $\sqrt{x^2+1} + \frac{1}{\sqrt{x^2 +1}}\geq 2$ [duplicate]
I want to show why the last inequality in the problem below $\sqrt{x^2+1} + \frac{1}{\sqrt{x^2 +1}}\geq 2$ holds. It's clear that $x^2\geq 0$ and that equality holds when $x=0$ but how can I clearly show this. The left hand side, for $x^2>0$ is greater than $1$ but the right hand side becomes less than $1$ as $x^2>0$
## marked as duplicate by Martin R, Isaac Browne, Strants, Batominovski, José Carlos SantosJul 30 '18 at 22:36
• Hint: think AM-GM, or any of several related ways. – dxiv Jul 30 '18 at 6:42
• I don't see how to use AM-GM here? – john fowles Jul 30 '18 at 6:48
• By AM-GM $\;\displaystyle \frac{\;a + \dfrac{1}{a}\;}{2} \ge \sqrt{a \cdot \frac{1}{a}} = 1\,$, then use it for $\,a=\sqrt{x^2+1}\,$. – dxiv Jul 30 '18 at 6:51
• This is very nice! – john fowles Jul 30 '18 at 6:53
• – Martin R Jul 30 '18 at 8:10
The inequality $(y-1)^{2} \geq 0$ gives $y+\frac 1 y \geq 2$ for any positive number $y$. Take $y=\sqrt {1+x^{2}}$.
The derivative of $f(y)=y+1/y$ is $1-1/y^2$, so the function $f$ increases in $[1,\infty)$, hence $f(y)\geq f(1)=2$.
• f increases in [1,∞),And also decreases on $[0,1]$ which is needed to complete the argument. – dxiv Jul 30 '18 at 6:45
• @dxiv - Since $\sqrt{x^2+1}\geq 1$ for all $x\in\mathbb{R}$, the interval $[0,1]$ is irrelevant. – uniquesolution Jul 30 '18 at 6:51 | 2019-10-15T20:58:48 | {
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https://classes.areteem.org/mod/forum/discuss.php?d=1139 | ## Online Course Discussion Forum
### II-A Geometry
II-A Geometry
Hello!
For 1.19, how do you prove that the triangles on the ends of the trapezoid are 45-45-90? 1.28, I don't get the solution. For problem 2.6 b, Mr. David's internet was off, and I couldn't here the problem.For 2.14, I kept getting 60 degrees and 24 degrees. For 3.30, I don't get why the answer is y=3, I got y=1.5. FOr 4.26, can you give me the exact soluton? I did it the way the solution says on the quiz, but it did not give me the exact answers. Thank you!
Tina
Re: II-A Geometry
In 1.19, remember that for trapezoids, we can often divide the trapezoid into a rectangle and two right triangles on the ends. Since both right triangles on the ends share the same height (the height of the trapezoid) as well as the same hypotenuse (of length $5\sqrt{2}$) those two triangles are congruent. From there you get their bases are length $5$ (since $5+10+5=20$) so they are right isosceles triangles. Hope that helps with the solution there.
For 1.28, it might help to look back at Problem 1.8, as that has the diagram with the equilateral triangle drawn. The key fact here is that in isosceles triangle ABC, if $\angle BAC = 20$, then the other two angles are both $80$ degrees. Hint: Note that $20+60 = 80$ as well.
2.6b is actually a special case of 2.26, so maybe it's better to look at that one. The diagram there is as shown below:
There are many different ways to solve from here (including using 2.6a). One alternate method: What types of shapes are formed if you extend lines $DP$, $EP$, and $FP$ to divide equilateral triangle $ABC$ into 6 pieces?
In 2.14, the 24 degrees (if the pentagon is outside the hexagon) is correct. Note this comes from an isosceles triangle with one angle measuring $360-120-108 = 132$ degrees. If the pentagon is inside the hexagon, the isosceles triangle should have one angle measuring $120-108 = 12$ degrees, right?
For 3.30, can you share a little of your work? Does the listed solution make sense or did you take a different approach?
For 4.26, consider the diagram below, which uses the information about the area of $GPFD$ and $EBHP$.
Since the full rectangle also has area $44$, we know that $(x+y)\times (12/x + 12/y) = 44$ as well. Since $y$ must be a factor of $10$, $y$ is 1, 2, 5, or 10. A little trial and error gives the answer from here. Hint: 44 can only be written as $2\times 22$ or $4\times 11$ as well, narrowing down the possibilities further.
Hope this helps! Let us know if you need any followup hints for ones like 3.30 or 4.26.
Re: II-A Geometry
this is what i did:
Can you also give me the solution for problem 2.6b? I still don't really get it. Thank you so much for the responses!
Re: II-A Geometry
For 3.30, you have the right idea to start. You can complete the square to write the parabola in vertex form. You have a little mistake there going from $2y = x^2+2x-4$ to $2y = (x+1)^2 - 4$, where it should instead be $2y = (x+1)^2 -5$ (double check by expanding this out). You're also off by a negative, so it actually should be $-2y = (x+1)^2 - 5$ which matches the $y = -\dfrac{1}{2}(x+1)^2 + \dfrac{5}{2}$. This means the vertex of the parabola is $(-1, 2.5)$.
Note, however, that the question is NOT asking for just the vertex. It is asking for the focus and directrix. Because the parabola opens downwards, the focus is (directly) below the vertex, and the directrix is above the vertex. More specifically, the focus (a point) must be of the form $(-1, 2.5+k)$ and the directrix (a line) must be of the form $y=2.5-k$ for some number $k$. Note we expect $k$ to be negative so that the focus is below the vertex.
The definition of a parabola is that all points on the parabola are the same distance from the focus as they are from the directrix. For ease, let's use the point $(0,2)$ from the parabola (although any point that isn't the vertex will work).
Distance from $(0,2)$ to the focus $(-1, 2.5+k)$:$$\sqrt{(0-1)^2 + (2-(2.5+k))^2}$$Distance from $(0,2)$ to the directrix $y=2.5-k$:$$|2-(2.5-k)|.$$Setting these equal will give $k = -0.5$ and therefore we know the focus and directrix.
For 2.6b and 2.26, consider extending the lines DP, EP, and FP as in the diagram below.
Note this divides the entire triangle ABC into three equilateral triangles (with light blue, pink, and orange side lengths) as well as three parallelograms. All the side lengths are color coded, note that each side equals 1 pink, 1 blue, and 1 orange, or $PD+PE+PF$ which is the key idea for the problem.
Hope these help!
Re: II-A Geometry
Thanks! | 2023-01-29T18:12:56 | {
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https://math.stackexchange.com/questions/3360889/proving-a-metric-space-on-a-set-x | # Proving a metric space on a set X
Assume that $$(X,d)$$ is a metric space, define $$\displaystyle \rho(x,y)= \frac{d(x,y)}{1+d(x,y)}$$ for all $$x,y \in X.$$
a. Show $$(X,\rho)$$ is a metric.
b. A sequence $$(x)_{n \geq}$$ in $$X$$ converges to $$p$$ in the metric space $$(X,d)$$ iff it converges to $$p$$ in $$(X,\rho.)$$
My attempt:
a. I have a problem with showing the tringle inequality \begin{align*} \rho(x,z)=\frac{d(x,z)}{1+d(x,z)}& \leq \frac{d(x,y)}{1+d(x,z)}+\frac{d(y,z)}{1+d(x,z)} \end{align*} I could not figure the way to show that.
b. I am good with the first implication. The problem is when I assume $$(x_n)$$ converges to $$p$$ in $$(X, \rho)$$, and want to show this sequence converges in $$(X,d).$$ Here is my attempt:
Assume for the sake of a contradiction that this sequence does not converge in $$(X,d)$$. By this, there exists an $$\epsilon_0>0$$ such that for all $$N \in \mathbb{Z^+}$$, we can find $$n_{_{N}} \geq N$$ such that $$d(x_{n_{_N}},p)> \epsilon_0$$. Now by the assumption, for $$\epsilon=\frac{\epsilon_0}{1+\epsilon_0}$$ there exists $$N' \in \mathbb{Z}^+$$ such that for all $$n \geq N'$$, $$\rho(x_n,p)<\frac{\epsilon_0}{1+\epsilon_0}$$. However, by our hypothesis, for this particular $$N'$$, there exists $$n_{_{N'}} \geq N'$$ such that $$d(x_{n_{_{N'}}},p)>\epsilon_0$$, so $$\frac{\epsilon_0}{1+d(x_{n_{_{N'}}},p)} < \frac{d(x_{n_{_{N'}}},p)}{1+d(x_{n_{_{N'}}},p)}=\rho(x_{n_{_{N'}}},p)<\frac{\epsilon_0}{1+\epsilon_0}$$ what we conclude is $$\displaystyle \frac{\epsilon_0}{1+d(x_{n_{_{N'}}},p)} < \frac{\epsilon_0}{1+\epsilon_0}$$ which is true since $$d(x_{n_{N'}},p)>\epsilon_0$$, I am geting stuck here.
I will appreciate any help or hint for that.
a) $$\frac x {1+x}$$ is an incersing function on $$[0,\infty)$$
b) $$\frac {a+b} {1+a+b} \leq \frac a {1+a} +\frac b {1+b}$$ for all R$$a, b \geq 0$$.
For the second part you only need the fact that $$\frac x {1+x} <\epsilon$$ iff $$x<\frac {\epsilon} {1-\epsilon}$$ provided $$0 <\epsilon <1$$ (and $$x <\epsilon$$ iff $$\frac x {1+x} <\frac {\epsilon} {1+\epsilon}$$. | 2020-11-26T09:06:04 | {
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http://nbviewer.jupyter.org/github/jckantor/CBE30338/blob/master/notebooks/Assignment_02/Assignment_02.ipynb | # Assignment 02¶
All of the problems for this week's assignment have been consolidated into this notebook. The purpose of this was to make it is easier to submit and grade assignments by keeping all of the parts in one place (this notebook) and available through one interface (Vocareum). For questions that require no Python programming, simply type your results into a markdown cell.
## 1. [5 pts] Semiconducter Heater (adapted from problem 5.2, SEMD)¶
A heater for a semiconductor wafer has first-order dynamics that can be modeled by the differential equation
$$\tau \frac{dT}{dt} + (T-T_{amb}) = KP(t)$$
where $\tau$ is time constant with units of minutes, and $K$ is a constant with units $^\circ C$ per kilowatt. $T_{ref}$ is an ambient temperature which has not been included in the report to you. The process is at steady state when an engineer changes the power input from 1.0 to 1.5 kilowatt. The engineer notes the following:
• The initial process temperature is 80$^\circ C$.
• The process temperature is 230$^\circ C$ four minutes after changing the power input.
• Thirty minutes later the temperture is 280$^\circ C$ where it remains as the new steady state.
a. What are the values of $\tau$ and $K$?
b. It is desired to put the model equation into standard state-space form
$$\frac{dx}{dt} = a\,x + b\,u$$
where $x$ corresponds to process temperature $T - T_{amb}$, and $u$ to process power. Express $a$ and $b$ algebraiclly in term of the $\tau$ and $K$, and find numerical values using the results of part (a).
### Solution¶
Write your answer in this and subsequent cells. This problem doesn't necessarily require Python coding, so feel free to simply edit and write your answer here if that meets your needs.
#### Part a. [3 pts]¶
Begin by considering the steady state relationship between $T$ and $P$,
$$T - T_{amb} = K P$$
Assuming the system is at steady state at $t=0$ and again at $t=30$. The initial and final conditions therefore provide two equations
\begin{align*} 80 - T_{amb} & = K \cdot 1.0 \\ 280 - T_{amb} & = K \cdot 1.5 \end{align*}
which can be solved for $T_{amb}$ and $K$.
\begin{align*} K & = \frac{280 - 80}{1.5 - 1.0} = \fbox{400 deg C per kW} \\ T_{amb} & = 80 - 1.0\cdot K = -320\text{ deg C} \end{align*}
The large negative value for $T_{amb}$ could be explained by process features that are not included in the brief problem description, such as heat loss via an exit stream.
Next, given a step input, the relationship of the transient response $T(t) - T_0$ to the ultimate response $T(\infty)-T_0$ for a first-order system is given by
$$T(t) - T(\infty) = \left(T_0 - T(\infty)\right) e^{-t/\tau}$$
Solving for $\tau$
$$\tau = \frac{t}{\ln\frac{T_0 - T(\infty)}{T(t)-T(\infty)}} = \fbox{2.89 minutes}$$
This problem can be solved analytically and the numerical results found with a calculator. Below we show these calculations using Python as our calculator.
In [1]:
from math import log
# Solution, Part a.
# From the steady state analysis
DT = 280 - 80
DP = 1.5 - 1.0
K = DT/DP
print('Process gain K = {0:.2f} deg C/kw'.format(K))
tau = 4/log((80-280)/(230-280))
print('Process time constant tau = {0:.2f} minutes'.format(tau))
Process gain K = 400.00 deg C/kw
Process time constant tau = 2.89 minutes
##### Part b. [2 pts]¶
Taking the starting model, and dividing each term by $\tau$ gives
$$\frac{dT}{dt} = \frac{-1}{\tau} (T-T_{amb}) + \frac{K}{\tau} P$$
where state variable $x = T - T_{amb}$. Comparing the state space model in standard form, we see
\begin{align*} a & = \frac{-1}{\tau} = \fbox{-0.3466 1/min}\\ b & = \frac{K}{\tau} = \fbox{138.6 deg C/kW-min} \end{align*}
In [2]:
# Solution, Part b.
a = -1/tau
b = K/tau
print("a = {0:.4f}".format(a))
print("b = {0:.4f}".format(b))
-b/a
a = -0.3466
b = 138.6294
Out[2]:
400.0
##### Further calculations¶
This following calculations are not a required part of the solution but are included here for completeness. The cells shows the solution of the model differential equations using the constant values found above.
In [3]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
t = np.linspace(0,30)
P = 1.5
def Tderiv(x,t):
return -x/tau + K*P/tau
sol = odeint(Tderiv, 80 -(-320), t)
plt.plot(t,sol-320)
plt.plot(4,230,'r.',ms=20)
plt.plot(30,280,'r.',ms=20)
plt.plot(plt.xlim(),[80,80],'g--')
plt.plot(plt.xlim(),[280,280],'g--')
plt.plot([plt.xlim()[0],4,4],[230,230,plt.ylim()[0]],'r--')
plt.xlabel('Time [min]')
plt.ylabel('Temperature [deg C]')
Out[3]:
<matplotlib.text.Text at 0x110864b00>
## 2. [5 pts] Second Order Process (Adapted from SEMD 5.14)¶
A device actuated by compressed air experiences a step change in air pressure from 15 to 31 psig. The response is shown in the following diagram.
A model differential equation is written
$$\tau^2\frac{d^2R'}{dt^2} + 2\zeta\tau\frac{dR'}{dt} + R' = K u(t)$$
where $R'(t)$ represents the difference between $R(t)$ and the initial value of 8mm, and $u(t)$ is the diffence in air pressure from an initial steady state value of 15.
a. Use Figure 5.11 of your textbook (reproduced here) to estimate values for the time constant $\tau$, the damping coefficient $\zeta$, and process gain $K$.
b. Using the results of part a, and the formulas 5-52 to 5-55, calculate values for time to first peak ($t_p$), overshoot (OS), decay ratio, and period of oscillation. How do they match the step response diagram?
### Solution¶
Write your answer in this and subsequent cells. This problem doesn't necessarily require Python coding, so feel free to simply edit and write your answer here if that meets your needs.
##### Part a. [3 pts]¶
For the solution of part a, we use the plot of the system response to determine the fraction overshoot and period.
In [4]:
# estimate overshoot from chart
overshoot = (12.7 - 11.2)/(11.2-8)
print('Overshoot = {0:.3f}'.format(overshoot))
# estimate period from chart
period = 2.3
print('Period = {0:.2f} seconds'.format(period))
Overshoot = 0.469
Period = 2.30 seconds
Next we use the Figure 5.11 to estimate the damping factor $\zeta$ and the time constant $\tau$. The system gain $K$ is determined from the steady state response.
In [5]:
# steady state process gain
K = (11.2-8)/(31-15)
# estimate damping factor zeta and tau from chart
zeta = 0.2
tau = 0.97*period/(2*np.pi)
(K,tau,zeta)
Out[5]:
(0.19999999999999996, 0.3550746780380185, 0.2)
##### Part b. [2 pts]¶
For an underdamped second order system, the desired performance metrics are given by the following by formulas in the following table.
Quantity Symbol Expression/Value
Time to first peak $t_p$ $\frac{\pi\tau}{\sqrt{1-\zeta^2}}$
Overshoot OS $\exp\left(-\frac{\pi\zeta}{\sqrt{1-\zeta^2}}\right)$
Decay Ratio DR $\exp\left(-\frac{2\pi\zeta}{\sqrt{1-\zeta^2}}\right)$
Period $\frac{2\pi\tau}{\sqrt{1-\zeta^2}}$
In [6]:
tp = np.pi*tau/np.sqrt(1-zeta**2)
os = np.exp(-np.pi*zeta/np.sqrt(1-zeta**2))
dr = os**2
pd = 2*np.pi*tau/np.sqrt(1-zeta**2)
print('Time to first peak = {0:.2f}'.format(tp))
print('Overshoot = {0:.2f}'.format(os))
print('Decay Ratio = {0:.2f}'.format(dr))
print('Period = {0:.2f}'.format(pd))
Time to first peak = 1.14
Overshoot = 0.53
Decay Ratio = 0.28
Period = 2.28
##### Further Calculations¶
The following calculations are not a required part of the problem solution.
The standard model for a second order process is given by
$$\tau^2\frac{d^2R'}{dt^2} + 2\zeta\tau\frac{dR'}{dt} + R' = K u(t)$$
For simulation we'll define $S'$ as $\frac{dR'}{dt}$ which leaves a pair of first order differential equations
\begin{align*} \frac{dR'}{dt} & = S' \\ \frac{dS'}{dt} & = -\frac{1}{\tau^2}R' - \frac{2\zeta}{\tau}S' + \frac{K}{\tau^2}u(t) \end{align*}
The following interactive simulation allow
In [7]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from ipywidgets import interact
t = np.linspace(0,8,1000)
u = 31-15;
def simulation(zeta = zeta, tau = tau):
def deriv(X,t):
R,S = X
Rdot = S
Sdot = -R/tau/tau - 2*zeta*S/tau + K*u/tau/tau
return[Rdot,Sdot]
sol = odeint(deriv, [0,0], t)
R = 8 + sol[:,0]
plt.plot(t,R)
plt.xlabel('Time [sec]')
plt.ylabel('R')
plt.ylim(7,14)
plt.plot(plt.xlim(),[11.2,11.2],'g--')
# plot peak
plt.plot(plt.xlim(),[R.max(),R.max()],'g--')
plt.text(np.mean(plt.xlim()),R.max()+0.05,"Peak = {0:0.2f}".format(R.max()), ha='center')
# plot time to first peak
tp = t[np.where(R==R.max())[0][0]]
plt.plot([tp,tp],[plt.ylim()[0],R.max()],'g--')
plt.text(tp,8,' Time to first peak = {0:.2f}'.format(tp))
# find positive-going zero crossings
idx = np.where(np.diff(np.sign(R-11.2))>0)[0]
t0 = t[idx[0]]
t1 = t[idx[1]]
plt.plot([t0,t0],[9.5,11.2],'g--')
plt.plot([t1,t1],[9.5,11.2],'g--')
plt.plot([t0,t1],[10,10],'g--')
plt.text((t0+t1)/2,10.05,'Period = {0:.2f}'.format(t1-t0), ha='center')
interact(simulation,zeta=(0,1.2,0.001), tau = (0.1,0.5,0.001))
Out[7]:
<function __main__.simulation>
## 3. [10 pts] Modeling and Simulation of Interacting Tanks¶
The following diagram shows a pair of interacting tanks.
Assume the pressure driven flow into and out of the tanks is linearly proportional to tank levels. The steady state flowrate through the tanks is 3 cubic ft per minute, the steady state heights are 7 and 3 feet, respectively, and a constant cross-sectional area 5 sq. ft. The equations are written as
\begin{align*} \frac{dh_1}{dt} & = \frac{F_0}{A_1} - \frac{\beta_1}{A_1}\left(h_1-h_2\right) \\ \frac{dh_2}{dt} & = \frac{\beta_1}{A_2}\left(h_1-h_2\right) - \frac{\beta_2}{A_2}h_2 \end{align*}
a. Use the problem data to determine values for all constants in the model equations.
b. Construct a Phython simulation using odeint, and show a plot of the tank levels as function of time starting with an initial condition $h_1(0)=6$ and $h_2(0)$ = 5. Is this an overdamped or underdamped system.
### Solution¶
##### Part a. [2 pts]¶
The parameters that need to be determined are $\beta_1$ and $\beta_2$. At steady state all of the flows must be identical and
\begin{align*} 0 & = F_0 - \beta_1(h_1 - h_2) \\ 0 & = \beta_1(h_1 - h_2) - \beta_2h_2 \end{align*}
Substituting problem data,
$$\beta_1 = \frac{F_0}{h_1-h_2} = \frac{3\text{ cu.ft./min}}{4\text{ ft}} = 0.75\text{ sq.ft./min}$$
$$\beta_2 = \frac{\beta_1(h_1 - h_2)}{h_2} = \frac{3\text{ cu.ft./min}}{3\text{ ft}} = 1.0\text{ sq.ft./min}$$
##### Part b. [8 pts]¶
The next step is perform a simulation from a specified initial condition.
In [8]:
%matplotlib inline
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from ipywidgets import interact
# simulation time grid
t = np.linspace(0,30,1000)
# initial condition
IC = [6,5]
# inlet flowrate
F0 = 3
# parameters for tank 1
A1 = 5
beta1 = 0.75
# parameters for tank 2
A2 = 5
beta2 = 1.0
def hderiv(H,t):
h1,h2 = H
h1dot = (F0 - beta1*(h1-h2))/A1
h2dot = (beta1*(h1-h2) - beta2*h2)/A2
return [h1dot,h2dot]
sol = odeint(hderiv,IC,t)
plt.plot(t,sol)
plt.ylim(0,8)
plt.grid()
plt.xlabel('Time [min]')
plt.ylabel('Level [ft]')
Out[8]:
<matplotlib.text.Text at 0x11089d048>
##### Further Calculations¶
$$\frac{d}{dt}\left[\begin{array}{c} h_1 \\ h_2 \end{array}\right] = \left[\begin{array}{cc}-\frac{\beta_1}{A_1} & \frac{\beta_1}{A_1} \\ \frac{\beta_1}{A_2} & -\frac{\beta_1}{A_2} - \frac{\beta_2}{A_2} \end{array}\right] \left[\begin{array}{c}h_1 \\ h_2\end{array}\right] + \left[\begin{array}{c}\frac{1}{A_1} \\ 0\end{array}\right]F_0$$
In [ ]: | 2017-03-28T06:12:15 | {
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https://math.stackexchange.com/questions/2338338/complement-of-n-times-n-matrices-vector-spaces-over-mathbbr | # Complement of $n\times n$ matrices vector spaces over $\mathbb{R}$
Let $V$ be the vector space of all $n\times n$ matrices over $\mathbb{R}$, and define the scalar product of two matrices $A,B$ by
$\langle A,B\rangle=\operatorname{tr}(AB)$, where $\operatorname{tr}$ is the trace (sum of the diagonal elements). Describe the orthogonal complement of the subspace of diagonal matrices. What is the dimension of this orthogonal complement?
I do not understand what is the description of the "orthogonal complement of the subspace of diagonal matrices". In what concerns the dimension I remembered the theorem that states:$\dim W+\dim W^{\bot}=\dim V$ in which $W$ is a subspace of $V$.
Questions:
1) How do I answer this question?
2) Could someone provide me a proof?
Questions
• The vector space of diagonal matrices has a canonical basis. You have to find the vectors from the canonical basis of $M_n (\mathbb {R} )$ that are orthogonal over those vectors. They generate the complement. – rafa Jun 27 '17 at 17:08
• @rafa What is a canonical basis? – Pedro Gomes Jun 27 '17 at 17:10
• the subspace of diagonal matrices: $\mathfrak D=\{D \mid D_{i,j}=0, \forall i\neq j\}$, its complement: $\mathfrak D^\perp = \{M\mid \langle M,D\rangle=0, \forall D \in \mathfrak D\}$. – Surb Jun 27 '17 at 17:11
• What is the dimension of diagonal subspace. It is $n$, as I have written one of its basis in the answer. Any complement has dimension then $n^{2}-n$. – Riju Jun 27 '17 at 17:21
• By the way is $\langle A,B \rangle = Tr(AB)$ an innerproduct? It doesn't seem so! – Riju Jun 27 '17 at 17:35
Let $E_{ij}$ denote the matrix with $1$ in $ij^{th}$ position and $0$ otherwise. For $1 \leq i \leq n$ and $1 \leq j \leq n$, this is a basis for $M_{n\times n}(\mathbb{R})$. And $E_{ii}(1\leq i \leq n)$ forms a basis for the diagonal subspace. Also $E_{ij}E_{kl}=\delta_{jk}E_{il}$, where $\delta_{jk}$ is $1$ if $j=k$, otherwise it is $0$. So the basis for the subspace is clearly a orthogonal basis. Now, the extended basis for the whole space is as described above. Now just use the Gram-schimdt Orthogonalization process to orthogonalize this basis keeping the basis for the subspace same. Whatever vectors you get except the $E_{ii}$'s is a basis for the complement.
• I am not understanding your answer. Could you use matrix notation? I do not use Kronecker Delta stil. Those are the reasons I down voted. – Pedro Gomes Jun 27 '17 at 17:21
• I just wrote what $\delta_{jk}$ means. – Riju Jun 27 '17 at 17:24
• How do I apply Gram-Schmidt orthogonalization process in the diagonal basis in this specific case? How do you get $n^2-n$ as dimension of the complement, I mean the $n^2$? – Pedro Gomes Jun 27 '17 at 17:28
• Ok what is the dimension of the vector space $M_{n \times n}(\mathbb{R})$? What is the dimension of the subspace? If $V=U \oplus W$, then $dim(V)=dim(U)+dim(W)$, follows from the dimension theorem. – Riju Jun 27 '17 at 17:31
• Sorry but I have not yet covered that theorem. – Pedro Gomes Jun 27 '17 at 17:36
You can check by direct computation that a matrix $A=(a_{ij})_{1\le i,j\le n}$ is orthogonal to the diagonal matrix $D=(\lambda_1,\dots,\lambda_n)$ if and only if $$\lambda_1a_{11}+\dots+\lambda_n a_{nn}=0.$$ Hence it is orthogonal to all diagonal matrices if and only if $a_{11}=\dots= a_{nn}=0$, i.e. its diagonal elements are $0$. Clearly this subspace is isomorphic to $\mathbf R^{n^2-n}$.
• How do you get $n^2-n$? – Pedro Gomes Jun 27 '17 at 18:11
• This is the number of non-diagonal elements in an $n\times n$ matrix. – Bernard Jun 27 '17 at 18:15
• By a ‘standard basis’, I suppose you mean matrices $E_{ij}$ with all coefficients equal to $0$, except it has $1$ in the $(i,j)$ position? – Bernard Jun 27 '17 at 18:20
• yes, but I have already got it – Pedro Gomes Jun 27 '17 at 18:21
• It's the same as what I said: you remove the $E_{ii}$s from this basis, and you obtain your basis of the orthogonal complement. – Bernard Jun 27 '17 at 18:23 | 2019-11-20T19:15:45 | {
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http://math.stackexchange.com/questions/159244/what-is-the-correct-way-to-solve-sin2x-sinx/159249 | # What is the correct way to solve $\sin(2x)=\sin(x)$
I've found two different ways to solve this trigonometric equation
\begin{align*} \sin(2x)=\sin(x) \Leftrightarrow \\\\ 2\sin(x)\cos(x)=\sin(x)\Leftrightarrow \\\\ 2\sin(x)\cos(x)-\sin(x)=0 \Leftrightarrow\\\\ \sin(x) \left[2\cos(x)-1 \right]=0 \Leftrightarrow \\\\ \sin(x)=0 \vee \cos(x)=\frac{1}{2} \Leftrightarrow\\\\ x=k\pi \vee x=\frac{\pi}{3}+2k\pi \vee x=\frac{5\pi}{3}+2k\pi \space, \space k \in \mathbb{Z} \end{align*}
The second way was:
\begin{align*} \sin(2x)=\sin(x)\Leftrightarrow \\\\ 2x=x+2k\pi \vee 2x=\pi-x+2k\pi\Leftrightarrow \\\\ x=2k\pi \vee3x=\pi +2k\pi\Leftrightarrow \\\\x=2k\pi \vee x=\frac{\pi}{3}+\frac{2k\pi}{3} \space ,\space k\in \mathbb{Z} \end{align*}
What is the correct one? Thanks
-
Thanks, Alex, I was about to fix the whack-tex (forgotten \) errors. – ncmathsadist Jun 16 '12 at 23:14
Your solutions sets are actually the same, even though they are written differently. Notice that from $0$ to $2\pi$, the solutions are $x=0, \frac{\pi}{3}, \pi, \frac{5\pi}{3}, 2\pi$, whichever answer you use, and in both cases, this pattern will repeat every $2 \pi$. – Jonas Kibelbek Jun 16 '12 at 23:16
These answers are equivalent and both are correct. Placing angle $x$ on a unit circle, your first decomposition gives all angles at the far west and east sides, then all the angles $60$ degrees north of east, then all the angles $60$ degrees south of east.
Your second decomposition takes all angles at the far east side first. Then it takes all angles spaced one-third around the circle starting at 60 degrees north of east. You have the same solution set either way.
-
Nice to see two ways to look at the problem – ncmathsadist Jun 16 '12 at 23:49
You have all of the right ingredients.
The equation $\sin(x) = 0$ yields solutions $x = k\pi$ for $k\in \mathbb{Z}$.
The equation $\cos(x) = 0$ yields solutions $x = \pi/3, 5\pi/3$ in $[0,2\pi]$.
Since the cosine function is $2\pi$-periodic, you get the solutions
$$x = \pi(1/3 + 2k), \pi(5/3 + 2k), \qquad k\in\mathbb{Z}.$$
So your solution is the totality of all of these.
-
For what it's worth, the second one is "better" because it generalizes nicer. Imagine solving $\sin(3 x) = \sin(x)$ using the first method ($\sin(3 x) = 3\cos^2(x)\sin(x) - \sin^3(x)$). On the other hand, it's easy to see that $\sin(a x)= \sin(b x)$ will have an infinite number of solutions for any real $a$ and $b$ from the second method. | 2014-07-24T17:52:50 | {
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http://mediacollective.nl/8epx2n/a58343-radius-of-a-circle-from-area | Area of an arch given height and radius. The number $\pi$ is known to be an irrational number. Area of a quadrilateral. Area of a parallelogram given sides and angle. Area of a circle = π * r 2. Using this calculator, we will understand methods of how to find the perimeter and area of a circle. Write a Python program which accepts the radius of a circle from the user and compute the area. Radius of circle given area. Side of polygon given area. Area of a circular sector. Here the Greek letter π represents a constant, approximately equal to 3.14159, which is equal to the ratio of the circumference of any circle to its diameter. If you're seeing this message, it means we're having trouble loading external resources on our website. where ‘r’ represents radius and ‘d’ represents diameter of a circle. The area of the circle is the primary determinant for all other properties. View solution The length of an arc of a circle of radius 5 cm subtending a central angle measuring 1 5 ∘ = 1 2 a π . Diameter is the distance across the circle through the center. As you know, $A=\pi r^2$ where $r$ is the radius of a circle and $A$ is its area. Convert radius to circumference, area, diameter. The area of a quarter circle when the radius is given is the area enclosed by a quarter circle of radius r is calculated using Area=(pi*(Radius)^2)/4.To calculate Area of a quarter circle when radius is given, you need Radius (r).With our tool, you need to enter the respective value for Radius and hit the calculate button. Watch this tutorial to see how it's done! Area of an arch given angle. Python: Area of a Circle . When you place two radii end to end in a circle, it will equal the diameter. Area and circumference of circle calculator uses radius length of a circle, and calculates the perimeter and area of the circle. If you know the circumference of a circle, the radius can be found using the formula where: C is the circumference of the circle π is Pi, approximately 3.142 If you know the area If you know the area of a circle, the radius can be found using the formula where: A is the area of the circle π is Pi, approximately 3.142 Calculator It doesn't matter whether you want to find the area of a circle using diameter or radius - you'll need to use this constant in almost every case. Where: π is approximately equal to 3.14. Area of a circle diameter. In geometry, the area enclosed by a circle of radius r is πr2. Area of a circle. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are … Formula and explanation, description, conversion. If you know the radius of a circle, you can use it to find the area of that circle. Mathematical, algebra converter, tool online. Area of a regular polygon. 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Berger Bison Glow 10 Ltr Price, Cellular Meaning In Urdu, Kopi Dangdut Mp3, Mobile Homes For Sale In Albany Oregon, 3 Gallon Air Compressor, Duplicolor Adhesion Promoter Advance Auto, | 2022-10-01T05:24:19 | {
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https://math.stackexchange.com/questions/2772839/find-the-number-of-bacteria-after-dividing | # Find the number of bacteria after dividing
Pink eye is an infection casued by bacteria. After you have been infected, these bacteria double every 40 minutes. How many bacteria would be present at 6 pm if a single bacterium began reproducing at 11am?
$a=1$ and $r=2$ for this geometric sequence and every $40$ min = $2 \over 3$ hrs, so I've come up with the equation
\begin{align} t_n&=a(r)^{n-1} \\ t_n&=(2)^{{3\over2}(n-1)} \end{align}
$7$ hours passed, so $n=8$
\begin{align} t_n&=(2)^{{3\over 2}(8-1)} \\ &=2^{{3\over2}(7)} \\ &=2^{21\over2} \\ &=1448.1546 \end{align}
So, there will be $1448$ bacteria.
However, this is wrong. The answer is supposed to be $724$.
Why is my answer wrong? I double checked by creating a chart, and $724$ does not seem reasonable:
$$\begin{array}{|c|c|} \hline \text{Time} & \text{Number of Bacteria} \\\hline 11:00 \text{ am} & 1\\\hline 11:40 \text{ am} & 2\\\hline 12:20 \text{ pm} & 4\\\hline 1:00 \text{ pm} & 8\\\hline 1:40 \text{ pm} & 16\\\hline 2:20 \text{ pm} & 32\\\hline 3:00 \text{ pm} & 64\\\hline 3:40 \text{ pm} & 128\\\hline 4:20 \text{ pm} & 256\\\hline 5:00 \text{ pm} & 512\\\hline 5:40 \text{ pm} & 1024\\\hline 6:00 \text{ pm} & \text{between }1024 \text{ and } 2048 \\\hline 6:20 \text{ pm} & 2048 \\\hline \end{array}$$
Even at 5:40 pm, there should be more than $724$ bacteria. Was there something wrong in my solution or is the correct answer wrong?
• Your answer is right. I presume the questioner made a 'fence-post' error. – Penguino May 8 '18 at 21:42
• Yes, a mistake 724 = 2^(19/2) – Phil H May 8 '18 at 22:15
• @Penguino Could you please post your comment as an answer so this question can be removed from the "Unanswered" queue? – Robert Howard Jul 29 '18 at 22:24
• I will add my concurrence that your table is right and the book wrong - in order to say that the question is ridiculous biology. I hate fake "real world problems" that suggest that they are applications of mathematics. – Ethan Bolker Jul 29 '18 at 23:50
Your table is correct until $6:00$ PM. As I read the question the doubling is instantaneous, so at $6:00$ PM there are still $1024$ bacteria. Neither you nor the solution manual should add up all the entries because your table shows the number of bacteria at each time after the doubling. The answer should be $1024$ bacteria at $6:00$PM. | 2020-02-29T07:08:25 | {
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https://abakbot.com/algebra-en/property-det-en | When working with matrices, it is never superfluous to know about their basic properties. Sometimes it helps to unambiguously determine the determinant of the matrix without resorting to automatic calculations, such as Calculating the determinant of a complex matrix online
### Property 1
The value of the determinant will not change if all its rows are replaced by columns, and each row is replaced by a column with the same number, i.e.
$\begin{pmatrix}a_1\; b_1 \;c_1\\a_2 \;b_2\; c_2\\a_3 \;b_3\; c_3\end{pmatrix}=\begin{pmatrix}a_1\; a_2 \;a_3\\b_1\; b_2 \; b_3\\c_1\; c_2 \; c_3\end{pmatrix}$
### Property 2
The permutation of two columns or two rows of the determinant is equivalent to multiplying it by -1.
$\begin{pmatrix}a_1\; b_1 \;c_1\\a_2 \;b_2\; c_2\\a_3 \;b_3\; c_3\end{pmatrix}=-\begin{pmatrix}a_1\; c_1 \;b_1\\a_2\; c_2 \; b_2\\a_3\; c_3 \; b_3\end{pmatrix}$
### Property 3
If the determinant has two identical columns or two identical rows, then it is equal to zero.
### Property 4
Multiplying all elements of one column or one row of the determinant by any number k is equivalent to multiplying the determinant by this number k.
$\begin{pmatrix}ka_1\; b_1 \;c_1\\ka_2 \;b_2\; c_2\\ka_3 \;b_3\; c_3\end{pmatrix}=k\begin{pmatrix}a_1\; b_1 \;c_1\\a_2 \;b_2\; c_2\\a_3 \;b_3\; c_3\end{pmatrix}$
### Property 5
If all elements of a column or a row are equal to zero, then the determinant itself is equal to zero. This property is a special case of the previous one (for k = 0 ).
### Property 6
If the corresponding elements of two columns or two rows of the determinant are proportional, then the determinant is zero. This is a consequence of two properties 4 and 3.
### Property 7
If each element of the n-ro column or nth row of the determinant is the sum of two terms, then the determinant can be represented as the sum of two determinants, of which one in the nth column, or respectively in the nth row, has the first of mentioned terms, and the second - the second; the elements standing in the remaining places are the same for all three determinants.
$\begin{pmatrix}a_1+m_1\; b_1 \;c_1\\a_2+m_2 \;b_2\; c_2\\a_3+m_3 \;b_3\; c_3\end{pmatrix}=\begin{pmatrix}a_1\; b_1 \;c_1\\a_2 \;b_2\; c_2\\a_3 \;b_3\; c_3\end{pmatrix}+\begin{pmatrix}m_1\; b_1 \;c_1\\m_2 \;b_2\; c_2\\m_3 \;b_3\; c_3\end{pmatrix}$
### Property 8
If we add to the elements of a column (or some row) the corresponding elements of another column (or another row) multiplied by any common factor, then the value of the determinant will not change.
$\begin{pmatrix}a_1+kb_1\; b_1 \;c_1\\a_2+kb_2 \;b_2\; c_2\\a_3+kc_3 \;b_3\; c_3\end{pmatrix}=\begin{pmatrix}a_1\; b_1 \;c_1\\a_2 \;b_2\; c_2\\a_3 \;b_3\; c_3\end{pmatrix}$
Further properties of the determinants are associated with the concept of algebraic complement and minor.
The minor of a certain element is the determinant obtained from this element by deleting the row and column at the intersection of which this element is located.
The algebraic complement of any element of the determinant is equal to the minor of this element, taken with its sign, if the sum of the row and column numbers at the intersection of which the element is located is an even number, and with the opposite sign if this number is odd.
In this article, we will denote the algebraic complement of an element by a capital letter of the same name and the same number as the letter that designates the element itself.
### Property 9
Determinant
$det=\begin{pmatrix}a_1\; b_1 \;c_1\\a_2 \;b_2\; c_2\\a_3 \;b_3\; c_3\end{pmatrix}$
equal to the sum of the products of the elements of any column (or row) by their algebraic additions /
In other words, the following equalities hold:
$det=a_1A_1+a_2A_2+a_3A_3$
$det=b_1B_1+b_2B_2+b_3B_3$
$det=c_1C_1+c_2C_2+c_3C_3$
$det=a_1A_1+b_1B_1+c_1C_1$
$det=a_2A_2+b_2B_2+c_2C_2$
$det=a_3A_3+b_3B_3+c_3C_3$
There is another interesting property of the matrix determinant in the matrix determinant material . Alternative look.
Copyright © 2021 AbakBot-online calculators. All Right Reserved. Author by Dmitry Varlamov | 2021-12-07T14:17:42 | {
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https://math.stackexchange.com/questions/1626905/region-on-the-complex-plane-z-z-1-z-z-2-intersection-of-two-unit-c | # Region on the complex plane: $|z-z_{1}| = |z-z_{2}|$. Intersection of two unit circles?
I have to draw the region on the complex plane defined by the following relation:
$|z-z_{1}|=|z-z_{2}|$.
After squaring both sides, we obtain the equality $(z-z_{1})\overline{(z-z_{1})} = (z-z_{2})\overline{(z-z_{2})}$. Then, letting $z = x + i y$, $z_{1} = x_{1} + iy_{1}$, and $z_{2} = x_{2} + i y_{2}$, multiplying out and collecting terms, we obtain the following expression:
$(x-x_{1})^{2} + (y-y_{1})^{2} = (x-x_{2})^{2}+(y-y_{2})^{2}$,
which, to me, looks like two unit circles set equal to each other, so I assume the region defined by the above relation is the area where the two circles intersect. Please let me know if I am correct.
However, I now have to sketch this. Is this going to require separate cases depending on which quadrant $(x_{1},y_{1})$ and $(x_{2},y_{2})$ are in, and then cases where they don't intersect at all? I'm going a bit crazy overthinking this. Please help me figure out what to sketch.
• Geometrically, this is the set of affixes of points equidistant from the points of affixes $z_1$ and $z_2$, that is... the line segment bisector of the segment $[z_1,z_2]$. Algebraically, you can develop the identity you arrived at, the terms $x^2$ and $y^2$ cancel out hence you are left with the equation of a line. – Did Jan 25 '16 at 21:30
• @Did, and isn't that then the overlap of the two circles centered at $z_{1}$ and $z_{2}$, respectively? – ALannister Jan 25 '16 at 21:31
• "and isn't that then the overlap of the two circles centered at z1 and z2, respectively?" Did you look at the link in my comment? – Did Jan 25 '16 at 21:32
• @Did, yes and the animated graphic looks an awful lot like the intersection of two circles, if you keep going with the arcs all the way around. – ALannister Jan 25 '16 at 21:34
Note that terms cancel and you end up with a straight line which is the perpendicular bisector of $z_1$ and $z_2$.
In simple language the equation $|z-z_1|=|z-z_2|$ says "$z$ moves in such a way that its distance to $z_1$ is equal to its distance to $z_2$".
• Multiplying everything out, I see what you mean about the $x^2$ and $y^2$ terms cancelling out. Eventually, you do get a line with equation $y = \frac{(2x_{1}-2x_{2})}{(2y_{2}-2y_{1}}x + \frac{(x_{2}^{2}-x_{1}^{2}+y_{2}^{2}-y_{1}^{2})}{2y_{2}-2y_{1}}$. Do I need to sketch this line exactly (no idea how to even begin), or just on the complex plane somewhere, WLOG perhaps in the first quadrant, sketch $z_{1}$ and $z_{2}$ and draw the perpendicular bisector? – ALannister Jan 25 '16 at 21:45
• Just choose any two points $z_1$ and $z_2$ anywhere you like and draw the perpendicular bisector – David Quinn Jan 25 '16 at 21:49
The equation of a circle is of the form
$$(x - a)^2 + (y - b)^2 = r^2$$
where $r$ is a constant radius equal to the distance from any point $(x,y)$ on the circle to the circle's center, $(a,b)$.
If you have $(x - a)^2 + (y - b)^2 =$ something that is not a constant, generally you do not have a circle. You have a circle only if you can reduce the equation to the form $(x - a)^2 + (y - b)^2 = r^2$ where $r$ is a constant.
What you actually have here is a set of points that are each equidistant from $z_1$ and $z_2$. In terms of plane geometry, suppose you have two points $A$ and $B$ and a point $P$ that is equidistant from them, that is, $PA = PB$. What is the shape of the set of all such points $P$?
Here's another example: let $z_1 = i$ and $z_2 = -i$. Is it possible to find a $z$ such that $|z - i| = |z - (-i)| = 10$? Approximately where on the complex plane would $z$ be? What about $|z - i| = |z - (-i)| = 100$? Or $|z - i| = |z - (-i)| = 1000$?
the points satisfying the constraint $$|z-z_{1}| = |z-z_{2}|$$ is on the perpendicular bisector of the line joining $z_1$ and $z_2.$ it is given by $$z = \frac12(z_1+z_2) + \frac12i(z_1-z_2)k, \text{ k any real number.}\tag 1$$
we can check the work by computing \begin{align}z-z_1 &= \frac12(z_2-z_1)+\frac12ik(z_1-z_2) = \frac12(z_1-z_2)(-1+ik)\\ z-z_2 &= \frac12(z_1-z_2)+\frac12ik(z_1-z_2) = \frac12(z_1-z_2)(1+ik)\\ \vert z-z_1\vert &=\vert z-z_2\vert = \frac12\vert z_1 - z_2 \vert(1+k^2)\end{align}
• that's the equation of the perpendicular bisector or the equation of the line joining $z_{1}$ and $z_{2}$? – ALannister Jan 25 '16 at 21:52
• the equation $(1)$ represents the perpendicular bisector with $k$ a real parameter. – abel Jan 25 '16 at 21:53
Hint:
$|z-z_1|$ is the distance of $z$ from $z_1$ and $|z-z_2|$ is the distance from $z_2$. Interpreting the complex numbers as point in a plane $|z-z_1|=|z-z_2|$ means that $z$ has the same distance from $z_1$ and from $z_2$. Do you know the locus of the points with such a property?
You are going in the right track except that the equation doesn't give you the intersection of two circle.
It basically states that the region is the locus of all points that are equidistant from $z_1$ and $z_2$ which is a line perpendicular to and passes through the midpoint of the segment joining $z_1$ and $z_2$. | 2020-09-28T06:32:50 | {
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https://mathematica.stackexchange.com/questions/158410/how-to-extract-a-single-image-from-the-output-of-the-discretewavelettransform | # How to extract a single image from the output of the DiscreteWaveletTransform[]?
So, I use the following code to carry out the discrete wavelet transform on an image of Lena. I am interested in then doing some changes to a single image out of the obtained 4. How do I extract any of the images? How do I then put it back into an object which I can then carry out the inverse wavelet transform on?
img = ExampleData[{"TestImage", "Lena"}];
dwd = DiscreteWaveletTransform[img, HaarWavelet[], 2];
• Did you look at the "Properties and Relations" section of the docs for WaveletImagePlot[]? It's explained there how to obtain separate images instead of a hierarchical grid. – J. M.'s technical difficulties Oct 23 '17 at 15:41
• I must have missed it. Thank you @J.M., I have looked into now and realized that I missed it. – user3318424 Nov 21 '17 at 14:33
As I noted in the comments, the docs mention how to obtain the component images used in WaveletImagePlot[]:
img = ExampleData[{"TestImage", "Lena"}];
dwd = DiscreteWaveletTransform[img, HaarWavelet[], 2];
dwd[Automatic, "Image"]
I'm not sure how to apply ImageAdjust[] to these component images to be consistent with the result of WaveletImagePlot[], however.
Why not use ImageTake?
img = ExampleData[{"TestImage", "Lena"}];
dwd = DiscreteWaveletTransform[img, HaarWavelet[], 2]; | 2020-06-02T19:06:48 | {
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https://math.stackexchange.com/questions/3369834/do-one-dimensional-vectors-exist-what-are-they-used-for | # Do one-dimensional vectors exist? What are they used for? [duplicate]
I'm taking a linear algebra course this semester, and so far we've only talked about vectors in $$\mathbb{R}^2, \mathbb{R}^3$$ and higher dimensions.
Does it makes sense to talk about one-dimensional vectors in $$\mathbb{R}^1$$?
Since we visualize $$\mathbb{R}^2$$ as a plane, would $$\mathbb{R}$$ be a simple number line?
If one-dimensional vectors are a thing, what are they used for?
• "one-dimensional vector" is an unlucky formulation beacuse a vector has no dimension, but a number of components. So, what you mean is a vector with one component, this behaves like a real number. Such a vector makes sense and is particular easy to handle. Sep 25, 2019 at 19:20
• By the way, a real number can also be considered as a $1\times 1$-matrix. Sep 25, 2019 at 19:23
• Sometimes it can make sense to view the complex numbers as 1d vector space over the reals and then consider $\mathbb{R}$-linear maps. Sep 25, 2019 at 20:06
• @Peter I don't have a problem with the formulation "$n$-dimensional vector". This seems to be a more concise phrasing of "an element of an $n$-dimensional vector space", which is an entirely reasonable object to discuss. Moreover, not all vectors have "components"---consider, for example, an element of $L^2(\mathbb{R})$, which is a vector, but which is not generally represented in terms of "components". I don't think that it is entirely unreasonable to state that an element of $L^2(\mathbb{R})$ is an "infinite dimensional vector". Jul 14 at 14:38
Yes. Not only are one dimensional vectors a thing, "zero dimensional" vectors are too! An example of a one dimensional vector would just be any real number, as you observed. A zero dimensional vector would be an element of the trivial vector space $$\{0\}$$.
This might seemingly conflict with uses of "vector" one typically learns in middle/high school, which states that a vector is a quantity with magnitude and direction, and quantities with only magnitude are called scalars. You can still think of elements of $$\mathbb R$$ this way, in drawing an arrow from $$0$$ to any $$x$$ on the real number line and having "right or left" (positive or negative) be your direction. But more generally, this is not actually a distinction we make in math, and the definition of a vector space over a field $$K$$ is abstract in terms of the defining axioms. The abstract definitions may mean that we call some weird things vectors that are hard to get used to at first, but eventually you'll realise the usefulness of our definitions.
The answer to your first question is yes; you may, if you wish, call any $$a \in \mathbb{R}$$ a vector. However, there isn’t much purpose, since they fulfill exactly the same purpose as what you may wish to call a scalar. For instance, the vector $$<3>$$ and the scalar $$3$$both produce the same vector if you take their dot product / product with any vector in $$\mathbb{R}^n$$.
• I think it's pretty useful that theorems and constructions about real vector spaces also apply to $\mathbb R$. Sep 25, 2019 at 19:54 | 2022-10-04T23:00:09 | {
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https://www.bionicturtle.com/forum/threads/value-at-risk.10624/ | What's new
# Value at Risk
#### FieryJam
##### New Member
Hi,
Just something fundamental that popped in my mind, I am thinking if value at risk ( Var) for a long portfolio composed of commodities assets will be higher in a high commodity price environment? Because 95th percentile vAR using delta normal approach will be 1.645*standard deviation* the value of the commodities portfolio. The commodities portfolio is equal to price of the commodity * commodity volume.
Do correct me if I am wrong.
#### QuantMan2318
##### Well-Known Member
Subscriber
Hi there @FieryJam
You may find this useful. It talks about the pro cyclical nature of VaR based on the approach employed based on the excellent discussions of @emilioalzamora1 and @David Harper CFA FRM
I would say that based on the number of observations (if you use Short Term observations, then the memory effect is amplified) that you use and the fact that Standard Deviation, which is a measure of volatility spikes in a crisis environment(like those of higher prices), the general trend is that in a higher price environment, VaR does tend to be higher
Thanks
#### ami44
##### Well-Known Member
Subscriber
Your calculation is correct, if by standard derivation you mean the standard derivation of the returns (also often called volatility).
If the standard derivation of the returns is constant your conjecture is true, that with increasing prices the VaR increases. If the standard derivation is e.g. 20%, independent of the price level, than the VaR increases with increasing price. Obviously 20% of a big number is more than 20% of a low number.
If the standard deviation is in fact constant depends on your underlying price dynamic. For a brownian motion, or a geometric brownian motion it's true. But other models exist (lookup CEV models if interested).
Of course the delta-normal method implicitly assumes a brownian motion for your risk factors i.e. the commodity price. But that is one of the weaknesses of the delta-normal method, we know that commodity prices are in reality seldom brownian motions.
In reality a positive correlation between prices and VaR exist as Quantman says in the previous message.
#### FieryJam
##### New Member
Your calculation is correct, if by standard derivation you mean the standard derivation of the returns (also often called volatility).
If the standard derivation of the returns is constant your conjecture is true, that with increasing prices the VaR increases. If the standard derivation is e.g. 20%, independent of the price level, than the VaR increases with increasing price. Obviously 20% of a big number is more than 20% of a low number.
If the standard deviation is in fact constant depends on your underlying price dynamic. For a brownian motion, or a geometric brownian motion it's true. But other models exist (lookup CEV models if interested).
Of course the delta-normal method implicitly assumes a brownian motion for your risk factors i.e. the commodity price. But that is one of the weaknesses of the delta-normal method, we know that commodity prices are in reality seldom brownian motions.
In reality a positive correlation between prices and VaR exist as Quantman says in the previous message.
#### FieryJam
##### New Member
Hi Quantman and ami44,
Fiery Jam
#### David Harper CFA FRM
##### David Harper CFA FRM
Staff member
Subscriber
@FieryJam it's a fascinating philosophical truth, I think, that you've identified. If I bought AAPL at ~ $70 (true story ) when it's VaR was 20%*α, now that the stock is over$150, has my dollar amount at risk doubled just because S*20%*α has doubled? I don't feel like it has! So many things to say about this, but here is just one thought: it could be a problem with standard deviation, right? Which notoriously is indifferent to the direction. I have not run a Sortino measure on AAPL's stock, but I bet you that the downside deviation is lower than the standard deviation (maybe my dollar downside deviation hasn't changed much even as the stock has doubled). In silly extremis, if an asset price mostly increases over a period, the Sortino could be nearly zero while the historical standard deviation is high. Put another way, we can imagine a growth scenario where a stock price doubles yet the downside deviation declines even as the standard deviation increases. So sometimes you get to a higher price and the percentage standard deviation, while mathematically fine, is not such a great input into a risk measure ... In any case, I agree with you too. Thanks!
Last edited:
#### Matthew Graves
##### Active Member
Subscriber
Just my 2 cents: It's important to remember that stock returns are heteroskedastic in general. When the stock price was \$70 AAPL was probably seen as riskier bet than now, hence higher volatility and higher VaR. The volatility has likely declined now that the price and stability of the returns has risen. Maybe an exponentially weighted volatility would be better when re-estimating your VaR?
#### FieryJam
##### New Member
Thank all for the replies. Interesting and thought provoking..
#### David Harper CFA FRM
##### David Harper CFA FRM
Staff member
Subscriber
Thank you @Matthew Graves (love that signature line "Certified FRM 2016")! My idea--to consider replacing standard deviation with downside deviation--was sort of narrow in comparison to Matthew's better, broader reaction: we have available the entire set of non-parametric alternatives and semi-parametric adjustments (adjustments to historical standard deviation, including age-weighting; e.g., we can measure VaR with volatility-weighted historical simulation which, in the current low VIX environment, would decrease the VaR as a function of lower current-regime volatility) that attempt to "remedy" the weakness of blindly calculating a standard deviation for some particular historical window. While Dowd is the primary FRM author on these topics, Carol Alexander's Volume IV is basically about many of the available choices (see https://www.bionicturtle.com/forum/resources/market-risk-analysis-value-at-risk-models-volume-iv.93/). Thanks!
Last edited:
#### kchalmers
##### New Member
Hi David,
I have a general question about when to include expected return and when not to include expected risk. As I understand it... absolute VaR does not include expected return whereas relative VaR does. Will the exam distinguish between the two?
#### emilioalzamora1
##### Well-Known Member
Hi @kchalmers,
It is the opposite way round:
1. Absolute VaR: - drift + (normal deviate * vol)
2. Relative VaR: (normal deviate * vol)
where drift = 0
https://www.bionicturtle.com/forum/...value-at-risk-var-dowd.3643/page-2#post-51392
No one knows what the exam will bring for exam takers but these two equations should not be too difficult to remember.
It depends: if you have a disproportionately large positive drift using the Absolute VaR notation then such a significant drift can somewhat bias the VaR figure (>>> make the VaR value smaller and therefore implying less risk).
In general this rule of thumb holds (should be applied) for daily return calculations: absolute VaR = relative VaR. In other words, daily returns should be assumed to be zero (at least if your sample is large).
Last edited:
#### kchalmers
##### New Member
Right sorry I mixed those up... thanks this is helpful! | 2021-05-11T14:44:05 | {
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http://ernakay.com/guild-wars-njsdnr/5ef7ae-scalene-obtuse-triangle | Find scalene triangle stock images in HD and millions of other royalty-free stock photos, illustrations and vectors in the Shutterstock collection. Introduction to obtuse triangle definition and obtuse triangle tutorials with a lot of examples. Obtuse Scalene Triangle Translation to prove SSS Congruence 1. Or look at the foot of this goose; it’s scalene and obtuse. Pythagorean triples. Obtuse triangle definition, a triangle with one obtuse angle. Here, the triangle ABC is an obtuse triangle, as ∠A measures more than 90 degrees. Perimeter: Semiperimeter: Area: Area: Base: Height: Angle Bisector of side a: Angle Bisector of side b: Yes! scalene triangle definition: 1. a triangle with three sides all of different lengths 2. a triangle with three sides all of…. A triangle is a polygon made up of 3 sides and 3 angles.. We can classify triangles according to the length of their sides. In this article, you will learn more about the Scalene triangle like its definition, properties, the formula of its perimeter and area along with some solved examples. Most triangles drawn at random would be scalene. An equiangular triangle is a kind of acute triangle, and is always equilateral. I did an obtuse scalene triangle translation by moving it 4 spaces to the right and 2 spaces up then placing a point. Side a: Side b: Side c: Area: Perimeter: For help with using this calculator, see the shape area help page. a) Scalene b) Isosceles c) Obtuse d) Right-angled e) Acute 4) What type of triangle is this? A scalene triangle may be right, obtuse, ... scalene triangle A triangle with three unequal sides. b) The lengths of all three sides. In an equiangular triangle, all the angles are equal—each one measures 60 degrees. Questionnaire. The scalene property of a triangle is linked to a comparison between the lenghts of its sides (or between its three angles), whereas obtuseness is linked to the value of its angles, one in particular. Triangle. select elements \) Customer Voice. Hypotenuse. FAQ. The sum of all the angles in any triangle is 180°. There are two ways to classify triangles: by their sides and their angles, like sails out on the high seas can be right or isosceles. Equilateral triangle. In any triangle, two of the interior angles are always acute (less than 90 degrees) *, so there are three possibilities for the third angle: . That triangle would also be called right if a ninety degree angle is inside. Learn more. While drawing an obtuse triangle, you can’t draw more than one obtuse angle. An obtuse triangle has one angle measuring more than 90º but less than 180º (an obtuse angle). A scalene triangle can be an obtuse, acute, or right triangle as long as none of its sides are equal in length. Thousands of new, high-quality pictures added every day. (i) Equilateral triangle: If all sides of a triangle are equal, then it is called an equilateral triangle. [insert scalene G U D with ∠ G = 154° ∠ U = 14.8° ∠ D = 11.8°; side G U = 17 cm, U D = 37 cm, D G = 21 cm] For G U D, no two sides are equal and one angle is greater than 90 °, so you know you have a scalene, obtuse (oblique) triangle. Calculates the other elements of a scalene triangle from the selected elements. Return to the Shape Area section. What is Obtuse Triangle? Area of Scalene Triangle With Base and Height Scalene Triangle Equations These equations apply to any type of triangle. c) has all 3 sides the same length and each inside angle is the same size. scattergram A graph with points plotted on a coordinate plane. Use details and coordinates to explain how the figure was transformed, including the translation rule you applied to your triangle. A triangle with one interior angle measuring more than 90° is an obtuse triangle or obtuse-angled triangle. Perimeter of a triangle. how to i find the length in a Scalene triangle? For the most general scalene triangle, click Insert > Shapes and select the Freeform tool. triangle [tri´ang-g'l] a three-cornered object, figure, or area, such as a delineated area on the surface of the body; called also trigone. A scalene triangle is one where none of the 3 sides are equal. Isosceles triangle. The triangles above have one angle greater than 90° Hence, they are called obtuse-angled triangle or simply obtuse triangle.. An obtuse-angled triangle can be scalene or isosceles, but never equilateral. It may come in handy. In geometry, Scalene Triangle is a triangle that has all its sides of different lengths. An obtuse triangle is one where one of the angles is greater than 90 degrees. a) Right-angled b) Scalene c) Acute d) Isosceles e) Obtuse 5) An equilateral triangle... a) has 3 different lengths and 3 different angles. Educational video for children to learn what a triangle is and how many types of triangles there are. b) has 3 sides of equal length. A complete and perfect idea of what is an obtuse triangle, obtuse scalene triangle and obtuse isosceles tringle; and how to solve obtuse angled triangle problems in real life. If c is the length of the longest side, then a 2 + b 2 < c 2, where a and b are the lengths of the other sides. Interior angles are all different. A triangle with an interior angle of 180° (and collinear vertices) is degenerate. Area of Scalene Triangle Formula. A scalene triangle may be right, obtuse, or acute (see below). carotid triangle, superior carotid trigone. Answer: 2 question Classify the triangle by the side and angle Equilateral Scalene , right Scalene , obtuse Isosceles, acute - the answers to estudyassistant.com Use details and coordinates to explain how the figure was transformed, including the translation rule you applied to your triangle. A scalene triangle is a triangle where all sides are unequal. Note: It is possible for an obtuse triangle to also be scalene or isosceles. In an obtuse triangle, if one angle measures more than 90°, then the sum of the remaining two angles is less than 90°. Area of a triangle. Drag the orange vertex to reshape the triangle. An obtuse triangle can also be an isosceles or scalene triangle but it can’t be an equilateral triangle. scientific notation A method for writing extremely large or small numbers compactly in which the number is shown as the product of two factors. Was this site helpful? Obtuse triangle. set A well-defined group of objects. Congruent triangles. This is one of the three types of triangles, based on sides.. We are going to discuss here its definition, formulas for perimeter and area and its properties. BookMark Us. Eugene Brennan (author) from Ireland on August 25, 2018: If two sides are given and the angle between them, use the cosine rule to find the remaining side, then the sine rule to find the other side. Acute triangle. Are you bored? Pythagorean Theorem. But the triangle you sketch should be a non-right-angle, scalene triangle (as opposed to an isosceles, equilateral, or right triangle). An obtuse triangle is a type of triangle where one of the vertex angles is greater than 90°. Step 1: Since it is an isosceles triangle it will have two equal angles. A triangle is:scalene if no sides are equal;isosceles if two sides are equal;equilateral if three sides are equal. A(6,8) B(-1,4) C(5,4) is the obtuse triangle … Describe the translation you performed on the original triangle. Click (but don't drag) the mouse cursor at the first, second, and third corners … (ii) Isosceles triangle: If two sides of a triangle are equal, then it is called an isosceles triangle. It is not possible to draw a triangle with more than one obtuse angle. The Formula for Scalene Triangle. Try the Fun Stuff. A triangle which has at least one angle which has measurement greater than 90° but less than 180° is known as an obtuse triangle. we konw only one angle and one length. What is a Scalene Triangle? The interior angles of a scalene triangle are always all different. The angles in the triangle may be an acute, obtuse or right angle. See more. To see why this is so, imagine two angles are the same. Describe the translation you performed on the original triangle. It means all the sides of a scalene triangle are unequal and all the three angles are also of different measures. Scalene Triangle: No sides have equal length No angles are equal. This is because scalene triangles, by definition, lack special properties such as congruent sides or right angles. Obtuse Scalene Triangle Translation to prove SSS Congruence 1. A triangle that has an angle greater than 90° See: Acute Triangle Triangles - Equilateral, Isosceles and Scalene Less than 90° - all three angles are acute and so the triangle is acute. The given 96º angle cannot be one of the equal pair because a triangle cannot have two obtuse angles. Types of Triangle by Angle. Obtuse Angled Triangle: A triangle whose one of the interior angles is more than 90°. Reduced equations for equilateral, right and isosceles are below. For finding out the area of a scalene triangle, you need the following measurements. Scalene triangle [1-10] /30: Disp-Num [1] 2020/12/16 13:45 Male / 60 years old level or over / A retired person / Very / Purpose of use To determine a canopy dimension. a) The length of one side and the perpendicular distance of that side to the opposite angle. An obtuse-angled triangle is a triangle in which one of the interior angles measures more than 90° degrees. with three different sides they’re called scalene. carotid triangle, inferior that between the median line of the neck in front, the sternocleidomastoid muscle, and the anterior belly of the omohyoid muscle. Some useful scalene triangle formula are as follows: Area of Triangle = $$\frac{1}{2} \times b \times h$$, where b is the base and h is the height. Step 2: Let x be one of the two equal angles. The converse of this is also true - If all three angles are different, then the triangle is scalene, and all the sides are different lengths. Scattergram a graph with points plotted on a coordinate plane 90° is an isosceles it! High-Quality pictures added every day and how many types of triangles there are ;. Is 180° angles is greater than 90° is an isosceles or scalene triangle translation by moving 4... Translation to prove SSS Congruence 1 of a scalene triangle are acute so! Is degenerate goose ; it ’ s scalene and obtuse did an obtuse, acute, obtuse or angles... Rule you applied to your triangle 180º ( an obtuse triangle, all the sides of scalene! To also be called right if a ninety degree angle is the.. The triangle is a kind of acute triangle triangles - equilateral, and. Angle greater than 90° B ( -1,4 ) c ( 5,4 ) the. Notation a method for writing extremely large or small numbers compactly in which one of the equal pair a. Right-Angled e ) acute 4 ) What type of triangle where one of vertex... Sum of all the sides of a scalene triangle most general scalene triangle, ∠A. In an equiangular triangle, you can ’ t draw more than 90 degrees different.. Of examples isosceles and scalene Yes why this is because scalene triangles, by definition, triangle! Angled triangle: No sides are unequal sides are equal, then it is an! No angles are equal—each one measures 60 degrees whose one of the angles. If three sides all of… coordinates to explain how the figure was transformed, including the translation you! You can ’ t be an equilateral triangle: a triangle can also be an isosceles triangle the., obtuse or right triangle as long as none of its sides are equal, then it is obtuse! A ) the length of one side and the perpendicular distance of that side the. Equilateral triangle illustrations and vectors in the triangle may be right, obtuse or right triangle as long as of... Has an angle greater than 90° lot of examples can be an equilateral triangle: if two sides unequal. Equations These equations apply to any type of triangle is 180° side to the opposite.. Is acute scalene and obtuse new, high-quality pictures added every day but! The three angles are acute and so the triangle ABC is an isosceles triangle not to. Isosceles if two sides of a scalene triangle: if two sides are and. Angles of a triangle are unequal Shutterstock collection to learn What a triangle which at! Be scalene or isosceles 90° is an obtuse triangle to also be right! And vectors in the triangle is and how many types of triangles there are inside angle is same. Coordinate plane and scalene Yes one obtuse angle known as an obtuse to! An interior angle measuring more than 90° side to the opposite angle as long as none of sides... As long as none of its sides of different measures find scalene from. Whose one of the equal pair because a triangle where all sides of a triangle where all sides scalene obtuse triangle. Means all the angles in any triangle is 180° for an obtuse scalene triangle equations These apply! Small numbers compactly in which one of the vertex angles is greater than degrees..., you need the following measurements as an obtuse triangle, you need the measurements! Here, the triangle is a triangle are equal 4 spaces to the opposite angle new high-quality!
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http://spacebd.biz/1nbqnla/bec524-equivalence-class-examples | This means that two equal sets will always be equivalent but the converse of the same may or may not be true. There are $$3$$ pairs with the first element $$c:$$ $${\left( {c,c} \right),}$$ $${\left( {c,d} \right),}$$ $${\left( {c,e} \right). Example-1: Let us consider an example of any college admission process. E.g. Let ∼ be an equivalence relation on a nonempty set A. If Boolean no. • If X is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class would consist of all green cars, and X/~ could be naturally identified with the set of all car colors. For each a ∈ A, the equivalence class of a determined by ∼ is the subset of A, denoted by [ a ], consisting of all the elements of A that are equivalent to a. Equivalence partitioning is a black box test design technique in which test cases are designed to execute representatives from equivalence partitions. The definition of equivalence classes and the related properties as those exemplified above can be described more precisely in terms of the following lemma. Hence, there are \(3$$ equivalence classes in this example: $\left[ 0 \right] = \left\{ { \ldots , – 9, – 6, – 3,0,3,6,9, \ldots } \right\}$, $\left[ 1 \right] = \left\{ { \ldots , – 8, – 5, – 2,1,4,7,10, \ldots } \right\}$, $\left[ 2 \right] = \left\{ { \ldots , – 7, – 4, – 1,2,5,8,11, \ldots } \right\}$, Similarly, one can show that the relation of congruence modulo $$n$$ has $$n$$ equivalence classes $$\left[ 0 \right],\left[ 1 \right],\left[ 2 \right], \ldots ,\left[ {n – 1} \right].$$, Let $$A$$ be a set and $${A_1},{A_2}, \ldots ,{A_n}$$ be its non-empty subsets. What is Equivalence Class Testing? {\left( {d,d} \right),\left( {e,e} \right)} \right\}.}\]. JavaTpoint offers college campus training on Core Java, Advance Java, .Net, Android, Hadoop, PHP, Web Technology and Python. 4.De ne the relation R on R by xRy if xy > 0. Let $$R$$ be an equivalence relation on a set $$A,$$ and let $$a \in A.$$ The equivalence class of $$a$$ is called the set of all elements of $$A$$ which are equivalent to $$a.$$. {\left( { – 11,9} \right),\left( { – 11, – 11} \right)} \right\}}\], As it can be seen, $${E_{2}} = {E_{- 2}},$$ $${E_{10}} = {E_{ – 10}}.$$ It follows from here that we can list all equivalence classes for $$R$$ by using non-negative integers $$n.$$. Test cases for input box accepting numbers between 1 and 1000 using Equivalence Partitioning: #1) One input data class with all valid inputs. These cookies will be stored in your browser only with your consent. What is an … If there is a possibility that the test data in a particular class can be treated differently then it is better to split that equivalence class e.g. {\left( {b,c} \right),\left( {c,a} \right),}\right.}\kern0pt{\left. Equivalence class testing (Equivalence class Partitioning) is a black-box testing technique used in software testing as a major step in the Software development life cycle (SDLC). Go through the equivalence relation examples and solutions provided here. Consider the elements related to $$a.$$ The relation $$R$$ contains the pairs $$\left( {a,a} \right)$$ and $$\left( {a,b} \right).$$ Hence $$a$$ and $$b$$ are related to $$a.$$ Similarly we find that $$a$$ and $$b$$ related to $$b.$$ There are no other pairs in $$R$$ containing $$a$$ or $$b.$$ So these items form the equivalence class $$\left\{ {a,b} \right\}.$$ Notice that the relation $$R$$ has $$2^2=4$$ ordered pairs within this class. The synonyms for the word are equal, same, identical etc. All these problems concern a set . Let R be the relation on the set A = {1,3,5,9,11,18} defined by the pairs (a,b) such that a - b is divisible by 4. Example: Let A = {1, 2, 3} Pick a single value from range 1 to 1000 as a valid test case. Examples of Equivalence Classes. $\forall\, a \in A,a \in \left[ a \right]$, Two elements $$a, b \in A$$ are equivalent if and only if they belong to the same equivalence class. If so, what are the equivalence classes of R? Examples. A set of class representatives is a subset of which contains exactly one element from each equivalence class. In any case, always remember that when we are working with any equivalence relation on a set A if $$a \in A$$, then the equivalence class [$$a$$] is a subset of $$A$$. Consider an equivalence class consisting of $$m$$ elements. maybe this example i found can help: If X is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class consists of all green cars. Next part of Equivalence Class Partitioning/Testing. The relation $$R$$ is reflexive. {\left( {9, – 11} \right),\left( {9,9} \right)} \right\}}\], ${n = – 10:\;{E_{ – 10}} = \left[ { – 11} \right] = \left\{ {9, – 11} \right\},\;}\kern0pt{{R_{ – 10}} = \left\{ {\left( {9,9} \right),\left( {9, – 11} \right),}\right.}\kern0pt{\left. The relation $$R$$ is symmetric and transitive. X/~ could be naturally identified with the set of all car colors. Each test case is representative of a respective class. Example: A = {1, 2, 3} An equivalence class can be represented by any element in that equivalence class. For any a A we define the equivalence class of a, written [a], by [a] = { x A : x R a}. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. The equivalence class of under the equivalence is the set of all elements of which are equivalent to. Boundary value analysis is a black-box testing technique, closely associated with equivalence class partitioning. It is mandatory to procure user consent prior to running these cookies on your website. Check below video to see “Equivalence Partitioning In Software Testing” Each … Duration: 1 week to 2 week. {\left( {c,b} \right),\left( {c,c} \right)} \right\}}$, So, the relation $$R$$ in roster form is given by, ${R = \left\{ {\left( {a,a} \right),\left( {a,b} \right),\left( {a,c} \right),}\right.}\kern0pt{\left. Then we will look into equivalence relations and equivalence classes. Two elements of the given set are equivalent to each other, if and only if they belong to the same equivalence class. In our earlier equivalence partitioning example, instead of checking one value for each partition, you will check the values at the partitions like 0, 1, 10, 11 and so on. R1 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} Boundary value analysis is usually a part of stress & negative testing. Boundary value analysis is based on testing at the boundaries between partitions. Note that $$a\in [a]_R$$ since $$R$$ is reflexive. If anyone could explain in better detail what defines an equivalence class, that would be great! Equivalence Class Testing: Boundary Value Analysis: 1. For the equivalence class $$[a]_R$$, we will call $$a$$ the representative for that equivalence class. It can be applied to any level of testing, like unit, integration, system, and more. The equivalence class of an element $$a$$ is denoted by $$\left[ a \right].$$ Thus, by definition, The subsets form a partition $$P$$ of $$A$$ if, There is a direct link between equivalence classes and partitions. Different forms of equivalence class testing Examples Triangle Problem Next Date Function Problem Testing Properties Testing Effort Guidelines & Observations. Answer: No. If a member of set is given as an input, then one valid and one invalid equivalence class is defined. Theorem 3.6: Let F be any partition of the set S. Define a relation on S by x R y iff there is a set in F which contains both x and y. Question 1: Let assume that F is a relation on the set R real numbers defined by xFy if and only if x-y is an integer. the equivalence classes of R form a partition of the set S. More interesting is the fact that the converse of this statement is true. It can be applied to any level of the software testing, designed to divide a sets of test conditions into the groups or sets that can be considered the same i.e. }\) This set of $$3^2 = 9$$ pairs corresponds to the equivalence class $$\left\{ {c,d,e} \right\}$$ of $$3$$ elements. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. Relation R is Reflexive, i.e. First we check that $$R$$ is an equivalence relation. {\left( { – 3,1} \right),\left( { – 3, – 3} \right)} \right\}}$, ${n = 10:\;{E_{10}} = \left[ { – 11} \right] = \left\{ { – 11,9} \right\},\;}\kern0pt{{R_{10}} = \left\{ {\left( { – 11, – 11} \right),\left( { – 11,9} \right),}\right.}\kern0pt{\left. For a positive integer, and integers, consider the congruence, then the equivalence classes are the sets, etc. R1 = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} The subsets $$\left\{{}\right\},\left\{ {0,2,1} \right\},\left\{ {4,3,5} \right\}$$ are not a partition because they have the empty set. R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)} R = {(1, 1), (2, 2), (1, 2), (2, 1), (2, 3), (3, 2)} So, in Example 6.3.2, $$[S_2] =[S_3]=[S_1] =\{S_1,S_2,S_3\}.$$ This equality of equivalence classes will be formalized in Lemma 6.3.1. When adding a new item to a stimulus equivalence class, the new item must be conditioned to at least one stimulus in the equivalence class. Read this as “the equivalence class of a consists of the set of all x in X such that a and x are related by ~ to each other”.. All rights reserved. Transcript. Developed by JavaTpoint. I'll leave the actual example below. Please mail your requirement at [email protected]. Question 1: Let assume that F is a relation on the set R real numbers defined by xFy if and only if x-y is an integer. It includes maximum, minimum, inside or outside boundaries, typical values and error values. {\left( {c,b} \right),\left( {c,c} \right),}\right.}\kern0pt{\left. Let R be any relation from set A to set B. Equivalence Class Testing. \[{A_i} \ne \varnothing \;\forall \,i$, The intersection of any distinct subsets in $$P$$ is empty. 2. Question 1 Let A ={1, 2, 3, 4}. Relation R is Symmetric, i.e., aRb ⟹ bRa Relation R is transitive, i.e., aRb and bRc ⟹ aRc. R-1 = {(1, 1), (2, 2), (3, 3), (2, 1), (3, 2), (2, 3)}. Hence, Reflexive or Symmetric are Equivalence Relation but transitive may or may not be an equivalence relation. Show that the distinct equivalence classes in example … For any equivalence relation on a set $$A,$$ the set of all its equivalence classes is a partition of $$A.$$, The converse is also true. B = {x, y, z}, Solution: R = {(1, y), (1, z), (3, y) The equivalence class testing, is also known as equivalence class portioning, which is used to subdivide or partition into multiple groups of test inputs that are of similar behavior. For each non-reflexive element its reverse also belongs to $$R:$$, ${\left( {a,b} \right),\left( {b,a} \right) \in R,\;\;}\kern0pt{\left( {c,d} \right),\left( {d,c} \right) \in R,\;\; \ldots }$. A relation R on a set A is called an equivalence relation if it satisfies following three properties: Example: Let A = {1, 2, 3, 4} and R = {(1, 1), (1, 3), (2, 2), (2, 4), (3, 1), (3, 3), (4, 2), (4, 4)}. Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. The inverse of R denoted by R-1 is the relations from B to A which consist of those ordered pairs which when reversed belong to R that is: Example1: A = {1, 2, 3} For example 1. if A is the set of people, and R is the "is a relative of" relation, then A/Ris the set of families 2. if A is the set of hash tables, and R is the "has the same entries as" relation, then A/Ris the set of functions with a finite d… The standard class representatives are taken to be 0, 1, 2,...,. This gives us $$m\left( {m – 1} \right)$$ edges or ordered pairs within one equivalence class. As you may observe, you test values at both valid and invalid boundaries. Equivalence Relation Examples. … All the null sets are equivalent to each other. In an Arbitrary Stimulus class, the stimuli do not look alike but the share the same response. Suppose X was the set of all children playing in a playground. It is well … This website uses cookies to improve your experience. The equivalence classes of $$R$$ are defined by the expression $$\left\{ { – 1 – n, – 1 + n} \right\},$$ where $$n$$ is an integer. In this video, we provide a definition of an equivalence class associated with an equivalence relation. To do so, take five minutes to solve the following problems on your own. $$R$$ is reflexive since it contains all identity elements $$\left( {a,a} \right),\left( {b,b} \right), \ldots ,\left( {e,e} \right).$$, $$R$$ is symmetric. This testing approach is used for other levels of testing such as unit testing, integration testing etc. if $$A$$ is the set of people, and $$R$$ is the "is a relative of" relation, then equivalence classes are families. The equivalence class of an element $$a$$ is denoted by $$\left[ a \right].$$ Thus, by definition, in the above example the application doesn’t work with numbers less than 10, instead of creating 1 class for numbers less then 10, we created two classes – numbers 0-9 and negative numbers. Find the equivalence class [(1, 3)]. JavaTpoint offers too many high quality services. $\left\{ 1 \right\},\left\{ 2 \right\},\left\{ 3 \right\}$ This black box testing technique complements equivalence partitioning. R-1 = {(y, 1), (z, 1), (y, 3)} Consider the relation on given by if. $\left\{ {1,2} \right\}$, The set $$B = \left\{ {1,2,3} \right\}$$ has $$5$$ partitions: $\left\{ 1 \right\},\left\{ 2 \right\}$ Equivalence Classes Definitions. Relation . aRa ∀ a∈A. 2. Boundary Value Analysis is also called range checking. }\) Similarly, we find pairs with the elements related to $$d$$ and $$e:$$ $${\left( {d,c} \right),}$$ $${\left( {d,d} \right),}$$ $${\left( {d,e} \right),}$$ $${\left( {e,c} \right),}$$ $${\left( {e,d} \right),}$$ and $${\left( {e,e} \right). Thus, the relation \(R$$ has $$2$$ equivalence classes $$\left\{ {a,b} \right\}$$ and $$\left\{ {c,d,e} \right\}.$$. Equivalence Partitioning is also known as Equivalence Class Partitioning. $\left\{ {1,2,3} \right\}$. > ISTQB – Equivalence Partitioning with Examples. Relation R is transitive, i.e., aRb and bRc ⟹ aRc. Equivalence Partitioning is a black box technique to identify test cases systematically and is often the first … Let $$R$$ be an equivalence relation on a set $$A,$$ and let $$a \in A.$$ The equivalence class of $$a$$ is called the set of all elements of $$A$$ which are equivalent to $$a.$$. Example of Equivalence Class Partitioning? Hence selecting one input from each group to design the test cases. For example, the relation contains the overlapping pairs $$\left( {a,b} \right),\left( {b,a} \right)$$ and the element $$\left( {a,a} \right).$$ Thus, we conclude that $$R$$ is an equivalence relation. We also use third-party cookies that help us analyze and understand how you use this website. The relation "is equal to" is the canonical example of an equivalence relation. … The partition $$P$$ includes $$3$$ subsets which correspond to $$3$$ equivalence classes of the relation $$R.$$ We can denote these classes by $$E_1,$$ $$E_2,$$ and $$E_3.$$ They contain the following pairs: ${{E_1} = \left\{ {\left( {a,a} \right),\left( {a,b} \right),\left( {a,c} \right),}\right.}\kern0pt{\left. The equivalence classes of this equivalence relation, for example: [1 1]={2 2, 3 3, ⋯, k k,⋯} [1 2]={2 4, 3 6, 4 8,⋯, k 2k,⋯} [4 5]={4 5, 8 10, 12 15,⋯,4 k 5 k ,⋯,} are called rational numbers. In equivalence partitioning, inputs to the software or system are divided into groups that are expected to exhibit similar behavior, so they are likely to be proposed in the same way. 1. Equivalence classes let us think of groups of related objects as objects in themselves. Equivalence Partitioning is also known as Equivalence Class Partitioning. The collection of subsets $$\left\{ {5,4,0,3} \right\},\left\{ 2 \right\},\left\{ 1 \right\}$$ is a partition of $$\left\{ {0,1,2,3,4,5} \right\}.$$. But as we have seen, there are really only three distinct equivalence classes. Below are some examples of the classes $$E_n$$ for specific values of $$n$$ and the corresponding pairs of the relation $$R$$ for each of the classes: \[{n = 0:\;{E_0} = \left[ { – 1} \right] = \left\{ { – 1} \right\},\;}\kern0pt{{R_0} = \left\{ {\left( { – 1, – 1} \right)} \right\}}$, ${n = 1:\;{E_1} = \left[ { – 2} \right] = \left\{ { – 2,0} \right\},\;}\kern0pt{{R_1} = \left\{ {\left( { – 2, – 2} \right),\left( { – 2,0} \right),}\right.}\kern0pt{\left. R-1 = {(1, 1), (2, 2), (3, 3), (2, 1), (1, 2)} Every element $$a \in A$$ is a member of the equivalence class $$\left[ a \right].$$ 3. This is because there is a possibility that the application may … Revision. the set of all real numbers and the set of integers. Therefore, all even integers are in the same equivalence class and all odd integers are in a di erent equivalence class, and these are the only two equivalence classes. The equivalence class of an element $$a$$ is denoted by $$\left[ a \right].$$ Thus, by definition, \[{\left[ a \right] = \left\{ {b \in A \mid aRb} \right\} }={ \left\{ {b \in A \mid a \sim b} \right\}.}$. You also have the option to opt-out of these cookies. R1∩ R2 = {(1, 1), (2, 2), (3, 3)}, Example: A = {1, 2, 3} An equivalence class is defined as a subset of the form {x in X:xRa}, where a is an element of X and the notation "xRy" is used to mean that there is an equivalence relation between x and y. R-1 is a Equivalence Relation. Linear Recurrence Relations with Constant Coefficients. The subsets $$\left\{ 2 \right\},\left\{ 1 \right\},\left\{ 5 \right\},\left\{ 3 \right\},\left\{ 0 \right\},\left\{ 4 \right\}$$ form a partition of the set $$\left\{ {0,1,2,3,4,5} \right\}.$$, The set $$A = \left\{ {1,2} \right\}$$ has $$2$$ partitions: $\forall\, a,b \in A,a \sim b \text{ iff } \left[ a \right] = \left[ b \right]$, Every two equivalence classes $$\left[ a \right]$$ and $$\left[ b \right]$$ are either equal or disjoint. system should handle them equivalently. Given a set A with an equivalence relation R on it, we can break up all elements in A … ${A_i} \cap {A_j} = \varnothing \;\forall \,i \ne j$, $$\left\{ {0,1,2} \right\},\left\{ {4,3} \right\},\left\{ {5,4} \right\}$$, $$\left\{{}\right\},\left\{ {0,2,1} \right\},\left\{ {4,3,5} \right\}$$, $$\left\{ {5,4,0,3} \right\},\left\{ 2 \right\},\left\{ 1 \right\}$$, $$\left\{ 5 \right\},\left\{ {4,3} \right\},\left\{ {0,2} \right\}$$, $$\left\{ 2 \right\},\left\{ 1 \right\},\left\{ 5 \right\},\left\{ 3 \right\},\left\{ 0 \right\},\left\{ 4 \right\}$$, The collection of subsets $$\left\{ {0,1,2} \right\},\left\{ {4,3} \right\},\left\{ {5,4} \right\}$$ is not a partition of $$\left\{ {0,1,2,3,4,5} \right\}$$ since the. Lemma Let A be a set and R an equivalence relation on A. {\left( {0, – 2} \right),\left( {0,0} \right)} \right\}}\], ${n = 2:\;{E_2} = \left[{ – 3} \right] = \left\{ { – 3,1} \right\},\;}\kern0pt{{R_2} = \left\{ {\left( { – 3, – 3} \right),\left( { – 3,1} \right),}\right.}\kern0pt{\left. Equivalence classes let us think of groups of related objects as objects in themselves. Necessary cookies are absolutely essential for the website to function properly. So in the above example, we can divide our test cases into three equivalence classes of some valid and invalid inputs. R2 = {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)} What is Equivalence Class Testing? Equivalence Classes Definitions. For example, “3+3”, “half a dozen” and “number of kids in the Brady Bunch” all equal 6! The equivalence class [a]_1 is a subset of [a]_2. {\left( {b,a} \right),\left( {b,b} \right),}\right.}\kern0pt{\left. \[\left\{ {1,2} \right\},\left\{ 3 \right\}$ It is generally seen that a large number of errors occur at the boundaries of the defined input values rather than the center. (iv) for the equivalence class {2,6,10} implies we can use either 2 or 6 or 10 to represent that same class, which is consistent with [2]=[6]=[10] observed in example 1. Two integers $$a$$ and $$b$$ are equivalent if they have the same remainder after dividing by $$n.$$, Consider, for example, the relation of congruence modulo $$3$$ on the set of integers $$\mathbb{Z}:$$, $R = \left\{ {\left( {a,b} \right) \mid a \equiv b\;\left( \kern-2pt{\bmod 3} \right)} \right\}.$. $\left\{ 1 \right\},\left\{ {2,3} \right\}$ At the time of testing, test 4 and 12 as invalid values … Similar observations can be made to the equivalence class {4,8}. This testing technique is better than many of the testing techniques like boundary value analysis, worst case testing, robust case testing and many more in terms of time consumption and terms of precision of the test … Click or tap a problem to see the solution. {\left( {b,c} \right),\left( {c,a} \right),}\right.}\kern0pt{\left. maybe this example i found can help: If X is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class consists of all green cars. A text field permits only numeric characters; Length must be 6-10 characters long; Partition according to the requirement should be like this: While evaluating Equivalence partitioning, values in all partitions are equivalent that’s why 0-5 are equivalent, 6 – 10 are equivalent and 11- 14 are equivalent. Objective of this Tutorial: To apply the four techniques of equivalence class partitioning one by one & generate appropriate test cases? Let $$R$$ be an equivalence relation on a set $$A,$$ and let $$a \in A.$$ The equivalence class of $$a$$ is called the set of all elements of $$A$$ which are equivalent to $$a.$$. R1∪ R2= {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}. Let us make sure we understand key concepts before we move on. $\require{AMSsymbols}{\forall\, a,b \in A,\left[ a \right] = \left[ b \right] \text{ or } \left[ a \right] \cap \left[ b \right] = \varnothing}$, The union of the subsets in $$P$$ is equal, The partition $$P$$ does not contain the empty set $$\varnothing.$$ We will see how an equivalence on a set partitions the set into equivalence classes. We know that each integer has an equivalence class for the equivalence relation of congruence modulo 3. In equivalence partitioning, inputs to the software or system are divided into groups that are expected to exhibit similar behavior, so they are likely to be proposed in the same way. {\left( {1, – 3} \right),\left( {1,1} \right)} \right\}}\], ${n = – 2:\;{E_{ – 2}} = \left[ 1 \right] = \left\{ {1, – 3} \right\},\;}\kern0pt{{R_{ – 2}} = \left\{ {\left( {1,1} \right),\left( {1, – 3} \right),}\right.}\kern0pt{\left. Let R be the equivalence relation on A × A defined by (a, b)R(c, d) iff a + d = b + c . Example: Let A = {1, 2, 3} One of the fields on a form contains a text box that accepts numeric values in the range of 18 to 25. Is R an equivalence relation? This adds $$m$$ more pairs, so the total number of ordered pairs within one equivalence class is, \[\require{cancel}{m\left( {m – 1} \right) + m }={ {m^2} – \cancel{m} + \cancel{m} }={ {m^2}. Equivalence Class Testing is a type of black box technique. The set of all the equivalence classes is denoted by ℚ. Partitions A partition of a set S is a family F of non-empty subsets of S such that (i) if A and B are in F then either A = B or A ∩ B = ∅, and (ii) union A∈F A= S. S. Partitions … We'll assume you're ok with this, but you can opt-out if you wish. These cookies do not store any personal information. This website uses cookies to improve your experience while you navigate through the website. Equivalence partitioning is also known as equivalence classes. It can be shown that any two equivalence classes are either equal or disjoint, hence the collection of equivalence classes forms a partition of X. Theorem: For an equivalence relation $$R$$, two equivalence classes are equal iff their representatives are related. Given a partition $$P$$ on set $$A,$$ we can define an equivalence relation induced by the partition such that $$a \sim b$$ if and only if the elements $$a$$ and $$b$$ are in the same block in $$P.$$. This is equivalent to (a/b) and (c/d) being equal if ad-bc=0. X/~ could be naturally identified with the set of all car colors. You are welcome to discuss your solutions with me after class. Example: The Below example best describes the equivalence class Partitioning: Assume that the application accepts an integer in the range 100 to 999 Valid Equivalence Class partition: 100 to 999 inclusive. Test Case ID: Side “a” Side “b” Side “c” Expected Output: WN1: 5: 5: 5: Equilateral Triangle: WN2: 2: 2: 3: Isosceles Triangle: WN3: 3: 4: 5: Scalene Triangle: WN4: 4: 1: 2: … I've come across an example on equivalence classes but struggling to grasp the concept. By Sita Sreeraman; ISTQB, Software Testing (QA) Equivalence Partitioning: The word Equivalence means the condition of being equal or equivalent in value, worth, function, etc. © Copyright 2011-2018 www.javatpoint.com. The subsets $$\left\{ 5 \right\},\left\{ {4,3} \right\},\left\{ {0,2} \right\}$$ are not a partition of $$\left\{ {0,1,2,3,4,5} \right\}$$ because the element $$1$$ is missing. The next step from boundary value testing Motivation of Equivalence class testing Robustness Single/Multiple fault assumption. A relation R on a set A is called an equivalence relation if it satisfies following three properties: Relation R is Reflexive, i.e. {\left( {b,a} \right),\left( {b,b} \right),}\right.}\kern0pt{\left. The equivalence class could equally well be represented by any other member. Notice an equivalence class is a set, so a collection of equivalence classes is a collection of sets. \[\left\{ {1,3} \right\},\left\{ 2 \right\}$ R2 = {(1, 1), (2, 2), (3, 3), (2, 3), (3, 2)} $$R$$ is transitive. Non-valid Equivalence Class partitions: less than 100, more than 999, decimal numbers and alphabets/non-numeric characters. For example, consider the partition formed by equivalence modulo 6, and by equivalence modulo 3. Technology and Python { 1, 2, 3, 4 } transitive may may!, two equivalence classes relation R is transitive, i.e., aRb and bRc aRc! Our test cases into three equivalence classes opt-out of these cookies and ⟹... The behavior of the same equivalence class third-party cookies that ensures basic functionalities and security of. M\ ) elements security features of the application with test data residing at the time of such., identical etc ( { m – 1 } \right ) \ ) edges or ordered pairs within one class... Of R underlying set into equivalence relations and equivalence classes step from boundary value:. 3, 4 } x/~ could be naturally identified with the set of all car colors us make we! Of which are equivalent to ( a/b ) and find all elements of which are equivalent (. R be any relation from set a only representated by its lowest reduced... What defines an equivalence relation examples and solutions provided here and gives a selection of test cases word. Suppose X was the set of all elements of which are equivalent to each other, if member... For other levels of testing, test 4 and 12 as invalid values … Transcript of &! Is generally seen that a = B, then the equivalence classes in example … equivalence Partitioning is known... All car colors classes is a black-box testing technique, we analyze the of! Only with your consent browsing experience 4,8 } the option to opt-out of these may... A ] _2 to running these cookies may affect your browsing experience rather than the.! The synonyms for the equivalence class testing: boundary value analysis:.. Unit testing, test 4 and 12 as invalid values … Transcript only with consent... Same, identical etc related to it if they belong to the same equivalence class equally! Normal equivalence class testing examples Triangle Problem next Date function Problem testing Properties testing Effort Guidelines & Observations classes us! The standard class representatives are taken to be 0, 1, 3 ) ] identical! The concept understand key concepts before we move on find the equivalence class and ( )! Valid test case the share the same equivalence class { 4,8 } any element in that equivalence testing! Only with your consent identical etc disjoint equivalence classes are equal, same, identical etc \ edges. Positive integer, and integers, consider the congruence, then one valid one. Your website such as unit testing, test 4 and 12 as invalid values … Transcript of these cookies be. Of stress & negative testing cookies on your website and equivalence classes are the equivalence class testing boundary. These cookies will be stored in your browser only with your consent positive integer, and by equivalence modulo,! For an equivalence class for the word are equal iff their representatives are taken equivalence class examples be,... Concepts before we move on a partition, [ a ] _R\ ) since \ ( R\ ) is and... Behavior of the same equivalence class testing is a black-box testing technique closely... Includes cookies that help us analyze and understand how you use this uses... Unit testing, like unit, integration testing etc be a set and R equivalence. Relation but transitive may or may not be an equivalence relation on the set of all related. Symmetric, i.e., aRb ⟹ bRa relation R is transitive,,. Security features of the underlying set into equivalence classes are equal iff their representatives are.! To discuss your solutions with me after class boundary value analysis is a!, identical etc values rather than the center car colors equivalence classes is a subset of [ a _2... Also use third-party cookies that ensures basic functionalities and security features of the with. A = { 1, 2,..., in that equivalence class is defined also! Includes cookies that help us analyze and understand how you use this website uses cookies to improve your experience you. Sets will always be equivalent but the share the same equivalence class: the four Normal. Class Partitioning Robustness Single/Multiple fault assumption other levels of testing, test 4 and 12 as invalid …... Black-Box testing technique, we can divide our test cases m\ ) elements one input from each group to the... Both, and let is used for other levels of testing, test 4 12! Cases can be applied to any level of testing, integration testing etc to the... Concepts before we move on will look into equivalence classes be an equivalence relation on a nonempty set.! Equivalence modulo 6, and since we have seen, there are really three... You may observe, you test values at both valid and invalid boundaries to get more information given! This, but you can opt-out if you wish and transitive negative testing any member. See the solution opting out of some valid and one invalid equivalence {. Easy to make sure we understand key concepts before we move on integer has an equivalence class is defined collection! ] _2 javatpoint offers college campus training on Core Java,.Net, Android, Hadoop, PHP, Technology! Testing examples Triangle Problem next Date function Problem testing Properties testing Effort &... Necessary cookies are absolutely essential for the website, Reflexive or Symmetric are equivalence relation lowest or reduced form of... Us think of groups of related objects as objects in themselves or reduced form one and! The application with test data residing at the boundaries of the defined input values than. 0, 1, 3 ) ] canonical example of an equivalence class testing: boundary value analysis is set... 'Ve come across an example of an equivalence relation but transitive may or may not be an equivalence testing... For example, we can divide our test cases into three equivalence.. Are equivalence relation \ ( 1\ ) to another element of an class! Any element in that equivalence class is a equivalence class examples testing technique, closely associated equivalence... ⟹ aRc on your own only representated by its lowest or reduced form Observations! B are two sets such that a = { 1, 3, 4 } we 'll assume you ok... For other levels of testing, like unit, integration, system, and since we have a partition the. | 2021-05-16T06:53:23 | {
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https://mathhothouse.me/2016/05/ | ## Monthly Archives: May 2016
### Pappus’s theorem
Problem:
Given a point on the circumference of a cyclic quadrilateral, prove that the product of the distances from the point to any pair of opposite sides or to the diagonals are equal.
Proof:
Let a, b, c, d be the coordinates of the vertices A, B, C, D of the quadrilateral and consider the complex plane with origin at the circumcenter of ABCD. Without loss of generality, assume that the circumradius equals 1.
The equation of line AB is
$\left | \begin{array}{ccc} a & \overline{a} & 1 \\ b & \overline{b} & 1 \\ z & \overline{z} & 1 \end{array} \right | = 0$.
This is equivalent to $z(\overline{a}-\overline{b})-\overline{z}(a-b)=\overline{a}b-a\overline{b}$, that is,
$z+ab\overline{z}=a+b$
Let point $M_{1}$ be the foot of the perpendicular from a point M on the circumcircle to the line AB. If m is the coordinate of the point M, then
$z_{M_{1}}=\frac{m-ab\overline{m}+a+b}{2}$
and
$d(M, AB)=|m-m_{1}|=|m-\frac{m-ab\overline{m}+a+b}{2}|=|\frac{(m-a)(m-b)}{2m}|$ since $m \overline{m} = 1$.
Likewise,
$d(M, BC)=|\frac{(m-b)(m-c)}{2m}|$, $d(M, CD)=|\frac{(m-c)(m-d)}{2m}|$
$d(M, DA)=|\frac{(m-d)(m-a)}{2m}|$, $d(M, AC)=|\frac{(m-a)(m-c)}{2m}|$
and $d(M, BD)=|\frac{(m-b)(m-d)}{2m}|$
Thus,
$d(M, AB).d(M, CD)=d(M, BC).d(M, DA)=d(M, AC).d(M, BD)$ as claimed.
QED.
More later,
Nalin Pithwa
PS: The above example indicates how easy it is prove many fascinating theorems of pure plane geometry using the tools and techniques of complex numbers.
### Archimedes, you old fraud!
“Give me a place to stand, and I will move the Earth.” So, famously, said Archimedes, dramatizing his newly discovered law of the lever. Which in this case takes the form
Force exerted by Archimedes distance from Archimedes to fulcrum equals
Mass of Earth distance from Earth to fulcrum.
Now, I don’t think Archimedes was interested in the position of the Earth in space, but he did want the fulcrum to be fixed. (I know he said ‘a place to stand’, but if the fulcrum moves, all bets are off, so presumably that’s what he meant.) He also needed a perfectly rigid lever of zero mass, and he probably did not realize that he also needed uniform gravity, contrary to astronomical fact, to convert mass to weight. No matter, I don’t want to get into discussions about inertia or other quibbles. Let’s grant him all those things. My question is: when the Earth moves, how far does it move? And, can Archimedes achieve the same result more easily?
(Ref: Prof Ian Stewart’s Cabinet of Mathematical Curiosities).
More later,
Nalin Pithwa
### ISI or Pre-RMO practice problems — I
Problem #1.
A man started from home at 14:30 hours and drove to a village, arriving there when the clock indicated 15:15 hours. After staying for 25 min. he drove back by a different route of length $5/4$ times the first route at a rate twice as fast, reaching home at 16:00 hours. As compared to the clock at home, the village clock is
(a) 10 min slow
(b) 5 min slow
(c) 5 min fast
(d) 20 min fast
Problem #2.
If $\frac{a+b}{b+c}=\frac{c+d}{d+a}$, then
(a) $a=c$
(b) either $a=c$ or $a+b+c+d=0$
(c) $a+b+c+d=0$
(d) $a=c$ and $b=d$.
Problem #3.
In an election, 10% of the voters on the voters list did not cast their votes and 60 voters cast their ballot papers blank. There were only two candidates. The winner was supported by 47% of all voters in the list and he got 308 votes more than his rival. The number of voters on the list was
(A) 3600
(B) 6200
(C) 4575
(D) 6028
I hope to post more such questions every week,
Nalin Pithwa
### Maxima and Minima using calculus
Problem:
The vertices of an $(n+1)$-gon lie on the sides of a regular n-gon and divide its perimeter into parts of equal length. How should one construct the $(n+1)-$ gon so that its area is :
(a) maximum
(b) minimum
Hint only:
[One of the golden rule of solving problems in math/physics is to draw diagrams, as had benn emphasized by the maverick American physics Nobel Laureate, Richard Feynman. He expounded this technique even in software development. So, in the present problem, first draw several diagrams.]
There exists a side $B_{1}B_{2}$ of the $(n+1)$ -gon that lies entirely on a side $A_{1}A_{2}$ of the n-gon. Let $b=B_{1}B_{2}$ and $b=A_{1}A_{2}$. Show that $b=\frac{n}{n+1}a$. Then, for $x=A_{1}B_{1}$, we have $0 \leq x \leq \frac{n}{n+1}$ and the area S of the $(n+1)$ -gon is given by
$S(x)=\frac{\sin{\phi}}{2}\Sigma_{i=1}^{n}(\frac{i-1}{n+1}a+x)(\frac{n-i+1}{n+1}a-x)$
where $\phi=\angle{A_{1}A_{2}A_{3}}$. Thus, $S(x)$ is a quadratic function of x. Show that $S(x)$ is a minimal when $x=0$ or $x=\frac{a}{n+1}$ and $S(x)$ is maximal when $x=\frac{a}{2(n+1)}$.
Let me know if you have any trouble when you attempt it,
-Nalin Pithwa
### A Brilliant Madness — Story of John Nash, Jr., Nobel Laureate, Abel Laureate, genius mathematician
The only thing greater than the power of the mind is the courage of the human heart. — John Nash, Jr.
Just sharing the following documentary with readers/students of my blog(s).
### The Missing Symbol
Place a standard mathematical symbol between 4 and 5 to get a number greater than 4 and less than 5.
(Another precious gem from Prof Ian Stewart).
(culled by)
Nalin Pithwa
### The most beautiful formula
Occasionally, people hold polls for the most beautiful mathematical formula of all time — yes, they really do, I am not making this up, honest —- and nearly, always the winner is a famous formula discovered by Euler, which uses complex numbers to link the two famous constants $e$ and $\pi$. The formula is
$e^{i \pi}=-1$
and it is extremely influential in a branch of mathematics known as complex analysis.
(Prof. Ian Stewart’s Cabinet of Mathematical Curiosities).
(culled by)
Nalin Pithwa
### MacMahon’s Squares
This puzzle was invented by the combinatorialist P.A. MacMahon in 1921. He was thinking about a square that has been divided into four triangular regions by diagonals. He wondered how many different ways there are to colour the various regions, using three colours. He discovered that if rotations and reflections are regarded as the same colouring, there are exactly 24 possibilities. Find them all.
Now, a 6 x 4 rectangle contains 24 1 x 1 squares. Can you fit the 24 squares together to make such a rectangle, so that adjacent regions have the same colour, and the entire perimeter of the rectangle has the same colour?
(Thanks to Prof. Ian Stewart for putting this in his cabinet and I pulled it out of it !! :-))
Nalin Pithwa
### Eight great reasons to do mathematics
A nice informative and inspirational article from a prominent applied mathematician, Chris Budd .
https://plus.maths.org/content/great-eight
Thanks Prof Budd. I take this opportunity to share your views with my motivated, talented maths students !
From,
Nalin Pithwa | 2018-07-17T01:57:17 | {
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https://math.stackexchange.com/questions/379555/recurrence-relation-for-strassen | # Recurrence Relation for Strassen
I'm trying to solve the following recurrence relation (Strassen's):- $$T(n) =\begin{cases} 7T(n/2) + 18n^2 & \text{if } n > 2\\ 1 & \text{if } n \leq 2 \end{cases}$$ So I multiplied the $7$ by $2$ several times and the 18n^2^2 several times and ended up with this general equation:- $$7k T(n/2^k) + 18n^2k$$ but, well, firstly, is this correct? and also, how do I find the value of k from that beast?!
• I reformatted your question. Please check and make sure that I didn't change what your are asking. – Thomas May 2 '13 at 19:22
• This relation doesn't appear to be well-defined. How can we calculate $T(3)$, for instance? – vadim123 May 2 '13 at 19:28
• thank you thomas, it reads much better now! – SexySarah May 2 '13 at 19:38
• Can't you use the Master Theorem for that? – skrtbhtngr Sep 15 '16 at 19:30
Write $n = 2^k$ and define $U_k = T(2^k)$. The relation beocmes
$$U_k - 7\, U_{k-1} = 18 \cdot 2^{2 k}$$
with the initial condition being
$$U_1 = 1$$
I broke the equation up into a homogeneous solution and an particular solution, then applied the initial condition, as follows:
$$U_k = H_k + P_k$$
$$H_k - 7 H_{k-1} = 0 \implies H_k = A \cdot 7^k$$
Choose $P_k = B \cdot 4^k$; then
$$B \cdot 4^k - \frac{7}{4} B \cdot 4^k = 18 \cdot 4^k \implies B = -24$$
Then $U_k = A \cdot 7^k - 24 \cdot 4^k$. Use $U_1=1$ to get that
$$7 A - 96 = 1 \implies A = \frac{97}{7}$$
The result is
$$U_k =97 \cdot 7^{k-1} -96 \cdot 4^{k-1}$$
To recover $T(n)$, substitute $k=\log_2{n}$ into $U_k$. The result is
$$T(n) = \frac{97}{7} n^{\log_2{7}} - 24 n^2$$
• Please could you go into more detail with the steps you took to get there? I just don't understand now – SexySarah May 2 '13 at 19:51
• @SexySarah: steps added – Ron Gordon May 2 '13 at 20:06
• @SexySarah: no problem. As you are new to M.SE, just remember that, if you find a solution useful, please accept it by clicking the checkmark underneath the number to the left of the solution. – Ron Gordon May 2 '13 at 20:42
• all done, hope that helps – SexySarah May 2 '13 at 21:48
We can actually obtain exact values for $T(n)$ for all values of $n$ and not just powers of two.
Start by working with the alternate recurrence relation $$S(n) = 7 S(\lfloor n/2 \rfloor) + 18n^2$$ where $S(0) = 0.$
Let the binary representation of $n$ be given by $$n = \sum_{k=0}^{\lfloor \log_2 n\rfloor} d_k 2^k.$$
Then it is not difficult to see that the exact value of $S(n)$ is given by $$S(n) = 18 \sum_{j=0}^{\lfloor \log_2 n\rfloor} 7^j \left(\sum_{k=j}^{\lfloor \log_2 n\rfloor} d_k 2^{k-j}\right)^2.$$
Now to get an upper bound on $S(n)$ consider the case where $n$ consists of all one digits, giving $$S(n) \le 18 \sum_{j=0}^{\lfloor \log_2 n\rfloor} 7^j \left( \sum_{k=j}^{\lfloor \log_2 n\rfloor} 2^{k-j} \right)^2 = \frac{441}{5} 7^{\lfloor \log_2 n\rfloor}-96 \times 4^{\lfloor \log_2 n\rfloor} + \frac{144}{5} 2^{\lfloor \log_2 n\rfloor} -3.$$ For a lower bound, take $n$ to be a one digit followed by zeros, giving $$S(n) \ge 18 \sum_{j=0}^{\lfloor \log_2 n\rfloor} 7^j (2^{\lfloor \log_2 n\rfloor-j})^2 = 42 \times 7^{\lfloor \log_2 n\rfloor} - 24 \times 4^{\lfloor \log_2 n\rfloor}.$$
We still need to account for the fact that $T(0)=1, T(1)=1$ and $T(2)=1.$ A simple calculation shows that $$T(n) = S(n) - \frac{197}{7} 7^{\lfloor \log_2 n\rfloor} d_{\lfloor \log_2 n\rfloor} + 78 \times 7^{\lfloor \log_2 n\rfloor-1} d_{\lfloor \log_2 n\rfloor-1}.$$ This formula is exact and holds for all $n\ge 3$.
Finally to get the asymptotics look at the two leading terms from the lower and upper bounds. The first is $$\Theta\left(7^{\lfloor \log_2 n\rfloor}\right) = \Theta(2^{\log_2 7 \log_2 n}) = \Theta(n^{\log_2 7}).$$
$T(n)$ fluctuates around this value with the coefficient at most for strings of ones $$\frac{1}{7} \left(\frac{441}{5} - \frac{197}{7} + \frac{78}{7} \right)= \frac{356}{35}$$ and at least $$42 - \frac{197}{7} = \frac{97}{7}.$$ The next term in the asymptotic expansion is $$\Theta\left(4^{\lfloor \log_2 n\rfloor}\right) = \Theta(n^2),$$ with the coefficient between $24$ and $96.$
• There is more on this method here. – Marko Riedel May 2 '13 at 23:45
• Marko: I have no idea whether your analysis is correct or not; it seems a bit above my pay grade. But I do have a bit of unsolicited advice. I have gone through all of your posts, and you consistently state your exact result in terms of a deux ex machina: "Then it is not difficult to see that the exact value of $S(n)$ is given by..." For whom are you presenting your results? Certainly not @SexySarah, who requested details of how I solved a simple, first-order linear recurrence. I get the impression that you need another audience. If not, then fill in details for us Neanderthals. – Ron Gordon May 3 '13 at 15:18
• @RonGordon. The part about it being easy to see above refers to the fact that the proof of the equation does not involve any theorems or require additional manipulation, it simply encapsulates and represents the facts. – Marko Riedel May 3 '13 at 19:12
• @RonGordon. Thanks very much for alerting me to this problem. The trouble with the tone is something that ocurrs frequently when translating from the German and I try to avoid this effect whenever possible. As for deus ex machina, that should refer to Maple, which I use to verify all my calculations. The availability of various CAS makes it possible to omit the more tedious part from a calculation, while giving the reader powerful tools to recapitulate them if needed. And finally, can you suggest alternate places? For first/second year comp. sci. level stackexchange is a good match. – Marko Riedel May 3 '13 at 19:17 | 2020-10-01T20:27:28 | {
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https://math.stackexchange.com/questions/2149812/the-two-daughter-problem/2151248 | # The two-daughter-problem [duplicate]
When hearing about the two-daughter problem, I first thought it to be quite clear (after, of course, at first falling into the trap like many of us), but on the second glance, I encountered some serious problems with my understanding.
The original problem seems to be quite easy: Assume that the only thing you know about a man with two kids is that at least one of the kids is a daughter. What is the probability that the other kid is a daughter as well? (Boys and girls are assumed to be born equally often.)
After the first impulse ("1/2 of course!"), it becomes clear that it is only 1/3. The problem can be mapped to a situation where from the multitude of families with two children, only those with M/M are ruled out, while the equally often cases F/F, F/M and M/F remain, making F/F only one third of all remaining cases.
But now, meet Mr. Smith. I don't know much about him (except that he has two children), but when he approached me, he told me: "I am so happy! Victoria just got the scholarship she wanted!"
Now what is the probability that Victoria has a sister?
Since I only know that Mr. Smith has two children, and one is obviously a girl, I am tempted to map this onto the two-daughter-problem, leading to the answer "1/3".
But wait! What if I ask Mr. Smith first, if Victoria is his elder daughter? Assume his answer is yes (and ignore any problems with twins - even then one is typically a few seconds "older" than the other). So now I know that from the cases (F/F, F/M, M/F), M/F also drops out. And now, the probability for F/F just rose to 1/2.
Okay, but what if his answer is no? Then Victoria is the younger one, and F/M drops out. Again, the probability rises to 1/2.
So I'm going to just ask him: "Well, Mr. Smith, is Victoria your elder daughter? Wait - don't answer, because whatever you may answer, it doesn't matter. The probability just rose from 1/3 to 1/2."
Or, even better, I do not even have to ask him, just thinking about the question will shift probabilities to 1/2, which means that the original probability for Victoria having a sister must already have been 1/2. But then the mapping to the two-daughter-problem is obviously false.
Where is my error?
Making things worse, I could also create a setup where Mr. Smith just tells me: "I have two kids, and at least one of them is a girl." I then ask him: "Oh, can you give me a name of a daughter of yours?" and he answers: "Sure. Victoria."
(Side note: I have a gut feeling that this has something to do with how to assume probability distributions behind situations, similar to the Two envelopes problem, but I can't figure this out completely yet.)
-------- UPDATE --------
It seems that my error is that the question "Is Victoria the older child?" does not change the probabilities. If I know for sure that Mr. Smith was picked from an equally distributed (M/F, F/M, F/F) sample, then the knowledge that Victoria is the older child does not change anything, as was pointed out here, and the probability for her having a sister is 1/3.
But it is very interesting that solely from the sentence "Victoria just got the scholarship she wanted!" I can NOT infer that Mr. Smith is indeed chosen from this uniform distribution.
Imagine that all kids have the same chance to get a scholarship, and the happy father will tell us if it is the case. Then it is actually twice as probable that Mr. Smith will tell us about his daughter's success if he has two girls, so the weighting of the four possibilities (M/M, F/M, M/F, F/F) is (0, 1, 1, 2). And in this case, the probability of Victoria having a sister is 1/2.
So another problem in my reasoning is the mapping of Mr. Smith's statement to the two-daughter-problem. Simply put, without knowing more about the circumstances that led to Mr. Smith telling me about Victoria, I simply can't say if the probability is 1/3 or 1/2.
• Forget the headache, have a pat on the back! You have reached the correct answer to the original question. Unless you are told how the Smith family was chosen in the first place it is unanswerable. (Although as you say in one comment, most plausible real-life scenarios that could lead to such a question give a probability of 1/2.) Feb 18 '17 at 14:47
• I didn't even read past the first 2 paragraphs. For me, probability theory is not applicable here in the "pure form": you cannot ask about probabilities of actual facts of life. What is the probability that Napoleon had blue eyes according to what you know? There is no "probability" here: he either had blue eyes, or he hadn't, regardless of what you know. What is the probability i am typing this from my office on Saturday evening? Feb 18 '17 at 15:27
• @Alexey I just flipped a fair coin. Is it meaningful to talk about the probability of it being heads? Feb 18 '17 at 15:54
• For more fun, suppose Mr. Smith tells you he has two children. You ask him to name a gender he has a child of. He answers female. What is the probability he has two girls? Assuming Mr. Smith makes his choices uniformly randomly, the answer is back to one half.
– user14972
Feb 19 '17 at 12:03
• But wait! What if I ask Mr. Smith first, if Victoria is his elder daughter? Assume his answer is yes (and ignore any problems with twins - even then one is typically a few seconds "older" than the other). So now I know that from the cases (F/F, F/M, M/F), M/F also drops out. And now, the probability for F/F just rose to 1/2. I would have said in that case it rose to 1. He can't have an elder daughter if he has only one daughter. Feb 19 '17 at 18:40
I think the confusion arises because the classical boy-girl problem is ambiguous:
'You know that Mr.Smith has two kids, one of which is a girl. What is the chance she has a sister?'
The ambiguity here is that from this description, it is not clear how we came to know that 'Mr.Smith has two kids, one of which is a daughter.'
Consider the following two scenarios:
Scenario 1:
You have never met Mr. Smith before, but one day you run into him in the store. He has a little girl with him, which he tells you is one of his two children.
Scenario 2:
You are a TV producer, and you decide to do a show on 'what is it like to raise a daughter?' and you put out a call for such parents to come on the show. Mr.Smith agrees to come on the show, and as you get talking he tells you that he has two children.
Now notice: the original description applies to both cases. That is, in both cases it is true that you know that 'Mr.Smith has two children, one of which is a daughter'.
However, in scenario 1, the chance of Mr. Smith having two daughters is $\frac{1}{2}$, but in scenario 2 it is $\frac{1}{3}$. The difference is that in the first scenario one specific child has been identified as female (and thus the chance of having two daughters amounts to her sibling being female, which is $\frac{1}{2}$), while in the second scenario no specific child is identified, so we can't talk about 'her sibling' anymore, and instead have to consider a conditional probability which turns out to be $\frac{1}{3}$.
Now, your original scenario, where you don't know anything about Mr. Smith other than that he has two children, and then Mr.Smith says 'I am so happy Victoria got a scholarship!' is like scenario 1, not scenario 2. That is, unless Mr. smith has two daughters called Victoria (which is possible, but extremely unlikely, and if he did one would have expected him to say something like 'my older Victoria'), with his statement Mr.Smith has singled out 1 of his two children, making it equivalent to scenario 1.
Indeed, I would bet that most real life cases where at some point it is true that 'you know of some parent to have two children, one of which is a girl' are logically isomorph to scenario 1, not scenario 2. That is, the classic two-girl problem is fun and all, but most of the time the description of the problem is ambiguous from the start, and if you are careful to phrase it in a way so that the answer is $\frac{1}{3}$, you will realize how uncommon it is for that kind of scenario to occur in real life. (Indeed, notice how I had to work pretty hard to come up with a real life scenario that is at least somewhat plausible).
Finally, all the variations of whether Victoria is the oldest, youngest, or whether you don't even know her name ('Mr. Smith tells you one his children got a scholarship to the All Girls Academy') do not change any of the probabilities (as you argued correctly): in most real life scenarios, the way you come to know that 'Mr.Smith has two children, one of which is a girl' (and I would say that includes your original scenario) means that the chance of the other child being a girl is $\frac{1}{2}$, not $\frac{1}{3}$.
So, when at the end of you original post you ask "where is my error?" I would reply: your 'error' is that you assumed that the correct answer should be $\frac{1}{3}$, and that since your argument implied that is would be $\frac{1}{2}$, you concluded that there must have been an error in your reasoning. But, as it turns out, there wasn't! For your scenario, the answer is indeed $\frac{1}{2}$, and not $\frac{1}{3}$. So your 'error' was to think that you had made an error!
Put a different way: you were temporarily blinded by the pure math ( and I say 'temporarily', because you ended up asking all the right citical questions, and later realized that the classic two-girl problem is ambiguous: good job!). But what I mean is: we have seen this two-girl problem so often, and we have been told that the solution is $\frac{1}{3}$ so many times, that you immediately assume that also in your descibed scenario that is the correct answer... When in fact that is not case because the initial assumptions are different: the classic problem assumes a Type 2 scenario, but the original scenario described in your post is a Type 1 scenario.
It's just like the Monty Hall problem ... We have seen it so often, that as soon as it 'smells' like the Monty Hall problem, we say 'switch!' ... when in fact there are all kinds of subtle variants in which switching is not any better, and sometimes even worse!
Also take a look at the Monkey Business Illusion: we have see that video of the gorilla appearing in the middle of people passing a basketball so many times that we can now surprise people on the basis of that!
• +1 I came to the same conclusion just now. It strongly depends on the circumstances of the setting. Unfortunately, I can't mark more than one answer as the correct one, but your answer and those of klajok and zoli explain the problems I have encountered. (And, interestingly, the first gut feeling of probability 1/2 seems to be mostly correct in typical real world scenarios.) Thanks everybody for your answers and ideas! Feb 18 '17 at 13:48
• @Nebr You're welcome! And yes, for your scenario, I would say 1/2 is the corrct answer. Feb 18 '17 at 14:04
• @Nebr I think what this example really demonstrates is that mathematical idealizations (or at least the assumptions going into them) don't always apply to real life. This particularly holds true for problems about incomplete information, e.g. In the Monty Hall problem, how can we possibly know that Monty behaves a certain way rather than some other way? And this is even more so when dealing with word problems about those situations, since it can be hard to separate between what is not stated but should nevertheless be assumed, and what is not stated because we really don't know. Feb 18 '17 at 14:13
• @Nebr Just looking at your addendum where at the end you say that you can't tell whether it should be $\frac{1}{3}$ or $\frac{1}{2}$ ... I would say that your * original description* where you assume that you don't know anything about Mr. Smith other than that he has two children (right before he makes the announcement that tells you that he has a daughter called Victoria) is definitely a Type 1 scenario, and thus the answer really is $\frac{1}{2}$ ... So what I am saying is that you had it right all along: it is $\frac{1}{2}$, not $\frac{1}{3}$. Your original reasoning was all correct! Feb 18 '17 at 18:47
• Only if I interpret "I know nothing about Mr. Smith except that he has two children" as "Mr. Smith has been chosen arbitrarily from a multiple of parents with two children of equally distributed sexes". But this interpretion is not necessarily correct, although it is plausible in a real-world scenario. If I would meet Mr. Smith by chance in reality, and he would make this statement, my confidence of his other child being a girl would indeed be 1/2, and I would also bet money this way. But in a strictly mathematical way, the problem is ambiguously formulated. Feb 18 '17 at 18:57
Let us take a pragmatic approach to this. For the first problem:
Step 1: Round up a million men, each of whom has two children.
Step 2: Tell all of the men who have no daughters to go home.
Step 3: Ask all of the remaining men who have two daughters to raise their hands.
Obviously, about one third of the remaining men will raise their hands: about 750,000 men remain, and about 250,000 of them have two daughters.
For the second problem:
Step 1: Round up a million men, each of whom has two children.
Step 2: Tell all of the men who don't have a daughter named Victoria to go home. (We can ignore the scholarship.)
Step 3: Ask all of the remaining men who have two daughters to raise their hands.
Now, suppose 1 in 100 girls are named Victoria. (The exact figure doesn't matter.) Then of the 500,000 fathers with a daughter and a son, 5,000 of them will have daughters named Victoria; and of the 250,000 fathers with two daughters, 5,000 of them will also have a daughter named Victoria (because they have 500,000 daughters in total). Therefore, of the 10,000 men remaining, 5,000 will raise their hands.
So the probability that Mr Smith has two daughters is $1/2$.
• Thank you for the answer. One main point I could deduce from the answers was that the mapping between Mr. Smith's statement and a probability distribution is the crucial problem. Simply put, it is not clear under which circumstances Mr. Smith's statement has happened. As soon as the probability distribution is known, the answer becomes straightforward. Since you provide a backstory for both examples that define a probability, your conclusion is right, but without that backstory, it is not defined. Feb 18 '17 at 19:49
• @TonyK Exactly! I used two different TV show scenarios to make it a little more realistic, but I made the same point: The OP was absolutely correct intially to say it is $\frac{1}{2}$, rather than $\frac{1}{3}$ Feb 18 '17 at 19:50
• @Nebr But you told us the backstory ... Which is that you don't know anything about Mr. Smith other than him having two kids. That makes it $\frac{1}{2}$. If the back story was that you already knew that Mr.Smith has a daughter, and then says 'I am so glad Victoria got a scholarship!', then it would be $\frac{1}{3}$. Feb 18 '17 at 19:53
• @Bram28 I am currently not sure if "not knowing anything" can be identified with a uniform distribution. Note that the "Two envelopes problem" (link in my original question) shows that it can be very wrong to assume a uniform distribution when nothing is known. Feb 18 '17 at 20:04
• @Nebr I think the issue with the two envelopes problem has nothing to do with uniform or non-uniform distributions. And in this case: if I genuinely don't know anything about Mr. Smith other than him having two kids, I would say it is perfectly reasonable to assume a uniform distribution between MM, MF,FM, and FF before he makes his comment about Victoria getting a scholarship. Feb 18 '17 at 20:13
Let $XY$ denote that the sex of the younger sibling is $X$ and that of the older sibling is $Y$. $X$ and $Y$ may be $M$ or $F$, male and female. We have the following three equally likely elementary events
$$\{FF, FM, MF, MM\}.$$
These are equally likely, so $P(\{XY\})=\frac14$ for all possible $X,Y$.
The event that at least one of the siblings is a girl is
$$\{FF, FM, MF\}.$$
The event that both siblings are females is
$$\{FF\}.$$
We want to calculate the following conditional probability
$$P(\{FF\}\mid \{FF, FM, MF\})=\frac{P(\{FF\})}{P(\{FF, FM, MF\})}=\frac{\frac14}{\frac34}=\frac13.$$
The question remains: Do we agree that the following two questions are the same questions?
• What is the probability that in a family both children are girls assuming that at least one of the children is a girl?
• Assume that in a family of two children one of the children is a girl. What is the probability then that the other child is also a girl?
EDIT
Assume that a father says that he has a daughter and that daughter is older than the other child of his. Then our question modifies:
Assume that the older kid is a girl, what is the probability that the younger child is also a girl. Our conditional probability is then:
$$P(\{FF\}\mid \{FF, MF\})=\frac{P(\{FF\})}{P(\{FF,MF\})}=\frac{\frac14}{\frac12}=\frac12.$$
So, there is no contradiction. The second question is simply another question.
EDIT 2
I am only thinking... I realize that whatever the most honorable father's answer is the probability changes to $\frac12$. Wrong! Let's see what if I don't get an answer. Then the answer is either yes or no. That is, we have the following conditional probability:
$$P(\{FF\}\mid \{FF, MF\}\cup\{FF, FM\})=\frac{P(\{FF\})}{P(\{FF, FM, MF\})}=\frac{\frac14}{\frac34}=\frac13.$$
• Thank you, but I know the explanation of the 1/3 probability. The question is where the error is in my logic. Since it is a step-by-step derivation, at least one step must be wrong. Feb 18 '17 at 11:49
• @Nebr: I did an edit, please take a look.
– zoli
Feb 18 '17 at 12:00
• Okay, but by simply not asking him, I change the question and the probabilities - and that is the paradox. Please note that in my example, I did not ask Mr. Smith (it was just a thought experiment) and thus did not modify the question. Feb 18 '17 at 12:06
• OK. Make me think! I am getting too lazy.
– zoli
Feb 18 '17 at 12:07
• @Nebr: I've edited...
– zoli
Feb 18 '17 at 12:15
One way to interpret an interview such as this with a particular person (while avoiding concerns such as whether we can define things like the probability that Napoleon had blue eyes) is to view the conversation as the result of some kind of sampling process. Then the question becomes one of the ratio of second-daughters to no-second-daughters in the population from which Mr. Smith was "sampled." As pointed out in other answers, however, we have information that restricts the sub-population to which Mr. Smith might belong, and the construction and composition of that subpopulation are what matters.
In the original two-daughter question, we have to be very careful to get the information we get in such a way that we have an unbiased sample of one family from the space of all two-child families in which at least one child is a girl. Essentially, we want something equivalent to where we ask a yes-no question whose answer is "yes" in the cases FF, FM, and MF and "no" in the cases MM and anything except exactly two children. Then we accept the first person who answers "yes" to this question as our sample of one.
Equivalently, we could list all the people in the world that have exactly two children in birth order FF, FM, or MF--that is, we remove from the list anyone with children MM or with more or fewer than two children--and randomly sample one person from that list.
When our sampling method consists of Mr. Smith volunteering the information about Victoria's scholarship, this is equivalent not to sampling from a list of people with children in birth order FF, FM, or MF, but to a much more limited list of people:
1. Starting with a list of all people in the world, first we remove everyone with more or fewer than two children or with two boys.
2. Now we remove anyone who doesn't have a child that recently received a scholarship.
3. Now we have to start making some reasonable assumptions, such as that if both of Mr. Smith's children had recently received scholarships, he would have mentioned both of them. So we remove anyone from the list whose children both received scholarships recently.
4. Now we remove anyone from the list who has not just now bragged to a comparative stranger about their child's recent scholarship.
5. Now we remove anyone from the list whose child who recently received a scholarship is not named Victoria.
The remaining list after these five steps is the list of people from whom Mr. Smith has been selected in an unbiased fashion.
Up through the end of Step 4, it seems reasonable to assume that the list of people has equal numbers of people from the population with children born in the order FF, FM, of MF. But Step 4 changes that. Assuming no boys are named Victoria, but that every girl has an equal (but small) chance to be named Victoria, we retain about twice as many people from the FF part of the list as from the FM part of the list. The precise proportions are, if each girl has a small probability $p$ to be named Victoria unless they already have a sister named Victoria, and if we have $N$ people in each of the sublists FF, FM, and MF, we keep $Np$ people from each of the sublists FM and MF and $N(1 - (1-p)^2) = N(2p - p^2) \approx 2Np$ people from the FF list.
We therefore have $2Np$ people remaining in the list with children in birth order FM or MF, and approximately $2Np$ people remaining in the list with two daughters. As Mr. Smith is sampled unbiased from this list, he has a second daughter with probability approximately $\frac12.$
If we know the probability of a random girl to be named Victoria, we could calculate a more exact probability, which would be slightly less than $\frac12.$ But the only way the probability would be $\frac13$ is if practically every girl is named Victoria except those who have older sisters named Victoria. (Blended families with half-sisters could also complicate this calculation a bit if we tried to account for them, but let's assume there are few enough of these that they have only a small effect on the probabilities.)
The reason that you get the probability $\frac12$ after asking Mr. Smith apparently useless questions such as whether Victoria is his older daughter, is because the probability was already $\frac12$ before you asked the useless questions.
• Nice! I was about to add something about the probability of girls being named Victoria myself but you just did. Excellent analysis! Feb 18 '17 at 22:53
• I like this approach. Although you can of course argue that this is only correct if Mr. Smith would have only told me about the scholarship of his daughter if she was named Victoria, and else would have remained silent - which is a quite unlikely assumption in a real-world scenario. (Else, the probability would be exactly 1/2.) Feb 19 '17 at 9:02
• It's kind of weird to talk about sampling when there doesn't need to be any sampling involved. Do you really need that? Feb 19 '17 at 9:02
• @Mehrdad Without knowing about the sampling method, there is no possibility to calculate a probability. Of course, this is also a valid approach: Stating that the sampling method is ultimately unkown, although reasonable (but not strictly provable) assumptions exist, and thus rejecting the question because of incomplete information. Feb 19 '17 at 9:06
• @Nebr: I'm not sure if you read my comment. I'm saying sampling is a red herring (i.e. it is irrelevant) and doesn't even need to occur here at all. Here's why: instead of picking out 1 man and asking about his children, you can just ask "what fraction of men ..." and talk about the composition of the entire population as a whole. You could even let the population be infinitely sized if you want. Now your entire answer breaks down because it relies on sampling, and yet nobody is sampling anything. So you might want to rewrite your answer. Feb 19 '17 at 9:15
Your error: in $FF, FM, MF$ of the two-daughter problem, Victoria rules out one of $MF, FM$.
• Thank you for your answer. I didn't understand it though. Why should Victoria rule out MF or FM? It is still perfectly possible that she has an older or younger brother. Feb 18 '17 at 11:48
• She isn't of male gender.
– user65203
Feb 18 '17 at 11:48
• Of course not. She is the F in MF or FM. I still didn't get you, sorry. Feb 18 '17 at 12:07
• @Nebr: so obviously the other combination is impossible.
– user65203
Feb 18 '17 at 12:49
## Summary
Mr. Smith has Victoria (i.e. has at least one daughter):
1. Victoria has a sister with probability $\frac{1}{3}$.
2. Knowing the above now we ask if Victoria is an older child:
• Victoria is older child - she has younger sister with (still the same) probability $\frac{1}{3}$.
• Victoria is younger child - she has older sister with (still the same) probability $\frac{1}{3}$.
## Details
As was noted, if Mr. Smith has at least one daughter then the probability of the following (ordered) child pairs FM, FF, MF is equal to $\frac{1}{3}$, so Victoria has a sister with probability $\frac{1}{3}$.
If Mr. Smith is talking about Victoria then it is talking about:
• younger daughter from the pair FM with probability $\frac{1}{3}$
• younger daughter from the pair FF with probability $\frac{1}{6}$
• older daughter from the pair FF with probability $\frac{1}{6}$
• older daughter from the pair MF with probability $\frac{1}{3}$
Now we ask if Victoria has a sister:
If Mr. Smith says that the Victoria was his younger daughter then with the probability $\frac{\frac{1}{3}}{\frac{1}{3} + \frac{1}{6} } = \frac{2}{3}$ she comes from the pair FM, and with the probability $\frac{\frac{1}{6}}{\frac{1}{3} + \frac{1}{6} } = \frac{1}{3}$ she comes from the pair FF. In other words Victoria has older sister with the probability $\frac{1}{3}$.
Similar analysis can be done for the opposite situation: if Victoria is older child then she has younger sister with the probability $\frac{1}{3}$.
## Rationale
Victoria has a sister with probability $\frac{1}{3}$. During initial analysis we can think about pairs FM,MF,FF as pairs of "child with smaller favorite bear, child with bigger favorite bear". So it doesn't matter if next we ask who is older, as age is independent (at least we can assume that :) ). So the answer for the question about age should not change probabilities.
I hope I was clear enough.
• +1 I have the feeling that this points into the right direction. If we take the situation that I absolutely know that Mr. Smith is chosen from a set of equally distributed families where simply the M/M cases have been removed, then indeed probabilities remain at 1/3 even if I ask if Victoria is the younger / older daughter. But I have to know that; solely from the information "Two kids, Victoria has got a scholarship" I can't directly infer this distribution. So it seems to boil down to the question which distribution is assumed. Feb 18 '17 at 11:59
• @Nebr Assumptions are 1) all pairs MM, MF, FM, FF have exactly the same probability $\frac{1}{4}$ 2) we know nothing more than Mr. Smith has "Two kids, one is Victoria". So we don't know that Mr. Smith is my neighbor and I know his family ;) Feb 18 '17 at 12:15
• @Nebr and the last one 3) there is no correlation between gender and relative age of children. Feb 18 '17 at 12:23
• And point 1) is exactly the critical one. We have to assume that a father with MF, FM and FF is equally likely to tell us about the scholarship of his daughter. But a father with FF is twice as likely to tell us about this! So the assumption that all cases are equally likely is at least questionable from the original talk with Mr. Smith. It depends on the circumstances. Feb 18 '17 at 13:21
• Still, it is absolutely correct that asking for the elder/younger daughter does not change probabilities, as you pointed out. Either the starting probability weights are (0,1,1,2) or (0,1,1,1), but whatever they are, the not-asking can't change this (as your calculation proved). Thanks for pointing out. Feb 18 '17 at 13:24
The error is that you consider the order of events for something where each event is independent. In other words - when you end up with three possibilities, i.e.
FF, FM and MF
you are wrong. The order doesn't matter.
You can look at like this.
There are 50% chance the known F is the oldest child. This give you two valid combinations:
FF and FM (with equal probability)
There are 50% chance the known F is the yongest child. This give you two valid combinations:
FF and MF (with equal probability)
So the probability for FF is:
P = 0.5*0.5 + 0.5*0.5 = 0.5
Now if you are told that the known F is the oldest, you'll instead get:
P = 1.0*0.5 = 0.5
As you can see that will not change the probability at all.
The only question here is whether the fact that Victoria is a girl made you more likely to hear about her.
Case 1: You live in a country where parents, if they have a daughter, will never say anything about their sons, but will instead talk about one of their daughters.
Case 2: Parents are equally likely to talk about kids of either gender.
Case 3: Parents are more likely to mention their daughters, but do sometimes talk about sons.
In case 1, the probability that Victoria's sibling is a sister is 1/3. In case 2, it's 1/2. In case 3, the probability is intermediate.
• A more striking case to add is a country where people always prefer to talk about sons if possible, in which case the fact you heard about Victoria means she has a sister with 100% probability!
– user14972
Feb 19 '17 at 16:38
• @Hurkyl, yes, absolutely right. Feb 19 '17 at 18:05
This problem actually bears a strong connection to the classic Monty Hall problem.
In this problem, the man tells you that at least one of his children is a girl - as the only other possibility is a boy, this gives three possibilities: FM, FF, or MF.
In the Monty Hall problem, Monty tells you that there are three doors, with one having a car and two having goats. Without loss of generality, let's say that the middle door actually has the car (but you don't know this) - so the order becomes GCG.
But in the Monty Hall problem, you choose one door, and then Monty reveals a goat behind one of the other two doors.
So if you look at the two remaining doors (other than the one you chose), you could have GC, GG, or CG - this maps perfectly onto the possibilities in this problem.
And then Monty revealing the goat behind one of these doors is equivalent to the man answering the question "is the girl you refer to the elder child?" - and the probabilities actually work exactly the same.
One of the interesting things about the Monty Hall problem is that the benefit of winning depends on the chances of the events playing out in certain ways - if Monty reveals a door randomly, then despite the actual result (Monty reveals a goat) being the same, the chances that you chose the car becomes 50% rather than the ~33% from the original formulation of the problem.
And the same applies to this problem - if you ask the man "Is the girl you refer to (Victoria) the elder child?" and he answers "yes", then the probability stays the same (~33% chance that the younger child is a girl). But if you ask "Is your elder child a girl?" and he answers "yes", then the probability that the younger child is a girl becomes 50%.
This isn't because of the information gained, but because of the underlying likelihood of the response given the question asked. In both cases, you learn that the elder child is a girl.
This is because you're not being asked to work out the chances of the man having two girls. You're asking for how confident you can be in the claim that the man has two girls given the information you have and the assumptions you take. And you're assuming a randomly chosen father of two children in a world where gender of each child is independent and equally likely to be male or female.
Then you're given further information. If you're simply informed, by another source, that his eldest child is a girl, without being told how that information was gathered, you're going to have to make an assumption about how the information was gathered.
If you assume that he was asked to identify one of the children (elder or younger) as a girl, then the chances that the other is a girl is 1/3. If you assume that he was asked whether his elder child was a girl, the chances that the other is a girl is 1/2.
And if you assume that he was asked if the younger child was a girl, then the chances that the younger is a girl is 0 - clearly, he said "no, the younger child is not a girl", and thus, in conjunction with "at least one child is a girl", it was concluded by your source that the older child must be a girl.
As I understand it, this is the principle behind Bayesian statistics - the "prior" is the assumption about how the information was obtained, in this case, and the prior is updated as appropriate.
Suppose that you were told that there were three possible questions, and one was selected from among those three - "Is your elder child a girl?", "Is your younger child a girl?", and "Please identify one of your children, elder or younger, as a girl" (assuming that the father would choose at random if both are girls).
Now, you're told that this man's elder daughter is a girl. You can immediately rule out the second question (no answer to the second question would allow your source to identify the elder daughter as a girl), and the first question is more likely than the third. This allows you to update your prior - your assumption about the information and its underlying distribution.
The crux of this problem, is whether or not your sampling method for the second sibling is biased in favour of sisters.
The reason your example is biased is that you are asking; "given that I have first selected and eliminated a daughter from your two children, what is the probability the 2nd is a daughter." This question is biased.
If you learn he has at least one daughter then the probability the other sibling is a sister, is always $\frac{1}{3}$ since the definition of the other is biased against sisters.
However if you first ask what his eldest child is, and he happens to answer "a daughter", or "a son" and you then ask the sex of the other, it will always be $50:50$, irrespective of the first answer, since you are simply taking a second unbiased sample from the population of children.
The way I see it, there're two ways to arrive at the (correct) conclusion that the probability of Victoria having a sister is 1 in 2.
Method 1 (The Genetics Method)
The probability of any given child being a daughter is 1 in 2. Therefore, the probability of Child 2 being a daughter is 1 in 2. This is unaffected by Child 1 (Victoria) happening to be a daughter.
Method 2 (The Four-Options Method)
For a family with two children, there are four possible configurations regarding the sex of the children, all of which are (for our purposes) equally likely.
Configuration A - Child 1 is a son; Child 2 is a son.
Configuration B - Child 1 is a son; Child 2 is a daughter.
Configuration C - Child 1 is a daughter; Child 2 is a son.
Configuration D - Child 1 is a daughter; Child 2 is a daughter.
When we are told that Child 1 (Victoria) is a daughter, this immediately eliminates configurations A and B, as each of these configurations involves Child 1 being a son, which we now know is not the case. Therefore, we are left with just two possible configurations:
Configuration C - Child 1 is a daughter; Child 2 is a son.
Configuration D - Child 1 is a daughter; Child 2 is a daughter.
These two configurations are equally likely, and, therefore, Child 2 (Victoria's sibling) is equally likely to be a son or a daughter.
• How do you determine that Victoria is child 1 rather than child 2?
– user14972
Feb 19 '17 at 16:44
• It also works equally well if Victoria is child 2. Jun 26 '17 at 20:56
• If Victoria is child 2, then the probability of child 2 being a daughter is 1 in 1, not 1 in 2 like the analysis assumes.
– user14972
Jun 27 '17 at 4:13
• Poor phrasing; if Victoria is child 2, then we're talking about the probability of child 1 being a daughter. Hope that clears things up. Jun 27 '17 at 22:07
• Which goes back to my question: how do you determine whether Victoria is child 1 or child 2? The phrasing is fine: there's simply an essential gap in the argument.
– user14972
Jun 28 '17 at 5:11
We make the simplifying assumptions:
• Equal chance of boy or girl.
• Victoria is a girl's name and is never given to boys.
• Mr Smith has two children, whom I've arbitrarily labeled Child A and Child B. Child A is girl with property Q or Child B is a girl with property Q. What is the chance that Child A is a girl and child B is a girl.
Let's rephrase the question to be more amenable to Bayesian Analysis.
• A priori, we know that Mr Smith has two children, whom I've arbitrarily labeled A and B. Given that (Child A is a girl and Child A has property Q) or (Child B is a girl and Child B has property Q), what is the chance that Child A is a girl and child B is a girl.
Let
• $A_G$ be the statement Child A is a girl
• $B_G$ be the statement Child B is a girl
• $A_Q$ be the statement Child A has property Q
• $B_Q$ be the statement Child B has property Q
\begin{align} P([A_G \text{ and } B_G] \text{ given } [(A_G \text{ and } A_Q) \text{ or } (B_G \text{ and } B_Q )]) \end{align}
Now, P (A \text{ given } B) = \frac{P(A \text{ and } B)}{P(B)}, so our expression above is equal to \begin{align} &\quad\frac{P([A_G \text{ and } B_G] \text{ and } [(A_G \text{ and } A_Q) \text{ or } (B_G \text{ and } B_Q )])}{P((A_G \text{ and } A_Q) \text{ or } (B_G \text{ and } B_Q ))}\\ & = \frac{P([A_G \text{ and } B_G \text { and } A_Q] \text{ or } [A_G \text{ and } B_G \text { and } B_Q])}{P((A_G \text{ and } A_Q) \text{ or } (B_G \text{ and } B_Q ))} \end{align} For now, let us assume that Q implies you are a girl. SoA_Q \implies A_G$, and$B_Q \implies B_G$. This means$P(A_Q \text{ and } A_G) = P(A_Q). We can then simplify the above statement to: \begin{align} \frac{P([B_G \text { and } A_Q] \text{ or } [A_G \text { and } B_Q])}{P(A_Q \text{ or } B_Q)} \end{align} Now,P(A \text{ or } B) = P(A \text{ and }\neg B) + P (\neg A \text{ and } B) + P(B \text {and} A)$. ($\negmeans not). So, expanding the top and the bottom, our formula becomes, \begin{align} \frac{P([B_G \text { and } A_Q] \text{ and } \neg [A_G \text { and } B_Q]) + P(\neg [B_G \text { and } A_Q] \text{ and } [A_G \text { and } B_Q]) + P([B_G \text { and } A_Q] \text{ and } [A_G \text { and } B_Q]) }{P(A_Q \text{ and } \neg B_Q) + P(\neg A_Q \text{ or } B_Q) + P(A_Q \text{ and } B_Q) } \end{align} Now our equation is a mess. But we did this so we can apply some symmetry arguments, and because it will be easier to understand. Let's simplify the top a bit first though. We note that \begin{align} &\quad[B_G \text { and } A_Q] \text{ and } \neg [A_G \text { and } B_Q]\\ & = [B_G \text { and } A_Q] \text{ and } [\neg A_G \text { or } \neg B_Q]\\ & = B_G \text { and } A_Q \text { and } \neg B_Q &\text{sinceA_Q \implies A_G} \end{align} Our formula then simplifies to \begin{align} \frac{P(B_G \text { and } A_Q \text{ and } \neg B_Q) + P( A_G \text { and } B_Q \text { and }\neg A_Q) + P([B_G \text { and } A_Q] \text{ and } [A_G \text { and } B_Q]) }{P(A_Q \text{ and } \neg B_Q) + P(\neg A_Q \text{ or } B_Q) + P(A_Q \text{ and } B_Q) } \end{align} We now apply the symmetry argument, noting that Child A and Child B are arbitrary labels, and as such \begin{align} P(B_G \text { and } A_Q \text{ and } \neg B_Q) &= P( A_G \text { and } B_Q \text { and }\neg A_Q)\\ P(A_Q \text{ and } \neg B_Q) &= P(\neg A_Q \text{ or } B_Q) \end{align} Our formula simplifies to \begin{align} &\quad\frac{2 P(B_G \text { and } A_Q \text{ and } \neg B_Q) + P([B_G \text { and } A_Q] \text{ and } [A_G \text { and } B_Q]) }{2 P(A_Q \text{ and } \neg B_Q) + P(A_Q \text{ and } B_Q) }\\ &=\frac{2 P( [A_Q \text{ and } B_G] \text { and } \neg B_Q ) + P(A_Q \text { and } B_Q ) }{2 P(A_Q \text{ and } \neg B_Q) + P(A_Q \text{ and } B_Q) }\\ \end{align} Now, we at last note thatP (A \text{ and } \neg B) = P(A) - P (A \text{ and } B)to rewrite the formula as: \begin{align} &\quad\frac{2 P( [A_Q \text{ and } B_G] \text { and } \neg B_Q ) - 2 P(A_Q \text{ and } B_G \text{ and } B_Q) + P(A_Q \text { and } B_Q ) }{2 P(A_Q \text{ and } \neg B_Q) + P(A_Q \text{ and } B_Q) }\\ &=\frac{2 P( A_Q \text{ and } B_G) - 2 P(A_Q \text{ and } B_Q) + P(A_Q \text { and } B_Q ) }{2 P(A_Q \text{ and } \neg B_Q) + P(A_Q \text{ and } B_Q) }\\ & = \frac{2 P( A_Q \text{ and } B_G ) - P(A_Q \text{ and } B_Q) }{2 P(A_Q \text{ and } \neg B_Q) + P(A_Q \text{ and } B_Q) } \end{align} In almost all cases,A_Q$and$B_G$are independent. That is, a priori,$A$having the property$Q$does not affect$B$'s sex. One of the only case where$A_Q$affects$B_G$is if$A_Q$states that A is the only girl. However, we cannot assume that$A_Q$is independent of$B_Q$. For example,$Abeing the older child means that B cannot be the older child. We then write our formula as \begin{align} &\quad\frac{2 P (A_Q)P( B_G ) - P(A_Q \text{ and } B_Q) }{2 P(A_Q \text{ and } \neg B_Q) + P(A_Q \text{ and } B_Q) }\\ &=\frac{2 P (A_Q)P( B_G ) - P(A_Q \text{ and } B_Q) }{2 P(A_Q \neg B_Q) - 2P (A_Q \text{ and } B_Q) + P(A_Q \text{ and } B_Q) }\\ & = \frac{2P (A_Q)P( B_G ) - P(A_Q \text{ and } B_Q) }{2 P(A_Q ) - P (A_Q \text{ and } B_Q) } \end{align} Finally, we note thatP(B_Q \text{ and } A_Q) = P(B_Q \text{ given } A_Q)P(A_Q)We finally have \begin{align} &\quad \frac{2P (A_Q)P( B_G ) - P(B_Q \text{ given } A_Q) P(A_Q) }{2 P(A_Q ) - P (B_Q \text{ given } A_Q) P(A_Q)}\\ & = \frac{2P( B_G ) - P(B_Q \text{ given } A_Q) }{2 - P (B_Q \text{ given } A_Q) } \end{align} Finally, We have, given the following assumptions: • Having the property Q implies being a girl. We can handle the case of Q not implying being a girl as well. Shown below. • Child A having property Q does not affect Child B's being a boy or a girl. \begin{align} \frac{P( B_G ) - \frac{1}{2} P(B_Q \text{ given } A_Q) }{1 - \frac{1}{2} P (B_Q \text{ given } A_Q) } \end{align} or if we setP(B_G) = 0.5\begin{align} \frac{1 - P(B_Q \text{ given } A_Q) }{2 - P (B_Q \text{ given } A_Q) } \end{align} We have done this to show that the probability ultimately depends on how likely it is that Child B has property Q given that Child A has property Q, a priori. Let us look at two extreme cases. We'll also use these cases to demonstrate how to handle the case when Q does imply being a girl. A)B_Q$cannot occur if$A_Q$is true. Mr Smith has two children, whom I've arbitrarily labeled Child A and Child B. Child A is girl and is oldest or Child B is a girl and is oldest. What is the chance that Child A is a girl and child B is a girl? Let$Q$be the property of being the oldest child. Q does imply being a girl. However, we can simply change$Q$to be the property of "being a girl, and being oldest". This does not change our question at all. Since only one child can be the oldest girl,$P(B_Q \text{ given } A_Q) = 0$. And our formula simplifies to$P(B_G) = 0.5$. B)$B_Q$and$A_Qare empty statements. Something that tells us nothing about A and B. For example • Mr Smith has two children, whom I've arbitrarily labeled Child A and Child B. Child A is girl (and is human) or Child B is a girl (and is human). What is the chance that Child A is a girl and child B is a girl? Once again, an empty statement (or being human) does not imply being a girl. So Q must therefore be the statement "is a girl (and is human)". That means Q=G. \begin{align} &\quad\frac{1 - P(B_Q \text{ given } A_Q) }{2 - P (B_Q \text{ given } A_Q) }\\ &=\frac{1 - P(B_G \text{ given } A_G) }{2 - P (B_G \text{ given } A_G)}\\ & = \frac{1 - P(B_G ) }{2 - P (B_G )} &\text{sinceB_G$and$A_Gare independent}\\ & = \frac{1}{3} \end{align} In summary, the more unlikelyP(B_Q \text{ given } A_Q)$is, the more the probability tends toward$\frac{1}{2}$. We should think of$P(B_Q \text{ given } A_Q)$as how unique the property$Q$is. If Q is super unique, then we can identify the child with property Q, and the probability tends toward$\frac{1}{2}$. If Q is vague, and it is very possible that both child has property Q, then we cannot use property Q to identify the child, and the probability tends toward$\frac{1}{3}$. To answer the question Mr Smith has two children, whom I've arbitrarily labeled Child A and Child B. Child A is a girl named Victoria or Child B is a girl named Victoria. What is the chance that Child A is a girl and child B is a girl? It depends how likely the parent named both girls Victoria. The difference is down to fact that the probability of a randomly chosen child of a man with two children, at least one of which is a girl, being a girl is not equal to the probability of a man with two children, a randomly chosen one of which is a girl, having at least one daughter. You probably read that last bit and though "well that's just a tautology, if he has a girl, then he must have at least one girl". Indeed, and I'm sure we can agree that him having a girl does not mean that if he picks a child at random that it will be a girl. This answer tries to address why you seem to have a paradox: it's simply because the information we have is different. I've ignored anything to do with children's names, we just assume that if a child is called Victoria then it is female. I use straight up probabilities, so there is no risk of introducing unintended assumptions (at least I hope not). Let's say he has two children, A and B. We express the probability of each being female as P(A) = 1/2 P(B) = 1/2 This encodes our assumption about any child having an equal and independent probability of being female. Now we define P(C) = 1/2, which is the probability that upon seeing this man with one child that it is child A (and so the probability of seeing child B is P(¬C) = 1-P(C) = 1/2), that is to say, there is an equal chance of seeing a man with two children with his elder or younger child when he only has one child with him. Duly, we can define random variable D, which is the probability that when we see this gent with a child that the child is female. For this, we simply compute the probability given all possible outcomes of C: P(D) = P(A|C) * P(C) + P(B|¬C) * P(¬C) = 1/2 * 1/2 + 1/2 * 1/2 = 1/2 We can also compute the probability of some man with two children having at least one daughter, which we all agree has an prior probability of 3/4. P(A or B) = P(A) + P(B) - (P and B) = 3/4 Intuitively, we state that the probability of A or B given D must be 1 (we can't have seen a girl if there was none to randomly select): P(A or B|D) = 1 Now we return to my first paragraph, and find the probability of D given A or B. Behold Bayes' theorem (it's a rearrangement of P(A and B) = P(A|B)*P(B) = P(B|A)*P(B)): P(D|A or B) = (P(A or B|D) * P(D)) / P(A or B) = (1 * 1/2) / P(3/4) = 2/3 In effect, you are more likely to have seen such a man with a girl if he has at least one daughter than you are to see him with a boy, but it isn't certain. For completeness: P(A and B) = 1/4 From the working above, we can quickly compute the 'answer' to original problem by invoking Bayes' again. The traditional riddle: P(A and B|A or B) = (P(A or B|A and B) * P(A and B)) / P(A or B) = (1 * 1/4) / 3/4 = 1/3 Your adaption: P(A and B|D) = (P(D|A and B) * P(A and B)) / P(D) = (1 * 1/4) / 1/2 = 1/2 Note that we can rewrite these to make P(A and B) the subject of each. P(A and B) = (P(A and B|A or B) * P(A or B)) / P(A or B|A and B) P(A and B) = (P(A and B|D) * P(D)) / P(D|A and B) As stated in your currently accepted answer, your mistake was to assume that P(A and B|D) = 1/3, likely a result of not realising that if a man has two daughters, you are more likely to see him with a daughter than if he has only one, and that this prior probability cannot be neglected (all humans are very good at neglecting prior probabilities (i.e. sampling biases)). A classic error for people starting out with probabilities to make is to say "I know T to be true, therefore P(T) = 1". Do not do this! Instead, you should consider situations given T. P(T) is called a 'prior' probability, and P(X|T) is a posterior probability. Your observations can never change these probabilities: you just have to work out which probability is the answer you are looking for. Again, this same neglect of prior probabilities is a massive trap that I personally have to climb out of every time I try to do stats. I'm writing this answer as much for my benefit as anyone else, because hopefully I'll not be lured into making this mistake again! Sorry this isn't so pretty as the other answers, I'm not clear on how to use MathJax, I'll try to format it in an edit. Edit: Having reformatted it as MathJax... I thought it was considerably more ugly and harder to read, so won't be committing the edit The key point as to the difference between the two problems is the hypothesis that a child is selected in a gender neutral fashion; so We can compute conditional probabilities $$P(\text{Victoria has a sister} | \text{ the announcement}) \\= \frac{P(FF \text{ and the announcement})}{P(\text{the announcement})} \\= \frac{P(\text{the announcement} | FF) P(FF)}{P(\text{the announcement})}$$ $$P(\text{Victoria has a brother} | \text{ the announcement}) \\= \frac{P(\text{the announcement} | MF) P(MF)}{P(\text{the announcement})}$$ where "the announcement" is the proposition that you heard Mr. Smith make that announcement. Now, we know that$P(FF) = 1/4$and$P(MF) = 1/2$(note that we still have$P(MM) = 1/4$; it's just that$P(MM | \text{the announcement}) = 0$). Thus, the relative probabilities of the two cases are: $$R = \frac{P(\text{Victoria has a sister} | \text{ the announcement})} {P(\text{Victoria has a brother} | \text{ the announcement})} = \frac{1}{2} \cdot \frac{ P(\text{the announcement} | FF)} { P(\text{the announcement} | MF)}$$ So, we have to judge how the sex of the children influence the probability that Mr. Smith would make that announcement. One way is to separate the announcement into two parts: • Scholarship - referring to Mr. Smith announcing a child got a scholarship • Victoria - that the child referred to is named Victoria We can rearrange the conditional probabilities into $$P(\text{the announcement} | FF) = P(\text{Victoria} | \text{FF and Scholarship}) P(\text{Scholarship} | FF)$$ and similarly for MF. Consequently, $$R = \frac{1}{2} \cdot \frac{P(\text{Victoria} | \text{FF and Scholarship}) P(\text{Scholarship} | FF)}{P(\text{Victoria} | \text{MF and Scholarship}) P(\text{Scholarship} | MF)}$$ At this point, the following hypotheses seem reasonable enough: Hypotheses: •$P(\text{Scholarship} | FF) = P(\text{Scholarship} | MF)$•$P(\text{Victoria} | \text{FF and Scholarship}) = 2 \cdot P(\text{Victoria} | \text{MF and Scholarship})$Therefore $$\frac{P(\text{Victoria has a sister} | \text{ the announcement})} {P(\text{Victoria has a brother} | \text{ the announcement})} = 1$$ thus confirming the probabilities should be one-half. Not sure , as there were so many answers, if it was answered or not. Neither 1/2 nor 1/3 are correct. The chance for him having a 2nd daughter is 1/4. This is about unordered sampling and you are confusing loads of various sample sets and dismissing certain things. Disregarding twins, etc, and assuming equal frequencies for boys and girls, Mr. daddy has the following 4 outcomes for his two children:bb, bg, gb, gg. So 1/4 for two boys, 1/2 for a boy and girl, (your questions about older/younger is about ordered sets) and 1/4 for two girls. You come along and ask him, and meet him and his daughter. This has no affliction upon his existing children and thus the probability of him having 2 girls, you can easily surmise to have been 1/4, so it still is 25% chance for being a girl. • I don't think that you may completely neglect the information that you just met a daughter of him. Imagine that afterwards you also met his other daughter - is the probability for two girls still 25% then (since meeting both daughters still has no affliction upon the existing children)? As David K pointed out in his answer, the term "probability" might not be completely correct here (since the sex of the children is already defined before any information is given), but statistics change if you get to know bits of information. Feb 20 '17 at 8:15 • No Nebr. You are thinking of something else. The difference of the two problems are easier understood as this: Two persons choose a number from 1to 100 mentally, write down the number in an envelope. What is the original chance that they chose the same number? Now , one of them gets to open the other envelope, and thus knows if they both did choose the same or not. He then proceeds to remove all but two balls. Both chosen numbers remain. The 2nd lad doesn't know if his number which remains is the same as the other or not but is NOW asked if he wishes to CHANGE his ball to the other ball to – Mal Feb 21 '17 at 5:50 • ...match the first one's ball, which was the aim. By not switching the chance 1 %, by accepting to swap, his chance is close to 1. This is what you are mixing up between the two situations. One AFFECTS the chances, (which is what you have been doing incorrectly), whereas the situation you posed gives the answer 25% that his other child is namely a girl. – Mal Feb 21 '17 at 5:57 • Your comment about seeing the full sample set and still wondering if [I think] it would be 25% is entirely wrong.. I am sure you can see that? That is no longer statistics. What you are asking then , is like as lim #samples -> inf, P(2 girls) -> 1/4. Yes. The more samples we take, statistics will assume probability of coin flip will converge to 1/2, just as the amount of girl/girl pairs will converge to 1/4. Just because people write endlessly complex and long answers, just goes to show how misunderstood simple things can be. – Mal Feb 21 '17 at 5:58 • So you are telling me, if I meet a man with 1000 children and I get to know 999 of his daughters, the probability that the last children is a girl is (1/2)^1000? Feb 21 '17 at 8:07 "But wait! What if I ask Mr. Smith first, if Victoria is his elder daughter? Assume his answer is yes (and ignore any problems with twins - even then one is typically a few seconds "older" than the other). So now I know that from the cases (F/F, F/M, M/F), M/F also drops out. And now, the probability for F/F just rose to 1/2." If Smith says Victoria is his elder daughter, the other child could be a boy of any age, or a girl that is younger than Victoria. If Smith says Victoria is NOT his elder daughter, the other child must be a girl that is older than Victoria. So a "yes" answer allows GG, GB, or BG. A no answer only allows GG. How many bits of information are transmitted with the answer to this question? • Why must the other child be a girl? If Victoria is NOT his elder daughter, she must be his younger daughter, which still leaves room for an elder brother. Feb 23 '17 at 7:41 For me, probability theory is not applicable here in the "pure form": you cannot ask about probabilities of actual facts of life. What is the probability that Napoleon had blue eyes according to what you know? There is no "probability" here: he either had blue eyes, or he hadn't, regardless of what you know. What is the probability I am typing this from my office on Saturday evening? What is the probability that Higgs boson exists? What is the probability that your wife/husband is cheating on you? The problem would make more sense in the following form (for example): suppose you know that someone has had a daughter, what is the probability that their next child, if there will be one, will be a daughter as well? You might argue that we can view the problem practically, as if someone tells you that someone else has two children, of which one is a daughter, and asks you to place a bet on the sex of the other child. But in this settings surely you should take into account the motives of the person who asks you this, because probably they wouldn't be asking if there were not good chance that you would lose your bet. More formally: the probability space in such questions about past events or actual facts of life is not well defined, and most definitions would be practically irrelevant. For past events and actual facts of life one uses statistics, not probability theory. Update. After much criticism my point of view and interpretation of the problem have received, let me mention that I am also aware of the following simple probabilistic interpretation. Consider the totality$X$of all people with 2 children of which at least one is a daughter. This is about 3/4 of the totality of all people with 2 children. Now choose a person in$X\$ at random. Then the probability that both their children are girls is approximately 1/3.
• what is the probability that their next child, if there will be one, will be a daughter as well is NOT the same question. Feb 18 '17 at 15:39
• Of course not, i explained why the original question makes no sense, while this one does. Feb 18 '17 at 15:41
• I can't believe I'm having to spell this out, but you introduced that statement with The problem would make more sense in the following form, but what followed is a different problem. So it doesn't "make more sense", because it changes the problem. You could say "A problem that would make more sense would be...", but again it's all irrelevant because it does nothing to answer the question that was asked. Feb 18 '17 at 15:52
• Well, IMO you cannot save the original problem, so if you want to make it meaningful, you'd have to restate it, for example in this form (if this vaguely resembles the practical question you are interested in). As the problem is meaningless, its different meaningful "forms" can be mutually inequivalent. Feb 18 '17 at 15:56
• Comments are not for extended discussion; this conversation has been moved to chat. Feb 19 '17 at 9:55
If the father is chosen randomly(among those with 2 children) the probability he's a parent of 2 daughters is obviously 1/3. (each of the combinations from F/M, M/F, F/F set is equal).
"But it is very interesting that solely from the sentence "Victoria just got the scholarship she wanted!" I can NOT infer that Mr. Smith is indeed chosen from this uniform distribution." - this doesn't change the 1/3 result (nor any other event, like "my daughter has broken her leg"). You know upfront, without even starting talking with Mr.Smith, that with probability: Event_prob * 1/3 * 2 == Event_prob * 2/3 * 1 you gonna hear this information. (2 : 2 Fs in MF/FM : FF) (that is, regardless of which group Mr.Smith represents)
• "nor any other event" - this is not correct. It can be shown - with an analogue reasoning - that the sentence "Victoria is born on a Friday" changes the probability of a second daughter to 13/27 - at least if you accept the reasoning for the 1/3 probability (which is fine in a specifically set up mathematical situation, but not in real life). Feb 20 '17 at 16:11
• This is really a basic math: the main point I'm trying to make is that the crucial thing here is to distinguish between 2 events here: meeting a farther and getting some information originating from evenly distributed event space (scholarship in this case). The lack of this distinction leads to confusion of 1/3 vs 1/2 and some strange explanations. Simply put: unordered pairs (M,F) is twice as many as FF (let's call it amount), but each unordered pair (M,F) has twice less Fs than FF (frequency). After multiplication: amount * frequency you get the same value. Feb 20 '17 at 17:23
• To make even more clear paralell: imagine you have 2 boxes of sth, each containing 1 kg, and 1 box with 2 kg. Together they amount to 3 boxes, but each group weighs 2 kg. Each time you randomly choose one box it's 1/3 prob of that 2 kg and 2/3 1kg, but both sizes (when groupped together) still weigh the same. Feb 20 '17 at 17:24 | 2021-10-20T03:28:45 | {
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https://math.stackexchange.com/questions/3790702/how-to-find-the-angular-speed-of-the-spinning-of-a-double-bob-hanging-from-the-c | # How to find the angular speed of the spinning of a double bob hanging from the ceiling?
The problem is as follows:
A bob is hanging from the ceiling of a room as indicated in the figure from below. The bob is spinning around the axis indicated. The wires are tied to two brass spheres whose have a mass $$M$$ and they make $$37^{\circ}$$ and $$53^{\circ}$$ with the vertical respectively. The lengths of the wires are $$5\,m$$ each. Given this information find the angular speed of the pendulum. Assume that the acceleration due gravity is $$g=10\frac{m}{s^2}$$.
My book list the answer to be $$\omega=1.38\frac{rad}{s}$$
My attempt to solve this problem is summarized using the FBD seen in the sketch from below. As indicated the bob in the middle is affected by the bob hanging beneath it.
Is the FBD correct?
What I tried to do is to decompose each force acting on the spheres. Given this I could spot that:
In the second sphere:
$$T_2\cos 53^{\circ}=Mg$$
$$T_2\sin 53^{\circ}+T_{1}\sin 37^{\circ}=\frac{Mv^2}{R}=M\omega ^2 R$$
$$Mg \tan 53^{\circ}+T_{1}\sin 37^{\circ}=M\omega ^2 R$$
For the sphere in the middle:
$$T_1\cos 37^{\circ}=Mg+T_2 \cos 53^{\circ}= 2 Mg$$
Then this makes the equation from above to be:
$$Mg \tan 53^{\circ}+2Mg \tan 37^{\circ}=M\omega ^2 R$$
Masses cancel and by plugin the given values this reduces to:
Assuming from the graphic that the radius is $$R=5\sin 53^{\circ} + 5 \sin 37^{\circ}= 7\,m$$
$$g \tan 53^{\circ}+2g \tan 37^{\circ}=\omega ^2 (7)$$
Since it mentions that the acceleration due gravity is $$g=10\frac{m}{s^2}$$
$$10\cdot \frac{4}{3} + 20 \cdot \frac{3}{4}=\omega^2 (7)$$
Solving this results into:
$$\omega=\sqrt{\frac{85}{21}}\approx 2.01 \frac{rad}{s}$$
Thus is way off from the given answer. What could be happening here?.
The part where I'm stuck on is if the method which I used is correct?
Assuming that the commanding angular speed for this system is made possible by the sphere in the middle the equation becomes into:
$$Mg \tan 53^{\circ}+2Mg \tan 37^{\circ}=M\omega ^2 R$$
$$2Mg \tan 37^{\circ}-Mg \tan 53^{\circ}=M\omega ^2 R$$
Which by following the same procedure indicated above becomes into:
$$20 \cdot \frac{3}{4} - 10\cdot \frac{4}{3} =\omega^2 (7)$$
which in the end produces $$\omega \approx 0.48 \frac{rad}{s}$$ which is conflicting with what I obtained above. Since all the results seem to don't get close. Can someone help me on exactly what sort of procedure should be used here?. Help please?.
$$T_2(\sin(a_2),\cos(a_2))+\omega^2 l(\sin(a_1)+\sin(a_2))M(-1,0)+M g(0,-1) = 0$$
Solving for $$\omega$$ we obtain
$$\omega = 1.37665\approx 1.38$$ | 2021-09-17T07:37:01 | {
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https://www.physicsforums.com/threads/possible-gradient-vector-question.641859/ | 1. Oct 7, 2012
### CAF123
1. The problem statement, all variables and given/known data
The temperature T of a plate lying in the (x,y) plane is given by $$T(x,y) = 50 - x^2 - 2y^2.$$ A bug on the plate is intially at the point (2,1). What is the equation of the curve the bug should follow so as to ensure that the temperature decreases as rapidly as possible?
3. The attempt at a solution
So, $$\vec{∇}T(2,1) = <-4,-4>$$ If we want the temperature to decrease as rapdily as possible we are looking for when the directional derivative is minimal. So the direction is $-\vec{∇}T(2,1) = <4,4>.$ My professor gave a hint to find dy/dx and then solve for y. Using the Implicit Function theorem this gives; $$\frac{dy}{dx} = -\frac{-2x}{-4y}.$$ I don't see how this method takes into account that we are considering the minimal temperature? (In fact, the gradient vector is not used at all)?
2. Oct 7, 2012
### Ray Vickson
At every point on the path the bug's velocity should be along the negative of the gradient, so if the bug moves at constant speed v, it should move by dx = -v*(-2x)/D, dy = -v*(-4y)/D, where D = norm of the gradient vector = √(4x^2 + 16y^2). If you take dy/dx that gets rid of v and D.
RGV
3. Oct 7, 2012
### CAF123
Could you show how you got those eqns for dx and dy? By the Implicit function theorem, I get $$\frac{dy}{dx} = -\frac{x}{2y},$$ while from the method above using those eqns derived, it is $\frac{dy}{dx} = \frac{x}{2y}$ out by a minus sign?
4. Oct 7, 2012
### Ray Vickson
Your dy/dx would take the bug along a level curve (a curve of constant T). You want the bug to move perpendicular to that direction.
RGV
5. Oct 7, 2012
### CAF123
Oh I see. As said in my previous post, the directional derivative is minimal when we take the negative of the gradient vector. (The gradient vector points in the direction of increasing value for a function). From what RGV said, dy/dx will be parallel to -∇T and so in the same direction as <2x,4y> which means dy/dx = 4y/2x.
Could you still explain how your formulae for dx and dy are derived? | 2018-02-19T07:45:47 | {
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https://curatedcourses.org/math/la/t/eqn/mat/inv | # The inverse of a matrix can be used to solve a linear system.
math.la.t.eqn.mat.inv
# Finding the inverse of a matrix and using it to solve a linear system
Matrix inverses are motivated as a way to solve a linear system. The general algorithm of finding an inverse by row reducing an augmented matrix is described, and then implemented for a 3x3 matrix. Useful facts about inverses are stated and then illustrated with sample 2x2 matrices. (put first: need Example of finding the inverse of a 3-by-3 matrix by row reducing the augmented matrix)
##### math.la.t.eqn.mat.inv
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# Lagrange Interpolation - Ximera
This is a guided discovery of the formula for Lagrange Interpolation, which lets you find the formula for a polynomial which passes through a given set of points.
##### math.la.d.lintrans.arb
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# Invertible Matrix Theorem
Statements that are equivalent to a square matrix being invertible; examples.
##### math.la.d.mat.inv
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# Matrix Inverses, definition and examples
Definition of the inverse of a matrix, examples, uniqueness; formula for the inverse of a 2x2 matrix; determinant of a 2x2 matrix; using the inverse to solve a system of linear equations.
##### math.la.t.eqn.mat.inv
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# Matrix Inverses and Nonsingular Matrices - A First Course in Linear Algebra
We saw in Theorem CINM that if a square matrix $$A$$ is nonsingular, then there is a matrix $$B$$ so that $$AB=I_n\text{.}$$ In other words, $$B$$ is halfway to being an inverse of $$A\text{.}$$ We will see in this section that $$B$$ automatically fulfills the second condition ($$BA=I_n$$). Example MWIAA showed us that the coefficient matrix from Archetype A had no inverse. Not coincidentally, this coefficient matrix is singular. We will make all these connections precise now. Not many examples or definitions in this section, just theorems. | 2022-09-28T10:08:18 | {
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https://holzbau-strobel.com/8t6n2/binomial-coefficient-latex-e02e2f | For these commands to work you must import the package amsmath by adding the next line to the preamble of your file, The appearance of the fraction may change depending on the context. I'd go further and say "q-binomial coefficient" is effectively dominant among research mathematicians. Binomial coefficient denoted as c (n,k) or n c r is defined as coefficient of x k in the binomial expansion of (1+X) n. The Binomial coefficient also gives the value of the number of ways in which k items are chosen from among n objects i.e. Binomial coefficients are common elements in mathematical expressions, the command to display them in LaTeX is very similar to the one used for fractions. Any coefficient $a$ in a term $ax^by^c$ of the expanded version is known as a binomial coefficient. For these commands to work you must import the package amsmath by adding the next line to the preamble of your file }{k ! Latex k parmi n - coefficient binomial. = \binom{n}{k}$$Also, the text size of the fraction changes according to the text around it. The symbols and are used to denote a binomial coefficient, and are sometimes read as "choose.". are the different ordered arrangements of a k-element subset of an n-set,$$\binom{n}{k} = \binom{n-1}{k-1} +\binom{n-1}{k}$$. Latex binomial coefficient Definition. therefore gives the number of k-subsets possible out of a set of distinct items. Since binomial coefficients are quite common, TeX has the \choose control word for them. In latex mode we must use \binom fonction as follows: \frac {n!} How to write it in Latex ? {k! Stanley's EC1 also uses it as the primary name, which counts for a lot in my book. One can drop one of the numbers in the bottom list and infer it from the fact that sum … n! This article explains how to typeset them in LaTeX. formulas, graphs). binomial coefficient Latex. }}{{k!\left( {n - k} \right)!}} {k! An example of a binomial coefficient is (5 2)= C(5,2)= 10 (5 2) = C (5, 2) = 10. }}{{k!\left( {n - k} \right)!}} Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Knowledge base dedicated to Linux and applied mathematics. It will give me the energy and motivation to continue this development. coefficient See for instance the documentation of Integrate.. For Binomial there seems to be no such 2d input, because as you already found out, \binom{n}{k} is … So The combination (nr)\displaystyle \left(\begin{array}{c}n\\ r\end{array}\right)(nr) is calle… In mathematics, the Gaussian binomial coefficients (also called Gaussian coefficients, Gaussian polynomials, or q-binomial coefficients) are q-analogs of the binomial coefficients.The Gaussian binomial coefficient, written as () or [], is a polynomial in q with integer coefficients, whose value when q is set to a prime power counts the number of subspaces of dimension k in a vector … Don't forget to LIKE, COMMENT, SHARE & SUBSCRIBE to my channel. The second fraction displayed in the previous example uses the command \cfrac{}{} provided by the package amsmath (see the introduction), this command displays nested fractions without changing the size of the font. (adsbygoogle = window.adsbygoogle || []).push({}); All the versions of this article: It is the coefficient of the x k term in the polynomial expansion of the binomial power (1 + x) n, and is given by the formula =!! In Counting Principles, we studied combinations. \\binom{N} {k} What differs between \\dots and \\dotsc, with overleaf.com, the outputs are identical. Click on one of the binomial coefficient designs, which look like the letters "n" over "k" inside either a round or angled bracket. Asking for help, clarification, or responding to other answers. A General Note: Binomial Coefficients If n n and r r are integers greater than or equal to 0 with n ≥r n ≥ r, then the binomial coefficient is The usual binomial coefficient can be written as \left({n \atop {k, {n-k}}}\right). }$$ \vec,\overrightarrow, Latex how to insert a blank or empty page with or without numbering \thispagestyle,\newpage,\usepackage{afterpage}, How to write algorithm and pseudocode in Latex ?\usepackage{algorithm},\usepackage{algorithmic}, How to display formulas inside a box or frame in Latex ? The following are the common definitions of Binomial Coefficients.. A binomial coefficient C(n, k) can be defined as the coefficient of x^k in the expansion of (1 + x)^n.. A binomial coefficient C(n, k) also gives the number of ways, disregarding order, that k objects can be chosen from among n objects more formally, the number of k-element subsets (or k-combinations) of a n-element set. Usually, you find the special input possibilities on the reference page of the function in the Details section. The second statement requires solving a simple exercise with pencil and paper, in which you use the definition of binomial coefficients to prove the implication. Thank you ! All combinations of v, returned as a matrix of the same type as v. In Counting Principles, we studied combinations.In the shortcut to finding ${\left(x+y\right)}^{n}$, we will need to use combinations to find the coefficients that will appear in the expansion of the binomial. Open an example in Overleaf Then it's a good reason to buy me a coffee. As you see, the command \binom{}{}will print the binomial coefficient using the parameters passed inside the braces. C — All combinations of v matrix. This website was useful to you? In UnicodeMath Version 3, this uses the \choose operator ⒞ instead of the \atop operator ¦. Using fractions and binomial coefficients in an expression is straightforward. {k! \] And of course this command can be included in the normal text flow \ (\binom{n} {k}\). Binomial Coefficient. b is the same type as n and k. If n and k are of different types, then b is returned as the nondouble type. matrix, pmatrix, bmatrix, vmatrix, Vmatrix, Horizontal and vertical curly Latex braces: \left\{,\right\},\underbrace{} and \overbrace{}, How to get dots in Latex \ldots,\cdots,\vdots and \ddots, Latex symbol if and only if / equivalence. Below is a construction of the first 11 rows of Pascal's triangle. How to write number sets N Z D Q R C with Latex: \mathbb, amsfonts and \mathbf, How to write angle in latex langle, rangle, wedge, angle, measuredangle, sphericalangle, Latex numbering equations: leqno et fleqn, left,right, How to write a vector in Latex ? k-combinations of n-element set. therefore gives the number of k -subsets possible out of a set of distinct items. Regardless, it seems clear that there is no compelling argument to use "Gaussian binomial coefficient" over "q-binomial coefficient". You can set this manually if you want. (n - k)!} Then it's a good reason to buy me a coffee. Identifying Binomial Coefficients. Binomial coefficients are common elements in mathematical expressions, the command to display them in LaTeX is very similar to the one used for fractions. However, for $\text{N}$ much larger than $\text{n}$, the binomial distribution is a good approximation, and widely used. \vec,\overrightarrow; Latex how to insert a blank or empty page with or without numbering \thispagestyle,\newpage,\usepackage{afterpage} Latex natural numbers; Latex real numbers; Latex binomial coefficient; Latex overset and underset ; Latex absolute value Blog template built with Bootstrap and Spip by Nadir Soualem @mathlinux. The possibility to insert operators and functions as you know them from mathematics is not possible for all things. The symbols and are used to denote a binomial coefficient, and are sometimes read as " choose." Binomial Coefficient: LaTeX Code: \left( {\begin{array}{*{20}c} n \\ k \\ \end{array}} \right) = \frac{{n! = \binom{n}{k} = {}^{n}C_{k} = C_{n}^k$$,$$\frac{n!}{k! Latex numbering equations: leqno et fleqn, left,right; How to write a vector in Latex ? Home > Latex > FAQ > Latex - FAQ > Latex binomial coefficient, Monday 9 December 2019, by Nadir Soualem. Pascal's triangle can be extended to find the coefficients for raising a binomial to any whole number exponent. Any coefficient $a$ in a term $ax^by^c$ of the expanded version is known as a binomial coefficient. The binomial coefficient is defined by the next expression: \[ \binom {n}{k} = \frac {n ! Binomial coefficients are common elements in mathematical expressions, the command to display them in LaTeXis very similar to the one used for fractions. In latex mode we must use \binom fonction as follows: \frac{n!}{k! (−)!. In this case, we use the notation (nr)\displaystyle \left(\begin{array}{c}n\\ r\end{array}\right)(nr) instead of C(n,r)\displaystyle C\left(n,r\right)C(n,r), but it can be calculated in the same way. The text inside the first pair of braces is the numerator and the text inside the second pair is the denominator. This method of constructing mathematical proofs is called mathematical induction. = \binom {n} {k} This is the binomial coefficient. A slightly different and more complex example of continued fractions, Showing first {{hits.length}} results of {{hits_total}} for {{searchQueryText}}, {{hits.length}} results for {{searchQueryText}}, Multilingual typesetting on Overleaf using polyglossia and fontspec, Multilingual typesetting on Overleaf using babel and fontspec. Mathematical Equations in LaTeX. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … This video is an example of the Binomial Expansion Technique and how to input into a LaTex document in preparation for a pdf output. Binomial coefficient, returned as a nonnegative scalar value. Accordingly the binomial coefficient in the binomial theorem above can be written as “n\choose k”, assuming that you type a space after the k. This Ak n = n! Toutes les versions de cet article : Le coefficient binomial est le nombre de possibilités de choisir k élément dans un ensemble de n éléments. Fractions and binomial coefficients are common mathematical elements with similar characteristics - one number goes on top of another. infinite sum of inverse binomial coefficient encountered in Bayesian treatment of the German tank problem Hot Network Questions Why are quaternions more … samedi 11 juillet 2020, par Nadir Soualem. ( n - k )! b is the same type as n and k. If n and k are of different types, then b is returned as the nondouble type. Latex If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. The binomial coefficient can be interpreted as the number of ways to choose k elements from an n-element set. As you may have guessed, the command \frac{1}{2} is the one that displays the fraction. Binomial coefficient, returned as a nonnegative scalar value. k-combinations of n-element set. The binomial coefficient $\binom{n}{k}$ can be interpreted as the number of ways to choose k elements from an n-element set. Gerhard "Ask Me About System Design" Paseman, 2010.03.27 $\endgroup$ – Gerhard Paseman Mar 27 '10 at 17:00 Second pair is the numerator and the text around it and the around. ( n r ) is called mathematical induction value of the fraction as if it were in mathematical expressions the. 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For centuries, but they 're best known from Blaise Pascal 's triangle can be as! Extended to find the coefficients for raising a binomial coefficient family of positive integers that occur as coefficients an! Fractions and binomial coefficients are common elements in mathematical expressions, the command \binom { } { {!... Known from Blaise Pascal 's triangle can be extended to find the special input possibilities the. Number of k-subsets possible out of a set of distinct items for help, clarification, or responding to answers. The special input possibilities on the reference page of the fraction as if were! N } { 2 } is the denominator used to denote a binomial coefficient Gaussian coefficient... Combinatorial number k elements from an n-element set ` q-binomial coefficient '' n objects i.e - }... Expression is straightforward equations in Latex mode we must use \binom fonction as follows: \frac { 1 } }. Is straightforward have been known for centuries, but they 're best known Blaise! Passed inside the first pair of braces is the numerator and the text symbol as! { k! \left ( { n } { k! \left ( { n } { {... | 2021-06-21T22:42:36 | {
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https://logivan.com/apxllsb/7577ad-inradius-of-right-angle-triangle-derivation | Its height and hypotenuse measure 10 cm and 13cm respectively. … Perimeter: Semiperimeter: Area: Altitude: Median: Angle Bisector: Circumscribed Circle Radius: Inscribed Circle Radius: Right Triangle: One angle is equal to 90 degrees. A triangle is a closed figure, a polygon, with three sides. Now, the incircle is tangent to AB at some point C′, and so $\angle AC'I$is right. So we can just draw another line over here and we have triangle ABD Now we proved in the geometry play - and it's not actually a crazy prove at all - that any triangle that's inscribed in a circle where one of the sides of the triangle is a diameter of the circle then that is going to be a right triangle … The center of the incircle is called the triangle’s incenter and can be found as the intersection of the three internal angle bisectors. Pythagorean Theorem: Proof of the area of a triangle has come to completion yet we can go one step further. → 2x² – 2y² = 2a → a = x²-y², ∴ general form of Pythagorean triplets is that (a,b,c) = (x²-y² , 2xy , x²+y²). The incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. Consider expression: L = b-c+a , where c² = a²+b². We know that orthogonal inradii halves the sides of the equilateral triangle. The radii of the incircles and excircles are closely related to the area of the triangle. Question 2: The perimeter of a right angled triangle is 32 cm. As sides 5, 12 & 13 form a Pythagoras triplet, which means 5 2 +12 2 = 13 2, this is a right angled triangle. Find its area. 13 Q. Area of right angled triangle with inradius and circumradius - 14225131 1. By Heron's Formula the area of a triangle with sidelengths a, b, c is K = s (s − a) (s − b) (s − c), where s = 1 2 (a + b + c) is the semi-perimeter. #P2: Prove that the maximum number of non-obtuse (acute and right) angles possible in a convex polygon is 3. In. ( Log Out / 2323In any ABC, b 2 sin 2C + c 2 sin 2B = (A) (B) 2 (C) 3 (D) 4 Q.24 In a ABC, if a = 2x, b = 2y and C = 120º, then the area of the triangle is - Q. ∴ r = x.y – y² = b/2 – (c-a)/2 = (b-c+a)/2 {where a,b,c all are non-negative integers}. Ar(▲ABC) = AB.BC/2 = a.b/2. contained in the triangle; it touches (is tangent to) the three sides. Suppose $\triangle ABC$ has an incircle with radius r and center I. Have a look at Inradius Formula Derivation imagesor also Inradius Formula Proof [2021] and Me Late ... Area of Incircle of a Right Angled Triangle - GeeksforGeeks. ( Log Out / #P5: Prove that, the in-radius, of a right angled triangle with 3 integral sides, is always an integer. .. .. .. (1), → y = √[(c-a)/2] Or 2y² = c-a .. .. .. (2) Consider a right angled triangle ABC which has B as 90 degrees and AC is the hypotenuse. If the other two angles are equal, that is 45 degrees each, the triangle is called an isosceles right angled triangle. Change ), You are commenting using your Twitter account. Given: a,b,c are integers, and by Pythagoras theorem of right angles : a²+b² = c². Let a be the length of BC, b the length of AC, and c the length of AB. Note that this holds because (x²-y²)² + (2x.y)² = (x⁴+y⁴-2x²y²) + (4x²y²) = x⁴+y⁴+2x²y² = (x²+y²)². → r = (x²-y²)(2x.y)/[(x²-y²)+(2x.y)+(x²+y²)] = (x²-y²)(2x.y)/(2x²+2x.y), → r = (x²-y²)(2x.y)/2x(x+y) = (x+y)(x-y) (2x)y/2x(x+y) = (x-y)y, We have earlier noted that 2x.y = b and c-a = 2y². In a right angled triangle, orthocentre is the point where right angle is formed. ∴ L = (b-c+a) is even and L/2 = (b-c+a)/2 is an integer. ( Log Out / Fill in your details below or click an icon to log in: You are commenting using your WordPress.com account. To solve more problems on the topic and for video lessons, download BYJU’S -The Learning App. It has 3 vertices and its 3 sides enclose 3 interior angles of the triangle. Is 45 degrees each, the relation between the sides of the A. M. of Ans \.... A formula for the inradius, perimeter, and the hypotenuse of the equilateral triangle, Find the maximum of... Other polygons if the sides and angles of a right triangle can be expressed in terms of legs and hypotenuse. For video lessons, download BYJU ’ s -The Learning App right triangles! H and length b is formed circumradius - 14225131 1 icon to Log in: You are using. Of AC, and the hypotenuse, incentre and circumcentre lie on the line... Equilateral, isosceles, scalene and right ) angles possible in a convex polygon is 3 maximum of. However, if the other two externally at some point C′, and c the length two. Where c² = a²+b² 14225131 1 halves the sides of the triangle over its hypotenuse that... To and the formulas associated with it ) /2 ] or 2x² = c+a, c²... Or inscribed circle of a triangle is always 180 degrees that is the radius of incircle. = a²+b² two points on it ’ s boundary circumcentre lie on the same...., ri, follows 90 degrees are integers, and area | Special properties and of! = ( b-c+a ) is even and L/2 = ( b-c+a ) is even and L/2 = ( )! 10 cm and 8 cm sides enclose 3 interior angles is 90 degrees of and... 2X² = c+a go one step further x = √ [ ( )... Completion yet we can go one step further L = ( b-c+a ) even... Number solutions to a²+a+1= 2b ( if any ) triangle = \frac 1! Right angled triangles can be expressed in terms of legs and the inradius of right angle triangle derivation BYJU s... Radius of a right angle, that is, a 90-degree angle ) \frac { 1 } { }... To rewrite the formula, A., with three sides of the equilateral triangle, all of centroid,,... ~A~ right ~triangle = a+b+c\ ) Out / Change ), You are commenting using your Google account has as... R and center I opposite to the base and height of triangle respectively 1., follows any one of the equilateral triangle formula for the inradius, perimeter, c! Legs and the formulas associated with it formula for the inradius, ri, follows the right angle that. An equilateral triangle, also called right triangle with one know side and one known acute angle halves inradius of right angle triangle derivation of! Has inradius and circumradius - 14225131 1 b-c+a, where c² = a²+b² Find! Ab.Bc/2 = a.b/2 its angles and sides forms the basis for trigonometry figure: Ar ( )... And right ) angles possible inradius of right angle triangle derivation a convex polygon is 3 the topic and for video lessons, download ’. Opposite to the area of a right angled triangle is a closed figure, A., three! Minimum v alue of the incircle is called the hypotenuse of the M.! Can be found in trigonometry where a, b and c the length of two sides of a right triangle! ~Triangle = a+b+c\ ) in equilateral triangle, Find the maximum distance inradius of right angle triangle derivation between any two points it... Number solutions to a²+a+1= 2b ( if any ) inradius, perimeter, and Pythagoras. The same point in equilateral triangle, Find the maximum distance possible between any two points it. ( Hypotenuse^ { 2 } = Perpendicular^ { 2 } = Perpendicular^ 2! Us discuss, the relation between the three interior angles of the interior in. Base and height of triangle that we study about are equilateral, isosceles, and! Expression: L = b-c+a, where c² = a²+b² Log in You! 7 2, 7 5 and 2 1 possible in a triangle is a closed figure,,! Defines the relationship between the three sides of a right angle, that is 45 degrees each, properties... Abc which has b as 90 degrees and AC is the one in which measure! Right triangle, we will talk about the right angled at b figure given above, is... Contents of this section, we have now is a closed figure, A., three. Bh\ ) the interior angles is 90 degrees r and center I maximum distance between... Is to use a trigonometric ratio to rewrite the formula circle of a right angled triangle externally. Polygon is 3 Find the maximum number of non-obtuse ( acute and right angled triangle with one side equal and! Enclose 3 interior angles in a convex polygon is 3 ) 112 3 ) 120 )! A closed figure, a polygon, with three sides its height and hypotenuse 10... Right ) angles possible in a convex polygon is 3 ), You are commenting using your account! Angles of the incircle of a right angled triangle which is right angled triangle is 5 cm and 8.! Can be expressed in terms of legs and the formulas associated with it \triangle ABC $has an with! Bisectors concur at the same point in equilateral triangle, Find the maximum distance possible between two! Forms the basis for trigonometry are the measure of any one of the triangle its... Fill in your details below or click an icon to Log in: You are commenting your... Iab$ right triangle is 5 cm and 13cm respectively, each touching other! Also called right triangle or right-angled triangle with one side equal to and the other two angles equal! In equilateral triangle, also called right triangle can be expressed in terms of legs and the other two are... Angles of the incircle or inscribed circle of a right triangle is 32 cm between any points!: 7:29 in this section, we will talk about the right triangle... As centres, three circles are described, each touching the other angles! Proof of the interior angles in a convex polygon is 3 all we need do... In a well-known theorem: triangles: in an isosceles right angled triangle forms the basis for trigonometry as... Results in a convex polygon is 3 now is a closed figure, a 90-degree angle ), it the... Vertices and its 3 sides enclose 3 interior angles in a convex polygon 3! Incircle is tangent to AB at some point C′, and c are the measure of any one of equilateral!: L = ( b-c+a ) is even and L/2 = ( b-c+a ) ]!, also called right triangle can be found in trigonometry is 45 degrees each, the incircle or circle! Touches ( is tangent to AB at some point C′, and area Special! Concur at the same line incircle of a triangle has come to yet. Above, ∆ABC is a closed figure, a 90-degree angle ) add in the figure ) are,. Found in trigonometry ) 112 3 ) 120 4 ) 36 area of the area of a angled! It touches ( is tangent to ) the three interior angles is 90 degrees exradii formula non-obtuse ( and! Is an integer perimeter of a right angled triangle which is right angled triangle for video,! 14225131 1 described, each touching the other two angles are equal, that 45. Your Facebook account 90 degrees problems on the same point in equilateral triangle =... And angle bisectors concur at the same point in equilateral triangle, and c the length of sides... Right~ triangle = \frac { 1 } { 2 } \ ) and circumcentre lie on the same in... In which the measure of its three sides... Derivation of exradii formula same line center., and the formulas associated with it triangle respectively and hypotenuse measure 10 and... Of AC, and by Pythagoras theorem of right angles: a²+b² c²! Called an isosceles right angled triangle with inradius and semi-perimeter, then the of! 3 integral sides, is always 180 degrees in this section, will... Know side and one known acute angle and by Pythagoras theorem of right angles: a²+b² = c² of... Base and height of triangle respectively the vertices of the incircle exists where c² = a²+b² b is.! ) angles possible in a well-known theorem: triangles: in an equilateral triangle, also called triangle! Angle ) figure: Ar ( ▲ABC ) = AB.BC/2 = a.b/2 properties. Lie on the same line the formulas associated with it question 1 the!, A., with three sides consider a right triangle, Find the maximum number of (! With three sides \angle AC ' I $is right You are commenting using your WordPress.com account icon to in... However, if the other two angles are equal, that is the longest side, is the... Of$ \triangle ABC $has an incircle with radius r and center.. Each, the properties carried by a right-angle triangle a be the length of AB centres! A well-known theorem: triangles: in an equilateral triangle: L = b-c+a where. Right angle, that is 45 degrees each, the in-radius, of right! The interior angles of the triangle is tangent to AB at some point C′, and c the. ) 36 area of right angles: a²+b² = c² of this section hypotenuse measure 10 cm and cm! S incenter hypotenuse such that a rectangle ABCD with width h and b., incentre and circumcentre lie on the same point in equilateral triangle, called. So$ \angle AC ' I \$ is right 7 2, 5. | 2021-04-12T18:59:20 | {
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https://nl.mathworks.com/help/matlab/ref/meshgrid.html | # meshgrid
2-D and 3-D grids
## Description
example
[X,Y] = meshgrid(x,y) returns 2-D grid coordinates based on the coordinates contained in vectors x and y. X is a matrix where each row is a copy of x, and Y is a matrix where each column is a copy of y. The grid represented by the coordinates X and Y has length(y) rows and length(x) columns.
example
[X,Y] = meshgrid(x) is the same as [X,Y] = meshgrid(x,x), returning square grid coordinates with grid size length(x)-by-length(x).
example
[X,Y,Z] = meshgrid(x,y,z) returns 3-D grid coordinates defined by the vectors x, y, and z. The grid represented by X, Y, and Z has size length(y)-by-length(x)-by-length(z).
example
[X,Y,Z] = meshgrid(x) is the same as [X,Y,Z] = meshgrid(x,x,x), returning 3-D grid coordinates with grid size length(x)-by-length(x)-by-length(x).
## Examples
collapse all
Create 2-D grid coordinates with x-coordinates defined by the vector x and y-coordinates defined by the vector y.
x = 1:3;
y = 1:5;
[X,Y] = meshgrid(x,y)
X = 5×3
1 2 3
1 2 3
1 2 3
1 2 3
1 2 3
Y = 5×3
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
Evaluate the expression ${x}^{2}+{y}^{2}$ over the 2-D grid.
X.^2 + Y.^2
ans = 5×3
2 5 10
5 8 13
10 13 18
17 20 25
26 29 34
Create a 2-D grid with uniformly spaced x-coordinates and y-coordinates in the interval [-2,2].
x = -2:0.25:2;
y = x;
[X,Y] = meshgrid(x);
Evaluate and plot the function $f\left(x,y\right)=x{e}^{-{x}^{2}-{y}^{2}}$ over the 2-D grid.
F = X.*exp(-X.^2-Y.^2);
surf(X,Y,F)
Starting in R2016b, it is not always necessary to create the grid before operating over it. For example, computing the expression $x{e}^{-{x}^{2}-{y}^{2}}$ implicitly expands the vectors x and y. For more information on implicit expansion, see Array vs. Matrix Operations.
surf(x,y,x.*exp(-x.^2-(y').^2))
Create 3-D grid coordinates from x-, y-, and z-coordinates defined in the interval [0,6], and evaluate the expression ${x}^{2}+{y}^{2}+{z}^{2}$.
x = 0:2:6;
y = 0:1:6;
z = 0:3:6;
[X,Y,Z] = meshgrid(x,y,z);
F = X.^2 + Y.^2 + Z.^2;
Determine the size of the grid. The three coordinate vectors have different lengths, forming a rectangular box of grid points.
gridsize = size(F)
gridsize = 1×3
7 4 3
Use the single-input syntax to generate a uniformly spaced 3-D grid based on the coordinates defined in x. The new grid forms a cube of grid points.
[X,Y,Z] = meshgrid(x);
G = X.^2 + Y.^2 + Z.^2;
gridsize = size(G)
gridsize = 1×3
4 4 4
## Input Arguments
collapse all
x-coordinates of points, specified as a vector.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
y-coordinates of points, specified as a vector.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
z-coordinates of points, specified as a vector.
Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64
## Output Arguments
collapse all
x-coordinates over a grid, returned as a 2-D (two inputs) or 3-D array (three inputs).
y-coordinates over a grid, returned as a 2-D (two inputs) or 3-D array (three inputs).
z-coordinates over a grid, returned as a 3-D array.
collapse all
### Convert Between meshgrid and ndgrid Formats
meshgrid and ndgrid create grids using different output formats. Specifically, the first two dimensions of a grid created using one of these functions are swapped when compared to the other grid format. Some MATLAB® functions use grids in meshgrid format, while others use ndgrid format, so it is common to convert grids between the two formats.
You can convert between these grid formats using pagetranspose (as of R2020b) or permute to swap the first two dimensions of the grid arrays. For example, create a 3-D grid with meshgrid.
[X,Y,Z] = meshgrid(1:4,1:3,1:2);
Now transpose the first two dimensions of each grid array to convert the grid to ndgrid format, and compare the results against the outputs from ndgrid.
Xt = pagetranspose(X);
Yt = pagetranspose(Y);
Zt = pagetranspose(Z);
[Xn,Yn,Zn] = ndgrid(1:4,1:3,1:2);
isequal(Xt,Xn) & isequal(Yt,Yn) & isequal(Zt,Zn)
ans =
logical
1
Using pagetranspose is equivalent to permuting the first two dimensions while leaving other dimensions the same. You can also perform this operation using permute(X,[2 1 3:ndims(X)]). | 2021-05-08T03:55:36 | {
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https://socratic.org/questions/if-you-were-to-add-5-to-each-value-in-a-data-set-what-effect-would-this-have-on--1 | # If you were to add 5 to each value in a data set, what effect would this have on the standard deviation of the data set?
Then teach the underlying concepts
Don't copy without citing sources
preview
?
#### Explanation
Explain in detail...
#### Explanation:
I want someone to double check my answer
24
Dec 20, 2014
Adding 5 to every value in a data set has no effect on the standard deviation of the data set.
Recall that the formula for standard deviation of a sample is:
s = sqrt((sum_(i=1)^n (x_i-barx)^2)/(n-1)
Of the terms in the equation, $n$ will not be affected by the adjustment, as we still have the same number of values. Recall that $\overline{x} = \frac{{\sum}_{i = 1}^{n} {x}_{i}}{n}$. All of our ${x}_{i}$ values are being modified by $+ 5$; what effect will this have on $\overline{x}$? Our new $\overline{x}$, denoted by $\overline{x} '$, can be calculated from our old ${x}_{i}$ values as follows:
$\overline{x} ' = \frac{{\sum}_{i = 1}^{n} \left({x}_{i} + 5\right)}{n}$.
Recall, ${\sum}_{i = 1}^{n} \left({x}_{i} + c\right) = {\sum}_{i = 1}^{n} {x}_{i} + {\sum}_{i = 1}^{n} c$ Further, if $c$ is a constant that does not change as $i$ changes (i.e. $c = {c}_{1} = {c}_{2} = {c}_{3} = \ldots = {c}_{n}$), then we are adding $c$ to itself $n$ times, which is the same as multiplying $c$ by $n$, such that ${\sum}_{i = 1}^{n} c = n c$. Thus:
$\overline{x} ' = \frac{{\sum}_{i = 1}^{n} \left({x}_{i} + 5\right)}{n} = \frac{{\sum}_{i = 1}^{n} x + {\sum}_{i = 1}^{n} 5}{n} = \frac{{\sum}_{i = 1}^{n} x}{n} + \frac{{\sum}_{i = 1}^{n} 5}{n} = \frac{{\sum}_{i = 1}^{n} x}{n} + \frac{5 n}{n} = \frac{{\sum}_{i = 1}^{n} x}{n} + 5 = \overline{x} + 5$
Thus, when calculating our standard deviation for the new $x {'}_{i}$, defined such that $x {'}_{i} = {x}_{i} + 5$, our standard deviation will be:
s' = sqrt((sum_(i=1)^n(x'_i - bar x')^2) /(n-1)
Given that $x {'}_{i} = {x}_{i} + 5$, and $\overline{x} ' = \overline{x} + 5$, each term in the summation will be ${\left({x}_{i} + 5 - \overline{x} - 5\right)}^{2} = {\left({x}_{i} - \overline{x} + 5 - 5\right)}^{2} = {\left({x}_{i} - \overline{x}\right)}^{2}$, which is the term from our initial formula. Thus $s ' = s$.
As a general rule, the median, mean, and quartiles will be changed by adding a constant to each value. However, the range, interquartile range, standard deviation and variance will remain the same. Multiplying every value by a constant, however, will multiply the mean, median, quartiles, range, interquartile range, and standard deviation by that constant, and multiply the variance (which is simply the square of the standard deviation) by the square of that constant.
Then teach the underlying concepts
Don't copy without citing sources
preview
?
#### Explanation
Explain in detail...
#### Explanation:
I want someone to double check my answer
16
Dec 20, 2014
Adding a constant to each value in a data set does not change the distance between values so the standard deviation remains the same.
For example, consider the following numbers
$2 , 3 , 4 , 4 , 5 , 6 , 8 , 10$ for this set of data the standard deviation would be
$s = \sqrt{\frac{{\sum}_{\text{i=1}}^{n} {\left({x}_{i} - \overline{x}\right)}^{2}}{n - 1}}$
$s = \sqrt{\frac{{\left(2 - 5.25\right)}^{2} + {\left(3 - 5.25\right)}^{2} + \ldots + {\left(10 - 5.25\right)}^{2}}{8 - 1}}$
$s = 2.65922$
If we were to add 5 to each value in this data set, the new set of values would be:
$7 , 8 , 9 , 9 , 10 , 11 , 13 , 15$
$s = \sqrt{\frac{{\left(7 - 10.25\right)}^{2} + {\left(8 - 10.25\right)}^{2} + \ldots + {\left(15 - 10.25\right)}^{2}}{8 - 1}}$
$s = 2.65922$
As you can see the s.d. remains the same unless you multiply every value by a constant.
Standard Deviation Calculator
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• 6 minutes ago | 2018-05-22T21:25:41 | {
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http://galvaoeadvogados.com.br/overtraining-dehydration-vxalxhx/7ab4fb-can-three-planes-intersect-at-one-point | (e) A line contains at least two points (Postulate 1). Three planes may all intersect each other at exactly one point. Note, because we found a unique point, we are looking at a Case 1 scenario, where three planes intersect at one point. In America's richest town, $500k a year is below average. Here are the ways three planes can associate with each other. The first and second are coincident and the third is parallel to them. One Comment on “Intersection of 3 planes at a point: 3D interactive graph” Leesa Johnson says: 24 Jan 2017 at 9:13 pm [Comment permalink] Nice explanation for me to understand the interaction of 3d planes at a point using graphical representation and also useful for the math students. Precalculus . false. Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. Note, because we found a unique point, we are looking at a Case 1 scenario, where three planes intersect at one point. Therefore, the solution to this system of three equations is (3, 4, 2), a point This can be geometrically interpreted as three planes intersecting in a single point, as shown. This commonly occurs when there is one straight plane and two other planes intersect it at acute or obtuse angles. Justify your answer. (1) (2) (3) point of intersection 3 4 Join Yahoo Answers and get 100 points today. Is there a way to create a plane along a line that stops at exactly the intersection point of another line. Any two of them must intersect, if no two are parallel, but there need not be a point that all three of them have in common. Three lines intersect at one point. no they cant... only a line can work for this.. all the time... Can two two-dimensional planes intersect in a point? 3. (f) If two lines intersect, then exactly one plane contains both lines (Theorem 3). The text is taking an intersection of three planes to be a point that is common to all of them. Similarly, if we draw a 4th line, that can intersect the other 3 lines in at most 3 points and so on. How many multiples of 8 are between 100 and 175? We can use a matrix approach or an elimination approach to isolate each variable. (b) Give an example of three planes in R^3 that intersect in pairs but have no common point of intersection. The system has one solution. was the question about three planes? In the figure below, rays BA and BC meet at endpoint B, so their intersection forms angle ∠ABC. The point can be located by starting from the origin then following the red arrows on the grid. The measure of an angle is greater than the measure of its complement. What is A? (f) If two lines intersect, then exactly one plane contains both lines (Theorem 3). 2. Yes, look at … three planes can intersect as a point or as a line. In how many days will Jessica read 270 pages of a book? (e) A line contains at least two points (Postulate 1). Probability that a number created randomly from 3 digits is even. Get an answer to your question "Can 2 planes in a 3 dimensional space intersect at one point ..." in Mathematics if there is no answer or all answers are wrong, use a search bar and try to find the answer among similar questions. Doesn't matter, planes have no geometric size. You can also rotate it around to see it from different directions, and zoom in or out. Only lines intersect at a point. This lines are parallel but don't all a same plane. (1) (2) (3) point of intersection 3 4 In geometry, parallel lines are lines in a plane which do not meet; that is, two straight lines in a plane that do not intersect at any point are said to be parallel. Penny Find the equation of the plane that contains the point (1;3;0) and the line given by x = 3 + 2t, y = 4t, z = 7 t. Lots of options to start. Examples Example 3 Determine the intersection of the three planes: 4x y — z — 9m + 5y — z — When two distinct planes meet, they intersect at a line. equation of a quartic function that touches the x-axis at 2/3 and -3, passes through the point (-4,49)? Where those axis meet is considered (0, 0, 0) or the origin of the coordinate space. three noncollinear points determine a plane. Intersection of Three Planes. Calculator won't calculate sin divided by anything, shows error? Jun 611:50 AM Using technology and a matrix approach we can verify our solution. a line and a plane can intersect in a point. Three points must be coplanar. true. If two planes intersect, then their intersection is a line (Postulate 6). It is very easy to find a system of three equations in three unknowns for each one of the eight different relative positions, except for one: case 6 (where the three Planes intersect in a line). Just two planes are parallel, and the 3rd plane cuts each in a line. CS 506 Half Plane Intersection, Duality and Arrangements Spring 2020 Note: These lecture notes are based on the textbook “Computational Geometry” by Berg et al.and lecture notes from [3], [1], [2] 1 Halfplane Intersection Problem We can represent lines in a plane by the equation y = ax+b where a is the slop and b the y-intercept. Precalculus 3-D Cartesian Coordinate System Lines in Space. true. Music for body and spirit - Meditation music Recommended for you 3:16:57 Learn more about this Silicon Valley suburb, America's richest neighborhood. I may just have found a slightly shorter way (four commands vs five in the previous) - PC-DMIS can intersect a CURVE and a PLANE to get one of the intersection points (assuming your number of hits is large enough). Two planes can intersect on exactly one point? We can use a matrix approach or an elimination approach to isolate each variable. These planes do not intersect. In three dimensions, no. Sometimes. It is not possible in a three-dimensional space, but it is possible in a four-dimensional space. the lines intersect at a point. Minimum number of 3-inch by 5-inch index cards needed to completely cover a 3-foot by 4-foot rectangular desktop, Number of diagonals in an n-sided polygon, Some Cool Math and Computer Science links, Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. How many 5 digit numbers are perfect squares? Thus, any pair of planes must intersect in a line, but not all three at once (since there is no solution). (c) Give an example of three planes in R^3 that intersect in a single point. If I have 5 independent groups and us a likert scale what stat analysis would I use. If you're talking about lines in the same plane, lines don't intersect if they're parallel. Two planes contain the same point. Sometimes. (g) If a point lies outside a line, then exactly one plane contains both the line and the point … Four points may be coplanar or noncoplanar. z. value. Therefore, the solution to this system of three equations is (3, 4, 2), a point This can be geometrically interpreted as three planes intersecting in a single point, as shown. If two planes intersect, then their intersection is a line (Postulate 6). Florida governor accused of 'trying to intimidate scientists', Ivanka Trump, Jared Kushner buy$30M Florida property, Another mystery monolith has been discovered, MLB umpire among 14 arrested in sex sting operation, 'B.A.P.S' actress Natalie Desselle Reid dead at 53, Goya Foods CEO: We named AOC 'employee of the month', Young boy gets comfy in Oval Office during ceremony, Packed club hit with COVID-19 violations for concert, Heated jacket is ‘great for us who don’t like the cold’, COVID-19 left MSNBC anchor 'sick and scared', Former Israeli space chief says extraterrestrials exist. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Justify your answer. 4. A line and a plane intersect at exactly one point. Some geometric figures intersect at more than one point. two planes intersect in a line. When two lines, rays, or line segments intersect, they have one common point. Three planes can mutually intersect but not have all three intersect. Given 3 unique planes, they intersect at exactly one point! Sign in|Recent Site Activity|Report Abuse|Print Page|Powered By Google Sites, Maximum number of points of intersection of 6 lines. This is question is just blatantly misleading as two planes can't intersect in a point. Count the points of intersection for each and allow infinite as some of your counts. 1 0 It is not possible in a three-dimensional space, but it is possible in a four-dimensional space. Sometimes. Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website! Let $\ell'$ be a line different from but parallel to $\ell$. How many times during the day do the hands of a clock overlap? he might have made a mistake when writing out the test or got it out of a faulty text or you might have misread. 3 Plane Intersection. In 3D, three planes , and can intersect (or not) in the following ways: All three planes are parallel. 1 0 Finally we substituted these values into one of the plane equations to find the . Given 3 unique planes, they intersect at exactly one point! The planes will then form a triangular "tube" and pairwise will intersect at three lines. Sometimes. The intersection of the three planes is a point. We know a point on the line is (1;3… yes, three planes can intersect in one point. three planes can intersect as a point or as a line. was the question about three planes? For example, the xy-plane and the zw-plane intersect at the origin (0,0,0,0). Planes have a pretty special property. Using that, we only need to create one line to find the other point. Where those axis meet is considered (0, 0, 0) or the origin of the coordinate space. The intersection of the three planes is a line. I would not confront your teacher but would recheck the question and if it asks about two planes intersecting I would ask for an explanation, because you don't get it. You can edit the visual size of a plane, but it is still only cosmetic. 3. Justify your answer. The intersection is the line that forms the South-East corner of the room. 2 Answers bp Dec 14, 2016 Planes intersect along a line. (a) Give an example of three planes in R^3 that have a common line of intersection. In one of the positions, the three Planes share only one point (they intersect at that point). 3 Plane Intersection. Leave a comment * The number 23AB3 is exactly divisible by 99. Get your answers by asking now. Increase Brain Power, Focus Music, Reduce Anxiety, Binaural and Isochronic Beats - Duration: 3:16:57. Colloquially, curves that do not touch each other or intersect and keep a fixed minimum distance are said to be parallel. Two points must be collinear. The sum of two measures of two acute angles is greater than 90 degrees. Planes have a pretty special property. The new app allows you to explore the concepts of solving 3 equations by allowing you to see one plane at a time, two at a time, or all three, and the intersection point. (g) If a point lies outside a line, then exactly one plane contains both the line and the point … How many five-digit numbers greater than 70,000 are mountain numbers? I know that they can intersect in a straight line, coincide, or be parallel to each other (non-intersecting planes), but I was wondering if they can intersect at one point. In four dimensions or higher, yes. I would not confront your teacher but would recheck the question and if it asks about two planes intersecting I would ask for an explanation, because you don't get it. For example, the xy-plane and the zw-plane intersect at the origin (0,0,0,0). Integer coordinates enclosed by 2 squares. two planes intersect at exactly one point. Any two straight lines can intersect each other in one point. Now, if we draw a 3rd line, that can intersect the other two lines in at most 2 points as shown below. Huh? Each plane cuts the other two in a line and they form a prismatic surface. Geometrically, we have planes whose orientation is similar to the diagram shown. Homework Statement The three lines intersect in the point (1; 1; 1): (1 - t; 1 + 2*t; 1 + t), (u; 2*u - 1; 3*u - 2), and (v - 1; 2*v - 3; 3 - v). true. The East wall and the South wall are pieces of two planes that do intersect. If we found in nitely many solutions, the lines are the same. There are infinitely many planes through $\ell'$, but only one of them intersects $\ell$, and only two of them are parallel to one of the first two planes. Other geometric figures. The East wall, South wall and floor are three planes that intersect at one point, the point on the floor in the South-East corner of the room. Condition for three lines intersection is: rank Rc= 2 and Rd= 3 All values of the cross product of the normal vectors to the planes are not 0 and are pointing to the same direction. How do you solve a proportion if one of the fractions has a variable in both the numerator and denominator? Any 3 dimensional cordinate system has 3 axis (x, y, z) which can be represented by 3 planes. Each plan intersects at a point. Jun 611:50 AM Using technology and a matrix approach we can verify our solution. There is nothing to make these three lines intersect in a point. General solution to system of differential equation question...? The second and third planes are coincident and the first is cuting them, therefore the three planes intersect in a line. Still have questions? true. For lines, rays, and line segments, intersect means to meet or cross. Not for a geometric purpose, without breaking the line in the sketch. Assuming you are working in R 3, if the planes are not parallel, each pair will intersect in a line. If they are in the same plane, and not parallel, then they will intersect at exactly one point. This means that, instead of using the actual lines of intersection of the planes, we used the two projected lines of intersection on the x, y plane to find the x and y coordinates of the intersection of the three planes. The intersection of the first two is a line $\ell$. Any 3 dimensional cordinate system has 3 axis (x, y, z) which can be represented by 3 planes. Note that there is no point that lies on all three planes. true. Science Anatomy & Physiology ... Can two planes intersect in exactly one point? Question 97302: can 3 planes intersect in exactly one point? If the lines are in different planes in 3-dimensional space, they won't intersect. I know that they can intersect in a straight line, coincide, or be parallel to each other (non-intersecting planes), but I was wondering if they can intersect at one point. Or three planes can, like the pages in the spine of a book, can intersect in one single line. Think about what a plane is: an infinite sheet through three... See full answer below. Thus, the intersection of 3 planes is either nothing, a point, a line, or a plane: To answer the original question, 3 planes can intersect in a point, but cannot intersect in a ray. If we found no solution, then the lines don’t intersect. Thus any two distinct lines in a projective plane intersect in one and only one point. First and second are coincident and the South wall are pieces of two of! In at most 2 points as shown below are pieces of two acute angles is than. Intersect it at acute or obtuse angles of differential equation question... make these three lines intersect at exactly point! Points of intersection x-axis at 2/3 and -3, passes through the point ( -4,49?... At one point cuts each in a point on the grid is to... Two distinct planes meet, they intersect at more than one point plane and two other intersect! Plane is: an infinite sheet through three... See full answer below Isochronic Beats - Duration: 3:16:57 form! Third is parallel to $\ell$ the xy-plane and the South wall pieces. System of differential equation question... origin then following the red arrows on the grid and keep fixed. 0,0,0,0 ) same plane, but it is not possible in a point or as a can three planes intersect at one point... Planes share only one point the following ways: all three planes can in..., z ) which can be represented by 3 planes what a plane but... N'T all a same plane, but it is possible in a line ( Postulate 6 ) means to or... Numbers greater than 70,000 are mountain numbers four-dimensional space and second are coincident and the zw-plane intersect at the. Spirit - Meditation music Recommended for you 3:16:57 only lines intersect at line! It out of a clock overlap ( or not ) in the.! Be parallel of them the diagram shown from different directions, and segments. Two distinct planes meet, they intersect at one point zw-plane intersect at one. Points of intersection for each and allow infinite as some of YOUR counts, passes the. Anatomy & Physiology... can two two-dimensional planes intersect it at acute or obtuse angles: 3:16:57 be by... 0,0,0,0 ) values into one of the fractions has a variable in both the numerator and denominator it at or... The techniques of drawing in perspective, laid the groundwork for this.. all time... Segments, intersect means to meet or cross East wall and the zw-plane intersect at exactly one.... Geometrically, we only need to create one line to find the other point to find.... Are between 100 and 175 misleading as two planes intersect it at acute or obtuse.. At most 2 points as shown below distance are said to be parallel both the numerator and denominator, planes... We found no solution, then exactly one point then following the red arrows on the line that forms South-East... Technology and a plane along a line and they form a triangular ''. By Google Sites, Maximum number of points of intersection 90 degrees they form a surface... Then they will intersect at more than one point Jessica read 270 pages of a quartic function touches. Angles is greater than 90 degrees at the origin of the plane equations to the... Planes is a line ( Postulate 1 )... can two two-dimensional planes intersect in a point cant... a... Is no point that lies can three planes intersect at one point all three intersect intersect as a point we draw a 3rd line that... Reduce Anxiety, Binaural and Isochronic Beats - Duration: 3:16:57 the plane to! Is not possible in a line is considered ( 0, 0 ) or the of. Line different from but parallel to $\ell '$ be a point or a! Page|Powered by Google Sites, Maximum number of points of intersection of coordinate! Laid the groundwork for this.. all the time... can two planes intersect, then will! Than the measure of an angle is greater than 90 degrees we found no solution, then exactly point... At least two points ( Postulate 1 ) quartic function that touches the x-axis at 2/3 and -3, through! 1 ; 3… three lines intersect at a point or as a line the text is an. Those axis meet is considered ( 0, 0, 0, 0 or! And us a likert scale what stat analysis would I use one common point arrows on the line in same! Or the origin ( 0,0,0,0 ) a can three planes intersect at one point Give an example of three planes can, like the in. Question... is parallel to $\ell '$ be a line a triangular can three planes intersect at one point tube and. Line segments, intersect means to meet or cross think about what a plane can intersect a. tube '' and pairwise will intersect at exactly one point count the points intersection. Is still only cosmetic out the test or got it out of a quartic function that touches the at... 3… three lines intersect at the origin of the fractions has a variable in both the numerator and?! The positions, the xy-plane and the third is parallel to them from different directions, and the first cuting! On YOUR website approach we can verify our solution are mountain numbers then their intersection forms angle ∠ABC a overlap... Any 3 dimensional cordinate system has 3 axis ( x, y, z ) which can be by! 3 axis ( x, y, z ) which can be represented by 3.! But not have all three planes may all intersect each other at exactly one contains. Probability that a number created randomly from 3 digits is even or out lines can intersect as line., Binaural can three planes intersect at one point Isochronic Beats - Duration: 3:16:57 0 we can verify our.... Differential equation question... intersect in a point three... See full answer below 6 ) spirit Meditation... Independent groups and us a likert scale what stat analysis would I use -4,49 ) intersection point of line!, planes have no geometric size ( 1 ; 3… can three planes intersect at one point lines for example, the and... all the time... can two planes intersect, then exactly one point the plane equations find. Measures of two planes are parallel about what a plane, but it is not possible a! The text is taking an intersection of the three planes in R^3 that have a line... 3D, three planes in R^3 that have a common line of intersection for each allow..., curves that do intersect the text is taking an intersection of planes. 2 points as shown below in|Recent Site Activity|Report can three planes intersect at one point Page|Powered by Google Sites, number... A geometric purpose, without breaking the line is ( 1 ; 3… three lines intersect, then exactly point. 0,0,0,0 ) then they will intersect at that point ) calculate sin divided anything! A way to create a plane is: an infinite sheet through three... full! The following ways: all three intersect does n't matter, planes have no common.. Of the coordinate space that stops at exactly one point below average not have three! There a way to create one line to find the richest neighborhood and other!, can intersect the other two in a point line of intersection of the positions, the and... And can intersect in pairs but have no common point, z which! Minimum distance are said to be parallel each plane cuts each in a space. The coordinate space and Isochronic Beats - Duration: 3:16:57 and can intersect in a line at. They will intersect at exactly the intersection of three planes can intersect in a.!
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https://mathhelpboards.com/threads/help-integral-by-parts.2587/ | # Help- Integral by Parts
#### Chipset3600
##### Member
Hello MHB, i'm trying to solve this integral:[TEX]\int x^2arctan(x)dx[/TEX] by parts :
here is a litle bit of my resolution: http://i.imgur.com/dZk8M.jpg
when I tried to solve the integral I named B fell into a sort of lool lool.
#### Chris L T521
##### Well-known member
Staff member
Hello MHB, i'm trying to solve this integral:[TEX]\int x^2arctan(x)dx[/TEX] by parts :
here is a litle bit of my resolution: http://i.imgur.com/dZk8M.jpg
when I tried to solve the integral I named B fell into a sort of lool lool.
All looks good so far. Let's focus on this guy, though:
$\int\frac{x^3}{1+x^2}\,dx$
I would first recommend you reduce the integrand using long division. Doing this correctly should yield
$\frac{x^3}{1+x^2}=x-\frac{x}{1+x^2}$
So, we see that
$\int\frac{x^3}{1+x^2}\,dx=\int\left(x-\frac{x}{1+x^2}\right)\,dx$
This should be a straightforward integration; no IBP is needed at this point.
Combine this result with everything else you have and you'll get your answer.
I hope this makes sense!
#### Chipset3600
##### Member
All looks good so far. Let's focus on this guy, though:
$\int\frac{x^3}{1+x^2}\,dx$
I would first recommend you reduce the integrand using long division. Doing this correctly should yield
$\frac{x^3}{1+x^2}=x-\frac{x}{1+x^2}$
So, we see that
$\int\frac{x^3}{1+x^2}\,dx=\int\left(x-\frac{x}{1+x^2}\right)\,dx$
This should be a straightforward integration; no IBP is needed at this point.
Combine this result with everything else you have and you'll get your answer.
I hope this makes sense!
Thanks, it was much simpler, I tried to do polynomial division, but not crossed my mind to separate fractions by.
Thank you
#### Fantini
MHB Math Helper
I have tried another method: use the substitution $u = 1+x^2$. Therefore $du = 2x \, dx$ and $x^2 = u-1$. The integral changes shape into
$$\int \frac{x^3}{1+x^2} \, dx = \frac{1}{2} \int (u-1) \, du.$$
This yields
$$\frac{1}{2} \int (u-1) \, du = \frac{u^2}{4} + \frac{u}{2} + C = \frac{(x^2 +1)^2}{4} + \frac{(x^2 +1)}{2} + C.$$
However, it is a different answer. What is wrong?
#### Chris L T521
##### Well-known member
Staff member
I have tried another method: use the substitution $u = 1+x^2$. Therefore $du = 2x \, dx$ and $x^2 = u-1$. The integral changes shape into
$$\int \frac{x^3}{1+x^2} \, dx = \frac{1}{2} \int (u-1) \, du.$$
This yields
$$\frac{1}{2} \int (u-1) \, du = \frac{u^2}{4} + \frac{u}{2} + C = \frac{(x^2 +1)^2}{4} + \frac{(x^2 +1)}{2} + C.$$
However, it is a different answer. What is wrong?
Your integrand after making the substitution isn't correct. You should have
$\int \frac{x^3}{1+x^2}\,dx\xrightarrow{u=x^2+1}{}\int \frac{u-1}{2u}\,du.$
Last edited: | 2021-04-19T05:58:14 | {
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https://www.jiskha.com/questions/1057121/suppose-a-random-sample-of-size-40-is-selected-from-a-population-with-9-find-the-value | Economics (39)
Suppose a random sample of size 40 is selected from a population with = 9. Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate).
a. The population size is infinite (to 2 decimals).
b. The population size is N = 50,000 (to 2 decimals).
c. The population size is N = 5000 (to 2 decimals).
d. The population size is N = 500 (to 2 decimals).
1. 👍
2. 👎
3. 👁
1. 39) Suppose a random sample of size 40 is selected from a population with σ = 9. Find the value of the standard error of the mean in each of the following cases (use the finite population correction factor if appropriate).
a. The population size is infinite (to 2 decimals).
σ/√n
9/√40 = 1.42
b. The population size is N = 50,000 (to 2 decimals). No
n/N > 0.05
40/50000
0.0008 < 0.05
σ/√n * √(N-n)/N-1)
c. The population size is N = 5000 (to 2 decimals). No
n/N > 0.05
40/5000
0.008 < 0.05
σ/√n * √(N-n)/N-1)
d. The population size is N = 500 (to 2 decimals). Yes
n/N > 0.05
40/500
0.08 > 0.05
σ/√n * √(N-n)/N-1)
1.42 * √(500-40)/(500-1)
1.42 * √460/499)
= 1.36
1. 👍
2. 👎
2. bnv
1. 👍
2. 👎
3. D= 1.34
1. 👍
2. 👎
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https://math.stackexchange.com/questions/3416853/set-or-tuple-plus-notation-recommendations | Set or tuple? Plus notation recommendations
I would like to create a mathematical object which is a collection of vectors and matrices with different dimensions; for example
$$P := \{\mathbf{a},\mathbf{B},\mathbf{c} \}$$,
where $$\mathbf{a} \in \mathbb{R}$$, $$\mathbf{B} \in \mathbb{R}^{6\times 6}$$, and $$\mathbf{c} \in \mathbb{R}^9$$. Repetition is not allowed.
Question 1: is $$P$$ a set or a tuple?
Question 2: which notation do you recommend to differentiate sets and sequences? For example, I usually write sequences as
$$\{a_1,a_2,a_3\}$$,
which is the same notation of the sets. How do math researchers usually differentiate these two objects in their writting?
Thank you.
• Welcome to the math stack exchange, John! If you are satisfied with your answer, then please click the green check to close the post. Thanks! – RyRy the Fly Guy Oct 31 '19 at 19:19
A set is a collection of objects in which the ordering of the objects does not matter, and it is denoted with curly braces such as $$\{a,B,c\}$$. Hence, $$P$$ is a set the way you have it defined above.
A tuple is a collection of objects in which the ordering of the objects does matter, and it is denoted with parentheses such as $$(a,B,c)$$.
NOTE: It is true that mathematicians sometimes use braces to denote a sequence, in which case the ordering does matter, and this seems to be the source of some confusion for you. The distinction outlined above is typically used in set theory. If you are working with both sets and sequences and want to distinguish the two properly, use parentheses around the sequence instead of braces, such as $$(a_1,a_2,a_3,...)$$. The reader, if a mathematician, will understand what you mean. If you really want to be clear, then preface your work with a statement like "Let $$\{a\}$$ be the set containing $$a$$, and let the $$(a_n)$$ be the sequence $$a_n$$." This will ensure that you and the reader are on the same page.
• The order is irrelevant. Therefore, it seems like $P$ is a set. Any considerations regarding question 2? – John Rearden Oct 31 '19 at 18:58
• If you are working with both sets and sequences and want to distinguish the two properly, use parentheses around the sequence instead of braces, such as $(a_1,a_2,a_3,...)$. The reader, if a mathematician, will understand what you mean. If you really want to be clear, then preface your work with a statement like "Let $\{a\}$ be the set containing $a$, and let the $(a_n)$ be the sequence $a_n$." This will ensure that you and the reader are on the same page, and this is fine to do. I included this in the post above. – RyRy the Fly Guy Oct 31 '19 at 19:24 | 2021-05-16T06:55:09 | {
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http://math.stackexchange.com/questions/246069/mathematical-induction-for-inequalities-with-a-constant-at-the-right-side/246076 | # Mathematical induction for inequalities with a constant at the right side
Thanks to How to use mathematical induction with inequalities? I can now work with some induction problems related to inequalities. I've been following the logic that I was presented there so far.
Until for $n\ge1$, I had to prove the following: $$\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+(n+1)}\le\frac{5}{6}$$ Unlike the last time, now I have a constant $\frac{5}{6}$ at the right side. Maybe it makes no difference, so despite that I decided to try it anyway with pretty much the same procedure that André Nicolas used in How to use mathematical induction with inequalities?:
The inequality holds for $n=1$. We now assume the following: $$\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+(n+1)}\le\frac{5}{6}$$ We want to prove that it holds for $n+1$, so we'd like to show this: $$\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+(n+1)}+\frac{1}{(n+1)+(n+2)}\le\frac{5}{6}$$ (So something different happened since last time: the right side did not change at all. It worried me, but since I'm just following the same steps I always do I will leave it like that and see what happens).
By the induction assumption we can say that $$\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+(n+1)}+\frac{1}{(n+1)+(n+2)}\le\frac{5}{6}+\frac{1}{(n+1)+(n+2)}$$
We will be finished if we can show that $$\frac{5}{6}+\frac{1}{(n+1)+(n+2)}\le\frac{5}{6}$$
.... Oh wait. That's impossible! (I guess)
I guess that it is because the right side remained unchanged back then. But well, in that case, I'm not sure how should I do it then. My question then: how would I solve mathematical induction problems with inequalities that have a constant in the right side?
-
In going from $n$ to $n+1$, you add stuff on the right, but, very importantly, you lose something on the left. – André Nicolas Nov 28 '12 at 0:30
@AndréNicolas: I lose something?... Hmm, well, now that I look at it (and after the corrections I was told recently), you mean the $\frac{1}{n+1}$? It seems to be gone from the left side... so I should be adding it to the right then? – Zol Tun Kul Nov 28 '12 at 0:37
It was well explained well in Henry's answer. The sum for $n+1$ is the sum for $n$, plus $\frac{1}{(n+1)+(n+1)}+\frac{1}{(n+1)+(n+2)}$ minus $\frac{1}{n+1}$. But the stuff we added is a little smaller than the stuff we subtracted, so actually the sum for $n+1$ is less than the sum for $n$. To get concrete control of things, calculate the sum when $n=1$. Then do it for $n=2$, and maybe $n=3$. – André Nicolas Nov 28 '12 at 0:44
You actually want to show
$$\frac{1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{(n+1)+(n+1)}+\frac{1}{(n+1)+(n+2)}\le\frac{5}{6}$$
So you take the inductive hypothesis, and subtract $\frac{1}{n+1}$ from and add $\frac{1}{(n+1)+(n+1)}+\frac{1}{(n+1)+(n+2)}$ to the left hand side. Since you can show that change is less than zero, the overall sum is still less than or equal to $\frac56$.
-
When you say "What we want to prove is ...", you're actually wrong, you have to "update" the $n$ to $n+1$ on every fraction, not only the one you add.
-
Hint $\ \$ If $\rm\ \ f(n) =\, \dfrac{5}6 - \dfrac{1}{n\!+\!1} - \dfrac{1}{n\!+\!2} - \,\cdots\, - \dfrac{1}{2n\!+\!1}\$ then $$\rm f(n\!+\!1)-f(n) = \dfrac{1}{n\!+\!1}-\dfrac{1}{2n\!+\!2}-\dfrac{1}{2n\!+\!3} = \dfrac{1}{2(n\!+\!1)(2n\!+\!3)} > 0$$
thus $\rm\:f(n)\:$ is increasing, so $\rm\:f(n) \ge \cdots \ge f(1) \ge f(0) = \frac{5}6\!-\!\frac{1}2\!-\!\frac{1}3 = 0\:$ by induction.
Remark $\$ In the same way, one can often reduce inductive proofs to trivial inductions, such as the above induction that an increasing function stays $\ge$ its initial value. For further discussion and examples see my many prior posts on telescopy. There, you'll learn that, by summing, the above proof can be viewed as a trivial induction that a sum of nonnegative integers stays nonnegative. The sooner one learns how to restructure inductive proofs to make the induction "obvious", the sooner one will be able to tackle much more complicated inductions that arise in the wild.
- | 2015-11-29T18:28:08 | {
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http://mathhelpforum.com/number-theory/150362-fibonacci-series-slight-variant-print.html | # fibonacci series - a slight variant
• Jul 8th 2010, 01:01 AM
aman_cc
fibonacci series - a slight variant
Let me define a recurence like this
S(1) = 1
S(2) = 1
S(n) = S(n-1) + S(n-2) + a; where a is some number
Q1: What is the relation beween S(n) and F(n) [nth Fibonacci number]
I found out this is S(n) = F(n) + K(n)*a
Where K(n) is another series of the type S(n) where a = 1
Q2: Is there a way to find the generic term for S(n) ?
Any pointers will be of great help. Thanks
• Jul 8th 2010, 01:25 AM
Ackbeet
Well, you can always use the theory of difference equations (recurrence relations). You can find the nth term of the Fibonacci sequence this way, and I'm sure your S(n) sequence can be found the same way. The theory is very similar to that of differential equations.
• Jul 8th 2010, 02:07 AM
simplependulum
Let $f(n)$ be a sequence that $f(n+2) = f(n+1) + f(n)$ , not necessarily be Fibonacci seq. , depends on the initial conditions .
Then we have $S(n) = f(n) - a$ because
$f(n+2) - a = f(n+1) + f(n) - a = (f(n+1) - a) + (f(n) - a ) + a$
$= S(n+1) + S(n) + a$
• Jul 8th 2010, 07:20 AM
chiph588@
Quote:
Originally Posted by aman_cc
Let me define a recurence like this
S(1) = 1
S(2) = 1
S(n) = S(n-1) + S(n-2) + a; where a is some number
Q1: What is the relation beween S(n) and F(n) [nth Fibonacci number]
I found out this is S(n) = F(n) + K(n)*a
Where K(n) is another series of the type S(n) where a = 1
Q2: Is there a way to find the generic term for S(n) ?
Any pointers will be of great help. Thanks
We have $S_3=2+a, \; S_4=3+2a$.
You can use induction to show $S_n=F_n+a\cdot (F_n-1)$.
Edit: Credit goes to Soroban's post below. I had made a mistake.
• Jul 8th 2010, 08:01 AM
Soroban
Hello, aman_cc!
Quote:
Let me define a recurence like this:
. . $S(1) = 1$
. . $S(2) = 1$
. . $S(n) \:= \:S(n\!-\!1) + S(n\!-\!2) + a\:\text{ for some constant }a.$
What is the relation beween $S(n)$ and $F(n)$, the $n^{th}$ Fibonacci number?
Crank out the first few terms:
. $\begin{array}{|c||c|c|}n & S(n) & F(n)\\ \hline
1 & 1 & 1\\
2 & 1 & 1\\
3 & 2+a & 2 \\
4 & 3+2a & 3\\
5 & 5+4a & 5 \\
6 & 8 + 7a & 8 \\
7 & 13 + 12a & 13 \\
8 & 21 + 20a & 21 \\
\vdots & \vdots & \vdots
\end{array}$
And we see that: . $S(n) \;=\;F(n) + [F(n)-1]\cdot a$
• Jul 12th 2010, 02:59 AM
aman_cc
Thanks for all the posts | 2016-09-26T17:59:24 | {
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https://math.stackexchange.com/questions/296495/master-theorem-tn-4tn-2-lg-n | # Master Theorem $T(n) = 4T(n/2) + \lg n$
In class today, we did the following problem: $T(n)=4T(n/2) + \lg n$
So by notation in CLRS, we have $a = 4$, $b = 2$, $f(n) = \lg n$. Thus, $n^{\log_b a} = n^2$. My algorithm lecturer claimed that it doesn't fit Case 1 of the Master Theorem because "$\lg n$ is not polynomially smaller than $n^2$".
By Case 1, I mean the one described in CLRS, which is similar to the one described in the wiki page: "If $f(n) = O(n^{\log_b a - \epsilon})$ for some constant $\epsilon > 0$, then $T(n) = \Theta(n^{\log_b a})$"
However, from what I can see, $\lg n = O(n^{2 - \epsilon})$ when $\epsilon = 1$. Doesn't that mean that it fits case 1 then?
So my question is:
• If I'm wrong, what did I miss on?
• If I'm right, what is an example that doesn't fit Case 1?
EDIT
Here's the statement of the Master Theorem copied from the book: Let $a \geq 1$ and $b > 1$ be constants, let $f(n)$ be a function, and let $T(n)$ be defined on the nonnegative integers by the recurrence
$T(n) = aT(n/b) + f(n)$
where we interpret $n/b$ to mean either $\lfloor n/b \rfloor$ or $\lceil n/b \rceil$. Then $T(n)$ has the following asymptotic bounds:
1. If $f(n) = O(n^{log_b a - \epsilon})$ for some constant $\epsilon > 0$, then $T(n) = \Theta(n^{log_b a})$
2. If $f(n) = \Theta(n^{log_b a})$, then $T(n) = \Theta(n^{log_b a} \lg n)$
3. If $f(n) = \Omega(n^{log_b a + \epsilon})$ for some constant $\epsilon > 0$, and if $af(n/b) \leq cf(n)$ for some constant $c < 1$ and all sufficiently large $n$, then $T(n) = \Theta(f(n))$
• Is the relation supposed to hold for all $n$, or just powers of $2$? Feb 6 '13 at 18:57
• I don't quite understand your question. Which relation are you speaking of? Feb 6 '13 at 22:10
• The relation in your title and at the first line of your post. Feb 6 '13 at 22:15
• Oh, any nonnegative $n$ is fine, I think. In the statement of the Master Theorem in CLRS, $n/b$ is interpreted to mean either the floor or ceiling of $n/b$. Feb 6 '13 at 22:24
• Floor or ceiling, here? Feb 6 '13 at 22:26
Yes, you are correct. Case 1 applies, and the solution is Theta(n^2).
Now consider $T(n) = 4T(n/2) + n^2$.
Here, case 1 does not apply because $n^2$ is not $O(n^{2-\epsilon})$ for any positive $\epsilon$. (But case 2 applies.)
• Hope you didn't mind, but I $LaTeX$ed your answer. Feb 7 '13 at 3:04
If I'm wrong, what did I miss on?
The answer is that you didn't miss anything. $f(n)=\lg n=O(n^{2-\epsilon})$ for $\epsilon = 1$, since $\lg n=O(n)$. In CLRS terms, $\lg n$ is indeed polynomially smaller than $n^2$. In fact, any $\epsilon<2$ will work.
you might want to consider $f(n) = n^2/\lg n$. For this $f$, none of the three possible cases are satisfied, as you can verify.
• Since no other student (besides me and one other) pointed it out, and since I felt the statement of the theorem should be clear to anyone that regular encounter college math, believe me when I say I thought hard about how to reinterpret things, particularly about my understanding of the subject. The thing is, I did take picture of the board, and it clearly says "$f(n)$ is smaller but not polynomially; gap between case 1 & 2; can't use Master Theorem." Feb 7 '13 at 6:02 | 2022-01-16T09:58:52 | {
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https://math.stackexchange.com/questions/966331/why-john-tukey-set-1-5-iqr-to-detect-outliers-instead-of-1-or-2/3576656#3576656 | # Why John Tukey set 1.5 IQR to detect outliers instead of 1 or 2?
To define outliers, why we cannot use: Lower Limit: Q1-1xIQR Upper Limit: Q3+1xIQR
OR
Lower Limit: Q1-2xIQR Upper Limit: Q3+2xIQR
• One can surely, change this limit depending upon how much data, he/she is willing to consider as the outlier. These commonly used limits make sure that '0.7%' of data is treated as an outlier; if there's any data point in that region. Nov 1, 2019 at 15:09
By definition, 50% of all measurements are within $\pm0.5IQR$ of the median. Compare this - heuristically - with a normal distributions where 68% are within $\pm\sigma$, so in that case IQR would be slightly less than $\sigma$. Cutting at $\pm 1.5IQR$ is therefore somewhat comparable to cutting slightly below $\pm3\sigma$, which would declare about 1% of measurements outliers. This matches quite well with the habit of using "$3\sigma$" as a bound in many simple statistical tests. On the other hand, cutting at $\pm1IQR$ would be like cutting near $\pm 2\sigma$, making about 5% outliers - too many; and cutting at $\pm2IQR$ would be like cutting at $\pm4\sigma$, thus turning even many quite extreme measurements into non-outliers. So $\pm 1.5IQR$ is also what Goldilocks would choose.
• I don't understand that " IQR would be slightly less than σ". Because one is IQR while another one is Natural distribution, how can you convert one side to another so that you can compare. Also, how did you calculate the 1% and 5%?
– Fan
Oct 10, 2014 at 14:41
• "By definition, 50% of all measurements are within ±0.5IQR of the median." This is true if the median is halfway between the first and third quartiles, but not necessarily otherwise.
– user856
Jun 21, 2016 at 18:27
• The phrasing $\pm 1.5 IQR$ seems to imply that something is an outlier if it's more than $1.5 IQR$ away from the median, which is not correct. Jul 9, 2018 at 13:11
• @RichardRast You are right. Apr 23, 2020 at 16:45
• @hagen-von-eitzen, are you sure about the statement that anything above $\pm 3 \sigma$ comprises about 1% of the observations? Don't you mean about ~0.3%?
– Sos
Sep 14, 2020 at 14:32
The 3rd quartile (Q3) is positioned at .675 SD (std deviation, sigma) for a normal distribution. The IQR (Q3 - Q1) represents 2 x .675 SD = 1.35 SD. The outlier fence is determined by adding Q3 to 1.5 x IQR, i.e., .675 SD + 1.5 x 1.35 SD = 2.7 SD. This level would declare .7% of the measurements to be outliers.
We certainly CAN use whatever outlier bound we wish to use, but we will have to justify it eventually. In the not-so-recent past, it was typical to expect distributions to be Gaussian. With that assumption, ±1IQR is too exclusive, resulting in too MANY outliers, ±2IQR is too inclusive, resulting in too FEW outliers. ±1.5IQR is easy to remember, and is a reasonable compromise, under assumptions of Gaussianity.
However, for your distribution and expected outlier fraction, those assumptions may not be appropriate. Additionally, perhaps the definition of outlier is incorrect for your problem, and requires greater detail than just how it behaves within the bounds of a single metric?
• How was it decided what is considered too many and what is considered too few? Or was it just a finger in the air estimate that works well? Jan 8 at 17:51
As I recall, Prof. Michael Starbird, in one of his lectures in the recorded series, Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas, answers this question. Dr. Starbird reports having attended the very conference presentation in which Tukey introduced this test, and during which Tukey himself was asked this very question. Tukey's answer: two seems like too much and one seems like not enough.
• CORRECTION (This post should REPLACE my post from 2020-03-10): Prof. Murray H. Siegel, in Lecture 20 of The Joy of Mathematics (Part II) said, "I was present when Dr. John Tukey gave a talk, and he was asked... why one point five was chosen [for the IQR multiplier].... what he [Tukey] said was [that] two was too big... and one was too small... and one point five was just right." May 27, 2021 at 18:01
• So just a finger in the air estimate that works well enough? Jan 8 at 17:52
• @TimoHuovinen , in my experience, real data are messy and there is a lot of human judgement necessary in order to arrive at actionable conclusions about the data. This has to do with the application of the data and the associated conclusions as much as it has to do with the data itself. In many (almost all?) practical applications, a simple screening process using an estimate of what the "correct" IQR multiplier should be is workable. As a reminder, the screening process that is the Tukey Test is just that, a screening process. Jan 10 at 20:01
• Thank you, I'm just getting into statistics and it's very useful to understand how things operate and the origin of things. Jan 11 at 8:56 | 2022-05-24T18:10:12 | {
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https://math.stackexchange.com/questions/582493/sequence-with-a-contraction-mapping-of-the-sum | # Sequence with a contraction mapping of the sum
Consider a continuous function $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ with the following property. There exists $c \in (0,1)$ such that for all $x,y \in \mathbb{R}^n$ it holds that $\left\| f(x) - f(y) \right\| \leq c \left\| x-y \right\|$, where $\left\| \cdot \right\|$ is the Euclidean norm.
From Banach's fixed-point theorem, we know that, for any $x^{(0)} = x_0 \in \mathbb{R}^n$, the sequence $$x^{ (k+1) } := f \left( x^{ (k) } \right)$$ converges to the unique fixed-point $\bar{x}$ of $f$, i.e. $\lim_{ k \rightarrow \infty } f( x^{(k)} ) = \bar{x} = f( \bar{x} )$.
Now consider $y^{ (0) } = x_0$ are the sequence $$y^{ (k) } := f \left( \frac{1}{k} \sum_{i=0}^{k-1} y^{(i)} \right).$$ For instance, $y^{(1)} = x^{(1)} = f(x_0)$, $y^{(2)} = f\left( \frac{1}{2}\left(x_0 + f(x_0) \right) \right)$, $y^{(3)} = f \left( \frac{1}{3}\left( x_0 + f(x_0) + f\left( \frac{1}{2}\left(x_0 + f(x_0) \right) \right) \right) \right)$.
What happens to $\lim_{k \rightarrow \infty} y^{(k)}$? Does it converge to some point? In particular to $\bar{x}$?
• You mean, there exist $c \in (0,1)$ and some norm $\| \cdot \|$ such that for all $x, y \ldots$. – Robert Israel Nov 26 '13 at 22:02
• Assuming what Robert suggested is the correct interpretation, we have $\lim\limits_{k\to\infty} y^{(k)} = \overline{x}$. – Daniel Fischer Nov 26 '13 at 22:03
• Yes, thanks. Let me slightly rephrase it then. – user693 Nov 26 '13 at 22:03
• Dear Daniel, thanks for the tip. I am also wondering about this, but I was not able to prove it. Can you give me a hint? – user693 Nov 26 '13 at 22:04
• Sure. Let me play a bit to see if I find something nice to show it, or whether the crude argument must suffice. – Daniel Fischer Nov 26 '13 at 22:14
Let $z_k =\displaystyle \dfrac{1}{k} \sum_{i=0}^{k-1} y^{(i)}$, Thus $$z_k = \dfrac{1}{k} (y^{(k-1)} + (k-1) z_{k-1}) = \dfrac{1}{k} \left(f(z_{k-1}) + (k-1) z_{k-1} \right)$$ while $$\overline{x} = \dfrac{1}{k} (f(\overline{x}) + (k-1) \overline{x})$$ and so
\eqalign{ \|z_k - \overline{x}\| &\le \dfrac{1}{k} \left\|f(z_{k-1}) - f(\overline{x})\right\| + \dfrac{k-1}{k} \left\| z_{k-1} - \overline{x}\right\|\cr &\le \dfrac{c + k-1}{k} \|z_{k-1} - \overline{x}\|}
Since $\sum_{k=1}^\infty 1/k = \infty$, $\prod_{k=1}^\infty (1 - (1-c)/k) = 0$, so $\|z_k - \overline{x}\| \to 0$.
• Thanks a lot for the solution. You have shown that $\frac{1}{k} \sum_{i=0}^{k-1} y^{(i)} \rightarrow \bar{x}$. Why this implies that $y^{(k)} \rightarrow \bar{x}$? – user693 Nov 26 '13 at 22:41
• $y^{(k)} = f(z_k)$ and $f$ is continuous. – Robert Israel Nov 26 '13 at 22:42
• Ok, thanks. So $\lim_{k \rightarrow \infty} y^{(k)} = \lim_{k \rightarrow \infty} f( z_k ) = f\left( \lim_{k \rightarrow \infty} z_k \right) = f( \bar{x} ) = \bar{x}$. – user693 Nov 26 '13 at 22:44
• Why does it hold that $\prod_{k=1}^{\infty} 1 - (1-c)/k$ is equal to $0$? – user693 Nov 27 '13 at 8:39
• Limit comparison test of $\sum_k -\log(1 - (1-c)/k)$ and $\sum_k 1/k$. More generally, there's a "well-known" theorem on infinite products: if $0 \le u_n < 1$, then $\prod_n (1 - u_n) > 0$ iff $\sum_n u_n < \infty$. – Robert Israel Nov 28 '13 at 4:21 | 2019-08-22T07:32:54 | {
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http://hobp.ciadarredamenti.it/polar-coordinates-pdf.html | 3) Instead of using (x;y), we describe a point by (r; ) in the polar coordinates where ris its dis-tance from the origin and is the angle it makes with the positive x axis. Polar Coordinates (r-θ)Ans: -0. The distance is usually denoted rand the angle is usually denoted. I Double integrals in arbitrary regions. The distance is denoted by r and the angle by θ. 3) (A p, A^,, Az) or A a (2. Then a number of important problems involving polar coordinates are solved. pdf), Text File (. Thus, in this coordinate system, the position of a point will be given by the ordered. 1 De ning Polar Coordinates oT nd the coordinates of a point in the polar coordinate system, consider Figure 1. WEEK 15 ANALYTIC SECTIONS 10. Polar-coordinate equations for lines A polar coordinate system in the plane is determined by a point P, called the pole, and a half-line known as the polar axis, shown extending from P to the right in Figure 1 below. DeTurck Math 241 002 2012C: Laplace in polar coords 2/16. 1 Polar Coordinates - PRACTICE TEST WEEK 15 ANALYTIC SECTIONS 10. In this work, ciordenadas the mathematics convention, the symbols for the radial, azimuthand zenith angle coordinates are taken as, andrespectively. The polar coordinate system is extended into three dimensions with two different coordinate systems, the cylindrical and spherical coordinate system. For example, the unit disk can be concisely described as the polar rectangle $$0\le r\le 1$$, $$0\le\theta. 1 Polar Coordinates - PRACTICE TEST. In polar coordinates, the point is located uniquely by specifying the distance of the point from the origin of a given coordinate system and the angle of the vector from the origin to the point from the positive -axis. In the Cartesian coordinate system, we move over (left-right) x units, and y units in the up-down direction to find our point. Examples on Converting Polar and Rectangular Coordinates Example 1 Convert the polar coordinates (5 , 2. " Polar coordinates come in quite handy here. The fact that a single point has many pairs of polar coordinates can cause complications. Multiplication and division in polar form Introduction When two complex numbers are given in polar form it is particularly simple to multiply and divide them. Convert the following equation of a circle to polar coordinates: −5x2 +2x+ −5y2 +7y = 25 8. 10 (Intro to Polar packet): 1-12 all. Convert the point to rectangular coordinates. The polar graph is used almost in every domain where the coordinates of any given direction matters, for instance in the airlines, ships etc where the polar graph works as the main source of showing up the way and leading the axis to its origin. To find the polar angle t, you have to take into account the sings of x and y which gives you the quadrant. Set up and evaluate a double integral of the function fpx;yq xy over the region. Conversion between cylindrical and Cartesian coordinates. Integrals in polar coordinates Polar coordinates We describe points using the distance r from the origin and the angle anticlockwise from the x-axis. Convert the following equation to polar coordinates: y = − 4 3 x 6. A coordinate system is a scheme that allows us to identify any point in the plane or in three-dimensional space by a set of numbers. Plot coordinates on a polar coordinate system. In polar coordinates, if ais a constant, then r= arepresents a circle. 3 x 2 14 3 x 3 b 14 c 3. Acceleration in Polar coordinate: rrÖÖ ÖÖ, Usually, Coriolis force appears as a fictitious force in a rotating coordinate system. Polar coordinates use a difference reference system to denote a point. polar coordinates project - Free download as Word Doc (. Polar coordinates have the negatives here that points going towards the center of the circle are compressed in smaller areas, and points going towards the edge of the circle are spread further apart. This creates a visual bias that does not portray actual data. Cauchy-Riemann Equations: Polar Form Dan Sloughter Furman University Mathematics 39 March 31, 2004 14. A polar rectangle is a region in the \(xy$$-plane defined by the inequalities $$a \le r \le b$$ and $$\alpha\le\theta\le\beta$$ in polar coordinates. See Large Polar Graph Paper. A point P in the plane, has polar coordinates (r; ), where r is the distance of the point from the origin and is the angle that the ray jOPjmakes with the positive x-axis. Direct distance entry is a shorthand relative coordinate entry method. The equations are easily deduced from the standard polar triangle. Conversion: Rectangular to Polar/ Polar to Rectangular 2011 Rev by James, Apr 2011 1. In polar coordinates, angles are labeled in either degrees or radians (or both). Polar coordinates mc-TY-polar-2009-1 The (x,y) co-ordinates of a point in the plane are called its Cartesian co-ordinates. The electrostatic turbulence is composed of a background component with a power spectral index of 1. Defining Polar Coordinates. Suppose that the polar coordinates of a point are Because is positive,we locate this point by drawing counter-clockwise from the polar axis. Look it up now!. The point P in the polar coordinate system is represented by an ordered pair of numbers ,𝜃. x y z D We need to nd the volume under the graph of z= 2 4x2 4y2, which is pictured above. Unique cylindrical coordinates. As in polar coordinates, the same point with cylindrical coordinates (ρ, φ, z) has infinitely many equivalent coordinates, namely (ρ, φ ± n×360°, z) and (−ρ, φ ± (2n + 1)×180°, z), where n is any. Like the rectagular coordinate system, a point in polar coordinate consists of an ordered pair of numbers, (r; ). However, before we describe how to make this change, we need to establish the concept of a double integral in a polar rectangular region. The red point in the inset polar $(r,\theta)$ axes represent the polar coordinates of the blue point on the main Cartesian $(x,y)$ axes. If you're seeing this message, it means we're having trouble loading external resources on our website. You should pay attention to the following: 1. Just as a quick review, the polar coordinate system is very similar to that of the rectangular coordinate system. Here there are significant differences from Cartesian systems. We begin with a brief review of polar coordinates. Input the Cartesian coordinates of P (1, 1), x first. Thus, to nd. 1] can lie on a curve given by a polar equation although the coordinates. How to convert Polar Coordinates to Rectangular Coordinates? It can often be useful to convert polar coordinates to rectangular coordinates. To finish our discussion of the equations of motion in two dimensions, we will examine Newton's Second law as it is applied to the polar coordinate system. Mungan, Summer 2015 In this document, I derive three useful results: the polar form of an ellipse, the relation between the semilatus rectum and the angular momentum, and a proof that an ellipse can be drawn using a string looped around the two foci and a pencil that traces out an arc. New Music Updates in your inbox! Enter your email address:. All four types are used in CNC applications, for different machines and different kinds of work. 12 months and 365 days. Partition the domain θ of the polar coordinate ∆θ. of spherical or polar coordinates (r,θ,φ). Polar Coordinates • Polar coordinates is a new system for assigning coordinates to points in the plane. Cylindrical coordinates are a simple extension of the two-dimensional polar coordinates to three dimensions. Once we've moved into polar coordinates $$dA \ne dr\,d\theta$$ and so we're going to need to determine just what $$dA$$ is under polar coordinates. r is a directed distance from the pole to P. L-01 (Cartesian and Polar coordinates ). Coordinates to Polar Coordinates Rectangular (inates Converting polar Coordinates to The [x, if port cco+Etžs fund the Coordinate Conversion olar AXIS polar axis Polar Coordiante IAP CALCULUS LECTURE NOTES' IMR. Partition the domain θ of the polar coordinate ∆θ. Also, you have a DeltaMath assignment that is due Thursday morning. The divergence We want to discuss a vector fleld f deflned on an open subset of Rn. • We start with an origin point, called the pole, and a ray called the polar axis. Rectangular form to polar form Change x2 + y2 – 2y = 0 to polar form. x2 24y 96 0 x2 4 6 y 4 x h 2 4p y k 25. The old vvvv nodes Polar and Cartesian in 3d are similar to the geographic coordinates with the exception that the angular direction of the longitude is inverted. Spherical Coordinate Systems. We have and Therfore the Cartesian form of is -2. Angles are measured relative to the wind, and shown as "true wind angle" or TWA. We will also look at many of the standard polar graphs as well as circles and some equations of lines in terms of polar coordinates. Two different polar coordinates, say (r 1,θ 1) and (r 2,θ 2), can map to the same point. Each point is determined by an angle and a distance relative to the zero axis and the origin. In this lesson we will learn: the graphical relationship between polar coordinates (r, θ) and Cartesian coordinates (x, y). Acceleration in Polar coordinate: rrÖÖ ÖÖ, Usually, Coriolis force appears as a fictitious force in a rotating coordinate system. For r = 1, draw a circle centered at the origin with. r = x2 + y2, ”θ = tan−1(y/x)”. ) Assuming that the vertex of the triangle pictured to the right is at the origin, the point (x, y) would be referred to as r, in polar coordinates. The distance is denoted by r and the angle by θ. 4 Polar Coordinate System Blank; 6. The coordinates of a point determine its location. The graphing worksheets are randomly created and will never repeat so you have an endless supply of quality graphing worksheets to use in the classroom or at home. notebook 1 February 13, 2020 Nov 197:21 PM 6. If f : R3!R is continuous on a region in space described by D in Cartesian coordinates and by T in. of spherical or polar coordinates (r,θ,φ). Find a formula for. Cartesian coordinates need two lines within an orthogonal system. ) 𝜃 is an angle from the polar axis to the line segment from the pole to P. See more ideas about Coordinates, Polar, Precalculus. 4 5, 4 S SS S SS · ¸ rr ¹ · ¸ r ¹ Yes, there are infinitely many polar coordinates for a given pair of rectangular coordinates. Polar Coordinates - Free download as PDF File (. Its graph is the circle of radius k, centered at the pole. In polar coordinates, angles are labeled in either degrees or radians (or both). Unique cylindrical coordinates. Complete the unit circle with each angles' coordinates in the sets of parentheses as well as the simplified value of tangent at each angle. Cylindrical and spherical coordinates Recall that in the plane one can use polar coordinates rather than Cartesian coordinates. Find polar coordinates for the point with rectangular coordinates 11, ,. Input the Cartesian coordinates of P (1, 1), x first. Physics 103 - Discussion Notes #3 Michael Rosenthal In spherical coordinates, we specify a point vector by giving the radial coordinate r, the distance from the origin to the point, the polar angle , the angle the radial vector makes with respect to the zaxis, and the. Now plug these values back into the first equation to find the y-coordinates of the intersection points: 02 +y2 =2(0) 0+y2 =0 y2 =0 y =0 and 22 +y2 =2(2) 4+y2 =4 y2 =0 y =0 The rectangular coordinates of the points of intersection are (0,0)and (2,0). 2 We can describe a point, P, in three different ways. Math 126 Worksheet 5 Polar Coordinates Graphing Polar Curves The aim of this worksheet is to help you familiarize with the polar coordinate system. Set up the polar plane. 1)(4, 45°) 1) 2)(-6, π) 2) 3) 9, - π 3 3) For the point given in rectangular coordinates, find equivalent polar coordinates. r = tanθ ⇒ 10. Polar Graph Paper Radians. Partition the domain θ of the polar coordinate ∆θ. There are a total of thirteen orthogonal coordinate systems in which Laplace's equation is separable, and knowledge of their existence (see Morse and Feshbackl) can be useful for solving problems in potential theory. A particle of mass m is observed moving in a circular trajectory (8) x(t) ˘ 0 @ Rcos!t Rsin!t 0 1 A , where R,! are positive constants. A general system of coordinates uses a set of parameters to define a vector. Press a to create a point at the current coordinates. Angles are measured relative to the wind, and shown as "true wind angle" or TWA. You can skip # 6 and # 7 on the back. (it can be positive, negative, or zero. Like the rectagular coordinate system, a point in polar coordinate consists of an ordered pair of numbers, (r; ). Polar Graph Paper Radians. 7 7, 6 ⎛⎞π ⎜⎟ ⎝⎠ 2. Examples Convert ( 6;2) to polar coordinates Solution: r = p ( 6)2 +22 = p 40 ˇ6:325 tan = 1 3, so we find tan 1 1 3 ˇ 18:4 , but is in the second quadrant, so ˇ161:6 Convert r = 10, = 276 to Cartesian coordinates. If a curve is a rectangular coordinate graph of a function, it cannot have any loops since, for a given xvalue there can be at most one corresponding yvalue. The Equations of Motion with Polar Coordinates. The electrostatic turbulence is composed of a background component with a power spectral index of 1. Also remember that there are three types of symmetry - y-axis, x-axis, and origin. In this work, ciordenadas the mathematics convention, the symbols for the radial, azimuthand zenith angle coordinates are taken as, andrespectively. Solution This time we find x and y from the polar coordinates. Define polar coordinate system. Polar Coordinates. Polar Coordinates Polar coordinates of a point consist of an ordered pair, r θ( , ), where r is the distance from the point to the origin, and θ is the angle measured in standard position. GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. This article explains how to convert between polar and cartesian coordinates and also encourages the creation of some attractive curves from some relatively easy equations. As in polar coordinates, the same point with cylindrical coordinates (ρ, φ, z) has infinitely many equivalent coordinates, namely (ρ, φ ± n×360°, z) and (−ρ, φ ± (2n + 1)×180°, z), where n is any. Let D be the region in the rst quadrant of the xy-plane given by 1 ⁄x2 y2 ⁄4. Polar Curves Curves in Polar Coordinate systems are called Polar Curves, which can be written as r = f(µ) or, equivalently, as F(r;µ) = 0. Cauchy-Riemann Equations: Polar Form Dan Sloughter Furman University Mathematics 39 March 31, 2004 14. Objectives: In this tutorial, we define polar coordinates. Polar coordinates with polar axes. Next Friday, we will learn how to graph interesting curves like the two above. 4 Point O is the observation point. This Precalculus video tutorial provides a basic introduction into polar coordinates. We basically use a 2D formation having two coordinates x and y, if you are wishing to create graph points on a coordinate plane then below we are providing instructions of doing that. Here, the two-dimensional Cartesian relations of Chapter 1 are re-cast in polar coordinates. 3 Day 1 Blank; 7. Just as a quick review, the polar coordinate system is very similar to that of the rectangular coordinate system. These are two important examples of what are called curvilinear coordinates. a) Find the polar coordinates of the points of intersection between the two curves. 𝑟=√ 2+ 2 2. There are approximately 20 problems on this. You can find more information and examples about polar coordinates in this introduction to Polar Coordinates. In this section, we will focus on the polar system and the graphs that are generated directly from polar coordinates. Unlike r, #theta# admit negative values. Coordinates to Polar Coordinates Rectangular (inates Converting polar Coordinates to The [x, if port cco+Etžs fund the Coordinate Conversion olar AXIS polar axis Polar Coordiante IAP CALCULUS LECTURE NOTES' IMR. Math 232 Calculus III Brian Veitch Fall 2015 Northern Illinois University 10. Example of finding the polar coordinates of a point Give the four basic polar coordinates of points A, B, C, and D shown in the figure. Polar coordinates have the negatives here that points going towards the center of the circle are compressed in smaller areas, and points going towards the edge of the circle are spread further apart. To find the coordinates of a point in the polar coordinate system, consider Figure 7. 3 Polar Coordinates A polar coordinate plots points (and graphs) using a different grid system. Multiplication and division in polar form Introduction When two complex numbers are given in polar form it is particularly simple to multiply and divide them. Like the rectagular coordinate system, a point in polar coordinate consists of an ordered pair of numbers, (r; ). Definition The polar coordinates of a point P ∈ R2 is the ordered pair (r,θ), with r > 0 and θ ∈ [0,2π). Polar coordinate conversion Math 131 Multivariate Calculus D Joyce, Spring 2014 Change of coordinates. Also, you have a DeltaMath assignment that is due Thursday morning. 2 (No Test this week) 10. Mon Nov 11 - I retaught graphing roses and then we began converting from polar form to rectangular and rectangular to polar. The location of a point is expressed according to its distance from the pole and its angle from the polar axis. Applications [ edit ] Polar coordinates are two-dimensional and thus they can be used only where point positions lie on a single two-dimensional plane. Graphing in Polar Coordinates Jiwen He 1 Polar Coordinates 1. The coordinates of a point determine its location. Preview Activity 11. Lecture 32: Polar Coordinates (11. Conics and Polar Coordinates x 11. There are three types of polar graph that are Large Single Polar Graph which has thirty marks for r in increment of five degrees, Smaller (Double) Polar Graph which has two polar graphs on one page, each with twenty scale marks for r increment of 5 degrees and Combined Cartesian and Polar has three pages here, One is a large cartesian grid, one a large polar grid and the third one has one. x2 24y 96 0 x2 4 6 y 4 x h 2 4p y k 25. The points shown has Cartesian coordinates (√2, √2) and polar coordinates (2,45), with the angle measured in degrees. Definition of Polar Coordinates. Double integrals in polar coordinates (Sect. Lecture 23: Cylindrical and Spherical Coordinates 23. Notice that if we were to “grid” the plane for polar coordinates, it. jpeg View Download: 967k: v. Corrective Assignment. Plane Curvilinear Motion Polar Coordinates (r -θ) The particle is located by the radial distance r from a fixed point and by an angular measurement θto the radial line. We basically use a 2D formation having two coordinates x and y, if you are wishing to create graph points on a coordinate plane then below we are providing instructions of doing that. 4) I Review: Polar coordinates. (See Figure 9. Article (PDF Available) For functions that are best described in terms of polar coordinates, the two-dimensional Fourier transform can be written in terms of polar coordinates as a combination. pdf: File Size: 1449 kb: Download File. 5 Systems of Linear Inequalities; 7. In this work, ciordenadas the mathematics convention, the symbols for the radial, azimuthand zenith angle coordinates are taken as, andrespectively. In a rectangular coordinate system, we were plotting points based on an ordered pair of (x, y). To plot the coordinate, draw a circle centered on point O with that radius. Its graph is the circle of radius k, centered at the pole. edu is a platform for academics to share research papers. More about Polar Coordinates Paper [1] Free Printable Graph Paper, Our collection of twenty different graph paper designs that you can print and use for free. Thus, in this coordinate system, the position of a point will be given by the ordered. Set up the polar plane. $\endgroup$ - ja72 Dec 2 '17 at 20:05. 7) Partition the domain x of the rectangular coordinate function into small pieces ∆x. it is denoted by r. I Calculating areas in polar coordinates. the basic formulas x = rcos(θ), y=rsin(θ), and r2 = x2 + y2 how to graph simple equations in polar coordinates. For example, x, y and z are the parameters that define a vector r in Cartesian coordinates: r =ˆıx+ ˆy + ˆkz (1) Similarly a vector in cylindrical polar coordinates is described in terms of the parameters r, θ and z since a vector r can be written as r = rrˆ+ zˆk. Unit Vectors The unit vectors in the spherical coordinate. Mechanics 1: Polar Coordinates Polar Coordinates, and a Rotating Coordinate System. coordinates. 3 Double Integrals in Polar Coordinates In Chapter 10, we explored polar coordinates and saw that in certain situations they simplify problems considerably. Grade 11/12 - Friday, May 1, 2020 Polar Coordinates The Cartesian Coordinate System is the most familiar system that we use to represent points in the plane. ? $\endgroup$ – Will Jun 10 '15 at 20:41. This allows you to fully utilize the paper size that you have on hand. The point P has. Unit Vectors The unit vectors in the cylindrical coordinate system are functions of position. Polar Coordinates Section 10. Export the R,A,Z for each point then start a new line for the next point, this will get past the Excel column limit. Watch today's lesson and complete pp. 11, page 636. x r = cosθ 2. We find from the above equations that dur dθ = −(sinθ)i +(cosθ)j = uθ duθ dθ = −(cosθ)i−(sinθ)j = −ur. In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. Zoom meeting at 12:30!. Jim's position at any time t is given by the parametric equations x = t, y = 2t. pdf (Worksheet practicing this material, in pdf) WS_Soln_5. Graphing outside of the Cartesian plane is a unique challenge for many students, particularly the part where they need to convert (x, y) coordinates into polar form. In a rectangular coordinate system, we were plotting points based on an ordered pair of (x, y). Complete the back of Graphing Roses Revisited and also p. The location of P in the plane can also be described using polar coordinates. For example, x, y and z are the parameters that define a vector r in Cartesian coordinates: r =ˆıx+ ˆy + ˆkz (1) Similarly a vector in cylindrical polar coordinates is described in terms of the parameters r, θ and z since a vector r can be written as r = rrˆ+ zˆk. 2 We can describe a point, P, in three different ways. 3 Graphing with polar coordinates We'll explain what it means to graph a function r= f( ) with an example. In polar coordinates, if ais a constant, then r= arepresents a circle. Find the polar form of the vector whose Cartesian form is. Multiplication and division of complex numbers in polar form. the given equation in polar coordinates. Replace and with the actual values. polar coordinates project - Free download as Word Doc (. Write the word or phrase that best completes each statement or answers the question. ) Assuming that the vertex of the triangle pictured to the right is at the origin, the point (x, y) would be referred to as r, in polar coordinates. (See Figure 9. L-01 (Cartesian and Polar coordinates ). 3D surface with polar coordinates¶ Demonstrates plotting a surface defined in polar coordinates. In particular, how the angle increases counter-clockwise and how the radius rincreases going away from the origin. Lengths in Polar CoordinatesAreas in Polar CoordinatesAreas of Region between two curvesWarning Lengths in Polar Coordinates Given a polar curve r = f( ), we can use the relationship between Cartesian coordinates and Polar coordinates to write parametric equations which describe the curve using the parameter x( ) = f( )cos y( ) = f( )sin. For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. In this work, ciordenadas the mathematics convention, the symbols for the radial, azimuthand zenith angle coordinates are taken as, andrespectively. Lecture Video and Notes Video Excerpts. The following steps can be used for graphing polar curves: 1. In particular, it means that a point [r. 1 De ning Polar Coordinates oT nd the coordinates of a point in the polar coordinate system, consider Figure 1. Find the polar form of the vector whose Cartesian form is. Like Cartesian coordinates, polar coordinates are used to identify the locations of points in the plane. PARAMETRIC EQUATIONS & POLAR COORDINATES. 686 CHAPTER 9 POLAR COORDINATES AND PLANE CURVES The simplest equation in polar coordinates has the form r= k, where kis a positive constant. In this section you will study a coordinate system called the polar coordinate system. If a curve is a rectangular coordinate graph of a function, it cannot have any loops since, for a given xvalue there can be at most one corresponding yvalue. 6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O;the rotating ray or half line from O with unit tick. Objective: To convert between polar and rectangular coordinates. The rst coordinate is the distance of the point from the origin (0;0), and the second coordinate is the angle, in standard. However, the Coriolis acceleration we are discussing here is a real acceleration and which is present when rand both change with time. A polar graph is a graph consists of many concentric lines passing out from the origin. Convert the following equation to polar coordinates: y = − 4 3 x 6. In this section we see that in some circumstances, polar coordinates can be more useful than rectangular coordinates. The ranges of the variables are 0 < p < °° 0 < < 27T-00 < Z < 00 A vector A in cylindrical coordinates can be written as (2. the given equation in polar coordinates. r2 = x2 + y2. The input values for x and y are read from the user using scanner object and these values are converted into corresponding polar coordinate values by following two equations. When we defined the double integral for a continuous function in rectangular coordinates—say, $$g$$ over a region $$R$$ in the $$xy$$-plane—we divided $$R$$ into subrectangles with sides parallel to the coordinate axes. To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas. The card10id is a special kind of limaçon. For example, the coordinates of [2, π] do not satisfy the equation. jpeg View Download: 967k: v. rectangular coordinates ⇒ polar coordinates polar coordinates ⇒ rectangular coordinates N=√ T2+ 2 U, 𝜃= P T= N K O𝜃 U= N O𝑖𝜃 The angle, θ, is measured from the polar axis to a line that passes through the point and the pole. Cylindrical Coordinates. Homework 2: Spherical Polar Coordinates Due Monday, January 27 Problem 1: Spherical Polar Coordinates Cartesian coordinates (x,y,z) and spherical polar coordinates (r,θ,ϕ) are related by x = r sinθ cosϕ y = r sinθ sinϕ z = r cosθ. =𝑟sin𝜃 Objective: As an admiral, you fire shots at your opponent’s armada until you have completely obliterated their naval force. 1 r =4secθ r =4secθ ⇒ r secθ =4 ⇒ 4cos(θ) ⇒ x =4 Thus,theCartesianequationisx =4. r=−2sinθ Identify the polar graph (line, circle, cardioid, limacon, rose): If a circle, name the center (in polar coordinates) and the radius. The equations of the 10 - and 20 - radius circles are r = 10 and r = 20, respectively. Polar Method: (@Distance >!. In the equation = 5ˇ 4, ris free, so we plot all of the points with polar representation r;5ˇ 4. 2 : Apr 24, 2019, 10:40 AM: Shawn Plassmann: Ċ: HPC - Reference Sheet - Polar Coordinates and Complex Numbers. 9) ( , ) 10) ( , ) Two points are specified using polar coordinates. Different microphones have different recording patterns depending on their purpose. The fact that a single point has many pairs of polar coordinates can cause complications. Angles are measured relative to the wind, and shown as "true wind angle" or TWA. The Jacobian determinant for this transformation is. There are approximately 20 problems on this. PHYS 419: Classical Mechanics Lecture Notes POLAR COORDINATES A vector in two dimensions can be written in Cartesian coordinates as r = xx^ +yy^ (1) where x^ and y^ are unit vectors in the direction of Cartesian axes and x and y are the components of the vector, see also the flgure. Jim's position at any time t is given by the parametric equations x = t, y = 2t. doc), PDF File (. (5, 960°) SOLUTION: Let P(r, θ) = (5, 960°). y2 4y 8x 20 0 y 2 2 4 2 x 3 2 23. ? $\endgroup$ – Will Jun 10 '15 at 20:41. For example, x, y and z are the parameters that define a vector r in Cartesian coordinates: r =ˆıx+ ˆy + ˆkz (1) Similarly a vector in cylindrical polar coordinates is described in terms of the parameters r, θ and z since a vector r can be written as r = rrˆ+ zˆk. You can find more information and examples about polar coordinates in this introduction to Polar Coordinates. " Polar coordinates come in quite handy here. In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. This printable worksheet of Polar Coordinates is tagged. 3 Graphing with polar coordinates We'll explain what it means to graph a function r= f( ) with an example. Many flows which involve rotation or radial motion are best described in Cylindrical Polar Coordinates. Convert the following equation to polar coordinates: y = − 4 3 x 6. For polar coordinates we have ∂f ∂θ = Du θ f. Find the polar form of the vector whose Cartesian form is. Convert to Polar Coordinates (1,1) Convert from rectangular coordinates to polar coordinates using the conversion formulas. Definition The polar coordinates of a point P ∈ R2 is the ordered pair (r,θ), with r > 0 and θ ∈ [0,2π). If we restrict rto be nonnegative, then = describes the. Cartesian/Polar Coordinates Junior high school The connection between Cartesian coordinates and Polar coordinates is established by basic trigonometry. Partition the domain θ of the polar coordinate ∆θ. This is the result of the conversion to polar coordinates in form. In many cases, it is convenient to represent the location of in an alternate set of coordinates, an example of which are the so-called polar coordinates. In a rectangular coordinate system, we were plotting points based on an ordered pair of (x, y). com by John Redden is licensed under a Creative Commons Attribution-ShareAlike 4. The point with rectangular coordinates (-1,0) has polar coordinates (1,pi) whereas the point with rectangular coordinates (3,-4) has polar coordinates (5,-0. Since r is equal to p x 2+ y, our ratio must be y/ p x 2+ y. This is a subtle point but you need to keep that in mind. Polar coordinates use r and , where represents the direction (as an angle) and r represents the distance in that direction. 7) Partition the domain x of the rectangular coordinate function into small pieces ∆x. Mon Nov 11 - I retaught graphing roses and then we began converting from polar form to rectangular and rectangular to polar. Polar Graph Paper Radians. Until now, we have worked in one coordinate system, the Cartesian coordinate system. Draw a horizontal line to the right to set up the polar axis. 3 Polar Coordinates 667 y x Polar axis and x-axis P = (r, u) = (x, y) Pole and origin x y r u Figure 6. We also know. tan y x θ = y r = sinθ 2 2 2 r x y = + Example 1: Convert the polar coordinate 2 2, 3 π to rectangular form. By converting to polar coordinates, or otherwise, evaluate the area of this loop. Polar coordinates have the negatives here that points going towards the center of the circle are compressed in smaller areas, and points going towards the edge of the circle are spread further apart. The Cartesian Coordinates of a point (x;y) and its polar coordinates (r; ) are related by the equations x= rcos , y= rsin. Cylindrical and spherical coordinates Recall that in the plane one can use polar coordinates rather than Cartesian coordinates. Polar-coordinate equations for lines A polar coordinate system in the plane is determined by a point P, called the pole, and a half-line known as the polar axis, shown extending from P to the right in Figure 1 below. 1 Polar Coordinates - PRACTICE TEST. 0 LUNAR COORDINATE SYSTEM SELECTION CRITERIA Members of the LDWG and PSWG worked to define a common Lunar Coordinate System for the LRO mission. the usual Cartesian coordinate system. The parametric arc length formula becomes Now and , so Square and add, using the fact that : Hence,. One way to do this is to use an angle and a distance r. Go Eagles! Go Eagles! Tutor Time Sign In. WEEK 15 ANALYTIC SECTIONS 10. The location of a point is expressed according to its distance from the pole and its angle from the polar axis. Math 232 Calculus III Brian Veitch Fall 2015 Northern Illinois University 10. This substitution would result in the Jacobian being multiplied by 1. Here, the two-dimensional Cartesian relations of Chapter 1 are re-cast in polar coordinates. 4 (Circular motion). We recall that the Dirichlet problem for for circular disk can be written in polar coordinates with 0 r R, ˇ ˇ as u= u rr+ 1 r u r+ 1 r2 u = 0 u(R; ) = f( ): 6. In polar coordinates, lines occur in two species. In this lesson we will learn: the graphical relationship between polar coordinates (r, θ) and Cartesian coordinates (x, y). Find the distance between the points. To view the value of θ. As shown in Figure 2(a), a point in polar coordinates can be represented by (ˆ, ), where ˆis the polar radius and is the polar angle of this point. Load the file antennaData. The coordinates of a point determine its location. 1 De ning Polar Coordinates oT nd the coordinates of a point in the polar coordinate system, consider Figure 1. A coordinate system is a scheme that allows us to identify any point in the plane or in three-dimensional space by a set of numbers. Definition of Polar Coordinates. pdf (Ken's lecture notes on polar coordinates, in pdf) WS_5_5_PolarCoordinates. The polar coordinates (r,θ) are related to the usual rectangular coordinates (x,y) by by x = r cos θ, y = r sin θ The figure below shows the standard polar triangle relating x, y, r and θ. In the Cartesian coordinate system, we move over (left-right) x units, and y units in the up-down direction to find our point. In the polar coordinate system, the ordered pair will now be (r, θ). pdf (Worksheet practicing this material, in pdf) WS_Soln_5. As in polar coordinates, the same point with cylindrical coordinates (ρ, φ, z) has infinitely many equivalent coordinates, namely (ρ, φ ± n×360°, z) and (−ρ, φ ± (2n + 1)×180°, z), where n is any. The axial coordinate or height z is the signed distance from the chosen plane to the point P. You can use absolute or relative polar coordinates (distance and angle) to locate points when creating objects. , the z coordinate is constant), then only the first two equations are used (as shown below). b) Show that the area of R is 1 (9 3 2) 16 − π. =𝑟sin𝜃 Objective: As an admiral, you fire shots at your opponent’s armada until you have completely obliterated their naval force. The formula for the area Aof a polar region Ris A= Z b a 1 2 [f( )]2 d = Z b a 1 2 r2 d : Caution: The fact that a single point has many representations in polar coordinates some-times makes it di cult to nd all the points of intersection of two polar curves. In certain problems, like those involving circles, it is easier to define the location of a point in terms of a distance and an angle. Definition of Polar Coordinates. 24 Polar and rectangular coordinate systems Check Point 2 Find another representation of in which a. Angle t is in the range [0 , 2Pi) or [0 , 360 degrees). Complete each of the following tables. Here is a graphic preview for all of the graphing worksheets. In spherical coordinates, we specify a point vector by giving the radial coordinate r, the distance from the origin to the point, the polar angle , the angle the radial vector makes with respect to the zaxis, and the. Then each point in the plane can be assigned polar coordinates as follows. First, fix an origin (called the pole) and an initial ray from O. Finally, the Coriolis acceleration 2r Ö. 3 Polar Coordinates 665 5 −5 Pole p P =()3, 4 d Figure 6. Polar coordinates use a distance and an angle to locate a point. Introduction to polar coordinates. the standard n-dimensional polar coordinates. For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. In the polar coordinate system, a circle centered at the origin with a radius a units has equation r = a Dead center has a radius of 1 meter. 2 Polar Coordinates 1) Rename the following point two different ways. If we restrict rto be nonnegative, then = describes the. The equations are easily deduced from the standard polar triangle. In this work, ciordenadas the mathematics convention, the symbols for the radial, azimuthand zenith angle coordinates are taken as, andrespectively. Zoom meeting at 12:30!. Step Four: Apply the Polar Coordinates Filter In the Layers panel, click on the top strip layer to activate it. Notice that this is just the reverse of the previous problem, included here to illustrate that care is needed to find the polar angle , especially when it's in the third quadrant. Pre-AP Pre-Calculus Name _____ Chapter 9 Polar Coordinates Study Guide Date _____ Period_____ 1. (b) The curve resembles an arch of the parabola 816yx 2. SYNOPSIS IntreatingtheHydrogenAtom'selectronquantumme-chanically, we normally convert the Hamiltonian from its Cartesian to its Spherical Polar form, since the problem is. (See Figure 9. I Double integrals in disk sections. 5 Polar Coordinates. That number is. In this way, both horizontal and oriented. Search Search. This curve is a linepassingthroughthe origin. To plot polar coordinates, set up the polar plane by drawing a dot labeled “O” on your graph at your point of origin. r = sin(3θ) ⇒ 22. In this section we see that in some circumstances, polar coordinates can be more useful than rectangular coordinates. Arc length and surface area of parametric equations. (it can be positive, negative, or zero. In the polar coordinate system, points are represented by ordered pairs of the form (r; ), where tells you the angle between the polar axis and the ray. Notice that this is just the reverse of the previous problem, included here to illustrate that care is needed to find the polar angle , especially when it's in the third quadrant. In spherical polar coordinates we describe a point (x;y;z) by giving the distance r from the origin, the angle anticlockwise from the xz plane, and the angle ˚from the z-axis. Polar Coordinates - Free download as PDF File (. y2 4y 8x 20 0 y 2 2 4 2 x 3 2 23. Intro to Polar Coordinates 1 , Intro to Polar. (See Figure 9. Let t be the number of seconds after the start of the race. Complete the unit circle with each angles' coordinates in the sets of parentheses as well as the simplified value of tangent at each angle. Definition The polar coordinates of a point P ∈ R2 is the ordered pair (r,θ), with r > 0 and θ ∈ [0,2π). Introduction to polar coordinates. Graph papers are being used in maths for a very long time by now. 5 Graphs of Polar Equations 937 x y <0 >0 x y 4 4 4 4 In r= 3 p 2, is free The graph of r= 3 p 2 3. Concentric Circles: 17 vs 13 Polar Radians. Parametric Equations and Polar Coordinates Topics: 1. You can select different variables to customize these graphing worksheets for your needs. Department of Mathematics - University of Houston. It is often convenient to use coordinate. Cylindrical coordinates are a simple extension of the two-dimensional polar coordinates to three dimensions. The fixed point is called the pole and the fixed line is called the polar axis. In order to fully grasp how to plot polar coordinates, you need to see what a polar coordinate plane looks like. theta# determines the direction. By converting to polar coordinates, or otherwise, evaluate the area of this loop. Its graph is the circle of radius k, centered at the pole. The origin is the vertex of the parabola. 9) ( , ) 10) ( , ) Two points are specified using polar coordinates. If you have to do several of these graphs at a time on blank graphs, use the Polar Circle Graph Paper or the Polar Plane Graph Paper. In this system coordinates for a point P are and , which are indicated in Fig. In a rectangular coordinate system, we were plotting points based on an ordered pair of (x, y). In this section, we will focus on the polar system and the graphs that are generated directly from polar coordinates. 3 Polar Coordinates 665 5 −5 Pole p P =()3, 4 d Figure 6. It has been accepted for inclusion in Chemistry Education Materials by an authorized administrator of [email protected] 24 Polar and rectangular coordinate systems Check Point 2 Find another representation of in which a. Many flows which involve rotation or radial motion are best described in Cylindrical Polar Coordinates. Example of finding the polar coordinates of a point Give the four basic polar coordinates of points A, B, C, and D shown in the figure. For example, the coordinates of [2, π] do not satisfy the equation. The course introduces the polar coordinate system and demonstrates how it can be used to describe locations in relation to the center of a circle. Practice solving problems involving coordinate conversions Example 1: Convert the rectangular coordinates (!7. In this polar coordinates worksheets, students change ordered pairs from rectangular form to polar form. Polar coordinates in the figure above: (3. The polar axis is usually horizontal and directed toward the right. Now all strip layers should be active. GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. In polar coordinates rectangles are clumsy to work with, and it is better to divide the region into wedges by using rays. In this section, we will focus on the polar system and the graphs that are generated directly from polar coordinates. For example in Lecture 15 we met spherical polar and cylindrical polar coordinates. 30 Coordinate Systems and Transformation azimuthal angle, is measured from the x-axis in the xy-plane; and z is the same as in the Cartesian system. Precalculus Examples. As in along with the polar paper the students will also get the radians inserted in it. We will derive formulas to convert between polar and Cartesian coordinate systems. (b) The curve resembles an arch of the parabola 816yx 2. This assignment offers a selection of questions on aspects of the STEP 3 polar coordinates specification and some other STEP 3 coordinate geometry questions. 6 Cylindrical and Spherical Coordinates A) Review on the Polar Coordinates The polar coordinate system consists of the origin O;the rotating ray or half line from O with unit tick. Plane Curvilinear Motion Polar Coordinates (r -θ) The particle is located by the radial distance r from a fixed point and by an angular measurement θto the radial line. Convert the following equation of a circle to polar coordinates: −5x2 +2x+ −5y2 +7y = 25 8. Getting Started To assist you, there is a worksheet associated with this lab that contains examples and even solutions to some of the exercises. Polar coordinates with polar axes. In polar coordinates we specify a point using the distance rfrom the origin and the angle with the x-axis. Now consider representing a region $$R$$ with polar coordinates. 31) Polar coordinates can be calculated from Cartesian coordinates like. A point P in the plane, has polar coordinates (r; ), where r is the distance of the point from the origin and is the angle that the ray jOPjmakes with the positive x-axis. Displacements in Curvilinear Coordinates. In the polar coordinate system, points are represented by ordered pairs of the form (r; ), where tells you the angle between the polar axis and the ray. Polar-coordinate equations for lines A polar coordinate system in the plane is determined by a point P, called the pole, and a half-line known as the polar axis, shown extending from P to the right in Figure 1 below. The polar coordinate system (r, θ) and the Cartesian system (x, y) are related by the following expressions: With reference to the two-dimensional equ ations or stress transformation. jpeg View Download: 967k: v. Counterclockwise is usually positive. Apr 27 - I was not able to post the entire week this time, but I should be updating soon. State three other pairs of polar coordinates for each point where —2m < 9 < 2m. Polar coord unit vectors and normal. View Test Prep - 10. In Cartesian coordinates this circle is a little harder to describe. Allows students to discover what polar coordinates are and how math and art can work together. 0 LUNAR COORDINATE SYSTEM SELECTION CRITERIA Members of the LDWG and PSWG worked to define a common Lunar Coordinate System for the LRO mission. Polar coordinates use a different kind of graph instead, based on circles: The center point of the graph (or "origin" in a rectangular grid) is the pole. I Double integrals in disk sections. r2 = x2 + y2. Find a formula for. is positive and b. The axial coordinate or height z is the signed distance from the chosen plane to the point P. PreCalculus. Lengths in Polar CoordinatesAreas in Polar CoordinatesAreas of Region between two curvesWarning Lengths in Polar Coordinates Given a polar curve r = f( ), we can use the relationship between Cartesian coordinates and Polar coordinates to write parametric equations which describe the curve using the parameter x( ) = f( )cos y( ) = f( )sin. Instead of using these rectangular coordinates, we can use a coordinate system to circular describe points on the plane: Polar Coordinates. Exploring Space Through Math. Recall from trigonometry that if x, y, r are real numbers and r 2 = x 2 + y 2, then there is a unique number θ with 0 ≤ θ < 2π such that. In polar coordinates, the point is located uniquely by specifying the distance of the point from the origin of a given coordinate system and the angle of the vector from the origin to the point from the positive -axis. It explains how to convert polar coordinates to rectangular coordinates and how to graph polar coordinates by. Introduction of Polar Coordinates. So let us first set us a diagram that will help us understand what we are talking about. The following steps can be used for graphing polar curves: 1. Write an equation for this curve in rectangular coordinates. Use double integrals in polar coordinates to calculate areas and volumes. The polar axis is usually horizontal and directed toward the right. You can find more information and examples about polar coordinates in this introduction to Polar Coordinates. units away from the last point entered. A particle of mass m is observed moving in a circular trajectory (8) x(t) ˘ 0 @ Rcos!t Rsin!t 0 1 A , where R,! are positive constants. TrigCheatSheet. But there is another way to specify the position of a point, and that is to use polar co-ordinates (r,θ). Let (r,θ) denote the polar coordinates describing the position of a particle. Spherical polar coordinates In spherical polar coordinates we describe a point (x;y;z) by giving the distance r from the origin, the angle anticlockwise from the xz plane, and the. Defining Polar Coordinates. In polar coordinates, if ais a constant, then r= arepresents a circle. is positive and b. You must know that x axis is always in the horizontal direction that is it goes from left to right and the y axis is in vertical direction. 5) I Review: Few curves in polar coordinates. theta# determines the direction. Polar coordinates in the figure above: (3. Cylindrical and spherical coordinates Recall that in the plane one can use polar coordinates rather than Cartesian coordinates. a 3, 5 3, 3 c, 4 9a 3b c, 11. Polar coordinates describe the distance from P to to a special point O, called the pole or origin and the angle that the line segment PO makes with a special ray called the polar axis. Introduction to polar coordinates. But many teachers might prefer that you measure angles by yourself using a protractor on blank paper. 1 Specifying points in spherical-polar coordinate s. Suppose that the polar coordinates of a point are Because is positive,we locate this point by drawing counter-clockwise from the polar axis. The variable rho is a measure of how intensely the antenna radiates for each value of theta. Polar Coordinates (r,θ) Polar Coordinates (r,θ) in the plane are described by r = distance from the origin and θ ∈ [0,2π) is the counter-clockwise angle. This is extremely frustrating to students who. Build rectangles on each piece of the domain. To find the polar angle t, you have to take into account the sings of x and y which gives you the quadrant. 5 Polar coordinates The two dimensional (planar) version of the the Cartesian coordinate system is the rectangular coordinate system and the two dimensional version of the spherical coordinate system is the polar coordinate system. 1 DEFINITION OF CYLINDRICAL COORDINATES A location in 3-space can be defined with (r, θ, z) where (r, θ) is a location in the xy plane defined in polar coordinates and z is the height in units over the location (r, θ)in the xy plane Example Exercise 11. In polar coordinates we specify a point using the distance rfrom the origin and the angle with the x-axis. Search this site. If you are looking for basic graph paper, then the Graph Paper Template is the resource you need. 4 The Reference 21 4. Arc length and surface area of parametric equations. We recall that the Dirichlet problem for for circular disk can be written in polar coordinates with 0 r R, ˇ ˇ as u= u rr+ 1 r u r+ 1 r2 u = 0 u(R; ) = f( ): 6. We use quotes around tan−1 to indicate it is not a single valued function. What we nd is that we are tracing out the line which contains the terminal side of = 5ˇ 4. 2 , 53 o) to rectangular coordinates to three decimal places. In this work, ciordenadas the mathematics convention, the symbols for the radial, azimuthand zenith angle coordinates are taken as, andrespectively. In polar coordinates rectangles are clumsy to work with, and it is better to divide the region into wedges by using rays. Multiplication and division of complex numbers in polar form. GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. 5 Test Review Polar. Lecture 19: Area between two curves; Polar coordinates Recall that our motivation to introduce the concept of a Riemann integral was to deflne (or to give a meaning to) the area of the region under the graph of a function. Precalculus: Polar Coordinates Practice Problems 3. In the equation = 5ˇ 4, ris free, so we plot all of the points with polar representation r;5ˇ 4. Math 232 Calculus III Brian Veitch Fall 2015 Northern Illinois University 10. Conversion between cylindrical and Cartesian coordinates. Lecture Video and Notes Video Excerpts. Article (PDF Available) For functions that are best described in terms of polar coordinates, the two-dimensional Fourier transform can be written in terms of polar coordinates as a combination. This is a free printable worksheet in PDF format and holds a printable version of the quiz Polar Coordinates. Spherical polar coordinates In spherical polar coordinates we describe a point (x;y;z) by giving the distance r from the origin, the angle anticlockwise from the xz plane, and the. Find the volume of the region bounded by the paraboloid z= 2 4x2 4y2 and the plane z= 0. There are three types of polar graph that are Large Single Polar Graph which has thirty marks for r in increment of five degrees, Smaller (Double) Polar Graph which has two polar graphs on one page, each with twenty scale marks for r increment of 5 degrees and Combined Cartesian and Polar has three pages here, One is a large cartesian grid, one a large polar grid and the third one has one. Polar Form of an Ellipse—C. Zoom meeting at 12:30!. Conversion: Rectangular to Polar/ Polar to Rectangular 2011 Rev by James, Apr 2011 1. Find the polar form of the vector whose Cartesian form is. For example, think of a circle of radius centred on the point. This article explains how to convert between polar and cartesian coordinates and also encourages the creation of some attractive curves from some relatively easy equations. 2 The naddplot Command: Coordinate Input. Graphing outside of the Cartesian plane is a unique challenge for many students, particularly the part where they need to convert (x, y) coordinates into polar form. Also, in order to deal with graphing a polar curve (ie plotting all points whose polar coordinates. As in polar coordinates, the same point with cylindrical coordinates (ρ, φ, z) has infinitely many equivalent coordinates, namely (ρ, φ ± n×360°, z) and (−ρ, φ ± (2n + 1)×180°, z), where n is any. In polar coordinates, every point is located around a central point, called the pole, and is named (r,nθ). Spherical polar coordinates are useful in cases where there is (approximate) spherical symmetry, in interactions or in boundary conditions (or in both).
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To enable students use algebra, graphs and tables to solve quadratic equations • To enable students form a quadratic equation to represent a given problem • To enable higher-level students form quadratic equations from their roots Prior Knowledge. 5 0 2 4 6 8 10 12 0 0. Create, edit, convert, annotate, protect, merge Quickly gather feedback from colleagues and customers. Put your answer in standard form. Flowchart of Pseudocode. For example, + − is a quadratic form in the variables x and y. Author: Timothy O. 1 Quadratic forms on the unit sphere In this section we deduce some properties of quadratic forms restricted to subsets of the unit sphere. A homogeneous polynomial. There is a reason for this. 8) Ax2 + By2 Cxy Dx Ey F = 0 If C = 0, then by completing the square in both x and y we are led to an equation which looks much like one of the standard forms, but with the center removed to a new point (x0; y0). Aug 29, 2020 quadratic and hermitian forms over rings grundlehren der mathematischen wissenschaften Posted By Agatha ChristiePublishing TEXT ID 386b361b Online PDF Ebook Epub Library. Student Activity Sheet Day 1_ Quadratic Functions for Projectile Motion. Maths-percentages composition, mathmatic equations, simplest radical form calculator, can ti 89 do quadratic formulas, quadratic cheat sheet. 1 Graph Quadratic Functions in Vertex Form A2. Central simple algebras over function flelds of quadratic forms 116 Chapter V. Quadratic Forms and Automorphic Forms Jonathan Hanke arXiv:1105. A quadratic form (V;q) is said to be isotropic if there is a nonzero v2V such that q(v) = 0. 1 Notes pages 48-51. Use the example below to guide your work. 2 + 2𝑎𝑎𝑏𝑏+ 𝑏𝑏. Steps to put quadratic function in standard form: 1. De nition 3. Posted on 28. For example, this quadratic. Quadratic formula activity pdf. definition and classification of quadratic forms. To graph a quadratic function, generate enough ordered pairs to see the shape of the parabola. f(x) = x 2 - 5x + 6. Write a function that computes the left-hand side of these two equations. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and we speak of a quadratic form over K. This line is the axis of symmetry. This is generally true when the roots, or answers, are not rational numbers. The BBC informs, educates and entertains - wherever you are, whatever your age. View Quadratic Forms Research Papers on Academia. Quadratic Forms and Canonical Forms. The (real) solutions of a quadratic equation are the real numbers x which satisfy the equation or make the statement true. Thus every quadratic form is diagonalizable. 2 + 𝑏𝑏𝑥𝑥+ 𝑐𝑐= 0 is called a quadratic equation, where 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 are real numbers and 𝑎𝑎 ≠0. This comes from the solution set of the KKT conditions having a particular structure, that can be exploited in a simplex-like fashion (see the seminal paper of Wolfe). Title: Write the quadratic equation, in vertex, form for each graph Author: Computer User Created Date: 1/25/2016 9:09:37 AM. Reliably create and distribute PDF forms. Distribution of Quadratic Forms 671 0 2 4 6 8 10 12 0 0. The general form of a quadratic polynomials is ax2 + bx + c, where a, b, c are real numbers such. Structure of the Witt ring 121 32. The value of a. 166 downloads 884 Views 2MB Size Report. 1) Identify the values of a, b, and c for the quadratic function in standard form y = -5x2 + 7x - 4 2) Why is the vertical line that passes through the vertex of a parabola call the axis of symmetry?. For example, 2x 2 + 3x + 2 = 0 is a quadratic equation while 3x + 2 is not a quadratic equation. key to algebra book 10 square roots and quadratic equations Aug 28, 2020 Posted By Barbara Cartland Library TEXT ID 75907704 Online PDF Ebook Epub Library 105 chapter viii 122 aug 17 2013 solving the book key to algebra book 10 square roots and quadratic equations it is extremely good to read there are a lot of folks that. We say that a2Z is a quadratic residue mod nif there exists b2Z such that a b2 mod n: If there is no such bwe say that ais a quadratic non-residue mod n. Linear & Quadratic Inequalities. Only use an electronic grapher to check your answers! Factored Form. A‐REI‐4b Solve quadratic equations by inspection (e. A quadratic form on V is a bilinear map Q: V V !R such that Qis. the first number of an ordered pair of numbers that. Witt's Chain Equivalence Theorem 15 38 §6. The class number, h(d), denotes the number of equivalence classes of binary quadratic forms of discriminant d. Factoring and Solving Quadratic Equations Worksheet Math Tutorial Lab Special Topic Example Problems Factor completely. Let us check the answers to our three examples in the "completing the square" section. 2020 by coqe. Basic Quadratic Forms is a great introduction to the theory of quadratic forms. Quadratic Form Theorem 4. To approach this question, we have studied the family of subsampled quadratic classifiers: linear classifiers augmented by the best k terms from the set of K = (d 2 + d)/2 second-order product terms available in d dimensions. Develops students' understanding of zeros and other key features from the factored form of a quadratic function ; Requires students to analyze and see the connection between quadratic functions represented graphically and algebraically; Requires students' use of precise course-appropriate mathematical language ; In the classroom:. Bilinear and quadratic forms under quadratic extensions 138 35. Then the quadratic form associated to A is the function QA defined by QA(�) := ��A� (� ∈ R�)� We have seen quadratic forms already, particularly in the context of positive-semidefinite matrices. Here we have provided you with a table showing examples of different forms of quadratic equations, such as vertex form and factor form. The Quadratic Formula (Quadratic formula in depth) Factoring (Factoring Method in depth) Completing the Square; Factor by Grouping; A quadratic equation is an equation that can be written as ax ² + bx + c where a ≠ 0. 3a Factor a quadratic expression to reveal the zeros of the function it defines. This is an intro to vertex form so rational "a"-values are not explored, nor is scaling. The goals today are: 1 Figure out how to graph a quadratic function written in standard form by hand, and 2 figure out how to write a given quadratic function written as f(x) = ax2 +bx +c in standard. They are listed in the order of Standard Form, Intercept Form, and Vertex Form. Plug in the coordinates for x and y into the general form. Topics include the theory of quadratic forms over local fields, forms with integral coefficients, genera and spinor genera, reduction theory for definite forms, and Gauss' composition. the process of writing a number or an algebraic expression as a product B. The optimal dual variable (Lagrange multiplier) for # a constraint is stored in constraint. Forms of Quadratic Functions – Factored Form Lesson 22 (C) Investigation – Investigating the Graphs of Quadratic Functions & Factored Form All of the quadratics you will graph are presented in the form of y = a(x – s)(x – t). Quadratic Formula. 2 - Quadratic Functions in Intercept & Standard Forms Subject: SMART Board Interactive Whiteboard Notes Keywords: Notes,Whiteboard,Whiteboard Page,Notebook software,Notebook,PDF,SMART,SMART Technologies ULC,SMART Board Interactive Whiteboard Created Date: 9/21/2018 5:53:33 PM. The "topograph " ofthe First Lecture makes the entire theory of binary quadratic forms so easy that we no longer need to think or prove theorems about these forms-justlook! In some sense. We say that a2Z is a quadratic residue mod nif there exists b2Z such that a b2 mod n: If there is no such bwe say that ais a quadratic non-residue mod n. a can't be 0. Author: Timothy O. Once the u sub is performed, you then solve it like a normal quadratic equation. Log InorSign Up. The total signature 133 34. To enable students use algebra, graphs and tables to solve quadratic equations • To enable students form a quadratic equation to represent a given problem • To enable higher-level students form quadratic equations from their roots Prior Knowledge. The graph of a quadratic function is called a parabola. 1 Notes pages 48-51. A few points near the vertex are: The graph is symmetrical about the line x = 0. 1 Reference Manual: Quadratic. Practice Worksheet: Graphing Quadratic Functions in Standard Form. 3) That is h = bb 2a and k = f(2a). But again, since 2 is a unit in K, it is purely a cosmetic matter. • A form is said to be primitive if gcd (a, b, c). • Quadratic norms. You may notice that the following examples of quadratic expressions each have a variable raised to the second. Shop Walmart. Graphing Quadratics from Standard Form Find the vertex, axis of symmetry, x-intercepts, y-intercept, value of the max/min, domain, and range of the following quadratics and then graph the parabola. Each section is given a point value and students must complete at least 30 points worth of questions so they can choose to do lots of easier questions or just a few difficult questions. Find two numbers whose product equals c and whose sum equals b. If a parabola is given in another form it must be converted to Standard Form. Post your questions for our community of 200+ million students and teachers. Any equation that equates a second order polynomial to zero is a quadratic equation. The graph of a quadratic function is a curve called a parabola. Given the quadratic function: = 2−8 +15 The equation for the axis of symmetry of a can be written in the form =ℎ. Use the example below to guide your work. - f(x;y;z) = x2 y2 z2 is a ternary quadratic form, whereas f(x;y;z) = x3 + xyz+ xyz2 is not a form at all. They are listed in the order of Standard Form, Intercept Form, and Vertex Form. Isotropy of quadratic forms and fleld invariants Detlev W. Answer to Fit a quadratic function of the form f(t) = C0 + C1t + C2t2 to the data points (0,1), (1, 2) Question: Fit A Quadratic Function Of The Form F(t) = C0 + C1t + C2t2 To The Data Points (0,1), (1. Improve your math knowledge with free questions in "Graph quadratic functions in vertex form" and thousands of other math skills. 6 is called a double root. Reliably create and distribute PDF forms. Actually, for reasons that will be clear shortly, it is prefer-able to put a factor 1 2 in front of the quadratic term, so that P(x. The technique finds broad use in operations research and is occasionally of use in statistical work. We have been working with quadratic equations in Vertex Form,. This calculator uses provided target function table data in form of points {x, f(x)} to build several regression models, namely, linear regression, quadratic regression, cubic regression, power. Note also that the cur. Free gamified quizzes for every subject to play in class and at home. CASSELS RATIONAL QUADRATIC FORMS PDF - Buy Rational Quadratic Forms (Dover Books on Mathematics) on FREE SHIPPING on qualified orders. In factored form, what seems to be the relationship between the vertex and the. 3 MATRICES AND QUADRATIC FORMS Best illustrated by a few examples. Here is a set of practice problems to accompany the Equations Reducible to Quadratic in Form section of the Solving Equations and Inequalities chapter of the notes for Paul Dawkins Algebra course at. Quadratic Forms. 4b Solve quadratic equations by inspection (e. A related technique can be found in [43]. Usually $R$ is the field $\mathbf C$, $\mathbf R$ or \mathbf Q. We have built a successful BI Practice by helping our clients align key performance. Search with the form below or the box above. is a quadratic form in the symmetric matrix ~ = (+) /, so the mean and variance expressions are the same, provided is replaced by ~ therein. quadratic-in-form. By a quadratic form, we understand a homogeneous quadratic polynomial in variables where the belong to a field or at least a commutative ring. Choose effi cient methods for solving quadratic equations. notebook October 05, 2015 To graph a quadratic, it is easier to convert from standard form to vertex form. 1 LEERGEDEELTE 3. Solve Equations in Quadratic Form. CHAPTER 9 QUADRATIC FORMS SECTION 9. We are now going to look at the Vertex Form of a Quadratic Expression. Quadratic programming (QP) is the problem of optimizing a quadratic objective function and is one of the simplests form of non-linear programming. In our example above, since the factored form of the function is y = (x - 1)(x + 3), the x-intercepts are x = 1 and x = -3. This operation is substantially more complicated [ citation needed ] than composition of forms, but arose first historically. Quadratic Functions Graphing quadratic functions Graphing quadratic inequalities Completing the square Solving quadratic equations -by taking square roots -by factoring -with the quadratic formula. The PDF format ensures that the recipient receives the document exactly as you saved it. Quadratic Forms and Automorphic Forms Jonathan Hanke arXiv:1105. 2 QUADRATIC EQUATIONS A quadratic equation in the variable x is an equation of the form ax 2 + bx + c = 0, where a, b, c are real number a ≠ 0. These take the general form: y = ax2 +bx+c. The graph of a quadratic function is called a parabola. You can see that a quadratic equation in the intercept form is synonymous with its factorized form. Notes: Quadratic Forms. To graph a quadratic function, generate enough ordered pairs to see the shape of the parabola. The class number, h(d), denotes the number of equivalence classes of binary quadratic forms of discriminant d. 2 Martin Kneser: Quadratic forms and arithmetic of algebraic groups 1955 - 1970 20 2. A quadratic is an equation in which the degree, or highest exponent, is a square. Which set of data is correct for the quadratic function: =−1( +2)( −4)? x-intercepts Axis of Symmetry. Such an NLP is called a Quadratic Programming (QP) problem. Value sets of quadratic forms. If a parabola is given in another form it must be converted to Standard Form. Quadratic form, inner product and norm on\mathbb{R}^3$2. Printable PDF version of. Example: Mul Given. It is a written summary of your academic qualifications, skill sets and previous work experience which you submit while applying for a job. A‐REI‐4b Solve quadratic equations by inspection (e. When you visit any web site, it may store or retrieve information on your browser, mostly in the form of cookies. Form contains no input fields? Select the 'Text' tool to type text and the 'Forms' tool for checkmarks and radio bullets. Lectures on Quadratic Fomrs By C. Write the equation in standard form. 1 2-4x x-2. 5759v3 [math. Method 3- Solving By Using The Quadratic Formula Step 1- get the values of a, b and c to use in the formula Solve x2 + 2x - 8 = 0 Solutions x = -4 or 2 ax2 + bx + c = 0 x2 + 2x - 8 = 0 Therefore a = 1, b = 2, c = -8 Step 2- substitute these values for a, b and c into the quadratic formula and go on to simplify and solve for x x = -b ± √(b2. Quadratic form. It has an infinite number of non-recurring decimals. QUADRATIC FORMS §The presence of in the quadratic form in Example 1(b) is due to the entries off the diagonal in the matrix A. The graph of a quadratic function is called a parabola. 0 = -x 2 + 2x + 3 = -(x 2 – 2x – 3). Quadratic forms with values in the real numbers. The examples given in the previous lesson were all given in Standard Form. This page will show you how to use the quadratic formula to get the two roots of a quadratic equation. Mathai and others published Quadratic Forms in Random Variables: Theory and Applications | Find, read and cite all the research you need on ResearchGate. 4 Quadratic Forms. The best quality PDF to Word conversion on the market - free and easy to use. Search phrases used on 2008-03-24: Students struggling with all kinds of algebra problems find out that our software is a life-saver. The degree also describes the number of possible solutions to the equation (therefore, the number of possible solutions for a quadratic is two). An integral form fof rank mis called n-regular if f represents all quadratic forms of rank nthat are represented. Search with the form below or the box above. a polynomial with three terms D. To see this recall that the exponent on the square root is one-half, then we can notice that the exponent on the first term is twice the exponent on the second term. Create, edit, convert, annotate, protect, merge Quickly gather feedback from colleagues and customers. Fractional terms. Previous / 361 / Quadratic Forms in Infinite Dimensional Vector Spaces. Derivative of a 2-form and divergence. Origin of quadratic form. Download: Adobe PDF, Microsoft Word (. Write the recursive equation for the patterns that are linear or quadratic. Quadratic Functions The graph of a quadratic function is a parabola. Examples: Q(x) = xTAx; with A = 2 4 3 2 2 7 3 5: We could rewrite this in the form Q(x) = 3x2 1 4x1x2 +7x 2 2. This operation is substantially more complicated [ citation needed ] than composition of forms, but arose first historically. With pieces of the equations missing, it requires students to be able to do some work to verify that a set of Equations matches a specific graph. Binary means the form has two variables, ternary means it has three, etc. 2 Symmetric bilinear forms and quadratic forms. Siegel Notes by K. Standard Form of a Quadratic Function ⃣Write an equation that describes how two things are related based on a real world context Vocabulary: standard form Definitions The Standard Form of a Quadratic Equation is y = Ax2 + Bx + C where A is not zero. 1 1 Graphing Quadratic Functions 1. Millions trust Grammarly's free writing app to make their online writing clear and effective. Note that every quadratic form is an integer multiple of a primitive quadratic form. Quadratic Equations. Tell whether each function is quadratic. Let us rst review the standard form of a QP (following CVXOPT The CVXOPT QP framework expects a problem of the above form, dened by the pa-rameters {P, q, G, h. An example of an equation in quadratic form would be x4 13 x2 36 0. Vertex Form of Quadratic Functions MathBitsNotebook. No watermarks or registration. Open, View, Edit, Save and Print PDF Documents without Adobe. 1 THE MATRIX OF A QUADRATIC FORM quadratic forms and their matrix notation Ifq=a 1 x 2 +a 2 y 2 +a 3 z 2 +a 4 xy+a 5 xz+a 6 yz then q is called a quadratic form (in variables x,y,z). Vertex Form of the Quadratic Function Any quadratic function (parabola) can be expressed in y = a(x - h)2 + k form. Hot Network Questions I was denied a flight from Alicante to. Binary Quadratic Forms Michael Drmota and Mariusz Skaªba 1 Abstract We study the number of divisors in residue classes modulo mand prove, for example, that there is an exact equidistribution if and only if m= 2kp 1p 2:::p s where kand s are non-negative integers and p j are distinct ermatF primes. Introduction to quadratic forms. Graphical analysis. Quadratic forms appear in di erential geometry, physics, economics, and statistics. If the matrix of ∇ is (a i,j) the formula for Q∇ is Q∇(x) = X i a i,ix. Derive the quadratic formula from this form. Symplectic bases for alternating bilinear forms are discussed in Section5. Pick an existing quiz or create your own for review, formative assessment, and more. import numpy import cvxopt. A constant function is a function of the form f(x) = b;. • Diagrams are NOT accurately drawn, unless otherwise indicated. Let A= " 5 1 1 2 # and x = " x 1 x 2 #. Given equation: (𝑥𝑥+ 1) 2 = 2(𝑥𝑥−3) Using the formula 𝑎𝑎( + 𝑏𝑏) 2 = 𝑎𝑎. finiteness theorems for quadratic forms (e. View Unit 1. Quadratic programming with interval coefficients developed to overcome cases in classic quadratic programming where the coefficient value is unknown and must be estimated. But again, since 2 is a unit in K, it is purely a cosmetic matter. Which set of data is correct for the quadratic function: =−1( +2)( −4)? x-intercepts Axis of Symmetry. Our PDF math worksheets are easy to print or download and free to use in your school or home. Later research focused on the general problem of determining the isomorphisms between classical groups. This is the most accessible form, and it provides a visual foundation for later work with standard or vertex form. are solutions of quadratic equations. If C 6 = 0, the situation is. NT] 27 Jun 2012 June 28, 2012. In general, for most students, quadratic equations create challenges in various ways such as diffi-culties in algebraic procedures, (particularly in factoring quadratic equations), and an inability. A quadratic form with integer coefficients buel called an integral binary quadratic formoften abbreviated to binary quadratic form. QUADRATIC FORMS AND DEFINITE MATRICES 3 1. To do this, we would perform the following steps:. 4x 2 7x 15 0 8. 1 Quadratic functions We rst looked at polynomials of simple form, of degree 1: f(x) = mx+ b:Now we move on to a more interesting case, polynomials of degree 2, the quadratics. attemptmg to compl ete the square x + using the k from the right hand Side. Mathai and others published Quadratic Forms in Random Variables: Theory and Applications | Find, read and cite all the research you need on ResearchGate. An integer mis represented by quadratic form fif. It should be included in the collection of any. Example 1 : Write the following quadratic function in factored form. VERTEX FORM OF A QUADRATIC FUNCTION f(x) = a(x – h)2 + k where h and k are real numbers and (h, k) is the vertex Example: 2Convert y = x + 12x + 32 into vertex form, and state the vertex. edu/math/documents/BiCoMathColloquiumJonathanHanke091514. January 24, 2020 I. Standard Form: f x ax bx c( )= ++2. Quadratic Formula noob quad form: quadform2003. y x The graph of a quadratic function is a parabola. 1 Quadratic forms on the unit sphere In this section we deduce some properties of quadratic forms restricted to subsets of the unit sphere. quadratic (not comparable) square-shaped (mathematics) of a polynomial, involving the second power (square) of a variable but no higher powers, as + +. 1 Determine whether a relationship is a function and identify independent and dependent variables, the domain, range, roots, asymptotes and any points of discontinuity of functions. Here we give several rational functions and the form of their partial fraction decompo-sitions. Elliptic Curves. PDF DOC TNS: Regents-Defining Functions 2 A2/B one-to-one & onto: 7/1: TST PDF DOC TNS: AI: Practice-Graphing Absolute Value Functions: 11: WS PDF: Practice-Graphing Quadratic Functions 1: 6: WS PDF: Practice-Graphing Quadratic Functions 2: 10: WS PDF: Practice-Graphing Quadratic Functions 3: 8: WS PDF: AII: Journal-Even and Odd Functions: 2. In doing this, the Derivative Calculator has to respect the order of operations. • Answer all questions. The addition of new quadratic forms that model a wider range of signal characteristics, and the extension to using intersections of. Standard Form. 5rs+25r 3s 15 14. Ernst Kani Queen's University. In general, for most students, quadratic equations create challenges in various ways such as diffi-culties in algebraic procedures, (particularly in factoring quadratic equations), and an inability. x 2 + y + y 2 = 1 and you were asked to sketch the set of points which satisfy this. In factored form, what seems to be the relationship between the vertex and the. If b or c is zero then these terms will not appear. Solving Using the Quadratic Formula Worksheet The Quadratic Formula: For quadratic equations: ax 2 bx c 0, a b b ac x 2 2 4 Solve each equation using the Quadratic Formula. Derive the quadratic formula from this form. PDF | In this work, we consider some properties of quadratic form F(x,y)=2x 2 +3xy+y 2. This information might be about you, your preferences or your device, and it can give you a. Remember y and f(x) represent the same. Khatri (1966. Pattern: b. t Use key characteristics of speci!c forms of quadratic functions to write equations. This thesis deals with quadratic programming (QP) algorithms for the use in fast model based predictive control applications. The names for the three forms are unimportant and may depend on the textbook, teacher or country. Reliably create and distribute PDF forms. Markscheme 4a + 2b = 20 a + b = 8 (A1) a – b = –4 (A1) (C2) Note: Award (A1)(A1) for any two of the given or equivalent equations. Quadratic Forms Overview. This work was begun in cooperation with R. Breaking news, sport, TV, radio and a whole lot more. This free and easy to use online tool allows to combine multiple PDF or images files into a single PDF document without having to install any software. De nition 3. Vertex Form of Quadratic Functions MathBitsNotebook. Roughly speaking, quadratic equations involve the square of the unknown. Binary means the form has two variables, ternary means it has three, etc. Assaf Naor §. Engineering Books Pdf have 305 Mathematics Pdf for Free Download. 1 Determine whether a relationship is a function and identify independent and dependent variables, the domain, range, roots, asymptotes and any points of discontinuity of functions. To graph a quadratic function, generate enough ordered pairs to see the shape of the parabola. As you make. We rst consider the use of quadratic forms in describing systems. With just a simple drag-and-drop, you can convert PDF to Word within seconds. Beyond that, he even de ned and calculated genus groups, which are essentially quotient groups, that explain which congruence classes of numbers can be represented by given sets of forms. Note also that the cur. Solve for x: 2x² + 9x − 5. While any equation with an x 2 {\displaystyle x^{2}} term in it qualifies as quadratic, the standard form sets everything to 0. To graph a quadratic function in standard form, follow these steps: Determine the axis of symmetry (this is also the x-coordinate of the vertex). 1968 edition. Our PDF math worksheets are easy to print or download and free to use in your school or home. integral quadratic form signed graph reduction algorithm roots and root systems radicals and their extensions Weyl group Coxeter matrix. Quadratic probing lies between the two in terms of cache performance and clustering. trinomial 4. denote the 2-dimensional hyperbolic quadratic form over k. Therefore, in this subsection, we discuss some results related to them. For a unitary U given as in (1), the outer product |xO xI. Lesson 9-1 Graphing Quadratic Functions 471 Graph Quadratic Functions The function describing the height of the rocket is an example of a quadratic function. Linear & Quadratic Inequalities. The proof of the theorem consists of showing that, if the in-equality fails to hold, we can find an equivalent form which reduces. Vertex Form of a Quadratic Function. ISBN 9781483256535, 9781483267678. It transforms it into a form that is better understandable by a computer, namely a tree (see figure below). Such a procedure has already been suggested by Nelson and Paris [10] (and applied to the case Fe L3/2). Describing a dynamical. The author is clearly an expert on the area as well as a masterful teacher. 6) In the classroom:. Many quadratic equations cannot be solved by factoring. This is the currently selected item. integral quadratic form signed graph reduction algorithm roots and root systems radicals and their extensions Weyl group Coxeter matrix. The number of tiles in each form quadratic sequences. 4x2 +16x 3. De nition 3. quadratic form （obviously, a real symmetric matrix） Def 4： Given quadratic form 1 2 Lf x x x n ( , , , ) =X T AX we say the rank of symmetric matrix A is the rank of quadratic form f. Title: Write the quadratic equation, in vertex, form for each graph Author: Computer User Created Date: 1/25/2016 9:09:37 AM. Write the equation in standard form. An integral form fof rank mis called n-regular if f represents all quadratic forms of rank nthat are represented. A second method of solving quadratic equations involves the use of the following formula: a, b, and. For example, 2x 2 + 3x + 2 = 0 is a quadratic equation while 3x + 2 is not a quadratic equation. It transforms it into a form that is better understandable by a computer, namely a tree (see figure below). We need to find the roots of the quadratic polynomial. The examples given in the previous lesson were all given in Standard Form. Elliptic curve. Quadratic Forms in Infinite Dimensional Vector Spaces. At quadratic we are passionate about making Business Intelligence easy. Support Form. On this page you can read or download quadratic equations in vertex form worksheet gina wilson 2012 answer key in PDF format. Linear & Quadratic Inequalities. Student Activity Sheet Day 1_ Quadratic Functions for Projectile Motion. Completely free. 5759v3 [math. Symplectic bases for alternating bilinear forms are discussed in Section5. Find two numbers whose product equals c and whose sum equals b. To see a parabola in the real world, throw a ball. 1A, Quadratic functions MATH 1410, (SOLUTIONS) 1. O'Meara, O. In both of the above formulas, the value of adetermines if the graph opens upward (a>0) or opens. 7a This is a (1) page PDF worksheet requires a student to convert a Quadratic Function in Standard Form to Vertex Form after identifying the vertex. Example B in your book solves a quadratic equation by “undoing”the order of operations. Kearton 135 Biography of Ernst Witt (1911{1991) Ina Kersten 155. Forms of Quadratic Functions Standard Form Factored Form Vertex Form y = ax22 + bx + c y = (x + a)(x + b) y = a(x - h) + k OR OR OR f(x) = ax22 + bx + c f(x) = (x - a)(x - b) f(x) = a(x - h) + k This form allows you to This form shows you the This form shows the quickly see the y-intercept. So try a solution of the form ˘(t) = A(t)ei!t: Plugging into the ODE, we nd that. 1 Notes pages 48-51. Quadratic form definition, a polynomial all of whose terms are of degree 2 in two or more variables, as 5x2 − 2xy + 3y2. CASSELS RATIONAL QUADRATIC FORMS PDF - Buy Rational Quadratic Forms (Dover Books on Mathematics) on FREE SHIPPING on qualified orders. QUADRATIC FORMULA Materials required for examination Items included with question papers Ruler graduated in centimetres and Nil millimetres, protractor, compasses, pen, HB pencil, eraser. In terms of this complex variable the equation of motion takes the form (exercise) ˘_ i!˘= F(t) m: You can easily see that, when F = 0, the solution is of the form Aei!t, where Ais a constant. Quadratic forms involving normal variables. The Quadratic Form. In the investigation, you approximated solutions to a quadratic equation using tables and graphs. Write a function that computes the left-hand side of these two equations. Siegel Notes by K. [2 marks]. Indeed, our work is motivated by the algorithmic problem of maximizing the quadratic form {u,v}∈E A(u, v)ϕ. of binary quadratic forms can be viewed as groups, at a time before group theory formally existed. • A form is said to be primitive if gcd (a, b, c). Flowchart of Pseudocode. This is the ‘completing the square’ form for a quadratic expression for which the coefficient of x2 is not 1. Learn more about how we can help at JotForm. 1) Identify the values of a, b, and c for the quadratic function in standard form y = -5x2 + 7x - 4 2) Why is the vertical line that passes through the vertex of a parabola call the axis of symmetry?. 1 2-4x x-2. The general form of the quadratic relationship is y = ax2 + bx + c where a, b and c are real numbers and a ! 0. Quadratic forms that are useful in describing a dynamical system. k Worksheet by Kuta Software LLC. Engineering Books Pdf have 305 Mathematics Pdf for Free Download. However, it is more common for quadratic equations to be given to us in Standard Form,. k=ℝk = \mathbb{R}. † Once in standard form, the vertex is given by (h;k). • Answer all questions. Some remarks are made about the group of equivalences of the quadratic forms. In fact, Brahmagupta (C. A quadratic equation is a polynomial of the form ax2 bx c, where a, b, and c are. A19-3 graphing quadraticsDocuments. Improve your math knowledge with free questions in "Graph quadratic functions in vertex form" and thousands of other math skills. 1 Example 1: a 2 2 quadratic form I Represent equation x2 +y2 = 1 in matrix form XTAX = 1 I Matrix A is a transformation matrix which represents the conic form of the equation. 4 QUADRATIC FORMS 1. The x-intercepts are -2 and 7 and the y-intercept is 6. Heading (click to edit). Quadratic Forms: Let V be a vector space over the field F. About This Quiz & Worksheet. Quadratics are written in three basic forms. It should be included in the collection of any quadratic forms enthusiast. Review A quadratic equation has the form a x 2 + b x + c = 0 with the coefficient a not equal to 0. 3 MATRICES AND QUADRATIC FORMS Best illustrated by a few examples. Construction of ˜2. over a commutative ring$ R \$ with an identity. A (0, 0); maximum C (0, 1); minimum B (0, 1); maximum D (0, 0); minimum ____ 2 Which of the quadratic functions has the narrowest graph? A y. 5 as the vector for his approach. • This fact is the foundation upon which many statistical tests rest. Factoring Quadratic Functions. WebLogic Server Infrastructure of the same version must be downloaded. Converting Quadratic Equations between Standard and Vertex Form Standard Form: y = ax2 + bx + c Vertex Form: y = a(x – h)2 + k Convert from Standard Form to Vertex Form: y = ax 2 + bx = c y = a(x – h) + k know a, b, c want a, h, k a = a = h Solve for y = k Substitute the values and rewrite. Quadratic forms involving normal variables. is a quadratic form in the symmetric matrix ~ = (+) /, so the mean and variance expressions are the same, provided is replaced by ~ therein. Formative is a web-app for classrooms that allows teachers to give live assignments to students, allowing instant teaching adjustments and long-term student growth tracking. ? 8 Ô Ö ; To write the program: Select: [PRGM] key, select New, type “QUAD” using letter keys, press [ENTER] (this. Convert the equations to the form. Our PDF math worksheets are easy to print or download and free to use in your school or home. The quadratic formula. Symmetric Matrices And Quadratic Forms. You can see that a quadratic equation in the intercept form is synonymous with its factorized form. It is useful to remember these results of expanding brackets: (x + a) 2 = x 2 + 2ax + a 2 (x - a) 2 = x 2 – 2ax + a 2 (x + a)(x – a) = x 2 – a 2. Furthermore, when the form given is a ‘reduced’ representative for its equivalence class, using any of the methods for reduction that have been proposed, the number a is the smallest natural number represented by the form. We developed an expression for the total parameter entropy, whose form shows that the capacity of an SQ classifier does. Pseudocode Example 7: Solve Quadratic Equation (Pseudocode If Else Example). The first four terms of a quadratic sequence are shown below Work out the next term. CLARK Contents 1. Quadratic programming (QP) is the problem of optimizing a quadratic objective function and is one of the simplests form of non-linear programming. quadratic form expressions. Symptoms of COVID-19 (PDF). Then the quadratic form associated to A is the function QA defined by QA(�) := ��A� (� ∈ R�)� We have seen quadratic forms already, particularly in the context of positive-semidefinite matrices. §In contrast, the quadratic form associated with the diagonal matrix A in Example 1(a) has no x1x2 cross-product term. Write the equation in standard form. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and we speak of a quadratic form over K. Its sign indicates the direction of the parabola. Witt's Chain Equivalence Theorem 15 38 §6. 2x3 +128y Solve the following. Choose effi cient methods for solving quadratic equations. x2 +4x 12 5. A y = 2 x 2 xy = 2x. The total signature 133 34. The width, direction, and vertex of the parabola can all be found from this. Here is the substitution. Siegel Notes by K. Example 1 : Write the following quadratic function in factored form. Open a Walmart Credit Card to Save Even More!. We don’t need to factor the quadratic equation because factoring is only a special case of finding the 2 real roots. t Analyze the different forms of quadratic functions. The examples given in the previous lesson were all given in Standard Form. these forms are derived. The graph of a quadratic function is called a parabola. CBSE Class 10 Maths Quadratic Equation Notes:-Download PDF Here. Roughly speaking, quadratic equations involve the square of the unknown. A quadratic form (V;q) is said to be universal if it represents every nonzero element of k. Notes: Quadratic Forms. Full text Full text is available as a scanned copy of the original print version. The way to. Here we have provided you with a table showing examples of different forms of quadratic equations, such as vertex form and factor form. e form is semi-de nite. Generalization of this notion to two variables is the quadratic form Q(x1;x2) = a11x 2 1 +a12x1x2 +a21x2x1 +a22x 2 2: Here each term has degree 2 (the sum of exponents is 2 for all summands). quadratic form on Q(A)r\Q(V) and then show this quadratic form is the quadratic form of some self-adjoint operator. pdf: File Size: 406 kb: File Type: pdf: Download File. Our discussion will be limited to integral quadratic forms (i. January 24, 2020 I. A quadratic form is not necessarily associated to a matrix. Practice the standard form of quadratic equations worksheets that consists of topics like converting quadratic equations to standard form and identifying the quadratic coefficients. To complete the square means to convert a quadratic to its standard form. New Approximation to Distribution of Positive RVs Applied to Gaussian Quadratic Forms Abstract: This letter introduces a new approach to the problem of approximating the probability density function (PDF) and the cumulative distribution function (CDF) of a positive random variable. x2 +y2 = (x;y) x y = (x;y) 1 0 0 1 x y = 1 (15) 10. Isotropy of quadratic forms and fleld invariants Detlev W. An example of a Quadratic Equation: Quadratic Equations make nice curves, like this one The Standard Form of a Quadratic Equation looks like this: a, b and c are known values. Factoring a quadratic polynomial can be frustrating, but the techniques. 6 is called a double root. 2 QUADRATIC EQUATIONS A quadratic equation in the variable x is an equation of the form ax 2 + bx + c = 0, where a, b, c are real number a ≠ 0. Exactness in R3 and conservation of energy. As noted above, this quadratic does not factor, so I can't solve the equation by factoring. Graphing Quadratics In Standard Form Worksheet Pdf 1 Math. On this page you can read or download quadratic equations in vertex form worksheet gina wilson 2012 answer key in PDF format. He gave the tower the equation of y = !!" (x - 45. Our PDF math worksheets are available on a broad range of topics including number sense, arithmetic. The graph of this relationship is a parabola. called the vertex form of a quadratic equation. Let us consider a simple example: min x;y 1 2 x2 + 3x+ 4y subject to x;y 0 x+ 3y 15 2x+ 5y 100 3x+ 4y 80 First, we rewrite the above in the given standard form:. Any second-degree, homogeneous polynomial. sesquilin-ear forms and quadratic forms and call indierently a quadratic form both functions q 1A quadratic form is an algebraic notion. We start with the former. section 10 in Analytic theory of modular forms (pdf). The following questions are about factored form: y=a (x - p)(x - q). Learn how to do anything with wikiHow, the world's most popular how-to website. Any strictly nondegenerate quadratic form of dimension twois one of these two types. To see this recall that the exponent on the square root is one-half, then we can notice that the exponent on the first term is twice the exponent on the second term. ISBN 9781483256535, 9781483267678. Equations of Quadratic Form An equation of the form au2 bu c 0 where u is an algebraic expression is called an equation is quadratic form. Discusses the distribution of quadratic forms involving normal random variables. ESSENTIAL IDEAS t The standard form of a quadratic function is written as f(x) 5 ax 2 1 bx 1 c, where a does not equal 0. New Approximation to Distribution of Positive RVs Applied to Gaussian Quadratic Forms Abstract: This letter introduces a new approach to the problem of approximating the probability density function (PDF) and the cumulative distribution function (CDF) of a positive random variable. Support Form. There are two forms of evaluation: supervised, which uses a ground truth class values for each sample. | 2021-04-19T21:11:54 | {
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https://lamotora.com.ec/rain-bird-mwsgw/4ed952-dividing-radicals-steps | Then divide by 3, 5, … Step 2. Key Steps: 1) Isolate the radical symbol on one side of the equation. Dividing Radical Expressions. When you have a root (square root for example) in the denominator of a fraction you can "remove" it multiplying and dividing the fraction for the same quantity. In particular, we will deal with the square root which is the consequence of having an exponent of {1 \over 2}. Dividing Radical Expressions. Vocabulary Refresher. The idea is to avoid an irrational number in the denominator. A common way of dividing the radical expression is to have the denominator that contain no radicals. Sometimes you will need to multiply multi-term expressions which contain only radicals. Students learn to divide radicals by dividing the numbers that are inside the radicals together. Start by dividing the number by the first prime number 2 and continue dividing by 2 until you get a decimal or remainder. Intro Simplify / Multiply Add / Subtract Conjugates / Dividing Rationalizing Higher Indices Et cetera. This is a situation for which vertical multiplication is … Conjugates & Dividing by Radicals. So the conjugate of (√3 − √2) is (√3 + √2). 2) Square both sides of the equation to eliminate the radical symbol For example, while you can think of as equivalent to since both the numerator and the denominator are square roots, notice that you … As you become more familiar with dividing and simplifying radical expressions, make sure you continue to pay attention to the roots of the radicals that you are dividing. The radicand refers to the number under the radical sign. The question requires us to divide 1 by (√3 − √2). If you don't know how to simplify radicals go to Simplifying Radical Expressions. We need to multiply top and bottom of the fraction by the conjugate of (√3 − √2). Students also learn that if there is a square root in the denominator of a fraction, the problem can be simplified by multiplying both the numerator and denominator by the square root that is in the denominator. An equation wherein the variable is contained inside a radical symbol or has a rational exponent. Improve your math knowledge with free questions in "Divide radical expressions" and thousands of other math skills. Consider: #3/sqrt2# you can remove the square root multiplying and dividing by #sqrt2#; #3/sqrt2*sqrt2/sqrt2# Example 1: Add or subtract to simplify radical expression: $2 \sqrt{12} + \sqrt{27}$ Solution: Step 1: Simplify radicals Dividing radical is based on rationalizing the denominator.Rationalizing is the process of starting with a fraction containing a radical in its denominator and determining fraction with no radical in its denominator. In the radical below, the radicand is the number '5'.. Refresher on an important rule involving dividing square roots: The rule explained below is a critical part of how we are going to divide square roots so make sure you take a second to brush up on this. Purplemath. Simplify radicals. Well, what if you are dealing with a quotient instead of a product? Combine like radicals. You can use the same ideas to help you figure out how to simplify and divide radical expressions. There is a rule for that, too. In this case, our minus becomes plus. Here are the steps required for Simplifying Radicals: Step 1: Find the prime factorization of the number inside the radical. Recall that the Product Raised to a Power Rule states that $\sqrt[x]{ab}=\sqrt[x]{a}\cdot \sqrt[x]{b}$. The steps in adding and subtracting Radical are: Step 1. The conjugate is easily found by reversing the sign in the middle of the radical expression. Numbers that are inside the radicals together first prime number 2 and continue dividing 2... Multiply top and bottom of the fraction by the first prime number 2 and continue dividing by 2 you! Multiply multi-term expressions which contain only radicals go to Simplifying radical expressions help you figure how... Particular, we will deal with the square root which is the consequence of having an exponent of { \over. + √2 ) of ( √3 + √2 ) ) Isolate the expression. To Simplifying radical expressions the fraction by the first prime number 2 and continue by... / dividing Rationalizing Higher Indices Et cetera Isolate the radical expression symbol or has a rational exponent product. The radical expression is to avoid an irrational number in the denominator radicals together found by reversing sign! The equation irrational number in the middle of the radical sign the is... What if you are dealing with a quotient instead of a product the radicand refers to the number under radical! 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Multiply multi-term expressions which contain only radicals a product Simplifying radical expressions multi-term expressions which contain only radicals 2.... | 2021-06-14T14:56:36 | {
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https://math.stackexchange.com/questions/1895485/can-someone-please-help-me-understand-implicit-differentiation | I have seen how people implicitly differentiate the equation $x^2 + y^2 = c$.
$$d/dx(x^2) + d/dx(y^2) = d/dx(c)$$
treating "$y$" as "$f(x)$" and using the chainrule we get
$$2x + 2y(y') = 0$$
and solving for $y'$
$$y'= -2x/2y$$
The problem is that I just don´t understand implicit differentiation, I do know the rules but they don´t make any sense to me. The fact that it is valid to differentiate both "$x$" and "$y$" on the same side of the equation is what´s bothering me and even if I see "$y$" as a function of "$x$" I just end up imagining
$$x^2 + (-x^2 + c) = c$$
which doesn´t help me. I also don´t know very much about partial derivatives but I´m willing to learn about them if that helps me understand implicit differentiation.
I really appreciate any thoughts or ideas. Thank you!
• Possible duplicate of Implicit differentiation – Git Gud Aug 17 '16 at 21:14
• Please format your question properly. Implicit differentiation is best understood by knowing what physics the Chain Rule is supposed to model. The following question should help. If you have a ball of ice, which melts so that the radius shrinks at the rate $r'(t) = m(t)$, at what rate does the volume shrink? – avs Aug 17 '16 at 21:15
• @GitGud: The answer provided there might be too technical for the current situation. – user66081 Aug 17 '16 at 21:20
• @user66081 "The" isn't right. There are several answers. Are you of the opinion that all of them are too technical? I did consider not closing as a duplicate, but since my vote is just one fifth of the closing contribution, I decided to vote to close. – Git Gud Aug 17 '16 at 21:22
• @Leo (cc. @GitGud): maybe you could comment if/why you find the answers to the previous question unhelpful. That would be of general interest. – user66081 Aug 17 '16 at 21:26
Maybe this helps. Take the function $f(x) = x^2$ and some other functions $g(x)$ and $h(x)$.
Let's differentiate the expression $$f(x) + (g(x))^2 = h(x)$$ with respect to $x$ (not: "differentiate $x$").
We use the fact that you can differentiate each summand individually, and the chain rule for $(g(x))^2$, to get $$f'(x) + 2 g(x) g'(x) = h'(x) .$$
In your case, $f(x) = x^2$, $g(x) = y(x)$, and $h(x) = c$, i.e.: $$2 x + 2 y(x) y'(x) = 0 .$$
Sometimes, the argument $(x)$ is omitted -- that's what you have there; but that's just notation (no deeper meaning, afaik).
• Now I get. Thank you so much! That was actually a really good explanation, at least for me. – Leo Aug 17 '16 at 21:26
• @Leo: Ok. Feel free to accept one of the answers. I think even better would be if you could write your own answer and explain the difficulties. – user66081 Aug 17 '16 at 21:29
We are just using the chain rule
If I said:
$f(x) = (2x +1) \implies \frac {df}{dx} = 2\\ g(x) = x^2\implies \frac {dg}{dx} = 2x\\ \frac {d}{dx}(g\circ f)(x) = \frac {dg}{df}\frac {df}{dx} = 2(2x+1)(2)$
Do have a problem with that?
So, $y$ is a function of $x.$ $y^2$ is a function of $y$
$\frac {d}{dx} y^2 = \frac {d}{dy}(y^2)\frac {dy}{dx}=2y\frac {dy}{dx}$
And now for the last little bit:
$\frac {d}{dx} (y^2 = c - x^2) \implies 2y\frac {dy}{dx} = -2x$
Is that so different from
$\frac {d}{dx} (x^2 + y^2 = c)\\ \frac {d}{dx} (x^2) + \frac {d}{dx} (y^2) = \frac {d}{dx} c \\ 2x + 2y\frac {dy}{dx} = 0\\ \frac{dy}{dx} = -\frac {x}{y}$
Does this help?
• Please, please, please do not write $dg/df$. – Ted Shifrin Aug 17 '16 at 21:36
• @TedShifrin why not? – Doug M Aug 17 '16 at 21:36
• Because $g$ is not a function of the variable $f$. – Ted Shifrin Aug 18 '16 at 1:15
• In the example above? Sure it is. It is a composition. g(f(x)) – Doug M Aug 18 '16 at 1:20
• No, that's not what function means. You mean $\dfrac{dg}{dx}(f(x))$, since you gave $g$ as a function of $x$ to start with. It would be far clearer to give $g(u)=u^2$ and write $\dfrac{dg}{du}\big|_{u=f(x)}$. – Ted Shifrin Aug 18 '16 at 1:24
When you differentiate, you differentiate with respect to a variable. As you remember, a derivative $dx/dt$ essentially means how much does x change when t is changed by an extremely small amount.
Recall the chain rule, which is the rule for differentiating composite functions: $$\frac{d}{dx}f(g(x)) = f'(g(x))g'(x)$$
In this case, you're differentiating with respect to $x$.
So for the $x^2$ term you can view this as the composition of these two functions $f(x) = x^2$ and $g(x) = x$. Applying the chain rule, you get $$\frac{d}{dx}x^2 = \frac{d}{dx}x^2(\frac{d}{dx}x) = 2x$$
Now we get to the $y^2$ term. For this, your functions are $f(x) = x^2$ and $g(x) = y$. Again, applying the chain rule gives $$\frac{d}{dx}y^2 = \frac{d}{dx}x^2(\frac{d}{dx}y) = 2x(y')$$
We have no information on the relationship between y and x, so all we can write is $\frac{dy}{dx}$ or $y'$ for short. | 2019-07-20T16:04:54 | {
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Write a numpy Monte Carlo simulation to estimate the expectation of X and also estimate the probability p that no women sit at table A. See screenshots, read the latest customer reviews, and compare ratings for Monte Carlo Pro. 3, Measurable Outcome 3. pdf), Text File (. It's really more of a simulation program that uses randomness as part of the whole experimental scenario. If you assume a uniform distribution of birthdays, the birthday-matching problem can be solved exactly. Tutorial on Monte Carlo 1 Monte Carlo: a tutorial Art B. Thus one can simulate games of one player, then randomly match pairs of games (or more if you are considering more players) and take the minimum. Sabri Pllana. MATLAB: Monte Carlo Simulation: five dice rolling. Monty-Hall Monte Carlo Simulation. This example shows how to perform Bayesian inference on a linear regression model using a Hamiltonian Monte Carlo (HMC) sampler. Monte Carlo Simulation is a technique used to provide a better degree of certainty on the probability of outcomes in financial, project management, cost, and other forecasting models. Monte-Carlo Methods in Practice Pseudo-Random Number Generation Anyone who attempts to generate random numbers by deterministic means is, of course, living in a state of sin. Monte Carlo method: Pouring out a box of coins on a table, and then computing the ratio of coins that land heads versus tails is a Monte Carlo method of determining the behavior of repeated coin tosses, but it is not a simulation. If you are programming this from scratch, rather than using software that already knows something about Monte Carlo, then your first step is to write functions for sampling probability distributions. This is true for both transient and steady-state studies. They can be used to model experimental or historical data in order to generate prediction estimates or analyze a large number of outcomes such as in Monte Carlo simulations. Owen probability that random variable X takes value x it is probably already in Matlab or R or python or. It then calculates results over and over, each time using a different set of random values from the probability functions. There is a solution for doing this using the Markov Chain Monte Carlo (MCMC). to test the null hypothesis using the dataset (for example, test that the mean = 70). Prakyathkantharaju/ Monte-Carlo-simulation. 3 Sampling from Probability Distribution Functions We now have all the basics necessary to build a simple Monte Carlo radiation transfer code { we emit a photon, send it a distance L, then either absorb (terminate) it or scatter it into a new direction. The first application to option pricing was by Phelim Boyle in 1977 (for European options ). It is a technique used to. Monte Carlo simulations model complex systems by generating random numbers. First, save a function on the MATLAB® path that returns the multivariate normal log probability density and its gradient. to save the results of the test (for example, “reject” or “fail to reject”). Markov Chain Monte Carlo. Monte Carlo Simulation of Conditional Mean Models What Is Monte Carlo Simulation? Monte Carlo simulation is the process of generating independent, random draws from a specified probabilistic model. pdf), Text File (. Create MCMC chains using a Hamiltonian Monte Carlo (HMC) sampler and compute MCMC diagnostics. Discrete Probability. Video created by University of Colorado Boulder for the course "Excel/VBA for Creative Problem Solving, Part 3 (Projects)". The ray-tracing base model is utilized. Although this might sound somewhat specific and not very promising, Monte Carlo methods are fundamental tools in many areas of modern science (ranging all the way from theoretical physics to political science). For a simple test we might look at the probability that a number u from the distribution is less than say 0. Botev (ISBN: 9780470177938) from Amazon's Book Store. Classical Monte Carlo: samples are drawn from a probability distribution, often the classical Boltzmann distribution, to obtain thermodynamic properties or minimum-energy structures; Quantum Monte Carlo: random walks are used to compute quantum-mechanical energies and wave functions, often to solve electronic structure problems, using. The problem-oriented approach makes it ideal for an applied course in basic probability and for a more specialized course in Monte Carlo methods. Monte Carlo simulation. Read Online Now eupdf an eulerian based monte carlo probability density function pdf solver users manual book Ebook PDF at our Library. You can't be so infinitely vague or else no one. Monte Carlo simulation (MCS) is a technique that incorporates the variability in PK among potential patients (between-patient variability) when predicting antibiotic exposures, and allows calculation of the probability for obtaining a critical target exposure that drives a specific microbiological effect for the range of possible MIC values [45, 46, 79-86]. Davidian, Spring 2005 Rationale: In statistics †. Series Explaines Monte Carlo Methods from Beggining "How to Generate a Random Number" to "Sampling Distributions in MATLAB", The Series contains 23 Video of 10 Minute Each, and includ the. How to perform a Monte Carlo simulation?. Monte-Carlo methods generally follow the following steps: 1. Monte Carlo simulations test your outcomes over a wide combination of possible market returns, and they typically deliver an answer in terms of your probability of success. The historical annualized return and standard deviation of the S&P 500 from 1871-2008 were used as inputs to this Monte Carlo simulation. You can't be so infinitely vague or else no one. The ray-tracing base model is utilized. Markov chain Monte Carlo (MCMC) is a technique for estimating by simulation the expectation of a statistic in a complex model. Performing Monte Carlo Sampling. However, the origin of Monte Carlo methods is older than the casino. Problem was first stated in 1777 by Georges-Louis Leclerc, comte de Buffon 2. and Beshir, M. The mathematical level of the course is somewhere between [RK] and [Liu]; [KW] is a good source for some of the topics we'll discuss. This app allows the user to graphically select blocks (such as gains and subsystems) to design a Monte Carlo simulation. Both MCMC and crude Monte Carlo techniques work as the long-run proportion of simulations that are equal to a given outcome will be equal* to the modelled probability of that outcome. As with the bestselling first edition, Computational Statistics Handbook with MATLAB, Second Edition covers some of the most commonly used contemporary techniques in computational statistics. Sampling and Monte Carlo Simulation It then returns to the topic of using probability and statistics to derive information from samples. What if you could take the monte carlo simulation and put it on steroids to in effect, have more combinations and what…. Parallel computing and Monte Carlo algorithms by Jeffrey S. This allows for the possibility of active. MATLAB Central Hi I am trying to formulate a Monte carlo simulation for corrosion initiation time and at the same time use Latin Hypercube. Download this app from Microsoft Store for Windows 10, Windows 10 Mobile, Windows 10 Team (Surface Hub), HoloLens, Xbox One. ) (Introduction to Monte Carlo simulation). I've never taken a probability course in my life and am having some heavy. pdf), Text File (. The Monte Carlo method has wide application in various scienti c areas such as nance, t-elecommunications and computer graphics [1]. • Monte-Carlo Integration – Probabilities and Variance – Analysis of Monte-Carlo Integration • Sampling • Advanced Monte-Carlo Rendering Monte-Carlo Computation of π • Take a random point (x,y) in unit square • Test if it is inside the ¼ disc –Is x 2 + y2 < 1? • Probability of being inside disc? – area of ¼ unit circle /. monte carlo simulation of Matlab implementation of the algorithm. In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features. 1118632168 Series Probability and Statistics)】 (Wiley and Method in Method Monte 【送料無料】【Simulation and Carlo the,カタログギフト Dolce【30800円コース】ルビーノ【送料無料】【出産祝い・内祝い】【メッセージカード1円】,【海外限定】ピュア タンクトップ レディース ルームウェア 【 PURE ESSENCE TANK GOWN 】. Note: The name Monte Carlo simulation comes from the computer simulations performed during the 1930s and 1940s to estimate the probability that the chain reaction needed for an atom bomb to detonate would work successfully. This MATLAB function generates a Markov chain by drawing samples using the Hamiltonian Monte Carlo sampler smp. To get around this problem, one can use quasi-random low-discrepancy sequences which are designed to fill a parameter space efficiently. In this example, that function is called normalDistGrad and is defined at the end of the example. This historical distribution overlays a Monte-Carlo simulated distribution of annualized returns over 30-year periods using a lognormal return distribution model. then the probability that. pptx from BUSA 542 at Texas A&M University, –Commerce. Carlo Simulation in Matlab that explains about. We argue that Monte Carlo algorithms are ideally suited to parallel computing, and that “parallel Monte Carlo” should be more widely used. 1 with a constant speed. Probability distributions are mathematical models that assign probability to a random variable. In future articles we will consider Metropolis-Hastings, the Gibbs Sampler, Hamiltonian MCMC and the No-U-Turn Sampler. 1 8 PAM – BER/SER Monte Carlo Simulation Matlab exercise. I am doing this for multiple monte carlo estimators (using different probability densitys). The phrase “Monte Carlo methods” was coined in the beginning of the 20th century, and refers to the famous casino in Monaco1—a place where random samples indeed play an important role. To perform Monte Carlo simulation of regression models with ARIMA errors:. Can anyone help me how. Instead of using point estimates to say we will have 4 loss events over the next year, and each one will cost us $300,000, we define ranges for these inputs and let the Monte Carlo simulation identify tens of thousands of possible outcomes. and Lopes, H. Monte Carlo Analysis runs thousands of scenarios and gives you the probability of a certain event occurring or not occurring. We suppose that for any given value of x, the probability density function f(x) can be computed, and further that. The comparison between numerical simulations and Monte Carlo simulations confirms the accuracy of our model. The number of times the event occurs divided by the number of times the conditions are generated should be approximately equal to P. The algorithm, however, is very complicated, and the output does not appear to follow a predictable pattern. In this example, that function is called normalDistGrad and is defined at the end of the example. Both MCMC and crude Monte Carlo techniques work as the long-run proportion of simulations that are equal to a given outcome will be equal* to the modelled probability of that outcome. In realist models, this probability is very hard to estimate, because exact simple analytical formulas are not available. Through the Monte Carlo simulation, 10. Diffusion via Monte Carlo Lab 13 Physics 430 d) What is the average distance of the walkers from the origin? How would you calculate that? e) What do your plots tell you about the average distance of the walkers from the origin as a function of step. Imagine that I have done it and move to the exciting part ;) Monte Carlo simulation. Evaluate the area of a circle of radius$1= \pi$using Monte Carlo method. Wilesco D18 Steam Engine with Dynamo (type 2) £ 399. Construct the graphical histogram of these results, which should confirm that the distribution is uniform (0,1). (a) What is the expected value and variance of the simple Monte Carlo estimate of I ? I wrote the following to calculate the Monte-Carlo approximation to the integral:. 60163 Note that the mean for a Bernoulli trial is p (0. 0For0example,0because0of0. Monte Carlo methods are mainly used in three distinct problem classes: optimization, numerical integration, and generating draws from probability distributions. Monte Carlo eXtreme (MCX) MCX is a Monte Carlo simulation software for static or time-resolved photon transport in 3D media. We are presumably trying to analyze a very large set X, but we cannot do so in a simple closed form. uk: Kindle Store. MATLAB is used for financial modeling, weather forecasting, operations analysis, and many other applications. We find that Bayesian Monte Carlo outperformed Annealed Importance Sampling, although for very high dimensional problems or problems with massive multimodality BMC may be less adequate. Monte Carlo Simulation/Markov Chains. Use Statgraphics 18 run Monte Carlo Simulations on this page. ABSTRACTIn this article, we introduce a new form of distribution whose components have the Poisson or Skellam marginal distributions. At any time instant, all the radioactive nuclides remaining the sample have equal decay probability. Discrete Probability. Learn more about monte, carlo, simulation, pdf, probability, density, function. NET, Visual Basic, Java, and MATLAB with uncertain variables and functions, running Monte Carlo simulations, and collecting statistics from the Monte Carlo samples. Learn more about monte carlo, distribution, probability. Typically, the First Order Reliability Method (FORM) is used in RBO for failure probability calculation and this is accurate enough for most practical cases. Monte Carlo method, also known as the computer random simulation method, which is based on results of a large number of events to some certainty calculation. Early History of Probability Theory and Monte Carlo Methods Early History of Probability Theory Early History of Monte Carlo: Before Los Alamos I Buffon Needle Problem: Early Monte Carlo (experimental mathematics) 1. Quantum computing is a paradigm shift in the world of cryptology, as well as major impact across many industries. Monte Carlo method: Pouring out a box of coins on a table, and then computing the ratio of coins that land heads versus tails is a Monte Carlo method of determining the behavior of repeated coin tosses, but it is not a simulation. Evaluate the area of a circle of radius$1= \pi$using Monte Carlo method. This may seem surprising at first, but the intuition is rather straight forward. Description. If you need to answer professionally, you can use the so called Monte Carlo method. Financial time series modelling. See screenshots, read the latest customer reviews, and compare ratings for Monte Carlo Pro. , testing whether the portfolio can sustain the planned withdrawals required for retirement or by an endowment fund. x=rand(m,n); To generate an U(a,b) uniform. Monte Carlo Simulation in matlab for finding mean and variance left or right with a probability of 0. at Los Alamos in the 1940s. The city of Monte Carlo in the country of Monaco has long served as a playground for the jet set, where rich gamblers who can afford to lose huge sums of money come to play for big stakes in games. Then the system EE is investigated and the influence of the primary system parameters on the system performance. Introduction to Monte Carlo Algorithms (1996) - Krauth - Download as PDF File (. As far as I know, market movements do not resemble any known distribution. Get MATLAB; Search File Exchange. 18 MB, 54 pages and we collected some download links, you can download this pdf book for free. Performing Monte Carlo Sampling. Monte Carlo Simulation- Simulating Sum of a DICE. Cody is a MATLAB problem-solving game that challenges you to expand your knowledge. Monte Carlo technique for normally distributed Learn more about monte carlo, normal distribution, randn MATLAB. Monte Carlo Analysis: Steps. Monte Carlo Method = a computer simulation that performs Monto Carlo experiments aimed to compute the above probability We will illustrate the Monto Carlo Method with a simple experiment to find Pi. Discrete Probability. 1118632168 Series Probability and Statistics)】 (Wiley and Method in Method Monte 【送料無料】【Simulation and Carlo the,カタログギフト Dolce【30800円コース】ルビーノ【送料無料】【出産祝い・内祝い】【メッセージカード1円】,【海外限定】ピュア タンクトップ レディース ルームウェア 【 PURE ESSENCE TANK GOWN 】. I want to estimate the value of pi using the Monte Carlo method, this is, A random number. Markov chain Monte Carlo (MCMC) is a technique for estimating by simulation the expectation of a statistic in a complex model. In this video I go through how to use MATLAB to simulate the Buffon Needle Problem in a Monte Carlo Simulation. But how can this be? There are 365 days in a year and half of that would be 182, so why wouldn’t you need at. The "Monte Carlo Method" is a method of solving problems using statistics. This tutorial shows a method for running a Monte Carlo simulation in the LibreOffice Calc spreadsheet package. Monte Carlo estimation refers to simulating hypothetical draws from a probability distribution, in order to calculate significant quantities of that distribution. Kroese: Amazon. Monte Carlo simulations are often used when the problem at hand has a probabilistic component. You only need to open an Excel file, model activity risks on cost and schedule, and produce two charts: If you…. Sampling From Probability Density Functions (PDFs) At the base of a Monte Carlo simulation are the PDFs, functions that define the range of possibilities and the relative probability of those possibilities for a given step in the simulation. You can use WealthTrace, the most accurate Monte Carlo analysis for consumers on the market, to calculate your probability of never running out of money. lookback options, asian options and spread options) or options where the payoff is dependent on a basket of underlying assets (rather than just a single asset). Through the Monte Carlo simulation, 10. Rubinstein Technion, Israel Institute of Technology Keywords Monte Carlo, simulation, MCMC, estimation, optimization Abstract Many quantitative problems in science, engineering, and economics are nowadays solved via statistical sampling. Monte Carlo Simulation - Download as Powerpoint Presentation (. The monte carlo simulation period will be 30 years and the number of experiments will be user defined -k. QRM with linear and quadratic programming. Take, for example, the abstract to the Markov Chain Monte Carlo article in the Encyclopedia of Biostatistics. Carlo Simulation in Matlab that explains about. The sample mean (or first moment) is the probability weighted result of the random process: x =1 n Σ i=0 n fi in MATLAB:-->m1 = sum(p)/100000 m1 = 0. Monte Carlo methods are numerical techniques which rely on random sampling to approximate their results. We find that Bayesian Monte Carlo outperformed Annealed Importance Sampling, although for very high dimensional problems or problems with massive multimodality BMC may be less adequate. Tutorial on Monte Carlo Techniques Gabriel A. Because they are so important, they shouldn't be buried into a very long lesson on Monte Carlo methods, but we will use them in the next coming chapters and thus, they need to be introduced at this point in the lesson. Instead of using point estimates to say we will have 4 loss events over the next year, and each one will cost us$300,000, we define ranges for these inputs and let the Monte Carlo simulation identify tens of thousands of possible outcomes. To get around this problem, one can use quasi-random low-discrepancy sequences which are designed to fill a parameter space efficiently. 2) is impractical. Besides being easily one of the most important laws of statistics, this is the basis for Monte Carlo simulations and allows us to build a stochastic model by the method of statistical trials. uk: Kindle Store. Handbook of Monte Carlo Methods is an excellent reference for applied statisticians and practitioners working in the fields of engineering and finance who use or would like to learn how to use Monte Carlo in their research. Monte Carlo simulation is a powerful statistical analysis. Learn more about monte carlo How to store outputs from Monte Carlo simulation and then calculate probability distribution function? I am beginner with MATLAB. 1 Introduction to Monte Carlo Simulaion Monte Carlo Option Price is a method often used in Mathematical - nance to calculate the value of an option with multiple sources of uncertain-ties and random features, such as changing interest rates, stock prices or exchange rates, etc. Take, for example, the abstract to the Markov Chain Monte Carlo article in the Encyclopedia of Biostatistics. Monte Carlo -- a bit of history •Credit for inventing the Monte Carlo method often goes to Stanislaw Ulam, a Polish born mathematician who worked for John von Neumann on the United States Manhattan Project during World War II. Let us take a look at how this works with a simple Monte Carlo simulation. The algorithm, however, is very complicated, and the output does not appear to follow a predictable pattern. The higher probability is 15,000, but it can be somewhere between. Monte Carlo Monte Carlo is a computational technique based on constructing a random process for a problem and carrying out a NUMERICAL EXPERIMENT by N-fold sampling from a random sequence of numbers with a PRESCRIBED probability distribution. The Solver SDK Platform includes complete facilities for creating models in C++, C#, VB. The monte carlo simulation period will be 30 years and the number of experiments will be user defined -k. Use the random number generator to make a bunch of random numbers and use those in a loop where, inside your loop, you do your experiment. to test the null hypothesis using the dataset (for example, test that the mean = 70). This technique for calculate VaR is very flexible, and can be extended to any other returns probability distribution. Each step of the analysis will be described in detail. In this particular case, there are 7776 combinations of dice rolls. Monte Carlo simulation is named after the city of Monte Carlo in Monaco, which is famous for gambling such s roulette, dice, and slot machines. Computational Statistics with Matlab Mark Steyvers May 13, 2011 2 Markov Chain Monte Carlo 15 Examples of Matlab functions for evaluating probability density. Monte Carlo simulation in Companion by Minitab can help you foresee the variety of outcomes likely to occur when you alter different variables. Main Menu Monte carlo steps 1 identify the areas. Monte Carlo Approximation for Optimization. Rubinstein Technion, Israel Institute of Technology Keywords Monte Carlo, simulation, MCMC, estimation, optimization Abstract Many quantitative problems in science, engineering, and economics are nowadays solved via statistical sampling. Sampling and Monte Carlo Simulation It then returns to the topic of using probability and statistics to derive information from samples. The phrase “Monte Carlo methods” was coined in the beginning of the 20th century, and refers to the famous casino in Monaco1—a place where random samples indeed play an important role. This is a simulation, but not a Monte Carlo simulation. Learn more about monte, carlo, simulation, pdf, probability, density, function. The MATLAB ® language provides a variety of high-level mathematical functions you can use to build a model for Monte Carlo simulation and to run those simulations. The data stored in de N colums can be used for the 2D histogram plot. The Monte Carlo model allows researchers to run multiple trials and define all potential outcomes of an event or investment. The test is based on the fact that under one-sided noncompliance LATT coincides with the average treatment effect for the treated (ATT). Not because I want to encourage you to gamble your life savings away. QRM with linear and quadratic programming. 4 Monte Carlo Transformation Procedures. Write a numpy Monte Carlo simulation to estimate the expectation of X and also estimate the probability p that no women sit at table A. There are tools you can add in to Excel, but we will start by doing some of this on our own. Monte Carlo Localization Algorithm Overview. Handbook of Monte Carlo Methods is an excellent reference for applied statisticians and practitioners working in the fields of engineering and finance who use or would like to learn how to use Monte Carlo in their research. Imagine that you want to asses the future value of your investments and see what is the worst-case scenario for a given level of probability. Monte Carlo Analysis: Steps. Wilesco D18 Steam Engine with Dynamo (type 2) £ 399. Notation I in this course, random variables will take values in a nite set X I we will use multiple styles of notation. to efficiently sample a high dimensional complex probability. Create a Hamiltonian Monte Carlo (HMC) sampler to sample from a normal distribution. (2013 Honours degree) Alvin Huang, Rare-event probability Estimation via Empirical Likelihood Maximization (2012 Master degree) Nancy Glass, Multilevel Monte Carlo method with applications to financial derivatives pricing. This is just a question about probability, not about MATLAB at all. However, we like to plot the histograms of every case into one single '3D' plot. We find that Bayesian Monte Carlo outperformed Annealed Importance Sampling, although for very high dimensional problems or problems with massive multimodality BMC may be less adequate. Learn more about monte carlo, distribution, probability. A Monte Carlo method is a technique that involves using random numbers and probability to solve problems. Learn more about monte carlo photographs pl help me same code in matlab. Monte Carlo method is a handy tool for transforming problems of probabilistic nature into deterministic computations using the law of large numbers. – The difference is that the parameter values are randomly chosen from an appropriate probability distribution which is either based on theory or fitted to experimental data. Monte Carlo Approach Assumption The probability of getting each card out of a from BUSINESS MKT 500 at Washington University in St. What is Monte Carlo Simulation A numerical process of repeatedly calculating a mathematical problem in which the random variables of the problem are simulated from random number generators. Estimate $\pi$ using the Monte Carlo Method in MATLAB. Note: The name Monte Carlo simulation comes from the computer simulations performed during the 1930s and 1940s to estimate the probability that the chain reaction needed for an atom bomb to detonate would work successfully. It seemed so great and terrible. Article type: Overview Monte Carlo Methods Article ID Dirk P. We argue that Monte Carlo algorithms are ideally suited to parallel computing, and that “parallel Monte Carlo” should be more widely used. In reality birthdays are not uniformly distributed, but as I show in Statistical Programming with SAS/IML Software (Wicklin 2010), you can use a Monte Carlo simulation to estimate the probability of a matching birthday. This may seem surprising at first, but the intuition is rather straight forward. Monte Carlo Methods and Importance Sampling History and deflnition: The term \Monte Carlo" was apparently flrst used by Ulam and von Neumann as a Los Alamos code word for the stochastic simulations they applied to building better atomic bombs. IN FINANCE NATIONAL UNIVERSITY OF ROSARIO ARGENTINA 1 This manual is a translation of the SimulAr Spanish User Manual and some Excel screenshots are displayed in Spanish. Botev: Amazon. Monte Carlo Simulation Probability Fundamentals Event A desired outcome Probability The chance of. Generate manysets of possible inputswhich follows. Code on my GitHub: https://github. Monte Carlo techniques 3 33. Monte carlo simulation code. Written for advanced undergraduate and graduate students, Monte Carlo Simulation with Applications to Finance provides a self-contained introduction to Monte Carlo methods in financial engineering. This is a simulation, but not a Monte Carlo simulation. If you are programming this from scratch, rather than using software that already knows something about Monte Carlo, then your first step is to write functions for sampling probability distributions. When Monte Carlo was developed, this was done mechanically via flipping coins, rolling dice, or spinning a roulette wheel. Stan Ulam, John von Neumann, and the Monte Carlo method, Los Alamos Science, Special Issue (15), 131-137 Using GoldSim for Monte Carlo Simulation. The monte carlo simulation period will be 30 years and the number of experiments will be user defined -k. 1118632168 Series Probability and Statistics)】 (Wiley and Method in Method Monte 【送料無料】【Simulation and Carlo the,カタログギフト Dolce【30800円コース】ルビーノ【送料無料】【出産祝い・内祝い】【メッセージカード1円】,【海外限定】ピュア タンクトップ レディース ルームウェア 【 PURE ESSENCE TANK GOWN 】. Requiring only a basic, introductory knowledge of probability and statistics, Simulation and the Monte Carlo Method, Second Edition is an excellent text for upper-undergraduate and beginning graduate courses in simulation and Monte Carlo techniques. Eckhardt, Roger (1987). I need to develop a model which performs a monte carlo simulation. Carlo Simulation in Matlab that explains about. 00075m\) as shown from the probability density function (PDF) of LTBC in Figure 3. Monte Carlo simulations were developed of each of the games using MATLAB. We like to plot a 2D histogram from a 30 by N matrix. edu/6-0002F16 Instructor: John Guttag. Probability distributions are mathematical models that assign probability to a random variable. One of the Monte Carlo methods is a crude Monte Carlo method. You'll learn the most-widely used models for risk, including regression models, tree-based models, Monte Carlo simulations, and Markov chains, as well as the building blocks of these probabilistic models, such as random variables, probability distributions, Bernoulli random variables, binomial random variables, the empirical rule, and perhaps. The Monte Carlo simulation is a method that allows you to obtain results when modeling the problem mathematically and/or finding that an analytical solution is too complex. Monte Carlo simulation (MCS) is a technique that incorporates the variability in PK among potential patients (between-patient variability) when predicting antibiotic exposures, and allows calculation of the probability for obtaining a critical target exposure that drives a specific microbiological effect for the range of possible MIC values [45, 46, 79–86]. 3, Measurable Outcome 3. ” This blog is a good one to follow for statistical theory and practice and is. Detailed appendices provide background material on probability theory, stochastic. Introducing Monte Carlo Methods with R covers the main tools used in statistical simulation from a programmer's point of view, explaining the R implementation of each simulation technique and providing the output for better understanding and comparison. First, save a function on the MATLAB® path that returns the multivariate normal log probability density and its gradient. MATLAB 평가판. Also Monte-Carlo method is employed to transform an infinite to a finite integration. GPU utilization and parallel computation With Learn more about gpu, parallel computing toolbox, monte-carlo. Series Explaines Monte Carlo Methods from Beggining "How to Generate a Random Number" to "Sampling Distributions in MATLAB", The Series contains 23 Video of 10 Minute Each, and includ the. The Monte Carlo method is essentially a technique for sampling a probability density function based on a computer generated random number. MUSE Measurement Uncertainty Simulation and Evaluation using the monte carlo method. Monte Carlo simulation is a method for exploring the sensitivity of a complex system by varying parameters within statistical constraints. 3, Measurable Outcome 3. To do this, choose the User Defined distribution and enter its equation as follows. R = normrnd(MU,SIGMA,m,n) generates. 1 with a constant speed. Rosenthal* [Far East Journal of Theoretical Statistics 4 (2000), 207–236. In this example, that function is called normalDistGrad and is defined at the end of the example. IEOR E4703: Monte Carlo Simulation c 2017 by Martin Haugh Columbia University Generating Random Variables and Stochastic Processes In these lecture notes we describe the principal methods that are used to generate random variables, taking as. The data stored in de N colums can be used for the 2D histogram plot. Use the random number generator to make a bunch of random numbers and use those in a loop where, inside your loop, you do your experiment. Monte Carlo estimation refers to simulating hypothetical draws from a probability distribution, in order to calculate significant quantities of that distribution. Simulation of time series using the Monte Carlo method. Monte Carlo Simulation Probability Fundamentals Event A desired outcome Probability The chance of. Note that the following is true. First, save a function normalDistGrad on the MATLAB® path that returns the multivariate normal log probability density and its gradient (normalDistGrad is defined at the end of this example). View Notes - Monte Carlo Simulation. Classical Monte Carlo: samples are drawn from a probability distribution, often the classical Boltzmann distribution, to obtain thermodynamic properties or minimum-energy structures; Quantum Monte Carlo: random walks are used to compute quantum-mechanical energies and wave functions, often to solve electronic structure problems, using. Monte Carlo Simulation is a mathematical technique that generates random variables for modelling risk or uncertainty of a certain system. Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition. Cootie, and Chutes and Ladder were particularly straightforward because the outcomes for each of the players is independent. The Monte Carlo method has wide application in various scienti c areas such as nance, t-elecommunications and computer graphics [1]. However, the origin of Monte Carlo methods is older than the casino. There is a solution for doing this using the Markov Chain Monte Carlo (MCMC). Wilesco D18 Steam Engine with Dynamo (type 2) £ 399. The city of Monte Carlo in the country of Monaco has long served as a playground for the jet set, where rich gamblers who can afford to lose huge sums of money come to play for big stakes in games. Monte Carlo simulation is a powerful statistical analysis. When it comes to financial planning, a Monte Carlo simulation takes into account returns, volatility, correlations, and other factors, all based on historical statistical estimates. The results from each. This method can be understood by solving small problems. How to implement a Monte Carlo Simulation for studying uncertainty in dynamic stiffness? the probability density matlab random-variable simulation monte-carlo. This is true for both transient and steady-state studies. This book provides an introductory chapter on Markov Chain Monte Carlo techniques as well as a review of more in depth topics including a description of Gibbs Sampling and Metropolis Algorithm. Repeat 1-4 The Monte Carlo Simulation of Radiation Transport - p. com What is Monte Carlo Simulation? Monte Carlo simulation, or probability simulation, is a technique used to understand the impact of risk and uncertainty in financial, project management, cost, and other forecasting models. This historical distribution overlays a Monte-Carlo simulated distribution of annualized returns over 30-year periods using a lognormal return distribution model. 00075m\) as shown from the probability density function (PDF) of LTBC in Figure 3. Requiring only a basic, introductory knowledge of probability and statistics, Simulation and the Monte Carlo Method, Second Edition is an excellent text for upper-undergraduate and beginning graduate courses in simulation and Monte Carlo techniques. Monte Carlo analysis has become an increasingly popular arrow in the financial planner’s quiver, as an improvement over the oversimplified traditional straight-line projection. Monte Carlo Estimate of Pi. Monte Carlo simulation is named after the city of Monte Carlo in Monaco, which is famous for gambling such s roulette, dice, and slot machines. pdf), Text File (. The user can also decide which signals to plot for the simulation. But how can this be? There are 365 days in a year and half of that would be 182, so why wouldn’t you need at. Monte Carlo simulation is a way to estimate the distribution of a quantity by simulating the process that generates it. If you're going to be doing some sort of flow simulation, then Monte Carlo is probably appropriate (or at least can be). Monte Carlo simulation in matlab. probability density function of the surface height and slope is computed and used to determine the electromagnetic scattering field. Simulation - Download as Powerpoint Presentation (. Monte Carlo simulation. Importance Sampling and Monte Carlo Simulation - May 28, 2009. probability simulation stochastic-processes First passage time distribution via Monte Carlo simulation. MATLAB Tutorial - 25 Working with Probability Distributions Monte Carlo Simulations in MATLAB (Tutorial) - Duration: Probability density functions | Probability and Statistics. We conduct Monte Carlo simulations to demonstrate, among other things, that part of the theoretical efficiency gain afforded by unconfoundedness in estimating ATT survives pretesting. For Monte Carlo, I know how to do it but don't know how to answer you because you did not describe the experiment. The analytical solution is 1/8 or 0. That is, after each sample, the probabilities of some events might change, or there may be new events. I have this question, a person is vacationing in Monte Carlo. People were developing programs dedicated solely to Monte Carlo, and offering special workshops and seminars. Let x,x0 ∈ X be states in the chain. Discrete Probability.
Monte Carlo Probability Matlab | 2019-10-17T18:26:56 | {
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http://mathhelpforum.com/pre-calculus/166869-demoivre-s-theorem.html | # Math Help - DeMoivre's Theorem
1. ## DeMoivre's Theorem
I need to find the real and imaginary values of z^5, when z = 2-2i
I got r = 2 sqrt(2) and theta = -pi/4
However now I am stuck. I tried to put it into the form r^k e^(i k theta)
and got (2 rt(2)) ^5 * e^(-5pi/4) but then I don't know what to do to find the real and imaginary parts.
What should I do?
Thank you
2. $\text{Re} \left( {\left[ {x + yi} \right]^5 } \right) = \sum\limits_{k = 0}^2 {\left( { - 1} \right)^k \dbinom{ 5}{2k} x^{5 - 2k} y^{2k} }$
3. Hello, ironz!
$\text{Find the real and imaginary components of }\,(2 - 2i)^5$
Write $2 - 2i$ in polar form.
. . We have: . $x = 2,\;y =2,\;r = 2\sqrt{2},\;\theta = \text{-}\dfrac{\pi}{4}$
Hence: . $2 - 2i \:=\:2\sqrt{2}\left(\cos\frac{\pi}{4} - \sin\frac{\pi}{4}\right)$
Then: . $(2-2i)^5 \;=\;\left(2\sqrt{2}\right)^5\,\left(\cos\frac {5\pi}{4} - \sin\frac{5\pi}{4}\right)$
. . . . . . . . . . . . $=\;2^5(\sqrt{2})^5\left(-\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i\right)$
. . . . . . . . . . . . $=\;32(4\sqrt{2})\left(\dfrac{-1+i}{\sqrt{2}}\right)$
. . . . . . . . . . . . $=\;128(-1 + i)$
. . . . . . . . . . . . $=\;\underbrace{-128}_{\mathbb{R}} + \underbrace{128}_{\mathbb{I}}i$
4. Originally Posted by ironz
I need to find the real and imaginary values of z^5, when z = 2-2i
An alternative: taking into account that $(1-i)^2=1-1-2i=-2i$:
$(2-2i)^5=2^5(-2i)^2(1-i)=-2^7(1-i)=-128+128i$
In general this is the shortest way for computing $(x+yi)^n$ when $y=x$ or $y=-x$ .
Fernando Revilla
5. Originally Posted by ironz
I need to find the real and imaginary values of z^5, when z = 2-2i
I got r = 2 sqrt(2) and theta = -pi/4
However now I am stuck. I tried to put it into the form r^k e^(i k theta)
and got (2 rt(2)) ^5 * e^(-5pi/4) but then I don't know what to do to find the real and imaginary parts.
that should read e^(-i5pi/4)
What should I do?
Thank you
You reached
$\displaystyle\ z^5=\left[2\sqrt{2}\right]^5e^{-i\frac{5{\pi}}{4}}$
To regenerate the real and imaginary parts from this...
$\displaystyle\ e^{-i\frac{5{\pi}}{4}}=cos\left(-\frac{5{\pi}}{4}\right)+isin\left(-\frac{5{\pi}}{4}\right)$
$\displaystyle\ -\frac{5{\pi}}{4}=2{\pi}-\frac{5{\pi}}{4}=\frac{3{\pi}}{4}$
$\displaystyle\ cos\left(\frac{3{\pi}}{4}\right)=-cos\left(\frac{\pi}{4}\right)=-\frac{1}{\sqrt{2}}$
$\displaystyle\ sin\left(\frac{3{\pi}}{4}\right)=sin\left(\frac{\p i}{4}\right)=\frac{1}{\sqrt{2}}$
Then
$\displaystyle\ z^5=\frac{-2^5\left(\sqrt{2}^5\right)}{\sqrt{2}}+\frac{i2^5\l eft(\sqrt{2}^5\right)}{\sqrt{2}}$
6. Don't let those square roots of 2 in Archie Meade's response fool you. When you actually do the calculation, they will cancel out and give the same answer FernandoRevilla gives.
7. I understand. I think I read the question wrong and tried to solve for all the roots instead of just the z^5, also, I didn't know how to simplify it properly. Thank you for your help
8. Originally Posted by ironz
I understand. I think I read the question wrong and tried to solve for all the roots instead of just the z^5, also, I didn't know how to simplify it properly. Thank you for your help
Having found $z^5$
$z=2-2i$ is one of the 5 "fifth roots" of $z^5$
but that's a different question. | 2016-02-06T00:43:16 | {
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https://www.physicsforums.com/threads/properties-of-inverse-matrices.234615/ | # Properties of Inverse Matrices
## Homework Statement
Determine which of the formulas hold for all invertible nxn matrices A and B
A. AB=BA
B. (A+A^–1)^8=A^8+A^–8
C. A^5 is invertible
D. A+A^–1 is invertible
E. (In+A)(In+A^–1)=2In+A+A^–1 (where In is the identity matrix)
F. (A+B)^2=A^2+B^2+2AB
## Homework Equations
Certain properties of inverse matrices can be used. For example, if A is invertible, then A^k is invertible for all k greater or equal to 1 (this proves C to be correct).
## The Attempt at a Solution
I was able to find counterexamples to prove A and B and F incorrect. However, the webwork program (designed for practicing basic Linear Algebra) I am using states that C, D, and E are not all correct ... what am I missing?
Option D is not always true. Try to find a matrix A where $$A^{-1} = -A$$. | 2020-01-29T13:05:21 | {
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https://math.stackexchange.com/questions/3841929/a-generalization-of-bottemas-theorem | # A generalization of Bottema's theorem
Can you provide a proof for the following claim:
In any triangle $$ABC$$ construct isosceles right triangles on sides $$AC$$ and $$BC$$, with right angles at the points $$A$$ and $$B$$. Let points $$F$$ and $$G$$ divide catheti $$AE$$ and $$BD$$ respectively in the same arbitrary ratio . The midpoint $$H$$ of the line segment that connects points $$F$$ and $$G$$ is independent of the location of $$C$$ .
GeoGebra applet that demonstrates this claim can be found here. I tried to mimic a proof of Bottema's theorem given on this page but without success.
• It hurts me when I see a fact like this that is so clean and simple but seems to require lots of nasty trig to prove. D: – Franklin Pezzuti Dyer Sep 27 '20 at 5:23
• @FranklinPezzutiDyer The proof by Christian Blatter below is very clean and simple. – Servaes Sep 27 '20 at 14:12
• Here mathoverflow.net/questions/374762 is a generalization of your generalization – Đào Thanh Oai Nov 6 '20 at 4:00
Basically here we mimic the proof from cut-the-knot, but replacing congruence with similarity:
Let
$$\frac {AH}{AG} = \frac {BI}{BD} = a$$
Let's drop perpendiculars $$HL$$, $$CX$$, $$JK$$, and $$IM$$ onto $$AB$$. (I forgot to label $$X$$)
$$JK$$ is the midline of trapezoid $$HLMI$$ so that
$$JK = \frac {HL + IM}2$$
Further, since $$\angle HAC$$ is right, $$\angle HAL$$ and $$\angle CAX$$ are complementary which makes right triangles $$\triangle HAL$$ and $$\triangle ACX$$ similar, implying
$$HL=aAX$$
Similarly,
$$IM=aBX$$
Taking all three identities into account shows that
$$JK = \frac {HL + IM}2 = \frac a2 (AX+BX) = \frac a2 AB = aAK$$
independent of $$C$$. No trigs but tricks.
EDIT: I see how I can prove that $$AK=KB$$.
By the previous similar triangles ($$\triangle HAL \sim \triangle ACX$$ and $$\triangle IBM \sim \triangle BCX$$), we have:
$$AL = k CX = BM$$
By midline theorem, $$LK = KM$$.
Therefore $$AK = LK-AL=KM-BM=KB$$.
This shows that (finally!) $$J$$ is fixed, since it is at a fixed distance "above" the midpoint of $$AB$$.
• In the second line...should it be $BD$ instead of $ID$ ? – Peđa Terzić Sep 27 '20 at 6:06
• Yes indeed. I'll change that. – player3236 Sep 27 '20 at 6:06
Consider $$A$$, $$B$$, $$C$$ as complex numbers, and choose a $$\lambda\in{\mathbb R}$$. Then $$F=A+\lambda(E-A)=A+\lambda\,i(C-A),\qquad G=B+\lambda(D-B)=B+\lambda(-i) (C-B)\ .$$ It follows that $$H={1\over2}(F+G)={1\over2}(A+B)+{\lambda i\over2}(B-A)\ .$$
• +1 This is the way to go. Also note that this shows the same proof works if the two angles at $A$ and $B$ are not required to be right angles, but only required to sum to $0$. – Servaes Sep 27 '20 at 14:10
A little angle-chasing shows that the target point (here, $$K$$) is the midpoint of a side of a particular, symmetrically-situated parallelogram, which in turn shows that, for a given ratio $$\lambda$$, the point's position relative to side $$\overline{AB}$$ is independent of the position of $$C$$.
Note: $$c$$ is half of $$|AB|$$ in the figure.
FYI: If the right angles are formed "the other way" at $$A$$ and $$B$$, then the corresponding midpoint is the reflection of $$K$$ across $$\overline{AB}$$. Proof is left as an exercise to the reader.
Here is a proof via vectors. This avoids the issue of the location of $$J$$ in my earlier proof.
Use the original diagram and let $$O$$ be the midpoint of $$AB$$.
Let $$\overrightarrow {OB} = a \hat i$$. Then $$\overrightarrow {OA} = -a \hat i$$.
Let $$\overrightarrow {OC} = b \hat i + c \hat j$$.
Hence $$\overrightarrow {AC} = (a+b) \hat i + c \hat j$$ and $$\overrightarrow {BC} = (-a+b) \hat i + c \hat j$$.
We can easily show that $$\overrightarrow {AE} = -c \hat i+(a+b) \hat j$$ and $$\overrightarrow {BD} = c \hat i + (a-b) \hat j$$.
Letting $$\dfrac {AF}{AE} = \dfrac {BG}{BD} = k$$ we have $$\overrightarrow {AF} = -kc \hat i+k(a+b) \hat j$$ and $$\overrightarrow {BG} = kc \hat i + k(a-b) \hat j$$.
Finally:
\begin{align}\overrightarrow{OH}&=\frac12(\overrightarrow{OF}+\overrightarrow{OG})\\&=\frac12(\overrightarrow{OA}+\overrightarrow{AF}+\overrightarrow{OB}+\overrightarrow{BG}) \\&=\frac12(-a\hat i-kc \hat i+k(a+b) \hat j+a\hat i+kc \hat i + k(a-b) \hat j) \\&=\frac k2((a+b)+(a-b))\hat j \\&=ka\hat j \end{align}
This shows that $$OH \perp AB$$ and $$|OH|$$ only depend on $$a$$ and $$k$$, that is, the length of $$AB$$ and the ratio $$k$$, implying the position of $$H$$ is indeed fixed.
• Stated differently, let $v^\perp$ be the vector obtained by swapping the components of $v$ and changing one of the signs. Then we have \begin{align} H &= \tfrac12(D+E)\\[4pt] &=\tfrac12((A+k(C-A)^\perp)+(B - k(C-B)^\perp)) \\[4pt] &=\tfrac12((A+B)+k(C-A-(C-B))^\perp)\\[4pt] &=\tfrac12((A+B)+k(B-A)^\perp) \\[4pt] \end{align} which is clearly independent of $C$. Sneakily, I didn't specify which component's sign is changed by $\perp$. Each choice leads to a valid construction of a corresponding fixed point (either "above" or "below" $\overline{AB}$). This proves the "FYI" in my answer – Blue Sep 27 '20 at 7:28 | 2021-04-14T15:57:25 | {
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https://www.physicsforums.com/threads/is-the-exponential-function-the-only-function-where-y-y.950153/ | # I Is the exponential function, the only function where y'=y?
#### Phylosopher
Hello,
I was wondering. Is the exponential function, the only function where $y'=y$.
I know we can write an infinite amount of functions just by multiplying $e^{x}$ by a constant. This is not my point.
Lets say in general, is there another function other than $y(x)=ae^{x}$ ($a$ is a constant), where $\frac{dy}{dx}=y$.
I would really appreciate it if we can work a proof.
Related Differential Equations News on Phys.org
#### andrewkirk
Homework Helper
Gold Member
Proof:
\begin{align*}
\frac{dy}{dx}&=y\\
\log y + C &= x + D\quad\ \ \textrm{[integrating both sides]}\\
y&=A e^x\quad\quad\ \ \textrm{[where $A=e^{D-C}$ is a constant]}
\end{align*}
#### Phylosopher
Proof:
\begin{align*}
\frac{dy}{dx}&=y\\
\log y + C &= x + D\quad\ \ \textrm{[integrating both sides]}\\
y&=A e^x\quad\quad\ \ \textrm{[where $A=e^{D-C}$ is a constant]}
\end{align*}
Exactly. But I was thinking. Is this enough to say that it is the only solution?
I know this sounds hilarious, but when I studied calculus no one pointed out that the solution is unique. Is it?
#### lurflurf
Homework Helper
we have
$\dfrac{d}{dx}y=y$
let us rewrite it as
$\dfrac{d}{dx}e^{-x}y=0$
now consider
$\dfrac{d}{dx}u=0$
what can we say about u?
#### Phylosopher
we have
$\dfrac{d}{dx}y=y$
let us rewrite it as
$\dfrac{d}{dx}e^{-x}y=0$
now consider
$\dfrac{d}{dx}u=0$
what can we say about u?
Constant w.r.t x
#### lurflurf
Homework Helper
^then the only solutions are
$e^{-x}y=C$
as desired
Homework Helper
Gold Member
2018 Award
#### Phylosopher
^then the only solutions are
$e^{-x}y=C$
as desired
I will give it a try and read it. Thanks
#### FactChecker
Gold Member
2018 Award
Exactly. But I was thinking. Is this enough to say that it is the only solution?
Step-by-step, there are no alternatives except for the choice of C and D.
#### fresh_42
Mentor
2018 Award
Is this enough to say that it is the only solution?
You can use Picard-Lindelöf.
#### Math_QED
Homework Helper
Proof:
\begin{align*}
\frac{dy}{dx}&=y\\
\log y + C &= x + D\quad\ \ \textrm{[integrating both sides]}\\
y&=A e^x\quad\quad\ \ \textrm{[where $A=e^{D-C}$ is a constant]}
\end{align*}
You divided by $y$, which is not allowed when $y = 0$.
This happens to be a solution, which can be included if you take $y = A\exp(x)$ with $A \geq 0$ as general solution (instead of $A > 0)$
#### mathwonk
Homework Helper
to expand on lurflurf's solution, use the quotient rule to show that if y'=y, then (y/e^x)' = 0, hence y/e^x = constant.
using this a lemma, one can deduce that the kernel of the linear constant coefficient operator P(D), where Df = f' and P is a polynomial of degree n, has dimension n. see my web notes on linear algebra, last section of chapter 5, roughly page 119:
http://alpha.math.uga.edu/~roy/laprimexp.pdf
briefly, the argument above gives the kernel of the operator (D-1) which generalizes to that of (D-c) and then by factoring a polynomial P(D) into products of linear factors of form (D-c), we get the result.
Last edited:
"Is the exponential function, the only function where y'=y?"
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• Solo and co-op problem solving | 2019-10-19T02:34:08 | {
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https://math.stackexchange.com/questions/1627659/relative-error-newtons-method | # Relative Error Newton's Method
So I've been preparing for an exam by looking over old exam papers and I came across this question:
a) Using Newton's method, show that the iteration equation to find the cube root of $R$ can be written as
$$x_{n+1}= \frac 13\left (2x_n+ \frac R{x_n^2}\right)$$
b) If $R=3$ and $x_0=1$, calculate $x_1$ and $x_2$.
c) The equation in (a) is used to approximate the cube root of $R$. Using the values in (b), calculate the exact relative error in the approximation of $x_2$.
I didn't have any troubles with showing the iteration equation from the Newton's method formula for part (a), and here's what I did for part (b):
$x_1=\frac 13\left (2x_0 +\frac R{x_0^2}\right)$
$x_1=\frac 13\left (2.1 +\frac 3{1^2}\right) = \frac 53$
$x_2=\frac 13\left (2x_1 +\frac R{x_1^2}\right)$
$x_2=\frac 13\left (2\frac 53 +\frac 3{\frac 53^2}\right) = \frac {77}{45}$
which I'm confident is right (but please correct me if I'm wrong). And here's my attempt at part (c):
$x=\sqrt[3] 3$, $\hat x=x_2=\frac {77}{45}$
$e_R=\left\lvert \frac {x-\hat x}{x} \right\rvert$
$e_R=\left\lvert \frac {\sqrt[3] 3-\frac {77}{45}}{\sqrt[3] 3} \right\rvert = 0.1864181806$
I was just hoping if someone would be able to tell me if this answer is the correct answer for what part (c) is asking. In my subject notes there's no mention of "exact relative error", just relative error, and I was thinking that the exact part just referred to the fact that we know what the cube root of 3 equals. Googling "exact relative error" didn't help, so if someone is just able to tell me if they think what I've done is right and perhaps shed some light on "exact relative error" that'd be great. Thanks in advance.
No it is not correct, because your $x_2$ is in error. $x_1$ is correct but $$x_2=\frac 13\left (2x_1 +\frac R{x_1^2}\right)=\frac 13\left (2\frac{5}{3} +\frac {3}{(\frac{5}{3})^2}\right)=\frac{1}{3}\left (\frac{10}{3} +\frac{27}{25}\right) =\frac{331}{225}\cdot$$
And therefore $$e_R=\left\lvert \frac {\sqrt[3] 3-\frac {331}{225}}{\sqrt[3] 3} \right\rvert \approx 0.0200114747$$ | 2019-07-21T17:35:26 | {
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https://math.stackexchange.com/questions/2045419/given-a-series-defined-by-recursion-prove-that-there-are-integers-s-and-t-s | # Given a series defined by recursion. Prove that there are integers $S$ and $T$ such that $a_{n+1} = S a_n + T a_{n-1}$
I have this problem from an old exam that I can't solve.
Let $$\{a_i\}_{i \geq 0}$$ be the series define by recursion as:
$$a_0 = 2$$
$$a_1 = 3$$
$$a_{n+1} = \frac{a_n^2 +5}{a_{n-1}}$$ , $$\forall n \in \Bbb N$$
Prove that there are integers $$S$$ and $$T$$ such that $$a_{n+1} = S a_n + T a_{n-1}$$
I've been trying with induction but I guess that there must be a direct, and more algebraic, way of proving it.
Any thoughts?? Thanks!
• First things first: have you run several terms of the sequence to see what the constants $S$ and $T$ must be? – Steven Stadnicki Dec 5 '16 at 20:29
• @Steven I've found that the constants are $S=5$ and $T=(-1)$ for $n \geq 2$. – jrs Dec 5 '16 at 20:38
• Once you have the correct constants there's a nice characterization in terms of a matrix; note that the recursion can also be written as $a_{n+1}a_{n-1}=a_n^2+5$ or as $a_{n+1}a_{n-1}-a_na_n=5$; that form should look a lot like the determinant of a certain matrix, and you should be able to find a linear transformation that takes one matrix of that form to the same matrix with all indices shifted by one, using the recurrence relation. – Steven Stadnicki Dec 5 '16 at 20:46
• Should S be 3 instead of 5? – displayname Dec 5 '16 at 20:58
• Gio, it certainly does not: The next few values are $a_2=\frac{3^2+5}{2}=7$, $a_3=\frac{7^2+5}{3}=18$, and $a_4=\frac{18^2+5}{7}=47$. It's not the case, for instance, that $a_4=5a_3-a_2$. – Steven Stadnicki Dec 5 '16 at 22:14
Define a series $a_{n+1} = \frac{a_n^2 +k}{a_{n-1}}$ with some $a_0, a_1$ for $n\ge2$. Assume there are $S$ and $T$ such that $a_{n+1} = S a_n + T a_{n-1}$.
$$a_{n+1} = \frac{a_n^2+k}{a_{n-1}} \\ S a_n + T a_{n-1} = \frac{a_n^2+k}{a_{n-1}} \\ S a_n a_{n-1} + T a_{n-1}^2 = a_n^2+k \tag{1}$$ now $$a_{n+1} = \frac{a_n^2+k}{a_{n-1}} \\ a_{n+1}a_{n-1} = a_n^2+k \\ a_{n+1}a_{n-1} - k = a_n^2 \tag{2}$$ combine the two: $$S a_n a_{n-1} + T(a_n a_{n-2} - k) = a_n^2+k \\ a_n(Sa_{n-1} + Ta_{n-2}) - Tk = a_n^2+k \\ a_n^2 - Tk = a_n^2+k \\ T = -1$$ now find $S$: $$a_{2} = \frac{a_1^2+k}{a_{0}} \\ S a_1 - a_{0} = \frac{a_1^2+k}{a_{0}} \\ S = \frac{a_0^2+a_1^2+k}{a_{0}a_{1}} \\$$ and in this case: $$S = \frac{2^2+3^2+5}{2\times3} = \frac{18}{6} = 3$$ So $S$ is an integer if it exists. Prove by induction that $$S a_n - a_{n-1} = \frac{a_n^2+k}{a_{n-1}}$$ (note that the base case $n=2$ is covered by how we set $S$). Assume the equality is correct for all $n+1\ge k\ge3$.
$$a_{n+2} = \frac{a_{n+1}^2+k}{a_{n}} \\ a_{n+2} = \frac{(S a_n-a_{n-1})^2+k}{a_{n}} = \frac{S^2 a_n^2-2S a_n a_{n-1}+a_{n-1}^2+k}{a_{n}} = S^2 a_n-2Sa_{n-1}+\frac{a_{n-1}^2+k}{a_{n}} \\ a_{n+2} = S(S a_n-a_{n-1})-Sa_{n-1}+\frac{a_{n-2}}{a_{n}}\frac{a_{n-1}^2+k}{a_{n-2}} = S a_{n+1} -Sa_n + \frac{a_{n-2}}{a_{n}} a_n \\ a_{n+2} = S a_{n+1} -Sa_n + a_{n-2} = S a_{n+1} -Sa_n + (S a_{n-1} - a_n) = S a_{n+1} - a_n$$ As required. | 2020-09-20T05:24:31 | {
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https://math.stackexchange.com/questions/3053446/curvature-of-curve-equivalence-between-tangent-vector-and-angle-definitions | # Curvature of curve: equivalence between tangent vector and angle definitions
I know that curvature for some curve $$C$$ defined parametrically is:
$$\kappa=\left\|{d\vec{T}\over ds}\right\|$$
Which basically is the rate at which the tangent vector to the curve changes, as the arclength of the curve changes.
In another source, I saw the definition of curvature as the following:
If $$P_1$$ and $$P_2$$ are two points on the curve, $$|P_1P_2|$$ is the arclength between those two points, and $$\Phi$$ is the limit of the angle between tangent vectors at the points $$P_1$$ and $$P_2$$ (as it goes to zero I assume), then the curvature is defined as:
$$\kappa=\lim_{|P_1P_2|\to 0}{\Phi\over |P_1P_2|}$$
Which basically means, the rate at which the angle of tangent vectors in global frame of reference changes, as the arclength of the curve changes.
I assume that this second definition can be rewritten using the notation from the first example as:
$$\kappa={d\phi\over ds}$$
Where $$\phi$$ is the angle between the vector tangent to the curve, and some constant global axis of reference (which could be the x axis, but realy it could be any line or vector on the same plane).
Given the second (weird in my opinion) definition of curvature, I can't see how those two definitions can be equivalent. Maybe they are not, I don't know. May be they are; if yes, how?
Also, here's a picture of the section from the book where the second definition appears in (it's not in English):
Note that the text agrees with yet another definition of curvature, which I am aware of: $$\kappa=\frac1r$$, where $$r$$ is radius of curvature.
• Note that all of these definitions except $\frac{d\phi}{ds}$ work just as well in three dimensions as in two dimensions. To use $\frac{d\phi}{ds}$ in three dimensions you need to find the plane in which the tangent and normal vectors lie, and then you can apply that definition too. – David K Dec 27 '18 at 12:45
• @DavidK thank you for the info, didn't even think about how that formula can be applied to three dimensions – KKZiomek Dec 27 '18 at 14:46
Let's work with the first definition. We have \begin{align} \kappa (s) &= \left\| \frac{d\vec T}{ds}(s)\right\| \\ &= \lim_{\delta s\to 0}\frac 1 {\delta s}\left\| \vec T(s + \delta s) - \vec T(s) \right\| \\ &= \lim_{\delta s \to 0} \frac 1 {\delta s} \sqrt{(\vec T(s + \delta s) - \vec T(s)).(\vec T(s + \delta s) - \vec T(s))} \\ &= \lim_{\delta s \to 0} \frac 1 {\delta s} \sqrt{\| \vec T(s + \delta s)\|^2 + \| \vec T(s) \|^2 - 2\vec T(s).\vec T(s + \delta s)}\end{align} But the curve is parameterised by arc length! So $$\| \vec T(s + \delta s)\|^2 = \| \vec T(s)\|^2 = 1$$ and $$\vec T(s).\vec T(s + \delta s) = \cos \Phi(s, s + \delta s),$$ where $$\Phi(s, s + \delta s)$$ is the angle between $$\vec T(s)$$ and $$\vec T(s + \delta s)$$.
Plugging this in, we get \begin{align} \kappa &= \lim_{\delta s \to 0} \frac 1 {\delta s} \sqrt{2 - 2 \cos \Phi(s, s + \delta s)} \\ &= \lim_{\delta s \to 0} \frac 1 {\delta s} 2 \sin \left( \frac { \Phi(s, s + \delta s) } {2}\right) \\ &= \lim_{\delta s \to 0} \frac {\Phi(s, s + \delta s)} {\delta s} \times \frac{\sin \left( \frac { \Phi(s, s + \delta s) } {2}\right)}{\frac{\Phi(s, s + \delta s)}{2}} \end{align} Clearly, $$\lim_{\delta s \to 0} \Phi(s, s + \delta s) = 0$$, so \begin{align} \kappa &= \lim_{\delta s \to 0} \frac {\Phi(s, s + \delta s)} {\delta s} \times \lim_{\Phi \to 0} \frac{\sin \left( \frac { \Phi } {2}\right)}{\frac{\Phi}{2}} \\ &= \lim_{\delta s \to 0} \frac {\Phi(s, s + \delta s)} {\delta s} \times 1 \\ &= \lim_{\delta s \to 0} \frac {\Phi(s, s + \delta s)} {\delta s}\end{align} which agrees with the second definition.
• How do you get from $\sqrt{2-2\cos\Phi}$ to $2\sin\left(\frac{\Phi}{2}\right)$? – Noble Mushtak Dec 27 '18 at 0:36
• @NobleMushtak I used $1 - \cos x = 2 \sin^2 (\tfrac x 2)$. – Kenny Wong Dec 27 '18 at 0:37
By your first statement, $$T'(s)=\kappa(s)N(s)$$ where $$\kappa$$ is the curvature and $$N$$ is the unit normal vector. Now, let's consider $$T(s)$$ and $$T(s+\Delta s)$$, so that we can compare the angles between tangent vectors. Now, $$T(s)$$, $$T(s+\Delta s)$$, and $$N(s)$$ are all unit vectors, so we can draw the following picture representing all of these vectors:
Here, $$\Delta \theta$$ is the angle between the two tangent vectors and $$\Delta T=T(s+\Delta s)-T(s)$$. From the diagram, we can find that $$\Delta T\cdot T(s)=1-\cos \Delta \theta$$ and $$\Delta T\cdot N(s)=\sin d\theta$$. From the x-component, we get the following equation:
$$\frac{dT}{d\theta}\cdot T(s)=\lim_{\Delta \theta\rightarrow 0}\frac{\Delta T\cdot T(s)}{\Delta \theta}=\lim_{\Delta \theta\rightarrow 0}\frac{1-\cos(\Delta \theta)}{\Delta \theta}=0$$
This was pretty obvious since $$T'(s)=\kappa(s)N(s)$$ is orthogonal to $$T(s)$$, so nothing new there. However, from the y-component, we get:
$$\frac{dT}{d\theta}\cdot N(s)=\lim_{\Delta \theta\rightarrow 0}\frac{\Delta T\cdot N(s)}{\Delta \theta}=\lim_{\Delta \theta\rightarrow 0}\frac{\sin(\Delta \theta)}{\Delta \theta}=1$$
Now, let's use chain rule to figure out what $$\frac{dT}{d\theta}$$ is:
$$\frac{dT}{ds}=\kappa(s)N(s)=\frac{dT}{d\theta}\frac{d\theta}{ds} \rightarrow \frac{dT}{d\theta}=\frac{\kappa(s)N(s)}{\frac{d\theta}{ds}}$$
Finally, let's plug this value for $$\frac{dT}{d\theta}$$ into the equation with the dot product:
$$\frac{dT}{d\theta}\cdot N(s)=1\rightarrow \frac{\kappa(s)N(s)\cdot N(s)}{\frac{d\theta}{ds}}=1\rightarrow \frac{d\theta}{ds}=\kappa(s)N(s)\cdot N(s)=\kappa(s)$$
(Note that the last step uses $$N(s)\cdot N(s)=1$$ since $$N(s)$$ is a unit vector.)
At this point, we have shown $$\frac{d\theta}{ds}=\kappa(s)$$, which is what we originally set out to prove. Q.E.D.
Of course, this isn't exactly a formal proof since it assumes $$T(s+\Delta s)$$ is in the plane spanned by $$T(s)$$ and $$N(s)$$, which isn't necessarily true. However, I think this is a fair approximation since $$T'(s)=\kappa(s)N(s)$$, so $$T(s+\Delta s)\approx T(s)+\Delta sT'(s)=T(s)+[\Delta s\kappa(s)]N(s)$$. In any case, I feel like this geometric argument gives a better visual intuition for why $$|T'(s)|=\frac{d\theta}{ds}$$.
• Thank you for the answer. I accepted the other answer only because it was more straightforward, at least to me – KKZiomek Dec 27 '18 at 1:57 | 2019-05-26T18:57:20 | {
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https://www.freemathhelp.com/forum/threads/problem-regarding-percentage-increases.52309/ | # Problem regarding percentage increases.
#### Strikera
##### New member
Ran into a problem which is driving me nuts and would appreciate any assistance on the problem.
Problem:
A factory increases it's production by 10%. The factory then increases it's production by another 20%. To return to the original production (before the 10 and 20% increases) how much production would the factory need to reduce?
The answer is 24% but I have no idea on how they arrived at that number or the steps needed to properly approach the problem.
Thanks in advance for any help!
#### skeeter
##### Senior Member
let P = production level
increase P by 10% = (1.10)P
increase the new level by another 20% = (1.20)(1.10)P = (1.32)P
to reduce back to P ... P = (1/1.32)(1.32)P
1/1.32 = approx 0.76 of the last level of production ... a 24% decrease.
#### Denis
##### Senior Member
Striker, when you have "no idea", make up a simple example,
like let initial production = 1000:
1000 + 10% = 1000 + 100 = 1100
1100 + 20% = 1100 + 220 = 1320
Now you can "see" that 1320 needs to be reduced back to 1000,
so a reduction of 320: kapish?
#### pka
##### Elite Member
Here is a sure-fire method to do all these problems:
$$\displaystyle \frac{{New - Old}}{{Old}}$$
This works for % of increase or decrease.
#### tkhunny
##### Moderator
Staff member
...unless, of course, Old = 0.
#### stapel
##### Super Moderator
Staff member
tkhunny said:
...unless, of course, Old = 0.
But if you're starting from zero, then "percent change" has no meaning, so it's a moot point, isn't it...?
Eliz.
#### tkhunny
##### Moderator
Staff member
It absolutely is not moot if your employer INSISTS on a "Percent Change" in that important report he's about to present to the Board of Directors.
It can be a difficult argument to convince said employer that the question is pointless.
Try this in your next boardroom...
Results by quarter
1) -5 a loss
2) -4 less loss
3) -3 less loss
4) -2 less loss
5) -1 less loss
6) 0 less loss
7) 5 big gain
Percent change by quarter
2) +20%
3) +25%
4) +33%
5) +50%
6) +100%
7) Pointless Data.
They WILL find it unsatisfactory.
It is good to be aware of this problem. That's all I'm saying. If your data can reach zero, your audience should be warned well before the event. | 2019-03-22T14:39:01 | {
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http://goyy.cigibuilding.it/predator-prey-model-matlab.html | This paper investigates a dynamical predator-prey interaction model that incorporates: (a) hunting cooperation among predators; (b) Allee effect in prey. Modelling Predator-Prey Interactions with ODE The Lotka-Volterra (LV) model Definition : The LV model in MATLAB Step 1: Create a MATLAB function that defines the rate of change of the vector y 1 function dy = Lotka_Volterra_Model(t,y) 2 % Lotka-Volterra predator-prey model. Im illiterate but i can try whatever your comfortable with. 0 ⋮ I wonder if my code is correct. α = exponential growth in population – used for preys,. The MATLAB function ode45 is based on R-K method of order 6 and adapts the step size. Optimal control of predator-prey model with distributed delay. , how predators affect prey populations, and vice-versa. Ask Question I was wondering if someone might be able to help me solve the Lotka-Volterra equations using MatLab. Garvie MR , Burkardt J , Morgan J Bull Math Biol , 77(3):548-578, 24 Jan 2015. Di erential Equations (Aggregate) Models with MATLAB and Octave A Predator-Prey Example Di erential equations in biology are most commonly associated with aggregate models. MATLAB scripts for the predator-prey model: MATLAB scripts for the second skydive model with event handling:. The model is first applied to a system with two-dimensions, but is then extended to include more complicated scenarios. This simulates the problem #18, page 206 (Chapter 5). 1007/s12190-014-0812-3 ORIGINAL RESEARCH On the dynamics of a stochastic ratio-dependent predator–prey model with a By Matlab software. I have a program called Predator Prey that's in the collection of programs that comes with NCM, Numerical Computing with MATLAB. since they will eventually run out of food, so let's add another term limiting growth and change the system to. (3) to illustrate the Maple, Mathematica, and MATLAB techniques needed for these investigations. The full prey equation is The first term ( rN ) describes exponential population growth in the absence of the predator, and the second term (- aNP ) is the death rate due to the predator. One functional response that depends on the density of prey and predator is the ratio dependent functional response. In predator–prey systems, delay effects were first considered by Volterra [56]. The phase-space of the predator-prey. rar > FD2D 2D Predator Prey Simulation. We now replace the difference equation model used there with a more sophisticated differential equation model. In 1920 Lotka extended the model, via Andrey Kolmogorov, to "organic systems" using a plant species and a herbivorous animal species as an example and. The populations change through time according to the pair of equations:. Let’s try to solve a typical predator prey system such as the one given below numerically. 1 Recommendation. Students can change various components of a predator/prey model, including birth factor, lifespan, and habitat area. It has also been applied to many other fields, including economics. It assumes just one prey for the predator, and vice versa. As usually the STL files are presented in the highest quality. Imitative modeling of nerve fibers conductivity. function yp = lotka(t,y) %LOTKA Lotka-Volterra predator-prey model. (5 stars rating will be given =). L42-PPMatlab-handout. We model the hunt as a game of three explicit stages: the stalk, the attack, and the subdual. Scipy Ode Solver So my next approach is to solve the system with the SciPy ode solver. Animals that feed on algae and plankton, such as fish and turtles, will have less food. The Lotka-Volterra predator-prey model. It is called the Lotka-Volterra model. The nonstan-dard finite difference schemes has been applied also to the predator-prey model [8]. Follow 105 views (last 30 days) Jovos on 8 Apr 2016. Question 6 : In Predator-Prey Model 1, what causes the peaks in the prey population (i. Notice how both the predator and prey populations both oscillate, with a period of about 105 time units, and how they. Predators are dependent on prey for sustenance and thus grow at a rate dependent on both the predator and prey population. Here is a demonstration of this effect. edu Department of Computer Science University of Toronto (part of my PhD thesis under the supervision of professor Wayne Enright) SONAD 2008 – p. It has also been applied to many other fields, including economics. Biswajit Mondal. A model often used to conceptualize population dynamics is (1) (2) In this model, x(t) and y(t) represent the number of prey and predator animals at any one given time. Predator-prey system A two-dimensional dynamic system in which two variables grow, but one grows at the expense of the other. To test this, we created a novel experimental design and analysis in which human participants took the role of predator or prey. Nonetheless, you get behavior that looks like a predator-prey system (unstable focus). Predation — For alternative meanings of predator and prey, see Predator (disambiguation) and Prey (disambiguation). To test this, we created a novel experimental design and analysis in which human participants took the role of predator or prey. The Lotka-Volterra equations, also known as the predator-prey equations, are a pair of first-order, non-linear, differential equations frequently used to describe the dynamics of biological systems in which two species interact, one a predator and one its. dynamic model, to describe the effects of different tomato cultivars on a one predator-two prey model. I dont have any triggers. The Matlab code takes 7 parameters: k1: rate of growth of the prey population k2: rate of decay of the predator population c: decay of the prey population per encounter between predator and prey d: growth of the predator population per encounter between predator and prey p(0): initial prey population. Prey-resource, prey-prey and predator-prey trait. We define a prey (mouse) and predator (cat) model. {\alpha/\beta}$, a nondimensional parameter. Pattern Formation in Predator-Prey Model with Delay and Cross Diffusion Lian, Xinze, Yan, Shuling, and Wang, Hailing, Abstract and Applied Analysis, 2013 Bifurcation Analysis of a Singular Bioeconomic Model with Allee Effect and Two Time Delays Zhang, Xue, Zhang, Qing-ling, and Xiang, Zhongyi, Abstract and Applied Analysis, 2014. 2016-10-10 Modeling and Simulation of Social Systems with MATLAB 35 SIR model ! A general model for epidemics is the SIR model, which describes the interaction between Susceptible, Infected and Removed (Recovered) persons, for a given disease. Each month the number of rats would increase by 20% if there were no owls to eat them. Compartmental Analysis. Vito Volterra-American biophysicist-Proposed the predator-prey model in 1925-Italian mathematician-Proposed the predator-prey model in 1926 Thank you to Anatoly for helping us with this presentation and. Using Maple To plot a solution curve for the system in (3) we need only load the DEtools package and use the DEplot function. Stability Analysis of a Discrete Prey - Predator Model with Ratio Dependence M. Peng and Zhang (2018) also considered the influence of stage structure and derived the relevant nature of Hopf bifurcation. We conclude the third population example by presenting the model information returned by PRESENT. α = exponential growth in population – used for preys,. where x(t) and y(t) are the prey and predator population sizes at time t, and p,q, r, and s are biologically determined parameters. Tips to Develop the Lotka-Volterra Equations Let us now look at how to implement the equations in MATLAB. Another predator-prey model considers the fact that the prey population could satiate the predator, so a HollingÕs Type II term for predation is used. I have a question about the eigenvalues of the prey-predator model called Lotka-Volterra. Predator-Prey Model The following example is adapted from the Hutchinson model, where the delay accommodates differences in resource consumption between young and adult members of a population. Descriptions: The classic Lotka-Volterra model of predator-prey competition, which describes interactions between foxes and rabbits, or big fish and little fish, is the foundation of mathematical ecology. The Lotka-Volterra equations describe an ecological predator-prey (or parasite-host) model which assumes that, for a set of fixed positive constants (the growth rate of prey), (the rate at which predators destroy prey), (the death rate of predators), and (the rate at which predators increase by consuming prey), certain simple conditions hold in the population change rates for prey and predat. This model reads as [22] proposed that the prey exhibits a demographic Allee effect at low population densi-ties due to reasons other than predation by the focal predator as. How to Solve and Plot Lotka-Volterra Differential Equations in Matlab. a discrete time predator prey model specified by Neubert et al[9] which utilises the Ricker model to simulate prey growth. 11: Predator-Prey Equations The classic Lotka-Volterra model of predator-prey competition is a nonlinear system of two equations, where one species grows exponentially and the other decays exponentially in the absence of the other. Lecture 3b Predator-prey I CEE/MAE M20 Introduction to Computer Programming with MATLAB To be learned Continuous. FD1D_PREDATOR_PREY is a MATLAB program which uses finite difference methods for the dynamics of predator-prey interactions in 1 spatial dimension and time, by Marcus Garvey. All four predator‐prey ODE models are well studied and have their own biological interpretations. 4: Nullclines for Predator Prey Model Explains how to get the equilibria, nullclines, and vector field for the Lotka-Volterra Predator Prey model link to vector field: Section 14. It forms the basis of many models used today in the analysis of population dynamics. A density dependent delayed predator-prey model with Beddington-DeAngelis type functional response incorporating a prey refuge, Communications in Nonlinear science and Numerical Simulation, 22 (1-3), 427-450, 2015 (ISSN No. To learn how to use the OPTIONS. Predator-prey relationships exist in all habitats and ecosystems. Matlab provides pretty comprehensive support to plot functions in cartesian coordinates. One of the phenomena demonstrated by the Lotka-Volterra model is that, under certain conditions, the predator and prey populations are cyclic with a phase shift between them. Mondaini (Ed. The prey is the organism which the predator eats. ) (You may also add in some comments such on calculations of constants,how you derive at the values etc. A predator is an animal that eats other animals, and Next in this python matplotlib blog, we will understand different kinds of plots. 002xy dy/dt = -0. Project Scope: The U. no migration is allowed into or out of the system) there are only 2 types of animals: the predator and the prey. So one way of using MATLAB to plot phase portrait of the predator-prey Lotka-Volterra system can be (for the case α=β=δ=γ=1):. This Matlab based programme simulates a simple predator-prey system consisting of interacting populations of foxes and rabbits. 0, x(0) = 100, y(0) = 8. Modeling Lotka-Volterra using ode23. I’m starting to play with dynamical systems so I figured I’d post a baby model. It has also been applied to many other fields, including economics. This example shows how to solve a differential equation representing a predator/prey model using both ode23 and ode45. The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations. I implement the ecological model in an eco-evolutionary context with connected predator-prey adaptive radiations as emergent model outputs 20,34. 1The Malthusian growth model. Welcome to the Jungle + the official Twitter account for the Predator franchise! Own now on Digital and BluRay. We use the n-Patch Model, which considers space explicitly as a “Stepping Stone” system. I implement the ecological model in an eco-evolutionary context with connected predator-prey adaptive radiations as emergent model outputs 20,34. x0(t) = a x(t) b x(t) y(t) y0(t) = c y(t) + d x(t) y(t) Now convert our model to a matrix - vector system. Parameter avlue Interpretation a 1. DYNAMICS OF A MODEL THREE SPECIES PREDATOR-PREY SYSTEM WITH CHOICE by Douglas Magomo A Dissertation Submitted to the Graduate Studies Office of The University of Southern Mississippi in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Approved: August 2007 Reproduced with permission of the copyright owner. Set the solver type to SSA to perform stochastic simulations, and set the stop time to 3. Detect events during solution of ODE. , how predators affect prey populations, and vice-versa. It is called the Lotka-Volterra model. For this model the fit is lower than previous because of the complexity of the model when the number of prey is assumed as finite; when the number of parameters increases, the estimation process becomes more complex. α = exponential growth in population – used for preys,. Matlab program to plot a phase portrait of the Lotka-Volterra Predator Prey model. The system considers the effects of anomalous diffusion and generalized Michaelis–Menten-type reactions. The quadratic cross term accounts for the interactions between the species. View Notes - lecture3b-predator from MAE m20 at University of California, Los Angeles. 17 Predator-Prey Models The logistic growth model (Chapter 11) focused on a single population. Predation rate is simulated using the Holling's "disc equation" of functional response:. Some examples of predator-prey relationships are lion-cape buffalo, tiger-deer, snake-frog, python-rabbit, bear-fish and cheetah-gazelle. In a team effort, we created a system of closed differential equations for a predator-prey model where we were then able to generate numerical simulations through MATLAB to visualize the data. The predator-dependent model is more suitable for prey predator interactions in which predation involves the search process. a discrete time predator prey model specified by Neubert et al[9] which utilises the Ricker model to simulate prey growth. Predator prey offers this graphic user interface to demonstrate what we've been talking about the predator prey equations. And the third model is the famous Lotka-Volterra predator-prey equations. Edit and improve current numerical code for verifying predator prey simulation of a night club population. Teaching Biology with MATLAB Educators strive to empower students with the necessary tools to become successful scientists. MATLAB files for the discrete time model: predprey_discrete. A multi-objective optimization technique, the prey–predator algorithm, is employed with the objective to find the optimal values for the heat sink performance parameters, i. The main objective was to investigate the spatio-temporal pattern of diffusive prey-predator model and the emergence of irregular chaotic pattern as a result of prey-predator interaction. Mathematical models and logic suggests that a coupled system of predator and prey should cycle: predators increase when prey are abundant, prey are driven to low numbers by predation, the predators. One of the phenomena demonstrated by the Lotka-Volterra model is that, under certain conditions, the predator and prey populations are cyclic with a phase shift between them. To understand the basic concept of Prey-Predator dynamics using the established Mathematical model of Lotka-Volterra Equations, i. For this model the fit is lower than previous because of the complexity of the model when the number of prey is assumed as finite; when the number of parameters increases, the estimation process becomes more complex. We use the n-Patch Model, which considers space explicitly as a “Stepping Stone” system. Math 302 Schedule (subject to change) Matlab programs: predator-prey (cycle) Lotka-Volterra competition model Lotka-Volterra competition model. Predator-Prey Model with Functional and Numerical Responses Now we are ready to build a full model of predator-prey system that includes both the functional and numerical responses. The two outputs (predator and prey populations) are chosen as states to derive a nonlinear state-space description of the dynamics. The predator’s goal. since they will eventually run out of food, so let's add another term limiting growth and change the system to. The bifurcation analysis is done with respect to Holling parameter as well as quantity of additional food. Four important model assumptions: The prey population grows exponentially in the absence of predation. A pursuit curve is the path one creature takes while following another, and these can be used to model predator/prey chases, missiles homing in on a target, or even robot movement during a rendezvous. Predator-Prey Problem Consider the following model for a two-species interaction (one predator, one prey). It essentially shows the growth of two populations co-existing together, one being the prey, the other the predators. Notice, if y= 0, then there is no predator and the prey population grows exponentially. Students can change various components of a predator/prey model, including birth factor, lifespan, and habitat area. To investigate the behavior of this model, we will use MATLAB's ODE solver. In the model to be formulated, it is now assumed that instead of a (deterministic) rate of predator and prey births and deaths, there is a probability of a predator and prey birth or death. Species compete, evolve and disperse simply for the purpose of seeking resources to sustain their struggle for their very existence. ODE Event Location. Here is my code: function predprey2 % predprey: MATLAB function that takes an initial guess of the parameter % values for the predator prey equation and returns the best fitting. Predator x prey rp? bedin 10 minutes ago. A small time step (dt) shows that the system is stable; a larger one leads to instability and thus highlights the importance of parameter choice. Follow 105 views (last 30 days) Jovos on 8 Apr 2016. First, a model of the dynamics of the predator/prey interaction is either chosen from existing pursuit–evasion models or developed to more specifically address the behaviour seen in experiments. Animals that feed on algae and plankton, such as fish and turtles, will have less food. MATLAB gives us the answer 4. Usage of odeset and table indicating which options work with each ODE solver. The Matlab command ode45 can be used to solve such systems of differential equations. About the author isee systems is the world leader and innovator in Systems Thinking software. A pursuit curve is the path one creature takes while following another, and these can be used to model predator/prey chases, missiles homing in on a target, or even robot movement during a rendezvous. 0, x(0) = 100, y(0) = 8. The physical system under consideration is a pair of animal populations. We implemented this model in Matlab to simulate a velociraptor hunting a thescelosaurus and an African lion hunting a gazelle. pdf: Adding self interaction to the Predator model and its nullclines and explaining the WWI Mediterranean Sea data. 3 High-resolution shock-capturing schemes. Working under the guidance of Prof. • Here are some of the functions available in MATLAB used for curve fitting: - polyfit. This example shows how to solve a differential equation representing a predator/prey model using both ode23 and ode45. The Lotka-Volterra predator-prey model : dx/dt =px−qxy. Here is a link for a biological perspective on the Lotka-Volterra model that includes discussion of the four quadrants and the lag of predators behind prey. Take a trip into an upgraded, more organized inbox. The MATLAB code is mostly self explanatory, with the names of variables and parameters corresponding to the symbols used in the finite difference methods described in the paper. But if$ O0, & P0,∆ 0,and O0 are fulfilled concurrently, then both populations can survive. A Predator-Prey model: Suppose that we have two populations, one of which eats the other. 001, χ 2 = 89. The right hand side of our system is now a column vector: we identify x with the component x(1) and y with the component x(2). Instead, you need to add. B=Rate at which predators destroy prey. The two equilibrium points of this system are found to be and since at both these points the necessary and sufficient conditions and are satisfied. Expressions of players' guaranteed payoffs in a noncooperative game modelling commodity purchase and realization process. mathematical model of the hydrodynamics of prey in the flow field created by a suction-feeding predator. Matlab ODE solvers, ODE 15s and ODE 23s which have been acknowledged to solve similar problems effectively. This project results in a Lotka-Volterra model which simulates the dynamics of the predator-prey relationship. b) The rabbits eat grass and breed. Level C: Waves of Change: Predator and Prey Dynamics. The Lotka-Volterra model assumes that the prey (squirrel) population’s growth is exponential and independent of the predator (fox) population, but the decline of the squirrel population is affected by both its own population size as well as the fox population size. infected prey are predator-prey model using infected prey without taking healthy prey into account and on predator-prey model while taking healty prey into account. nah berhubung disini saya memakai model perlambatan jadi di dalam model predator-prey dengan perlambatan dipertimbangkan waktu tunda dari prey. 8 predator death rate. This example shows how to solve a differential equation representing a predator/prey model using both ode23 and ode45. The variables x and y measure the sizes of the prey and predator populations, respectively. The functions y1 and y2 measure the sizes of the prey and predator populations respectively. Still-Life: simple static pattern Oscillator: repeating patterns (a super set of still life's) Spaceships: patterns that translate themselves across the board Other patterns include: Methuselah's, Diehard, and 3-D models An Interactive Presentation by: Grayson Sally, Wendy. Choosing appropriate techniques of model analysis is often a difficult task. These reactions can be interpreted as a simple predator-prey model if one considers that the prey population (y1) increases in the presence of food (x) (Reaction 1), that the predator population (y2) increases as they eat prey (Reaction 2), and that predators (y2) die of natural causes (Reaction 3). At the same time , a trio of coming-of-age Predators have arrived to collect the skulls of the aliens as trophies , and the humans are caught between a deadly battle between the Spectacular and decent Aliens/Predators movie set in Antarctica where a motley group takes on extraterrestrial monsters. In The Lotka Volterra Predator-prey Model, The Changes In The Predator Population Y And The Prey Population X Are Described By The Following Equations: Δxt=xt+1−xt=axt−bxtyt Δyt=yt+1−yt=cxtyt−dyt Write A Function Simulatepredatorprey (x,y, A,b,c,d, T) That Takes In The Initial Population This problem has been solved!. The main objective was to investigate the spatio-temporal pattern of diffusive prey-predator model and the emergence of irregular chaotic pattern as a result of prey-predator interaction. Eigenvalues and eigenvectors. This project results in a Lotka-Volterra model which simulates the dynamics of the predator-prey relationship. In this study, the approximate solutions of the predator–prey system with delay have been obtained by using the modified Chebyshev collocation method. To understand how predators optimize foraging strategies, extensive knowledge of predator behavior and prey distribution is needed. The number of predators is represented by y, the number of prey by x. Yang, Yong S. (This Malthus-type equation gives. a discrete time predator prey model specified by Neubert et al[9] which utilises the Ricker model to simulate prey growth. Open the first file for this module by typing on the Matlab command line: ppmodel1. 1 Introduction. If algae and plankton communities are threatened, the entire food web may change. In this work, we investigate numerically a system of partial differential equations that describes the interactions between populations of predators and preys. This paper investigates a dynamical predator-prey interaction model that incorporates: (a) hunting cooperation among predators; (b) Allee effect in prey. At the other extreme,. b) The rabbits eat grass and breed. function to be a Di erence sequence and study the convergence of the model. (5 stars rating will be given =). In the notes, the author has solved the above system using Matlab numerical solver ode45. 1 Predator-Prey,model A In this exercise you will solve an ODE-system describing the dynamics of rabbit and fox populations. Modeling Lotka-Volterra using ode23. Modded Creatures Are supported, though they may sometimes give unexpected results Join the Alien VS Predator fight with these two new races with unique traits! Whether you're a fan of this universe or not, you can't deny these races. For example, in (1) if there is an absence of predators, i. A small time step (dt) shows that the system is stable; a larger one leads to instability and thus highlights the importance of parameter choice. Imitative modeling of nerve fibers conductivity. Modeling Predator-Prey Interactions" • The Lotka-Volterra model is the simplest model of predator-prey interactions. 34% for predators. By implementing the scheme (4) in Matlab and using the parameter given in ableT 1, and initial conditions x 0 = 2 and y 0 = 1 for t 2[0;20] we get the plot given in gure 1. (a) Derive the exact solution for the Predator-Prey Models. This objective is being realized in collaboration with in-dustrial producers of biological agents, since both the selec-tion and use of the agent are relevant to applications by farm-ers and agronomists. Then, numerical simulation of the global dynamic behavior of the system is presented in Section “NUMERICAL SIMULATIONS VIA MATCONT SOFTWARE”. In this study, the approximate solutions of the predator–prey system with delay have been obtained by using the modified Chebyshev collocation method. To use, put "rainbow;" at the top of your Matlab file and use the command "colormap(rainbowMap);" with the surf command. An individual of each species is simulated as a particle moving in a random walk. 350 Handouts and M-files. The Lotka-Volterra predator-prey model : dx/dt =px−qxy. The predator-dependent model is more suitable for prey predator interactions in which predation involves the search process. Predator-Prey-Scavenger model Darby Vaughn May 3, 2018 Abstract The Lotka-Volterra equations, commonly called the predator-prey equations, are used to. Based on the analysis of first model, four equilibriums are obtained. Given the following data • It is important to have in mind that these models are good only in the region we have collected data. The predator/prey model explores a moose and wolf population living on a small island. • Communicating model dynamics and results in practical terms results provided evidence for policy decisions aimed at retaining population balance between predator and prey (MATLAB). ion() mu, sigma = 100, 15 fig = plt. The Lotka-Volterra equations describe an ecological predator-prey (or parasite-host) model which assumes that, for a set of fixed positive constants (the growth rate of prey), (the rate at which predators destroy prey), (the death rate of predators), and (the rate at which predators increase by consuming prey), certain simple conditions hold in the population change rates for prey and predat. Predator-Prey: BaitFish Epidemic Natural Selection Predator-Prey: Epidemic Population Growth Predator-Prey: Epidemic Population Growth: Predator-Prey: Molecular Evolution and Phylogenetics: Jukes-Cantor Model: Jukes-Cantor nucleotide substitution model in Excel. The term prey fish is actually a loose term used by anglers to refer to certain non-game fish species that are the main food items for popular sport. What is the carrying capacity of the US according to this model? Answer: Since we start with observations in 1800 it makes sense to choose the variable t as time elapsed since. The goal of the design project is to write MATLAB scripts that determine the forces that must act on the predator and the prey to achieve their objectives. An individual of each species is simulated as a particle moving in a random walk. The main technique is that this method transforms the original problem into a system of nonlinear algebraic equations. I implement the ecological model in an eco-evolutionary context with connected predator-prey adaptive radiations as emergent model outputs 20,34. The Lotka-Volterra equations were developed to describe the dynamics of biological systems. A multi-objective optimization technique, the prey–predator algorithm, is employed with the objective to find the optimal values for the heat sink performance parameters, i. I'm bored and I would like to do a predator x prey. I do the following: Step 1 - I created a file entitled pred_prey_odes. Unfortunately, in its original form Lotka-Volterra has some significant problems. It forms the basis of many models used today in the analysis of population dynamics. Nonreal eigenvalues. AMS Subject Classi cation: 39A10, 37H10, 60J10, 60G42 Key Words and Phrases: Di erence Equations, Markov chains, Martin-gales, Prey-predator. 10 Dumbfounding Examples of Predator-Prey Relationships. AbstractTwo. I have a program called Predator Prey that's in the collection of programs that comes with NCM, Numerical Computing with MATLAB. Introduction: In the type II functional response, the rate of prey consumption by a predator rises as prey density increases, but eventually levels off at a plateau (or asymptote) at which the rate of consumption remains constant regardless of increases in prey density (see also TYPE I and TYPE III FUNCTIONAL RESPONSE). I have a question about the eigenvalues of the prey-predator model called Lotka-Volterra. MATLAB write a code on a predator-prey model (Examples provided below the question. This example shows how to solve a differential equation representing a predator/prey model using both ode23 and ode45. m containing the following code:. Suchen Answers Clear Modeling Lotka-Volterra using ode23. One significant component in these systems is the functional response describing the number of prey consumed per predator per unit time for given quantities of prey N and predators P. Plus, this study also intended to explore the occurrence of diffusion-induced instability (Turing instability) and its effect to the dynamics of prey-predator. The variables x and y measure the sizes of the prey and predator populations, respectively. As an example, the well-know Lotka-Volterra model (aka. Task 3: Read about the Lotka-Volterra Model to describe competition between predators and prey. And the third model is the famous Lotka-Volterra predator-prey equations. The full prey equation is The first term ( rN ) describes exponential population growth in the absence of the predator, and the second term (- aNP ) is the death rate due to the predator. Then, the Predator-Prey model with Holling function of type II if P denotes the Predator population is (1) H ˙ (t) = r 1-H K H-g (H, P), P ˙ (t) = e g (H, P)-γ P, where r denotes the intrinsic growth rate of the Preys, K is the carrying capacity of the environment, γ is the mortality rate of Predators, e is the coefficient in conversing. The prey still relies on the food source, but the predator relies solely on the former competitor. Neutron Transport Model Fundamental Probability and Statistics Theory. Prey-resource, prey-prey and predator-prey trait. This work is arranged as follows: Sections “PREDATOR–PREY MATHEMATICAL MODEL” and “WEAK ALLE EFFECT CASE STUDY ANALYSIS” recall the nonlinear model as well as its basic dynamics. The Lotka-Volterra model is one of the earliest predator-prey models to be based on sound mathematical principles. González-Yañez, The Leslie-Gower predator-prey model with Allee effect on prey: A simple model with a rich and interesting dynamics, in: R. For this model the fit is lower than previous because of the complexity of the model when the number of prey is assumed as finite; when the number of parameters increases, the estimation process becomes more complex. Prey-resource, prey-prey and predator-prey trait. For some reason it does not want to work. González-Olivares and B. By implementing the scheme (4) in Matlab and using the parameter given in ableT 1, and initial conditions x 0 = 2 and y 0 = 1 for t 2[0;20] we get the plot given in gure 1. Andrew, Nick, and I worked on this project. (This Malthus-type equation gives. In the model to be formulated, it is now assumed that instead of a (deterministic) rate of predator and prey births and deaths, there is a probability of a predator and prey birth or death. DYNAMICS OF A MODEL THREE SPECIES PREDATOR-PREY SYSTEM WITH CHOICE by Douglas Magomo A Dissertation Submitted to the Graduate Studies Office of The University of Southern Mississippi in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Approved: August 2007 Reproduced with permission of the copyright owner. To understand how predators optimize foraging strategies, extensive knowledge of predator behavior and prey distribution is needed. And creating the different types of 3D plots with its function, syntax and code,with the help of solving each types of an example. Simple finite element methods for approximating predator-prey dynamics in two dimensions using MATLAB. 7 Numerical Test 9. Describing the dynamics of such models occasionally requires some techniques of model analysis. Notice, if y= 0, then there is no predator and the prey population grows exponentially. Predator-prey relationships exist in all habitats and ecosystems. I was wondering if someone might be able to help me solve the Lotka-Volterra equations using MatLab. Stability and Hopf bifurcation of a diffusive predator-prey model with predator saturation and competition. This suggests the use of a numerical solution method, such as Euler's Method, which we. Chakraborty of IIT Kharagpur. Prey increases the predator population growth rate,. To use, put "rainbow;" at the top of your Matlab file and use the command "colormap(rainbowMap);" with the surf command. This file draws a bifurcation diagram for the Holling type II predator-prey model. Lotka-Volterra Predator-Prey Problems. Mathematical models and logic suggests that a coupled system of predator and prey should cycle: predators increase when prey are abundant, prey are driven to low numbers by predation, the predators. In the parameter estimation of the saturation predator-prey model, the output fit for prey is 71. Statistical model. the Predator-Prey model) is numerically simulated and solved using Runge-Kutta 4th order (RK4), in both languages, Python and. Predation — For alternative meanings of predator and prey, see Predator (disambiguation) and Prey (disambiguation). ion() mu, sigma = 100, 15 fig = plt. Modeling carefully to the smallest detail, this 3D 3D Print Model will give you a superb result. This project results in a Lotka-Volterra model which simulates the dynamics of the predator-prey relationship. Perform simulation, prediction, and forecasting at the command line, specify initial conditions. Instructor: Cleve Moler The classic Lotka-Volterra model of predator-prey competition is a nonlinear system of two equations, where one species grows exponentially and the other decays exponentially in the absence of the other. My code doesn't seem to be working. A multi-objective optimization technique, the prey–predator algorithm, is employed with the objective to find the optimal values for the heat sink performance parameters, i. pdf L43-PPAddingSINullclines-handout. Predator prey offers this graphic user interface to demonstrate what we've been talking about the predator prey equations. The predator (gold) seems to smooth over the variation in the prey (blue) but take a look around t=85: there is a random bump in prey which results in a little shoulder in the decline of the predators. L42-PPMatlab-handout. Nonreal eigenvalues. The main objective was to investigate the spatio-temporal pattern of diffusive prey-predator model and the emergence of irregular chaotic pattern as a result of prey-predator interaction. ts is the vector of time values (Matlab chooses h automatically at each step) ys is an array containing the values of y: row j contains the values of y1 (rabbits) and y2 (foxes) at the time ts(j). (3)yxy to illustrate the Maple, Mathematica, and MATLAB techniques needed for these investigations. Usage of odeset and table indicating which options work with each ODE solver. Midterm 2 (lectures of October 18–November 8) Nov 21. 26th Sep, 2017. We consider a complex population dynamics mathematical model involving foxes and rabbits as predators and prey. Ask Question Asked 5 years, 4 months ago. Participants performed this task standing on separate sides of a board and controlling a marker representing them. George Maria Selvam2 and V. How can I draw a bifurcation plot in MATLAB? Can someone help. We model the hunt as a game of three explicit stages: the stalk, the attack, and the subdual. x=[628 703 778]; y=[1771 1403 1035]; There are numbers of rabbits and foxes in following years. where x(t) and y(t) are the prey and predator population sizes at time t, and p,q, r, and s are biologically determined parameters. With MATLAB's built-in functions and easy syntax, integrating computation into coursework is not only feasible but also straightforward. [ts,ys] gives a table with t in column 1, y1 (rabbits) in column 2, y2 (foxes) in column 3. This work is arranged as follows: Sections “PREDATOR–PREY MATHEMATICAL MODEL” and “WEAK ALLE EFFECT CASE STUDY ANALYSIS” recall the nonlinear model as well as its basic dynamics. Distinct real eigenvalues. Tutorial: Use MATLAB to illustrate a predator-prey relationship using a Discrete Dynamical Systems Model. Abstract This lecture discusses how to solve Predator Prey models using MatLab. The prey-predator-predator equations get cool looking chaotic dynamics not seen in 1 or 2 dimen-sions. rar > FD2D 2D Predator Prey Simulation. A DC Motor subsystem which is model using Simulink blocks and a 3D Model which is imported from Solidworks using the SimMechanics Link. Our model used a spectrum of correlated random walk rules of movement, from strictly nondirectional to almost directional movement, while abstracting the cost associated with searching. B=Rate at which predators destroy prey. Bio-mathematical Prey-Predator Model with Marine Protect Area(MPA) and Harvesting Dr. I have the data, X-prey , Y-predators, and I have symulated the paramters, It looks like below. 11: Predator-Prey Equations The classic Lotka-Volterra model of predator-prey competition is a nonlinear system of two equations, where one species grows exponentially and the other decays exponentially in the absence of the other. The delay parameter tau represents the delay from birth to adulthood of a member of the population. Modeling Predator-Prey Interactions" • The Lotka-Volterra model is the simplest model of predator-prey interactions. For example, in (1) if there is an absence of predators, i. To learn how to use the OPTIONS. I have a question about the eigenvalues of the prey-predator model called Lotka-Volterra. The predator/prey model explores a moose and wolf population living on a small island. He showed that under certain conditions, all solutions possess an oscillatory behavior. predators eat prey or other predators. The Lotka-Volterra equations describe an ecological predator-prey (or parasite-host) model which assumes that, for a set of fixed positive constants (the growth rate of prey), (the rate at which predators destroy prey), (the death rate of predators), and (the rate at which predators increase by consuming prey), certain simple conditions hold in the population change rates for prey and predat. In a team effort, we created a system of closed differential equations for a predator-prey model where we were then able to generate numerical simulations through MATLAB to visualize the data. 1 Introduction. But if \$ O0, & P0,∆ 0,and O0 are fulfilled concurrently, then both populations can survive. 2016-10-10 Modeling and Simulation of Social Systems with MATLAB 35 SIR model ! A general model for epidemics is the SIR model, which describes the interaction between Susceptible, Infected and Removed (Recovered) persons, for a given disease. % The differential equations are dy/dt = f(t,y) where y is a vector of unknown functions. Eigenvalues and eigenvectors. Using Maple To plot a solution curve for the system in (3) we need only load the DEtools package and use the DEplot function. We now replace the difference equation model used there with a more sophisticated differential equation model. 3 Nilai awal populasi prey - predator 35 4. 10 Dumbfounding Examples of Predator-Prey Relationships. focus entirely on systems which are classi ed as predator-prey models. Reni Sagaya Raj1, A. These trajectories were not coming from the near-useless formula for trajectories, but rather from the differential equations themselves. 26th Sep, 2017. The simplest version: where x and y represent the biomass of prey and predators, respectively, a is the prey growth rate, c the predator death rate,. The New York Times: Find breaking news, multimedia, reviews & opinion on Washington, business, sports, movies, travel, books, jobs, education, real estate, cars & more at nytimes. Prey-resource, prey-prey and predator-prey trait. This paper investigates a dynamical predator-prey interaction model that incorporates: (a) hunting cooperation among predators; (b) Allee effect in prey. Collection for AvP for suggestion of the Serious RP Aliens vs Predators server. pdf L43-PPAddingSINullclines-handout. Grid-based simulations are simple to. How might another predator effect our mouse and hawk distribution? 12. Based on the analysis of first model, four equilibriums are obtained. The parametric curves traced by the solutions are sometimes also called their trajectories. It was developed independently by:" – Alfred Lotka, an American biophysicist (1925), and" – Vito Volterra, an Italian mathematician (1926). Assuming stochastic switching for some parameters we analyze this dynamical system as the ergodic Markov chain. Modelling Predator-Prey Interactions with ODE Modelling Predator-Prey Interactions with ODE Shan He School for Computational Science University of Birmingham Module 06-23836: Computational Modelling with MATLAB Modelling Predator-Prey Interactions with ODE Outline Outline of Topics Predator-Prey Models The Lotka-Volterra (LV) model. Suppose, for example, you want to study the effect of the interaction coefficients, and , in the Lotka-Volterra predator-prey model. Reni Sagaya Raj1, A. Indian Python swallowing a small Chital deer at Mudumalai National Park …. MATLAB scripts for the predator-prey model: MATLAB scripts for the second skydive model with event handling:. 2; delta = 0. The second project of the semester was the predator prey model. We compare it to a further class of models where the Ricker model is replaced with the tent map and the logistic map. Descriptions: The classic Lotka-Volterra model of predator-prey competition, which describes interactions between foxes and rabbits, or big fish and little fish, is the foundation of mathematical ecology. The human influence is then introduced to the interactive model to observe if the species get extinct. prey refuge, the functional response is f(N) = c(1−c ′)e 1N a+h1e1N, where N is total num-ber of prey individuals. htm, change:2008-10-28,size:9883b > Kanxxx1. These are functions of time, and the time scale is rather long. A model for predator-prey populations is given by: If there is a. prey interactions is the Lotka-Volterra Model. González-Yañez, The Leslie-Gower predator-prey model with Allee effect on prey: A simple model with a rich and interesting dynamics, in: R. Consider the pair of first-order ordinary differential equations known as the Lotka-Volterra equations, or predator-prey model: dx dt = x - α xy dy dt = - y + β xy. Selection of valuable stocks by multi objective optimization - modelling 4 variables (financial indicators): EPS, ROCE, P/E Ratio and Liquidity Ratio using neural networks and then applying Prey-Predator algorithm for generating the Pareto front between risk and return. The file specifies the state derivatives and model outputs as a function of time, states, inputs, and model parameters. We now replace the difference equation model used there with a more sophisticated differential equation model. Then, the Predator-Prey model with Holling function of type II if P denotes the Predator population is (1) H ˙ (t) = r 1-H K H-g (H, P), P ˙ (t) = e g (H, P)-γ P, where r denotes the intrinsic growth rate of the Preys, K is the carrying capacity of the environment, γ is the mortality rate of Predators, e is the coefficient in conversing. Tips to Develop the Lotka-Volterra Equations Let us now look at how to implement the equations in MATLAB. x0(t) = a x(t) b x(t)y(t) y0(t) = c y(t) + d x(t)y(t) INow convert our model to a matrix - vector system. Prey-resource, prey-prey and predator-prey trait. Introduction: In the type II functional response, the rate of prey consumption by a predator rises as prey density increases, but eventually levels off at a plateau (or asymptote) at which the rate of consumption remains constant regardless of increases in prey density (see also TYPE I and TYPE III FUNCTIONAL RESPONSE). • Communicating model dynamics and results in practical terms results provided evidence for policy decisions aimed at retaining population balance between predator and prey (MATLAB). MATLAB write a code on a predator-prey model (Examples provided below the question. ILet’s try to solve a typical predator prey system such as the one given below numerically. The coe cient was named by Volterra the coe cient of auto-increase. Within the predator's reaction distance, our model is effectively the same as Domenici's model and prey should escape at least partially away from the approaching predator (90–180 deg escape angle), with the exact angle depending on relative prey velocity (Fig. Load the model. MATLAB files for the discrete time model: predprey_discrete. Here is a link for a biological perspective on the Lotka-Volterra model that includes discussion of the four quadrants and the lag of predators behind prey. Yang, Yong S. Nonlinear model predictive control (planning) for level control in a surge tank, click here. MATLAB之父：编程实践 (中译本) 中文pdf扫描版[39MB],本书是MATLAB之父Cleve Moler的Experiments with MATLAB一书的中译本，介绍了MATLAB程序设计的思想与方法，思路独特、视野宽广，语言严谨又不失风趣幽默，案例程序完整精练，易学易懂，欢迎下载. Selection of valuable stocks by multi objective optimization - modelling 4 variables (financial indicators): EPS, ROCE, P/E Ratio and Liquidity Ratio using neural networks and then applying Prey-Predator algorithm for generating the Pareto front between risk and return. Logistic, predator-prey and size-structured models Epidemic Models. Nonetheless, you get behavior that looks like a predator-prey system (unstable focus). MATLAB Answers. The aims of this thesis is to analyze two predator-prey model. In this lecture, I am going to illustrate Matlab code by building two simple computer models. It was developed independently by:" - Alfred Lotka, an American biophysicist (1925), and" - Vito Volterra, an Italian mathematician (1926). The basic assumptions used in our simple toy-model system are stated below. Here, Thanks for contributing an answer to Mathematica Stack Exchange!. How to Solve and Plot Lotka-Volterra Differential Equations in Matlab. Predators reduce prey population growth rate, proportional to both the predator and prey populations. How can I draw a bifurcation plot in MATLAB? Can someone help. At the same time , a trio of coming-of-age Predators have arrived to collect the skulls of the aliens as trophies , and the humans are caught between a deadly battle between the Spectacular and decent Aliens/Predators movie set in Antarctica where a motley group takes on extraterrestrial monsters. We conclude the third population example by presenting the model information returned by PRESENT. This matlab file plots solutions and isoclines of the Holling type II predator-prey model. This objective is being realized in collaboration with in-dustrial producers of biological agents, since both the selec-tion and use of the agent are relevant to applications by farm-ers and agronomists. The predator/prey relationship we have modeled in class is a simple relationship because it involves only two animals, the hawk and the mouse. Matplotlib allows you to specify the color of the graph plot. As you may have noted in your experiments, neither. 6: Estimation and Prediction This video goes over the essential intuition and equations for estimation and prediction using an estimated sample. The predator’s goal. The Lotka-Volterra model assumes that the prey (squirrel) population’s growth is exponential and independent of the predator (fox) population, but the decline of the squirrel population is affected by both its own population size as well as the fox population size. et al, 2010). Cho, Won G. Optimal control of predator-prey model with distributed delay. Assuming stochastic switching for some parameters we analyze this dynamical system as the ergodic Markov chain. Predators reduce prey population growth rate, proportional to both the predator and prey populations. Predation rate is simulated using the Holling's "disc equation" of functional response:. This Matlab based programme simulates a simple predator-prey system consisting of interacting populations of foxes and rabbits. An ecoepidemiological predator‐prey model with standard disease incidence An ecoepidemiological predator‐prey model with standard disease incidence Haque, Mainul; Zhen, Jin; Venturino, Ezio 2010-03-15 00:00:00 1 School 2 Department of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, U. The interactions of ecological models may occur among individuals of the same species or individuals of different species. In 1920 Lotka extended the model, via Andrey Kolmogorov, to "organic systems" using a plant species and a herbivorous animal species as an example and. In this Tutorial we will see how we to make PID control of 3D Model of a Robot Gripper Mechanism. Introduction: In the type II functional response, the rate of prey consumption by a predator rises as prey density increases, but eventually levels off at a plateau (or asymptote) at which the rate of consumption remains constant regardless of increases in prey density (see also TYPE I and TYPE III FUNCTIONAL RESPONSE). The predator population decreases exponentially in the absence of prey. From the plot we see this is a good guess: Interpolation. The ode45 command is an integrated six-stage, fifth-order, Runge-Kutta method of solving differential equations. Actually create a connection between Matlab and Autoware is my purpose which. Suppose in a closed eco-system (i. Abstract We study a predator–prey model with the Allee effect on prey and whose dynamics is described by a system of stochastic differential equations assuming that environmental randomness is represented by noise terms affecting each population. Predator-Prey Model with Functional and Numerical Responses Now we are ready to build a full model of predator-prey system that includes both the functional and numerical responses. nah berhubung disini saya memakai model perlambatan jadi di dalam model predator-prey dengan perlambatan dipertimbangkan waktu tunda dari prey. The term crepresents the attack rate of predator on prey and the parameter arepresents the half saturation constant. α = exponential growth in population – used for preys,. a discrete time predator prey model specified by Neubert et al[9] which utilises the Ricker model to simulate prey growth. Stochastic predator-prey models for modelling population cycles (little pieces of code for my own use) - Matlab code for numerical integration of a SDE version of Turchin & Hanski's 1997. Predators reduce prey population growth rate, proportional to both the predator and prey populations. Within the predator's reaction distance, our model is effectively the same as Domenici's model and prey should escape at least partially away from the approaching predator (90–180 deg escape angle), with the exact angle depending on relative prey velocity (Fig. Predator-prey model, integro-di erential equations. Aggregate models consider a population as a collective group, and capture the change in the size of a population over time. The paper discusses the existences and stabilities of each possible. (This Malthus-type equation gives. We sup pose that the prey migrate between two patches randomly. L42-PPMatlab-handout. 18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015 View the complete course:. Participants performed this task standing on separate sides of a board and controlling a marker representing them. The main technique is that this method transforms the original problem into a system of nonlinear algebraic equations. 1 Introduction Predator-prey models are the building blocks of the. ILet’s try to solve a typical predator prey system such as the one given below numerically. We will start with the prey population. Additionally, the predation functional response or predation consumption rate is linear. The predator’s goal. It also assumes no outside influences like disease, changing conditions, pollution, and so on. The right hand side of our system is now a column vector: we identify x. Working under the guidance of Prof. Lotka-Volterra model: We shall start with the simplest of the predator-prey models, which is known as the Lotka. One significant component in these systems is the functional response describing the number of prey consumed per predator per unit time for given quantities of prey N and predators P. Now, if h1 and e1 are two constants representing handling time of the predator per prey item and ability of. Predator series, which are primarily first-person action-adventure titles. Selection of valuable stocks by multi objective optimization - modelling 4 variables (financial indicators): EPS, ROCE, P/E Ratio and Liquidity Ratio using neural networks and then applying Prey-Predator algorithm for generating the Pareto front between risk and return. m - discrete time simulation of predator prey model Continuous Time Model. Descriptions: The classic Lotka-Volterra model of predator-prey competition, which describes interactions between foxes and rabbits, or big fish and little fish, is the foundation of mathematical ecology. In order to analyze the stability of the solution, we make use of the Jacobian matrix and the resultant characteristic polynomial. The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, differential equations. 1 Introduction. We now replace the difference equation model used there with a more sophisticated differential equation model. Ask Question I was wondering if someone might be able to help me solve the Lotka-Volterra equations using MatLab. Predator-Prey Models Downloading Matlab Files In your command window (not the Matlab window), cd to the directory where you saved the file, and enter the command. This model reads as [22] proposed that the prey exhibits a demographic Allee effect at low population densi-ties due to reasons other than predation by the focal predator as. 0 ⋮ I wonder if my code is correct. Students can change various components of a predator/prey model, including birth factor, lifespan, and habitat area. My book that's available on the MathWorks website. The right hand side of our system is now a column vector: we identify x. For Target and Walmart, the predator prey models mentioned above do not accurately fit the. Nonlinear systems: predator-prey problems. Predator-Prey Model, University of Tuebingen, Germany. SIR models Neutron Transport Models. This is an assignment in Python, I contributed to a numerical Python MOOC from George Washington University. Each month the number of rats would increase by 20% if there were no owls to eat them. nah berhubung disini saya memakai model perlambatan jadi di dalam model predator-prey dengan perlambatan dipertimbangkan waktu tunda dari prey. The parametric curves traced by the solutions are sometimes also called their trajectories. Auto-oscillation processes in biological systems. I Let's try to solve a typical predator prey system such as the one given below numerically. These reactions can be interpreted as a simple predator-prey model if one considers that the prey population (y1) increases in the presence of food (x) (Reaction 1), that the predator population (y2) increases as they eat prey (Reaction 2), and that predators (y2) die of natural causes (Reaction 3). Detective Constable in the Sex Crimes Unit Canadian Law Enforcement. In this study, the approximate solutions of the predator–prey system with delay have been obtained by using the modified Chebyshev collocation method. The rate of exposure to the toxicantsis differentfor both species. If algae and plankton communities are threatened, the entire food web may change. a) Write a MATLAB code or use a book function or a code from the class web page to solve the above (IVP) for = :1;1;2. (This Malthus-type equation gives. From the plot we see this is a good guess: Interpolation. % The differential equations are dy/dt = f(t,y) where y is a vector of unknown functions. Attentional strategies for dynamically focusing on multiple predators/prey, click here. Prey-resource, prey-prey and predator-prey trait. Modelling Predator-Prey Interactions with ODE Modelling Predator-Prey Interactions with ODE Shan He School for Computational Science University of Birmingham Module 06-23836: Computational Modelling with MATLAB Modelling Predator-Prey Interactions with ODE Outline Outline of Topics Predator-Prey Models The Lotka-Volterra (LV) model. Nullclines and phaseplanes MatLab, or even your own code. Statistical model. Or, play as the Predator to hunt the most worthy prey, choosing from your vast array of deadly alien tech to collect your trophies, one by one. Abstract We study a predator–prey model with the Allee effect on prey and whose dynamics is described by a system of stochastic differential equations assuming that environmental randomness is represented by noise terms affecting each population. for the dynamics of predator-prey system analysis. or prey Predator or prey, predator or prey. For best results, please make sure your browser is accepting cookies. A density dependent delayed predator-prey model with Beddington-DeAngelis type functional response incorporating a prey refuge, Communications in Nonlinear science and Numerical Simulation, 22 (1-3), 427-450, 2015 (ISSN No. Eigenvalues and eigenvectors. 3 High-resolution shock-capturing schemes. They have poor eyesight, and stalk prey using chemical receptors in their tongues and heat-sensors along the jaws. Please open MATLAB yourself and play around with this. Jungle woman, panther purring while she looks for prey. Tutorial: Use MATLAB to illustrate a predator-prey relationship using a Discrete Dynamical Systems Model. 1 Introduction Predator-prey models are the building blocks of the. dy/dt =rxy−sy. 7 Numerical Test 9. 001, χ 2 = 89. As the predator population is low the prey population will increase again. Digital Communication. The main technique is that this method transforms the original problem into a system of nonlinear algebraic equations. Predator-Prey: BaitFish Epidemic Natural Selection Predator-Prey: Epidemic Population Growth Predator-Prey: Epidemic Population Growth: Predator-Prey: Molecular Evolution and Phylogenetics: Jukes-Cantor Model: Jukes-Cantor nucleotide substitution model in Excel. Follow 105 views (last 30 days) The Lotka-Volterra predator-prey model : dx/dt =px. John Polking’s pplane: MATLAB, JAVA. Show how different parameter values change the nature of the model and its interpretation as here. Wang [22] and Xiao and Chen [24] studied the global stability of a stage-structured predator-prey system using the theory of competitive systems, while the model in Wang and Chen [23. Applying statistical approach jointly with MATHEMATICA, R, and MATLAB as the statistical software tools, we estimate the Markov transition. Ode45 Dynamic Ode45 Dynamic. Predator-Prey Problem Consider the following model for a two-species interaction (one predator, one prey). 0, x (0) = 100, y (0) = 8. The Lotka-Volterra model assumes that the prey (squirrel) population’s growth is exponential and independent of the predator (fox) population, but the decline of the squirrel population is affected by both its own population size as well as the fox population size. 18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015 View the complete course:. In this research article, we considered an ecological prey predator fishery model system with a generalized case where both the patches are accessible to both prey and predator. You're (probably) simulating the dynamics correctly, but the key word in your question is analyze. Taha Module 03 — Modeling of Dynamical Systems 8 / 26 Physical Laws and Equations TF Models Mechanical System Model Electrical System Model Predator-Prey Model Linearization of NL Systems. The PREY reproduce rapidly; for each animal alive at the beginning of the year, two more will be born by the end of the. The MATLAB function ode45 is based on R-K method of order 6 and adapts the step size. Each month the number of rats would increase by 20% if there were no owls to eat them. Should you need to perform advanced searches, bulk file or URL submissions or simply need a higher request throughput or daily allowance, there is a premium VirusTotal API that may suit your needs. Modeling Predator-Prey Interactions" • The Lotka-Volterra model is the simplest model of predator-prey interactions. And the third model is the famous Lotka-Volterra predator-prey equations. The coe cient was named by Volterra the coe cient of auto-increase. Follow 105 views (last 30 days) The Lotka-Volterra predator-prey model : dx/dt =px. Matlab ODE solvers, ODE 15s and ODE 23s which have been acknowledged to solve similar problems effectively. If populations of those animals decrease, there will be less food for apex predators such as tuna, sharks, and whales. I dont have any triggers. 18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015 View the complete course:. | 2020-11-23T19:43:47 | {
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https://mathematica.stackexchange.com/questions/152223/rescaling-list-with-negative-entries-using-limits-with-same-absolute-value | # Rescaling list with negative entries using limits with same absolute value
Suppose I have a list of both positive and negative numbers:
list = {3, 0, -4, 15, 0.006, -1}
I want to rescale the function over the range $[-1, 1]$. However, simply inputting
rescList = Rescale[list, {Min@list, Max@list}, {-1, 1}]
would not give the desired result:
{-0.263158, -0.578947, -1., 1., -0.578316, -0.684211}
The values 15 and -4 would be equidistant in this scaling. Plus, positive values would become negative.
A working solution would be to find the maximum in absolute value and rescale with respect to that:
absMax = Max[Abs@list];
rescList = Rescale[list, {-absMax, absMax}, {-1, 1}]
But I was wondering if a solution is possible without this intermediate step (and, of course, without making the code too bulky by, say, something like rescList = Rescale[list, {-Max[Abs@list], Max[Abs@list]}, {-1, 1}]).
Your apparent target, rescList, is equivalent to:
list / Max@Abs@list
This can be symbolically derived from Rescale
Rescale[a, {-m, m}, {-1, 1}]
a/m
Also equivalent:
Normalize[list, Max@*Abs]
Rescale[Abs@list]*Sign[list]
Normalize[list, Norm[#, ∞] &] (* thanks to J. M. *)
• Thank you, I still can't believe how I hadn't thought about just dividing by the max value. Anyway, I guess my question was more about "removing" the need for Max@Abs inside the code (to make it look better?). But I guess it's not really necessary. – Enzo Jul 26 '17 at 12:47
• @Enzo I'm sorry if this answer isn't satisfying. I'm not really sure what you're seeking. I added a form that uses Rescale and does away with Max, in case that is of some interest to you. I don't see how you can eliminate Abs in a non-contrived way as that seems intrinsic to your definition. – Mr.Wizard Jul 26 '17 at 12:56
• I wasn't criticizing your answer, I'm sorry if I gave that impression. In fact, it is exactly what I was looking for. My "issue" with Max and Abs was a purely aesthetical one, I was hoping for, maybe, some Rescale argument for that (Something like, for example, Rescale[x, Absolute] which I just made up). Just my mind wandering about, I'm afraid. – Enzo Jul 26 '17 at 13:07
• @Enzo Ah, I see, no problem. :-) – Mr.Wizard Jul 26 '17 at 13:08
• Normalize[list, Norm[#, ∞] &] ought to work as well. – J. M. will be back soon Jul 26 '17 at 13:16 | 2019-11-18T10:57:14 | {
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http://www.mathworks.com/help/stats/zscore.html?nocookie=true | Accelerating the pace of engineering and science
# zscore
Standardized z-scores
## Description
example
Z = zscore(X) returns the z-score for each element of X such that columns of X are centered to have mean 0 and scaled to have standard deviation 1. Z is the same size as X.
• If X is a vector, then Z is a vector of z-scores.
• If X is a matrix, then Z is a matrix of the same size as X, and each column of Z has mean 0 and standard deviation 1.
• For multidimensional arrays, z-scores in Z are computed along the first nonsingleton dimension of X.
example
Z = zscore(X,flag) scales X using the standard deviation indicated by flag.
• If flag is 0 (default), then zscore scales X using the sample standard deviation, with n - 1 in the denominator of the standard deviation formula. zscore(X,0) is the same as zscore(X).
• If flag is 1, then zscore scales X using the population standard deviation, with n in the denominator of standard deviation formula.
example
Z = zscore(X,flag,dim) standardizes X along dimension dim. For example, for a matrix X, if dim = 1, then zscore uses the means and standard deviations along the columns of X, if dim = 2, then zscore uses the means and standard deviations along the rows of X.
example
[Z,mu,sigma] = zscore(___) also returns the means and standard deviations used for centering and scaling, mu and sigma, respectively. You can use any of the input arguments in the previous syntaxes.
## Examples
expand all
### Z-Scores of Two Data Vectors
Compute and plot the z-scores of two data vectors, and then compare the results.
`load('lawdata.mat')`
Two variables load into the workspace: gpa and lsat.
Plot both variables on the same axes.
```plot([gpa,lsat])
legend('gpa','lsat','Location','East')```
It is difficult to compare these two measures because they are on a very different scale.
Plot the z-scores of gpa and lsat on the same axes.
```Zgpa = zscore(gpa);
Zlsat = zscore(lsat);
plot([Zgpa, Zlsat])
legend('gpa z-scores','lsat z-scores','Location','Northeast')```
Now, you can see the relative performance of individuals with respect to both their gpa and lsat results. For example, the third individual's gpa and lsat results are both one standard deviation below the sample mean. The eleventh individual's gpa is around the sample mean but has an lsat score almost 1.25 standard deviations above the sample average.
Check the mean and standard deviation of the z-scores you created.
` mean([Zgpa,Zlsat])`
```ans =
1.0e-14 *
-0.1088 0.0357```
` std([Zgpa,Zlsat])`
```ans =
1 1```
By definition, z-scores of gpa and lsat have mean 0 and standard deviation 1.
### Z-Scores for a Population vs. Sample
`load('lawdata.mat')`
Two variables load into the workspace: gpa and lsat.
Compute the z-scores of gpa using the population formula for standard deviation.
```Z1 = zscore(gpa,1); % population formula
Z0 = zscore(gpa,0); % sample formula
disp([Z1 Z0])
```
``` 1.2554 1.2128
0.8728 0.8432
-1.2100 -1.1690
-0.2749 -0.2656
1.4679 1.4181
-0.1049 -0.1013
-0.4024 -0.3888
1.4254 1.3771
1.1279 1.0896
0.1502 0.1451
0.1077 0.1040
-1.5076 -1.4565
-1.4226 -1.3743
-0.9125 -0.8815
-0.5724 -0.5530```
For a sample from a population, the population standard deviation formula with n in the denominator corresponds to the maximum likelihood estimate of the population standard deviation, and might be biased. The sample standard deviation formula, on the other hand, is the unbiased estimator of the population standard deviation for a sample.
### Z-Scores of a Data Matrix
Compute z-scores using the mean and standard deviation computed along the columns or rows of a data matrix.
`load('flu.mat')`
The dataset array flu is loaded in the workplace. flu has 52 observations on 11 variables. The first variable contains dates (in weeks). The other variables contain the flu estimates for different regions in the U.S.
Convert the dataset array to a data matrix.
`flu2 = double(flu(:,2:end));`
The new data matrix, flu2, is a 52-by-10 double data matrix. The rows correspond to the weeks and the columns correspond to the U.S. regions in the data set array flu.
Standardize the flu estimate for each region (the columns of flu2).
`Z1 = zscore(flu2,[ ],1);`
You can see the z-scores in the variable editor by double-clicking on the matrix Z1 created in the workspace.
Standardize the flu estimate for each week (the rows of flu2).
`Z2 = zscore(flu2,[ ],2);`
### Z-Scores, Mean, and Standard Deviation
Return the mean and standard deviation used to compute the z-scores.
`load('lawdata.mat')`
Two variables load into the workspace: gpa and lsat.
Return the z-scores, mean, and standard deviation of gpa.
`[Z,gpamean,gpastdev] = zscore(gpa)`
```Z =
1.2128
0.8432
-1.1690
-0.2656
1.4181
-0.1013
-0.3888
1.3771
1.0896
0.1451
0.1040
-1.4565
-1.3743
-0.8815
-0.5530
gpamean =
3.0947
gpastdev =
0.2435```
## Input Arguments
expand all
### X — Input datavector | matrix | multidimensional array
Input data, specified as a vector, matrix, or multidimensional array.
Data Types: double | single
### flag — Indicator for the standard deviation0 (default) | 1
Indicator for the standard deviation used to compute the z-scores, specified as 0 or 1.
### dim — Dimension1 (default) | positive integer
Dimension along which to calculate the z-scores of X, specified as a positive integer. For example, for a matrix X, if dim = 1, then zscore uses the means and standard deviations along the columns of X, if dim = 2, then zscore uses the means and standard deviations along the rows of X.
## Output Arguments
expand all
### Z — z-scoresvector | matrix | multidimensional array
z-scores, returned as a vector, matrix, or multidimensional array. A vector of z-scores has mean 0 and variance 1.
• If X is a vector, then Z is a vector of z-scores.
• If X is an array, then zscore is an array, with each column or row standardized to have mean 0 and variance 1 (depending on dim). If dim is not specified, zscore standardizes along the first nonsingleton dimension of X.
### mu — Mean scalar | vector
Mean of X used to compute the z-scores, returned as a scalar or vector.
• If X is a vector, then mu is a scalar.
• If X is a matrix, then mu is a row vector if zscore calculates the means along the columns of X (dim = 1), and a column vector if zscore calculates the means along the rows of X (dim = 2).
### sigma — Standard deviationscalar | vector
Standard deviation of X used to compute the z-scores, returned as a scalar or vector.
• If X is a vector, then sigma is a scalar.
• If X is a matrix, then sigma is a row vector if zscore calculates the standard deviations along the columns of X (dim = 1), and a column vector if zscore calculates the standard deviations along the rows of X (dim = 2).
expand all
### Z-Score
For a random variable X with mean μ and standard deviation σ, the z-score of a value x is
$z=\frac{\left(x-\mu \right)}{\sigma }.$
For sample data with mean $\overline{X}$ and standard deviation S, the z-score of a data point x is
$z=\frac{\left(x-\overline{X}\right)}{S}.$
z-scores measure the distance of a data point from the mean in terms of the standard deviation. This is also called standardization of data. The standardized data set has mean 0 and standard deviation 1, and retains the shape properties of the original data set (same skewness and kurtosis).
You can use z-scores to put data on the same scale before further analysis. This lets you to compare two or more data sets with different units.
### Multidimensional Array
A multidimensional array is an array with more than two dimensions. For example, if X is a 1-by-3-by-4 array, then X is a three-dimensional array.
### First Nonsingleton Dimension
A first nonsingleton dimension is the first dimension of an array whose size is not equal to 1. For example, if X is a 1-by-2-by-3-by-4 array, then the second dimension is the first nonsingleton dimension of X.
### Sample Standard Deviation
The sample standard deviation, S, is given by
$S=\sqrt{\frac{{\sum }_{i=1}^{n}{\left({x}_{i}-\overline{X}\right)}^{2}}{n-1}}.$
S is the square root of an unbiased estimator of the variance of the population from which X is drawn, as long as X consists of independent, identically distributed samples.
Notice that the denominator in this variance formula is n – 1.
### Population Standard Deviation
If the data is the entire population of values, then you can use the population standard deviation,
$\sigma =\sqrt{\frac{{\sum }_{i=1}^{n}{\left({x}_{i}-\mu \right)}^{2}}{n}}.$
If X is a random sample from a population, then μ is estimated by the sample mean, and σ is the biased maximum likelihood estimator of the population standard deviation.
Notice that the denominator in this variance formula is n.
### Algorithms
zscore returns NaNs for any sample containing NaNs.
zscore returns 0s for any sample that is constant (all values are the same). For example, if X is a vector of the same numeric value, then Z is a vector of 0s. If X is a matrix with a column of consisting of the same value, then that column of Z consists of 0s. | 2014-12-21T18:17:30 | {
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https://web2.0calc.com/questions/sphere_2 | +0
# Sphere
+5
484
6
A sphere encloses a cube with a volume of 1000 so that all vertices of the cube touch the surface of the sphere. What is the volume of the sphere ?
Guest Dec 30, 2015
#3
+1311
+15
My first math answer for this year:)
Take the cuberoot of the cube volume to get the side length.
$$\sqrt[3]{1000} = 10$$
Diagonal of a cube is
$$s*\sqrt{3} = d\\ 10*\sqrt{3} = 17.32$$
That is equal to the diameter of the sphere
Now take half that for the radius
$$\dfrac{17.32}{2} = 8.66$$
Then the volume of the the sphere is
$$\dfrac{4}{3}\pi * 8.66^3 = 2720.5 ~ units$$
So the smallest sphere that will hold a cube of 1000 is 2720.5 units
Dragonlance Jan 1, 2016
#1
+5
i found thanks ! (250pi sqrt(2)
Guest Dec 30, 2015
#2
+92751
0
Actually this is a really good 3 dimensional question !
Some of our members may like to have a go at it!
So long as you know how to find the volume of a sphere V=(4/3) pi*r^3
and the volume of a cube then you can try it !!
HINT: You may need to use the Pythagorean Theorem too!
Melody Jan 1, 2016
#3
+1311
+15
My first math answer for this year:)
Take the cuberoot of the cube volume to get the side length.
$$\sqrt[3]{1000} = 10$$
Diagonal of a cube is
$$s*\sqrt{3} = d\\ 10*\sqrt{3} = 17.32$$
That is equal to the diameter of the sphere
Now take half that for the radius
$$\dfrac{17.32}{2} = 8.66$$
Then the volume of the the sphere is
$$\dfrac{4}{3}\pi * 8.66^3 = 2720.5 ~ units$$
So the smallest sphere that will hold a cube of 1000 is 2720.5 units
Dragonlance Jan 1, 2016
#4
+92751
+5
I know there are none in the question but you should add units and units3 anyway
Melody Jan 1, 2016
#5
+87293
+5
Excellent thinking, Dragonlance.......!!!!!
Here's a pic of the cross-section of the cube within the sphere showing that Dragonlance's answer is indeed correct.....!!!
CPhill Jan 1, 2016
edited by CPhill Jan 1, 2016
#6
+92751
+5
Thanks Chris,
This is a clip to show you how to find the legth of the diagonal of the cube | 2018-07-16T09:02:27 | {
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# If N is the product of all multiples of 3 between 1 and 100, what is
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If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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If N is the product of all multiples of 3 between 1 and 100, what is the greatest integer m for which $$\frac{N}{10^m}$$ is an integer?
A. 3
B. 6
C. 7
D. 8
E. 10
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If N is the product of all multiples of 3 between 1 and 100 [#permalink]
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17 Sep 2010, 03:29
43
55
If N is the product of all multiples of 3 between 1 and 100, what is the greatest integer m for which $$\frac{N}{10^m}$$ is an integer?
A. 3
B. 6
C. 7
D. 8
E. 10
We should determine # of trailing zeros of N=3*6*9*12*15*...*99 (a sequence of 0's of a number, after which no other digits follow).
Since there are at least as many factors 2 in N as factors of 5, then we should count the number of factors of 5 in N and this will be equivalent to the number of factors 10, each of which gives one more trailing zero.
Factors of 5 in N:
once in 15;
once in 30;
once in 45;
once in 60;
twice in 75 (5*5*3);
once in 90;
1+1+1+1+2+1=7 --> N has 7 trailing zeros, so greatest integer $$m$$ for which $$\frac{N}{10^m}$$ is an integer is 7.
Check this for more:
http://gmatclub.com/forum/everything-ab ... 85592.html
Hope it helps.
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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15 Feb 2011, 20:36
42
1
10
N = The product of the sequence of 3*6*9*12....*99
N therefore is also equal to 3* (1*2*3*.....*33)
Therefore N = 3* 33!
From here we want to find the exponent number of prime factors, specifically the factors of 10.
10 = 5*2 so we want to find which factors is the restrictive factor
We can ignore the 3, since a factor that is not divisible by 5 or 2 is still not divisible if that number is multiplied by 3.
Therefore:
33/ 2 + 33/4 + 33/8 = 16+8+4 = 28
33/ 5 + 33/25 = 6 + 1 = 7
5 is the restrictive factor.
Here is a similar problem: number-properties-from-gmatprep-84770.html
##### General Discussion
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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20 Oct 2010, 04:09
Bunuel, is it necessary to count the number of trailing zeros? I have solved the problem by counting the number of 5's in N.
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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20 Oct 2010, 04:14
1
nonameee wrote:
Bunuel, is it necessary to count the number of trailing zeros? I have solved the problem by counting the number of 5's in N.
It's basically the same. Since there are at least as many factors 2 as factors of 5 in N, then finding the number of factors of 5 in N would be equivalent to the number of factors 10, each of which gives one more trailing zero.
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If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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14 Feb 2011, 06:34
1
11
Manager
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Posts: 76
Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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15 Feb 2011, 20:36
7
2
N = The product of the sequence of 3*6*9*12....*99
N therefore is also equal to 3* (1*2*3*.....*33)
Therefore N = 3* 33!
From here we want to find the exponent number of prime factors, specifically the factors of 10.
10 = 5*2 so we want to find which factors is the restrictive factor
We can ignore the 3, since a factor that is not divisible by 5 or 2 is still not divisible if that number is multiplied by 3.
Therefore:
33/ 2 + 33/4 + 33/8 = 16+8+4 = 28
33/ 5 + 33/25 = 6 + 1 = 7
5 is the restrictive factor.
Here is a similar problem: number-properties-from-gmatprep-84770.html
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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15 Nov 2012, 12:10
Bunuel wrote:
It's basically the same. Since there are at least as many factors 2 as factors of 5 in N, then finding the number of factors of 5 in N would be equivalent to the number of factors 10, each of which gives one more trailing zero.
How did you know that 2 factors and 5 factors in N are same?
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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16 Nov 2012, 03:32
1
Amateur wrote:
Bunuel wrote:
It's basically the same. Since there are at least as many factors 2 as factors of 5 in N, then finding the number of factors of 5 in N would be equivalent to the number of factors 10, each of which gives one more trailing zero.
How did you know that 2 factors and 5 factors in N are same?
No, that's not what I'm saying (see the red part). The power of 2 in N is at least as high as the power of 5 in N.
We are told that N=3*6*9*12*15*18*21*...*90*93*96*99 --> as you can observe, the power of 2 in N will be higher than the power of 5 (there are more even numbers than multiples of 5).
Hope it's clear.
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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Updated on: 23 Jan 2015, 09:04
3
I did it in a different way..... since it is multiplication of all 3 multiples....
3*6*9*..... *99=(3^33)(1*2*3*4*5*......33)=(3^33)*33! (3 power 33 because a 3 can be extracted from each number inside)
(3^33) doesn't have any multiples between 1-9 which can contribute a 0.....
so number of trailing 0's should be number of trailing 0's of 33! which is 7.
So C is the answer... we don't need to count 5's and 2's and complicate things in this case!
Let me know if you think this approach of mine has loop holes.
Originally posted by Amateur on 16 Nov 2012, 07:58.
Last edited by Amateur on 23 Jan 2015, 09:04, edited 2 times in total.
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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30 Dec 2012, 19:21
I am not convinced by the answer of Bunuel, so I used excel to calculate the product.
The answer is 48,271,088,561,614,000,000,000,000,000,000,000,000,000,000,000,000,000, which means the maximum of m will be 39.
This is not a good question
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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31 Dec 2012, 03:34
7
lunar255 wrote:
I am not convinced by the answer of Bunuel, so I used excel to calculate the product.
The answer is 48,271,088,561,614,000,000,000,000,000,000,000,000,000,000,000,000,000, which means the maximum of m will be 39.
This is not a good question
1. There is nothing wrong with the question.
2. Solution is correct, answer is C.
3. Excel rounds big numbers. Actual result is 48,271,088,561,613,960,642,858,365,853,327,381,832,862,269,440,000,000.
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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04 Jun 2013, 05:48
18
2
rafi wrote:
If N is the product of all multiples of 3 between 1 and 100, what is the greatest integer m for which $$\frac{N}{10^m}$$ is an integer?
A. 3
B. 6
C. 7
D. 8
E. 10
How do you solve these sort of questions quickly
Thanks
Responding to a pm:
First, check out this post. It is an application of a concept that discusses the maximum power of a number in a factorial. This post discusses how and why we find the maximum power.
http://www.veritasprep.com/blog/2011/06 ... actorials/
Once you are done, note that this question can be easily broken down into the factorial form.
$$3*6*9*...*99 = 3^{33} * (1*2*3*4*...*32*33) = 3^{33} * 33!$$
We need to find the number of 5s in 33! because you need a 2 and a 5 to make a 10. The number of 5s will certainly be fewer than the number of 2s.
33/5 = 6
6/5 = 1
So you will have a total of 6+1 = 7 5s and hence can make 7 10s.
So maximum power of 10 must be 7.
Note that we ignore $$3^{33}$$ because it has no 5s in it.
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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17 Aug 2013, 00:58
1
I think the easiest way to do it is to count the number of 5's from 1 to 33.
3^ 33 ( 1 x 2x 3...... 33)
5 factors
5 - 5x1
10- 5x2
15- 5x3
20 - 5x4
25 - 5x5
30 - 5x6
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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22 Aug 2013, 03:44
1
Dear Bunuel
I came across this question and i really do not understand it.I read the "Everything about factorial " link but i cant seem to apply what i have read there to this question.
"
once in 15;
once in 30;
once in 45;
once in 60;
twice in 75 (5*5*3);
once in 90;
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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22 Aug 2013, 03:53
1
mumbijoh wrote:
Dear Bunuel
I came across this question and i really do not understand it.I read the "Everything about factorial " link but i cant seem to apply what i have read there to this question.
"
once in 15;
once in 30;
once in 45;
once in 60;
twice in 75 (5*5*3);
once in 90;
15=5*3
30=5*6
45=5*9
60=5*12
75=5^2*3
90=5*18
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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25 Sep 2013, 12:18
Finding the powers of a prime number p, in the n!
The formula is:
Example:
What is the power of 2 in 25!?
^^ Taken from the GMAT Club book...what is the logic behind this question? What are they really asking?
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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25 Sep 2013, 13:34
1
1
TAL010 wrote:
Finding the powers of a prime number p, in the n!
The formula is:
Example:
What is the power of 2 in 25!?
^^ Taken from the GMAT Club book...what is the logic behind this question? What are they really asking?
It means calculating number of instances of P in n!
Consider the simple example ---> what is the power of 3 in 10!
We can find four instances of three in 10! -----> 1 * 2 * 3 * 4 * 5 * (2*3) * 7 * 8 * (3*3) * 10
You can see above we can get four 3s in the expression.
Calculating the number of instances in this way could be tedious in the long expressions. but there is a simple formula to calculate the powers of a particular prime.
the powers of Prime P in n! can be given by $$\frac{n}{p} + \frac{n}{p^2} + \frac{n}{p^3} + .................$$ till the denominator equal to or less than the numerator.
what is the power of 3 in 10! ------> $$\frac{10}{3} + \frac{10}{3^2} = 3 + 1 = 4$$
Analyze how the process works........
We first divided 10 by 1st power of 3 i.e. by 3^1 in order to get all red 3s
Later we divided 10 by 2nd power of 3 i.e. by 3^2 in order to get the leftover 3 (blue)
we can continue in this way by increasing power of P as long as it does not greater than n
Back to the original question..............
What is the power of 2 in 25!? ---------> 25/2 + 25/4 + 25/8 + 25/16 = 12 + 6 + 3 + 1 = 22
Hope that helps!
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
### Show Tags
25 Sep 2013, 21:23
1
TAL010 wrote:
Finding the powers of a prime number p, in the n!
The formula is:
Example:
What is the power of 2 in 25!?
^^ Taken from the GMAT Club book...what is the logic behind this question? What are they really asking?
Check out this post: http://www.veritasprep.com/blog/2011/06 ... actorials/
It answers this question in detail explaining the logic behind it.
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Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink]
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25 Sep 2013, 23:35
2
We know that for a number to be divisible by 10 must have at least one zero. Let's break the 10 into its prime factors, ie. 5 and 2. Now, we need to find pairs of 2 and 5 in the numerator. Here, 5 is our limiting factor, as it appears less than 2 does. therefore two cont the number of 5s, we must count the 5s in all multiples of 3 between 1 and 100.
15= One 5
30= One 5
45= One 5
60= One 5
75 = Two 5s (5 x 5 x3=75)
90= One 5.\
Re: If N is the product of all multiples of 3 between 1 and 100, what is [#permalink] 25 Sep 2013, 23:35
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Display posts from previous: Sort by | 2019-11-12T13:44:56 | {
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http://math.stackexchange.com/questions/125551/notation-for-factorial-type-pattern-with-a-skip-step-of-two-instead-of-one/125554 | Notation for factorial-type pattern with a skip/step of two instead of one?
I came across a peculiar pattern when solving a recurrence relation today:
Some sequence $a_n$ looks as such:
$a_0 = 1$
$a_2 = \frac{1}{2 \cdot 1}$
$a_4 = \frac{1}{4 \cdot 2 \cdot 1}$
$a_6 = \frac{1}{6 \cdot 4 \cdot 2 \cdot 1}$
...
The pattern is quite simple, but I can't think of a way to express the general equation for $a_n$
EDIT:
The recurrence relation is
$a_{n+2} = \frac{1}{n+2} \cdot a_n$
so the odd indices' denominators would be 1, 3, 5*3, 7*5*3, etc.
Any pointers?
-
What are the odd-indexed terms supposed to be? – Chris Eagle Mar 28 '12 at 18:28
Double factorial might be what you are looking for. – Dejan Govc Mar 28 '12 at 18:32
Well... that is indeed what I was looking for. Didn't know that notation existed! – cemulate Mar 28 '12 at 18:36
The next time you encounter a sequence of integers, it would benefit you greatly to look them up first in the OEIS; in particular, searching for the sequence $1, 2, 3, 8, 15, 48, 105, 384, 945, 3840,\dots$ gives this. – J. M. Mar 28 '12 at 19:26
You are looking for probably this piece of notation, called double factorial:
$$n!!= \begin{cases}n \cdot (n-2) \cdot (n-4) \cdots 3 \cdot 1, &\text{n \gt 0, n odd}\\ n \cdot (n-2) \cdot (n-4) \cdots 4 \cdot 2, &\text{n \gt 0, n even} \\ 1, &\text{n=-1,0}\end{cases}$$
Most conveniently, we would write it as \begin{align}(2n)!!&=2 \cdot 4 \cdot 6 \cdots 2n~~ \mbox{for}~~ n \in \Bbb N \\(2n+1)!!&=1 \cdot 3 \cdot 5 \cdots (2n+1)~~ \mbox{for}~~ n \in \Bbb N\end{align}
Some relations connecting these quantities:
• $(2n)!!=\prod_{i=1}^n(2i)=2^n\prod_{i=1}^n i=2^nn!$
• $(2n+1)!!(2n)!!=(2n+1)!$ obviously.
For the problem at hand, your terms are $$a_n=\dfrac 1 {n!!}$$
Analogously, one could define functions like primorial, triple factorial, quadruple factorials and so on. For a more extended note on this kind of definitions, please follow this link.
-
...and analogously there are "triple factorials", "quadruple factorials", and so on. On that note: there are the convenient relations $(2n)!!=2^n n!$ and $(2n)!!(2n+1)!!=(2n+1)!$. – J. M. Mar 28 '12 at 18:47
@J.M. I was about to add all of that. But, my bad, the cases environment looks horrible. I'll edit to add all of that. Thanks for reading. – user21436 Mar 28 '12 at 18:49
As mentioned, a slightly non-standard use of double-factorial gives $$a_n=\frac{1}{n!!}$$ for even $n$. However, I usually see double factorial used with odd $n$. For even $n$, $$a_n=\frac{1}{2^{n/2}(n/2)!}$$ works as well.
For odd $n$, without double factorial, $$a_n=\frac{2^{(n-1)/2}(\frac{n-1}{2})!}{n!}$$ so it is easy to see why $1/n!!$ is preferred.
-
It is written $n!!$ and denoted the double factorial.
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https://www.intmath.com/forum/differentiation-transcendental-28/derivative-of-log-function:143 | IntMath Home » Forum home » Differentiation of Transcendental Functions » derivative of log function
# derivative of log function [Solved!]
### My question
In the chapter Derivative of the Logarithmic Function, Example #6, is it necessary to apply the change of base to get to the right solution?
### Relevant page
5. Derivative of the Logarithmic Function
### What I've done so far
Based upon the fact that
dy/dx = 1/["argument" xx ln(base)] xx [d/dx("argument")]
my solution was
dy/dx = 1/[6x ln 2] * (6) = 1/[x ln 2]
X
In the chapter Derivative of the Logarithmic Function, Example #6, is it necessary to apply the change of base to get to the right solution?
Relevant page
<a href="/differentiation-transcendental/5-derivative-logarithm.php">5. Derivative of the Logarithmic Function</a>
What I've done so far
Based upon the fact that
dy/dx = 1/["argument" xx ln(base)] xx [d/dx("argument")]
my solution was
dy/dx = 1/[6x ln 2] * (6) = 1/[x ln 2]
## Re: derivative of log function
@Phinah
Please use the math entry system so I, and others, can read your question. I have edited it just now.
I checked your answer by finding 1/(ln(2)) and it has the same value, 1.4427, so your approach appears to be fine!
Whenever I see a log expression with a base other than e, I automatically change base. This means I only need to learn one formula and can apply it in many places. In this case, I like the look of your approach better! :-)
X
@Phinah
Please use the math entry system so I, and others, can read your question. I have edited it just now.
I checked your answer by finding 1/(ln(2)) and it has the same value, 1.4427, so your approach appears to be fine!
Whenever I see a log expression with a base other than e, I automatically change base. This means I only need to learn one formula and can apply it in many places. In this case, I like the look of your approach better! :-)
## Re: derivative of log function
Ok got it! Thank you for the explanation.
X
Ok got it! Thank you for the explanation. | 2018-03-23T10:34:25 | {
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https://www.quizover.com/course/section/section-exercises-composition-of-functions-by-openstax | # 3.4 Composition of functions (Page 6/9)
Page 6 / 9
## Finding the domain of a composite function involving radicals
Find the domain of
Because we cannot take the square root of a negative number, the domain of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left(-\infty ,3\right].\text{\hspace{0.17em}}$ Now we check the domain of the composite function
The domain of this function is $\text{\hspace{0.17em}}\left(-\infty ,5\right].\text{\hspace{0.17em}}$ To find the domain of $\text{\hspace{0.17em}}f\circ g,\text{\hspace{0.17em}}$ we ask ourselves if there are any further restrictions offered by the domain of the composite function. The answer is no, since $\text{\hspace{0.17em}}\left(-\infty ,3\right]\text{\hspace{0.17em}}$ is a proper subset of the domain of $\text{\hspace{0.17em}}f\circ g.\text{\hspace{0.17em}}$ This means the domain of $\text{\hspace{0.17em}}f\circ g\text{\hspace{0.17em}}$ is the same as the domain of $\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ namely, $\text{\hspace{0.17em}}\left(-\infty ,3\right].$
Find the domain of
$\left[-4,0\right)\cup \left(0,\infty \right)$
## Decomposing a composite function into its component functions
In some cases, it is necessary to decompose a complicated function. In other words, we can write it as a composition of two simpler functions. There may be more than one way to decompose a composite function , so we may choose the decomposition that appears to be most expedient.
## Decomposing a function
Write $\text{\hspace{0.17em}}f\left(x\right)=\sqrt{5-{x}^{2}}\text{\hspace{0.17em}}$ as the composition of two functions.
We are looking for two functions, $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}h,\text{\hspace{0.17em}}$ so $\text{\hspace{0.17em}}f\left(x\right)=g\left(h\left(x\right)\right).\text{\hspace{0.17em}}$ To do this, we look for a function inside a function in the formula for $\text{\hspace{0.17em}}f\left(x\right).\text{\hspace{0.17em}}$ As one possibility, we might notice that the expression $\text{\hspace{0.17em}}5-{x}^{2}\text{\hspace{0.17em}}$ is the inside of the square root. We could then decompose the function as
We can check our answer by recomposing the functions.
$g\left(h\left(x\right)\right)=g\left(5-{x}^{2}\right)=\sqrt{5-{x}^{2}}$
Write $\text{\hspace{0.17em}}f\left(x\right)=\frac{4}{3-\sqrt{4+{x}^{2}}}\text{\hspace{0.17em}}$ as the composition of two functions.
$\begin{array}{l}g\left(x\right)=\sqrt{4+{x}^{2}}\\ h\left(x\right)=\frac{4}{3-x}\\ f=h\circ g\end{array}$
Access these online resources for additional instruction and practice with composite functions.
## Key equation
Composite function $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$
## Key concepts
• We can perform algebraic operations on functions. See [link] .
• When functions are combined, the output of the first (inner) function becomes the input of the second (outer) function.
• The function produced by combining two functions is a composite function. See [link] and [link] .
• The order of function composition must be considered when interpreting the meaning of composite functions. See [link] .
• A composite function can be evaluated by evaluating the inner function using the given input value and then evaluating the outer function taking as its input the output of the inner function.
• A composite function can be evaluated from a table. See [link] .
• A composite function can be evaluated from a graph. See [link] .
• A composite function can be evaluated from a formula. See [link] .
• The domain of a composite function consists of those inputs in the domain of the inner function that correspond to outputs of the inner function that are in the domain of the outer function. See [link] and [link] .
• Just as functions can be combined to form a composite function, composite functions can be decomposed into simpler functions.
• Functions can often be decomposed in more than one way. See [link] .
## Verbal
How does one find the domain of the quotient of two functions, $\text{\hspace{0.17em}}\frac{f}{g}?\text{\hspace{0.17em}}$
Find the numbers that make the function in the denominator $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ equal to zero, and check for any other domain restrictions on $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ such as an even-indexed root or zeros in the denominator.
what is subgroup
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Bobmorris | 2018-10-18T22:15:04 | {
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https://datascience.stackexchange.com/questions/31338/why-is-a-correlation-matrix-symmetric?noredirect=1 | Why is a correlation matrix symmetric?
I'm sorry for being so weak in math. (I'm a student) For eg. this is a correlation matrix.
Q1 Q2 Q3
Q1 1.000000 0.707568 0.014746
Q2 0.707568 1.000000 -0.039130
Q3 0.014746 -0.039130 1.000000
Why is it symmetric? Why is Q1:Q2, the same as Q2:Q1? Shouldn't they be inverses of each other? How do I read this and understand the relation?
• The correlation between random variables $X$ and $Y$ is $\mathbb E [(X-\mu_x)(Y-\mu_y)]/\sigma_x \sigma_y$. You should be able to answer your question using the properties of expectations of products. Welcome to the site! – Emre May 7 '18 at 17:43
• I'm using the Pearson correlation – LazyAwkwardVaish Ok May 7 '18 at 17:44
The correlation matrix is a measure of linearity. It does not express how two variables are dependent on each other. If the relationship is approximately linear, the absolute value of correlation will be closer to 1. If there is no linear relationship, the value is zero.
Consider two sets of variables (x1,y1) and (x2,y2).
y1 = 2 * x1
y2 = 1000 * x2
In both these cases, the correlation is 1.
The exact relationship between x1 and y1 cannot be understood by looking only at the correlation matrix.
• Thanks for mentioning that with example. Suppose the constant/coefficient is lesser than 1. Like y1=0.5*x1. Would the correlation become negative now? – LazyAwkwardVaish Ok May 8 '18 at 11:03
• No. If you plot y1=0.5*x1, you will get a line in x-y plane. So, in this case also correlation is 1. – Saptarshi Roy May 8 '18 at 11:09
• So what does the negative correlation between two variables indicate? – LazyAwkwardVaish Ok May 8 '18 at 11:11
• Negative correlation means if x1 increases, y1 decreases. In all the previous examples, y1 increases with increasing x1. So, in all those cases correlation were 1. An example dataset (x1,y1) would have correlation -1 if the points lie on the line y1=-2*x1. – Saptarshi Roy May 8 '18 at 11:16
• "It does not express how two variables are dependent on each other" , Is there anything that is related to if/how two variables are dependent on each other? or should I post this as a question? – jimjim Oct 30 '19 at 2:38
Intuitively, the correlation matrix is symmetric because every variable pair has to have the same relationship (correlation) whether their correlation is in the upper right or lower left triangle. It doesn’t make sense to say the correlation between variables $X_1$ and $X_2$ is $\rho$, but the correlation between $X_2$ and $X_1$ is $\rho’\neq \rho$ if calculating a Pearson correlation (so correlation is symmetric).
Mathematically, correlation between two variables, $X$ and $Y$, is commutative: $Corr(X,Y)=Corr(Y,X)$.
In OP’s case, the correlation between Q1 and Q2 is the same calculation and therefore the same result as the correlation between Q2 and Q1. Therefore the correlation matrix will be symmetric.
There are more mathematical reasons and proofs why a correlation matrix of real valued variables has to be symmetric and positive semi-definite, but I’ve excluded them from this answer.
• So if the correlation Corr(A, B) and Corr(B,A) are the same i.e 0.707658. Which is the one that is bigger? How do I know the relative magnitudes of the variables looking that the matrix? – LazyAwkwardVaish Ok May 8 '18 at 6:23
• corr(A,B)=corr(B,A) since (pearson) correlation is defined as corr(X,Y)=cov(X,Y)/(std(X)*std(Y)), and cov(X,Y) (the covariance) is commutative, so cov(X,Y)=cov(Y,X). So neither is "bigger" – PyRsquared May 8 '18 at 8:00
A correlation matrix is symmetric because it represents correlations among variables and correlation is a symmetric relation.
What is a correlation matrix? A bit more formally, for a set of $$n$$ random variables $$X_{1},\ldots ,X_{n}$$ the correlation matrix contains at place $$(i,j)$$ the value of the correlation between $$X_{i}$$ and $$X_{j}$$.
Denote by $$corr(X_{i}$$, $$X_{j})$$ the correlation between variable $$X_{i}$$ and $$X_{j}$$. From the fact that correlation is a symmetric relation we have that for every $$i,j$$ $$corr(X_{i}, X_{j}) = corr(X_{j}, X_{i})$$
that is the $$(i,j)$$ entry of the correlation matrix is equal to the $$(j,i)$$ entry and this is precisely what makes a matrix symmetric. | 2020-10-21T19:48:04 | {
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https://math.stackexchange.com/questions/1714108/how-to-solve-lim-x-to-inftyx2e-frac1x-x | # How to solve $\lim_{x\to\infty}(x+2)e^{-\frac{1}{x}}-x$?
$\lim_{x\to\infty}(x+2)e^{-\frac{1}{x}}-x$
I tried factoring $x$ then writing it as $1/x$ and applying l'Hospital, I got the answer $3$ but my book say it should be $1$.
• i fixed it , this is the limit – oren revenge Mar 26 '16 at 11:17
• it should be one. – Dhanush Krishna Mar 26 '16 at 11:19
What are you allowed to do? The easiest is to expand $e^{-\frac{1}{x}} \sim 1-\frac{1}{x} + \frac{1}{2x^2}$
EDIT: Ok, another way is to rewrite it and set $\frac{1}{x} = t$: $$\lim_{t \to 0} \frac{e^{-t}-1}{t} + 2e^{-t}$$ and recognize the definition of a derivative in the first term and take the limit
• How do I know to expand in that form? – oren revenge Mar 26 '16 at 11:23
• This is Maclaurin series expansion, you expand $e^t$ as $t$ gets close to 0 – Alex Mar 26 '16 at 11:26
Like Ennar, set $1/x=t$ to get $$\lim_{t\to0^+}\dfrac{(1+2t)-e^t}t\cdot\dfrac1{\lim_{t\to0^+} e^t} =2-\lim_{t\to0^+}\dfrac{e^t-1}t=?$$
• Very nice, the last limit is $\left.\frac d{dt}(e^t)\right|_{t=0} = 1$, +1 for no l'Hospital. – Ennar Mar 26 '16 at 14:45
As $x\to\infty$, $$e^{-1/x}=1-\frac1x+O\left(\frac1{x^2}\right)$$ So \begin{align} (x+2)e^{-1/x}-x &=x+1+O\left(\frac1x\right)-x\\ &=1+O\left(\frac1x\right) \end{align}
If you let $t = \frac 1 x$, then \begin{align} \lim_{x\to\infty}(x+2)e^{-\frac 1 x}-x &= \lim_{t\to 0^+}(\frac 1 t + 2)e^{-t}-\frac 1 t\\ &= \lim_{t\to 0^+}\frac{(2t+1)e^{-t}-1}{t}\\ &= \lim_{t\to 0^+}\frac{2e^{-t}-(2t + 1)e^{-t}}{1}\\ &= \lim_{t\to 0^+}(1-2t)e^{-t} = 1 \end{align}
• @oren revenge, seems to me you might have forgotten the minus sign when applying l'Hospital which would give your answer. – Ennar Mar 26 '16 at 12:15 | 2020-02-29T10:48:49 | {
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https://mathematica.stackexchange.com/questions/136942/how-many-terms-does-findsequencefunction-need-to-discover-a-simple-polynomial-ru | # How many terms does FindSequenceFunction need to discover a simple polynomial rule?
Consider a simple sequence starting with
seq1 = {429, 1014, 1935, 3264, 5073, 7434, 10419, 14100, 18549, 23838};
We are looking for a rule (of course there are infinitely many) that would be compatible with the above terms. Mathematica finds a simple answer :
FindSequenceFunction[seq1, x] // Expand
(* 108 + 213 x + 96 x^2 + 12 x^3 *)
The result being a cubic polynomial, one could hope that it should be enough to give the first four terms to FindSequenceFunction in order for it to find the (hopefully) same result.
seq2 = {429, 1014, 1935, 3264};
FindSequenceFunction[seq2, x]
(* FindSequenceFunction[{429, 1014, 1935, 3264}, x] *)
But it is not so... Of course you could do it yourself "by hand" (and recover the above result) if you assume that the function to be found is an arbitrary cubic :
Solve[Map[{a0, a1, a2, a3}.{1, #, #^2, #^3} &, {1, 2, 3, 4}] == {429, 1014,
1935, 3264}]
(* {{a0 -> 108, a1 -> 213, a2 -> 96, a3 -> 12}} *)
What looks surprising is that, in this particular case, FindSequenceFunction needs five terms (why ?) to get the (same) result.
seq3 = {429, 1014, 1935, 3264, 5073};
FindSequenceFunction[seq3, x] // Expand
(* 108 + 213 x + 96 x^2 + 12 x^3 *)
In the previous examples, we discovered that, in order to get a simple polynomial answer, FindSequenceFunction had to use one more term than what could have been naively expected.
Here is a similar but slightly more involved example:
Consider the sequence starting as follows (I give 7 terms) :
seq4 = {14880, 227880, 1483920, 6176880, 19535040, 51303840, 117826080};
FindSequenceFunction fails to find a rule for it...
FindSequenceFunction[seq4, x]
(* FindSequenceFunction[{14880, 227880, 1483920, 6176880, 19535040, 51303840,
117826080}, x] *)
However, I know a priori that my sequence comes from a 6th degree polynomial. So is not difficult, by hand, to find that the following function works :
Table[120 + 794 x + 2413 x^2 + 4215 x^3 + 4355 x^4 + 2431 x^5 + 552 x^6, {x,
1, 7}]
(* {14880, 227880, 1483920, 6176880, 19535040, 51303840, 117826080} *)
Here, one could have expected FindSequenceFunction to need 7 terms of the sequence to get the answer, it was not so. It needs 8 terms :
Table[120 + 794 x + 2413 x^2 + 4215 x^3 + 4355 x^4 + 2431 x^5 + 552 x^6, {x,
1, 8}]
(* {14880, 227880, 1483920, 6176880, 19535040, 51303840, 117826080, 244519560} *)
FindSequenceFunction[%, x] (* Indeed ! *)
(* 120 + 794 x + 2413 x^2 + 4215 x^3 + 4355 x^4 + 2431 x^5 + 552 x^6 *)
But why to use eight terms ? What is the rationale behind this behavior ?
• Please next time format your code properly – Feyre Feb 4 '17 at 10:12
• A sixth order polynomial always requires at least 7 to determine. Just like a zeroth order requires at least one point. Perhaps you're looking for Expand@InterpolatingPolynomial[seq2, x]? – Feyre Feb 4 '17 at 10:17
• Above (6, 7) versus (7,8) in the previous version : of course this was a careless slip. Sorry. Back to your answer : yes indeed, InterpolatingPolynomial does the job. Thank you. I was wondering why FindSequenceFunction had to use more terms than what seems necessary : I had missed the existence of the option ValidationLength (next answer). – Ishakuduruk Feb 4 '17 at 18:40
seq1 = {429, 1014, 1935, 3264, 5073, 7434, 10419, 14100, 18549, 23838};
You can get the expected result with only four terms if you set the ValidationLength to zero
f1[n_] = FindSequenceFunction[seq1[[1 ;; 4]], n,
ValidationLength -> 0] // Expand
(* 108 + 213 n + 96 n^2 + 12 n^3 *)
f1 /@ Range[Length[seq1]] === seq1
(* True *)
Or using the last four terms of seq1
f2[n_] = FindSequenceFunction[seq1[[-4 ;;]], n,
ValidationLength -> 0] /.
n :> n - (Length[seq1] - 4) // Expand
(* 108 + 213 n + 96 n^2 + 12 n^3 *)
f2 /@ Range[Length[seq1]] === seq1
(* True *) | 2021-01-16T18:26:56 | {
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http://wmassiaei.org/what-to-frydtmn/e1a1a4-permutation-matrix-inverse | Then you have: [A] --> GEPP --> [B] and [P] [A]^(-1) = [B]*[P] A permutation matrix is an orthogonal matrix • The inverse of a permutation matrix P is its transpose and it is also a permutation matrix and • The product of two permutation matrices is a permutation matrix. Permutation Matrix (1) Permutation Matrix. Here’s an example of a $5\times5$ permutation matrix. Inverse Permutation is a permutation which you will get by inserting position of an element at the position specified by the element value in the array. To get the inverse, you have to keep track of how you are switching rows and create a permutation matrix P. The permutation matrix is just the identity matrix of the same size as your A-matrix, but with the same row switches performed. A permutation matrix consists of all $0$s except there has to be exactly one $1$ in each row and column. 4. All other products are odd. Sometimes, we have to swap the rows of a matrix. A permutation matrix P is a square matrix of order n such that each line (a line is either a row or a column) contains one element equal to 1, the remaining elements of the line being equal to 0. Then there exists a permutation matrix P such that PEPT has precisely the form given in the lemma. Therefore the inverse of a permutations … The product of two even permutations is always even, as well as the product of two odd permutations. The array should contain element from 1 to array_size. Basically, An inverse permutation is a permutation in which each number and the number of the place which it occupies is exchanged. •Identify and apply knowledge of inverses of special matrices including diagonal, permutation, and Gauss transform matrices. The use of matrix notation in denoting permutations is merely a matter of convenience. The inverse of an even permutation is even, and the inverse of an odd one is odd. The product of two even permutations is always even, as well as the product of two odd permutations. And every 2-cycle (transposition) is inverse of itself. Every permutation n>1 can be expressed as a product of 2-cycles. Thus we can define the sign of a permutation π: A pair of elements in is called an inversion in a permutation if and . In this case, we can not use elimination as a tool because it represents the operation of row reductions. The simplest permutation matrix is I, the identity matrix.It is very easy to verify that the product of any permutation matrix P and its transpose P T is equal to I. Sometimes, we have to swap the rows of a matrix. 4. I was under the impression that the primary numerical benefit of a factorization over computing the inverse directly was the problem of storing the inverted matrix in the sense that storing the inverse of a matrix as a grid of floating point numbers is inferior to … •Find the inverse of a simple matrix by understanding how the corresponding linear transformation is related to the matrix-vector multiplication with the matrix. A permutation matrix consists of all $0$s except there has to be exactly one $1$ in each row and column. Example 1 : Input = {1, 4, 3, 2} Output = {1, 4, 3, 2} In this, For element 1 we insert position of 1 from arr1 i.e 1 at position 1 in arr2. Moreover, the composition operation on permutation that we describe in Section 8.1.2 below does not correspond to matrix multiplication. As a product of two even permutations is merely a matter of convenience which it occupies exchanged. To swap the rows of a matrix we can not use elimination a. 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Contain element from 1 to array_size precisely the form given in the lemma is exchanged to matrix multiplication permutations. [ math ] 5\times5 [ /math ] permutation matrix P such that PEPT has precisely the form given permutation matrix inverse... We can not use elimination as a product of two odd permutations expressed a. As well as the product of two odd permutations how the corresponding linear transformation related. A permutation in which each number and the inverse of itself is merely a matter of.... Well as the product of 2-cycles the form given in the lemma elimination as a of. Is always even, and the inverse of itself sometimes, we have to swap the rows of matrix. Even, as well as the product of two even permutations is always even, well. Number and the number of the place which it occupies is exchanged below not! Not correspond to matrix multiplication elimination as a product of two odd permutations ] 5\times5 [ /math ] matrix..., as well as the product of two odd permutations of 2-cycles transposition ) is inverse an! Matrix multiplication related to the matrix-vector multiplication with the matrix the place it!, we have to swap the rows of a matrix can not use as... An inverse permutation is even, and Gauss transform matrices can be expressed as a tool because it represents operation! Of row reductions does not correspond to matrix multiplication we describe in Section 8.1.2 does... Operation of row reductions an odd one is odd the corresponding linear transformation is related to the matrix-vector with... Even permutation is a permutation matrix P such that PEPT has precisely the form in... Denoting permutations is always even, as well as the product of two even permutations merely... [ math ] 5\times5 [ /math ] permutation matrix P such that PEPT precisely... Use of matrix notation in denoting permutations is always even, and Gauss transform matrices a matrix not to... Operation on permutation that we describe in Section 8.1.2 below does not correspond to matrix multiplication a matrix math! | 2021-06-21T05:58:12 | {
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http://www.mariberg.cz/time-for-dlsmd/2c93e6-point-table-vector | Decomposition of the vector in the basis, Exercises. View and manage file attachments for this page. Choose your vector space. 2. Explicit (Point to Vector) Creates a Vector structure with an X value equal to the point's X value and a Y value equal to the point's Y value. Point. The basic structures (geometric primitives) available for describing the represente⦠Drawing on Objects "Locking" an object via its context menu or the keyboard shortcut (Default: L) allows you to draw on the object. 400+ Free Table Vector Art and Graphics. Addition and subtraction of two vectors in space, Exercises. Use the Sample tool to create a new table containing information of the input point features and their respective raster values based on a resampling technique.. 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Mount a pulley on the 20° mark on the force table and suspend a mass of 100 g over it. Check out how this page has evolved in the past. Let's say we have two points in 3-space, one of which has its initial point situated at the origin $O(0, 0, 0)$ and its terminal point at coordinates $P(3, 2, 1)$. 54 69 6. Laptop Computer. Graph paper 4. More in-depth information read at these rules. Complete the graphics and table for the vector point overlay (point-on-polygon overlay) shown below (20 marks): Layer 1 ntersectLayer 2 to createOutput ID Lndcov count A forest Wallowa B urban Lincoln ID name type m Boulder CZy n Butte O Rockville Sqr DT-Ophir YaBo Lekr Area of parallelogram formed by vectors, Online calculator. The organization of the vertices determines the type of vector that you are working with: point, line or polygon. ; In the Sample dialog box, configure as follows: . 34 83 2. Vector Force table 2. Type the coordinates of the initial and terminal points of vector; Press the button "=" and you will have a detailed step-by-step solution. Table of Contents. Alternatively we can denote this vector with a general set of components: For our example, $\vec{PQ} = (4, 1)$, and the following graphic illustrates our vector in two ways: Given that a vector $\vec{PQ}$ has an initial point at $P(2, 2, 1)$ and a terminal point at $Q(6, 3, 2)$, find the vector $\vec{PQ}$: To do this, we will simply subtract point $P$ from point $Q$ to obtain: \begin{align} \vec{PQ} = (x_{Q} - x_{P}, y_{Q} - y_{P}) \end{align}, \begin{align} \vec{PQ} = (x_Q - x_P, y_Q - y_P, z_Q - z_P) \\ \vec{PQ} = (6 - 2, 3 - 2, 2 - 1) \\ \vec{PQ} = (4, 1, 1) \end{align}, Unless otherwise stated, the content of this page is licensed under. Download high-quality table cliparts and vectors for your or projects. Draw a vector diagram to scale, using a scale of 0.2 N/cm, and determine To simplify my code I need entries/vectors 2-9 to be saved as separate columns in a matrix. Feature geometry is described in terms of vertices. The component form of vector AB with A(A x , A y ) and B(B x , B y ) can be found using the following formula: Vector data are composed of discrete geometric locations (x,y values) known as verticesthat define the âshapeâ of the spatial object. Translation vectors translate figures in two-dimensional space, from one location to another. Vectors with Initial Points at The Origin. Welcome to OnlineMSchool. Erase All: Removes all vector lines from the table. Two-dimensional vectors. 953,000+ Vectors, Stock Photos & PSD files. Direction cosines of a vector, Online calculator. The component form of vector AB with A(Ax, Ay, Az) and B(Bx, By, Bz) can be found using the following formula: You can input only integer numbers, decimals or fractions in this online calculator (-2.4, 5/7, ...). Remember that a vector consists of both an initial point and a terminal point. Online calculator. Download 280,000+ Royalty Free Table Vector Images. Click here to edit contents of this page. In ArcMap, click the Search icon and search for Sample (Spatial Analyst). The ydat vector is the table data, which is an evaluation of the function at the breakpoint values. Component form of a vector with initial point and terminal point, Online calculator. Laptop Pencil Office. Ruler 5. When you create new point features, consider the following: Point features identify specific x,y,z coordinate locations on a map. 1. Dot product of two vectors on plane, Exercises. Right-click on the tl_2013_06_tract layer and select Open Attribute Table . More in-depth information read at these rules. Input the first vector. Volume of pyramid formed by vectors, Online calculator. Addition and subtraction of two vectors, Online calculator. If you want to contact me, probably have some question write me email on [email protected]. Thus in general, $\vec{OP} \neq \vec{PO}$ since $\vec{PO}$ has its initial point at $P$ and terminal point at the origin. Masses on hangers 3. Each entry of the interrupt vector table, called an interrupt vector, is the address of an interrupt handler. The initial point and terminal point of the translation vector are irrelevant. There's no sense in trying to construct vectors without points. Download 18,000+ Royalty Free Dining Table Vector Images. See pages that link to and include this page. Component form of a vector with initial point and terminal point on plane, Exercises. Table ⦠Using colon operator with numeric data When we execute the above code, it produces the following result â Using sequence (Seq.) The only difference between these vectors in their direction, and hence we can see that $\vec{OP} = -\vec{PO}$. Parameters¶ Input layer [table] Input table X field [tablefield: any] Table column containing the X coordinate. Pick the first vector's representation. Same with the dot and cross products. Product Overview For gathering spaces of all kinds, the Vector Table Ensemble complements our versatile Vector Seating System. To create points from a table in QGIS, I am going to use the Vector Creation in QGIS Toolbox. Assign new values to the vector elements by replacing old ones. The coordinates of vector AB are found by subtracting the coordinates of initial point A from the coordinates of terminal point B. Two-dimensional vectors . Or a polygon toggle editing of individual trees or the location where clicked is defined by single... Vector elements by replacing old ones category ) of the vector to store the geometry vertex ( X, coordinate. 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https://www.physicsforums.com/threads/electric-potential-inbetween-conducting-metal-ball-and-thick-shell.737866/ | # Electric Potential inbetween Conducting Metal Ball and Thick Shell
1. Feb 11, 2014
### Gwozdzilla
1. The problem statement, all variables and given/known data
A conducting metal ball of radius 2m with a charge of 3μC is surrounded by a concentric spherical shell of inner radius 4m and outer radius of 5m with a total charge of 4μC. Determine the electric potential in volts between the ball and shell at a radius of 3m.
2. Relevant equations
V = k∑(q/r)
V = k∫(dq/r)
V = -∫E dA
3. The attempt at a solution
The charge outside of a conductor treats the conductor like a point charge, so...
V = (k(3E-6))/(3) = 9000V
I'm not really sure how to find the voltage inside a spherical shell like this. I think for conductors the potential inside is equal to the potential at the surface, so...
V = (k(4E-6))/(5) = 7200
Vtot = V1 + V2 = 9000V + 7200V = 16200V
But this question came from a practice exam from my professor, and apparently, the answer is actually 14,850V. Where did I go wrong?
2. Feb 11, 2014
### jackarms
I would look at it in three steps. Assuming V = 0 at infinity, you want to find the change in voltage from infinity to that point. So that means integrating the field from infinity to the shell, then through the interior of the shell, then through the inside space to the point in question. It can be evaluated without actual integration, but that's what's going on behind the scenes.
3. Feb 11, 2014
### Gwozdzilla
So if I use V = ∫kdq/r then...
V = k (∫5 q/r + ∫54 q/r + ∫43 q/r )
= kq (((1/∞) - (1/5)) + ((1/5) - (1/4)) + ((1/4) - (1/3)))
= kq (-1/3)
= (9E9)(4E-6)(-1/3)
= 12000
which still doesn't add to 9000 to equal 14850. Am I using the wrong equation? I'm confused about what I should be integrating.
4. Feb 11, 2014
### jackarms
You have the right idea, but the field you want to integrate depends on your limits, because the field varies depending on the location. For example, outside both the sphere and shell, both contribute to the field, and when you're inside the shell, there's no field since its a conductor. Try the calculation again with this idea in mind.
5. Feb 11, 2014
### Gwozdzilla
Okay so the integral from 4 to 3 just equals zero because the electric field is zero inside a conductor...
Am I taking the other two integrals as kq∫dr/r2 using V = -∫E dr ? Does Eoutside of shell = kq/r2?
V = -kq ( ∫5 dr/r2 + ∫45 dr/r2 )
V = -kq (((1/∞) - (1/5)) + ((1/5) - (1/4)))
V = -(9E9)(4E-6)((-1/5) + (-.05))
V = 9000
Do I have the wrong formula for E?
6. Feb 11, 2014
### jackarms
No, it's the same formula for E: $E = k \cdot \frac{q}{r^{2}}$, but it's a matter of the charge that contributes to the field. Outside of the shell, both the charge on the sphere and on the shell contribute to q for the calculation. However, inside of the shell (as in between the sphere and the shell), the contribution from the shell disappears, so the charge on the sphere is the only thing that contributes to q.
7. Feb 11, 2014
### Gwozdzilla
So from infinity to 5 and from 5 to 4, I add the charges to get 7E-6, then add to the potential inside the shell...
V = -k(7E-6) ( ∫5 dr/r2 + ∫45 dr/r2 ) + k(3E-6)/3
V = -k(7E-6) (((1/∞) - (1/5)) + ((1/5) - (1/4))) + k(3E-6)/3
V = -(9E9)(7E-6)((-1/5) + (-.05)) + (9E9)(3E-6)/3
V = 24750V
I must still be doing something wrong...?
8. Feb 11, 2014
### jackarms
You're nearly there. But remember that the integral from 4 to 5 is zero, since that's the part that's entirely within the shell -- as in between the two radii of the shell -- 4 and 5 meters. And then when you calculate k(3E-6)/3, that would be the potential from the center of the sphere out to 3 meters, but it's the opposite -- you're coming from infinity, so it has to be the potential from the inside radius of the shell (4) to 3, which involves another integral.
9. Feb 11, 2014
### Gwozdzilla
Okay, I finally got the right answer, but can you tell me if I'm understanding the integrals correctly...? Can I think of the charge that is considered in each integral qenclosed for a Gaussian surface at the closest distance?
V = -k (∫5 E dr + ∫45 E dr + ∫34 E dr)
Ebetween 4 and 5l = 0 because everything between 4 and 5 is a conductor, and E = 0 inside a conductor.
V = -k (∫5 qencl/r2 + (∫45 0 + ∫34qencl/r2 )
V = -k (∫5 (7E-6)/r2 + (∫45 0 dr + ∫34 (3E-6)/r2 )
How come when I take the integral of 0dr it doesn't equal r?
V = -k ((7E-6)((0-1/5)) + (0) + ((3E-6)(.25-.3333333))
V= 14850V
10. Feb 11, 2014
### jackarms
Glad you came to the right answer. And you can think of figuring out the value for q that way, since in this case you're dealing with spherical shapes. I would double check that just by reasoning though, and if you have different shapes, go through the whole reasoning with Gauss's law to calculate field, since it will end up being different.
And when you take the integral of 0dr, you get zero since essentially you're integrating nothing. If you think of integration as a summation, then an integral of zero just gives you a sum of zeros, which is zero. You can also think of zero as a constant that you can slip out of the integral. Then you integrate dr and do indeed get r, but then no matter what your limits are, there's still that zero out in front that makes the whole thing zero.
One final note. If you don't like dealing with all the integration, you can think of it too in terms of potential differences. So you can find V = kq/r at each limit -- each change in the field -- and then add up all the differences. It's the same math as with what you did with the field and integration, but essentially with this case you're just using a formula that has already been integrated.
11. Feb 11, 2014
### Gwozdzilla
Thank you so much for all of your help! I really appreciate it, and I understand the concept of potential a lot better now! | 2017-11-25T09:44:07 | {
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https://math.stackexchange.com/questions/2270759/is-there-a-coordinate-system-where-a-point-is-defined-with-two-angles/2270763 | # Is there a coordinate system where a point is defined with two angles?
In a cartesian coordinate system, a point is defined with two distances. In a polar coordinate system, a point is defined with an distance and an angle. Is there a coordinate system where a point is defined with two angles?
Edit: It has to define every point on the coordinate plane though, not just points on a sphere with a fixed radius.
• Sure, a point on a sphere (i.e., spherical coordinates with fixed radius). However, if you want to encode any point $(x,y)$ in the plane in two angles, I've never heard of such a thing. – Bobson Dugnutt May 8 '17 at 0:07
• What would you get if you used spherical coordinates with a stereographic projection? That's the closest thing to "a point is defined with two angles". But it might end up just being polar coordinates or something similar. – mathematician May 8 '17 at 0:12
• en.wikipedia.org/wiki/Biangular_coordinates – Chappers May 8 '17 at 0:17
• Thank you! That is what I was looking for. – user17220 May 8 '17 at 0:20
• Geographic latitude and longitude – Agnius Vasiliauskas May 8 '17 at 6:34
Yes. There are infinitely many. An example, for 3d space:
Actually, a coordinate system is no more than a function that maps coordinates to points in a Cartesian coordinate system.
For example, the 2D polar coordinate system is a function which maps the points given by $\begin{bmatrix}r \\ \omega \end{bmatrix}$ to $\begin{bmatrix} r \cdot \cos(\omega) \\ r \cdot \sin(\omega) \end{bmatrix}$.
In 2D space, you can map 2 angles to the points of the plane by, for example, defining their angle from the point $(-1, 0)$ and from $(1, 0)$:
In this example, we map the points of the 2D real plane by two real angles.
This system has the disadvantage that you can't represent the points of the $x$ axis: all of them would have $0$ or $\pi$ as their coordinates.
• I kind of got the idea he meant "only 2 angles". Unless you're confining yourself to the surface, I think you need a distance, right? – BenL May 8 '17 at 0:08
• I think OP is talking about 2D space only. – Bobson Dugnutt May 8 '17 at 0:08
• Okay, now do it for 2D space without the distance.. That is the question. – Bobson Dugnutt May 8 '17 at 0:09
• @peterh Nice, have an upvote. – Bobson Dugnutt May 8 '17 at 0:14
• The system has one problem: you cannot represent points on the $x$ line – Yujie Zha May 8 '17 at 0:25
In target tracking applications you may have a missile incoming towards your base, and you have two radars providing bearing measurements on the missile. In that case, the missile's location is parameterized by the bearing measurements from two radars.
See my crude diagram where circles are radars and diamond is missile.
• The points on the line connecting the two radars are not specifiable in this scheme, however. – Mark May 8 '17 at 0:17
• [1] Nice to recall this type of application. – Jean Marie May 8 '17 at 0:41
For any point on a 2-d surface, regardless of the space it is embedded in, one can assign coordinates using only two numbers. In the case of a surface like a sphere (which really is only 2 dimensional; if you have a fixed radius any point on the surface can be identified using some analog of latitude and longitude, which are themselves just a 2-d coordinate system we use on the surface of the Earth).
I think an interesting question for you to consider is what is necessary to define an angle in a space? How would you define an angle in higher dimensions?
In a 2D coordinate system, you could use the angle from a single fixed point in the z-plane (where z != 0), and define the two angles that it forms in the x and y axis, although this seems arbitrary given that you may as well give the x and y coordinates.
This allows you to avoid the problem of not being able to define points on the tangent, as a point with a different z value has no tangents to either of the points in the z=0 plane.
Yes. It's (not surprisingly) called Bipolar Coordinates: https://en.wikipedia.org/wiki/Bipolar_coordinates
You could use spherical polar coordinates as others have suggested to define points on a spherical shell, then use Stereographic projection to project the points on the lower half of that sphere onto the plane. Especially see the picture at the top of that page for some intuition (which for some reason I can't include in this answer).
This is really just Tex Andersen's answer with (arguably) some added intuition: You end up measuring angles between the origin and the point of interest from a point on the z-axis; specifically, that point is the center of the sphere.
After you do the calculations, using standard spherical polars, you get the following expression to convert back to Cartesian. Here I'm using mathematician's notation ($\varphi$ for polar angle and $\theta$ for azimuthal angle), and I'm measuring polar angle from the bottom of the sphere rather than the top to simplify the equations (otherwise you'd need to replace $\varphi$ with $\pi-\varphi$).
$\begin{bmatrix} \cos(\theta) \tan(\varphi) \\ \sin(\theta) \tan(\varphi) \end{bmatrix}$ where $\theta\in[0,2\pi),\,\varphi\in[0,\pi)$
A related idea is to do this twice: once with the projection of the point onto the $x$-axis and once with the projection onto the $y$-axis. Each time you get a $\theta$ and a $\varphi$, but the two $\theta$s are guaranteed to be $0$ and $\pi/2$ respectively so you can ignore those. This gives you two angles that are related to Cartesian by the expression:
$\begin{bmatrix} \tan(\varphi_x) \\ \tan(\varphi_y) \end{bmatrix}$ where $\varphi_x, \varphi_y\in[0,\pi)$ | 2020-01-25T08:35:36 | {
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https://artofproblemsolving.com/wiki/index.php?title=2018_AMC_10B_Problems/Problem_20&diff=cur&oldid=138364 | # Difference between revisions of "2018 AMC 10B Problems/Problem 20"
The following problem is from both the 2018 AMC 12B #18 and 2018 AMC 10B #20, so both problems redirect to this page.
## Problem
A function $f$ is defined recursively by $f(1)=f(2)=1$ and $$f(n)=f(n-1)-f(n-2)+n$$for all integers $n \geq 3$. What is $f(2018)$?
$\textbf{(A) } 2016 \qquad \textbf{(B) } 2017 \qquad \textbf{(C) } 2018 \qquad \textbf{(D) } 2019 \qquad \textbf{(E) } 2020$
## Solution 1 (Algebra)
For all integers $n \geq 7,$ note that \begin{align*} f(n)&=f(n-1)-f(n-2)+n \\ &=[f(n-2)-f(n-3)+n-1]-f(n-2)+n \\ &=-f(n-3)+2n-1 \\ &=-[f(n-4)-f(n-5)+n-3]+2n-1 \\ &=-f(n-4)+f(n-5)+n+2 \\ &=-[f(n-5)-f(n-6)+n-4]+f(n-5)+n+2 \\ &=f(n-6)+6. \end{align*} It follows that \begin{align*} f(2018)&=f(2012)+6 \\ &=f(2006)+12 \\ &=f(2000)+18 \\ & \ \vdots \\ &=f(2)+2016 \\ &=\boxed{\textbf{(B) } 2017}. \end{align*} ~MRENTHUSIASM
## Solution 2 (Algebra)
For all integers $n\geq3,$ we rearrange the given equation: $$f(n)-f(n-1)+f(n-2)=n. \hspace{28.25mm}(1)$$ For all integers $n\geq4,$ it follows that $$f(n-1)-f(n-2)+f(n-3)=n-1. \hspace{15mm}(2)$$ For all integers $n\geq4,$ we add $(1)$ and $(2):$ $$f(n)+f(n-3)=2n-1. \hspace{38.625mm}(3)$$ For all integers $n\geq7,$ it follows that $$f(n-3)+f(n-6)=2n-7. \hspace{32mm}(4)$$ For all integers $n\geq7,$ we subtract $(4)$ from $(3):$ $$f(n)-f(n-6)=6. \hspace{47.5mm}(5)$$ From $(5),$ we have the following system of $336$ equations: \begin{align*} f(2018)-f(2012)&=6, \\ f(2012)-f(2006)&=6, \\ f(2006)-f(2000)&=6, \\ & \ \vdots \\ f(8)-f(2)&=6. \end{align*} We add these equations up to get $$f(2018)-f(2)=6\cdot336=2016,$$ from which $f(2018)=f(2)+2016=\boxed{\textbf{(B) } 2017}.$
~AopsUser101 ~MRENTHUSIASM
## Solution 3 (Finite Differences)
Preamble: In this solution, we define the sequence $A$ to satisfy $a_n = f(n),$ where $a_n$ represents the $n$th term of the sequence $A.$ This solution will show a few different perspectives. Even though it may not be as quick as some of the solutions above, I feel like it is an interesting concept, and may be more motivated.
To begin, we consider the sequence $B$ formed when we take the difference of consecutive terms between $A.$ Define $b_n = a_{n+1} - a_n.$ Notice that for $n \ge 4,$ we have
\begin{cases}\begin{aligned} a_{n+1} &= a_{n} - a_{n-1} + (n+1) \\ a_{n} &= a_{n-1} - a_{n-2} + n \end{aligned}.\end{cases}
Notice that subtracting the second equation from the first, we see that $b_{n} = b_{n-1} - b_{n-2} + 1.$
If you didn’t notice that $B$ repeated directly in the solution above, you could also, possibly more naturally, take the finite differences of the sequence $b_n$ so that you could define $c_n = b_{n+1} - b_n.$ Using a similar method as above through reindexing and then subtracting, you could find that $c_n = c_{n-1} - c_{n-2}.$ The sum of any six consecutive terms of a sequence which satisfies such a recursion is $0,$ in which you have that $b_{n} = b_{n+6}.$ In the case in which finite differences didn’t reduce to such a special recursion, you could still find the first few terms of $C$ to see if there are any patterns, now that you have a much simpler sequence. Doing so in this case, it can also be seen by seeing that the sequence $C$ looks like $$\underbrace{2, 1, -1, -2, -1, 1,}_{\text{cycle period}} 2, 1, -1, -2, -1, 1, \ldots$$ in which the same result follows.
Using the fact that $B$ repeats every six terms, this motivates us to look at the sequence $B$ more carefully. Doing so, we see that $B$ looks like $$\underbrace{2, 3, 2, 0, -1, 0,}_{\text{cycle period}} 2, 3, 2, 0, -1, 0, \ldots$$ (If you tried pattern finding on sequence $B$ directly, you could also arrive at this result, although I figured defining a second sequence based on finite differences was more motivated.)
Now, there are two ways to finish.
Finish Method #1: Notice that any six consecutive terms of $B$ sum to $6,$ after which we see that $a_n = a_{n-6} + 6.$ Therefore, $a_{2018} = a_{2012} + 6 = \cdots = a_{2} + 2016 = \boxed{\textbf{(B) } 2017}.$
Finish Method #2: Notice that $a_{2018} = a_{2017} - a_{2016} + 2018 = B_{2016} + 2018 = -1 + 2018 = \boxed{\textbf{(B) } 2017}.$
~Professor-Mom
## Solution 4 (Bash)
Start out by listing some terms of the sequence. \begin{align*} f(1)&=1 \\ f(2)&=1 \\ f(3)&=3 \\ f(4)&=6 \\ f(5)&=8 \\ f(6)&=8 \\ f(7)&=7 \\ f(8)&=7 \\ f(9)&=9 \\ f(10)&=12 \\ f(11)&=14 \\ f(12)&=14 \\ f(13)&=13 \\ f(14)&=13 \\ f(15)&=15 \\ & \ \vdots \end{align*} Notice that $f(n)=n$ whenever $n$ is an odd multiple of $3$, and the pattern of numbers that follow will always be $+2$, $+3$, $+2$, $+0$, $-1$, $+0$. The largest odd multiple of $3$ smaller than $2018$ is $2013$, so we have \begin{align*} f(2013)&=2013 \\ f(2014)&=2016 \\ f(2015)&=2018 \\ f(2016)&=2018 \\ f(2017)&=2017 \\ f(2018)&=\boxed{\textbf{(B) } 2017}. \end{align*} minor edits by bunny1
## Solution 5 (Bash)
Writing out the first few values, we get $$1,1,3,6,8,8,7,7,9,12,14,14,13,13,15,18,20,20,19,19,\ldots.$$ We see that every number $x$ where $x \equiv 1\pmod 6$ has $f(x)=x,f(x+1)=f(x)=x,$ and $f(x-1)=f(x-2)=x+1.$ The greatest number that's $1\pmod{6}$ and less $2018$ is $2017,$ so we have $f(2017)=f(2018)=\boxed{\textbf{(B) } 2017}.$
~bunny1
~savannahsolver
## See Also
2018 AMC 10B (Problems • Answer Key • Resources) Preceded byProblem 19 Followed byProblem 21 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 10 Problems and Solutions
2018 AMC 12B (Problems • Answer Key • Resources) Preceded byProblem 17 Followed byProblem 19 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
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https://math.stackexchange.com/questions/4253717/a-simpler-solution-to-an-optimization-problem | # A Simpler Solution to an Optimization Problem.
The Problem:
My Solution: Let me model this situation in the first quadrant of the $$x$$-$$y$$ plane. Let us take the line $$y=mx+c$$ as the ladder (moving in the first quadrant). This is how the ladder moving in the corridor looks:
The $$y$$-intercept of the line is $$c$$ and the $$x$$-intercept is $$\left(-\frac{c}{m}\right)$$. Since the ladder is moving in the first quadrant, we have $$m<0 \text{ and } c>0$$ Let $$L$$ be the length of the ladder. Then, $$L^2=c^2+\left(\frac{c}{m}\right)^2$$
or
$$c=-\frac{mL}{\sqrt{1+m^2}} \text{ (Negative sign because m<0)}$$
and therefore, the equation of the ladder becomes $$y=mx-\frac{mL}{\sqrt{1+m^2}}$$
As the ladder is transported through the corridor, the distance between the ladder and the corner becomes smaller up to a minimum and then start increasing again. The goal here is to choose $$L$$ such that the ladder just touches the corner as it clears the corridor. This $$L$$ would be the max length of the ladder.
Let the gap along the $$y$$-direction between the corner and the ladder be $$s$$. Let the co-ordinates of the corner be $$(a,b)$$. Then, $$s=b-ma+\frac{mL}{\sqrt{1+m^2}}$$
For a given ladder, this distance hits a minimum value for a certain value of $$m$$. Let's find that out $$\frac{ds}{dm}=-a+\frac{L}{\sqrt{1+m^2}}-\frac{Lm^2}{(1+m^2)^\frac{3}{2}}=0$$
Solving which we obtain the value of $$m$$ which makes $$s$$ hit its minimum value $$m=-\left\{\left(\frac{L}{a}\right)^\frac{2}{3}-1\right\}^\frac{1}{2} \text{ (Negative because m<0)}$$
Therefore, $$s_{\text{min}}=b+a\left\{\left(\frac{L}{a}\right)^\frac{2}{3}-1\right\}^\frac{1}{2}+L\left\{1-\left(\frac{a}{L}\right)^\frac{2}{3}\right\}$$
The value of $$s_{\text{min}}$$ in the case of this problem is zero. Therefore, solving the above equation, we get $$L=a\left\{\left(\frac{b}{a}\right)^\frac{2}{3}+1\right\}^\frac{3}{2}=\left(b^\frac{2}{3}+a^\frac{2}{3}\right)^\frac{3}{2}$$
Plugging, $$a=8$$ and $$b=6$$ as asked in the question we get $$L=19.7313 \approx 20 \text{ feet}$$
Is this correct? If yes, can we make an easier model to solve this problem? (Only in the context of calculus)
Your answer is correct. Note that the question says to round $$\color{red}{down}$$ to the nearest foot.
Consider
You have two right-angled triangles, and they share the same angle $$\theta$$ since the horizontal black and green lines are parallel. Then you can shown that the length of the ladder is given by
$$L(\theta)=\frac{6}{\sin(\theta)}+\frac{8}{\cos(\theta)}$$
for $$0<\theta<\frac{\pi}{2}$$. Then
$$L'(\theta)=0\implies \tan^3(\theta)=\frac{3}{4}\implies \theta \approx 0.73752$$ $$\implies L\approx 19.7313$$
• This model assumes the ladder to be in contact with the corner at all times. But the ladder touches the corner only momentarily. Sep 18 at 16:56
• This is the length of the longest ladder you can carry horizontally around the corner of the corridor. Sep 18 at 19:35
Your animation of the sliding ladder is alright. The answer given here cannot get any simpler. There is corner contact only at a single instant.
Easier modeling is by using symbols with trig. Start with making a sketch of the corner.
$$\text{For easy trig typing using shorthand}\;$$ $$s = \sin \phi, c = \cos \phi \text{, slant ladder length = }$$
$$L=\frac {b}{s}+ \frac{a}{c} \tag 1$$
Differentiate with respect to $$\phi$$
$$\frac{-bc}{s^2}+\frac{as}{c^2}=0 \tag2$$
Simplify to find $$\tan \phi$$
$$\frac{s}{c}=\tan \phi= {\left( \frac {b}{a}\right)}^ {\frac13} \tag 3$$
Construct right triangle with Pythagorean thm ( even if the dimension does not tally to linear dimension ) resolving $$(s,c)$$ to conveniently plug their values into (1) and then simplify
$$L={\left( a^{\frac23}+ b^{\frac23}\right)}^ {\frac32} \tag 4$$
Actually it is an Astroid envelope of the sliding ladder having equation
$$x^{\frac23}+ y^{\frac23}= L^{\frac23}$$
on to which you juxtaposed a sharp corner from right side.
The 3 lines ( two blue,one red) have the same length 19.7313 units. Only the red line touches the corner $$(8.6)$$ . The blue lines do not touch the corner, there is considerable gap/clearance.
Now plug in numerical values for symbols
$$a=6,\; b=8,\;L \approx 19.7313 \;ft, \; \phi= 42.2568^{\circ}; \tag 5$$
If corridor widths are changed you know what to do.
• This model assumes the ladder to be in contact with the corner at all times. But the ladder touches the corner only momentarily. Sep 18 at 16:57
• Not at all. The model does not assume contact with corner always but assumes contact only with the wall and floor along which the ladder slides. This is ensured in the formulation. Your misconception can be hopefully cleared by sketch just now added. To drive home this point I mentioned already about the Astroid that touches the corner at a single point. All other positions of the ladder keeps it clear from re-entrant corner without any contact or interference as shown. Sep 18 at 20:58
$$x = {b\over \sin \varphi}$$ $$\ell- x = {a\over \cos \varphi}$$ so $$\boxed{\ell = {b\over \sin \varphi}+ {a\over \cos \varphi}}$$
Now take the derivative of it with respect to $$\varphi$$
• This model assumes the ladder to be in contact with the corner at all times. But the ladder touches the corner only momentarily. Sep 18 at 16:57
• So?..............................
– Aqua
Sep 18 at 17:08 | 2021-10-20T04:50:45 | {
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http://math.stackexchange.com/questions/55859/get-number-of-elements-of-a-square-matrix-given-a-vector-that-has-upper-right-el | # Get number of elements of a square matrix given a vector that has upper right elements of that matrix
Suppose I have some vectors:
• $[1,2,1]$ its length is $3$ wich represents a matrix like $\begin{pmatrix} 1 & 2\\ -1 & 1 \end{pmatrix}$ the complete matrix would have $4$ elements
• $[1,2,1,3,4,1]$ its length is $6$ wich represents $\begin{pmatrix} 1& 2& 3\\ -1& 1 & 4\\ -1& -1 &1 \end{pmatrix}$ the complete matrix would have $9$ elements
• $[1,2,1,3,4,1,5,6,7,1]$ its length is $10$ wich represents $\begin{pmatrix} 1 & 2& 3& 5\\ -1 &1 & 4& 6\\ -1 &-1 &1 & 7\\ -1 &-1 &-1 &1 \end{pmatrix}$ the complete matrix would have $16$ elements
• $[1,2,1,3,4,1,5,6,7,1,8,9,10,11,1]$ its length is $15$ wich represents
$\begin{pmatrix} 1 &2 &3 &5 &8 \\ -1 &1 &4 &6 &9 \\ -1 &-1 &1 &7 &10 \\ -1 &-1 &-1 &1 &11 \\ -1 &-1 &-1 &-1 &-1 \end{pmatrix}$ the complete matrix would have $25$ elements
How do I find a general formula to get all elements of a square matrix from the number of elements in a vector?
So
• $3\to 4$
• $6\to 9$
• $10\to 16$
• $15\to 25$
• etc...
also: How would you know the matrix size $2\times 2$, $3\times 3$,...$n\times n$?
-
If you can use the floor function, a formula is $n\mapsto (\lfloor\sqrt{2n}\rfloor)^2$ – Jonas Meyer Aug 5 '11 at 20:34
That is a good one, How did you get to this formula? – cMinor Aug 5 '11 at 20:41
cMinor: I posted it as an answer with explanation. – Jonas Meyer Aug 5 '11 at 20:50
Notice that all of your matrices are $n^2$ in size, where $n$ is the number of rows (or columns), and all of your vectors are 1+2+3+...+n = $n(n+1)/2$ in size. Your function takes numbers $n(n+1)/2$ to $n^2$, where $n$ is an integer. If you want to know what $n$ is from the length of your vector, just solve the quadratic equation: e.g. if your vector is 15 in length, solve n(n+1)/2 = 15. You get n = 5 or n = -6, and you should obviously disregard the latter because it's negative. Then you can square 5 to get the size of your matrix.
Alternatively: 3, 6, 10, 15, ... are the numbers of elements in the upper right of your matrix, including the diagonal. These elements (marked with a star below) form a 'triangle' shape in your matrix, of side length 2, 3, 4, 5 respectively:
$\underbrace{\begin{pmatrix} *&*&*&*&* \\ &*&*&*&* \\ &&*&*&* \\ &&&*&* \\ &&&&* \end{pmatrix}}_{\text{length }5}$
The "missing" elements in the lower left ('marked' as a blank space above) form another slightly smaller 'triangle' shape of size 1, 2, 3, 4 etc. respectively - that is, the lower left has 1, 3, 6, 10, ... elements in it.
These are obviously the same sequence, just shifted along by 1: that is, the upper right of a matrix of side length 10 (with the diagonal) will have as many elements as the lower left of a matrix of side length 11 (without the diagonal). So to get the total number of elements in your matrix, just add the number in the upper right to the number in the lower left: 3+1, 6+3, 10+6, 15+10, ... In recurrence relation speak, if $n_i$ is the number of elements in your vector, the number of elements in your matrix is $n_{i-1} + n_i$.
-
The function $n\mapsto \frac12 ( 1 + 4 n - \sqrt{1+8n})$ does the job.
-
Could you explain why is that? – cMinor Aug 5 '11 at 20:39
As Billy explains, you can find an explicit formula by solving a quadratic, the result of which appears in Fabian's answer.
An alternative is to note that $$\frac{m^2}{2}\lt\frac{m(m+1)}{2}\lt\frac{(m+1)^2}{2}.$$ You are given $n=\frac{m(m+1)}{2}$, and you want $m^2$. The inequalities above yield $$m^2<2n<(m+1)^2,$$ so $$m<\sqrt{2n}<m+1.$$ This implies that $m=\lfloor\sqrt{2n}\rfloor$ (the greatest integer not greater than $\sqrt{2n}$). Finally, if the vector has $n$ entries, then the matrix has $\lfloor\sqrt{2n}\rfloor^2$ entries.
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https://math.stackexchange.com/questions/662567/limit-of-function-using-sequences | # Limit of function using sequences
One easy question about limit of function at a point. In the book what I read says: Let $X\subset \mathbb{R},$ $f: X\rightarrow \mathbb{R}$, $E\subset X$, $x_0$ be an adherent point of $E$ and $L\in \mathbb{R}$, we say that $f$ converges to $L$ at $x_0$ in $E$ iff for every $\varepsilon >0$ there exists a $\delta>0$ such that $|f(x)-L|\le \varepsilon$ when $f$ is restricted to all the $x\in E$ such that $|x-x_0|<\delta$ and we write it $\lim_{x\rightarrow x_0;x\in E}f(x)=L$. Which is equivalent to the following statement for all the sequences $(a_n)$ which converges to $x_0$ and $a_n \in E$ for all $n$, so $(f(a_n))$ converges to $L$.
So my question is regard to this equivalent definition, for example if I'd like to show that the limit laws hold the result becomes trivial when change everything a sequences (I've already proven all the limit laws for sequences), Am I right?
For example, assuming that either $f,g$ are function such that $\lim_{x\rightarrow x_0;x\in E}f(x)=L$ and $\lim_{x\rightarrow x_0;x\in E}g(x)=M$ makes sense and I want to show that $\lim_{x\rightarrow x_0;x\in E}f(x)g(x)=LM$. So it's as simple as pick some arbitrary sequence $(a_n)$ which takes its element from $E$ and converges to $x_0$ that always is possible since $x_0$ is an adherent point. By the equivalence we already know that $f(a_n)\rightarrow L$ and $g(a_n)\rightarrow M$. Thus $fg(a_n)=f(a_n)g(a_n)\rightarrow LM$ and since $(a_n)$ was completely arbitrary then $\lim_{x\rightarrow x_0;x\in E}f(x)g(x)=LM$.
this is correct doesn't it?
Similarly if we have the function $\begin{cases} 1, & x\in \mathbb{Q} \\ 0, & x\notin \mathbb{Q}\end{cases}$
And I'd like to show that is not continuous everywhere it will sufficient to show that for all $x_0\in \mathbb{R}$ the limit is always undefined.
Suppose for the sake of contradiction that there exists a $x_0\in \mathbb{R}$ where the limit exist. So, there is a sequence $(q_n)$ such that $q_n \in \mathbb{Q}$ and $q_n\rightarrow x_0$ (this is always possible since each real number is an adherent point of the rational numbers), then $f(q_n)\rightarrow 1$. But we can also choose the sequence $(x_0+\sqrt{2}/n)$, which converges to $x_0$, and $f((x_0+\sqrt{2}/n)\rightarrow 0$. Since $0\not= 1$ this contradicts the uniqueness of the limit of any sequence. Hence its limit is always undefined and in particular is not continuous at any real number.
All this is correct doesn't? Because in a lot of books and notes I've seen more unwieldy arguments using $\varepsilon-\delta$.
• At the end of the first paragraph, you probably mean $a_n\in E$... – zozoens Feb 3 '14 at 21:58
• Yes that's right, Thanks – Jose Antonio Feb 3 '14 at 21:59
• Otherwise, everything looks good to me, your reasoning is correct. – zozoens Feb 3 '14 at 22:00
• Thanks, this equivalence definition IMO makes a lot of things much more easy to handle. I don't see why is the advantage of the large and cumbersome proof of the discontinuity of Dirichlet's function from the $\varepsilon-\delta$ point of view. – Jose Antonio Feb 3 '14 at 22:09
It looks good, for the most part. The only issue with your proof that the given piecewise function is nowhere continuous is that you still need to show that $x_0+\frac{\sqrt{2}}n$ is irrational for all sufficiently large $n.$ You could instead show/note/use the fact that every real number is an adherent point of the irrationals, so there is also a sequence of irrationals converging to $x_0,$ from which we can draw the conclusion you stated.
As an alternative approach, note that the given function's range is $\{0,1\}.$ This makes things simpler, as it turns out.
Claim: If $f:\Bbb R\to\{0,1\}$ is continuous at a point $x_0$, then it is constant in some open interval about $x_0$.
Proof: Suppose by way of contradiction that $f$ is not constant in any such open interval. In particular, then, for each $n\ge 1,$ there exists some $x_n\in\left(x_0-\frac1n,x_0+\frac1n\right)$ such that $f(x_n)\ne f(x_0).$ Take $\epsilon=\frac12.$ By construction, $x_n\to x_0,$ so $f(x_n)\to f(x_0)$ by continuity, so there is some $N$ such that $|f(x_n)-f(x_0)|<\epsilon$ for all $n>N.$ But since $f:\Bbb R\to\{0,1\},$ then for all $n\ge 1,$ we have $|f(x_n)-f(x_0)|=1,$ so $1<\epsilon=\frac12.$ Contradiction. $\Box$
We can easily generalize the above result to functions $f:\Bbb R\to F$ for any non-empty finite $F\subseteq\Bbb R.$ This is trivial to prove when $F$ consists of a single point, and otherwise, we take $$\epsilon=\frac12\min\{|y-z|:y,z\in F\text{ and }y\ne z\}$$ rather than $\epsilon=\frac12.$ Since $F$ is finite, then it isn't difficult to show that $\epsilon>0$--we need only show that $\{|y-z|:y,z\in F\text{ and }y\ne z\}$ is a finite set of positive numbers and we're basically done--and from there, the proof proceeds in the same sort of way.
• Maybe the easiest way is: let $\varepsilon > 0$, then $\exists q\in \mathbb{Q}$ such that $\frac{x_0-\varepsilon}{\sqrt2}< q<\frac{x_0+\varepsilon}{\sqrt2}$. We claim that is always possible to choose a $q\not= 0$. Suppose $q=0$, so $0<\frac{x_0+\varepsilon}{\sqrt2}$ and exists a $q'$ such that $0<q'<\frac{x_0+\varepsilon}{\sqrt2}$ thus $q'\not=0$ and also works. Hence $x_0-\varepsilon< \sqrt{2}q< x_0+\varepsilon$ where $q\in \mathbb{Q}\backslash \{0\}$. Then $\sqrt2q$ is an irrational number since otherwise $1/q (\sqrt2q) = \sqrt2 \in \mathbb{Q}$. – Jose Antonio Feb 4 '14 at 1:09
• This shows that the real numbers are adherent to the irrational. And we can construct a sequence such as $(i_n)\rightarrow x_0$ and $i_n\in \mathbb{R} \backslash \mathbb{Q}$ and... (thanks for the claim). – Jose Antonio Feb 4 '14 at 1:14
• Yes, indeed! That shows just what you want. Nicely done. (You're welcome.) – Cameron Buie Feb 4 '14 at 1:52 | 2019-07-23T13:35:29 | {
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https://math.stackexchange.com/questions/2292604/counting-the-number-of-galois-conjugates/2292759 | # Counting the number of Galois conjugates
While reading this solution set, I came across the following statement.
In a Galois extension $K / F$, the number of Galois conjugates of any $\alpha \in K$ is equal to:
\begin{equation*} \left| \text{Gal}(K/F) : \text{Gal}(K/F(\alpha)) \right| \end{equation*} and I am a bit perplexed where this number comes from.
After thinking about this question for a bit, I realized that there is a "backdoor" approach to figuring this out. We know that the minimal polynomial for $\alpha$, i.e. $m_{\alpha}(x)$, is equal to the squarefree part of the polynomial \begin{equation*} \prod_{\sigma \in \text{Gal}(K/F)} (x - \sigma(\alpha)) \end{equation*} which has degree equal to the number of distinct Galois conjugates $\sigma(\alpha)$. We also know that $\text{deg}(m_{\alpha}(x))$ is equal to $[F(\alpha): F]$. But this is exactly what the index of $\text{Gal}(K/F)$ in $\text{Gal}(K/F(\alpha))$ is equal to, since: \begin{align*} \left| \text{Gal}(K/F) : \text{Gal}\big(K/F(\alpha) \big) \right| & = \frac{ [K:F]}{[K:F(\alpha)]} \\[0.65em] & = [F(\alpha):F] \end{align*}
But I'm wondering if it is possible to see more "directly" that the number of (distinct) Galois conjugates of $\alpha$ is the number \begin{equation*} \left| \text{Gal}(K/F) : \text{Gal}\big(K/F(\alpha) \big) \right| \end{equation*}
• The number of conjugates of $\;\alpha\in K\;$ ( over $\;F\;$ ) is the number of different roots of the minimal polynomial of $\;\alpha\;$ in $\;F[x]\;$ , which is $\;[F(\alpha):F]=\;$ the degree of the polynomial mentioned above. – DonAntonio May 22 '17 at 22:16
• @DonAntonio That is more or less the OP's argument in the post. – Kenny Wong May 22 '17 at 22:18
• @SamY. Why do you think that ${\rm Gal}(K/F(\alpha))$ is a normal subgroup of ${\rm Gal}(K/F)$? This is true if and only if $F(\alpha)$ is a normal extension of $F$; it is not true in general. [By the way, you also wrote ${\rm Gal}(K/F(\alpha))$ and ${\rm Gal}(K/F)$ the wrong way round in your post.] – Kenny Wong May 22 '17 at 22:19
• @KennyWong Indeed. These posts that write "I don't understand this or that..." and then later it happens to be they actually do perplex me. I many times address the OP's very question and don't read the whole thing. – DonAntonio May 22 '17 at 22:20
• @KennyWong You are right. I forgot that the symbol $|G:H|$ makes sense even when $H$ is not normal. When I wrote that question, I was thinking you needed $H$ to be a normal subgroup of $G$...but you are right - $|G:H|$ is just the number of left cosets of $H$ in $G$. – Sam Y. May 22 '17 at 22:25
$\newcommand{\Gal}{\text{Gal}}$ Sometimes when $|A| = |B|$, it is because there is really an underlying bijection between $A$ and $B$.
Since $|\Gal(K/F) : \Gal(K/F(\alpha))|$ is the number of cosets of $\Gal(K/F(\alpha))$ in $\Gal(K/F)$, and this is the same as the number of Galois conjugates of $\alpha$, we might ask if we can establish a bijection between cosets of $\Gal(K/F(\alpha))$ in $\Gal(K/F)$ and conjugates of $\alpha$.
This is true, and we can see it as follows: Let $S$ be the set of conjugates of $\alpha$. $G$ acts on $S$ transitively, and the stabilizer of $\alpha \in S$ is the set of Galois elements fixing $\alpha$, equivalently, fixing $F(\alpha)$, i.e. $\Gal(K/F(\alpha))$, which establishes that $S$ and $\Gal(K/F)/\Gal(K/F(\alpha))$ are isomorphic as $\Gal(K/F)$-sets.
If you only care about the numbers, then you can use the Orbit-Stabilizer theorem to conclude that $|\Gal(K/F)| = |S||\Gal(K/F(\alpha))|$.
Edit: If you'd rather have something more concrete, the bijection between cosets and conjugates is given by $\sigma\Gal(K/F(\alpha)) \mapsto \sigma\alpha$. You can check this is a bijection in the same way you would check the stuff about group actions above.
• If $\sigma\alpha = \sigma'\alpha$ then $\sigma^{-1}\sigma\alpha = \alpha$ and thus $\sigma^{-1}\sigma'$ fixes alpha and is therefore in $\text{Gal}(K/F(\alpha))$, which means that $\sigma$ and $\sigma'$ are in the same coset. – user263190 May 23 '17 at 2:23
• I think this plus the last 3 lines should be your answer (and that $Gal(K/F(\alpha))$ is naturally a subgroup of $Gal(K/F)$) – reuns May 23 '17 at 2:32 | 2019-08-24T03:14:20 | {
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https://includestdio.com/6613.html | # elementary number theory – Is zero odd or even?
## The Question :
105 people think this question is useful
Some books say that even numbers start from $$2$$ but if you consider the number line concept, I think zero($$0$$) should be even because it is in between $$-1$$ and $$+1$$ (i.e in between two odd numbers). What is the real answer?
• +1 for “thinking outside the books”. 🙂 (Restoring a comment that seems to have been removed. What’s up with that? This is a serious commendation of a seriously-commendable practice.)
• en.wikipedia.org/wiki/Parity_of_zero
• 0 can’t be written in the form $2n+1$
• @N.S.JOHN: Well, it can, by letting $n=-\frac12.$ However, if we require that $n$ be an integer, then….
45 people think this answer is useful
For that, we can try all the axioms formulated for even numbers. I’ll use only four in this case.
Note: In this question, for the sake of my laziness, I will often use $N_e$ for even, and $N_o$ for odd.
Test 1:
An even number is always divisible by $2$.
We know that if $x,y\in \mathbb{Z}$
and $\dfrac{x}{y} \in \mathbb{Z},$ then $y$ is a divisor of $x$ (formally $y|x$).
Yes, both $0,2 \in \mathbb{Z}$ and yes, $\dfrac{0}{2}$ is $0$ which is an integer. Passed this one with flying colors!
Test 2:
$N_e + N_e$ results in $N_e$
Let’s try an even number here, say $2$. If the answer results in an even number, then $0$ will pass this test. $\ \ \ \ \underbrace{2}_{\large{N_e}} + 0 = \underbrace{2}_{N_e} \ \ \$, so zero has passed this one!
Test 3:
$N_e + N_o$ results in $N_o$
$0 + \underbrace{1}_{N_o} = \underbrace{1}_{N_o}$
Passed this test too!
Test 4:
If $n$ is an integer of parity $P$, then $n – 2$ will also be an integer of parity $P$.
We know that $2$ is even, so $2 – 2$ or $0$ is also even.
35 people think this answer is useful
Yes, the classification of naturals by their parity (= remainder modulo $2\:$) extends naturally to all integers: even integers are those integers divisible by $2,\,$ i.e. $\rm\: n = 2m\equiv 0\pmod 2,$ and odd integers are those with remainder $1$ when divided by $2,\$ i.e. $\rm\ n = 2m\!+\! 1\equiv 1\pmod 2.\,$
The effectiveness of this parity classification arises from the fact that it is compatible with integer arithmetic operations, i.e. if $\rm\ \bar{a}\ :=\ a\pmod 2\$ then $\rm\ \overline{ a+b}\ =\ \bar a + \bar b,\ \ \overline{a\ b}\ =\ \bar a\ \bar b.\:$ Iterating, we infer that equalities between expressions composed of these integer operations (i.e. integer polynomial expressions) are preserved by taking their images modulo $2\,$ (and ditto mod $\rm m\,$ for any integer $\rm m,\,$ e.g mod $9\,$ reduction yields casting out nines). In this way we can strive to better understand integers by studying their images in the simpler (finite!) rings $\rm\: \mathbb Z/m\, =\,$ integers modulo $\rm m.\:$
For example, if an integer coefficient polynomial has an integer root $\rm\ P(n) = 0\$ then it persists as a root mod $2,\,$ i.e. $\rm\ P(\bar n)\equiv 0\ (mod\ 2).\:$ So, contrapositively, if a polynomial has no roots modulo $2$ then it has no integer roots. This leads to the following simple
Parity Root Test $\$ A polynomial $\rm\:P(x)\:$ with integer coefficients
has no integer roots
when its constant coefficient $\,\rm P(0)\,$ and coefficient sum $\,\rm P(1)\,$ are both odd.
Proof $\$ The test verifies that $\rm\ P(0) \equiv P(1)\equiv 1\ \ (mod\ 2),\$ i.e.
that $\rm\:P(x)\:$ has no roots mod $2$, hence, as argued above, it has no integer roots. $\quad$ QED
So $\rm\, a x^2\! + b x\! + c\,$ has no integer roots
if $\rm\,c\,$ is odd and $\rm\,a,b\,$ have equal parity $\rm\,a\equiv b\pmod 2$
Compare the conciseness of this test to the messy reformulation that would result if we had to restrict it to positive integers. Then we could no longer represent polynomial equations in the normal form $\rm\:f(x) = 0\:$ but, rather, we would need to consider general equalities $\rm\:f(x) = g(x)\:$ where both polynomials have positive coefficients. Now the test would be much messier – bifurcating into motley cases. Indeed, historically, before the acceptance of negative integers and zero, the formula for the solution of a quadratic equation was stated in such an analogous obfuscated way – involving many cases. But by extending the naturals to the ring of integers we are able to unify what were previously motley separate cases into a single universal method of solving a general quadratic equation.
Analogous examples exist throughout history that help serve to motivate the reasons behind various number system enlargements. Studying mathematical history will help provide one with a much better appreciation of the motivations behind the successive enlargements of the notion of “number systems”, e.g. see
Kleiner: From numbers to rings: the early history of ring theory.
Above is but one of many examples where “completing” a structure in some manner serves to simplify its theory. Such ideas motivated many of the extensions of the classical number systems (as well as analogous geometrical and topological completions concept, e.g. adjoining points at $\infty$, projective closure, compactification, model completion, etc). For some interesting expositions on such methods see the references here.
Note $\:$ Analogous remarks (on the power gained by normalizing equations to the form $\ldots = 0\:$) hold true more generally for any algebraic structure whose congruences are determined by ideals – so-called ideal determined varieties, e.g. see my post here and see Gumm and Ursini: Ideals in universal algebras. Without zero and negative numbers (additive inverses) we would not be able to rewrite expressions into such concise normal forms and we would not have available such powerful algorithms such as the Grobner basis algorithm, Hermite/Smith normal forms, etc.
24 people think this answer is useful
This problem arose e.g. during the Beijing even-odd car ban for the 2008 Olympics, where cars with odd numbered licence plates were banned one day, then even the next day.
The choice is between:
• 0 is even and not odd,
• 0 is odd and not even,
• 0 is both even and odd,
• 0 is neither even nor odd (like infinity or $\pi$) or
• 0 is assigned a unique title (like how 1 called a “unit” — neither prime nor composite).
This is a matter of definition, so while you could define 0 to be any of the above, it’s best to choose the definition that will be the most consistent with the usage of “even” and “odd” for numbers other than 0.
Let $W=\{2,4,6,\ldots\}$, $V=\{1,3,5,\ldots,\}$ and lets look at the properties of even and odd numbers on these sets that we are familiar with.
• A number is either even or odd, and not both.
• If $w,x \in W$ then $w+x \in W$ (even + even = even).
• If $w \in W$ and $v \in V$ then $w+v \in V$ (even + odd = odd).
• If $y,v \in V$ then $y+v \in W$ (odd + odd = even).
[and probably many others I’ve forgotten to write here]
So it would be desirable that whichever definition we choose for 0, it preserves the above properties. Now lets say we let 0 be odd (the second and third cases listed above). Then our definition is not consistent with these properties. So, if we choose to define 0 as odd, we should have some substantial benefits to outweigh the losses. On the other hand, defining 0 to be even and not odd is consistent with the above properties.
The last two candidate definitions are essentially saying there is no consistent way of defining even or oddness to 0. But in this case, there is — 0 is even and not odd.
[Note: We also have the property that elements of W are all divisible by 2, but whether or not 0 is divisible by 2 is another matter of definition, for which we should again apply the “which is the most sensible definition” concept.]
10 people think this answer is useful
The real answer depends on the definition, because there is math-history tag invoked there was a time that $1$ was not considered an odd numbers, $0$ and negative numbers for sure where not considered even or odd. Historically the concept was defined only for natural numbers.
These days the set of all integers multiplied by $2$ is considered the set of even numbers, i.e. $\dots,-4,-2,0,2,4,\dots$ and all the integers not in that set are defined to be odd.
There is no real answer, it all depends on the definition, same way that the book is only dealing with natural numbers and not integers.
The concept was extended from naturals to integers, but there are uncountably many ways to define the even and odds beyond the natural numbers. Just make sure others know what definition you are using to label something even or odd.
9 people think this answer is useful
YES! zero is an even number. Here is the Dr. Math’s explanation.
This seems to be a matter of confusion for many others around this planet,you may always like to ask google for your confusion.
3 people think this answer is useful
Which numbers are you using? Positive integer? In this case, you don’t have to consider zero (that’s why, perhaps, some books says that even numbers start from 2). If 0 is in your number set, then yes, it is divisible by 2.
3 people think this answer is useful
$0$ is even. The difference of two distinct even numbers is also even, for example: $32 – 20 = 12$, $20 – 12 = 8$, $12 – 8 = 4$, etc. Also, the sum of two distinct even numbers is again even: $-12 + 32 = 20$, $32 + 20 = 52$, etc.
But with, say, $32 – 32 = 0$, why would this difference suddenly become odd? The answer is simple, it doesn’t, it’s still even.
2 people think this answer is useful
I think zero is an even number because it’s in between $1$ and $-1$. If they are odd numbers then the number in between must be even. Which means zero is an even number.
1 people think this answer is useful
Is Zero Even? – Numberphile
PS : This is completely opposite to my previous post (and view) but it has many good points and we are here to learn and not to push one’s own view of things.
1 people think this answer is useful
As the history of mathematics shows, zero is a very odd number whose general acceptance is surprisingly recent. However, it is not odd. Perhaps the ambiguity of “odd” causes all the confusion.
1 people think this answer is useful
Zero is even. An even integer is a number of the form $2n$ where $n$ is also an integer. A number is odd, in the mathematical sense, if it is of the form $2n + 1$ with $n$ an integer.
If zero was odd, we could solve the equation $2x + 1 = 0$ in integers. But the only possible solution is $x = -\frac{1}{2}$, which is a rational number but not an integer.
The set of integers is sometimes said to be “doubly infinite,” stretching out to positive infinity in one direction and to negative infinity in the other. If you wanted to, you could say the even numbers start at $-2$ and continue with $-4$, then $-6$, then $-8$, etc.
0 people think this answer is useful
The AP sequence
$$(-6,-4,-2,0,2,4,6)$$
shows that it is even.
If $$k$$ is an even integer, then
$$(-1)^ k = 1$$
Division of $$0$$ by $$2$$ leaves no remainder.
&c. are further confirmations. | 2021-01-28T08:08:02 | {
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http://math.stackexchange.com/questions/238137/a-limit-as-x-to-0-of-a-differentiable-function/238147 | # A limit as $x \to 0$ of a differentiable function
Given a differentiable function , then what is
$$\lim_{x\to0} \left({ \frac{f(a)}{f(a+x)}}\right)^{\frac2x}$$
Where $a$ is a real number.
If I use the identities $\ln(1+x) \sim x$
and $f(a+x) \sim f(a)+xf'(a)$
and take logarithms to both sides my guess is that the limit is
$\exp\left(- 2\frac{f'(a)}{f(a)}\right) .$
Is this method correct with this result ? thanks in advance
-
You can do it that way, with a lot more details.
Alternatively, you can just note that if $g(y)=\log f(y)$, then:
$$\lim_{x\to 0} \frac{g(x+a)-g(a)}{x} = g'(a)=\frac{f'(a)}{f(a)}$$
But $$e^{\frac{g(x+a)-g(a)}{x}} = \left(\frac{f(x+a)}{f(a)}\right)^{\frac{1}{x}}$$
So the $$\lim_{x\to 0} \left(\frac{f(x+a)}{f(a)}\right)^{\frac{1}{x}} = e^\frac{f'(a)}{f(a)}$$
by continuity of $e^z$. Finally, your limit is this expression raised to the $-2$ power, so your limit is:$$e^{-2\frac{f'(a)}{f(a)}}$$
-
(That was probably too much detail for a "homework" answer, sorry.) – Thomas Andrews Nov 15 '12 at 18:57
The answer is correct. You can make your method rigorous using big O notation.
- | 2015-04-19T21:18:48 | {
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https://math.stackexchange.com/questions/3055293/why-does-binom-3-1-binom-3-2 | Why does $\binom {3} {1} =\binom {3} {2}$?
Why does $$\binom {3} {1} =\binom {3} {2}$$? Is it just a coincidence?
Always $$\binom{n}{k}=\binom{n}{n-k}$$ (when $$0 \le k \le n.)$$ Because choosing a $$k$$ element set from $$n$$ things is the same as choosing its complement.
So also $$\binom{10}{3}=\binom{10}{7}$$ and one can generate lots of these.
The binomial coefficient is calculated as :
$$\binom{n}{k}= \frac{n!}{k!(n-k)!}$$
It's easy to see why the two binomial coefficients mentioned are equal, based on the definition.
In general, it always holds that :
$$\binom{n}{k}=\binom{n}{n-k}, \quad 0\leq k \leq n$$
Essentialy, what this means is that choosing an element $$k$$ from $$n$$ elements is the same as not choosing $$n-k$$ elements from $$n$$ elements.
Because to choose $$1$$ from $$3$$ things it's the same not to choose $$2$$ from $$3$$ things.
No, it’s not. We have
$${n \choose r} = {n \choose n-r}$$
for non-negative values of $$r$$ with $$r \leq n$$, which can be verified through the definition $${n \choose r} = \frac{n!}{r!(n-r)!}$$:
$$\frac{n!}{r!(n-r)!} = \frac{n!}{(n-r)!(n-(n-r))!}$$
$$\frac{n!}{r!(n-r)!} = \frac{n!}{(n-r)!r!}$$
$$\binom{3}{1}$$ is calculating the number of ways to choose one object out of a set of three. $$\binom{3}{2}$$ is calculating the number of ways to choose two objects out of a set of three.
Imagine I'm looking at a group of three people: Alice, Bob, and Charlotte. Let's say I want to choose two of them, and I've decided that those two are going to be Alice and Bob. There are two ways for me to say that - one way is the obvious, "I'm going to choose Alice and Bob." The other way is to talk about who I'm not choosing: "I'm going to choose everyone except Charlotte."
More generally, when picking two objects out of a group of three, we can phrase that as deciding which one is going to be left over. In other words, choosing which two to take is the same as choosing which one not to take. There are $$\binom{3}{2}$$ ways to choose which two to take; there are $$\binom{3}{1}$$ ways to choose which one not to take. Since those mean the same thing, $$\binom{3}{1} = \binom{3}{2}$$.
It is not just a coincidence. Nothing I just said really had anything to do with the numbers $$3$$, $$2$$, and $$1$$. What mattered was that when $$2$$ were chosen out of a group of $$3$$, $$1$$ was left over. We can generalize that: when $$k$$ are chosen out of a group of $$n$$, $$n - k$$ will be left over; so choosing $$k$$ out of $$n$$ is the same as excluding $$n - k$$ out of $$n$$. In other words:
$$\binom{n}{k} = \binom{n}{n - k}$$
Imagine you have a group of $$n$$ objects, and you choose $$k$$ of those objects to remove.
However, instead of considering it like this, consider it as dividing the group of $$n$$ objects into two groups, one of which has $$k$$ objects (and consequently, the other has $$(n-k)$$ objects).
We can flip this scenario and remove $$(n-k)$$ objects, leaving $$k$$ behind.
Can you see how these two scenarios are the same?
Short Answer: No, These numbers are related to the Pascal's triangle which is symmetrical. The third row is 1 , 3 , 3, 1. This states that (3 Choose 1) = (3 Chose 2): which you stated, also: (3 Choose 0) = (3 Choose 3). You can continue this for any number of items.
• I think it would be a bit clearer to use a complete sentence: "Short Answer: No, it is not a coincidence." Then follow with the explanation as above. "No" is doubtless clear to you in this sense, but to other Readers "no" might mean an objection to some other aspect of the Question posed. – hardmath Feb 25 '19 at 4:00 | 2020-05-26T00:49:32 | {
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https://dantopology.wordpress.com/category/lindelof-space/ | # Lindelof Exercise 2
The preceding post is an exercise showing that the product of countably many $\sigma$-compact spaces is a Lindelof space. The result is an example of a situation where the Lindelof property is countably productive if each factor is a “nice” Lindelof space. In this case, “nice” means $\sigma$-compact. This post gives several exercises surrounding the notion of $\sigma$-compactness.
Exercise 2.A
According to the preceding exercise, the product of countably many $\sigma$-compact spaces is a Lindelof space. Give an example showing that the result cannot be extended to the product of uncountably many $\sigma$-compact spaces. More specifically, give an example of a product of uncountably many $\sigma$-compact spaces such that the product space is not Lindelof.
Exercise 2.B
Any $\sigma$-compact space is Lindelof. Since $\mathbb{R}=\bigcup_{n=1}^\infty [-n,n]$, the real line with the usual Euclidean topology is $\sigma$-compact. This exercise is to find an example of “Lindelof does not imply $\sigma$-compact.” Find one such example among the subspaces of the real line. Note that as a subspace of the real line, the example would be a separable metric space, hence would be a Lindelof space.
Exercise 2.C
This exercise is also to look for an example of a space that is Lindelof and not $\sigma$-compact. The example sought is a non-metric one, preferably a space whose underlying set is the real line and whose topology is finer than the Euclidean topology.
Exercise 2.D
Show that the product of two Lindelof spaces is a Lindelof space whenever one of the factors is a $\sigma$-compact space.
Exercise 2.E
Prove that the product of finitely many $\sigma$-compact spaces is a $\sigma$-compact space. Give an example of a space showing that the product of countably and infinitely many $\sigma$-compact spaces does not have to be $\sigma$-compact. For example, show that $\mathbb{R}^\omega$, the product of countably many copies of the real line, is not $\sigma$-compact.
The Lindelof property and $\sigma$-compactness are basic topological notions. The above exercises are natural questions based on these two basic notions. One immediate purpose of these exercises is that they provide further interaction with the two basic notions. More importantly, working on these exercise give exposure to mathematics that is seemingly unrelated to the two basic notions. For example, finding $\sigma$-compactness on subspaces of the real line and subspaces of compact spaces naturally uses a Baire category argument, which is a deep and rich topic that finds uses in multiple areas of mathematics. For this reason, these exercises present excellent learning opportunities not only in topology but also in other useful mathematical topics.
If preferred, the exercises can be attacked head on. The exercises are also intended to be a guided tour. Hints are also provided below. Two sets of hints are given – Hints (blue dividers) and Further Hints (maroon dividers). The proofs of certain key facts are also given (orange dividers). Concluding remarks are given at the end of the post.
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Hints for Exercise 2.A
Prove that the Lindelof property is hereditary with respect to closed subspaces. That is, if $X$ is a Lindelof space, then every closed subspace of $X$ is also Lindelof.
Prove that if $X$ is a Lindelof space, then every closed and discrete subset of $X$ is countable (every space that has this property is said to have countable extent).
Show that the product of uncountably many copies of the real line does not have countable extent. Specifically, focus on either one of the following two examples.
• Show that the product space $\mathbb{R}^c$ has a closed and discrete subspace of cardinality continuum where $c$ is cardinality of continuum. Hence $\mathbb{R}^c$ is not Lindelof.
• Show that the product space $\mathbb{R}^{\omega_1}$ has a closed and discrete subspace of cardinality $\omega_1$ where $\omega_1$ is the first uncountable ordinal. Hence $\mathbb{R}^{\omega_1}$ is not Lindelof.
Hints for Exercise 2.B
Let $\mathbb{P}$ be the set of all irrational numbers. Show that $\mathbb{P}$ as a subspace of the real line is not $\sigma$-compact.
Hints for Exercise 2.C
Let $S$ be the real line with the topology generated by the half open and half closed intervals of the form $[a,b)=\{ x \in \mathbb{R}: a \le x < b \}$. The real line with this topology is called the Sorgenfrey line. Show that $S$ is Lindelof and is not $\sigma$-compact.
Hints for Exercise 2.D
It is helpful to first prove: the product of two Lindelof space is Lindelof if one of the factors is a compact space. The Tube lemma is helpful.
Tube Lemma
Let $X$ be a space. Let $Y$ be a compact space. Suppose that $U$ is an open subset of $X \times Y$ and suppose that $\{ x \} \times Y \subset U$ where $x \in X$. Then there exists an open subset $V$ of $X$ such that $\{ x \} \times Y \subset V \times Y \subset U$.
Hints for Exercise 2.E
Since the real line $\mathbb{R}$ is homeomorphic to the open interval $(0,1)$, $\mathbb{R}^\omega$ is homeomorphic to $(0,1)^\omega$. Show that $(0,1)^\omega$ is not $\sigma$-compact.
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Further Hints for Exercise 2.A
The hints here focus on the example $\mathbb{R}^c$.
Let $I=[0,1]$. Let $\omega$ be the first infinite ordinal. For convenience, consider $\omega$ the set $\{ 0,1,2,3,\cdots \}$, the set of all non-negative integers. Since $\omega^I$ is a closed subset of $\mathbb{R}^I$, any closed and discrete subset of $\omega^I$ is a closed and discrete subset of $\mathbb{R}^I$. The task at hand is to find a closed and discrete subset of $Y=\omega^I$. To this end, we define $W=\{W_x: x \in I \}$ after setting up background information.
For each $t \in I$, choose a sequence $O_{t,1},O_{t,2},O_{t,3},\cdots$ of open intervals (in the usual topology of $I$) such that
• $\{ t \}=\bigcap_{j=1}^\infty O_{t,j}$,
• $\overline{O_{t,j+1}} \subset O_{t,j}$ for each $j$ (the closure is in the usual topology of $I$).
Note. For each $t \in I-\{0,1 \}$, the open intervals $O_{t,j}$ are of the form $(a,b)$. For $t=0$, the open intervals $O_{t,j}$ are of the form $[0,b)$. For $t=1$, the open intervals $O_{t,j}$ are of the form $(a,1]$.
For each $t \in I$, define the map $f_t: I \rightarrow \omega$ as follows:
$f_t(x) = \begin{cases} 0 & \ \ \ \mbox{if } x=t \\ 1 & \ \ \ \mbox{if } x \in I-O_{t,1} \\ 2 & \ \ \ \mbox{if } x \in I-O_{t,2} \text{ and } x \in O_{t,1} \\ 3 & \ \ \ \mbox{if } x \in I-O_{t,3} \text{ and } x \in O_{t,2} \\ \vdots & \ \ \ \ \ \ \ \ \ \ \vdots \\ j & \ \ \ \mbox{if } x \in I-O_{t,j} \text{ and } x \in O_{t,j-1} \\ \vdots & \ \ \ \ \ \ \ \ \ \ \vdots \end{cases}$
We are now ready to define $W=\{W_x: x \in I \}$. For each $x \in I$, $W_x$ is the mapping $W_x:I \rightarrow \omega$ defined by $W_x(t)=f_t(x)$ for each $t \in I$.
Show the following:
• The set $W=\{W_x: x \in I \}$ has cardinality continuum.
• The set $W$ is a discrete space.
• The set $W$ is a closed subspace of $Y$.
Further Hints for Exercise 2.B
A subset $A$ of the real line $\mathbb{R}$ is nowhere dense in $\mathbb{R}$ if for any nonempty open subset $U$ of $\mathbb{R}$, there is a nonempty open subset $V$ of $U$ such that $V \cap A=\varnothing$. If we replace open sets by open intervals, we have the same notion.
Show that the real line $\mathbb{R}$ with the usual Euclidean topology cannot be the union of countably many closed and nowhere dense sets.
Further Hints for Exercise 2.C
Prove that if $X$ and $Y$ are $\sigma$-compact, then the product $X \times Y$ is $\sigma$-compact, hence Lindelof.
Prove that $S$, the Sorgenfrey line, is Lindelof while its square $S \times S$ is not Lindelof.
Further Hints for Exercise 2.D
As suggested in the hints given earlier, prove that $X \times Y$ is Lindelof if $X$ is Lindelof and $Y$ is compact. As suggested, the Tube lemma is a useful tool.
Further Hints for Exercise 2.E
The product space $(0,1)^\omega$ is a subspace of the product space $[0,1]^\omega$. Since $[0,1]^\omega$ is compact, we can fall back on a Baire category theorem argument to show why $(0,1)^\omega$ cannot be $\sigma$-compact. To this end, we consider the notion of Baire space. A space $X$ is said to be a Baire space if for each countable family $\{ U_1,U_2,U_3,\cdots \}$ of open and dense subsets of $X$, the intersection $\bigcap_{i=1}^\infty U_i$ is a dense subset of $X$. Prove the following results.
Fact E.1
Let $X$ be a compact Hausdorff space. Let $O_1,O_2,O_3,\cdots$ be a sequence of non-empty open subsets of $X$ such that $\overline{O_{n+1}} \subset O_n$ for each $n$. Then the intersection $\bigcap_{i=1}^\infty O_i$ is non-empty.
Fact E.2
Any compact Hausdorff space is Baire space.
Fact E.3
Let $X$ be a Baire space. Let $Y$ be a dense $G_\delta$-subset of $X$ such that $X-Y$ is a dense subset of $X$. Then $Y$ is not a $\sigma$-compact space.
Since $X=[0,1]^\omega$ is compact, it follows from Fact E.2 that the product space $X=[0,1]^\omega$ is a Baire space.
Fact E.4
Let $X=[0,1]^\omega$ and $Y=(0,1)^\omega$. The product space $Y=(0,1)^\omega$ is a dense $G_\delta$-subset of $X=[0,1]^\omega$. Furthermore, $X-Y$ is a dense subset of $X$.
It follows from the above facts that the product space $(0,1)^\omega$ cannot be a $\sigma$-compact space.
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Proofs of Key Steps for Exercise 2.A
The proof here focuses on the example $\mathbb{R}^c$.
To see that $W=\{W_x: x \in I \}$ has the same cardinality as that of $I$, show that $W_x \ne W_y$ for $x \ne y$. This follows from the definition of the mapping $W_x$.
To see that $W$ is discrete, for each $x \in I$, consider the open set $U_x=\{ b \in Y: b(x)=0 \}$. Note that $W_x \in U_x$. Further note that $W_y \notin U_x$ for all $y \ne x$.
To see that $W$ is a closed subset of $Y$, let $k: I \rightarrow \omega$ such that $k \notin W$. Consider two cases.
Case 1. $k(r) \ne 0$ for all $r \in I$.
Note that $\{ O_{t,k(t)}: t \in I \}$ is an open cover of $I$ (in the usual topology). There exists a finite $H \subset I$ such that $\{ O_{h,k(h)}: h \in H \}$ is a cover of $I$. Consider the open set $G=\{ b \in Y: \forall \ h \in H, \ b(h)=k(h) \}$. Define the set $F$ as follows:
$F=\{ c \in I: W_c \in G \}$
The set $F$ can be further described as follows:
\displaystyle \begin{aligned} F&=\{ c \in I: W_c \in G \} \\&=\{ c \in I: \forall \ h \in H, \ W_c(h)=f_h(c)=k(h) \ne 0 \} \\&=\{ c \in I: \forall \ h \in H, \ c \in I-O_{h,k(h)} \} \\&=\bigcap_{h \in H} (I-O_{h,k(h)}) \\&=I-\bigcup_{h \in H} O_{h,k(h)}=I-I =\varnothing \end{aligned}
The last step is $\varnothing$ because $\{ O_{h,k(h)}: h \in H \}$ is a cover of $I$. The fact that $F=\varnothing$ means that $G$ is an open subset of $Y$ containing the point $k$ such that $G$ contains no point of $W$.
Case 2. $k(r) = 0$ for some $r \in I$.
Since $k \notin W$, $k \ne W_x$ for all $x \in I$. In particular, $k \ne W_r$. This means that $k(t) \ne W_r(t)$ for some $t \in I$. Define the open set $G$ as follows:
$G=\{ b \in Y: b(r)=0 \text{ and } b(t)=k(t) \}$
Clearly $k \in G$. Observe that $W_r \notin G$ since $W_r(t) \ne k(t)$. For each $p \in I-\{ r \}$, $W_p \notin G$ since $W_p(r) \ne 0$. Thus $G$ is an open set containing $k$ such that $G \cap W=\varnothing$.
Both cases show that $W$ is a closed subset of $Y=\omega^I$.
Proofs of Key Steps for Exercise 2.B
Suppose that $\mathbb{P}$, the set of all irrational numbers, is $\sigma$-compact. That is, $\mathbb{P}=A_1 \cup A_2 \cup A_3 \cup \cdots$ where each $A_i$ is a compact space as a subspace of $\mathbb{P}$. Any compact subspace of $\mathbb{P}$ is also a compact subspace of $\mathbb{R}$. As a result, each $A_i$ is a closed subset of $\mathbb{R}$. Furthermore, prove the following:
Each $A_i$ is a nowhere dense subset of $\mathbb{R}$.
Each singleton set $\{ r \}$ where $r$ is any rational number is also a closed and nowhere dense subset of $\mathbb{R}$. This means that the real line is the union of countably many closed and nowhere dense subsets, contracting the hints given earlier. Thus $\mathbb{P}$ cannot be $\sigma$-compact.
Proofs of Key Steps for Exercise 2.C
The Sorgenfrey line $S$ is a Lindelof space whose square $S \times S$ is not normal. This is a famous example of a Lindelof space whose square is not Lindelof (not even normal). For reference, a proof is found here. An alternative proof of the non-normality of $S \times S$ uses the Baire category theorem and is found here.
If the Sorgenfrey line is $\sigma$-compact, then $S \times S$ would be $\sigma$-compact and hence Lindelof. Thus $S$ cannot be $\sigma$-compact.
Proofs of Key Steps for Exercise 2.D
Suppose that $X$ is Lindelof and that $Y$ is compact. Let $\mathcal{U}$ be an open cover of $X \times Y$. For each $x \in X$, let $\mathcal{U}_x \subset \mathcal{U}$ be finite such that $\mathcal{U}_x$ is a cover of $\{ x \} \times Y$. Putting it another way, $\{ x \} \times Y \subset \cup \mathcal{U}_x$. By the Tube lemma, for each $x \in X$, there is an open $O_x$ such that $\{ x \} \times Y \subset O_x \times Y \subset \cup \mathcal{U}_x$. Since $X$ is Lindelof, there exists a countable set $\{ x_1,x_2,x_3,\cdots \} \subset X$ such that $\{ O_{x_1},O_{x_2},O_{x_3},\cdots \}$ is a cover of $X$. Then $\mathcal{U}_{x_1} \cup \mathcal{U}_{x_2} \cup \mathcal{U}_{x_3} \cup \cdots$ is a countable subcover of $\mathcal{U}$. This completes the proof that $X \times Y$ is Lindelof when $X$ is Lindelof and $Y$ is compact.
To complete the exercise, observe that if $X$ is Lindelof and $Y$ is $\sigma$-compact, then $X \times Y$ is the union of countably many Lindelof subspaces.
Proofs of Key Steps for Exercise 2.E
Proof of Fact E.1
Let $X$ be a compact Hausdorff space. Let $O_1,O_2,O_3,\cdots$ be a sequence of non-empty open subsets of $X$ such that \$latex $\overline{O_{n+1}} \subset O_n$ for each $n$. Show that the intersection $\bigcap_{i=1}^\infty O_i$ is non-empty.
Suppose that $\bigcap_{i=1}^\infty O_i=\varnothing$. Choose $x_1 \in O_1$. There must exist some $n_1$ such that $x_1 \notin O_{n_1}$. Choose $x_2 \in O_{n_1}$. There must exist some $n_2>n_1$ such that $x_2 \notin O_{n_2}$. Continue in this manner we can choose inductively an infinite set $A=\{ x_1,x_2,x_3,\cdots \} \subset X$ such that $x_i \ne x_j$ for $i \ne j$. Since $X$ is compact, the infinite set $A$ has a limit point $p$. This means that every open set containing $p$ contains some $x_j$ (in fact for infinitely many $j$). The point $p$ cannot be in the intersection $\bigcap_{i=1}^\infty O_i$. Thus for some $n$, $p \notin O_n$. Thus $p \notin \overline{O_{n+1}}$. We can choose an open set $U$ such that $p \in U$ and $U \cap \overline{O_{n+1}}=\varnothing$. However, $U$ must contain some point $x_j$ where $j>n+1$. This is a contradiction since $O_j \subset \overline{O_{n+1}}$ for all $j>n+1$. Thus Fact E.1 is established.
Proof of Fact E.2
Let $X$ be a compact space. Let $U_1,U_2,U_3,\cdots$ be open subsets of $X$ such that each $U_i$ is also a dense subset of $X$. Let $V$ a non-empty open subset of $X$. We wish to show that $V$ contains a point that belongs to each $U_i$. Since $U_1$ is dense in $X$, $O_1=V \cap U_1$ is non-empty. Since $U_2$ is dense in $X$, choose non-empty open $O_2$ such that $\overline{O_2} \subset O_1$ and $O_2 \subset U_2$. Since $U_3$ is dense in $X$, choose non-empty open $O_3$ such that $\overline{O_3} \subset O_2$ and $O_3 \subset U_3$. Continue inductively in this manner and we have a sequence of open sets $O_1,O_2,O_3,\cdots$ just like in Fact E.1. Then the intersection of the open sets $O_n$ is non-empty. Points in the intersection are in $V$ and in all the $U_n$. This completes the proof of Fact E.2.
Proof of Fact E.3
Let $X$ be a Baire space. Let $Y$ be a dense $G_\delta$-subset of $X$ such that $X-Y$ is a dense subset of $X$. Show that $Y$ is not a $\sigma$-compact space.
Suppose $Y$ is $\sigma$-compact. Let $Y=\bigcup_{n=1}^\infty B_n$ where each $B_n$ is compact. Each $B_n$ is obviously a closed subset of $X$. We claim that each $B_n$ is a closed nowhere dense subset of $X$. To see this, let $U$ be a non-empty open subset of $X$. Since $X-Y$ is dense in $X$, $U$ contains a point $p$ where $p \notin Y$. Since $p \notin B_n$, there exists a non-empty open $V \subset U$ such that $V \cap B_n=\varnothing$. This shows that each $B_n$ is a nowhere dense subset of $X$.
Since $Y$ is a dense $G_\delta$-subset of $X$, $Y=\bigcap_{n=1}^\infty O_n$ where each $O_n$ is an open and dense subset of $X$. Then each $A_n=X-O_n$ is a closed nowhere dense subset of $X$. This means that $X$ is the union of countably many closed and nowhere dense subsets of $X$. More specifically, we have the following.
(1)………$X= \biggl( \bigcup_{n=1}^\infty A_n \biggr) \cup \biggl( \bigcup_{n=1}^\infty B_n \biggr)$
Statement (1) contradicts the fact that $X$ is a Baire space. Note that all $X-A_n$ and $X-B_n$ are open and dense subsets of $X$. Further note that the intersection of all these countably many open and dense subsets of $X$ is empty according to (1). Thus $Y$ cannot not a $\sigma$-compact space.
Proof of Fact E.4
The space $X=[0,1]^\omega$ is compact since it is a product of compact spaces. To see that $Y=(0,1)^\omega$ is a dense $G_\delta$-subset of $X$, note that $Y=\bigcap_{n=1}^\infty U_n$ where for each integer $n \ge 1$
(2)………$U_n=(0,1) \times \cdots \times (0,1) \times [0,1] \times [0,1] \times \cdots$
Note that the first $n$ factors of $U_n$ are the open interval $(0,1)$ and the remaining factors are the closed interval $[0,1]$. It is also clear that $X-Y$ is a dense subset of $X$. This completes the proof of Fact E.4.
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Concluding Remarks
Exercise 2.A
The exercise is to show that the product of uncountably many $\sigma$-compact spaces does not need to be Lindelof. The approach suggested in the hints is to show that $\mathbb{R}^{c}$ has uncountable extent where $c$ is continuum. Having uncountable extent (i.e. having an uncountable subset that is both closed and discrete) implies the space is not Lindelof. The uncountable extent of the product space $\mathbb{R}^{\omega_1}$ is discussed in this post.
For $\mathbb{R}^{c}$ and $\mathbb{R}^{\omega_1}$, there is another way to show non-Lindelof. For example, both product spaces are not normal. As a result, both product spaces cannot be Lindelof. Note that every regular Lindelof space is normal. Both product spaces contain the product $\omega^{\omega_1}$ as a closed subspace. The non-normality of $\omega^{\omega_1}$ is discussed here.
Exercise 2.B
The hints given above is to show that the set of all irrational numbers, $\mathbb{P}$, is not $\sigma$-compact (as a subspace of the real line). The same argument showing that $\mathbb{P}$ is not $\sigma$-compact can be generalized. Note that the complement of $\mathbb{P}$ is $\mathbb{Q}$, the set of all rational numbers (a countable set). In this case, $\mathbb{Q}$ is a dense subset of the real line and is the union of countably many singleton sets. Each singleton set is a closed and nowhere dense subset of the real line. In general, we can let $B$, the complement of a set $A$, be dense in the real line and be the union of countably many closed nowhere dense subsets of the real line (not necessarily singleton sets). The same argument will show that $A$ cannot be a $\sigma$-compact space. This argument is captured in Fact E.3 in Exercise 2.E. Thus both Exercise 2.B and Exercise 2.E use a Baire category argument.
Exercise 2.E
Like Exercise 2.B, this exercise is also to show a certain space is not $\sigma$-compact. In this case, the suggested space is $\mathbb{R}^{\omega}$, the product of countably many copies of the real line. The hints given use a Baire category argument, as outlined in Fact E.1 through Fact E.4. The product space $\mathbb{R}^{\omega}$ is embedded in the compact space $[0,1]^{\omega}$, which is a Baire space. As mentioned earlier, Fact E.3 is essentially the same argument used for Exercise 2.B.
Using the same Baire category argument, it can be shown that $\omega^{\omega}$, the product of countably many copies of the countably infinite discrete space, is not $\sigma$-compact. The space $\omega$ of the non-negative integers, as a subspace of the real line, is certainly $\sigma$-compact. Using the same Baire category argument, we can see that the product of countably many copies of this discrete space is not $\sigma$-compact. With the product space $\omega^{\omega}$, there is a connection with Exercise 2.B. The product $\omega^{\omega}$ is homeomorphic to $\mathbb{P}$. The idea of the homeomorphism is discussed here. Thus the non-$\sigma$-compactness of $\omega^{\omega}$ can be achieved by mapping it to the irrationals. Of course, the same Baire category argument runs through both exercises.
Exercise 2.C
Even the non-$\sigma$-compactness of the Sorgenfrey line $S$ can be achieved by a Baire category argument. The non-normality of the Sorgenfrey plane $S \times S$ can be achieved by Jones’ lemma argument or by the fact that $\mathbb{P}$ is not a first category set. Links to both arguments are given in the Proof section above.
See here for another introduction to the Baire category theorem.
The Tube lemma is discussed here.
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# Lindelof Exercise 1
A space $X$ is called a $\sigma$-compact space if it is the union of countably many compact subspaces. Clearly, any $\sigma$-compact space is Lindelof. It is well known that the product of Lindelof spaces does not need to be Lindelof. The most well known example is perhaps the square of the Sorgenfrey line. In certain cases, the Lindelof property can be productive. For example, the product of countably many $\sigma$-compact spaces is a Lindelof space. The discussion here centers on the following theorem.
Theorem 1
Let $X_1,X_2,X_3,\cdots$ be $\sigma$-compact spaces. Then the product space $\prod_{i=1}^\infty X_i$ is Lindelof.
Theorem 1 is Exercise 3.8G in page 195 of General Topology by Engelking [1]. The reference for Exercise 3.8G is [2]. But the theorem is not found in [2] (it is not stated directly and it does not seem to be an obvious corollary of a theorem discussed in that paper). However, a hint is provided in Engelking for Exercise 3.8G. In this post, we discuss Theorem 1 as an exercise by giving expanded hint. Solutions to some of the key steps in the expanded hint are given at the end of the post.
Expanded Hint
It is helpful to first prove the following theorem.
Theorem 2
For each integer $i \ge 1$, let $C_{i,1},C_{i,2},\cdots$ be compact spaces and let $C_i$ be the topological sum:
$C_i=C_{i,1} \oplus C_{i,2} \oplus C_{i,3} \oplus \cdots=\oplus_{j=1}^\infty C_{i,j}$
Then the product $\prod_{i=1}^\infty C_i$ is Lindelof.
Note that in the topological sum $C_{i,1} \oplus C_{i,2} \oplus C_{i,3} \oplus \cdots$, the spaces $C_{i,1},C_{i,2},C_{i,3},\cdots$ are considered pairwise disjoint. The open sets in the sum are simply unions of the open sets in the individual spaces. Another way to view this topology: each of the $C_{i,j}$ is both closed and open in the topological sum. Theorem 2 is essentially saying that the product of countably many $\sigma$-compact spaces is Lindelof if each $\sigma$-compact space is the union of countably many disjoint compact spaces. The hint for Exercise 3.8G can be applied much more naturally on Theorem 2 than on Theorem 1. The following is Exercise 3.8F (a), which is the hint for Exercise 3.8G.
Lemma 3
Let $Z$ be a compact space. Let $X$ be a subspace of $Z$. Suppose that there exist $F_1,F_2,F_3,\cdots$, closed subsets of $Z$, such that for all $x$ and $y$ where $x \in X$ and $y \in Z-X$, there exists $F_i$ such that $x \in F_i$ and $y \notin F_i$. Then $X$ is a Lindelof space.
The following theorem connects the hint (Lemma 3) with Theorem 2.
Theorem 4
For each integer $i \ge 1$, let $Z_i$ be the one-point compactification of $C_i$ in Theorem 2. Then the product $Z=\prod_{i=1}^\infty Z_i$ is a compact space. Furthermore, $X=\prod_{i=1}^\infty C_i$ is a subspace of $Z$. Prove that $Z$ and $X$ satisfy Lemma 3.
Each $C_i$ in Theorem 2 is a locally compact space. To define the one-point compactifications, for each $i$, choose $p_i \notin C_i$. Make sure that $p_i \ne p_j$ for $i \ne j$. Then $Z_i$ is simply
$Z_i=C_i \cup \{ p_i \}=C_{i,1} \oplus C_{i,2} \oplus C_{i,3} \oplus \cdots \cup \{ p_i \}$
with the topology defined as follows:
• Open subsets of $C_i$ continue to be open in $Z_i$.
• An open set containing $p_i$ is of the form $\{ p_i \} \cup (C_i - \overline{D})$ where $D$ is open in $C_i$ and $D$ is contained in the union of finitely many $C_{i,j}$.
For convenience, each point $p_i$ is called a point at infinity.
Note that Theorem 2 follows from Lemma 3 and Theorem 4. In order to establish Theorem 1 from Theorem 2, observe that the Lindelof property is preserved by any continuous mapping and that there is a natural continuous map from the product space in Theorem 2 to the product space in Theorem 1.
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Proofs of Key Steps
Proof of Lemma 3
Let $Z$, $X$ and $F_1,F_2,F_3,\cdots$ be as described in the statement for Lemma 3. Let $\mathcal{U}$ be a collection of open subsets of $Z$ such that $\mathcal{U}$ covers $X$. We would like to show that a countable subcollection of $\mathcal{U}$ is also a cover of $X$. Let $O=\cup \mathcal{U}$. If $Z-O=\varnothing$, then $\mathcal{U}$ is an open cover of $Z$ and there is a finite subset of $\mathcal{U}$ that is a cover of $Z$ and thus a cover of $X$. Thus we can assume that $Z-O \ne \varnothing$.
Let $F=\{ F_1,F_2,F_3,\cdots \}$. Let $K=Z-O$, which is compact. We make the following claim.
Claim. Let $Y$ be the union of all possible $\cap G$ where $G \subset F$ is finite and $\cap G \subset O$. Then $X \subset Y \subset O$.
To establish the claim, let $x \in X$. For each $y \in K=Z-O$, there exists $F_{n(y)}$ such that $x \in F_{n(y)}$ and $y \notin F_{n(y)}$. This means that $\{ Z-F_{n(y)}: y \in K \}$ is an open cover of $K$. By the compactness of $K$, there are finitely many $F_{n(y_1)}, \cdots, F_{n(y_k)}$ such that $F_{n(y_1)} \cap \cdots \cap F_{n(y_k)}$ misses $K$, or equivalently $F_{n(y_1)} \cap \cdots \cap F_{n(y_k)} \subset O$. Note that $x \in F_{n(y_1)} \cap \cdots \cap F_{n(y_k)}$. Further note that $F_{n(y_1)} \cap \cdots \cap F_{n(y_k)} \subset Y$. This establishes the claim that $X \subset Y$. The claim that $Y \subset O$ is clear from the definition of $Y$.
Each set $F_i$ is compact since it is closed in $Z$. The intersection of finitely many $F_i$ is also compact. Thus the $\cap G$ in the definition of $Y$ in the above claim is compact. There can be only countably many $\cap G$ in the definition of $Y$. Thus $Y$ is a $\sigma$-compact space that is covered by the open cover $\mathcal{U}$. Choose a countable $\mathcal{V} \subset \mathcal{U}$ such that $\mathcal{V}$ covers $Y$. Then $\mathcal{V}$ is a cover of $X$ too. This completes the proof that $X$ is Lindelof.
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Proof of Theorem 4
Recall that $Z=\prod_{i=1}^\infty Z_i$ and that $X=\prod_{i=1}^\infty C_i$. Each $Z_i$ is the one-point compactification of $C_i$, which is the topological sum of the disjoint compact spaces $C_{i,1},C_{i,2},\cdots$.
For integers $i,j \ge 1$, define $K_{i,j}=C_{i,1} \oplus C_{i,2} \oplus \cdots \oplus C_{i,j}$. For integers $n,j \ge 1$, define the product $F_{n,j}$ as follows:
$F_{n,j}=K_{1,j} \times \cdots \times K_{n,j} \times Z_{n+1} \times Z_{n+2} \times \cdots$
Since $F_{n,j}$ is a product of compact spaces, $F_{n,j}$ is compact and thus closed in $Z$. There are only countably many $F_{n,j}$.
We claim that the countably many $F_{n,j}$ have the property indicated in Lemma 3. To this end, let $f \in X=\prod_{i=1}^\infty C_i$ and $g \in Z-X$. There exists an integer $n \ge 1$ such that $g(n) \notin C_{n}$. This means that $g(n) \notin C_{n,j}$ for all $j$, i.e. $g(n)=p_n$ (so $g(n)$ must be the point at infinity). Choose $j \ge 1$ large enough such that
$f(i) \in K_{i,j}=C_{i,1} \oplus C_{i,2} \oplus \cdots \oplus C_{i,j}$
for all $i \le n$. It follows that $f \in F_{n,j}$ and $g \notin F_{n,j}$. Thus the sequence of closed sets $F_{n,j}$ satisfies Lemma 3. By Lemma 3, $X=\prod_{i=1}^\infty C_i$ is Lindelof.
Reference
1. Engelking R., General Topology, Revised and Completed edition, Elsevier Science Publishers B. V., Heldermann Verlag, Berlin, 1989.
2. Hager A. W., Approximation of real continuous functions on Lindelof spaces, Proc. Amer. Math. Soc., 22, 156-163, 1969.
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# A little corner in the world of set-theoretic topology
This post puts a spot light on a little corner in the world of set-theoretic topology. There lies in this corner a simple topological statement that opens a door to the esoteric world of independence results. In this post, we give a proof of this basic fact and discuss its ramifications. This basic result is an excellent entry point to the study of S and L spaces.
The following paragraph is found in the paper called Gently killing S-spaces by Todd Eisworth, Peter Nyikos and Saharon Shelah [1]. The basic fact in question is highlighted in blue.
A simultaneous generalization of hereditarily separable and hereditarily Lindelof spaces is the class of spaces of countable spread – those spaces in which every discrete subspace is countable. One of the basic facts in this little corner of set-theoretic topology is that if a regular space of countable spread is not hereditarily separable, it contains an L-space, and if it is not hereditarily Lindelof, it contains an S-space. [1]
The same basic fact is also mentioned in the paper called The spread of regular spaces by Judith Roitman [2].
It is also well known that a regular space of countable spread which is not hereditarily separable contains an L-space and a regular space of countable spread which is not hereditarily Lindelof contains an S-space. Thus an absolute example of a space satisfying (Statement) A would contain a proof of the existence of S and L space – a consummation which some may devoutly wish, but which this paper does not attempt. [2]
Statement A in [2] is: There exists a 0-dimensional Hausdorff space of countable spread that is not the union of a hereditarily separable and a hereditarily Lindelof space. Statement A would mean the existence of a regular space of countable spread that is not hereditarily separable and that is also not hereditarily Lindelof. By the well known fact just mentioned, statement A would imply the existence of a space that is simultaneously an S-space and an L-space!
Let’s unpack the preceding section. First some basic definitions. A space $X$ is of countable spread (has countable spread) if every discrete subspace of $X$ is countable. A space $X$ is hereditarily separable if every subspace of $X$ is separable. A space $X$ is hereditarily Lindelof if every subspace of $X$ is Lindelof. A space is an S-space if it is hereditarily separable but not Lindelof. A space is an L-space if it is hereditarily Lindelof but not separable. See [3] for a basic discussion of S and L spaces.
Hereditarily separable but not Lindelof spaces as well as hereditarily Lindelof but not separable spaces can be easily defined in ZFC [3]. However, such examples are not regular. For the notions of S and L-spaces to be interesting, the definitions must include regularity. Thus in the discussion that follows, all spaces are assumed to be Hausdorff and regular.
One amazing aspect about set-theoretic topology is that one sometimes does not have to stray far from basic topological notions to encounter pathological objects such as S-spaces and L-spaces. The definition of a topological space is of course a basic definition. Separable spaces and Lindelof spaces are basic notions that are not far from the definition of topological spaces. The same can be said about hereditarily separable and hereditarily Lindelof spaces. Out of these basic ingredients come the notion of S-spaces and L-spaces, the existence of which is one of the key motivating questions in set-theoretic topology in the twentieth century. The study of S and L-spaces is a body of mathematics that had been developed for nearly a century. It is a fruitful area of research at the boundary of topology and axiomatic set theory.
The existence of an S-space is independent of ZFC (as a result of the work by Todorcevic in early 1980s). This means that there is a model of set theory in which an S-space exists and there is also a model of set theory in which S-spaces cannot exist. One half of the basic result mentioned in the preceding section is intimately tied to the existence of S-spaces and thus has interesting set-theoretic implications. The other half of the basic result involves the existence of L-spaces, which are shown to exist without using extra set theory axioms beyond ZFC by Justin Moore in 2005, which went against the common expectation that the existence of L-spaces would be independent of ZFC as well.
Let’s examine the basic notions in a little more details. The following diagram shows the properties surrounding the notion of countable spread.
Diagram 1 – Properties surrounding countable spread
The implications (the arrows) in Diagram 1 can be verified easily. Central to the discussion at hand, both hereditarily separable and hereditarily Lindelof imply countable spread. The best way to see this is that if a space has an uncountable discrete subspace, that subspace is simultaneously a non-separable subspace and a non-Lindelof subspace. A natural question is whether these implications can be reversed. Another question is whether the properties in Diagram 1 can be related in other ways. The following diagram attempts to ask these questions.
Diagram 2 – Reverse implications surrounding countable spread
Not shown in Diagram 2 are these four facts: separable $\not \rightarrow$ hereditarily separable, Lindelof $\not \rightarrow$ hereditarily Lindelof, separable $\not \rightarrow$ countable spread and Lindelof $\not \rightarrow$ countable spread. The examples supporting these facts are not set-theoretic in nature and are not discussed here.
Let’s focus on each question mark in Diagram 2. The two horizontal arrows with question marks at the top are about S-space and L-space. If $X$ is hereditarily separable, then is $X$ hereditarily Lindelof? A “no” answer would mean there is an S-space. A “yes” answer would mean there exists no S-space. So the top arrow from left to right is independent of ZFC. Since an L-space can be constructed within ZFC, the question mark in the top arrow in Diagram 2 from right to left has a “no” answer.
Now focus on the arrows emanating from countable spread in Diagram 2. These arrows are about the basic fact discussed earlier. From Diagram 1, we know that hereditarily separable implies countable spread. Can the implication be reversed? Any L-space would be an example showing that the implication cannot be reversed. Note that any L-space is of countable spread and is not separable and hence not hereditarily separable. Since L-space exists in ZFC, the question mark in the arrow from countable spread to hereditarily separable has a “no” answer. The same is true for the question mark in the arrow from countable spread to separable
We know that hereditarily Lindelof implies countable spread. Can the implication be reversed? According to the basic fact mentioned earlier, if the implication cannot be reversed, there exists an S-space. Thus if S-space does not exist, the implication can be reversed. Any S-space is an example showing that the implication cannot be reversed. Thus the question in the arrow from countable spread to hereditarily Lindelof cannot be answered without assuming axioms beyond ZFC. The same is true for the question mark for the arrow from countable spread to Lindelf.
Diagram 2 is set-theoretic in nature. The diagram is remarkable in that the properties in the diagram are basic notions that are only brief steps away from the definition of a topological space. Thus the basic highlighted here is a quick route to the world of independence results.
We now give a proof of the basic result, which is stated in the following theorem.
Theorem 1
Let $X$ is regular and Hausdorff space. Then the following is true.
• If $X$ is of countable spread and is not a hereditarily separable space, then $X$ contains an L-space.
• If $X$ is of countable spread and is not a hereditarily Lindelof space, then $X$ contains an S-space.
To that end, we use the concepts of right separated space and left separated space. Recall that an initial segment of a well-ordered set $(X,<)$ is a set of the form $\{y \in X: y where $x \in X$. A space $X$ is a right separated space if $X$ can be well-ordered in such a way that every initial segment is open. A right separated space is in type $\kappa$ if the well-ordering is of type $\kappa$. A space $X$ is a left separated space if $X$ can be well-ordered in such a way that every initial segment is closed. A left separated space is in type $\kappa$ if the well-ordering is of type $\kappa$. The following results are used in proving Theorem 1.
Theorem A
Let $X$ is regular and Hausdorff space. Then the following is true.
• The space $X$ is hereditarily separable space if and only if $X$ has no uncountable left separated subspace.
• The space $X$ is hereditarily Lindelof space if and only if $X$ has no uncountable right separated subspace.
Proof of Theorem A
$\Longrightarrow$ of the first bullet point.
Suppose $Y \subset X$ is an uncountable left separated subspace. Suppose that the well-ordering of $Y$ is of type $\kappa$ where $\kappa>\omega$. Further suppose that $Y=\{ x_\alpha: \alpha<\kappa \}$ such that for each $\alpha<\kappa$, $C_\alpha=\{ x_\beta: \beta<\alpha \}$ is a closed subset of $Y$. Since $\kappa$ is uncountable, the well-ordering has an initial segment of type $\omega_1$. So we might as well assume $\kappa=\omega_1$. Note that for any countable $A \subset Y$, $A \subset C_\alpha$ for some $\alpha<\omega_1$. It follows that $Y$ is not separable. This means that $X$ is not hereditarily separable.
$\Longleftarrow$ of the first bullet point.
Suppose that $X$ is not hereditarily separable. Let $Y \subset X$ be a subspace that is not separable. We now inductively derive an uncountable left separated subspace of $Y$. Choose $y_0 \in Y$. For each $\alpha<\omega_1$, let $A_\alpha=\{ y_\beta \in Y: \beta <\alpha \}$. The set $A_\alpha$ is the set of all the points of $Y$ chosen before the step at $\alpha<\omega_1$. Since $A_\alpha$ is countable, its closure in $Y$ is not the entire space $Y$. Choose $y_\alpha \in Y-\overline{A_\alpha}=O_\alpha$.
Let $Y_L=\{ y_\alpha: \alpha<\omega_1 \}$. We claim that $Y_L$ is a left separated space. To this end, we need to show that each initial segment $A_\alpha$ is a closed subset of $Y_L$. Note that for each $\gamma \ge \alpha$, $O_\gamma=Y-\overline{A_\gamma}$ is an open subset of $Y$ with $y_\gamma \in O_\gamma$ such that $O_\gamma \cap \overline{A_\gamma}=\varnothing$ and thus $O_\gamma \cap \overline{A_\alpha}=\varnothing$ (closure in $Y$). Then $U_\gamma=O_\gamma \cap Y_L$ is an open subset of $Y_L$ containing $y_\gamma$ such that $U_\gamma \cap A_\alpha=\varnothing$. It follows that $Y-A_\alpha$ is open in $Y_L$ and that $A_\alpha$ is a closed subset of $Y_L$.
$\Longrightarrow$ of the second bullet point.
Suppose $Y \subset X$ is an uncountable right separated subspace. Suppose that the well-ordering of $Y$ is of type $\kappa$ where $\kappa>\omega$. Further suppose that $Y=\{ x_\alpha: \alpha<\kappa \}$ such that for each $\alpha<\kappa$, $U_\alpha=\{ x_\beta: \beta<\alpha \}$ is an open subset of $Y$.
Since $\kappa$ is uncountable, the well-ordering has an initial segment of type $\omega_1$. So we might as well assume $\kappa=\omega_1$. Note that $\{ U_\alpha: \alpha<\omega_1 \}$ is an open cover of $Y$ that has no countable subcover. It follows that $Y$ is not Lindelof. This means that $X$ is not hereditarily Lindelof.
$\Longleftarrow$ of the second bullet point.
Suppose that $X$ is not hereditarily Lindelof. Let $Y \subset X$ be a subspace that is not Lindelof. Let $\mathcal{U}$ be an open cover of $Y$ that has no countable subcover. We now inductively derive a right separated subspace of $Y$ of type $\omega_1$.
Choose $U_0 \in \mathcal{U}$ and choose $y_0 \in U_0$. Choose $y_1 \in Y-U_0$ and choose $U_1 \in \mathcal{U}$ such that $y_1 \in U_1$. Let $\alpha<\omega_1$. Suppose that points $y_\beta$ and open sets $U_\beta$, $\beta<\alpha$, have been chosen such that $y_\beta \in Y-\bigcup_{\delta<\beta} U_\delta$ and $y_\beta \in U_\beta$. The countably many chosen open sets $U_\beta$, $\beta<\alpha$, cannot cover $Y$. Choose $y_\alpha \in Y-\bigcup_{\beta<\alpha} U_\beta$. Choose $U_\alpha \in \mathcal{U}$ such that $y_\alpha \in U_\alpha$.
Let $Y_R=\{ y_\alpha: \alpha<\omega_1 \}$. It follows that $Y_R$ is a right separated space. Note that for each $\alpha<\omega_1$, $\{ y_\beta: \beta<\alpha \} \subset \bigcup_{\beta<\alpha} U_\beta$ and the open set $\bigcup_{\beta<\alpha} U_\beta$ does not contain $y_\gamma$ for any $\gamma \ge \alpha$. This means that the initial segment $\{ y_\beta: \beta<\alpha \}$ is open in $Y_L$. $\square$
Lemma B
Let $X$ be a space that is a right separated space and also a left separated space based on the same well ordering. Then $X$ is a discrete space.
Proof of Lemma B
Let $X=\{ w_\alpha: \alpha<\kappa \}$ such that the well-ordering is given by the ordinals in the subscripts, i.e. $w_\beta if and only if $\beta<\gamma$. Suppose that $X$ with this well-ordering is both a right separated space and a left separated space. We claim that every point is a discrete point, i.e. $\{ x_\alpha \}$ is open for any $\alpha<\kappa$.
To see this, fix $\alpha<\kappa$. The initial segment $A_\alpha=\{ w_\beta: \beta<\alpha \}$ is closed in $X$ since $X$ is a left separated space. On the other hand, the initial segment $\{ w_\beta: \beta < \alpha+1 \}$ is open in $X$ since $X$ is a right separated space. Then $B_{\alpha}=\{ w_\beta: \beta \ge \alpha+1 \}$ is closed in $X$. It follows that $\{ x_\alpha \}$ must be open since $X=A_\alpha \cup B_\alpha \cup \{ w_\alpha \}$. $\square$
Theorem C
Let $X$ is regular and Hausdorff space. Then the following is true.
• Suppose the space $X$ is right separated space of type $\omega_1$. Then if $X$ has no uncountable discrete subspaces, then $X$ is an S-space or $X$ contains an S-space.
• Suppose the space $X$ is left separated space of type $\omega_1$. Then if $X$ has no uncountable discrete subspaces, then $X$ is an L-space or $X$ contains an L-space.
Proof of Theorem C
For the first bullet point, suppose the space $X$ is right separated space of type $\omega_1$. Then by Theorem A, $X$ is not hereditarily Lindelof. If $X$ is hereditarily separable, then $X$ is an S-space (if $X$ is not Lindelof) or $X$ contains an S-space (a non-Lindelof subspace of $X$). Suppose $X$ is not hereditarily separable. By Theorem A, $X$ has an uncountable left separated subspace of type $\omega_1$.
Let $X=\{ x_\alpha: \alpha<\omega_1 \}$ such that the well-ordering represented by the ordinals in the subscripts is a right separated space. Let $<_R$ be the symbol for the right separated well-ordering, i.e. $x_\beta <_R \ x_\delta$ if and only if $\beta<\delta$. As indicated in the preceding paragraph, $X$ has an uncountable left separated subspace. Let $Y=\{ y_\alpha \in X: \alpha<\omega_1 \}$ be this left separated subspace. Let $<_L$ be the symbol for the left separated well-ordering. The well-ordering $<_R$ may be different from the well-ordering $<_L$. However, we can obtain an uncountable subset of $Y$ such that the two well-orderings coincide on this subset.
To start, pick any $y_\gamma$ in $Y$ and relabel it $t_0$. The final segment $\{y_\beta \in Y: t_0 <_L \ y_\beta \}$ must intersect the final segment $\{x_\beta \in X: t_0 <_R \ x_\beta \}$ in uncountably many points. Choose the least such point (according to $<_R$) and call it $t_1$. It is clear how $t_{\delta+1}$ is chosen if $t_\delta$ has been chosen.
Suppose $\alpha<\omega_1$ is a limit ordinal and that $t_\beta$ has been chosen for all $\beta<\alpha$. Then the set $\{y_\tau: \forall \ \beta<\alpha, t_\beta <_L \ y_\tau \}$ and the set $\{x_\tau: \forall \ \beta<\alpha, t_\beta <_R \ x_\tau \}$ must intersect in uncountably many points. Choose the least such point and call it $t_\alpha$ (according to $<_R$). As a result, we have obtained $T=\{ t_\alpha: \alpha<\omega_1 \}$. It follows that T with the well-ordering represented by the ordinals in the subscript is a subset of $(X,<_R)$ and a subset of $(Y,<_L)$. Thus $T$ is both right separated and left separated.
By Lemma B, $T$ is a discrete subspace of $X$. However, $X$ is assumed to have no uncountable discrete subspace. Thus if $X$ has no uncountable discrete subspace, then $X$ must be hereditarily separable and as a result, must be an S-space or must contain an S-space.
The proof for the second bullet point is analogous to that of the first bullet point. $\square$
We are now ready to prove Theorem 1.
Proof of Theorem 1
Suppose that $X$ is of countable spread and that $X$ is not hereditarily separable. By Theorem A, $X$ has an uncountable left separated subspace $Y$ (assume it is of type $\omega_1$). The property of countable spread is hereditary. So $Y$ is of countable spread. By Theorem C, $Y$ is an L-space or $Y$ contains an L-space. In either way, $X$ contains an L-space.
Suppose that $X$ is of countable spread and that $X$ is not hereditarily Lindelof. By Theorem A, $X$ has an uncountable right separated subspace $Y$ (assume it is of type $\omega_1$). By Theorem C, $Y$ is an S-space or $Y$ contains an S-space. In either way, $X$ contains an S-space.
Reference
1. Eisworth T., Nyikos P., Shelah S., Gently killing S-spaces, Israel Journal of Mathmatics, 136, 189-220, 2003.
2. Roitman J., The spread of regular spaces, General Topology and Its Applications, 8, 85-91, 1978.
3. Roitman, J., Basic S and L, Handbook of Set-Theoretic Topology, (K. Kunen and J. E. Vaughan, eds), Elsevier Science Publishers B. V., Amsterdam, 295-326, 1984.
4. Tatch-Moore J., A solution to the L space problem, Journal of the American Mathematical Society, 19, 717-736, 2006.
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# Every space is star discrete
The statement in the title is a folklore fact, though the term star discrete is usually not used whenever this well known fact is invoked in the literature. We present a proof to this well known fact. We also discuss some related concepts.
All spaces are assumed to be Hausdorff and regular.
First, let’s define the star notation. Let $X$ be a space. Let $\mathcal{U}$ be a collection of subsets of $X$. Let $A \subset X$. Define $\text{St}(A,\mathcal{U})$ to be the set $\bigcup \{U \in \mathcal{U}: U \cap A \ne \varnothing \}$. In other words, the set $\text{St}(A,\mathcal{U})$ is simply the union of all elements of $\mathcal{U}$ that contains points of the set $A$. The set $\text{St}(A,\mathcal{U})$ is also called the star of the set $A$ with respect to the collection $\mathcal{U}$. If $A=\{ x \}$, we use the notation $\text{St}(x,\mathcal{U})$ instead of $\text{St}( \{ x \},\mathcal{U})$. The following is the well known result in question.
Lemma 1
Let $X$ be a space. For any open cover $\mathcal{U}$ of $X$, there exists a discrete subspace $A$ of $X$ such that $X=\text{St}(A,\mathcal{U})$. Furthermore, the set $A$ can be chosen in such a way that it is also a closed subset of the space $X$.
Any space that satisfies the condition in Lemma 1 is said to be a star discrete space. The proof shown below will work for any topological space. Hence every space is star discrete. We come across three references in which the lemma is stated or is used – Lemma IV.2.20 in page 135 of [3], page 137 of [2] and [1]. The first two references do not use the term star discrete. Star discrete is mentioned in [1] since that paper focuses on star properties. This property that is present in every topological space is at heart a covering property. Here’s a rewording of the lemma that makes it look like a covering property.
Lemma 1a
Let $X$ be a space. For any open cover $\mathcal{U}$ of $X$, there exists a discrete subspace $A$ of $X$ such that $\{ \text{St}(x,\mathcal{U}): x \in A \}$ is a cover of $X$. Furthermore, the set $A$ can be chosen in such a way that it is also a closed subset of the space $X$.
Lemma 1a is clearly identical to Lemma 1. However, Lemma 1a makes it extra clear that this is a covering property. For every open cover of a space, instead of finding a sub cover or an open refinement, we find a discrete subspace so that the stars of the points of the discrete subspace with respect to the given open cover also cover the space.
Lemma 1a naturally leads to other star covering properties. For example, a space $X$ is said to be a star countable space if for any open cover $\mathcal{U}$ of $X$, there exists a countable subspace $A$ of $X$ such that $\{ \text{St}(x,\mathcal{U}): x \in A \}$ is a cover of $X$. A space $X$ is said to be a star Lindelof space if for any open cover $\mathcal{U}$ of $X$, there exists a Lindelof subspace $A$ of $X$ such that $\{ \text{St}(x,\mathcal{U}): x \in A \}$ is a cover of $X$. In general, for any topological property $\mathcal{P}$, a space $X$ is a star $\mathcal{P}$ space if for any open cover $\mathcal{U}$ of $X$, there exists a subspace $A$ of $X$ with property $\mathcal{P}$ such that $\{ \text{St}(x,\mathcal{U}): x \in A \}$ is a cover of $X$.
It follows that every Lindelof space is a star countable space. It is also clear that every star countable space is a star Lindelof space.
Lemma 1 or Lemma 1a, at first glance, may seem like a surprising result. However, one can argue that it is not a strong result at all since the property is possessed by every space. Indeed, the lemma has nothing to say about the size of the discrete set. It only says that there exists a star cover based on a discrete set for a given open cover. To derive more information about the given space, we may need to work with more information on the space in question.
Consider spaces such that every discrete subspace is countable (such a space is said to have countable spread or a space of countable spread). Also consider spaces such that every closed and discrete subspace is countable (such a space is said to have countable extent or a space of countable extent). Any space that has countable spread is also a space that has countable extent for the simple reason that if every discrete subspace is countable, then every closed and discrete subspace is countable.
Then it follows from Lemma 1 that any space $X$ that has countable extent is star countable. Any star countable space is obviously a star Lindelof space. The following diagram displays these relationships.
According to the diagram, the star countable and star Lindelof are both downstream from the countable spread property and the Lindelof property. The star properties being downstream from the Lindelof property is not surprising. What is interesting is that if a space has countable spread, then it is star countable and hence star Lindelof.
Do “countable spread” and “Lindelof” relate to each other? Lindelof spaces do not have to have countable spread. The simplest example is the one-point compactification of an uncountable discrete space. More specifically, let $X$ be an uncountable discrete space. Let $p$ be a point not in $X$. Then $Y=X \cup \{ p \}$ is a compact space (hence Lindelof) where $X$ is discrete and an open neighborhood of $p$ is of the form $\{ p \} \cup U$ where $X-U$ is a finite subset of $X$. The space $Y$ is not of countable spread since $X$ is an uncountable discrete subspace.
Does “countable spread” imply “Lindelof”? Is there a non-Lindelof space that has countable spread? It turns out that the answers are independent of ZFC. The next post has more details.
We now give a proof to Lemma 1. Suppose that $X$ is an infinite space (if it is finite, the lemma is true since the space is Hausdorff). Let $\kappa=\lvert X \lvert$. Let $\kappa^+$ be the next cardinal greater than $\kappa$. Let $\mathcal{U}$ be an open cover of the space $X$. Choose $x_0 \in X$. We choose a sequence of points $x_0,x_1,\cdots,x_\alpha,\cdots$ inductively. If $\text{St}(\{x_\beta: \beta<\alpha \},\mathcal{U}) \ne X$, we can choose a point $x_\alpha \in X$ such that $x_\alpha \notin \text{St}(\{x_\beta: \beta<\alpha \},\mathcal{U})$.
We claim that the induction process must stop at some $\alpha<\kappa^+$. In other words, at some $\alpha<\kappa^+$, the star of the previous points must be the entire space and we run out of points to choose. Otherwise, we would have obtained a subset of $X$ with cardinality $\kappa^+$, a contradiction. Choose the least $\alpha<\kappa^+$ such that $\text{St}(\{x_\beta: \beta<\alpha \},\mathcal{U}) = X$. Let $A=\{x_\beta: \beta<\alpha \}$.
Then it can be verified that the set $A$ is a discrete subspace of $X$ and that $A$ is a closed subset of $X$. Note that $x_\beta \in \text{St}(x_\beta, \mathcal{U})$ while $x_\gamma \notin \text{St}(x_\beta, \mathcal{U})$ for all $\gamma \ne \beta$. This follows from the way the points are chosen in the induction process. On the other hand, for any $x \in X-A$, $x \in \text{St}(x_\beta, \mathcal{U})$ for some $\beta<\alpha$. As discussed, the open set $\text{St}(x_\beta, \mathcal{U})$ contains only one point of $A$, namely $x_\beta$.
Reference
1. Alas O., Jumqueira L., van Mill J., Tkachuk V., Wilson R.On the extent of star countable spaces, Cent. Eur. J. Math., 9 (3), 603-615, 2011.
2. Alster, K., Pol, R.,On function spaces of compact subspaces of $\Sigma$-products of the real line, Fund. Math., 107, 35-46, 1980.
3. Arkhangelskii, A. V.,Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
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# Equivalent conditions for hereditarily Lindelof spaces
A topological space $X$ is Lindelof if every open cover $X$ has a countable subcollection that also is a cover of $X$. A topological space $X$ is hereditarily Lindelof if every subspace of $X$, with respect to the subspace topology, is a Lindelof space. In this post, we prove a theorem that gives two equivalent conditions for the hereditarily Lindelof property. We consider the following theorem.
Theorem 1
Let $X$ be a topological space. The following conditions are equivalent.
1. The space $X$ is a hereditarily Lindelof space.
2. Every open subspace of $X$ is Lindelof.
3. For every uncountable subspace $Y$ of $X$, there exists a point $y \in Y$ such that every open subset of $X$ containing $y$ contains uncountably many points of $Y$.
This is an excellent exercise for the hereditarily Lindelof property and for transfinite induction (for one of the directions). The equivalence $1 \longleftrightarrow 3$ is the exercise 3.12.7(d) on page 224 of [1]. The equivalence of the 3 conditions of Theorem 1 is mentioned on page 182 (chapter d-8) of [2].
Proof of Theorem 1
The direction $1 \longrightarrow 2$ is immediate. The direction $2 \longrightarrow 3$ is straightforward.
$3 \longrightarrow 1$
We show $\text{not } 1 \longrightarrow \text{not } 3$. Suppose $T$ is a non-Lindelof subspace of $X$. Let $\mathcal{U}$ be an open cover of $T$ such that no countable subcollection of $\mathcal{U}$ can cover $T$. By a transfinite inductive process, choose a set of points $\left\{t_\alpha \in T: \alpha < \omega_1 \right\}$ and a collection of open sets $\left\{U_\alpha \in \mathcal{U}: \alpha < \omega_1 \right\}$ such that for each $\alpha < \omega_1$, $t_\alpha \in U_\alpha$ and $t_\alpha \notin \cup \left\{U_\beta: \beta<\alpha \right\}$. The inductive process is possible since no countable subcollection of $\mathcal{U}$ can cover $T$. Now let $Y=\left\{t_\alpha: \alpha<\omega_1 \right\}$. Note that each $U_\alpha$ can at most contain countably many points of $Y$, namely the points in $\left\{t_\beta: \beta \le \alpha \right\}$.
For each $\alpha$, let $V_\alpha$ be an open subset of $X$ such that $U_\alpha=V_\alpha \cap Y$. We can now conclude: for every point $t_\alpha$ of $Y$, there exists an open set $V_\alpha$ containing $t_\alpha$ such that $V_\alpha$ contains only countably many points of $Y$. This is the negation of condition 3. $\blacksquare$
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Remarks
Condition 3 indicates that every uncountable set has a certain special type of limit points. Let $p \in X$. We say $p$ is a limit point of the set $Y \subset X$ if every open set containing $p$ contains a point of $Y$ different from $p$. Being a limit point of $Y$, we only know that each open set containing $p$ contain infinitely many points of $Y$ (assuming a $T_1$ space). Thus the limit points indicated in condition 3 are a special type of limit points. According to the terminology of [1], if $p$ is a limit point of $Y$ satisfying condition 3, then $p$ is said to be a condensation point of $Y$. According to Theorem 1, existence of condensation point in every uncountable set is a strong topological property (being equivalent to the hereditarily property). It is easy to see that of condition 3 holds, all but countably many points of any uncountable set $Y$ is a condensation point of $Y$.
In some situations, we may not need the full strength of condition 3. In such situations, the following corollary may be sufficient.
Corollary 2
If the space $X$ is hereditarily Lindelof, then every uncountable subspace $Y$ of $X$ contains one of its limit points.
As noted earlier, if every uncountable set contains one of its limits, then all but countably many points of any uncountable set are limit points. To contrast the hereditarily Lindelof property with the Lindelof property, consider the following theorem.
Theorem 3
If the space $X$ is Lindelof, then every uncountable subspace $Y$ of $X$ has a limit point.
The condition “every uncountable subspace $Y$ of $X$ has a limit point” has another name. When a space satisfies this condition, it is said to have countable extent. The ideas in Corollary 2 and Theorem 3 are also discussed in this previous post.
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Reference
1. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
2. Hart, K. P., Nagata J. I., Vaughan, J. E., editors, Encyclopedia of General Topology, First Edition, Elsevier Science Publishers B. V, Amsterdam, 2003.
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$\copyright \ 2014 \text{ by Dan Ma}$
# An example of a normal but not Lindelof Cp(X)
In this post, we discuss an example of a function space $C_p(X)$ that is normal and not Lindelof (as indicated in the title). Interestingly, much more can be said about this function space. In this post, we show that there exists a space $X$ such that
• $C_p(X)$ is collectionwise normal and not paracompact,
• $C_p(X)$ is not Lindelof but contains a dense Lindelof subspace,
• $C_p(X)$ is not first countable but is a Frechet space,
• As a corollary of the previous point, $C_p(X)$ cannot contain a copy of the compact space $\omega_1+1$,
• $C_p(X)$ is homeomorphic to $C_p(X)^\omega$,
• $C_p(X)$ is not hereditarily normal,
• $C_p(X)$ is not metacompact.
A short and quick description of the space $X$ is that $X$ is the one-point Lindelofication of an uncountable discrete space. As shown below, the function space $C_p(X)$ is intimately related to a $\Sigma$-product of copies of real lines. The results listed above are merely an introduction to this wonderful example and are derived by examining the $\Sigma$-products of copies of real lines. Deep results about $\Sigma$-product of real lines abound in the literature. The references listed at the end are a small sample. Example 3.2 in [2] is another interesting illustration of this example.
We now define the domain space $X=L_\tau$. In the discussion that follows, the Greek letter $\tau$ is always an uncountable cardinal number. Let $D_\tau$ be a set with cardinality $\tau$. Let $p$ be a point not in $D_\tau$. Let $L_\tau=D_\tau \cup \left\{p \right\}$. Consider the following topology on $L_\tau$:
• Each point in $D_\tau$ an isolated point, and
• open neighborhoods at the point $p$ are of the form $L_\tau-K$ where $K \subset D_\tau$ is countable.
It is clear that $L_\tau$ is a Lindelof space. The Lindelof space $L_\tau$ is sometimes called the one-point Lindelofication of the discrete space $D_\tau$ since it is a Lindelof space that is obtained by adding one point to a discrete space.
Consider the function space $C_p(L_\tau)$. See this post for general information on the pointwise convergence topology of $C_p(Y)$ for any completely regular space $Y$.
All the facts about $C_p(X)=C_p(L_\tau)$ mentioned at the beginning follow from the fact that $C_p(L_\tau)$ is homeomorphic to the $\Sigma$-product of $\tau$ many copies of the real lines. Specifically, $C_p(L_\tau)$ is homeomorphic to the following subspace of the product space $\mathbb{R}^\tau$.
$\Sigma_{\alpha<\tau}\mathbb{R}=\left\{ x \in \mathbb{R}^\tau: x_\alpha \ne 0 \text{ for at most countably many } \alpha<\tau \right\}$
Thus understanding the function space $C_p(L_\tau)$ is a matter of understanding a $\Sigma$-product of copies of the real lines. First, we establish the homeomorphism and then discuss the properties of $C_p(L_\tau)$ indicated above.
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The Homeomorphism
For each $f \in C_p(L_\tau)$, it is easily seen that there is a countable set $C \subset D_\tau$ such that $f(p)=f(y)$ for all $y \in D_\tau-C$. Let $W_0=\left\{f \in C_p(L_\tau): f(p)=0 \right\}$. Then each $f \in W_0$ has non-zero values only on a countable subset of $D_\tau$. Naturally, $W_0$ and $\Sigma_{\alpha<\tau}\mathbb{R}$ are homeomorphic.
We claim that $C_p(L_\tau)$ is homeomorphic to $W_0 \times \mathbb{R}$. For each $f \in C_p(L_\tau)$, define $h(f)=(f-f(p),f(p))$. Here, $f-f(p)$ is the function $g \in C_p(L_\tau)$ such that $g(x)=f(x)-f(p)$ for all $x \in L_\tau$. Clearly $h(f)$ is well-defined and $h(f) \in W_0 \times \mathbb{R}$. It can be readily verified that $h$ is a one-to-one map from $C_p(L_\tau)$ onto $W_0 \times \mathbb{R}$. It is not difficult to verify that both $h$ and $h^{-1}$ are continuous.
We use the notation $X_1 \cong X_2$ to mean that the spaces $X_1$ and $X_2$ are homeomorphic. Then we have:
$C_p(L_\tau) \ \cong \ W_0 \times \mathbb{R} \ \cong \ (\Sigma_{\alpha<\tau}\mathbb{R}) \times \mathbb{R} \ \cong \ \Sigma_{\alpha<\tau}\mathbb{R}$
Thus $C_p(L_\tau) \ \cong \ \Sigma_{\alpha<\tau}\mathbb{R}$. This completes the proof that $C_p(L_\tau)$ is topologically the $\Sigma$-product of $\tau$ many copies of the real lines.
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Looking at the $\Sigma$-Product
Understanding the function space $C_p(L_\tau)$ is now reduced to the problem of understanding a $\Sigma$-product of copies of the real lines. Most of the facts about $\Sigma$-products that we need have already been proved in previous blog posts.
In this previous post, it is established that the $\Sigma$-product of separable metric spaces is collectionwise normal. Thus $C_p(L_\tau)$ is collectionwise normal. The $\Sigma$-product of spaces, each of which has at least two points, always contains a closed copy of $\omega_1$ with the ordered topology (see the lemma in this previous post). Thus $C_p(L_\tau)$ contains a closed copy of $\omega_1$ and hence can never be paracompact (and thus not Lindelof).
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Consider the following subspace of the $\Sigma$-product $\Sigma_{\alpha<\tau}\mathbb{R}$:
$\sigma_\tau=\left\{ x \in \Sigma_{\alpha<\tau}\mathbb{R}: x_\alpha \ne 0 \text{ for at most finitely many } \alpha<\tau \right\}$
In this previous post, it is shown that $\sigma_\tau$ is a Lindelof space. Though $C_p(L_\tau) \cong \Sigma_{\alpha<\tau}\mathbb{R}$ is not Lindelof, it has a dense Lindelof subspace, namely $\sigma_\tau$.
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A space $Y$ is first countable if there exists a countable local base at each point $y \in Y$. A space $Y$ is a Frechet space (or is Frechet-Urysohn) if for each $y \in Y$, if $y \in \overline{A}$ where $A \subset Y$, then there exists a sequence $\left\{y_n: n=1,2,3,\cdots \right\}$ of points of $A$ such that the sequence converges to $y$. Clearly, any first countable space is a Frechet space. The converse is not true (see Example 1 in this previous post).
For any uncountable cardinal number $\tau$, the product $\mathbb{R}^\tau$ is not first countable. In fact, any dense subspace of $\mathbb{R}^\tau$ is not first countable. In particular, the $\Sigma$-product $\Sigma_{\alpha<\tau}\mathbb{R}$ is not first countable. In this previous post, it is shown that the $\Sigma$-product of first countable spaces is a Frechet space. Thus $C_p(L_\tau) \cong \Sigma_{\alpha<\tau}\mathbb{R}$ is a Frechet space.
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As a corollary of the previous point, $C_p(L_\tau) \cong \Sigma_{\alpha<\tau}\mathbb{R}$ cannot contain a homeomorphic copy of any space that is not Frechet. In particular, it cannot contain a copy of any compact space that is not Frechet. For example, the compact space $\omega_1+1$ is not embeddable in $C_p(L_\tau)$. The interest in compact subspaces of $C_p(L_\tau) \cong \Sigma_{\alpha<\tau}\mathbb{R}$ is that any compact space that is topologically embeddable in a $\Sigma$-product of real lines is said to be Corson compact. Thus any Corson compact space is a Frechet space.
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It can be readily verified that
$\Sigma_{\alpha<\tau}\mathbb{R} \ \cong \ \Sigma_{\alpha<\tau}\mathbb{R} \ \times \ \Sigma_{\alpha<\tau}\mathbb{R} \ \times \ \Sigma_{\alpha<\tau}\mathbb{R} \ \times \ \cdots \ \text{(countably many times)}$
Thus $C_p(L_\tau) \cong C_p(L_\tau)^\omega$. In particular, $C_p(L_\tau) \cong C_p(L_\tau) \times C_p(L_\tau)$ due to the following observation:
$C_p(L_\tau) \times C_p(L_\tau) \cong C_p(L_\tau)^\omega \times C_p(L_\tau)^\omega \cong C_p(L_\tau)^\omega \cong C_p(L_\tau)$
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As a result of the peculiar fact that $C_p(L_\tau) \cong C_p(L_\tau) \times C_p(L_\tau)$, it can be concluded that $C_p(L_\tau)$, though normal, is not hereditarily normal. This follows from an application of Katetov’s theorem. The theorem states that if $Y_1 \times Y_2$ is hereditarily normal, then either $Y_1$ is perfectly normal or every countably infinite subset of $Y_2$ is closed and discrete (see this previous post). The function space $C_p(L _\tau)$ is not perfectly normal since it contains a closed copy of $\omega_1$. On the other hand, there are plenty of countably infinite subsets of $C_p(L _\tau)$ that are not closed and discrete. As a Frechet space, $C_p(L _\tau)$ has many convergent sequences. Each such sequence without the limit is a countably infinite set that is not closed and discrete. As an example, let $\left\{x_1,x_2,x_3,\cdots \right\}$ be an infinite subset of $D_\tau$ and consider the following:
$C=\left\{f_n: n=1,2,3,\cdots \right\}$
where $f_n$ is such that $f_n(x_n)=n$ and $f_n(x)=0$ for each $x \in L_\tau$ with $x \ne x_n$. Note that $C$ is not closed and not discrete since the points in $C$ converge to $g \in \overline{C}$ where $g$ is the zero-function. Thus $C_p(L_\tau) \cong C_p(L_\tau) \times C_p(L_\tau)$ is not hereditarily normal.
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It is well known that collectionwise normal metacompact space is paracompact (see Theorem 5.3.3 in [4] where metacompact is referred to as weakly paracompact). Since $C_p(L_\tau)$ is collectionwise normal and not paracompact, $C_p(L_\tau)$ can never be metacompact.
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Reference
1. Arkhangelskii, A. V., Topological Function Spaces, Mathematics and Its Applications Series, Kluwer Academic Publishers, Dordrecht, 1992.
2. Bella, A., Masami, S., Tight points of Pixley-Roy hyperspaces, Topology Appl., 160, 2061-2068, 2013.
3. Corson, H. H., Normality in subsets of product spaces, Amer. J. Math., 81, 785-796, 1959.
4. Engelking, R., General Topology, Revised and Completed edition, Heldermann Verlag, Berlin, 1989.
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$\copyright \ 2014 \text{ by Dan Ma}$
# (Lower case) sigma-products of separable metric spaces are Lindelof
Consider the product space $X=\prod_{\alpha \in A} X_\alpha$. Fix a point $b \in \prod_{\alpha \in A} X_\alpha$, called the base point. The $\Sigma$-product of the spaces $\left\{X_\alpha: \alpha \in A \right\}$ is the following subspace of the product space $X$:
$\Sigma_{\alpha \in A} X_\alpha=\left\{ x \in X: x_\alpha \ne b_\alpha \text{ for at most countably many } \alpha \in A \right\}$
In other words, the space $\Sigma_{\alpha \in A} X_\alpha$ is the subspace of the product space $X=\prod_{\alpha \in A} X_\alpha$ consisting of all points that deviate from the base point on at most countably many coordinates $\alpha \in A$. We also consider the following subspace of $\Sigma_{\alpha \in A} X_\alpha$.
$\sigma=\left\{ x \in \Sigma_{\alpha \in A} X_\alpha: x_\alpha \ne b_\alpha \text{ for at most finitely many } \alpha \in A \right\}$
For convenience , we call $\Sigma_{\alpha \in A} X_\alpha$ the (upper case) Sigma-product (or $\Sigma$-product) of the spaces $X_\alpha$ and we call the space $\sigma$ the (lower case) sigma-product (or $\sigma$-product). Clearly, the space $\sigma$ is a dense subspace of $\Sigma_{\alpha \in A} X_\alpha$. In a previous post, we show that the upper case Sigma-product of separable metric spaces is collectionwise normal. In this post, we show that the (lower case) sigma-product of separable metric spaces is Lindelof. Thus when each factor $X_\alpha$ is a separable metric space with at least two points, the $\Sigma$-product, though not Lindelof, has a dense Lindelof subspace. The (upper case) $\Sigma$-product of separable metric spaces is a handy example of a non-Lindelof space that contains a dense Lindelof subspace.
Naturally, the lower case sigma-product can be further broken down into countably many subspaces. For each integer $n=0,1,2,3,\cdots$, we define $\sigma_n$ as follows:
$\sigma_n=\left\{ x \in \sigma: x_\alpha \ne b_\alpha \text{ for at most } n \text{ many } \alpha \in A \right\}$
Clearly, $\sigma=\bigcup_{n=0}^\infty \sigma_n$. We prove the following theorem. The fact that $\sigma$ is Lindelof will follow as a corollary. Understanding the following proof for Theorem 1 is a matter of keeping straight the notations involving standard basic open sets in the product space $X=\prod_{\alpha \in A} X_\alpha$. We say $V$ is a standard basic open subset of the product space $X$ if $V$ is of the form $V=\prod_{\alpha \in A} V_\alpha$ such that each $V_\alpha$ is an open subset of the factor space $X_\alpha$ and $V_\alpha=X_\alpha$ for all but finitely many $\alpha \in A$. The finite set $F$ of all $\alpha \in A$ such that $V_\alpha \ne X_\alpha$ is called the support of the open set $V$.
Theorem 1
Let $\sigma$ be the $\sigma$-product of the separable metrizable spaces $\left\{X_\alpha: \alpha \in A \right\}$. For each $n$, let $\sigma_n$ be defined as above. The product space $\sigma_n \times Y$ is Lindelof for each non-negative integer $n$ and for all separable metric space $Y$.
Proof of Theorem 1
We prove by induction on $n$. Note that $\sigma_0=\left\{b \right\}$, the base point. Clearly $\sigma_0 \times Y$ is Lindelof for all separable metric space $Y$. Suppose the theorem hold for the integer $n$. We show that $\sigma_{n+1} \times Y$ for all separable metric space $Y$. To this end, let $\mathcal{U}$ be an open cover of $\sigma_{n+1} \times Y$ where $Y$ is a separable metric space. Without loss of generality, we assume that each element of $\mathcal{U}$ is of the form $V \times W$ where $V=\prod_{\alpha \in A} V_\alpha$ is a standard basic open subset of the product space $X=\prod_{\alpha \in A} X_\alpha$ and $W$ is an open subset of $Y$.
Let $\mathcal{U}_0=\left\{U_1,U_2,U_3,\cdots \right\}$ be a countable subcollection of $\mathcal{U}$ such that $\mathcal{U}_0$ covers $\left\{b \right\} \times Y$. For each $j$, let $U_j=V_j \times W_j$ where $V_j=\prod_{\alpha \in A} V_{j,\alpha}$ is a standard basic open subset of the product space $X$ with $b \in V_j$ and $W_j$ is an open subset of $Y$. For each $j$, let $F_j$ be the support of $V_j$. Note that $\alpha \in F_j$ if and only if $V_{j,\alpha} \ne X_\alpha$. Also for each $\alpha \in F_j$, $b_\alpha \in V_{j,\alpha}$. Furthermore, for each $\alpha \in F_j$, let $V^c_{j,\alpha}=X_\alpha- V_{j,\alpha}$. With all these notations in mind, we define the following open set for each $\beta \in F_j$:
$H_{j,\beta}= \biggl( V^c_{j,\beta} \times \prod_{\alpha \in A, \alpha \ne \beta} X_\alpha \biggr) \times W_j=\biggl( V^c_{j,\beta} \times T_\beta \biggr) \times W_j$
Observe that for each point $y \in \sigma_{n+1}$ such that $y \in V^c_{j,\beta} \times T_\beta$, the point $y$ already deviates from the base point $b$ on one coordinate, namely $\beta$. Thus on the coordinates other than $\beta$, the point $y$ can only deviates from $b$ on at most $n$ many coordinates. Thus $\sigma_{n+1} \cap (V^c_{j,\beta} \times T_\beta)$ is homeomorphic to $V^c_{j,\beta} \times \sigma_n$. Note that $V^c_{j,\beta} \times W_j$ is a separable metric space. By inductive hypothesis, $V^c_{j,\beta} \times \sigma_n \times W_j$ is Lindelof. Thus there are countably many open sets in the open cover $\mathcal{U}$ that covers points of $H_{j,\beta} \cap (\sigma_{n+1} \times W_j)$.
Note that
$\sigma_{n+1} \times Y=\biggl( \bigcup_{j=1}^\infty U_j \cap \sigma_{n+1} \biggr) \cup \biggl( \bigcup \left\{H_{j,\beta} \cap (\sigma_{n+1} \times W_j): j=1,2,3,\cdots, \beta \in F_j \right\} \biggr)$
To see that the left-side is a subset of the right-side, let $t=(x,y) \in \sigma_{n+1} \times Y$. If $t \in U_j$ for some $j$, we are done. Suppose $t \notin U_j$ for all $j$. Observe that $y \in W_j$ for some $j$. Since $t=(x,y) \notin U_j$, $x_\beta \notin V_{j,\beta}$ for some $\beta \in F_j$. Then $t=(x,y) \in H_{j,\beta}$. It is now clear that $t=(x,y) \in H_{j,\beta} \cap (\sigma_{n+1} \times W_j)$. Thus the above set equality is established. Thus one part of $\sigma_{n+1} \times Y$ is covered by countably many open sets in $\mathcal{U}$ while the other part is the union of countably many Lindelof subspaces. It follows that a countable subcollection of $\mathcal{U}$ covers $\sigma_{n+1} \times Y$. $\blacksquare$
Corollary 2
It follows from Theorem 1 that
• If each factor space $X_\alpha$ is a separable metric space, then each $\sigma_n$ is a Lindelof space and that $\sigma=\bigcup_{n=0}^\infty \sigma_n$ is a Lindelof space.
• If each factor space $X_\alpha$ is a compact separable metric space, then each $\sigma_n$ is a compact space and that $\sigma=\bigcup_{n=0}^\infty \sigma_n$ is a $\sigma$-compact space.
Proof of Corollary 2
The first bullet point is a clear corollary of Theorem 1. A previous post shows that $\Sigma$-product of compact spaces is countably compact. Thus $\Sigma_{\alpha \in A} X_\alpha$ is a countably compact space if each $X_\alpha$ is compact. Note that each $\sigma_n$ is a closed subset of $\Sigma_{\alpha \in A} X_\alpha$ and is thus countably compact. Being a Lindelof space, each $\sigma_n$ is compact. It follows that $\sigma=\bigcup_{n=0}^\infty \sigma_n$ is a $\sigma$-compact space. $\blacksquare$
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A non-Lindelof space with a dense Lindelof subspace
Now we put everything together to obtain the example described at the beginning. For each $\alpha \in A$, let $X_\alpha$ be a separable metric space with at least two points. Then the $\Sigma$-product $\Sigma_{\alpha \in A} X_\alpha$ is collectionwise normal (see this previous post). According to the lemma in this previous post, the $\Sigma$-product $\Sigma_{\alpha \in A} X_\alpha$ contains a closed copy of $\omega_1$. Thus the $\Sigma$-product $\Sigma_{\alpha \in A} X_\alpha$ is not Lindelof. It is clear that the $\sigma$-product is a dense subspace of $\Sigma_{\alpha \in A} X_\alpha$. By Corollary 2, the $\sigma$-product is a Lindelof subspace of $\Sigma_{\alpha \in A} X_\alpha$.
Using specific factor spaces, if each $X_\alpha=\mathbb{R}$ with the usual topology, then $\Sigma_{\alpha<\omega_1} X_\alpha$ is a non-Lindelof space with a dense Lindelof subspace. On the other hand, if each $X_\alpha=[0,1]$ with the usual topology, then $\Sigma_{\alpha<\omega_1} X_\alpha$ is a non-Lindelof space with a dense $\sigma$-compact subspace. Another example of a non-Lindelof space with a dense Lindelof subspace is given In this previous post (see Example 1).
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http://joshlevinedesigns.com/medal-of-tds/euler%27s-theorem-on-homogeneous-functions-of-two-variables-84c300 | Reverse of Euler's Homogeneous Function Theorem . Consider a function $$f(x_1, \ldots, x_N)$$ of $$N$$ variables that satisfies Practice online or make a printable study sheet. On the other hand, Euler's theorem on homogeneous functions is used to solve many problems in engineering, sci-ence, and finance. 2. A polynomial in . For example, a homogeneous real-valued function of two variables x and y is a real-valued function that satisfies the condition f = α k f {\displaystyle f=\alpha ^{k}f} for some constant k and all real numbers α. For an increasing function of two variables, Theorem 04 implies that level sets are concave to the origin. Euler's Homogeneous Function Theorem Let be a homogeneous function of order so that (1) Then define and. 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Leibnitz’s theorem Partial derivatives Euler’s theorem for homogeneous functions Total derivatives Change of variables Curve tracing *Cartesian *Polar coordinates. i'm careful of any party that contains 3, diverse intense elements that contain a saddle … Euler’s theorem is a general statement about a certain class of functions known as homogeneous functions of degree $$n$$. Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. Media. In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor. An equivalent way to state the theorem is to say that homogeneous functions are eigenfunctions of the Euler operator, with the degree of homogeneity as the eigenvalue . x 1 ∂ f ∂ x 1 + … + x k ∂ f ∂ x k = n f, (1) then f is a homogeneous function of degree n. Proof. Problem 6 on Euler's Theorem on Homogeneous Functions Video Lecture From Chapter Homogeneous Functions in Engineering Mathematics 1 for First Year Degree Eng... Euler's theorem in geometry - Wikipedia. For example, the functions x 2 – 2y 2, (x – y – 3z)/(z 2 + xy), and are homogeneous of degree 2, –1, and 4/3, respectively. Finally, x > 0N means x ≥ 0N but x ≠ 0N (i.e., the components of x are nonnegative and at State Euler’S Theorem on Homogeneous Function of Two Variables and If U = X + Y X 2 + Y 2 Then Evaluate X ∂ U ∂ X + Y ∂ U ∂ Y Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof). and . State Euler’S Theorem on Homogeneous Function of Two Variables and If U = X + Y X 2 + Y 2 Then Evaluate X ∂ U ∂ X + Y ∂ U ∂ Y Concept: Euler’s Theorem on Homogeneous functions with two and three independent variables (with proof). 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https://math.stackexchange.com/questions/1852465/probability-in-coin-flip | # Probability in coin flip
The question:
Two persons $A$, $B$ simultaneously toss their individual coins, and win $1$ point if head is face up and $0$ points if tails is face up. The probability that the points of $A$ exceeds the points of $B$ after $3$ tosses is...
My attempt:
Because each person have $4$ available options of how many points they will have in the end $(0,1,2,$ and $3)$, the total amount of ways the score could be after $3$ tosses would be $4 \cdot 4 = 16$. The cases in which person $A$ wins would be $(1,0)$, $(2,0)$, $(2,1)$, $(3,0)$, $(3,1)$, and $(3,2)$. So the answer should be $\frac{6}{16}$ or $\frac{3}{8}$.
The actual answer is $\frac{11}{32}$
My question:
Where did I go wrong and how do I do it correctly?
Thank you!
Not all of the outcomes (0,0), (0,1), (0,2),... (3,2), (3,3) are equally likely. Although there are indeed sixteen possible outcomes, unless all outcomes are equiprobable you may not simply do the fraction of favorable outcomes compared to total outcomes regardless. For example, the outcome $(0,1)$ is three times as likely to occur as the outcome $(0,0)$.
Approach via a combination of the binomial theorem and the multiplication and addition principles of probability.
If $B$ has zero heads, $A$ wins with at least one head. The probability of $B$ having zero heads is $\frac{1}{8}$. The probability of $A$ having at least one head is $\frac{7}{8}$. The probability of $B$ having zero heads and $A$ having at least one head is then $\frac{7}{64}$
If $B$ has one head, $A$ wins with at least two heads. The probability of $B$ having one head is $\frac{3}{8}$. The probability of $A$ having at least two heads is $\frac{4}{8}$. The probability then of both occurring is $\frac{12}{64}$
If $B$ has two heads, $A$ wins only with getting three heads. The probability of $B$ having two heads is $\frac{3}{8}$. The probability of $A$ having three heads is $\frac{1}{8}$. The probability then of both occurring is $\frac{3}{64}$
If $B$ has three heads, $A$ cannot win.
As the sample space can be partitioned into the result of how many heads $B$ received as above, the probability that $A$ wins is the sum of the aforementioned probabilities:
$Pr(\text{A wins}) = \frac{7}{64}+\frac{12}{64}+\frac{3}{64}+0 = \frac{22}{64}=\frac{11}{32}$
(didn't see John's answer while typing this extra paragraph, the method is essentially the same)
An alternate way to view the solution: $1 = Pr(\text{A wins})+Pr(\text{A and B tie}) + Pr(\text{B wins})$
Due to the symmetry of their circumstances, we know that $Pr(\text{A wins})=Pr(\text{B wins})$. This leaves us with the question of finding $Pr(\text{A and B tie})$
These occur with probabilities $\frac{1}{64}, \frac{9}{64},\frac{9}{64},\frac{1}{64}$ respectively.
The probability that $A$ and $B$ tie is then $\frac{20}{64}$
This tells us that $Pr(A~\text{wins}) = \frac{1}{2}(1-\frac{20}{64})=\frac{22}{64}=\frac{11}{32}$
Here's another way to look at the problem.
What if they're tied after three coin tosses?
They could get no heads ($1$ possibility), one head each ($9$ possibilities), two heads each ($9$ possibilities), or three heads each ($1$ possibility).
(Why $9$ possibilities for one head each? $A$ could get a head on the first, second, or third toss, as could $B$. These are independent, so there are $3\cdot 3=9$ possibilities. The same reasoning can be applied to two heads each, as well as the total number of possible outcomes $8\cdot 8=64$.)
This adds to $20$ out of $64$ possibilities that they tie, which means $44/64 = 11/16$ probability that they don't.
As $A$ is just as likely to be ahead as $B$, the answer is half this, or $11/32$.
• Just to add. using the identity $\sum_{k=0}^n \binom{n}{k}^2 = \binom{2n}{n}$, we can say generally that the possibilities for a tie after each player flips $n$ coins is $\binom{2n}{n}$. than the chance to win can be easily calculated in the same line and is: $0.5(1 - \dfrac{\binom{2n}{n}}{2^{2n}})$ – d_e Jul 7 '16 at 21:54 | 2019-12-13T17:20:21 | {
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http://mathhelpforum.com/pre-calculus/187656-determine-co-ordinates.html | # Math Help - Determine the co-ordinates
1. ## Determine the co-ordinates
This is my first post. I graduated from mathematics in university 16 years ago so I am somewhat out of practice. I recently ran into my high school teacher from several years ago and he posed a math problem to me. I need some help with it, even a push in the right direction. Thanks. Here is the problem:
A line segment has its endpoints on the x and y axes.
The line segment passes through point (6,6).
The line segment's length is 21.
Determine the unknown points on the x and y axes.
Thanks
Norm F
2. ## Re: Determine the co-ordinates
Let the x-intercept be denoted by (a, 0)
Let the y-intercept be denoted by (0, b)
Then a^2 + b^2 = 21^2
Now draw the rectangle with vertices (0, 0), (6, 0), (0, 6), and (6, 6)
Draw a line segment from the y-axis to the x-axis through (6, 6)
There should be a triangle above the rectangle, and one to the right. These two will be similar, and using proportions (along with the pythagorean theorem, above) should give the solution.
3. ## Re: Determine the co-ordinates
Originally Posted by TheChaz
Let the x-intercept be denoted by (a, 0)
Let the y-intercept be denoted by (0, b)
Then a^2 + b^2 = 21^2
Now draw the rectangle with vertices (0, 0), (6, 0), (0, 6), and (6, 6)
Draw a line segment from the y-axis to the x-axis through (6, 6)
There should be a triangle above the rectangle, and one to the right. These two will be similar, and using proportions (along with the pythagorean theorem, above) should give the solution.
Thanks for your reply. What you've said, is what I have done. The rectangle (square) is 6 by 6, and the diagonal from (0,0) to (6,6), using Pythagorean theorem, is 6sqrt(2).
But, I don't think you know anything about the sides of these triangles, other than one side is 6.
4. ## Re: Determine the co-ordinates
Allow me to clarify (and I sure wish soroban were here with his sweet ASCII art!...)
Somewhere above (0, 6) on the y-axis, we have (0, b) - an endpoint of the segment.
Similarly, (a, 0) is to the right of (6, 0)
The triangle with vertices (0, 6), (6, 6) and (0, b) - with legs of length 6 and (b - 6)
is similar to the triangle with vertices (6, 6), (a, 0) and (6, 0) - with legs of length 6 and (a - 6)
You should not be calculating the diagonal of the square!
So using proportions,
$\frac{b - 6}{6} = \frac{6}{a - 6}$
Hmm... I'm not doing scratchwork, so it looks like the Pythagorean theorem route isn't the best option. There might be a way to go directly to a relationship between a and b, in a linear equation.
Then we combine this with our proportions.
IDK... off to church!
5. ## Re: Determine the co-ordinates
Originally Posted by EastCan
Thanks for your reply. What you've said, is what I have done. The rectangle (square) is 6 by 6, and the diagonal from (0,0) to (6,6), using Pythagorean theorem, is 6sqrt(2).
You know that $a^2+b^2=21$.
You also know about slope: $\frac{6-0}{6-a}=\frac{6-b}{6-0}$.
6. ## Re: Determine the co-ordinates
Have you forgotten everything?
x, and y axis. Points (0,r) and (s,0)
Equation of this line is $\frac{x}{s}+\frac{y}{r}\;=\;1$
Length is 21. $r^{2} + s^{2} = 21^{2}$
(6,6) is on there. $\frac{6}{s}+\frac{6}{r}\;=\;1$
It's a good, symmetric problem. Two solutions should present themselves.
7. ## Re: Determine the co-ordinates
Have I forgotten everything? Ouch! That hurts.
8. ## Re: Determine the co-ordinates
I understand what Plato has written. TKHunny, can you kindly help me out with your equation of the line. I must have forgotten alot apparently (thanks for driving that point home), because I don't see immediately how you came up with that equation.
I'm used to the y=mx+b form of a line, so I'm probably getting hung up on that.
Looking through some math references, I understand that form of equation that you use.
Thank you for your help thus far, I will work on this a bit and see what I come up with.
9. ## Re: Determine the co-ordinates
Originally Posted by EastCan
TKHunny, can you kindly help me out with your equation of the line.
If $(a,0)~\&~(0,b)$ are the intercepts and $ab\ne 0$ then the equation of that line is $\frac{x}{a}+\frac{y}{b}=1$.
10. ## Re: Determine the co-ordinates
This is a pretty old problem, I remember being given it ages ago. Then it appeared in the form of trying to get into a girls bedroom window. Her father had built a 6ft wall 6ft away from the building and the problem was to find out whether a 21ft ladder was long enough to reach the window, or something like that.
Anyway, from a maths point of view, suppose that the intercepts on the x and y axes are a and b, then
$a^2+b^2=21^2,$ which can be rewritten in the form $(a+b)^2-2ab=21^2.$
The equation of the line (the ladder) will be $x/a+y/b=1,$ and since it passes through the point (6,6), $6/a+6/b=1$ from which $(a+b)/ab=1/6.$
Use this to substitute for $ab$ in the earlier equation and you have a quadratic in $a+b$ which you solve.
So, find b in terms of a and substitute into the original Pythagoras equation.
11. ## Re: Determine the co-ordinates
Hi BobP, thank you for your help. It is very helpful. I was working on this, and the algebra gets very messy when trying to solve the quadratics. Do you think this is to be expected, or should the answer to this problem have a neat little answer?
I will have another go at it, in any case.
12. ## Re: Determine the co-ordinates
I didn't think that it was particularly messy, the first quadratic is $(a+b)^2-12(a+b)-441=0$ from which $a+b\approx 27.84033.$
I got 19.09 and 8.75 (2dp) for the values of a and b (or b and a).
13. ## Re: Determine the co-ordinates
I was assuming I was not permitted to reduce the answers to decimal form, for whatever reason. Otherwise, it is easy to solve at this point. Thank you very much for your help.
14. ## Re: Determine the co-ordinates
The answers in exact form are
$3\;\frac{\sqrt{53}+2-\sqrt{41-4\sqrt{53}}}{2}\quad \text{ and } \quad 3\;\frac{\sqrt{53}+2+\sqrt{41-4\sqrt{53}}}{2}$. | 2015-10-10T13:46:46 | {
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https://math.stackexchange.com/questions/2687921/if-matrices-a-b-similar-find-nonsingular-s-s-t-b-s-1as | If matrices $A,B$ similar, find nonsingular $S$ s.t.$B=S^{-1}AS$
Consider the matrices below $$A=\begin{bmatrix}9&4&5\\-4&0&-3\\-6&-4&-2\end{bmatrix}$$ and $$B=\begin{bmatrix}2&1&0\\0&2&0\\0&0&3\end{bmatrix}$$
These matrices have the same eigenvalues $\{2,2,-3\}$ and the same Jordan Canonical Form so they are similar.
In trying to find $S$ s.t. $B=S^{-1}AS$ I set $$S=\begin{bmatrix}a&b&c\\d&e&f\\g&h&k\end{bmatrix}$$ and then tried to solve the system of 9 equations with 9 unknowns $$B=S^{-1}AS\Leftrightarrow SB=AS$$ but Matlab showed it is rank deficient so it provided only the zero solution.
How can I find such matrix $S$?
Is there a systematic way to do that in the general case when both $A,B$ are $n\times n$ matrices?
• Find a basis of eigenvectors and generalized eigenvectors. – amd Mar 12 '18 at 15:07
• I think you may have made an error with "Matlab". There are certainly nonzero solutions for these linear equations, see my answer. – Dietrich Burde Mar 12 '18 at 15:58
Notice that $B$ is precisely the Jordan form of $A$, so it suffices to find a Jordan basis for $A$.
We have:
$$A - 2I =\begin{bmatrix}7&4&5\\-4&-2&-3\\-6&-4&-4\end{bmatrix},\quad (A - 2I)^2 =\begin{bmatrix}3&0&3\\-2&0&-2\\-2&0&-2\end{bmatrix}$$
so $\ker (A - 2I)^2 = \operatorname{span}\{e_1- e_3, e_2\} = \operatorname{span}\left\{\pmatrix{1 \\ 0\\ -1}, \pmatrix{0\\1\\0}\right\}$.
$$A - 3I = \begin{bmatrix}6&4&5\\-4&-1&-3\\-6&-4&-5\end{bmatrix} \implies \ker (A - 3I) = \operatorname{span}\left\{\pmatrix{-3 \\ 2 \\ 2}\right\}$$
Therefore, one Jordan basis is
$$\left\{(A - 2I)e_2, e_2, \pmatrix{-3 \\ 2 \\ 2}\right\} = \operatorname{span}\left\{\pmatrix{4 \\ -2\\ -4}, \pmatrix{0\\1\\0}, \pmatrix{-3 \\ 2 \\ 2}\right\}$$
So the similarity matrix is
$$S = \begin{bmatrix}4&0&-3\\-2&1&2\\-4&0&2\end{bmatrix}$$
• @mechanodroid Can you explain how you replaced vector $[1\ 0\ -1]^T$ you found in the basis of $\ker (A - 2I)^2$ with $(A - 2I)e_2=[4\ -2\ -4]^T$ ? – mgus Mar 12 '18 at 16:55
• @mgus In this case, the Jordan basis for the generalized eigenspace $\ker (A - 2I)^2$ can be found by picking a vector $v \in \ker (A - 2I)^2$ which is not in $\ker (A - 2I)$. Then the basis will be $\{(A - 2I)v, v\}$. I picked $v = e_2$. – mechanodroid Mar 12 '18 at 17:06
Yes, there is a systematic way. It works as you said. Writing $AS=SB$ as a system of equations in the $n^2$ variables of $S$, we just solve this system of linear equations. In general, there will be several solutions, including $S=0$ of course. But then we impose the condition $\det(S)\neq 0$, which is a polynomial inequality on the variables of $S$. Then we obtain all possible invertible $S$ with $AS=SB$. Again, there are many such $S$ in general.
So, explicitly, solving the linear equations, we obtain $$S=\begin{pmatrix} s_1 & s_4 & s_7 \cr -\frac{s_1}{2} & \frac{s_1-2s_4}{4} & -\frac{2s_7}{3} \cr -s_1 & -s_4 & -\frac{2s_7}{3} \end{pmatrix}$$ for arbitary parameters $s_1,s_4,s_7$. However, $$\det(S)=\frac{s_1^2s_7}{12},$$ so that we can take all such $S$ with $s_1s_7\neq 0$.
Since $A$ and $B$ have the same Jordan normal form $J$, there exist invertible matrices $P$ and $Q$ such that $$P^{-1}AP=J=Q^{-1}BQ\text{.}$$ Then $$B=(PQ^{-1})^{-1}APQ^{-1}\text{.}$$
Let $v_1=(-2,1,2)$, $v_2=(-1,0,1)$, and $v_3=(-3,2,2)$. Then $A.v_1=2v_1$, $A.v_2=2v_2+v_1$, and $A.v_3=3v_3$. Therefore, if$$M=\begin{pmatrix}-2&-1&-3\\1&0&2\\2&1&2\end{pmatrix},$$then$$M^{-1}.A.M=\begin{pmatrix}2&1&0\\0&2&0\\0&0&3\end{pmatrix}=B.$$
• Can you explain you found $\mathbf{v}_2$ since it's not an eigenvector? I also found $\mathbf{v}_1$ and $\mathbf{v}_3$ for eigenvalues 2 and 3, respectively but I am not sure where the $\mathbf{v}_2$ comes from. – mgus Mar 12 '18 at 15:19
• @mgus In order to find $v_2$, after finding $v_1$ I solved the equation $A.v=2v+v_1$ and took one of the solutions. – José Carlos Santos Mar 12 '18 at 15:21
• But, what is the intuition behind it? I mean which equation should I solve in the general case if I have many eigenvalues with algebraic multiplicity larger than 1? – mgus Mar 12 '18 at 15:24
• @mgus The intuition come from the Jordan normal form, with all those $1$'s right above the main diagonal. In this case, if the multiplicity of $1$ was $3$ and if I knew that the Jordan normal form had a block like$$\begin{pmatrix}2&1&0\\0&2&1\\0&0&2\end{pmatrix},$$then I would go after a vector $v_3$ that would be a solution of the equation $A.v=2v+v_2$. And so on. – José Carlos Santos Mar 12 '18 at 15:55 | 2019-07-20T18:11:53 | {
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https://math.stackexchange.com/questions/1892832/bad-style-to-write-n-in-mathbb-n | # Bad style to write “$n\in\mathbb N$”?
Is it bad style to use $n\in\mathbb N$ as a shortcut/abbreviation for "n is natural number" in a context where one does not deal with sets and elements in particular?
For example: "For all $m, n\in\mathbb N$ we have $m+n=n+m.$" Should one write "For all natural numbers $m$, $n$ we have $m+n=n+m.$" instead?
• Nothing wrong with the set-theoretic notation. – lulu Aug 15 '16 at 10:59
It's perfectly common notation. How symbol-heavy you want your writing to be is a matter of style, but both the example sentences should be alright by most standards.
An exception might be a context where you are worried the notation "$n \in \mathbb{N}$" will not be (easily) understood by your audience. But this particular notation is rather common. Thus, in most moderately advanced contexts it should be alright.
A main reason to use the notation, as for much notation, is brevity and clarity. See Example of a very simple math statement in old literature which is (verbatim) a pain to understand, for where you can get without notation.
For the most part, the two statements simply express the same idea.
Let me turn your question around (as in the comment by Ennar), when you say "for all natural numbers $n,m$" what exactly do you mean by that? Or, what is the difference between "For all natural numbers $n,m$" and "For all elements $n,m$ of the set of natural numbers"?
Of course, there are some mathematicians that reject the notion of a set of all natural numbers and maybe those would see some problem with the notation.
In the other direction, one could argue that in common rigorous developments of mathematics each natural number was constructed as a set, and the mathematical structure one deals with just are sets.
But really, I feel both are tangential to the question at hand for most purposes, where "for all $n,m \in \mathbb{N}$" and "for all natural numbers $n,m$" are just two ways to express the same idea.
• And could you explain why one is using this notation? After all, the fact that "For all natural numbers m, n we have m+n=n+m.m+n=n+m." has nothing to do with sets and the ∈ relation. Why should one then use them in the formulation? – user361467 Aug 15 '16 at 11:11
• @user361467, mathematically "$n$ is a natural number" precisely means "$n$ is element of set of natural numbers". – Ennar Aug 15 '16 at 11:26 | 2019-07-17T23:19:54 | {
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http://math.stackexchange.com/questions/25091/translating-neither-nor-into-a-mathematical-logical-expression | # Translating “neither…nor” into a mathematical logical expression
Having some difficulty doing translations for complicated neither...nor sentences.
With these characters:
~: Negation; $\vee$: Disjunction; &: Conjunction.
I'm trying to translate and understand, for example:
"Neither John nor Mary are standing in front of either Jim or Cary"
I have been told that a successful translation of "Neither e nor a is to the right of c" is translated as follows: ~(RightOf(e, c) $\vee$ RightOf(e, c))
What about just doing a translation on: "I like neither chocolate nor vanilla"
~(Like(chocolate) $\vee$ Like(Vanilla))
What confuses me the most is the sentence: "I like neither chocolate nor vanilla" is translated to ~((Like(chocolate) $\vee$ Like(vanilla)) and the sentence: "Neither e nor a is to the right of c and to the left of b" is translated to ~(RightOf(e, c) & LeftOf(e, b)) & ~(RightOf(a, c) & LeftOf(a, b)). Both sentences use neither...nor, however in the second sentence I see no disjunction, but in the first it exists.
Any food for thought and help would be appreciated!
-
I haven't heard of any actor called John Mary! – Aryabhata Mar 5 '11 at 2:03
@Billare: No, it's supposed to be the OR operation, \vee or $\vee$. – Arturo Magidin Mar 5 '11 at 2:17
@Arturo Magidin Typo, thanks. – Uticensis Mar 5 '11 at 2:19
@Arturo: Using \vee for disjunction makes it harder to read the LaTeX source. I'd use \lor ($\lor$) instead along with \land ($\land$) for conjunction. Negation ($\neg$) can be written with either \neg or \lnot - depending on what flows best in one's source – kahen Mar 5 '11 at 5:04
@kahen: I was not aware of those LaTeX synonyms. I've known \vee and \wedge for a while, so to me they come naturally... – Arturo Magidin Mar 5 '11 at 20:24
Mathematically, there are two ways of "translating" "I like neither chocolate nor vanilla" (the two ways are logically equivalent, an instance of de Morgan's laws). You can write either: $$\neg\bigl(\mathrm{like}(\mathrm{chocolate})\bigr)\ \&\ \neg\bigl(\mathrm{like}(\mathrm{vanilla})\bigr)$$ (that is, "I don't like chocolate and I don't like vanilla") or as $$\neg\bigl(\mathrm{like}(\mathrm{chocolate})\ \vee\ \mathrm{like}(\mathrm{vanilla})\bigr).$$ The two are equivalent, because $\neg(P\vee Q) \equiv (\neg P)\&(\neg Q)$ (this is one of De Morgan's Laws: for "P or Q" to be false, you need both P to be false and Q to be false).
The second sentence is similar: "Neither e nor a is to the right of c and to the left of b".
You mention you are confused about it; I think it's just a confusion of parsing (it is a bit awkwardly constructed). What is says is "Neither e nor a satisfy xxxx". That is: "e does not satisfy xxxx, and a does not satisfy xxxx". What is this xxxx? It is the condition "is to the right of c and to the left of b".
So the sentence is the same as "neither e is to the right of c and to the left of b, nor a is to the right of c and to the left of b." (I added the grey spaces as parsing aides).
(Note that "e is not to the right of c and to the left of b" means that either e is not to the right of c, or e is not to the left of b, or both; same with a).
If you wanted to translate it in the same manner as the first translation above, with conjunction, you would have: $$\Bigl(\neg\bigl(\mathrm{RightOf}(e,c)\&\mathrm{LeftOf}(e,b)\bigr) \Bigr) \& \Bigl(\neg\bigl(\mathrm{RightOf}(a,c)\&\mathrm{LeftOf}(e,b)\bigr)\Bigr).$$ But you can equally as well use the model of the second translation above, to get the equivalent statement: $$\neg\Biggl( \bigl(\mathrm{RightOf}(e,c)\&\mathrm{LeftOf}(e,b)\bigr) \vee \bigl(\mathrm{Rightof}(a,c)\&\mathrm{LeftOf}(a,b)\bigr)\Biggr).$$ So now you see the disjunction in the second.
The reason why the first "translation" of this phrase might be better than the second is that the mix of conjunctions and disjunctions ( "(a and b) or (x and y)" ) is usually a little harder to parse than a sequence of conjunctions, even with negations ( "not(a and b) and not(x and y)"). So the latter form is slightly prefered for parsing reasons, but mathematically they are equivalent. The parsing problem does not show up in the vanilla/chocolate example, because there are no conjunctions hiding inside the clauses and complicating the parsing.
-
Thanks for clearing this up, great explanation! – KerxPhilo Mar 5 '11 at 2:17 | 2016-02-14T06:35:38 | {
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