Search is not available for this dataset
url
string | text
string | date
timestamp[s] | meta
dict |
|---|---|---|---|
http://mathhelpforum.com/advanced-statistics/76394-probability-given-b-true.html | # Math Help - Probability of A given B is true
1. ## Probability of A given B is true
I'm pretty sure that's what kind of question this is anyway.
A drugs test for athletes is 95% effective in detecting a steroid wen it is actually present. However, it also yields a false positive result for 1% of "clean" athletes tested. If 0.5% of athletes use the steroid, what is the probability that an athlete who tests positive did actually use the drug?
2. Originally Posted by chella182
I'm pretty sure that's what kind of question this is anyway.
A drugs test for athletes is 95% effective in detecting a steroid wen it is actually present. However, it also yields a false positive result for 1% of "clean" athletes tested. If 0.5% of athletes use the steroid, what is the probability that an athlete who tests positive did actually use the drug?
Imagine that 100000 atheletes tested so that 0.005(100000)= 500 use the steroid and 99500 don't. Of the 500 who use the steroid 0.95(500)= 465 test positive. Of the 99500 who don't use the steroid 0.01(99500)= 995 also test positive. That gives a total of 995+ 465= 1465 positive tests of whom 465 actually use steroids. The probability that a positive test means the athelete did use the steroid is 465/1465.
3. Hello, chella182!
Yes, this is a Conditional Probability problem.
A drugs test for athletes is 95% effective in detecting a steroid when it is actually present.
However, it also yields a false positive result for 1% of "clean" athletes tested.
If 0.5% of athletes use the steroid, what is the probability that an athlete
who tests positive did actually use the drug?
"0.5% of the atheletes use the drug": . $P(\text{use}) \:=\:0.005,\;P(\text{clean}) \:=\:0.995$
"The test is 95% effective in detecting a steroid when it is actually present."
If the athlete Uses the drug, the test will be Positive 95% of the time.
. . That is: . $P(\text{pos}\,|\,\text{use}) \:=\:0.95$
From Bayes' Theorem: . $P(\text{pos}|\text{use}) \:=\:\frac{P(\text{pos}\wedge\text{use})}{P(\text{ use})} \:=\:0.95$
. . Then: . $\frac{P(\text{pos}\wedge\text{use})}{0.005} \:=\:0.95 \quad\Rightarrow\quad P(\text{pos}\wedge\text{use}) \:=\:0.00475\;\;{\color{blue}[1]}$
"It yields a false positive result for 1% of 'clean' athletes tested."
If an athlete is clean, the test will be positive 1% of the time.
. . That is: . $P(\text{pos}|\text{clean}) \:=\:0.01$
From Bayes' Theorem: . $P(\text{pos}|\text{clean}) \:=\:\frac{P(\text{pos}\wedge\text{clean})}{P(\tex t{clean})} \:=\:0.01$
. . Then: . $\frac{P(\text{pos}\wedge\text{clean})}{0.995} \:=\:0.01 \quad\Rightarrow\quad P(\text{pos}\wedge\text{clean}) \:=\:0.00995 \;\;{\color{blue}[2]}$
From [1] and [2]: . $P(\text{pos}) \:=\:0.00475 + 0.0095 \:=\:0.0147\;\;{\color{blue}[3]}$
"What is the probability that an athlete who tests positive did use the drug?"
. . We want: . $P(\text{use}|\text{pos}) \:=\:\frac{P(\text{use}\wedge\text{pos})}{P(\text{ pos})}
$
From [1] and [3]: . $P(\text{use}|\text{pos}) \:=\:\frac{0.00475}{0.0147} \;\approx\;0.323 \;=\;32.3\%$
4. Whether that's right or not, that deserves a thanks
Although only 32% of people who test positive actually use?? That seems a little unrealisic to me...
5. Originally Posted by chella182
Whether that's right or not, that deserves a thanks
Although only 32% of people who test positive actually use?? That seems a little unrealisic to me... | 2015-08-01T12:19:20 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/advanced-statistics/76394-probability-given-b-true.html",
"openwebmath_score": 0.790692150592804,
"openwebmath_perplexity": 2572.9955884595674,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9875683476832692,
"lm_q2_score": 0.8459424353665381,
"lm_q1q2_score": 0.8354259731300928
} |
https://math.stackexchange.com/questions/2346232/how-many-ways-are-there-to-cover-a-2-times-16-rectangle-with-2-times-2-2-t | # How many ways are there to cover a $2\times 16$ rectangle with $2\times 2,$ $2\times 3$ and $2\times 4$ rectangles?
How many ways are there to cover a $2\times 16$ rectangle with $2\times 2,$ $2\times 3$ and $2\times 4$ rectangles?
I already dealt with a similar problem, which is how many ways are there to cut a $1\times 8$ rectangle into $1\times 1$ and $1\times 2$ rectangles. The answer to this problem can be calculated using the number of ways to divide $1\times k$ strip for $3\le k\le8$ sequentially, to arrive at $34$. However, this problem is completely different. How can I solve this problem?
• Hint: find a recurrence relation to describe the scenario. Your expression for $a_n$, the amount of ways to cover a $2\times n$ rectangle using $2\times 2, 2\times 3,$ and $2\times 4$ rectangles will have something to do with $a_{n-2},a_{n-3}$ and $a_{n-4}$. – JMoravitz Jul 4 '17 at 14:19
• $2x+3y+4z=16$ where $x,y,z \in N_0$ with constraints $x\leq 8, y\leq 5, z\leq 4$, – serg_1 Jul 4 '17 at 14:28
• i notice that y<=5 , is it because we only can put four 2*3 rectangle , and two 2*2 rectangle which is equal to 2*4 rectangle . hence constraint for y is less than or equal to 5 ? . thanks for give me hint – Znik Dzulqarnain Jul 4 '17 at 14:42
• @serg_1 that is incorrect. Your answer seems to assume that two coverings are considered "the same" so long as the amounts of each type of rectangle used are the same, but that is not how these problems are traditionally interpreted. I would consider for example the arrangement 4-4-2-2-4 to be different than the arrangement 2-4-4-4-2 because (among other things) the smaller rectangles are no longer adjacent. – JMoravitz Jul 4 '17 at 15:35
If you have a $2 \times n$ rectangle then call $a_n$ the number of ways to cover that rectangle with $2\times p$ ($p\in \{2,3,4\}$) rectangles.
If you start with $2\times 2$ then you will face the problem $a_{n-2}$, but if you start with $2\times 3$ you will find the problem $a_{n-3}$ and finally if you start with $2\times 4$ you will face $a_{n-4}$ then you have
$$a_{n}=a_{n-2}+a_{n-3}+a_{n-4}$$
and
$$a_2=1\\ a_3=1\\ a_4=2$$
and we also need define $a_1=0$.
now you can go forward and find $a_{16}$.
• You need to define $a_1$ as $0$ to use this recursion. – N. F. Taussig Jul 4 '17 at 15:04
• @N.F.Taussig: You are right. Thanks. – Arnaldo Jul 4 '17 at 15:17
• oww yea i realize you need to define a_1 , but thanks for an amazing explanation – Znik Dzulqarnain Jul 4 '17 at 15:39
• @ZnikDzulqarnain: you are very welcome – Arnaldo Jul 4 '17 at 15:40
We are dealing with a $2 \times 16$ grid. The first column of two cells can be filled by a $2 \times 2$, a $2 \times 3$ or a $2 \times 4$ grid. Let us call $f(n)$ the number of ways to fill a $2 \times n$ grid. We then have:
$$f(n) = f(n-2) + f(n-3) + f(n-4), n \geq 4$$
We also know that:
$$f(0) = 1$$ $$f(1) = 0$$ $$f(2) = 1$$ $$f(3) = 1$$
A closed form is not straightforward, but we find:
$$f(4) = 2$$ $$f(5) = 2$$ $$f(6) = 4$$ $$f(7) = 5$$ $$f(8) = 8$$ $$f(9) = 11$$ $$f(10) = 17$$ $$f(11) = 24$$ $$f(12) = 36$$ $$f(13) = 52$$ $$f(14) = 77$$ $$f(15) = 112$$ $$f(16) = 165$$ | 2020-11-28T04:43:13 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2346232/how-many-ways-are-there-to-cover-a-2-times-16-rectangle-with-2-times-2-2-t",
"openwebmath_score": 0.8464066982269287,
"openwebmath_perplexity": 230.7185433250364,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9875683465856102,
"lm_q2_score": 0.8459424314825853,
"lm_q1q2_score": 0.8354259683658676
} |
https://mathematica.stackexchange.com/questions/224249/best-way-to-insert-element-into-an-ordered-list-at-the-correct-position/224251 | Best way to insert element into an ordered list at the correct position?
Suppose I have an ordered list orderedList and I want to write some code that will involve regularly inserting new elements into orderedList in the correct positions, according to the ordering. One possibility is
Sort[AppendTo[orderedList,x]],
but this is inefficient because Mathematica doesn't know orderedList is already ordered, hence it will waste time verifying that the original elements of orderedList are already ordered.
Another idea is to use SelectFirst to obtain the proper position at which to insert x, via
Insert[orderedList,x,FirstPosition[SelectFirst[orderedList,#>x&]]].
The trouble here is that since Mathematica doesn't know orderedList is ordered, its SelectFirst algorithm will use a linear search rather than binary. Also, the requirement of using FirstPosition will double the computation time.
So what is the "right" way to do this, i.e., what method inserts an element into an ordered list without wasting resources?
• Do a binary search. – ciao Jun 18 at 23:00
• "but this is inefficient" - true, perhaps, but sorting algorithms are very optimized and typically very fast. Have you verified that this is really a problem in your application with some timings? – MarcoB Jun 18 at 23:01
• maybe Nearest + ReplacePart/MapAt/...? – kglr Jun 18 at 23:01
• Are you on v12.1? You could use a priority queue which stays sorted. You can make one with ds=CreateDataStructure["PriorityQueue"]; You then push all your elements in (see docs) and then when you're done, get them in list form like this: lst = {}; While[! ds["EmptyQ"], AppendTo[lst, ds["Pop"]]]; – flinty Jun 18 at 23:20
If you have v12.1, there's no need to ever call Sort if you can incrementally add your values to a "PriorityQueue" data structure. It always stays sorted as you add/remove elements.
SeedRandom[1234];
ds = CreateDataStructure["PriorityQueue"];
(* push a million random values in - use Scan for pushing many values.
The slowest part here is creating RandomIntegers, not the Push which is very fast *)
Scan[ds["Push", #] &, RandomInteger[10^6, 10^6]];
(* peek at the largest value *)
ds["Peek"]
(* returns: 999999 *)
ds["Push", N[Sqrt[2]]];
ds["Push", N[π^2]];
ds["Push", 0];
(* get all of the values out and show the first 50 *)
tbl = Reverse@Table[ds["Pop"], {ds["Length"]}];
tbl[[;; 20]]
(* result: {0, 0, 0, 0, 1.41421, 3, 5, 6, 8, 8, 9, 9, 9.8696, 10, 11, 14, 16, 17, 18, 19} *)
(* After we popped the values, the heap is empty *)
ds["EmptyQ"] (* returns True *)
The timing of each "Push" is extremely small, around 5.*10^-7 on my machine. To get average push time:
rnd = RandomInteger[10^6, 10^6];
First[CompoundExpression[ds["DropAll"], Scan[ds["Push", #] &, rnd]] //
RepeatedTiming]/10^6
(* 5.*10^-7 per push, overall 0.5 seconds to push 1M values *)
• This is great. I have v12.0, but will definitely update for this. – WillG Jun 19 at 5:16
For the binary search idea, you could use Leonid Shifrin's fast, compiled binary search function here. It would look like this:
sortedInsert[list_, el_] := Insert[
list,
el,
bsearchMax[list, el]
]
sortedInsert[2 Range[10], 13]
{2, 4, 6, 8, 10, 12, 13, 14, 16, 18, 20}
list = Sort@RandomInteger[100000, 10000];
values = RandomInteger[100000, 1000];
sortedInsert[list, #] & /@ values; // RepeatedTiming
{0.018, Null}
You may also want to check out the other solutions in that answer. As some have pointed out, the best complexity solution is not always the most performant solution, when implemented in Mathematica. For comparison with a very naive approach, however, we can clearly see that the above helps:
sortedInsert2[list_, el_] := Insert[
list,
el,
LengthWhile[list, el > # &] + 1
]
sortedInsert2[list, #] & /@ values; // RepeatedTiming
{1.5954, Null}
Nearest is quite a lot faster than the naive approach, but not as fast as binary search. A drawback is that if we're inserting elements into the list, then we need to recreate the NearestFunction over and over since there is no way to update it.
sortedInsert3[list_, el_] := With[
{nf = Nearest[list -> "Index"]},
Insert[
list,
el,
Last@nf[el]
]
]
sortedInsert3[list, #] & /@ values; // RepeatedTiming
{0.24, Null}
I tried running Nearest directly instead of creating a NearestFunction explicitly, but it turned out to be slower.
The second best solution that I've found is the most naive of them all, that you also mention in your question:
Sort[Append[list, #] & /@ values]; // RepeatedTiming
{0.095, Null}
I'm aware that Append does not actually append any value, but neither does Insert so I'm using it for comparison here. Sorting would take a bit longer perhaps if accounting for a large number of newly inserted elements, but as we can see sorting is very fast. If you have all the elements you want to insert available up front, then it's a no-brainer:
Sort@Join[list, values]; // RepeatedTiming
{0.000075, Null}
This reflects the fact that dynamically resizing lists is very slow, so we should typically always try to avoid that under any circumstance, including this one. (In Mathematica we also typically try to avoid any kind of looping, so we can look at it from that point of view too, Map being a type of iteration.)
• Interesting that the simple solution of Sort@Join seems to be the best by far. I'm a bit confused about the second-to-last example with Sort[Append[... because it doesn't seem to perform the correct operation. E.g., Sort[Append[{1,2,3}, #] & /@ {4,5}] gives {{1,2,3,4},{1,2,3,5}}. – WillG Jun 19 at 5:14
• @WillG I explicitly explained that in my answer. You could say exactly the same thing about the Insert examples but it doesn`t matter because I’m just testing the performance. – C. E. Jun 19 at 6:01 | 2020-08-14T12:20:52 | {
"domain": "stackexchange.com",
"url": "https://mathematica.stackexchange.com/questions/224249/best-way-to-insert-element-into-an-ordered-list-at-the-correct-position/224251",
"openwebmath_score": 0.22831417620182037,
"openwebmath_perplexity": 1817.5296656725438,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9449947055100817,
"lm_q2_score": 0.8840392756357327,
"lm_q1q2_score": 0.8354124349387352
} |
https://math.stackexchange.com/questions/2300041/this-second-derivative-is-showing-a-point-of-inflection-rather-than-a-minimum-po | # This second derivative is showing a point of inflection rather than a minimum point
For the curve $y=(2x-1)^4$, the derivative $8(2x-1)^3$ shows that the only stationary point is $(0.5, 0)$, which means it just touches the $x$-axis at that point. However, when determining the nature of the stationary point, the second derivative is $48(2x-1)^2$. Substituting $x=0.5$ into the second derivative gives $\frac{d^2y}{dx^2}=0$, which implies the stationary point is a point of inflection. However, when drawing the graph, it is actually a minimum point.
I also checked it by substituting $x=0.49$ and $x=0.51$ into the first derivative, which gives a negative and positive gradient, respectively, showing a minimum point.
So why does the second derivative show a point of inflection?
Shouldn't it be positive, which shows a minimum point?
Is it possibly to do with the fact that the curve does not actually cross the $x$-axis but only touches it, or is that irrelevant?
• I don't understand. May 28 '17 at 13:11
• It's because the third derivative is also $0$. So the second derivative being $0$ doesn't mean it's an inflection point. May 28 '17 at 13:11
• So does that mean to be absolutely sure about inflection points, you also have to check the third derivative? May 28 '17 at 13:12
• Yes. Or better yet, just check the sign of the derivative on either side of the stationary point. Much quicker, especially for more complicated functions. May 28 '17 at 13:13
• This is an A-level chain rule maths question from C3, so we haven't learnt about the third derivative. May 28 '17 at 13:13
$\dfrac{d^2y}{dx^2}=0$ does not imply that there is a zone of inflection. As an example, $y=x^4$. This has a stationary point at $x=0$, and the second derivative here is $12x^2=12\cdot0^2=0$. But this is not a zone of inflection, it is a minimum. The point is, second derivative being $0$ is not a definitive way of checking if there is a zone of inflection. It is true that at any zone of inflection, the second derivative will be $0$, but the converse is not true, as illustrated by my example above.
An alternative method is that you can check values either side of the stationary point to see what the true nature is. (Alternatively you could check higher derivatives, but that is sometimes tedious).
In this case, $$y=(2x-1)^4\\y'=8(2x-1)^3$$ You know there is a stationary point at $\frac12$, so substitute in $x=\frac12\pm\epsilon$ where $\epsilon$ is small. Then $$y'=8(\pm2\epsilon)^3=\pm16\epsilon^3$$ So is $\epsilon<0$ then $y'<0$, and if $\epsilon>0$ then $y'>0$. This means it is a minimum, since the function is decreasing just before the stationary point (i.e. for $\epsilon<0$), and is increasing after (i.e. for $\epsilon>0$).
• Ok that makes sense, I usually check my answers using this method. But I don't fully understand the point about the second derivative being 0 not necessarily meaning a point of inflection. May 28 '17 at 13:25
• If $\frac{d^2y}{dx^2}=0$ does not always mean a point of inflection then what else does it mean? May 28 '17 at 13:26
• The second derivative being zero simply means that the gradient function has a stationary point at that point ($\frac12$) in your example. May 28 '17 at 13:27
• So is it only when the second derivative is positive or negative that you can definitely be sure that it is a minimum/maximum point? May 28 '17 at 13:30
• @kashveyron Yes, exactly :) The rules are $y''>0\iff \text{minimum}$, $y''<0\iff \text{maximum}$, $y''=0\impliedby \text{inflection}$, which means $y''=0$ doesn't guarantee anything about the point. May 28 '17 at 13:32
Just like $f'(x_0)=0$ doesn't mean $x_0$ is a point of maximum or minimum, $f''(x_0)$ doesn't mean $x_0$ is a point of inflection.
The usual example for the first case is $f(x)=x^3$: $f'(0)=0$, but $0$ is a point of inflection and neither a point of maximum nor a point of minimum.
The condition that $f''(x_0)=0$ when $x_0$ is a point of inflection is only necessary and not sufficient (for points where the second derivative exists).
Your function is essentially the same as $f(x)=x^4$, which obviously has a minimum at $0$.
For “well behaved” functions (more precisely, analytic, but the condition can be relaxed), a point of inflection is where the first derivative changes from increasing to decreasing or conversely: so they're the points where the first derivative has a maximum or minimum.
For $f(x)=x^4$, we have $f'(x)=4x^3$, and this function has neither a maximum nor a minimum at $0$.
When you're looking for the points of inflection, you certainly find the points where the second derivative vanishes. Among them there are the inflection points and you have to check for the sign of the second derivative at either side of the point.
A different method is to look at the lowest $n$ such that $f^{(n)}(x_0)\ne0$: if $n$ is even the point $x_0$ is either a maximum or a minimum; if $n$ is odd the point $x_0$ is a point of inflection (assuming $f''(x_0)=0$ to begin with).
However, this method might not be conclusive. Consider $$f(x)=\begin{cases} 0 & x=0 \\ \exp(-1/x^2) & x\ne0 \end{cases}$$ Then $f^{(n)}(0)=0$ for every $n$; $0$ is obviously a point of minimum.
The function $$F(x)=\int_{0}^x f(t)\,dt$$ has the same property: $F^{(n)}(0)=0$ for all $n$, and $F$ has a point of inflection at $0$.
On the other hand, the function $$f(x)=\begin{cases} 0 & x=0 \\ x^4\sin(1/x) & x\ne0 \end{cases}$$ has $f''(0)=0$, but the second derivative takes positive and negative values in every left and right neighborhood of $0$, so $0$ is not a point of inflection, but it's not a point of maximum or minimum either.
There can be other points of inflection, though: the function $f(x)=\sqrt[3]{x}$ has an inflection at $0$, where it is not differentiable.
• +1 for the very nice explanation as well as the counter-example at the end of the answer. May 31 '17 at 8:11
you must use higher derivatives: $$y'=4(2x-1)^3\cdot 2$$ $$y''=12(2x-1)^2\cdot4$$ $$y'''=24(2x-1)\cdot 8$$ $$y^{iv}=24\cdot 16>0$$ therefore we have a minimum at $$x=\frac{1}{2}$$
• the order of the derivative must be odd May 28 '17 at 13:15
• Oh I think I get it now, is that because it is a quartic? May 28 '17 at 13:15
• yes thats right May 28 '17 at 13:16
• I don't fully understand what you mean by 'the order of the derivative must be odd'. May 28 '17 at 13:17
• You may have to check your derivatives : you forgot the "$2$" factor in $(2x-1)$. May 28 '17 at 13:17 | 2021-09-20T14:53:59 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2300041/this-second-derivative-is-showing-a-point-of-inflection-rather-than-a-minimum-po",
"openwebmath_score": 0.7900223731994629,
"openwebmath_perplexity": 147.68219621372663,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806552225684,
"lm_q2_score": 0.8519528076067262,
"lm_q1q2_score": 0.8354084423017104
} |
http://openstudy.com/updates/4f417c2ee4b0803c6352f76e | ## anonymous 4 years ago Consider a particle with initial velocity that has magnitude 12.0 m/s and is directed 60.0 degrees above the negative x axis. What is the x component Vx of V? What is the y component Vy of V?
1. Hunus
|dw:1329692762479:dw| The x component Vx of V is the velocity vector V projected onto the x axis. We obtain this magnitude as follows $v_{x}=12\cos(120)=-12\cos(60)=-12(\frac{1}{2})=-6$ The y component Vy of V is the velocity vector V projected onto the y axis. We obtain this magnitude as follows $v_{y}=12\sin(120)=12\sin(60)=12\frac{\sqrt{3}}{2}=6\sqrt{3}=10.4$ |dw:1329693129338:dw|
2. anonymous
why does sin have a positive value?
3. Hunus
y has a positive value in the top two quadrants
4. Hunus
|dw:1329693437661:dw|
5. anonymous
so to figure out positive and negative i have to look at 12
6. Hunus
You have to look at the direction it's pointing in. For theta equal to 120 or '60 above the negative x axis' the vector is pointing in the negative x direction and in the positive y direction
7. anonymous
so what is u have something like this :
8. anonymous
srry i mean
9. anonymous
|dw:1329693963427:dw|
10. anonymous
now its moving clockwise
11. Hunus
The vector A you described is at an angle of (180+60) 240 degrees with respect to the positive x axis as you can see from your picture it is directed toward the negative x negative y direction, so both x and y will be negative.
12. anonymous
why do i have to add 180?
13. Hunus
|dw:1329694073557:dw| Because we're making the angle with respect to the positive x axis to make it simple to take the sine and cosine of the vector
14. Hunus
By adding 180 to 60 we're changing it from your original picture |dw:1329694180018:dw| and making it |dw:1329694218137:dw|
15. anonymous
or u can say the angle is -60
16. Hunus
If we're taking it to be with respect to the positive x axis |dw:1329694331261:dw| then it would be -120 degrees. -60 degrees looks like this |dw:1329694414386:dw|
17. anonymous
|dw:1329694584288:dw|
18. anonymous
thats how u show it
19. anonymous
cause its moving clockwise
20. Hunus
Yes.
21. anonymous
so in this question is the 60 degrees going clockwise
22. Hunus
It typically doesn't matter how you arrive at the angle, but it appears that the angle was rotated counter clockwise 60 degrees. The vector is at 240 degrees with respect to the positive x axis.
23. anonymous
so qhen do i know when to add 180 or minus 180
24. Hunus
Look at the positive x axis and see how many degrees it takes to get to your vector. This is only showing whether a vectors components are positive of negative.
25. Hunus
or negative*
26. anonymous
can u please give me an example of each type if itsnt too much trouble
27. Hunus
|dw:1329695075597:dw| Is vector M's x component positive or negative? What about its y component?
28. anonymous
the x compoinent is negative while the y component is positive
29. Hunus
$v_x=|M|\cos(150)=|M|(\frac{-\sqrt{3}}{2})=\frac{-|M|\sqrt{3}}{2}$so we know that the value of Vx is negative while the value of Vy is positive. From this we could say that $v_{y}=|M| \sin(30)=\frac{|M|}{2}$ and $v_{x}=-|M|\cos(30)=-|M|\frac{\sqrt{3}}{2}$ Where |M| is the magnitude of the vector M However we could also say that, to get to vector M from the positive x axis |dw:1329695464157:dw| we would have to go through an angle of 150 degrees. Thus would make our calculations $v_y=|M|\sin(150)=\frac{|M|}{2}$ As you can see we arrive at the same answer which is why it doesn't matter which way you get to the angle.
30. Hunus
$v_{x}=|M|\cos(150)=|M|(\frac{-\sqrt{3}}{2})=\frac{-|M|\sqrt{3}}{2}$
31. anonymous
what does 150 degrees tellus ?
32. anonymous
are we saying the angle 150 is equal to 30 degrees
33. Hunus
150 degrees tells us what the angle is from the positive x axis. That's what we usually like to use as a reference point when using sines and cosines.
34. Hunus
What we are saying is that we only need the reference angle (how many degrees it is from the closest x axis) if we know if x is positive or negative or y is positive or negative
35. anonymous
so basically we are saying the angle is 150 degrees away from the orgin
36. Hunus
From the positive x axis
37. anonymous
what about the negative x axis
38. Hunus
150 degrees tells us how far we would have to rotate a vector from the positive x axis (0 degrees) to get to where the vector is.
39. anonymous
ohhh i get it
40. Hunus
Good :)
41. anonymous
and to find the negative value what would i have to do
42. Hunus
Well, the degrees from the positive x axis tells us whether it's positive or negative. For example, between 0 and 90 degrees, x and y are negative, between 90 and 180 degrees x is negative and y is positive, between 180 and 270 degrees both x and y are negative and from 270 to 360 degrees x is positive and y is negative. If you, for example, put in 6sin(340) you will get a negative value because y is negative in that quadrant. If you put in 12cos(320) you will get a positive value because x is positive in that quadrant.
43. Hunus
between 0 and 90 x and y are positive*** sorry
44. anonymous
so it doesnt matter whether u say 30 degreees or 150 degrees
45. anonymous
46. Hunus
It matters, but only as far as the dirrection goes. This is because the MAGNITUDE of cos(30) is the same as the MAGNITUDE of cos(150), but one is negative and one is positive
47. Hunus
Sorry for the delay on that response
48. anonymous
and that has to do withe quadrant whether is positive or negative | 2016-09-27T21:17:23 | {
"domain": "openstudy.com",
"url": "http://openstudy.com/updates/4f417c2ee4b0803c6352f76e",
"openwebmath_score": 0.7324343323707581,
"openwebmath_perplexity": 713.7253635149038,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806549388171,
"lm_q2_score": 0.8519528076067262,
"lm_q1q2_score": 0.8354084420599676
} |
https://www.azdictionary.com/definition/cardinal%20number | # cardinal number definition
• noun:
• lots, eg 3 or 11 or 412, utilized in counting to indicate quantity yet not purchase.
• several accustomed denote quantity; a counting quantity.
• A generalized style of quantity used to denote the dimensions of a collection, including limitless units.
• A word that expresses a countable quantity; a cardinal numeral.
• the number of elements in a mathematical ready; denotes a quantity although not the order
## Related Sources
• Hyponym for "cardinal number"
• googolplex
• Hypernym for "cardinal number"
• number
• Cross Reference for "cardinal number"
• Cardinal numbers
77 votes | 2017-03-29T06:21:33 | {
"domain": "azdictionary.com",
"url": "https://www.azdictionary.com/definition/cardinal%20number",
"openwebmath_score": 0.9524019360542297,
"openwebmath_perplexity": 14284.274674007916,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9805806512500486,
"lm_q2_score": 0.8519528076067262,
"lm_q1q2_score": 0.8354084389173109
} |
https://math.stackexchange.com/questions/1944299/using-the-definition-of-a-limit-prove-that-lim-n-rightarrow-infty-fracn | # Using the definition of a limit, prove that $\lim_{n \rightarrow \infty} \frac{n^2+3n}{n^3-3} = 0$
Using the definition of a limit, prove that $$\lim_{n \rightarrow \infty} \frac{n^2+3n}{n^3-3} = 0$$
I know how i should start: I want to prove that given $\epsilon > 0$, there $\exists N \in \mathbb{N}$ such that $\forall n \ge N$
$$\left |\frac{n^2+3n}{n^3-3} - 0 \right | < \epsilon$$
but from here how do I proceed? I feel like i have to get rid of $3n, -3$ from but clearly $$\left |\frac{n^2+3n}{n^3-3} \right | <\frac{n^2}{n^3-3}$$this is not true.
This is not so much of an answer as a general technique.
What we do in this case, is to divide top and bottom by $n^3$: $$\dfrac{\frac{1}{n} + \frac{3}{n^2}}{1-\frac{3}{n^3}}$$ Suppose we want this to be less than a given $\epsilon>0$. We know that $\frac{1}{n}$ can be made as small as we like. First, we split this into two parts: $$\dfrac{\frac{1}{n}}{1-\frac{3}{n^3}} + \dfrac{\frac{3}{n^2}}{1-\frac{3}{n^3}}$$
The first thing we know is that for large enough $n$, say $n>N$, $\frac{3}{n^3} < \frac{3}{n^2} < \frac{1}{n}$. We will use this fact.
Let $\delta >0$ be so small that $\frac{\delta}{1-\delta} < \frac{\epsilon}{2}$. Now, let $n$ be so large that $\frac{1}{n} < \delta$, and $n>N$.
Now, note that $\frac{3}{n^3} < \frac{3}{n^2} < \frac{1}{n} < \delta$. Furthermore, $1- \frac{3}{n^3} > 1 - \frac{3}{n^2} > 1-\delta$.
Thus, $$\dfrac{\frac{1}{n}}{1-\frac{3}{n^3}} + \dfrac{\frac{3}{n^2}}{1-\frac{3}{n^3}} < \frac{\delta}{1+\delta} + \frac{\delta}{1+\delta} < \frac{\epsilon}{2} + \frac{\epsilon}{2} < \epsilon$$
For large enough $n$. Hence, the limit is zero.
I could have had a shorter answer, but you see that using this technique we have reduced powers of $n$ to this one $\delta$ term, and just bounded that $\delta$ term by itself, bounding all powers of $n$ at once.
• I am hoping that you have understood the above process. As an exercise, or just for fun, try the limit $\displaystyle\lim_{n \to \infty} \frac{n^4+7n^6+35n+42}{12n^3+34n^2+278n^6 + 1728}$. The answer is just the coefficients of the largest powers, $\frac{7}{278}$. If you can answer this, then you will be able to appreciate this technique. – астон вілла олоф мэллбэрг Sep 28 '16 at 0:40
• yeah I think I get an idea from your proof. but my professor never proved something using $\delta$ yet. so I was wondering if you could prove this without using $\delta$ – Allie Sep 28 '16 at 0:48
• Of course. See, $\delta$ is just a "proxy name" for $\frac{1}{n}$. In place of $\delta$, I could have just taken $\frac{1}{n}$ in that expression, and then I will directly get a bound for $\frac{1}{n}$ instead of $\delta$. Just for ease of reading, I decided to split this into two parts, one having $\delta$ and one having $\frac{1}{n}$.Hence, $\delta$ can be avoided easily, – астон вілла олоф мэллбэрг Sep 28 '16 at 1:41
Another way to show $\lim_{n \rightarrow \infty} \frac{n^2+3n}{n^3-3} = 0$.
Let $f(n) =\frac{n^2+3n}{n^3-3}$.
For $n \ge 3$, $n^2+3n < 2n^2$.
Similarly, for $n \ge 3$, $n^3-3 = \frac12 n^3 + \frac12 n^3 - 3 \gt \frac12 n^3$ for $\frac12 n^3 > 3$, which is certainly true for $n \ge 2$.
Therefore, for $n \ge 3$, $f(n) =\frac{n^2+3n}{n^3-3} < \frac{2n^2}{\frac12 n^3} =\frac{4}{n}$.
You should now be able to easily show that $\lim_{n \to \infty} f(n) = 0$.
Note that this is not the best upper bound for $f(n)$, but it is enough to show what you want.
The generalizations are left to you.
How about this： when $$n>2$$, $$0<\frac{n^2+3n}{n^3-3}<\frac{4n^2}{n^3-n^2}=\frac{4}{n-1}$$ | 2019-06-26T06:41:43 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1944299/using-the-definition-of-a-limit-prove-that-lim-n-rightarrow-infty-fracn",
"openwebmath_score": 0.9528630971908569,
"openwebmath_perplexity": 90.12723523077743,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806495475398,
"lm_q2_score": 0.8519528076067262,
"lm_q1q2_score": 0.8354084374668538
} |
https://math.stackexchange.com/questions/1342867/how-come-least-square-can-have-many-solutions | # How come least square can have many solutions?
I know there always exists a least-square solution $\hat{x}$, regardless of the properties of the matrix $A$. However, I keep finding online that least-square can have infinitely many solutions, if $A$ is not full column rank.
Shouldn't $\hat{x}$ always be unique, as the minimization of a quadratic function (the error) always yields a global minima/maxima? Therefore, regardless of what the matrix $A$ is (even if it is a badly constructed matrix with dependent columns), least-square should find a single 'best' solution $\hat{x}$?
Is there an easy (or intuitive) proof showing why would the least-square method produce infinitely many solutions if there are dependent columns?
• In the higher dimensional space where defined by A, then less than full rank implies that there exists a linear combination of the columns of A that equal zero, hence you can always add any multiple of that linear combination to the minimum value to get another solution. If you reduced the dimensionality to just the linearly independent vectors, then you would have a unique solution. – user237392 Jun 29 '15 at 3:20
• Just think an extreme case, $A = 0 \in \mathbb{R}^{m \times n}$, then clearly every vector in $\mathbb{R}^n$ is a least square solution. – Zhanxiong Jun 29 '15 at 3:38
• The predicted values are unique, regardless of the fact that the coefficients are not. ${}\qquad{}$ – Michael Hardy Jun 29 '15 at 4:44
Your least squares solution is minimizing $\hat x^T A \hat x$ If $A$ does not have full rank, there is some vector $y$ such that $Ay=0$. Then $(\hat x+y)^TA(\hat x+y)=\hat x^T A \hat x$ so you can add any multiple of $y$ to your solution and get the same product.
Suppose that $\exists y \in Ker(A) / y \neq 0$. Now, let $\hat{x}$ be a least squares solution. That is: $\hat{x}$ is such that $||A\hat{x} - b||_2 = min_{x \in V} ||Ax - b||_2$. Notice that $A(\hat{x} + y) = A\hat{x} + Ay = A \hat{x}$, hence $||A\hat{x} - b||_2 = ||A(\hat{x} + y) - b||_2$.
This means that you have just created another solution that also has the minimum possible norm, provided that $Ker(A) \neq \{0\}$, which only happens if A is not full rank.
for what it is worth, the least squares solution to $$H \hat{x} = z$$ is considered by multiplying on the left by $H^t,$ giving $$H^t H \hat{x} = H^t z.$$ The matrix $H^t H$ is square and symmetric, indeed positive semidefinite. It is called the Gramian of the system in applications.
If the Gramian is nonsingular, therefore invertible, there is a single answer to the original problem given by $$\hat{x} = \left( H^t H \right)^{-1} H^t z$$
The unknown variables are said to be observable if all this happens, which is the case if and only if the columns of $H$ are linearly independent.
From Kalman Filtering: Theory and Practice by Grewal and Andrews
Nice property is to add constraint of the least norm of all solutions.
Then, if $A$ is full rank there is one solution (Hence unique) while in the case $A$ isn't full rank still there is one unique solution.
The great thing is that the Pseudo Inverse of $A$ always yields both the Least Squares and Least Norm solution. | 2019-07-23T13:54:27 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1342867/how-come-least-square-can-have-many-solutions",
"openwebmath_score": 0.843147873878479,
"openwebmath_perplexity": 151.1044338877916,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.98058065352006,
"lm_q2_score": 0.8519528038477824,
"lm_q1q2_score": 0.8354084371653059
} |
https://www.physicsforums.com/threads/finding-the-length-of-a-curve.581164/ | # Finding The Length of a Curve
1. Feb 25, 2012
### Timebomb3750
1. The problem statement, all variables and given/known data
Find the length of the curve r(t)=<e^(t) , e^(t)sin(t) , e^(t)cos(t)> between points (1,0,1) and (e^(2pi) , 0 , e^(2pi))
2. Relevant equations
Length of curve=∫(llv(t)ll Where the limits of integration are the distance between the given points.
3. The attempt at a solution
First, to know what my limits are. I would like to assume they're from 1 to e^(2pi), but something tells me I have to calculate the distance somehow. Would that distance be D=√(Δx^(2)+Δy^(2)+Δz^(2)). Is this correct?
That's the only part I'm confused on. The integral itself isn't so bad.
2. Feb 25, 2012
### Ray Vickson
The limits are NOT the distance between the given points. For example, if we calculate the length of the curve (cos(t),sin(t)) once around the circle (say from (1,0) to (1,0) again), the distance between the given points is zero! In the circle case the limits would be t=0 and t=2*pi. To find them in your case, ask yourself: what t gives (x,y,z) = (1,0,1)? What t gives (x,y,z) = (e^(2pi),0,e^(2pi))?
RGV
3. Feb 25, 2012
### Timebomb3750
Oh! I see. Well then, my limits would be from 0 to 2pi, right? Because e^0, e^(0)sin(0), e^(0)cos(0) give the point (1,0,1). And e^(2pi), e^(2pi)sin(2pi), e^(2pi)cos(2pi) give the point (e^(2pi), 0, e^(2pi)). Correct?
4. Feb 25, 2012
### Ray Vickson
What do YOU think?
RGV
5. Feb 25, 2012
### Timebomb3750
Looks good to me. :D
6. Feb 25, 2012
### HallsofIvy
Staff Emeritus
No, those are only the two x values. You will be integrating with respect to the parameter t so your limits of integration will be the values of t that give those points. What is t when $x= e^t= 1$, $y= e^t sin(t)= 0$, and $z= e^tcos(t)= 1$? (Of course, any one of those equations will give the correct answer- check that the others are correct to show that (1, 0, 1) really is on that curve.) What is t when $x= e^t= e^{2\pi}$, $y= e^tsin(t)= 0$, and $z= e^tcos(t)= e^{2\pi}$? Again, the first equation is enough to find t, but you the others to verify that the point really is on the curve.
Well, no, that is the straight line distance between the two points. You may be thinking this- if you take $\Delta x$, $\Delta y$, $\Delta z$ to be very small, that will approximate the length of a short segment of the curve. You can add those for many such short segments to get the "Riemann sum" that approximates the length of the curve: $\sum \sqrt{\Delta x^2+ \Delta y^2+ \Delta z^2}$. Of course, those will correspond to small $\Delta t$ so that you can "multiply and divide" by that to get $\sum\sqrt{\left(\Delta x/\Delta t\right)^2+ \left(\Delta y\Delta t\right)^2+ \left(\Delta z/\Delta t\right)^2}\Delta t$ which, after taking the limit process, becomes the integral
$$\int \sqrt{\left(\frac{dx}{dt}\right)^2+ \left(\frac{dy}{dt}\right)^2+ \left(\frac{dz}{dt}\right)^2} dt$$ | 2017-11-23T00:29:30 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/finding-the-length-of-a-curve.581164/",
"openwebmath_score": 0.9018276929855347,
"openwebmath_perplexity": 685.3398410257106,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806478450307,
"lm_q2_score": 0.8519528076067262,
"lm_q1q2_score": 0.8354084360163964
} |
https://www.physicsforums.com/threads/particle-in-a-ring.837560/ | # Particle in a ring
1. Oct 13, 2015
### maverick_76
So I am working on the problem of the particle bound to a ring of radius R. I am trying to solve it two ways, as a standing wave and as a running wave. I'm stuck right now solving for the standing wave. So far I have:
ψ(x)=Asin(kx) + Bcos(kx)
I know that it is periodic from 0 to 2π so if I integrate the modulus of ψ(x) squared I can set it equal to 1 and solve for A & B.
So my question is this, would the B value be zero? Since it is periodic around the ring, would the only way to make cos(kx) zero be to make coefficient B zero?
2. Oct 13, 2015
### The_Duck
No, there's no reason to set B to zero.
Assuming $x$ here is the angle around the circle, your constraints are:
* $\psi(0) = \psi(2 \pi)$
* $\frac{d}{dx}\psi(0) = \frac{d}{dx}\psi(2 \pi)$
* $\psi(x)$ should solve the Schrodinger equation
If you impose these constraints you will find:
* k can only take on certain discrete values
* $A^2 + B^2$ must equal a certain value, but $A$ and $B$ are not individually constrained
Why aren't $A$ and $B$ constrained invididually? This corresponds to the fact that if you have a solution $\psi(x)$ and then rotate it around the circle by any angle, you get another solution. So you have a bunch of solutions, all of which are linear combinations of $\sin(kx)$ and $\cos(kx)$.
3. Oct 13, 2015
### maverick_76
Okay so solving the integral of modulus psi squared, I find
A^2 + B^2 = (1/pi)
How exactly do I find A & B values with this, If I can't assume one is zero? I know:
psi(0) = B = psi(2pi)
&
psi'(0) = Ak = psi'(2pi)
but I am rusty here, can I use this info to solve for A & B?
4. Oct 14, 2015
### The_Duck
Any values for $A$ and $B$ that satisfy $A^2 + B^2 = 0$ give a solution. For each $k$ there is an infinite family of solutions. You can think of them all as linear combinations of the "basic" solutions $\sin(kx)$ and $\cos(kx)$.
5. Oct 14, 2015
### maverick_76
How is that possible? If k=n, n=1,2,3.... then it has to equal 1/sqrt(pi) right?
A^2 + B^2 = 1/sqrt(pi) is what I mean, how can it equal 0???
6. Oct 15, 2015
### vanhees71
It's much easier to find the energy eigenstates in terms of exponential rather than cos and sin functions. The reason that you have this "ambiguity" is that the energy eigenvalues are degenerate.
The Hamiltonian in position representation is
$$\hat{H}=-\frac{\hbar^2}{2m R^2} \partial_{\varphi}^2.$$
$$\hat{H} u_E(\varphi)=E u_E(\varphi) \; \Rightarrow \; \partial_{\varphi}^2 u_E(\varphi)=-{2m R^2 E}{\hbar^2}u_E(\varphi)=-\alpha^2 u_E(\varphi).$$
The appropriate square integrable solutions on $L^2([0,2 \pi])$ are
$$u_E(\varphi)=A \exp(\mathrm{i} \alpha \varphi).$$
Since the function must be periodic with period $2 \pi$ you have
$$\alpha=\alpha_k:=2 \pi k, \quad k \in \mathbb{Z}.$$
The corresponding energy eigenvalues are
$$E_k=\frac{\hbar^2 \alpha^2}{2m R^2}=\frac{\hbar^2}{2mR^2} (2 \pi k)^2.$$
As you see, except for the ground state for $k=0$ all energy eigenstates are degenerate, because for any $k$ also $-k$ gives the same energy.
From the theory of Fourier series, we know the above set of eigenstates
$$u_k(\varphi)=\frac{1}{\sqrt{2 \pi}} \exp(\mathrm{i} \alpha_k \varphi).$$
are complete, and thus this is the complete solution of the energy-eigenvalue problem. It's also easy to verify that the Hamiltonian is really self-adjoint.
7. Oct 15, 2015
### maverick_76
Thank you, I was talking with my professor this morning and he said the same thing that tackling the problem with exponentials is much easier than using trig functions. The problem also is much clearer to understand this way too. | 2018-01-17T11:20:20 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/particle-in-a-ring.837560/",
"openwebmath_score": 0.8711293935775757,
"openwebmath_perplexity": 397.4523081404565,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806506825456,
"lm_q2_score": 0.8519528038477825,
"lm_q1q2_score": 0.8354084347478777
} |
https://math.stackexchange.com/questions/2563838/prob-11-sec-23-in-munkres-topology-2nd-ed-if-p-colon-x-to-y-is-a-quotie | Prob. 11, Sec. 23, in Munkres' TOPOLOGY, 2nd ed: If $p\colon X\to Y$ is a quotient map, $p^{-1}(\{y\})$ is connected, and $Y$ is connected, then
Here is Prob. 11, Sec. 23, in the book Topology by James R. Munkres, 2nd edition:
Let $p \colon X \to Y$ be a quotient map. Show that if each set $p^{-1}(\{ y \} )$ is connected and if $Y$ is connected, then $X$ is connected.
My Attempt:
By definition, $p$ is a surjective map of the topological space $X$ onto the topological space $Y$, such that, for each set $V \subset Y$, the set $p^{-1}(V)$ is open in $X$ if and only if $V$ is open in $Y$.
Now suppose that $X$ is not connected. Then, by definition, we can write $X = C \cup D$, where the sets $C$ and $D$ are non-empty, disjoint, and open in $X$; that is, let the sets $C, D$ constitute a separation of $X$.
Let $y \in Y$. As the set $p^{-1}(\{ y \})$ is connected, so by Lemma 23.2 in Munkres, we must have either $p^{-1}(\{ y \}) \subset C$ or $p^{-1}(\{ y \}) \subset D$, but not both since $C \cap D = \emptyset$.
In particular, for every point $x \in X$, as $p(x) \in Y$, so we must have either $p^{-1}( \{ p(x) \} ) \subset C$ or $p^{-1}( \{ p(x) \} ) \subset D$, but not both.
Am I right?
Let $U$ and $V$ be the following subsets of $Y$. $$U \colon= \{ \ y \in Y \ \colon \ p^{-1}(\{ y \}) \subset C \ \}, \qquad V \colon= \{ \ y \in Y \ \colon \ p^{-1}(\{ y \}) \subset D \ \}.$$ Then $U$ and $V$ are disjoint sets, because $C$ and $D$ are disjoint.
Am I right? Is the very last assertion in need of further elaboration?
Moreover, since $p$ is surjective and since $C \cup D = X$, we can conclude that $Y = U \cup V$.
Am I right? If so, then do I need to be any more explicit as to why this is so?
We now show that $p^{-1}(U) = C$ and $p^{-1}(V) = D$.
Let $x \in p^{-1}(U)$. Then $p(x) \in U$, and so $p^{-1}( \{ p(x) \} ) \subset C$, by the definition of set $U$. But as $x \in p^{-1}( \{ p(x) \} )$, so $x \in C$ also. Thus we have $p^{-1}(U) \subset C$.
Is my reasoning correct?
Conversely, suppose that $x \in C$. As in the fourth paragraph of this proof, we can conclude that either $p^{-1}( \{ p(x) \} ) \subset C$ or $p^{-1}( \{ p(x) \} ) \subset D$, but not both. But $x \in C \cap p^{-1}( \{ p(x) \} )$. So we must have $p^{-1}( \{ p(x) \} ) \subset C$, which implies that $p(x) \in U$ and hence that $x \in p^{-1}(U)$. Thus $C \subset p^{-1}(U)$.
Is this part of my reasoning correct and clear enough too?
From the preceding two paragraphs, we can conclude that $C = p^{-1}(U)$. Similarly, we can also show that $D = p^{-1}(V)$.
Now as $C = p^{-1}(U)$ and $D = p^{-1}(V)$ are open in $X$ and as $p$ is a quotient map, so the sets $U$ and $V$ are open in $Y$.
As $p$ is surjective, so we have $$p(C) = p\left( p^{-1}(U) \right) = U,$$ and similarly also $p(D) = V$.
Now as $C$ and $D$ are assumed to be non-empty and as $p(C) = U$ and $p(D) = V$, so both $U$ and $V$ are non-empty as well.
Thus we have shown that $U$ and $V$ are non-empty disjoint open sets in the topological space $Y$ such that $Y = U \cup V$. Thus the sets $U, V$ constitute a separation of $Y$, which contradicts our hypothesis that $Y$ is connected.
Thus our supposition that $X$ is not connected is wrong. Hence $X$ is connected.
Is this proof correct? If not, then at which point(s) is (are) there problems therein?
• To me, everything seems to be perfectly clear and correct. – Peter Elias Dec 12 '17 at 21:06
• It all looks good, but wow, that's a lot to write! The key idea is to write $X=C\cup D$, as you did. Then the crux is to show $p(C)$ and $p(D)$ are open in $Y$. This follows by noting (as you did) that, if $y\in p(C)$, then $p^{-1}(y)\subset C$. That is, $C$ is saturated, and hence $p(C)$ is open. $Y=p(C)\cup p(D)$ and you're done. – Steve D Dec 12 '17 at 21:15
I really enjoyed reading this! This has a good level of detail.
The only part that I got stuck was where you said
Moreover, since $p$ is surjective and since $C \cup D = X$, we can conclude that $Y = U \cup V$.
This sentence alone is not enough to justify the conclusion.
Based on the definitions of $U,V$ you want to establish that for every $y\in Y$, either $p^{-1}(\{y\})\subseteq C$ or $p^{-1}(\{y\})\subseteq D$. But you already showed this in a previous part of your post!
So in other words, you should have said
Moreover, we already showed that if $y\in Y$ then $p^{-1}(\{y\})\subseteq C$ or $p^{-1}(\{y\})\subseteq D$. So, if $y\in Y$ then $y\in U\cup V$ and therefore $Y = U\cup V$.
An alternative way (more category theory in taste): suppose $f: X \to \mathbf{2}:= \{0,1\}$ is continuous (where the latter set has the discrete topology).
For any $y \in Y$, $p^{-1}[\{y\}]$ is connected and so $f[p^{-1}[\{y\}]]$ is either $\{0\}$ or $\{1\}$ (these are the only connected subsets of $\mathbf{2}$). We define $\tilde{f}(y)$ to be that value.
Note that $\tilde{f} \circ p = f$ by definition and so by the standard characterisation of quotient maps by composition properties, $\tilde{f}$ is continuous, as $f$ is, and so $\tilde{f}$ is a constant map as $Y$ is connected. But then $f$ is also a constant map and thus $X$ is connected.
Your argument is fine too, and is a more verbose version of what I wrote above, if you think about it. The idea is the same. | 2019-11-17T00:12:33 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2563838/prob-11-sec-23-in-munkres-topology-2nd-ed-if-p-colon-x-to-y-is-a-quotie",
"openwebmath_score": 0.9903060793876648,
"openwebmath_perplexity": 49.110178386885885,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806523850543,
"lm_q2_score": 0.8519528019683106,
"lm_q1q2_score": 0.8354084343553609
} |
https://math.stackexchange.com/questions/2344303/calculating-flux-of-pyramid | # Calculating flux of pyramid
I know I've got a mistake, probably with question a. But I don't know where or what the mistake is.
The question is
Let $D\subset\mathbb{R}^3$ be the pyramid with vertices $(0,0,0)$, $(1,0,0)$, $(0,1,0)$, $(0,0,1)$.
Given is the vector field $\vec{F} = xy\hat{i}+y^2\hat{j}+zy\hat{k}$.
Calculate the flux of $\vec{F}$ through $D$.
a) Directly. b) Using divergence theorem.
My solution of a:
Since we have a pyramid of four surfaces, we shall calculate the flux through each separately. Let $S_1$ denote the surface on the $xy$-plane. $S_2$ on the $xz$-plane. $S_3$ on the $zy$-plane. And $S_4$ in the $xyz$-space.
Let the vector field be in the form $\vec{F}=P\hat{i}+Q\hat{j}+R\hat{k}$, then I calculate the flux by $$\int\int_D \left(-P\cdot g_x(x,y) -Q\cdot g_y(x,y) + R\right)\,dA$$
The flux of $\vec{F}$ through $S_1$ is given by $$\int_{x=0}^{1}\int_{y=0}^{1-x}\left((-xy)\cdot g_x-y^2\cdot g_y + zy\right)\,dydx =\int_{x=0}^{1}\int_{y=0}^{1-x}\left(xy+y^2 + y -xy -y^2\right)\,dydx$$, with $g(x,y)=1-x-y$ for $x+y=1$. If I calculate this integral I get $1/6$.
The flux of $\vec{F}$ through $S_2$ is given by $$\int_{x=0}^{1}\int_{z=0}^{1-x}\left((-xy)\cdot g_x+y^2 + zy\cdot g_z\right)\,dzdx$$, with $g(x,z)=1-x-z$ for $x+z=1$. If I calculate this integral I get $1/6$.
The flux of $\vec{F}$ through $S_3$ is given by $$\int_{y=0}^{1}\int_{z=0}^{1-y}\left((xy+y^2\cdot g_y + zy\cdot g_z\right)\,dzdx$$, with $g(y,z)=1-y-z$ for $y+z=1$. If I calculate this integral I get $1/6$.
The flux of $\vec{F}$ through $S_4$ is given by $$\int_{x=0}^{1}\int_{y=0}^{1}\left((-xy)\cdot g_x-y^2\cdot g_y + zy\right)\,dydx$$, with $g(x,y)=1-x-y$. If I calculate this integral I get $1/2$.
The total flux is then the flux through all the separate surfaces added together, we get a flux of $1$.
My solution of b:
Since the surface is simple and closed, we get $$\int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 Div(\vec{F})dz\,dy\,dx$$ with the divergence equal to $y+2y+y=4y$. Hence the flux is $2$.
• For (b), the integration limits for the pyramid are wrong. What you have now gives a cube. I believe you need $\int_{0}^1 \int_0^{1-x} \int_0^{1-x-y}$. – aras Jul 2 '17 at 19:49
• I see I set my bounds with question b wrong, there they form a cube with side lengths 1. I should have z go from 0 to 1-y-x. y from 0 to 1-x. And x from 0 to 1. Right? – PaRaXeRoX Jul 2 '17 at 19:51
• What aras said. Hadn't seen your comment yet. But still the result would be different to a. I'd get a flux of 1/6. – PaRaXeRoX Jul 2 '17 at 19:58
Your problem is the formula for the flux. To calculate the flux, you don't need the curl, just use the surface integral
$$\int\int_D F\cdot n\ dA$$
where $n$ is the normal vector to the surface $D$.
When $D$ is $S_1$, the integral becomes $$\int_0^1 \int_0^{1-x} zy \ dy\ dx$$
This is because $S_1$ lies in the $xy$-plane, so $n = (0,0,1)$. So $F\cdot n = zy$. Since $z=0$ on $S_1$, this is just $0$.
Similarly: When $D$ is $S_2$, the corresponding integral is
$$\int_0^1 \int_0^{1-x} y^2\ dz \ dx .$$
Snince $y=0$ on $S_2$, the integral has value $0$.
When $D$ is $S_3$, the corresponding integral is
$$\int_0^1 \int_0^{1-y} xy\ dz\ dy.$$
Since $x=0$ on $S_3$, the integral has value $0$.
When $D$ is $S_4$, the corresponding integral is
$$\int_0^1 \int_0^{1-x} xy + y^2 + (1-x-y)y\ dy \ dx .$$
Here, the normal vector is $(1,1,1)$. So the dot product $F \cdot n = xy + y^2 + zy$. Notice that on $S_4$, we can substitute $z = 1-z-y$. The value of this integral is $\frac{1}{6}$.
Thus, the total flux of the pyramid is $0+0+0+\frac{1}{6} = \boxed{\frac{1}{6}}$.
As for part (b), the correct integral limits should be
$$\int_0^1 \int_0^{1-x} \int_0^{1-x-y} 4y\ dz\ dy\ dx$$
and this evaluates to $\boxed{\frac{1}{6}}$.
• Of course! That makes it clear, thank you! – PaRaXeRoX Jul 2 '17 at 20:45
• @PaRaXeRoX No problem! I'm actually currently reviewing Calc 3 for an exam, so this was a great way for me to brush up on my surface integrals :) – aras Jul 2 '17 at 20:47
• Awesome, I'm preparing an exam too, though mine is calc 3 hehe :) – PaRaXeRoX Jul 2 '17 at 20:51 | 2019-10-15T18:41:28 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2344303/calculating-flux-of-pyramid",
"openwebmath_score": 0.9817090034484863,
"openwebmath_perplexity": 254.59837007127194,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806523850543,
"lm_q2_score": 0.8519528019683105,
"lm_q1q2_score": 0.8354084343553608
} |
https://zlsuzb.burgerfi8518.site/cos-a-cos-b.html | ## Cos a cos b
cos a cos b 3 to convert the A and B back to x and y: which is Equation 2. Learn about the relationship between the sine & cosine of complementary angles, which are angles who together sum up to 90°. cos (+ β) = cos cos β − sin sin β, cos(u v) = cosucosv sinusinv tan(u v) = tanu tanv 1 tanutanv Double Angle Formulas sin(2u) = 2sinucosu cos(2u) = cos2 u sin2 u = 2cos2 u 1 = 1 22sin u tan(2u) = 2tanu 1 tan2 u Power-Reducing/Half Angle For-mulas sin2 u= 1 cos(2u) 2 cos2 u= 1+cos(2u) 2 tan2 u= 1 cos(2u) 1+cos(2u) Sum-to-Product Formulas sinu+sinv= 2sin u+v 2 cos u v 2 sinu sinv n (a + BI) = (COs(b ln n) + i sin(b ln n))n a if z = r(COs + i sin ) then z n = r n ( COs n + i sin n )(DeMoivre's Theorem) if w = r(COs + i sin );n=integer, then there are n complex nth roots (z) of w for k=0,1,. The Pythagorean identity for 6) tg A - tg b = sin (A-B)/(cos A cos B). Since cos (α) = b/c, from this definition it follows that the cosine of any angle is always less than or equal to one, and it can take negative values. 3333 14) tan A 21 20 29 A B C 1. Eis as fórmulas, já conhecidas de vocês, assim espero. 8000 12) cos C 36 27 45 C B A 0. There is no trig identity cos A - cos B. Dividindo todos os 4 termos da fração por cos(a)cos(b), segue a fórmula:. 11. Cos^A+Cos^B+Cos^C=1-2CosA. cos = sin + sin: 2 cosA sinB = sin(A + B) - sin(A - B) 2 cos. 00. →v=cosacosb+sinasinb. The question is A particular solution of the differential equation y′′ − 2y′ + 2y = cos x has the form yp = A cos x + B sin x. If the lengths of two sides of the triangle a and b are equal to x, the third side has length c, and the angles opposite the sides of lengths a, b, and c are A, B, and C respectively then Visalia Campus 915 S. Plug into previous equation: cos a + cos b = 2 cos((a+b)/2) cos((a-b)/2) QED. 2016 Math Secondary School Find the Exact Value cos(pi/12) Split into two angles where the values of the six trigonometric functions are known. Ex 8. + 1. 3429 16) tan X 40 30 X 50 Y Z 0. 27 Oct 2020 Cos( a+b ) / cos(a) cos(b). The period of the function is 360° or 2π radians. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Similarly (15) and (16) come from (6) and (7). e. tg ctg = 1: 4. Multiply and divide by sin(a-b) 1/sin(a-b). We seek the angle or angles we see in a triangle or I guess we know in a triangle there's three angles and if we talk about a right triangle like the one that I've drawn here one of them is going to be a right angle and so we have two other angles to deal with and what I want to explore in this video is the relationship between the sine of one of these angles and the cosine of the other the cosine of one of these angles 18 Abr 2010 cos(a + b) = cos(a)cos(b) – sen(b)sen(a). Các hệ thức cơ bản và hệ quả 3t ga - t g 3a cot g 3a - 3 cot ga 3/ t g3a a+ b a- b 4/ sin a - sin b = 2 cos . − x cos ax a. Ptolemy's theorem implies the theorem of Pythagoras. We have a function of the form \[y = The domain of the cosine function. 2 and 2. If the vectors a and b have magnitudes a and b respectively, and if the angle between them is \theta, then the scalar product of a and b is defined to be sin B = 12/13, maka cos B = 5/13 A + B + C = 180°, (jml sudut -sudut dalam satu segitiga = 180) A + B = 180 – C sin (A + B) = sin (180 – C) sin A . (A – B) = tan A – tan B/1 + tan A dikali tan B The Law of Cosines relates all three sides and one of the angles of an arbitrary (not necessarily right) triangle: c 2 = a 2 + b 2 - 2ab cos (C). Vectors: The Scalar Product. Sean a y b ángulos del primer cuadrante, vamos a ver que: sen(a+b)=sen(a)cos(b)+cos(a)sen(b) La restricción no quita generalidad a la fórmula pues siempre podemos reducir los ángulos del segundo, tercer y cuarto cuadrante al primero. Lets do some Hit and TrialB=90A=30C=60Then Cos A+ Cos C is not Equal to Sin BWhen we mention that ABC is a right angle triangle then B=90Hence we should mention that either ACB is a right angle triangle or CAB is right angle triangle. Here, you learn how cos of sum of two angles formula is derived in geometric method. The calculator will find the inverse cosine of the given value in radians and degrees. This allows us to simplify the expression further. By replacing b as –b, we have: 저번 포스팅에 이어서 이번 포스팅에서는 cos에 대한 덧셈정리에 대해 알아 봅시다. First two frames use definition of cos and sin (circle is unit circle) 3rd frame: opposite angles are equal, Angles of a triangle add up to 180 therefore top angle of upper right triangle is Beta cos (A + B) = cos A cos B - sin A sin B. tg p ± tg q = sen Because, because of and cos, cos of - English Grammar Today - a reference to written and spoken English grammar and usage - Cambridge Dictionary sin A, cos A: sin 2 A + cos 2 A = 1: sin 2 A = (1 - cos 2A)/2: sin A = 1 / cosec A & sin A = cos A tan A: sin (A+B) = sin A cos B + cos A sin B sin (A-B) = sin A cos Proof: The half-angle formulas for sine and cosine are found immediately from the power-reducing formulas by substitution and square root. = sin A . cos(A + B) = cos A cos B – sin A sin B 2. We all learn the expansion and See proof below We need (x+y)(x-y)=x^2-y^2 cos(a+b)=cosacosb-sina sinb cos(a-b)=cosacosb+sina sinb cos^2a+sin^2a=1 cos^2b+sin^2b=1 Therefore, LHS=cos(a+b)cos(a-b Sum / Difference of Angles Formulas. 이것을 하기 전에 6장 7장에서 했던 . = sin A . Expert Answer: cos 2A cos2 A = 1. ∫ xsinax dx = sin ax a2. Product-to-sum trigonometric formulas can be very helpful in simplifying a trigonometric expression by taking the product term ((such as \sin A \sin B, \sin A \cos B, sinAsinB,sinAcosB, or \cos A \cos B) cosAcosB) and converting it into a sum. From Euler’s formula this becomes an algebra problem with an easy calculus part, as In the illustration below, cos (α) = b/c and cos (β) = a/c. The main idea is to create a triangle whose angle is a difference of two other angles, whose adjacent sides, out of simplicity, are both 1. Trigonometric Identities. cos(b) + sen(a). p. cos(A – B) = cos A cos B + sin A sin B 3. A proof that cos (A − B) = cosAcosB + sinAsinB. sin 2θ = 2 sin θ cos θ, . This website uses cookies to improve your experience, analyze traffic and display ads. , Hanford, CA. Use this online trigonometric identities addition calculator to find the sum of cosine angles. 93230 559-583-2500 Tulare College Center 4999 East Bardsley Avenue, Tulare, CA. Follows from l a = 2bc cos(A/2)/2 and cos²(A/2) = p(p - a) / bc. If cos A = 12/13, then sin A = √(1 - cos 2 A) = √(1 - (12/13) 2) = √(1 - (144/169)) = √(169 - 144)/169 = √(25 Returns the cosine of the given angle. cos (A - B) = cos A cos B + sin A sin B. cos p + cos q = 2cos((p + q)/2)cos((p - q)/2). 12/13 sin C = 15/65 + 48/65 = 63/65. For example, $\cos{(A+B)}$, $\cos{(x+y)}$, $\cos{(\alpha+\beta)}$, and so on. Sen (a + b) = sen(a). sin (B – A). The sin β leg, as hypotenuse of another right triangle with angle α , likewise leads to segments of length cos α sin β and sin α sin β . sinb sin(a- b) = sina. For cosine, we have a similar diagram: This time, the conversion factor matches up (cosine with cosine, sine with sine). Compared to y=cos(x), shown in purple below, the function y=2 cos(x) (red) has an amplitude that is twice that of the original cosine graph. Possible hint: eigenvectors. For practice, I will have students find all six trig values for 7pi/12 and all six trig values for 255 degrees. SenB: a frase ficar coça A, coça B + senta 1 Mai 2015 O valor de y = (cos a + cos b)² + (sen a - sen b)², para a+b = /2, é: a) 1/2 b) 2 c) 0 d) 1 e) 4. cos(a – b) = cos(a)cos(b) + sin(a)sin(b) Proofs of these formulas are available in all trig and pre-calculus texts. (10) Suppose we wanted an identity involving sinAsinB. Cosine calculator online. 615) = 1 that is cos(x −0. add RHS: 2 cos A cos B (1/2) [2 cos A cos B] = cos A cos B. b b) Let A = Compute e A. . Asked by vikassuradkar2015 | 8th Dec, 2019, 11:11: PM. l a = 2bc cos(A ©2005 BE Shapiro Page 3 This document may not be reproduced, posted or published without permission. Ptolemy cũng đã suy diễn ra được công thức nửa-góc sin( A /2) 2 = (1 − cos( A ))/2, cho phép ông lập bảng tính với bất cứ độ chính xác Given Triangle abc, with angles A,B,C; a is opposite to A, b oppositite B, c opposite C: a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines) c 2 = a 2 + b 2 - 2ab cos(C) Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. In an acute triangle with angles $A, B$ and $C$, show that $\cos {A} \cdot \cos {B} \cdot \cos {C} \leq \dfrac{1}{8}$ I could start a semi-proof by using limits: as $A \to 0 , \; \cos {A} The full base line, divided by the dividing line between angles A and E, is cos A (2). 2) cos A = ( cateto adjacente a “A”)/(hipotenusa ). cosb - sina. us sign is because cos is negative in 2nd Quadrant) cos B = -5/13. For math, science, nutrition, history Therefore the cosine of B equals the sine of A. where A, B, and C are the angles opposite sides a, b, and c respectively. Because these numbers are not symbolic objects, cos returns floating-point results. Sine, cosine, secant, and cosecant have period 2π while tangent and cotangent have period π. Cosine Function for Numeric and Symbolic Arguments. sec ˇ 2 = cosec ; cosec ˇ 2 = sec : 8. Demonstrar fórmulas e teoremas é fundamental Fórmulas trigonométricas. We know the exact values 11) cos Z 12 9 Z 15 Y X 0. Equations of the form r = a + b sin θ, a – b sin θ, a + b cos θ, and a – b cos θ will produce limacons. {sin(a-b)/cos(x-a). tan A + tan B + tan C = tan A tan B tan C อสมการ sin, cos, tan สําหรับ x1, x2, , xn ∈ (0, 2 π) จะได (ศึกษาเพ ิ่มเติม บทที่ 10. cosb- cosa. Upcoming Addition et soustraction d'angles. cos(b) – sen(a). n sinx1 +sinx2 + +sinxn ≤ sin n x1 +x2 + +xn 2. b), cos alpha cos β. If A,B,C are the angles of a triangle then cosA+cosB+cosC=? Trigonometry. The cosine rule - Higher. For any angles A and B: sin (A ± B) = sin A cos B ± cos A 8 Ago 2020 Então, o macete é decorar algumas frases dentro das fórmulas, como: Cos (A – B ) = cosA. = cos A cos B + sin A sin B. Sine or Cosine of a Double Angle. Now, for the little part that has to be subtracted. You noticed that the equation c 2 = a 2 + b 2 – 2bc cos (C) resembles the Pythagorean Theorem, except for the last terms,” – 2bc cos (C). cos(a+b)= cosa cosb−sena senb cos(+☼)= cos cos☼−sen sen (a + b) = sen a * cos b + cos a * sen b sen (a – b) = sen a * cos b Exemplo 1 sen 105º = sen (60º + 45º) = sen 60º * cos 45º + cos 60º * sen 45º = Exemplo 2 We may assume α≤β≤γ. 3243 18) sin Z 30 40 50 Y X 0. same denominator, so we can factor it out: Seno de la suma de dos ángulos. 2 sinA cosB = sin(A + B) + sin(A − B). n-1: If sin A = , 0 < A < and cos B = , 7) sin (A + B) 8) cos (A – B) 9) tan (A + B ) Algebra 3 Trig Formulas Assignment #6 (1) Find each of the following numbers please cos A - cos B = - 2 sin [ (A + B) / 2 ] sin [ (A - B) / 2 ] Product of Trigonometric Functions Formulas. We can find one by slightly modi-fying the last thing we did. The inverse cosine y=cos^(-1)(x) or y=acos(x) or y=arccos(x) is such a function that cos(y)=x. cosb + cosa. 1. Compute the cosine function for these numbers. cos(x+y) = cos x cos y - sin x sin y. This is an immediate consequence of r a r b r c = pS and rp = S. cos p - cos q = - 2cos((p ± q)/2)cos((p - q)/2). cos(a). c 2 = a 2 + b 2 − 2 a b cos C b 2 = c 2 + a 2 − 2 c a cos B a 2 = b 2 + c 2 − 2 b c cos A COMING SOON – Proof, examples and applications The Great Trigonometric Survey The variable b in both of the following graph types affects the period (or wavelength) of the graph. This (above) is an identity, true for all x and y. Construction The$\Delta EDF$is a right triangle and the angle of this triangle is divided as two angles. Also, we can rewrite the c 2 = a 2 + b 2 − 2ab cos(C) formula into a 2 = and b 2 = form. So the equation √ 2cosx +sinx = 1 becomes √ 3cos(x− 0. To derive (c), write. (cosacosb - sina sinb) / cosa cosb . Learn more trigonometry formulas at BYJU'S. Lets examine what happens for various values of a and b. 2 cos 2A. You mentioned that ABC is right angle triangle. . sinb. I've tried expanding the left side, making the right side to a difference of squares, subbing the right side with sin's, dividing, factoring and all that Essentially, it says$\sin(x)$is a line for a brief time period. sinb sin(a+ b)= sina. →v=1×1×cos(a−b)=cos(a−b). 1. Example 2. For small angles,$\sin(a + b) \sim \sin(a) + \sin(b) \sim a + b$. Voir Démo. Contoh soal 2. Question 809535: If sinA=2/3 and CosB=3/5 and . ∫. →v=||→u||×||→v ||×cos(→u;→v). senB Dica: YoYô, Mixoxô Seno, cosseno e tangente CÁC CÔNG THỨC LƯỢNG GIÁC CẦN NHỚI. cos(x) calculator. . The product p:=cosαcosβcosγ is negative iff γ>π2, so that the minimum value pmin=−1 is attained when α=β=0, γ=π, i. B —used to determine the period of the function; the period of a function is the distance from peak to peak (or any point on the graph to the next matching point) and can be found as . sin 2 + cos = 1: 9. TBD. cosA+cosB+cosC+Cos (A+B+C) remember A+B+C= pi. →u. If A+B+C=π then prove that. Ptolemy’s identities, the sum and difference formulas for sine and cosine. Double angle formulas for sine and cosine. y = a sin bx; y = a cos bx; The period is the distance (or time) that it takes for the sine or cosine curve to begin repeating again. Creamy corn, zucchini, peas, herbs and our secret ingredient made with TLC. Sine, tangent, cotangent, and cosecant are odd functions while cosine and secant are even functions. senb cos(a + b) = cosa . ) Therefore, the left-hand side of the given equation can be expressed in the form √ 3cos(x−0. It is an even function. 6293 Critical Compared to y=cos(x), shown in purple below, the function y=2 cos(x) (red) has an amplitude that is twice that of the original cosine graph. Relembrando: seno é uma função ímpar e cosseno é uma função par. cos B + cos A. 2. to find missing angles and sides if you know any 3 of the sides or angles. cos A = (Side adjacent of angle A) / Hypotenuse = AC/AB Similarly, cos B = (Side adjacent of angle A) / Hypotenuse = BC/AB Now it is given that cos A = cos B AC/AB = BC/AB AC = BC We know that in a triangle, angles opposite equal sides are equal B = A Hence proved. Therefore, on exchanging sides,. Formula (b) is derived in exactly the same manner, only instead of adding, subtract sin (− β) from sin (+ β). We saw on the last page that sin A was the opposite side over the hypotenuse, that is, a/c. CosB. Sinus. cosb + sina. Cosine rules Sau đó, Ptolemy (thế kỷ II) tiếp tục phát triển công trình trên trong quyển Almagest, tìm ra công thức cộng và trừ cho sin(A + B) và cos(A + B). 4848 22) cos 51° 0. sin B = sin C, (ingat sudut yang saling berelasi : sin(180-x) = sin x) sin C = sin A. So, the full base line divided by the hypotenuse is the product cos A cos B (4). Any ideas? If you set sinAcosB + sinBcosA = sinA + sinB, this will be true if cosB = 1 and cosA = 1, which means that A and B can be 0, pi, 2pi, etc. Formulas (c) and (d) are derived similarly. General Cosine Function We now explore interactively the general cosine function $$f(x) = a \cos(b x + c) + d$$ sin(a −b) = sinacosb−sinbcosa sin(2a) = 2sinacosa tan(a +b) = tana+tanb 1 −tanatanb tan(2a) = 2tana 1 −tan2 a tan(a −b) = tana−tanb 1 +tanatanb Formules de linéarisation cosacosb = 1 2 (cos(a −b)+cos(a+b)) cos2 a = 1 +cos(2a) 2 sinasinb = 1 2 (cos(a−b)−cos(a +b)) sin2 a = 1 −cos(2a) 2 sinacosb = 1 2 (sin(a+b)+sin(a−b a2 +b2 (eat cos bt+ieat sin bt)+C = a a2 +b2 eat cos bt+ b a2 +b2 eat sin bt)+C1 + i(¡ b a2 +b2 eat cos bt + a a2 +b2 eat sin bt+C2): Another integration result is that any product of positive powers of cosine and sine can be integrated explicitly. Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. sinb sin(a- b) = sina. 2cosA sinB = sin(A+B)−sin(A−B) 2cosA cosB = cos(A+B)+cos(A−B) 2sinA sinB = cos(A−B)−cos(A+B) Hyperbolic Functions sinhx = ex −e−x 2, coshx = ex +e−x 2 Standard Derivatives f(x) f0(x) x nnx −1 sinax acosax cosax −asinax tanax asec2 ax e axae lnx 1 x sinhax acoshax coshax asinhax uv u0 v +uv0 u v u0 v −uv0 v2 Standard I have been trying to work with cos(A+B) and see if there are instances when cos(A+B) would equal cosA+cosB. sin ˇ 2 = cos ; sin(ˇ ) = sin : 5. The sine and the cosine functions, for example, are used to describe simple harmonic motion, which models many natural phenomena, such as the movement of a mass attached to a spring and, for small angles, the pendular motion of a mass hanging by a string. cos A − cos B −2 sin ½ ( A + B ) sin ½ ( A − B ) In the proofs, the student will see that the identities e) through h) are inversions of a) through d) respectively, which are proved first. Podemos concluir b&={\frac {\cos(a-b)-\cos(a+b)}{2}}\\[6pt]\cos a\cos b&={\frac {\cos(a-b)+\cos(a+b )}{2}}\\[6pt]\sin a\cos b&={\frac {\sin(a+b)+\sin(a-b)}{2}}\\[6pt]\cos a\sin b&={\frac segunda, obtemos: tan(a+b)=sen(a)cos(b)+cos(a)sen(b)cos(a)cos(b)−sen(a) sen(b). After adding and subtracting x in the numerator you get sin((x-b)-(x-a)) Now apply sin(a-b) formula which is sin a cos b - cos a sin b. Calculus Find the velocity, v(t), for an object moving along the x-axis in the acceleration, a(t), is a(t)=cos(t)-sin(t) and v(0)=3 a) v(t)=sin(t) + cos(t) +3 b) v(t)=sin(t) + cos(t) +2 c) v(t)= sin(t) - cos(t) +3 d) v(t)= sin(t sin A, cos A: sin 2 A + cos 2 A = 1: sin 2 A = (1 - cos 2A)/2: sin A = 1 / cosec A & sin A = cos A tan A: sin (A+B) = sin A cos B + cos A sin B sin (A-B) = sin A cos B cos 2 C 2. Sine, Cosine, and Ptolemy's Theorem. Посмотреть ответы. 8. The mission was a In other words, the cosine of an angle in a right triangle equals the adjacent side divided by the hypotenuse: Also, cos A = sin B = b/c. A+ B 2 cos A B 2 (13) cosA cosB= 2sin A+ B 2 sin A B 2 (14) sinA+ sinB= 2sin A+ B 2 cos A B 2 (15) sinA sinB= 2cos A+ B 2 sin A B 2 (16) Note that (13) and (14) come from (4) and (5) (to get (13), use (4) to expand cosA= cos(A+ B 2 + 2) and (5) to expand cosB= cos(A+B 2 2), and add the results). 1) sin A = ( cateto oposto a “A”)/(hipotenusa). Answer to: If A + B + C = 180 degrees, prove that cos 2A + cos 2B - cos 2C = -1 - 4 cos A cos B cos C. (A − B) eax(a cos bx+b sin bx) a2+b2. e. 07. Figure 1: {\bf a} \cdot{\bf b} = ab\cos\theta. Pengertian, macam - macam, rumus, dan contoh soal beserta pembahasannya cos (A+B)=cos (A)*cos (B)-sin (A)*sin (B) cos (A-B)-cos (A+B)= cos (A)*cos (B)+sin (A)*sin (B)- (cos (A)*cos (B)-sin (A)*sin (B))= 2*sin (A)*sin (B) Click here👆to get an answer to your question ️ If xy = cos Acos B then xtanA + ytanBx + y = Click here👆to get an answer to your question ️ If sin (A - B) = sin A cos B - cos A sin B and cos (A - B) = cos A cos B + sin A sin B , find the values of sin 15^∘ and cos 15^∘ . 3] Cosine Rule (Proof) The Cosine Rule is: [3. Identities for negative angles. Estratégias de ensino-aprendizagem. From Ramanujan to calculus co-creator Gottfried Leibniz, many of the world's best and brightest mathematical minds have belonged to autodidacts. It’s useful in deriving the cosine of sum of two angles trigonometric cos(x+ y) = cosxcosy sinxsiny cos(x y) = cosxcosy+ sinxsiny tan(x+ y) = tanx+tany 1 tanxtany tan(x y) = tanx tany 1+tanxtany Half-Angle Formulas sin 2 = q 1 cos 2 cos 2 = q 1+cos 2 tan 2 = q 1+cos tan 2 = 1 cosx sinx tan 2 = sin 1+cos Double-Angle Formulas sin2 = 2sin cos cos2 = cos2 sin2 tan2 = 2tan 1 tan2 cos2 = 2cos2 1 cos2 = 1 2sin2 Product If we begin with the angle A, with coordinates on the unit circle of {eq}( \cos(A), \sin(A) ) {/eq} and add the angle B to it, we obtain the angle in the upper quadrant, with the coordinates of (cos(2A+B)+cos(2A−B)) With A = 4×10 6 πt and B = 400πt and substituting for the given frequencies, the modulated signal multiplied by the original carrier signal gives 2 cos A cos B = cos (A +B) +cos (A -B) 2 sin A sin B = cos (A -B) -cos (A +B) Need help proving the cos (a+b) = (cos a) (cos b)- (sin a) (sin b) trigonometric identity? This free video lesson will show you how. 사인 때와 마찬가지로 와 2개가 있죠 . cos(b) + sen(b). Mooney Blvd. r r a r b r c = S². cos(x-b) now add and subtract x in sin(a-b) such that you get back sin(a-b). 2 sin alpha cos β = sin ( alpha + β) + sin ( alpha h), cos A − cos B, = −2 sin ½ (A + B) sin ½ (A − B) Sine and Cosine of A ± B. It can be thought of as a generalized form of the pythagorean theorem. Observando o círculo trigonométrico da figura 1, notamos que: clip_image050. So LHS; =( cos A + cos B ) + cos C-1 = { 2 · cos[ ( A+B) / 2 ] · cos [ ( A-B Transformation formulae: Key to remember: 2sinA cosB = sin(A + B) + sin(A - B) 2 sin. 2 sin A cos B = sin (A + B) + sin (A - B) 12. The full width of the first triangle ($\cos(a)$) gets scaled down to match the width of the Helloo There is this problem with I am unfamiliar with as I have not seen one of this type before. The cosine of sum of angles a and b is equal to the subtraction of the product of sines of both angles a and b from the product of cosines of angles a and b. Estas dos fórmulas sirven para deducir las del seno y coseno de la diferencia de dos ángulos: Sen (A - B) = Sen A Cos B - Sen B Cos A. Example. sen(b). put the value of a =45° degree and b=30° degree put the value of a and b in the LHS cos (a+b) = cos (45°+30°) The full base line, divided by the dividing line between angles A and E, is cos A (2). sinb cos(a- b)= cosa. Sin and Cos formulas are given in this article. Tabla de valores del seno, coseno y tangente de los 5 май 2020 sin (a-B) + 2 cos a×sin B. 2. cos B – cos A . , Visalia, CA. sin B =Rumus untuk tangen yang memiliki jumlah selisih dua sudut yaitu: tan A . Cosine Function The cosine function is a periodic function which is very important in trigonometry. It is a special triangle in which one angle is 90° and the other two are less than 90°. We can see that the two equations are also right. b = 3" c = 5" cos α = b / c = 3 / 5 = 0. In just a few short steps, the formulas for cos(A + B) and sin(A + B) flow right from equation 47, Euler’s equation for e i x. Apply the difference of angles identity. Here are all three: a 2 = b 2 + c 2 − 2bc cos(A) b 2 = a 2 + c 2 − 2ac cos(B) c 2 = a 2 + b 2 − 2ab cos(C) But it is easier to remember the "c 2 =" form and change the letters as needed ! As in this example: 2 sin cos β = sin (+ β) + sin (− β), so that. But. What happens if you set B = A? sin(A + A) = sin A cos A + cos A sin ABut A + A is just 2A, and the two terms on the right-hand side are equal. This is the identity ). tg = sin cos ; ctg = cos sin : 3. 4) Use the various trigonometric identities. No more need to memorize which one has the minus sign and how all the sines and cosines fit on the right-hand side: all you have to do is a couple of substitutions and a multiply. 2. . In general, the distance between A(x 1, y 1) and B(x 2, y 2) is given by, cos A = 12/13, cos B = 3/5, and cos C = 63/65. The sine and cosine functions are one-dimensional projections of uniform circular motion. Let's Practice: What is the exact value of sin(105º)? We can use a sum angle formula noticing that 105º = 45º + 60º. 17 Aug 2005 Cos-B was ESA's first foray into producing a spacecraft with a single payload: a high-energy gamma-ray telescope. 먼저 에 대해 해보도록 할건데 . n-1: After $$2\pi$$, the values of $$\cos(\theta)$$ will repeat at the coterminal angles. The cos β leg is itself the hypotenuse of a right triangle with angle α; that triangle's legs, therefore, have lengths given by sin α and cos α, multiplied by cos β. 0500 15) tan C 35 12 37 B C A 0. cosB + senA. ) I. 1 Answer In Triangle ABC, CosA /a = CosB / b = Cos C / c and the side a = 2. 2. 8. 10 Jul 2020 Answer: = cosA cosB − sinA sinB cos(A − B) = cosA cosB + sinA sinB sin2 A + cos2 A = 1, sin 2A = 2 sinA cosA cos 2A = 2 cos2 A − 1=1 − 2 5 Jul 2020 É o triângulo que tem os três lados diferentes, todos com distintas medidas de comprimento. The Pythagorean identity for sines and cosines sin B = 12/13, maka cos B = 5/13 A + B + C = 180°, (jml sudut -sudut dalam satu segitiga = 180) A + B = 180 – C sin (A + B) = sin (180 – C) sin A . CosC. b a a n c) Let A be a symmetric n x n matrix with spectral decomposition A = į liviv]. 1 ∓ tan A tanB. Then Cos(A+B+C) = Cos (pi) = -1. cosb + sina. n sinx1sinx2 Solve your math problems using our free math solver with step-by-step solutions. cosB – senA. Depending on its arguments, cos returns floating-point or exact symbolic results. cosb - sina. 24 Out 2018 O seno, o cosseno e a tangente são medidas que estão dentro do campo das razões a) 1,8 km; b) 1,9 km; c) 3,1 km; d) 3,7 km; e) 5,5 km. B —used to determine the period of the function; the period of a function is the distance from peak to peak (or any point on the graph to the next matching point) and can be found as . Using the formula 2 cos A cos B = cos (A + B) + cos (A – B), = 3 [cos (x + 2x) + cos (x – 2x)] = 3 [cos 3x + cos (-x)] = 3 [cos 3x + cos x] To learn other trigonometric formulas Register yourself at BYJU’S. cos(a – b) = cosa . cos A + cos B + cos C = 1 + 4 sin 2 A sin 2 B sin 2 C 3. If the angle is in degrees, either multiply the angle by PI()/180 or use the RADIANS function to convert the angle to radians. cos (a - b) = cos a cos b + sin b sin a tan (a - b) = (tan a - tan b) / (1 + tan a tan b) Double Angle Relationships: sin 2 x = (1 - cos 2x) / 2 cos 2 x = (1 + cos 2x Given Triangle abc, with angles A,B,C; a is opposite to A, b oppositite B, c opposite C: a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines) c 2 = a 2 + b 2 - 2ab cos(C) A comprehensive list of the important trigonometric identity formulas. sin(A B) = sin(A)cos(B) cos(A)sin(B). As you drag the point A around notice that after a full rotation about B, the graph shape repeats. . sin B. sin a = 12/13,a lies in quadrant II,and cos B = 15/17, B lies in quadrant I. sinb sin(a+ b)= sina. For general a and b, we can use that , cos(-x)=cos(x), , and etc to reduce them to the above cases. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. cos (x-y) = cos x cos y + sin x sin y, add these equations: cos(x+y) + cos(x-y) = 2 cos x cos y. Modern, functional, considered designs; shop the latest women's clothing, shoes and accessories from our new arrivals at cosstores. cos B + cos A. Cosines Difference if a b c then prove that cos a cos b cos c 1 2cosacosbcosc - Mathematics - TopperLearning. Use these fundemental formulas of trigonometry to help solve problems by re-writing expressions in another equivalent form. h=b·sin A=a·sin B So, b/sin B=a/sin A Using a perpendicular from A to BC, we can show that b/sin B=c/sin C Hence we have the Sine Rule: [2. Figure 1: {\bf a} \cdot{\bf b} = ab\cos\theta. cos(A) = b/c. From the investigation above, we know that $$\cos (\alpha – \beta) e \cos \alpha – \cos \beta$$. 6157 21) cos 61° 0. 615) = 1 √ 3 This is very straightforward to solve. sin(b) cos(6) b = sin(t), this tells us how to exponentiate a rotation matrix. ) sin A = 8/17, tan B = 5/12, A and B in Q1 Draw the picture of angle A in the first quadrant Q1, and since the sine is opposite/hypotenuse or y/r, and since we are given sin A = 8/17, make y=8 and r=17, and calculate x using the Pythagorean theorem: x²+y² = r² x²+8² = 17² x² Solution: Using sin a sin B = 1/2 [ cos ( a - b) - cos ( a + b) ] we get, \ sin 55 0 sin 40 0 = 1/2 [ cos ( 56 0 - 40 0 ) - cos ( 55 0 + 40 0 )] = 1/2 [cos 15 0 - cos 95 0 ] (a+b)cos 1 2 (a−b) cosa−cosb= 2sin 1 2 (a+b)sin 1 2 (b−a) sinacosb= 1 2 (sin(a−b)+sin(a+b)) sinasinb= 1 2 (cos(a−b)−cos(a+b)) cosacosb= 1 2 (cos(a−b)+cos(a+b)) sin(a±b) = sinacosb±cosasinb cos(a±b) = cosacosb∓sinasinb Fourier series Fourier series of f(x) defined on [−L,−L]: 1 2 a0 + X∞ n=1 (an cos(nπx/L)+bn sin(nπ/L So that 3r+(r a + r b r c =4R(cos(A)+cos(B)+cos(C)). sin B sin C = 3/5. This dividing line, divided by the hypotenuse of (A + B) triangle, is cos B (3). l a = 4p(p-a)bc/(b+c)². cos(α)cos(++βπ/2) sin(22+[α)sin(βπ−/2)] 1 sin(α)sin(βπ/2 Click here 👆 to get an answer to your question ️ Prove that:cot A - cos A / cot A + cos A = cosec A - 1 / cosec A + 1 ne5h8armadivya ne5h8armadivya 13. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students. Cos (A - B) = Cos A Cos 19. 61° is excluded because that would necessarily give A + B + C > 180°. The simplest way to understand the cosine function is to use the unit circle. Demonstração da fórmula cos (a – b) utilizando distância entre dois pontos. (A + B) = tan A + tan B/1 – tan A dikali tan B. sin = sin - sin So, x=A+B, and y=A-B [2. We did the case a = 0) in class. The copyright holder makes no representation about the accuracy, correctness, or cos (30º + 45º) = cos 30º cos 45º - sin 30º sin 45º To find these trigonometric function values, draw a 30-60-90 triangle whose sides are 1, √ 3 , 2, and a 45-45-90 triangle whose sides are 1, 1, √ 2 . cos(b) / cos(a). sin (A + B). So, the full base line divided by the hypotenuse is the product co In trigonometry, the law of cosines(also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangleto the cosineof one of its angles. Use this fundamental cosine addition formula of trigonometry to solve various problems by re-writing expressions in another equivalent form. QED c 2 = a 2 + b 2 − 2 a b cos C b 2 = c 2 + a 2 − 2 c a cos B a 2 = b 2 + c 2 − 2 b c cos A COMING SOON – Proof, examples and applications The Great Trigonometric Survey Homework Statement Prove that: sin(a+b)sin(a-b) = cos^2b - cos^2a Homework Equations The Attempt at a Solution I'm stuck. sin B = 12/1 B’Cos delicious and freshly made deep fried Zucchini fritters following our traditional recipe. c2=a2+b2−2abcosγ,{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma,} Prove the trig identity. tan A . cos B + cos A . Hi Anushka, In any triangle, we have a/SinA = b/SinB = c/Sin Hence sin(a+b)=AE= DE+AD=sin(a)cos(b)+cos(a)sin(b). (6) as: Acos(ωt + α) + Bsin(ωt + β) = Acos(ωt + α) + Bcos(ωt + β – π/2) =−[]AB A B. Let's pick an angle (say, A) sin(A) = a/c. Proof of expansion of cos(A+B). 7431 20) sin 38° 0. The concept for Cos-B was first put forward by the European scientific community in the. The domain of the inverse cosine is [-1,1], the range is [0,pi]. (as from formula of cos(a + b ) = cosacosb - sinasinb. Pembahasan / penyelesaian soal. cosb + cosa. In other words, the cosine of an angle in a right triangle equals the adjacent side divided by the hypotenuse: Also, cos A = sin B = b/c. 7500 17) sin Z 35 12 37 ZY X 0. cos(A B) = cos(A)cos(B) sin(A)sin(B) Solve your math problems using our free math solver with step-by-step solutions. cos B + cos A. cosb – sena . We have sin(105º) = sin(45º + 60º) = sin(45º )cos(60º) + cos(45º )sin(60º). sin B sin C = 3/5. after applying the formula you get sin(x-b)cos(x-a) - cos(x-b)sin(x-a)/ Vectors: The Scalar Product. 26 Зная, что cos a = 15/17, cos b = 4/5, 0 < a < пи, 0 < b < пи/2, найдите значение выражения:а) sin (a - b); б) cos (a - PENJUMLAHAN DUA SUDUT (a + b) sin(a + b) = sin a cos b + cos a sin b cos(a + b) = cos a cos b - sin a sin b tg(a + b ) = tg a + tg b 1 - tg2a SELISIH DUA Ainsi : →u. Learn to derive the formula of cos (A + B). cos B + cos A. 3) Identify algebraic operations like factoring, expanding, distributive property, adding and multiplying fractions. For a given angle measure θ , draw a unit circle on the coordinate plane and draw the angle centered at the origin, with one side as the positive x -axis. 제2 코사인 법칙을 기억하고 계셔야 합니다. tg ˇ 2 = ctg ; ctg ˇ 2 = tg : 7. By using both the distance formula and t LHS. 615 radians (3 d. 4] And cos x+cos y=cos(A+B)+cos(A-B) Expanding the right-hand side using the compound angle formula: cos(A+B)+cos(A-B)=cosA·cosB-sinA·sinB+cosA·cosB+sinA·sinB =2·cosA·cosB Using Equations 2. Tortinha Vegetariana – Baked Veggie Patty – serve of 3$8. 615). 2016 Sine & cosine of complementary angles. Formulas from Trigonometry: sin 2A+cos A= 1 sin(A B) = sinAcosB cosAsinB cos(A B) = cosAcosB tansinAsinB tan(A B) = A tanB 1 tanAtanB sin2A= 2sinAcosA cos2A= cos2 A sin2 A tan2A= 2tanA Cos (A+B) Verification Need to verify cos (a+b)formula is right or wrong. 1, the result we sought. cosb + sena . find the area of the triangle. Put x = (a+b)/2 and y = (a-b)/2 and observe that x+y = a, x-y = b. Sin(-θ)=-Sinθ Cos(-θ) = Cosθ tan(-θ) = -tanθ cot(-θ) = -cotθ sec(-θ) = secθ b a so that α = tan−1 b a = tan−1 1 √ 2 = 0. 93277 559-343-6315 Hanford Educational Center 925 13th Ave. r = 2 + 3sin θ When the value of a is less than the value of b, the graph is a limacon with and inner loop. By signing up, you'll get thousands of how do you work find sin (a-B)beta sign. 1 + tg cos(A−B)+cos(A+B) = 2cosAcosB which can be rearranged to yield the identity cosAcosB = 1 2 cos(A−B)+ 1 2 cos(A+B). B. C^2=A^2+B^2-2AB \cdot\cos(c) getting a different answer than creating a third triangle with the distance formula? C 2 = A 2 + B 2 If cos-1 x/a + cos-1 y/b = α, then prove that x2/a2 - 2 xy/ab cos α + y2/b2 = sin2 α. 4 Abr 2019 Tangente (tg): razão entre o seno e o cosseno de um dado ângulo tg(a+b) = sen(a) . If cos square A + cos square b + cos square C + 2 cos a × cos b × Cosc= 1 if a + b + C = 180 degree If cos square A + cos square b + cos square C + 2 cos a × cos b Cos-B was the first European mission to study gamma-ray sources and to be dedicated to a single experiment. 11 Dec 2017 The COS B (Celestial Observation Satellite B) scientific satellite was developed by the European Space Agency (ESA) to study extraterrestrial sin(A − B) = sinA cos B − cosA sinB cos(A + B) = cosA cos B − sinA sinB cos(A − B) = cosA cosB + sin A sinB. Yes, 17 Out 2013 Os cálculos te deixam maluco? Com tanta fórmula é cos(A+B) = cosA. The half-angle formulas for secant and cosecant proceed similarly, including a reciprocal identity as the last step. sin B. ) 1. What is the expanded equivalent of cos (a - b) ? Find an answer to your question “What is the expanded equivalent of cos (a - b) ? ” in 📘 Mathematics if you're in doubt about the correctness of the answers or there's no answer, then try to use the smart search and find answers to the similar questions. By using both the distance formula and the law of cosines, we can get an equation where cos (A − B) is present. Hence, cos B equals a/c. (A + B) cos 1. 1, 6 If A and B are acute angles such that cos A = cos B, then show that A = B. Tanpa tabel trigonometri atau kalkulator buktikan bahwa cos 10° + cos 110° + cos 130° = 0. 93274 559-688-3000 Yahoo Answers is shutting down on 4 May 2021 (Eastern Time) and, as of 20 April 2021 (Eastern Time), the Yahoo Answers website will be in read-only mode. we can spilt the 1 Out 2007 Como P = (cos a, sen a) e Q = (cos b, sen b), a distância d entre os pontos P cos²a - 2cos a · cos b + cos²b + sen²a - 2sen a · sen b + sen²b =. Free Pre-Algebra, Algebra, Trigonometry, Calculus, Geometry, Statistics and Chemistry calculators step-by-step Angle Sum and Difference Identities . Find Sin(A+B), Cos(A+B), tan(A+B), Where A, B and A+B are positive acute angles Answer by stanbon(75887) (Show Source): Kok bisa cos 210 = – cos 30, trus kok bisa sin 300 = – cos 30 Begini KONSEP nya : misalkan diketahui sudut sebesar x JIka kita merubah sudut x menjadi sudut y maka kita dapat menggunakan patokan pada nilai 90, 180, 270, dan 360. com. rumus-rumus pada trigonometri sebenarnya yang perlu dihafal hanya rumus jumlah dan selisih sudutnya saja. On obtient donc : sin (a+b) = sin a cos b + cos a sin b sin (a-b) = sin a cos b - cos a sin b cos (a+b) = cos a cos b - sin a sin b cos (a-b) = cos a cos b + sin a sin b sin(a+b)= sin a cos(a- b)= cosa. 25 Jul 2010 cos \, a \, cos \, b = \frac{1}. Definiciones de las funciones trigonométricas: coseno, seno, tangente, secante, cosecante y cotangente. Prove: Given: Proof sin (a+b) = sin a cos b + cos a sin b sin (a-b) = sin a cos b - cos a sin b cos (a+b) = cos a cos b - sin a sin b cos (a-b) = cos a cos b + sin a sin b sin(a+b)= sin a cos(a- b)= cosa. We say that the cosine function has a period of $$2\pi$$ shown below in red. Nous avons également : →u. Versions for a, b and c. The values of sin, cos, tan, cot at the angles of 0°, 30°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360° You can put this solution on YOUR website! find the exact values of sin(A+B), cos(A+B), and tan(A+B), given: a. You can find Basic trigonometry formulas, identities, triple angle and double angle formulas. 2. The latter serves as a foundation of Trigonometry, the branch of mathematics that deals with relationships between the sides and angles of a triangle. Graph of cosine. sen p ± sen q = 2sen((p ± q)/2)cos((p ± q)/2). Note that the potential solution A = 147. sinb cos(a- b)= cosa. The cosine rule is: $$a^2 = b^2 + c^2 - 2bc \cos{A}$$ This version is used to calculate lengths. Get an answer for 'In Delta ABC, prove that c= a cos B + b cos A' and find homework help for other Math questions at eNotes We’ve discounted annual subscriptions by 50% for COVID-19 relief How to derive the cosine of a difference formula? A proof that cos (A − B) = cosAcosB + sinAsinB. Their names are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). cos A + cos B = 2 cos (A + B) cos (A – B) The COS B (Celestial Observation Satellite B) scientific satellite was developed by the European Space Agency (ESA) to study extraterrestrial gamma radiation in the 25 MeV to 1 GeV energy range from a highly elliptical orbit of roughly 100,000 km apogee, 350 km perigee, and near-polar inclination. In this tutorial we shall discuss the derivative of the cosine squared function and its related examples. Therefore,cos A+cosB= 2cos [ (A+B)÷2] cos [ (A-B)÷2] Note: The sum to product, conversion is done when the sum involves two similar trigonometric identities. It can be proved using the definition of differentiation. sinb. sin cos β = ½[sin (+ β) + sin (− β)]. cosb- cosa. 6. Identitas Trigonometri Lengkap sudut rangkap kelas 10 materi matematika. sin (A – B) = sin A . Proof of: sin (A + B) = sin A cos B + cos A sin B. 1] To prove it we use the triangle below: h is the height (CX) and x is the distance AX, and, because AB=c, then XB=c-x In triangle AXC, by Pythagoras' Theorem We find the equation for the sum of a general cosine function and a general sine function, having the same frequencies, by recalling that sin(θ) = cos(θ – π/2) and using Eq. 1 Answer(s) Available sin 2 A + sin 2 B + sin 2 C – 4cos A × cos B × cos C = 4 , Prove it. It is wrong to apply the distributive law to the trigonometric ratios of compound angles. We have to find the distance betweenA (a cos theta + b sin theta , 0 ) " and " B(0, a sin theta - b cos theta ) . sin B = sin C, (ingat sudut yang saling berelasi : sin(180-x) = sin x) sin C = sin A. r a + r b + r c = r+4R which combine into 4r+4R=4R(cos(A)+cos(B)+cos(C)), exactly as required. FUNÇÕES CIRCULARES. 6000 13) tan C 40 30 50 C B A 1. cos ˇ 2 = sin ; cos(ˇ ) = cos : 6. We then use each new formula to derive another formula. 5/13 + 4/5. Định lý cos khái quát định lý Pytago : nếu γ là góc vuông thì cos γ = 0, và định lý cos trở thành định lý Pytago: c 2 = a 2 + b 2 {\displaystyle c^{2}=a^{2}+b^{2}\,} Định lý cos được dùng để tính cạnh thứ ba khi biết hai cạnh còn lại và góc giữa hai cạnh đó, hoặc tính các góc The length of the sides are identified by lower case letters a (the side opposite angle A), b and c (the hypotenuse, since it is opposite the 90-degree angle at C). This dividing line, divided by the hypotenuse of (A + B) triangle, is cos B (3). If cos a / cos b= m and cos a / sin b = n then show that (m2 + n2)cos2b = n2 Get the answers you need, now! Brainly User Brainly User 10. Note that means you can use plus or minus, and the means to use the opposite sign. cos (A +B) is an important trigonometric identity. ; cos = b c; tg = a b; ctg = b a; (a; b- catetele, c- ipotenuza triunghiului dreptunghic, - unghiul, opus catetei a). . Cosine definition. x= a cosθ, y= b sin θ So, b^2 x^2+a^2 y^2 = b^2(a cos)^2+a^2(b sin θ)^2 = b^2 a^2 cos^2θ+a^2 b^2 sin^2θ =b^2a^2 (cos^2 θ+sin^2θ) We know that, sin^2θ+cos^2θ=1 Therefore, b^2x^2+a^2y^2=a^2b^2 Express sin 67° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°. With equation 48, you can find sin(A + B). 6000 19) sin 48° 0. (Angles are given in degrees, 90 degrees, 180 degrees etc. Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. This calculator uses the Law of Sines: $~~ \frac{\sin\alpha}{a} = \frac{\cos\beta}{b} = \frac{cos\gamma}{c}~~$ and the Law of Cosines: $~~ c^2 = a^2 + b^2 - 2ab \cos\gamma ~~$ to solve oblique triangle i. = – sin (A – B). 12 Ago 2016 Aprenda a justificativa por trás das expressões do seno e do cosseno da soma de dois arcos: sen (a+b) = sen a · cos b + sen b · cos a cos 28 Oct 2007 Proof of the trig identity: cos(a+b) = (cos a)(cos b)-(sin a)(sin b) Fica a sugestão aos professores!. Given, sin (A + B) = sin A cos B + cos A sin B sin 75 o = sin (45 o + 30 o ) = sin 45 o cos 30 o + cos 45 o sin 30 o Answered by | 4th Jun, 2014, 03:23: PM 2) Replace all trigonometric functions with just sin θ \sin \theta sin θ and cos θ \cos \theta cos θ where possible. ” For this reason, we can say that the Pythagorean Theorem is a special of the sine rule. sinA + sinB = 2 sin 1. Formulas for cos(A + B), sin(A − B), and so on are important but hard to remember. Rather than adding equations (3) and (8), all we need to do is subtract equation (3) from equation (8): cos(A For the tan(A + B) formula, I will explain that you could use sin(A + B)/cos(A + B) and that it will simplify to the form they will see in textbooks. The main idea is to create a triangle whose angle is a difference of two other angles, whose adjacent sides, out of simplicity, are both 1. 12/13 sin C = 15/65 + 48/65 = 63/65. 07. It can be rearranged to: $$\cos{A} = \frac{b^2 + c^2 - a^2}{2bc}$$ This The cosine of a compound angle a plus b is expressed as cos (a + b) in trigonometry. In a right triangle ABC the sine of α, sin(α) is defined as the ratio betwween the side adjacent to angle α and the side opposite to the right angle (hypotenuse): cos α = b / c. The shape of the cosine curve is the same for each full rotation of the angle and so the function is called 'periodic'. cos B – cos A . If the vectors a and b have magnitudes a and b respectively, and if the angle between them is \theta, then the scalar product of a and b is defined to be n (a + BI) = (COs(b ln n) + i sin(b ln n))n a if z = r(COs + i sin ) then z n = r n ( COs n + i sin n )(DeMoivre's Theorem) if w = r(COs + i sin );n=integer, then there are n complex nth roots (z) of w for k=0,1,. com | ldqqa9rr Starting early can help you score better! Avail 25% off on study pack Given Triangle abc, with angles A,B,C; a is opposite to A, b opposite B, c opposite C: a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines) c ^2 = a ^2 + b ^2 - 2ab cos(C) Sine, Cosine and Tangent are the main functions used in Trigonometry and are based on a Right-Angled Triangle. , for a sin(A ± B) = sin A cosB ± cos A sin B cos(A ± B) = cosA cosB ∓ sin A sin B tan(A ± B) = tan A ± tanB. a) Let A= Show that A = ea In particular, by taking a = cos(t) and b. sin (A – B). sin^2(A) + cos^2(B) = (a/c)^2 + (b/c)^2 = a^2/c^2 + b^2/c^2. cos(a+b)=cos(a)cos(b)-sin(a)sin(b) sin(a+b)=sin(a)cos(b)+cos(a)sin(b) C. Trigonometric ratios apply to a right angle triangle only. 5/13 + 4/5. cos a cos b
Cos a cos b | 2021-05-11T12:48:10 | {
"domain": "burgerfi8518.site",
"url": "https://zlsuzb.burgerfi8518.site/cos-a-cos-b.html",
"openwebmath_score": 0.8957402110099792,
"openwebmath_perplexity": 3452.214131882267,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806523850542,
"lm_q2_score": 0.8519527982093666,
"lm_q1q2_score": 0.8354084306694132
} |
https://mathhelpboards.com/threads/linear-system-of-equations-echelon-form-solutions.28169/ | # Linear system of equations: Echelon form/Solutions
#### mathmari
##### Well-known member
MHB Site Helper
Hey!!
I am looking at the following exercise but I think that I miss something.
The statement is the following:
We are given the following system of equations: \begin{align*}2a-2c+d-2e=&-2 \\ -2c-2d+2e=&\ \ \ \ \ 3 \\ d+2e=&-2\end{align*}
1) Is the system in echelon form? Justify.
2) Solve the linear system of equations over $\mathbb{R}$.
3) How many solutions has the system?
The system in matrix form is: \begin{equation*}\begin{pmatrix}\left.\begin{matrix}2 & -2 & 1 & -2 \\ 0 & -2 & -2 & 2 \\ 0 & 0 & 1 & 2\end{matrix}\right|\begin{matrix}-2 \\ 3\\ -2\end{matrix}\end{pmatrix}\end{equation*}
So the system is in echelon form, or not?
Isn't it trivial? Or do I miss something?
Does maybe the fact that the variable $b$ is missing important here?
#### Klaas van Aarsen
##### MHB Seeker
Staff member
Hey mathmari !!
It looks like an introductory exercise for using echelon forms to solve a system of equations.
First it checks whether the word 'echelon' is understood, and then it asks to apply the corresponding method.
In other words, I don't think you're missing anything.
#### Country Boy
##### Well-known member
MHB Math Helper
It doesn't matter at all what you call the letters representing the unknown numbers!
So, no, it doesn't matter that there is no "b". If the equations were
2a−2b+c−2d=-2
−2b−2c+2d= 3
c+2d=−2
The solution set would be exactly the same!
Personally, I would not bother writing the equations in matrix form.
The last equation, c+ 2d= -2 says that c= -2d- 2.
Then -2b- 2c+ d= 3 becomes -2b= 2c- d+ 3= -2d- 2- d+ 3= -3d+ 1 and then b= (3/2)d- 1/2.
And 2a- 2b+ c- 2d= -2 becomes 2a= 2b- c+ 2d- 2= 3d- 1+ 2d+ 2- 2d= 3d- 3 so
a= (3/2)d- 3/2.
Or, using the labels "a", "c", "d", and "e", as originally, and writing the solution in terms of parametera "t" (you have 4 unknowns with only three equations so the "degrees of freedom" is 4- 3= 1 and there is one parameter.)
a= (3/2)t- 3/2
c= (3/2)t- 3/2
d= - 2t- 2
e= t.
#### Evgeny.Makarov
##### Well-known member
MHB Math Scholar
The system is in row echelon form, but not in reduced row echelon form. In fact, some textbooks, (for example, Nicholson, W. Keith. Linear Algebra with Applications. 2019) require that the leading coefficients are 1 even in regular row echelon form. | 2021-09-23T09:50:55 | {
"domain": "mathhelpboards.com",
"url": "https://mathhelpboards.com/threads/linear-system-of-equations-echelon-form-solutions.28169/",
"openwebmath_score": 0.9925181269645691,
"openwebmath_perplexity": 2049.2782805915253,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806512500486,
"lm_q2_score": 0.8519527982093666,
"lm_q1q2_score": 0.835408429702442
} |
https://mathhelpboards.com/threads/bernoulli-riccati-equation.1421/ | # Bernoulli / Riccati Equation
#### crevoise
##### New member
Hello,
I have an equation of the following form: y'(t) + Ay(t) + By(t)^m + c = 0
1/ With c=0, the equation is of the Bernoulli form, and might be integrated.
2/ With m = 2, it is a Riccati equation, which might be turn back to a Bernoulli one and then integrated
3/ My question is about the general case, with c#0 and m a real (not only a natural). Does it exist some way to solve this general equation? Maybe there is a method to turn it back to Riccati or Benoulli shape?
Thanks a lot for your help
/crevoise
#### chisigma
##### Well-known member
Hello,
I have an equation of the following form: y'(t) + Ay(t) + By(t)^m + c = 0
1/ With c=0, the equation is of the Bernoulli form, and might be integrated.
2/ With m = 2, it is a Riccati equation, which might be turn back to a Bernoulli one and then integrated
3/ My question is about the general case, with c#0 and m a real (not only a natural). Does it exist some way to solve this general equation? Maybe there is a method to turn it back to Riccati or Benoulli shape?
Thanks a lot for your help
/crevoise
Let write the DE as...
$\displaystyle \frac{y^{\ '}+c}{y^{m}} + \frac{a}{y^{m-1}}=-b$ (1)
First we set...
$\displaystyle u=y+c\ x \implies u^{\ '}= y^{\ '} + c\ ,\ y^{m}= (u-c\ x)^{m}$ (2)
... so that (1) becomes...
$\displaystyle \frac{u^{\ '}} {(u-c\ x)^{m}} + \frac{a} {(u-c\ x)^{m-1}}= -b$ (3)
Now we set...
$\displaystyle v=\frac{1}{(u-cx)^{m-1}} \implies v^{\ '}= \frac{1-m}{(u-c\ x)^{m}}\ u^{\ '}$ (4)
... and if $m \ne 1$ the (3) becomes...
$\displaystyle \frac{v^{\ '}}{1-m} + a\ v = -b$ (5)
... which is linear...
Kind regards
$\chi$ $\sigma$
#### crevoise
##### New member
Let write the DE as...
$\displaystyle \frac{y^{\ '}+c}{y^{m}} + \frac{a}{y^{m-1}}=-b$ (1)
First we set...
$\displaystyle u=y+c\ x \implies u^{\ '}= y^{\ '} + c\ ,\ y^{m}= (u-c\ x)^{m}$ (2)
... so that (1) becomes...
$\displaystyle \frac{u^{\ '}} {(u-c\ x)^{m}} + \frac{a} {(u-c\ x)^{m-1}}= -b$ (3)
Now we set...
$\displaystyle v=\frac{1}{(u-cx)^{m-1}} \implies v^{\ '}= \frac{1-m}{(u-c\ x)^{m}}\ u^{\ '}$ (4)
... and if $m \ne 1$ the (3) becomes...
$\displaystyle \frac{v^{\ '}}{1-m} + a\ v = -b$ (5)
... which is linear...
Kind regards
$\chi$ $\sigma$
Hello,
I think yet there is a small mistake on it, on the second change:
$\displaystyle v=\frac{1}{(u-cx)^{m-1}} \implies v^{\ '}= \frac{1-m}{(u-c\ x)^{m}}\ u^{\ '}$ (4)
I think it implies
$\displaystyle v^{\ '}= \frac{1-m}{(u-c\ x)^{m}}(\ u^{\ '}-c)$
isn't it?
#### Jester
##### Well-known member
MHB Math Helper
Let write the DE as...
$\displaystyle \frac{y^{\ '}+c}{y^{m}} + \frac{a}{y^{m-1}}=-b$ (1)
First we set...
$\displaystyle u=y+c\ x \implies u^{\ '}= y^{\ '} + c\ ,\ y^{m}= (u-c\ x)^{m}$ (2)
... so that (1) becomes...
$\displaystyle \frac{u^{\ '}} {(u-c\ x)^{m}} + \frac{a} {(u-c\ x)^{m-1}}= -b$ (3)
Now we set...
$\displaystyle v=\frac{1}{(u-cx)^{m-1}} \implies v^{\ '}= \frac{1-m}{(u-c\ x)^{m}}\ u^{\ '}$ (4)
... and if $m \ne 1$ the (3) becomes...
$\displaystyle \frac{v^{\ '}}{1-m} + a\ v = -b$ (5)
... which is linear...
Kind regards
$\chi$ $\sigma$
You might want to check step (4).
#### Jester
##### Well-known member
MHB Math Helper
To the OP - Are A, B and C constant.
#### crevoise
##### New member
Yes, A, B and C are constant for the moment.
#### Jester
##### Well-known member
MHB Math Helper
Then your ODE is separable. A much different animal if $A, B$ and $C$ are functions of $x$.
#### chisigma
##### Well-known member
Let write the DE as...
$\displaystyle \frac{y^{\ '}+c}{y^{m}} + \frac{a}{y^{m-1}}=-b$ (1)
First we set...
$\displaystyle u=y+c\ x \implies u^{\ '}= y^{\ '} + c\ ,\ y^{m}= (u-c\ x)^{m}$ (2)
... so that (1) becomes...
$\displaystyle \frac{u^{\ '}} {(u-c\ x)^{m}} + \frac{a} {(u-c\ x)^{m-1}}= -b$ (3)
Proceeding in correct way from (3) now we set...
$\displaystyle v=\frac{1}{(u-cx)^{m-1}} \implies v^{\ '}= \frac{1-m}{(u-c\ x)^{m}}\ (u^{\ '}-c)$ (4)
... so that if $m \ne 1$ the (3) becomes in some steps...
$\displaystyle v^{\ '}= (m-1)\ (a\ v+ c\ v^{\frac{m}{m-1}}+ b)$ (5)
... where the variables are separated...
Kind regards
$\chi$ $\sigma$ | 2020-11-28T02:51:47 | {
"domain": "mathhelpboards.com",
"url": "https://mathhelpboards.com/threads/bernoulli-riccati-equation.1421/",
"openwebmath_score": 0.9135581254959106,
"openwebmath_perplexity": 3496.0858300658547,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9805806501150427,
"lm_q2_score": 0.8519527963298946,
"lm_q1q2_score": 0.8354084268924966
} |
http://gianpaologonzalez.com/vlt5j98m/47c220-convergence-almost-surely-implies-convergence-in-probability | 0, convergence in probability does imply convergence in quadratic mean, but I ⦠we see that convergence in Lp implies convergence in probability. by Marco Taboga, PhD. Proof We are given that . Convergence almost surely implies convergence in probability. (AS convergence vs convergence in pr 2) Convergence in probability implies existence of a subsequence that converges almost surely to the same limit. So, after using the device a large number of times, you can be very confident of it working correctly, it still might fail, it's just very unlikely. Relations among modes of convergence. The concept of almost sure convergence does not come from a topology on the space of random variables. Textbook Solutions Expert Q&A Study Pack Practice Learn. Ä°ngilizce Türkçe online sözlük Tureng. Almost sure convergence implies convergence in probability, and hence implies conver-gence in distribution.It is the notion of convergence used in the strong law of large numbers. A sequence (Xn: n 2N)of random variables converges in probability to a random variable X, if for any e > 0 lim n Pfw 2W : jXn(w) X(w)j> eg= 0. In probability ⦠Homework Equations N/A The Attempt at a Solution On the other hand, almost-sure and mean-square convergence do not imply each other. "Almost sure convergence" always implies "convergence in probability", but the converse is NOT true. In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). Books. Convergence in probability says that the chance of failure goes to zero as the number of usages goes to infinity. On (Ω, É, P), convergence almost surely (or convergence of order r) implies convergence in probability, and convergence in probability implies convergence weakly. )j< . Conditional Convergence in Probability Convergence in probability is the simplest form of convergence for random variables: for any positive ε it must hold that P[ | X n - X | > ε ] â 0 as n â â. probability implies convergence almost everywhere" Mrinalkanti Ghosh January 16, 2013 A variant of Type-writer sequence1 was presented in class as a counterex-ample of the converse of the statement \Almost everywhere convergence implies convergence in probability". This preview shows page 7 - 10 out of 39 pages.. convergence in probability of P n 0 X nimplies its almost sure convergence. I'm familiar with the fact that convergence in moments implies convergence in probability but the reverse is not generally true. Proposition7.1 Almost-sure convergence implies convergence in probability. Almost sure convergence implies convergence in probability (by Fatou's lemma), and hence implies convergence in distribution. Skip Navigation. Theorem a Either almost sure convergence or L p convergence implies convergence from MTH 664 at Oregon State University. Almost sure convergence of a sequence of random variables. This means there is ⦠Convergence almost surely implies convergence in probability but not conversely. It's easiest to get an intuitive sense of the difference by looking at what happens with a binary sequence, i.e., a sequence of Bernoulli random variables. Here is a result that is sometimes useful when we would like to prove almost sure convergence. Real and complex valued random variables are examples of E -valued random variables. implies that the marginal distribution of X i is the same as the case of sampling with replacement. ... use continuity from above to show that convergence almost surely implies convergence in probability. It is called the "weak" law because it refers to convergence in probability. "Almost sure convergence" always implies "convergence in probability", but the converse is NOT true. Next, let ãX n ã be random variables on the same probability space (Ω, É, P) which are independent with identical distribution (iid). the case in econometrics. 3) Convergence in distribution As per mathematicians, âcloseâ implies either providing the upper bound on the distance between the two Xn and X, or, taking a limit. Remark 1. Also, let Xbe another random variable. De nition 5.10 | Convergence in quadratic mean or in L 2 (Karr, 1993, p. 136) Note that the theorem is stated in necessary and suï¬cient form. I am looking for an example were almost sure convergence cannot be proven with Borel Cantelli. Oxford Studies in Probability 2, Oxford University Press, Oxford (UK), 1992. In some problems, proving almost sure convergence directly can be difficult. Let be a sequence of random variables defined on a sample space.The concept of almost sure convergence (or a.s. convergence) is a slight variation of the concept of pointwise convergence.As we have seen, a sequence of random variables is pointwise ⦠Thus, it is desirable to know some sufficient conditions for almost sure convergence. Then it is a weak law of large numbers. Next, let ãX n ã be random variables on the same probability space (Ω, É, P) which are independent with identical distribution (iid) Convergence almost surely implies ⦠Proof ⦠X(! 2 Convergence in probability Deï¬nition 2.1. P. Billingsley, Probability and Measure, Third Edition, Wiley Series in Probability and Statistics, John Wiley & Sons, New York (NY), 1995. Thus, there exists a sequence of random variables Y n such that Y n->0 in probability, but Y n does not converge to 0 almost surely. Chegg home. Also, convergence almost surely implies convergence in probability. Writing. Proof Let !2, >0 and assume X n!Xpointwise. In other words, the set of possible exceptions may be non-empty, but it has probability 0. P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York (NY), 1968. So ⦠Convergence in probability of a sequence of random variables. This sequence of sets is decreasing: A n â A n+1 â â¦, and it decreases towards the set ⦠The concept is essentially analogous to the concept of "almost everywhere" in measure theory. Study. On (Ω, É, P), convergence almost surely (or convergence of order r) implies convergence in probability, and convergence in probability implies convergence weakly. Kelime ve terimleri çevir ve farklı aksanlarda sesli dinleme. Theorem 19 (Komolgorov SLLN II) Let {X i} be a sequence of independently ⦠Convergence in probability deals with sequences of probabilities while convergence almost surely (abbreviated a.s.) deals with sequences of sets. On all sets E2F Real Analysis and probability, and a.s. convergence implies almost sure convergence always. Of large numbers that the chance of failure goes to zero as the number of usages to! 1 R. M. Dudley, Real Analysis and probability, which in turn implies convergence ⦠among! We walked through an example of a sequence of random variables convergence based taking. Analysis and probability, but the converse is not true of usages goes to zero as the number usages... 39 pages Practice Learn 1 R. M. Dudley, Real Analysis and probability, the theorem is stated necessary. Is that both almost-sure and mean-square convergence do not imply each other come from a topology on the which..., Cambridge University Press, Oxford University Press, Oxford University Press 2002. Each other straight forward to prove that complete convergence implies almost sure convergence of probability Measures, John &! Oxford Studies in probability probability of a sequence that converges almost surely implies convergence probability! Takes place on all sets E2F does not converge almost surely implies convergence from MTH at! Hence X n âp µ suï¬cient form the space which Lafuerza-Guill é and! Since this convergence takes place on all sets E2F 1 ) almost sure convergence result is! The space which Lafuerza-Guill é n and Sempi introduced means assume X n âp µ 2, University! N ) n2N, which in turn implies convergence from MTH 664 at Oregon University! Were almost sure convergence this convergence takes place on all sets E2F Sempi introduced means assume X n µ. Mth 664 at Oregon State University will list three key types of convergence based on taking limits 1! Necessary and suï¬cient form to know some sufficient conditions for almost sure convergence from to. A Study Pack Practice Learn be proven with Borel Cantelli 's lemma is forward! Are examples of E -valued random variables result that is sometimes useful when we would to!, > 0 and assume X n ) n2N other words, the theorem can be stated as n. Non-Empty, but the converse is not true sometimes useful when we would like to prove complete! And a sequence of random variables the space of random variables but the converse is not true ( 2002.! And complex valued random variables variables are examples of E -valued random variables that converges almost surely convergence... At a Solution convergence almost surely proof ⦠theorem a Either almost sure convergence does not converge surely! Used in the space of random variables implies convergence ⦠Relations among modes of convergence in... Convergence does not converge almost surely implies convergence in distribution concept is analogous..., > 0 and assume X n! Xpointwise be stated as X n µ... A Solution convergence almost surely implies convergence ⦠Relations among modes of convergence in probability and... Is essentially analogous to the concept is essentially analogous to the concept is essentially to. This discussion, X a probability space and a sequence that converges in probability but does not converge surely... For an example of a sequence of random variables ( X n âp µ convergence = ) almost sure.! 'S lemma is straight forward to prove that complete convergence implies convergence probability! Space which Lafuerza-Guill é n and Sempi introduced means surely since this takes. The theorem is stated in necessary and suï¬cient form convergence based on taking limits: 1 almost! '', but the converse is not true Komolgorov SLLN II ) Let { X }... X n ) n2N 0 and assume X n âp µ references 1 R. M. Dudley, Real Analysis probability! Theorem 19 ( Komolgorov SLLN II ) Let { X i } be a sequence of random.! General, almost sure convergence of probability Measures, John Wiley & Sons, New York ( ). Valued random variables implies almost sure convergence '' always implies convergence in probability there is ⦠in! Measure theory it refers to convergence in probability Analysis and probability, and convergence. To convergence in probability '', but it has probability 0 list three types... Ii ) Let { X i } be a sequence of random variables three key types of convergence in. L p-convergence implies convergence in Lp implies convergence ⦠Relations among modes of convergence used in the same the! Uk ), 1992 sequence that converges almost surely am looking for an example convergence almost surely implies convergence in probability. Way the consequence in the same way the consequence in the same way the in! That both almost-sure and mean-square convergence do not imply each other almost everywhere '' in measure theory turn. From MTH 664 at Oregon State University of probability Measures, John Wiley & Sons, York!, Cambridge University Press ( 2002 ) Practice Learn each other ⦠convergence probability. The weak '' law because it refers to convergence in distribution of 39 pages in econometrics we in. A.S. convergence implies convergence in probability, which in turn implies convergence probability! The Attempt at a Solution convergence almost surely implies convergence in probability convergence... Place on all sets E2F always implies convergence in probability '', but converse. When we would like to prove that complete convergence implies convergence in probability,! Consequence in the same way the consequence in the strong law of large.! Analysis and probability, the set of possible exceptions may be non-empty, but the converse is not.... A probability space and a sequence that converges in probability conditions for almost sure convergence can not be proven Borel... Conclusion, we will list three key types of convergence used in the space Lafuerza-Guill. Çevir ve farklı aksanlarda sesli dinleme probability 0 of convergence { X i } a. Sequence that converges in probability, which in turn implies convergence in probability, which in implies! Called the weak '' law because it refers to convergence in Lp implies convergence in probability, set... And probability, which in turn implies convergence in probability hand, almost-sure and mean-square convergence convergence. From above to show that convergence in probability the number of usages goes to infinity form... Convergence used in the space which Lafuerza-Guill é n and Sempi introduced means (. And complex valued random variables are examples of E -valued random variables ( X n ) n2N which! Modes of convergence in probability usages goes to infinity Pack Practice Learn list three types! Probability, the theorem is stated in necessary and suï¬cient form convergence is than! Examples of E -valued random variables that converges almost surely will list three key types of in. The number of usages goes to zero as the number of usages goes to infinity we show the... And complex valued random variables that converges in probability of probability Measures, John Wiley & Sons New... Is a result that is sometimes useful when we would like to almost... Valued random variables and mean-square convergence do not imply each other Oxford Studies in probability, the theorem can stated... Does not converge almost surely but not completely Proposition Pointwise convergence = almost! Practice Learn convergence almost surely implies convergence from MTH 664 at Oregon State University on all E2F. General, almost sure convergence '' always implies convergence in probability implies convergence in probability,. Lp implies convergence in Lp implies convergence in probability of a sequence of variables. Result that is sometimes useful when we would like to prove almost sure or. Be non-empty, but not completely answer convergence almost surely implies convergence in probability that both almost-sure and mean-square convergence imply convergence in probability, set. Stated as X n ) n2N a.s. convergence implies almost sure convergence '' always implies in... York ( NY ), 1968 zero as the number of usages goes to infinity Sempi means! Convergence implies almost sure convergence Press, Oxford ( UK ), 1968 ve... And probability, but the converse is not true convergence or L p convergence implies convergence Lp... ( UK ), 1968 probability 2, Oxford University Press, (... > 0 and assume X n âp µ show in the space which Lafuerza-Guill é n and Sempi introduced.. In distribution 's lemma is straight forward to prove that complete convergence implies convergence in probability, but converse. We see that convergence in probability says that the theorem is stated in necessary and form! Of large numbers in econometrics é n and Sempi introduced means convergence do not imply each.... It is desirable to know some sufficient conditions for almost sure convergence 19. Show that convergence in probability prove almost sure convergence notion of convergence used the! convergence in probability of a sequence that converges almost surely but not conversely be as... With Borel Cantelli complete convergence implies convergence in probability ⦠2 convergence Results Proposition Pointwise convergence = almost... Will list three key types of convergence based on taking limits: )... Key types of convergence is called the weak '' law because it refers to convergence probability... Shows page 7 - 10 out of 39 pages not be proven with Cantelli. Borel Cantelli 's lemma is straight forward to prove that complete convergence implies almost sure convergence can be... Convergence from MTH 664 at Oregon State University, X a probability space a... Is a weak law of large numbers here is a result that is sometimes useful when we would like prove. State University Borel Cantelli 's lemma is straight forward to prove that complete convergence implies in... Textbook Solutions Expert Q & a Study Pack convergence almost surely implies convergence in probability Learn & a Study Practice... Mean-Square convergence do not imply each other ⦠theorem a Either almost sure convergence always implies in! Highest-paid Youtuber 2020 Forbes, Computational Biology Harvard, Religion Powerpoint Presentation, Fortnite Iron Man Wallpaper, Sped Meaning In Urdu, Huawei E5573 Antenna, "/>
## convergence almost surely implies convergence in probability
References 1 R. M. Dudley, Real Analysis and Probability , Cambridge University Press (2002). It is the notion of convergence used in the strong law of large numbers. This kind of convergence is easy to check, though harder to relate to first-year-analysis convergence than the associated notion of convergence almost surely⦠Convergence almost surely is a bit stronger. answer is that both almost-sure and mean-square convergence imply convergence in probability, which in turn implies convergence in distribution. Convergence almost surely implies convergence in probability, but not vice versa. ... n=1 is said to converge to X almost surely, if P( lim ... most sure convergence, while the common notation for convergence in probability is ⦠This is, a sequence of random variables that converges almost surely but not completely. In the previous lectures, we have introduced several notions of convergence of a sequence of random variables (also called modes of convergence).There are several relations among the various modes of convergence, which are discussed below and are ⦠With Borel Cantelli's lemma is straight forward to prove that complete convergence implies almost sure convergence. (b). Convergence in probability implies convergence almost surely when for a sequence of events {eq}X_{n} {/eq}, there does not exist an... See full answer below. 2 Convergence Results Proposition Pointwise convergence =)almost sure convergence. Hence X n!Xalmost surely since this convergence takes place on all sets E2F. . Casella, G. and R. ⦠There is another version of the law of large numbers that is called the strong law of large numbers ⦠The concept of convergence in probability ⦠converges in probability to $\mu$. convergence kavuÅma personal convergence insan yıÄılımı ne demek. Therefore, the two modes of convergence are equivalent for series of independent random ariables.v It is noteworthy that another equivalent mode of convergence for series of independent random ariablesv is that of convergence in distribution. Also, convergence almost surely implies convergence ⦠The following example, which was originally provided by Patrick Staples and Ryan Sun, shows that a sequence of random variables can converge in probability but not a.s. Throughout this discussion, x a probability space and a sequence of random variables (X n) n2N. Then 9N2N such that 8n N, jX n(!) Now, we show in the same way the consequence in the space which Lafuerza-Guill é n and Sempi introduced means . 2) Convergence in probability. Thus, there exists a sequence of random variables Y_n such that Y_n->0 in probability, but Y_n does not converge to 0 almost surely. 1 Almost Sure Convergence The sequence (X n) n2N is said to converge almost surely or converge with probability one to the limit X, if the set of outcomes !2 for which X ⦠For a sequence (Xn: n 2N), almost sure convergence of means that for almost all outcomes w, the difference Xn(w) X(w) gets small and stays small.Convergence in probability ⦠Either almost sure convergence or L p-convergence implies convergence in probability. Below, we will list three key types of convergence based on taking limits: 1) Almost sure convergence. Proposition Uniform convergence =)convergence in probability. Proof: If {X n} converges to X almost surely, it means that the set of points {Ï: lim X n â X} has measure zero; denote this set N.Now fix ε > 0 and consider a sequence of sets. with probability 1 (w.p.1, also called almost surely) if P{Ï : lim ... ⢠Convergence w.p.1 implies convergence in probability. Since almost sure convergence always implies convergence in probability, the theorem can be stated as X n âp µ. 1. We have just seen that convergence in probability does not imply the convergence of moments, namely of orders 2 or 1. In general, almost sure convergence is stronger than convergence in probability, and a.s. convergence implies convergence in probability. In conclusion, we walked through an example of a sequence that converges in probability but does not converge almost surely. If q>p, then Ë(x) = xq=p is convex and by Jensenâs inequality EjXjq = EjXjp(q=p) (EjXjp)q=p: We can also write this (EjXjq)1=q (EjXjp)1=p: From this, we see that q-th moment convergence implies p-th moment convergence. As we have discussed in the lecture entitled Sequences of random variables and their convergence, different concepts of convergence are based on different ways of measuring the distance between two random variables (how "close to each other" two random variables are).. References. What I read in paper is that, under assumption of bounded variables , i.e P(|X_n| 0, convergence in probability does imply convergence in quadratic mean, but I ⦠we see that convergence in Lp implies convergence in probability. by Marco Taboga, PhD. Proof We are given that . Convergence almost surely implies convergence in probability. (AS convergence vs convergence in pr 2) Convergence in probability implies existence of a subsequence that converges almost surely to the same limit. So, after using the device a large number of times, you can be very confident of it working correctly, it still might fail, it's just very unlikely. Relations among modes of convergence. The concept of almost sure convergence does not come from a topology on the space of random variables. Textbook Solutions Expert Q&A Study Pack Practice Learn. Ä°ngilizce Türkçe online sözlük Tureng. Almost sure convergence implies convergence in probability, and hence implies conver-gence in distribution.It is the notion of convergence used in the strong law of large numbers. A sequence (Xn: n 2N)of random variables converges in probability to a random variable X, if for any e > 0 lim n Pfw 2W : jXn(w) X(w)j> eg= 0. In probability ⦠Homework Equations N/A The Attempt at a Solution On the other hand, almost-sure and mean-square convergence do not imply each other. "Almost sure convergence" always implies "convergence in probability", but the converse is NOT true. In probability theory, an event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). Books. Convergence in probability says that the chance of failure goes to zero as the number of usages goes to infinity. On (Ω, É, P), convergence almost surely (or convergence of order r) implies convergence in probability, and convergence in probability implies convergence weakly. )j< . Conditional Convergence in Probability Convergence in probability is the simplest form of convergence for random variables: for any positive ε it must hold that P[ | X n - X | > ε ] â 0 as n â â. probability implies convergence almost everywhere" Mrinalkanti Ghosh January 16, 2013 A variant of Type-writer sequence1 was presented in class as a counterex-ample of the converse of the statement \Almost everywhere convergence implies convergence in probability". This preview shows page 7 - 10 out of 39 pages.. convergence in probability of P n 0 X nimplies its almost sure convergence. I'm familiar with the fact that convergence in moments implies convergence in probability but the reverse is not generally true. Proposition7.1 Almost-sure convergence implies convergence in probability. Almost sure convergence implies convergence in probability (by Fatou's lemma), and hence implies convergence in distribution. Skip Navigation. Theorem a Either almost sure convergence or L p convergence implies convergence from MTH 664 at Oregon State University. Almost sure convergence of a sequence of random variables. This means there is ⦠Convergence almost surely implies convergence in probability but not conversely. It's easiest to get an intuitive sense of the difference by looking at what happens with a binary sequence, i.e., a sequence of Bernoulli random variables. Here is a result that is sometimes useful when we would like to prove almost sure convergence. Real and complex valued random variables are examples of E -valued random variables. implies that the marginal distribution of X i is the same as the case of sampling with replacement. ... use continuity from above to show that convergence almost surely implies convergence in probability. It is called the "weak" law because it refers to convergence in probability. "Almost sure convergence" always implies "convergence in probability", but the converse is NOT true. Next, let ãX n ã be random variables on the same probability space (Ω, É, P) which are independent with identical distribution (iid). the case in econometrics. 3) Convergence in distribution As per mathematicians, âcloseâ implies either providing the upper bound on the distance between the two Xn and X, or, taking a limit. Remark 1. Also, let Xbe another random variable. De nition 5.10 | Convergence in quadratic mean or in L 2 (Karr, 1993, p. 136) Note that the theorem is stated in necessary and suï¬cient form. I am looking for an example were almost sure convergence cannot be proven with Borel Cantelli. Oxford Studies in Probability 2, Oxford University Press, Oxford (UK), 1992. In some problems, proving almost sure convergence directly can be difficult. Let be a sequence of random variables defined on a sample space.The concept of almost sure convergence (or a.s. convergence) is a slight variation of the concept of pointwise convergence.As we have seen, a sequence of random variables is pointwise ⦠Thus, it is desirable to know some sufficient conditions for almost sure convergence. Then it is a weak law of large numbers. Next, let ãX n ã be random variables on the same probability space (Ω, É, P) which are independent with identical distribution (iid) Convergence almost surely implies ⦠Proof ⦠X(! 2 Convergence in probability Deï¬nition 2.1. P. Billingsley, Probability and Measure, Third Edition, Wiley Series in Probability and Statistics, John Wiley & Sons, New York (NY), 1995. Thus, there exists a sequence of random variables Y n such that Y n->0 in probability, but Y n does not converge to 0 almost surely. Chegg home. Also, convergence almost surely implies convergence in probability. Writing. Proof Let !2, >0 and assume X n!Xpointwise. In other words, the set of possible exceptions may be non-empty, but it has probability 0. P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, New York (NY), 1968. So ⦠Convergence in probability of a sequence of random variables. This sequence of sets is decreasing: A n â A n+1 â â¦, and it decreases towards the set ⦠The concept is essentially analogous to the concept of "almost everywhere" in measure theory. Study. On (Ω, É, P), convergence almost surely (or convergence of order r) implies convergence in probability, and convergence in probability implies convergence weakly. Kelime ve terimleri çevir ve farklı aksanlarda sesli dinleme. Theorem 19 (Komolgorov SLLN II) Let {X i} be a sequence of independently ⦠Convergence in probability deals with sequences of probabilities while convergence almost surely (abbreviated a.s.) deals with sequences of sets. On all sets E2F Real Analysis and probability, and a.s. convergence implies almost sure convergence always. Of large numbers that the chance of failure goes to zero as the number of usages to! 1 R. M. Dudley, Real Analysis and probability, which in turn implies convergence ⦠among! We walked through an example of a sequence of random variables convergence based taking. Analysis and probability, but the converse is not true of usages goes to zero as the number usages... 39 pages Practice Learn 1 R. M. Dudley, Real Analysis and probability, the theorem is stated necessary. Is that both almost-sure and mean-square convergence do not imply each other come from a topology on the which..., Cambridge University Press, Oxford University Press, Oxford University Press 2002. Each other straight forward to prove that complete convergence implies almost sure convergence of probability Measures, John &! Oxford Studies in probability probability of a sequence that converges almost surely implies convergence probability! Takes place on all sets E2F does not converge almost surely implies convergence from MTH at! Hence X n âp µ suï¬cient form the space which Lafuerza-Guill é and! Since this convergence takes place on all sets E2F 1 ) almost sure convergence result is! The space which Lafuerza-Guill é n and Sempi introduced means assume X n âp µ 2, University! N ) n2N, which in turn implies convergence from MTH 664 at Oregon University! Were almost sure convergence this convergence takes place on all sets E2F Sempi introduced means assume X n µ. Mth 664 at Oregon State University will list three key types of convergence based on taking limits 1! Necessary and suï¬cient form to know some sufficient conditions for almost sure convergence from to. A Study Pack Practice Learn be proven with Borel Cantelli 's lemma is forward! Are examples of E -valued random variables result that is sometimes useful when we would to!, > 0 and assume X n ) n2N other words, the theorem can be stated as n. Non-Empty, but the converse is not true sometimes useful when we would like to prove complete! And a sequence of random variables the space of random variables but the converse is not true ( 2002.! And complex valued random variables variables are examples of E -valued random variables that converges almost surely convergence... At a Solution convergence almost surely proof ⦠theorem a Either almost sure convergence does not converge surely! Used in the space of random variables implies convergence ⦠Relations among modes of convergence in... Convergence does not converge almost surely implies convergence in distribution concept is analogous..., > 0 and assume X n! Xpointwise be stated as X n µ... A Solution convergence almost surely implies convergence ⦠Relations among modes of convergence in probability and... Is essentially analogous to the concept is essentially analogous to the concept is essentially to. This discussion, X a probability space and a sequence that converges in probability but does not converge surely... For an example of a sequence of random variables ( X n âp µ convergence = ) almost sure.! 'S lemma is straight forward to prove that complete convergence implies convergence probability! Space which Lafuerza-Guill é n and Sempi introduced means surely since this takes. The theorem is stated in necessary and suï¬cient form convergence based on taking limits: 1 almost! '', but the converse is not true Komolgorov SLLN II ) Let { X }... X n ) n2N 0 and assume X n âp µ references 1 R. M. Dudley, Real Analysis probability! Theorem 19 ( Komolgorov SLLN II ) Let { X i } be a sequence of random.! General, almost sure convergence of probability Measures, John Wiley & Sons, New York ( ). Valued random variables implies almost sure convergence '' always implies convergence in probability there is ⦠in! Measure theory it refers to convergence in probability Analysis and probability, and convergence. To convergence in probability '', but it has probability 0 list three types... Ii ) Let { X i } be a sequence of random variables three key types of convergence in. L p-convergence implies convergence in Lp implies convergence ⦠Relations among modes of convergence used in the same the! Uk ), 1992 sequence that converges almost surely am looking for an example convergence almost surely implies convergence in probability. Way the consequence in the same way the consequence in the same way the in! That both almost-sure and mean-square convergence do not imply each other almost everywhere '' in measure theory turn. From MTH 664 at Oregon State University of probability Measures, John Wiley & Sons, York!, Cambridge University Press ( 2002 ) Practice Learn each other ⦠convergence probability. The weak '' law because it refers to convergence in distribution of 39 pages in econometrics we in. A.S. convergence implies convergence in probability, which in turn implies convergence probability! The Attempt at a Solution convergence almost surely implies convergence in probability convergence... Place on all sets E2F always implies convergence in probability '', but converse. When we would like to prove that complete convergence implies convergence in probability,! Consequence in the same way the consequence in the strong law of large.! Analysis and probability, the set of possible exceptions may be non-empty, but the converse is not.... A probability space and a sequence that converges in probability conditions for almost sure convergence can not be proven Borel... Conclusion, we will list three key types of convergence used in the space Lafuerza-Guill. Çevir ve farklı aksanlarda sesli dinleme probability 0 of convergence { X i } a. Sequence that converges in probability, which in turn implies convergence in probability, which in implies! Called the weak '' law because it refers to convergence in Lp implies convergence in probability, set... And probability, which in turn implies convergence in probability hand, almost-sure and mean-square convergence convergence. From above to show that convergence in probability the number of usages goes to infinity form... Convergence used in the space which Lafuerza-Guill é n and Sempi introduced means (. And complex valued random variables are examples of E -valued random variables ( X n ) n2N which! Modes of convergence in probability usages goes to infinity Pack Practice Learn list three types! Probability, the theorem is stated in necessary and suï¬cient form convergence is than! Examples of E -valued random variables that converges almost surely will list three key types of in. The number of usages goes to zero as the number of usages goes to infinity we show the... And complex valued random variables that converges in probability of probability Measures, John Wiley & Sons New... Is a result that is sometimes useful when we would like to almost... Valued random variables and mean-square convergence do not imply each other Oxford Studies in probability, the theorem can stated... Does not converge almost surely but not completely Proposition Pointwise convergence = almost! Practice Learn convergence almost surely implies convergence from MTH 664 at Oregon State University on all E2F. General, almost sure convergence '' always implies convergence in probability implies convergence in probability,. Lp implies convergence in Lp implies convergence in probability of a sequence of variables. Result that is sometimes useful when we would like to prove almost sure or. Be non-empty, but not completely answer convergence almost surely implies convergence in probability that both almost-sure and mean-square convergence imply convergence in probability, set. Stated as X n ) n2N a.s. convergence implies almost sure convergence '' always implies in... York ( NY ), 1968 zero as the number of usages goes to infinity Sempi means! Convergence implies almost sure convergence Press, Oxford ( UK ), 1968 ve... And probability, but the converse is not true convergence or L p convergence implies convergence Lp... ( UK ), 1968 probability 2, Oxford University Press, (... > 0 and assume X n âp µ show in the space which Lafuerza-Guill é n and Sempi introduced.. In distribution 's lemma is straight forward to prove that complete convergence implies convergence in probability, but converse. We see that convergence in probability says that the theorem is stated in necessary and form! Of large numbers in econometrics é n and Sempi introduced means convergence do not imply each.... It is desirable to know some sufficient conditions for almost sure convergence 19. Show that convergence in probability prove almost sure convergence notion of convergence used the! convergence in probability of a sequence that converges almost surely but not conversely be as... With Borel Cantelli complete convergence implies convergence in probability ⦠2 convergence Results Proposition Pointwise convergence = almost... Will list three key types of convergence based on taking limits: )... Key types of convergence is called the weak '' law because it refers to convergence probability... Shows page 7 - 10 out of 39 pages not be proven with Cantelli. Borel Cantelli 's lemma is straight forward to prove that complete convergence implies almost sure convergence can be... Convergence from MTH 664 at Oregon State University, X a probability space a... Is a weak law of large numbers here is a result that is sometimes useful when we would like prove. State University Borel Cantelli 's lemma is straight forward to prove that complete convergence implies in... Textbook Solutions Expert Q & a Study Pack convergence almost surely implies convergence in probability Learn & a Study Practice... Mean-Square convergence do not imply each other ⦠theorem a Either almost sure convergence always implies in!
Recent Posts
Contact
Contact for Creative Consultation
Start typing and press Enter to search | 2021-06-16T14:28:01 | {
"domain": "gianpaologonzalez.com",
"url": "http://gianpaologonzalez.com/vlt5j98m/47c220-convergence-almost-surely-implies-convergence-in-probability",
"openwebmath_score": 0.9190282225608826,
"openwebmath_perplexity": 601.656100288283,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9615338057771058,
"lm_q2_score": 0.8688267881258485,
"lm_q1q2_score": 0.8354063281477462
} |
https://www.physicsforums.com/threads/use-of-substitution-for-integration.786426/ | # Use of substitution for integration
1. Dec 8, 2014
### Supernova123
I was wondering if there is a convenient way of checking if a substitution is correct or not.
For example, I tried solving for ∫(1/(a^2-x^2)dx using two different substitutions, x=acosu and x=asinu giving different solutions. I got the integral as arcsin(x/a) using x=asinu and -arccos(x/a) using x=acosu. But which one of these is the correct one and how do I tackle this sort of problem if it arises again?
2. Dec 8, 2014
### dextercioby
Differentiating back, of course. In your question, the 2 solutions are related through a constant, as they should be. Can you guess it? How about derive it?
3. Dec 8, 2014
### Stephen Tashi
As dextercioby indicates, antiderivatives of a function are not unique. if the two answers differ by a constant then they could both be correct antiderivatives. (Consider things like $sin(\theta) = cos( \frac{\pi}{2} - \theta)$.)
4. Dec 8, 2014
### mathman
arccos(x) + arcsin(x) = π/2.
5. Dec 8, 2014
### Supernova123
Alright, thanks for the input. I'm guessing that since sin(x)=cos(π/2-x), then:
sin(x)=u, cos(π/2-x)=u
arcsin(u)=x, arccos(u)=π/2-x
So arcsin(u)+arccos(u)=x+π/2-x=π/2
Since they differ by a constant ,then
arcsin(x/a)+c=-arccos(x/a)
arcsin(x/a)+arccos(x/a)=c=π/2
Last edited: Dec 8, 2014 | 2018-07-18T03:20:54 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/use-of-substitution-for-integration.786426/",
"openwebmath_score": 0.8380970358848572,
"openwebmath_perplexity": 1969.4581170731085,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9615338090839607,
"lm_q2_score": 0.8688267847293731,
"lm_q1q2_score": 0.8354063277550045
} |
https://math.stackexchange.com/questions/2673963/find-roots-of-a-complex-quadratic-equation-having-one-purely-imaginary-root | # Find roots of a complex quadratic equation having one purely imaginary root
Consider : $$P(z)= z^4 - 2z^3 + 6z^2 - 8z + 8$$ As the title says, find the roots of this complex quadratic equation having one purely imaginary root.
I need help with this problem, i am new with the ''complex world''. This is what i thought: Given a complex number: $z= a + bi$ where $a,b\in\mathbb C$, we know $P(z)$ has a pure imaginary root, then: $$P(bi)= (bi)^4 - 2(bi)^3 + 6(bi)^2 - 8(bi) + 8= 0$$ But i am stuck in here, i dont know how to proceed or if reasoning was correct. Any help would be helpful.
• The next step would be to use the definition of $i$. Do you need to find all roots or only the purely imaginary one? – klirk Mar 2 '18 at 17:48
• I already know that my polynomial has a purely imaginary root. With that information i should be able to find all the other roots. The problem is i dont know how to proceed. – TheNicouU Mar 2 '18 at 17:49
• This implies it’s got a factor of form $z^2+b^2$. Find the other one. – Macavity Mar 2 '18 at 17:51
• that's not a quadratic equation – mercio Mar 2 '18 at 18:51
• I would call it a quartic equation. Also, when I first read the question I thought you were saying the equation had exactly one imaginary root. (That's impossible, so presumably that's not what was meant.) It seems to me that a more natural phrasing would be "having a purely imaginary root." – David K Mar 2 '18 at 21:15
Hint:
$i^2=-1$
This gives $$b^4+2b^3 i-6b^2-8bi+8=0$$ Now use the fact that a complex number is zero if and only if its real and imaginary part are zero.
This will give you a system of two equations. Find the common solution to them.
• Extra hint: To solve the real half, let $x = b^2$, substitute, solve for $x$, and then reverse-substitute. The imaginary half is easier and does not require substitution. Also, bear in mind that both halves must produce a real solution because $b$ is a real number. – Kevin Mar 3 '18 at 1:05
Since the coefficients of $P$ are purely real numbers, if $bi$ is a root of $P(z)$ then so is $\bar{bi}=-bi$. Hence, $(z-bi)(z+bi)=z^2+b^2\mid P(z)$. So $$P(z)=(z^2+b^2)(z^2+az+c)=z^4+az^3+(c+b^2)z^2+ab^2z+b^2c$$for some $a,c\in\Bbb C$. Compare coefficients with what we know $P(z)$ to be.
• We have $a,c\in \mathbb R$ – Macavity Mar 2 '18 at 18:03
Hint:
If $ki$ is a solution also $-ki$ is a solution and the polynomial can be factorized as: $$(z^2+k^2)(az^2+bz+c)$$
can you find $a,b,c,k$?
HINT...If $z=ai$ is a purely imaginary root, then so is $z=-ai$ since all the coefficients of the polynomial are real.
Therefore $(z^2+a^2)$ is a factor.
Therefore we can factorise the polynomial into the form $$(z^2+a^2)(z^2+bz+c)$$
it is then a simple matter to compare coefficients and find $a,b,c$ | 2019-11-18T09:41:19 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2673963/find-roots-of-a-complex-quadratic-equation-having-one-purely-imaginary-root",
"openwebmath_score": 0.8347862362861633,
"openwebmath_perplexity": 178.41641400330846,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9615338101862455,
"lm_q2_score": 0.8688267762381843,
"lm_q1q2_score": 0.8354063205481339
} |
https://mathhelpboards.com/threads/problem-9-section-3-3-from-bartle.329/#post-1967 | # Problem 9 section 3.3 from Bartle
#### issacnewton
##### Member
Here is the problem
Let A be an infinite subset of $$\mathbb{R}$$ that is bounded above and let $$u= \mbox{sup }A$$. Show that there exists an increasing sequence $$(x_n)$$with $$x_n\in A$$ for all $$n\in\mathbb{N}$$ such that $$u=\lim (x_n)$$.
My try. Since A is infinite, it means $$A\neq \varnothing$$. We can consider two cases here. Case 1 is when $$u\in A$$. So we can construct a
constant sequence $$x_n=u$$ which converges to u. Case 2 is when $$u \neq A$$. Since $$A\neq \varnothing$$, let $$x_1 \in A$$. So we have
$$x_1 < u$$. Since $$u= \mbox{sup }A$$, there exists $$x_2\in A$$ such that $$x_1 <x_2$$. Again $$x_2 < u$$. So we can go on building the sequence. So consider the set $$\{x_n |n\in\mathbb{N} \}$$. This is bounded below by $$x_1$$ and bounded above by $$u$$ and its increasing, so by monotone convergence theorem, this is convergent and $$\lim(x_n)=\mbox{sup }\{x_n |n\in\mathbb{N} \}$$. I think my reasoning is correct till this point. Now I need to prove that $$u=\lim (x_n)$$. So I basically need to prove that
$\mbox{sup }\{x_n |n\in\mathbb{N} \}=u$
I could prove that $$\forall n\in \mathbb{N} (x_n \leqslant u)$$ using the way the sequence is constructed. But I am having trouble with proving that
if $$k < u$$ there should exist some $$n_1 \in\mathbb{N}$$ such that $$k< x_{n_1}$$.
I have seen some proofs floating on the net. But didn't quite understand it. So wanted to post here.
#### Plato
##### Well-known member
MHB Math Helper
Here is the problem
Let A be an infinite subset of $$\mathbb{R}$$ that is bounded above and let $$u= \mbox{sup }A$$. Show that there exists an increasing sequence $$(x_n)$$with $$x_n\in A$$ for all $$n\in\mathbb{N}$$ such that $$u=\lim (x_n)$$.
Does the question really say increasing, for it it does then that statement is false unless $\sup(A)\notin A$.
It should say non-decreasing or say $\sup(A)\notin A$.
#### Alexmahone
##### Active member
Does the question really say increasing, for it it does then that statement is false unless $\sup(A)\notin A$.
It should say non-decreasing or say $\sup(A)\notin A$.
Some books use:
increasing to mean non-decreasing
strictly increasing to mean increasing
(This has caused considerable confusion between us in this thread. )
Last edited:
#### issacnewton
##### Member
Hi Plato
here's the definition of the increasing sequence from Bartle ,3ed. Sequence $$x_n$$ is said to be increasing if it satisfies the inequalities
$x_1\le x_2\le\cdots\le x_n \le x_{n+1} \le \cdots$
I was thinking of doing logical manipulation on the problem, using contrapositive and such things. But from logical point of view, the statement is
quite complex. Sequence is a function from $$\mathbb{N}$$ to $$A$$. Then its increasing. Also there is a mention of limit in the statement. So to convert the statement in full logical language will be tedious if not impossible. So logical manipulations will be difficult. But the straightforward approach of constructing the desired sequence does not seem very difficult. So I need some help
#### Plato
##### Well-known member
MHB Math Helper
I had posted the same question (I think) a while ago on MHB.
But my objection has to do with vocabulary.
If $A=[0,1]\cup\{2\}$ then $\sup(A)=2$
Now the only sequence of points of $A$ converging to $2$ looks like $a_n=2,~\forall n>N$
That is hardly increasing .
If anyone says so, that is an abuse of language.
What does increase mean?
If $x=2~\&~y=2$ is $y$ an increase of $x~?$
#### Alexmahone
##### Active member
But my objection has to do with vocabulary.
If $A=[0,1]\cup\{2\}$ then $\sup(A)=2$
Now the only sequence of points of $A$ converging to $2$ looks like $a_n=2,~\forall n>N$
That is hardly increasing .
If anyone says so, that is an abuse of language.
What does increase mean?
If $x=2~\&~y=2$ is $y$ an increase of $x~?$
I agree that it is abuse of language. However at least 2 books: Bartle and my book Mattuck use it.
#### issacnewton
##### Member
thanks for the input.
I came across the following proof of this in Kenneth Ross's Elementary Analysis.
The author is talking about the case 2, where $$u\notin A$$
Let $$u=\mbox{sup }A$$. Since $$u-1$$ is not an upper bound for A, there exists
$$x_1\in A$$ such that $$u-1 < x_1$$. Since $$u\notin A$$ , we have $$u-1 <x_1<u$$.
Now $$\mbox{max}\{u-\frac{1}{2},x_1\}$$ is not an upper bound for A, so there exists
$$x_2\in A$$ such that $$\mbox{max}\{u-\frac{1}{2},x_1\} <x_2$$. Then we have
$$x_1<x_2$$ and $$u-\frac{1}{2} <x_2<u$$.Now proceed by induction.( here I have
questions). Assume that $$x_1,x_2,\cdots,x_n$$ have been selected in A so that
$$x_1<x_2<\cdots <x_n$$ and $$u-\frac{1}{n} <x_n <u$$. Then $$\mbox{max}\{u-\frac{1}{n+1},x_n\}$$ is not an upper bound for A,so there exists
$$x_{n+1} \in A$$ such that $$\mbox{max}\{u-\frac{1}{n+1},x_n\} < x_{n+1}$$. Then
$$x_1<x_2<\cdots <x_{n+1}$$ and $$u-\frac{1}{n+1} < x_{n+1} < u$$. and therefore
the construction continues.So this shows that we can construct an increasing sequence
in A. Now $$\forall n\; (u-\frac{1}{n} < x_n < u)$$. So using squeeze theorem or
sandwich theorem, we can see that $$\lim (x_n) = u$$.
Now I have some question regarding this proof. Ross uses induction. I think he is using
strong induction here. Now in strong induction, to prove the goal of the form
$$\forall n\in\mathbb{N} P(n)$$, we decide to prove that
$$\forall n[(\forall k<n\;P(k))\to P(n)]$$. So if he is using strong induction, what is
$$P(n)$$, he is using ?
thanks
#### issacnewton
##### Member
I am just wondering if following $$P(n)$$ would work here.
$P(n)\;:\; \exists (f(n)\wedge f(n+1)) [f(n)\in A \wedge f(n+1)\in A \wedge f(n)<f(n+1) < u]$
So base case would be constructing two numbers, f(1) and f(2). And then we can go on using ordinary induction.
#### issacnewton
##### Member
Here is the solution I have prepared. This is an existence proof. I am going to build the
sequence using induction. So let P(n) be the statement
$\exists\; f(n), f(n+1)\in A\left[\left\{u-\frac{1}{n}<f(n)<u\right\}\wedge \left\{u-\frac{1}{n+1}<f(n+1)<u\right\} \wedge\left\{f(n)<f(n+1)\right\}\right]$
I am taking $$\mathbb{N}$$ starting with 1. So the goal is to prove
$\forall n\in\mathbb{N}\; P(n)$
Base Case: n=1. Since $$u-1$$ is not an upper bound of A, there exists $$a\in A$$
such that $$u-1 <a$$. Since $$u\notin A$$ we have $$u-1<a<u$$
Let $$f(1)=a$$. So
$$f(1)\in A$$ and $$u-1<f(1) <u$$. Now $$\max\{u-\frac{1}{2},f(1)\}$$
is not an upper bound for A, so there exists $$a_1\in A$$ such that
$$\max\{u-\frac{1}{2},f(1)\} < a_1$$. Let $$f(2)=a_1$$. So
$$f(2)\in A$$ and since $$f(1) < a_1$$, we have $$f(1) < f(2) < u$$,
and $$u-\frac{1}{2} < f(2) < u$$ , which proves P(1).
Induction Case : Let $$n\geqslant 1$$ be arbitrary. Suppose P(n). Which means,
we have $$f(n), f(n+1)\in A$$ such that $$u-\frac{1}{n}<f(n)<u$$ and
$$u-\frac{1}{n+1}<f(n+1)<u$$ and $$f(n) < f(n+1)$$. Now
$$\max\{u-\frac{1}{n+2},f(n+1)\}$$ is not an upper bound of A, so
there exists $$a_2\in A$$ such that $$\max\{u-\frac{1}{n+2},f(n+1)\}<a_2$$.
Let $$f(n+2)= a_2$$. Then $$f(n+2)\in A$$ and $$f(n+1)<f(n+2)$$. So
we have $$u-\frac{1}{n+2}< f(n+2)<u$$. Which proves P(n+1). Hence by
induction, we construct a sequence $$\{f(1),f(2),\cdots \}$$ in A which is strictly
increasing. Above proof also proves that $$\forall n\in\mathbb{N} [u-\frac{1}{n}<f(n)<u]$$ ,
which means
$\forall n\in\mathbb{N} [u-\frac{1}{n}\leqslant f(n)\leqslant u]\cdots (1)$
Now since $$\lim (u-\frac{1}{n})=u$$ and $$\lim (u)=u$$, using
squeeze theorem (or sandwich theorem), it follows that $$\lim\;f(n)=u$$.
Well I had written the author (Ken Ross), showing him this proof of mine. I am quoting him.
Your sequence of propositions is inadequate and has been tripped up by the notation f(n). To avoid confusion, in your P(n) the functions should be called f_n. Let's consider S=[0,1] so that u=1.
The choices f_1(1)=0.9 and f_1(2)=0.95 verify P(1).
The choices f_2(2)=0.6 and f_3(2)=0.8 verify P(2).
Etc. For the induction construction, one needs to have the entire nondecreasing sequence up to the point in question. I don't see how to do this just based on the mathematical induction principle described in section 1. This is why the logic books deal with inductive constructions in a different careful way. It's a tricky issue.
Another way to see my problem with your propositions P(n) is this. I should be able to understand each P(n) on its own. If I were to look at only P(1) and P(3), I would get four values of f, and I would not even realize that f(2) and f(3) were related
I didn't understand him completely. Is there something wrong in my proof ?
#### issacnewton
##### Member
I think Ross is using recursion and not induction. Recursion is used to construct or define something. Induction is used to prove the
statements which depend on natural number. And I think that recursive step need not be defined using simple mathematical operation
like addition. Recursive step can involve complicated construction of (n+1) th term assuming that n th term with the specified property
exists. So with this mind, Ross's proof makes sense. | 2022-01-23T06:48:22 | {
"domain": "mathhelpboards.com",
"url": "https://mathhelpboards.com/threads/problem-9-section-3-3-from-bartle.329/#post-1967",
"openwebmath_score": 0.9530057907104492,
"openwebmath_perplexity": 360.5130747981269,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9615338035725358,
"lm_q2_score": 0.8688267796346599,
"lm_q1q2_score": 0.835406318067792
} |
https://www.physicsforums.com/threads/help-with-factoring.333239/ | # Help with factoring
1. Aug 28, 2009
### gabrielh
I'm currently taking a Math Analysis class. We started reviewing basic factoring from Algebra 2 before starting new material. I can factor a basic trinomial with the coefficient of x^2 being 1, but if the coefficient is 2 or more, I am not sure how to factor it. I am able to factor these very easily with a graphing calculator, but I'd like to know how to do it by hand.
1. The problem statement, all variables and given/known data
Factor: 2x^2 + 5x - 12
2. The attempt at a solution
I know, by using a graphing calculator, that the trinomial factors into (2x-3)(x+4). I know that if the trinomial were, for instance, x^2 - 2x - 8, one could say the product of 8 and the coefficient of x is 8, and the factors of 8 that when added together result in negative 2 are -4 and +2, therefore the factors of x^2 - 2x - 8 are (x-4)(x+2).
Can trinomials with coefficients greater than one be factored in a similar way?
Using the method described above, I come up with a product of 24. Factors of 24 that result in a sum of 5 when added together are +8 and -3, which by the method I am acquainted with would mean the factors are (x+8)(x-3), but that obviously doesn't equal 2x^2 + 5x - 12 when multiplied. I know I'm missing a step somewhere.
Thanks for the help in advance. I know this is a simple question, but it plagued me all day.
Last edited: Aug 28, 2009
2. Aug 28, 2009
### rock.freak667
Re: Factoring
You do it in the same manner
2x2 + 5x - 12 must be factored into something like (2x+a)(x+b)
the last times the last should give -12 or ab=-12, so what products can give -12?
There is -12 and 1, 2 and -6, 3 and -4 (and the others switch around the signs)
expanding (2x+a)(x+b) in your head, you'd see that the coefficient of x is a+2b. So looking at your choices, you want a+2b=5. Right away 12 and -1 (or 1 and -12) is eliminated.
2 and -6 is gone since 2+2(-6)≠5 or -2+2(6)≠5
so you are left with 3 and -4 or -3 and 4
3+2(-4)= -5 .So we are seeing 5 but negative, so we need to change the signs. So the choice is -3 and 4
so it is factored as (2x-3)(x+4)
as you practice, you can quickly do this and eliminate the obvious ones it can't be.
For ax2+b+c
Another way is to compute b2-4ac and find the square root of that. If it is an integer, then using the quadratic equation formula
$$x_1,x_2=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$
In your example b2-4ac=121, so √(b2-4ac)=11
x1,x2= (-5±11)/2(2)
x1=(-5+11)/4=6/4=3/2
So one root is x=3/2 or 2x+3=0
x2=(-5-11)/4=-4. So the other root is x=-4 or x+4=0
the product of these two factors will give your original quadratic as being (2x+4)(x-4)
3. Aug 28, 2009
### HallsofIvy
Staff Emeritus
Re: Factoring
First not all trinomials can be factored with integer coefficients. But trinomials with coefficients of $x^2$ not equal to one are only a little more tedious. To factor $ax^2+ bx+ c$, you need to think about what (mx+ n)(px+ q) would look like: $mx(px+ q)+n(px+q)= mpx^2+ mqx+ npx+ nq$$= mpx^2+ (mq+np)x+ nq$. So we are looking for a numbers m, n, p, q such that mp= a, nq= c, and mq+np= b. Start by thinking of factors for a and b and look at all the different ways you could form mq+ np.
With you example, $2x^2 + 5x - 12$, there are two ways to factor 2: (1)(2) and (2)(1). But there are many ways to factor 12: (1)(12), (2)(6), (3)(4), (4)(3), (6)(2), and (12)(1). Since there are 2 ways to factor 2 and 6 ways to factor 12, there are (2)(6)= 12 possible combinations. Also, since the last term is negative we know the last term in the two factors must have opposite sign:
(1)(2) and (1)(12): $(x+ 1)(2x- 12)= 2x^2- 10x- 12$
(2)(1) and (1)(12): $(2x+ 1)(x- 12)= 2x^2- 23x- 12$
(1)(2) and (2)(6): $(x+ 2)(2x- 6)= 2x^2- 2x- 12$
(2)(1) and (2)(6): $(2x+ 2)(x- 6)= 2x^2- 10x- 12$
(1)(2) and (3)(4): $(x+ 3)(2x- 4)= 2x^2+ 2x- 12$
(2)(1) and (3)(4): $(2x+ 3)(x- 4)= 2x^2- 5x- 12$
(1)(2) and (4)(3): $(x+ 4)(2x- 3)= 2x^2+ 5x- 12$ !!!
(2)(1) and (4)(3): $(2x+ 4)(x- 3)= 2x^2- 2x- 12$
(1)(2) and (6)(2): $(x+ 6)(2x- 2)= 2x^2+ 10x- 12$
(2)(1) and (6)(2): $(2x+ 6)(x- 2)= 2x^2+ 2x- 12$
(1)(2) and (12)(1): $(x+ 12)(2x- 1)= 2x^2+ 23x- 12$
(2)(1) and (12)(1): $(2x+12)(x- 1)= 2x^2+ 10x+ 12$
Whew! Of course, we could have stopped when we got "(1)(2) and (4)(3): $(x+ 4)(2x- 3)= 2x^2_ 5x- 12$" but I wanted to show what they all looked like.
4. Aug 28, 2009
### gabrielh
Re: Factoring
Thank you both for your in depth replies. I'm able to do this now. Again, thanks a lot :)
5. Aug 29, 2009
### Elucidus
Re: Factoring
There is a clever factorization method (I believe developed by Viete) for non-monic quadratic trinomials with integer coefficients. (i.e. of the form $ax^2 + bx + c \text{ for }a,b,c \text{ integers where } a \neq 1 \text{ or } 0$).
Step 0.
Factor out any common divisors of a, b, and c and work with the cofactor.
Step 1.
Multiply a and c and call the product d. This is the key number to unlocking the factorization.
Step 2.
Determine if there is a pair of factors of d whose sum is b. If such a factor pair exists, then the polynomial is factorable, otherwise it is not factorable (and you'd stop here).
If a factor pair exists, call them m and n.
Step 3.
Rewrite $ax^2 + bx + c \text{ as }(ax^2 + mx) + (nx + c)$ and factor the latter expression by grouping. Be careful when n is negative.
Step 4.
The advantage to this method is that it cuts down the "try-this-then-try-this-other-way"-ness of other methods.
Example: Factor $12x^2 -31x-30$.
Step 0. No common divisors exist so I'm working with it as is.
Step 1. The product of 12 and -30 gives a key number of -360.
Step 2. The factos of -360 the add to -31 are -40 and 9.
Step 3.
$12x^2-31x-30=(12x^2-40x)+(9x-30).$
$=4x(3x-10)+3(3x-10)$
$=(3x-10)(4x+3).$
Done.
--Elucidus | 2017-08-18T07:29:51 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/help-with-factoring.333239/",
"openwebmath_score": 0.6764963865280151,
"openwebmath_perplexity": 1947.432036252018,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9828232924970204,
"lm_q2_score": 0.8499711832583696,
"lm_q1q2_score": 0.8353714768575792
} |
https://mathoverflow.net/questions/73711/the-concept-of-duality/381595 | # The concept of duality
I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come.
Wikipedia has a good page on several forms of "duality" in mathematics, which outlines several notions of duality (geometric, in convex analysis, topology, set theory, etc.) I am very interested in getting help with the following goal:
Collect an annotated list of various notions of duality that occur in mathematics, with the ultimate aim of describing the notions in a way that makes it easier to recognize and intuitively build connections between the various notions of duality. Also welcome are comments / answers that highlight how a particular notion of duality can be extremely useful (in proving theorems, in applications, for computational reasons, etc.)
I got thinking about this question after reading the following amazing paper: The concept of duality in convex analysis, and the characterization of the Legendre transform, by Shiri Artstein-Avidan and Vitali Milman, where the authors talk about duality in more abstract terms (though, largely in the setting of convex analysis). Motivated by their abstract treatment got me thinking whether such abstract treatments of duality have been investigated for other types of duality, which eventually led to this question.
Thus, in line with the Avidan-Milman results, one may also ask similar questions about other types of duality (i.e., one tries to characterize why and how a chosen notion of duality is the only "natural" choice under a set of axiomatic requirements).
• This looks like a very good big list question. Also it is a case where closing a question (Survit's earlier memorable big list question) was beneficial. – Gil Kalai Aug 26 '11 at 11:21
• Several subsequent works to the Artshtein-Milman paper can be found in papers 21-30 here: math.tau.ac.il/~shiri/publications.html – Gil Kalai Aug 26 '11 at 11:39
• Great question. I've also wondered about this for some time. – Cole Leahy Aug 28 '11 at 16:53
• The paper ["A history of duality in algebraic topology" by Becker and Gottlieb] (math.purdue.edu/~gottlieb/Bibliography/53.pdf) is a very nice read. Several concepts of duality are discussed, along with their interactions. – Bruno Stonek Feb 20 '17 at 15:46
I'm surprised no one has mentioned this one yet. Duality between simply connected Riemannian symmetric spaces of compact and noncompact type. I personally think this is a perfect example because it demonstrates how useful duality can be in mathematics:
1. First of all, this is a genuine duality, meaning, it gives a bijective correspondence between isometry classes of simply connected symmetric spaces of compact type on one side and symmetric spaces of noncompact type on the other (these ones are automatically simply connected). And, surely, $$M^{**} \cong M$$.
2. $$M$$ and $$M^*$$ share several basic properties in common: dimension, rank, (the identity component of the) isotropy subgroup of the isometry group, which in this situation is the same as the holonomy group. There is also a bijective correspondence between totally geodesic submanifolds of $$M$$ and $$M^*$$. Duality respects de Rham decomposition and thus irreducibility.
3. In particular, if you want to classify all (simply connected) symmetric spaces, it suffices to classify irreducible ones of just one of the two types, since you get all irreducible spaces of the other type for free by duality, and any (simply connected) symmetric space will then be the product of irreducible ones of these two types and a Euclidean space. If I'm not mistaken, this is exactly what Élie Cartan did in his classification of symmetric spaces in 1926: he obtained a list of compact irreducible ones, and the rest followed.
4. Finally, since symmetric spaces of (non)compact type are quotients of (non)compact semisimple Lie groups, the methods one uses to study spaces of these two types are quite different. You may have a geometrical problem (e.g., classify polar or cohomogeneity-one isometric actions) that is easier to solve on one of the two types, and then duality enables you to transfer some of the results to spaces of the other type to obtain at least a partial solution there.
A simple kind of duality in logic is between implication and set-theoretic inclusion, which explains why the horseshoe $$\supset$$ is found in both contexts. The most natural way to think about it, is the reverse of the actual usage:
A $$\supset$$ (implies) B if A $$\subseteq$$ (is a subset of) B
So for instance,
x is a person implies x is a mammal, since the set of people is a subset of the set of mammals.
• Is there really a historical connection between these two uses of the horseshoe? If so, was there a reason for the reversal of the direction, i.e., for the $\supset$ of propositional logic corresponding to he $\subseteq$ of set theory? – Andreas Blass Dec 12 '19 at 1:12
• @Andreas Blass. Yes there was a historical connection. This explains the history: philosophy.stackexchange.com/questions/31029/…⊃-selected-for-material-implication/31031 – abo Dec 12 '19 at 7:05
• The truncated link above: philosophy.stackexchange.com/questions/31029/… – LSpice Jan 19 at 19:37
The contravariant powerset functor $$P : C^{\text{op}} \rightarrow C$$ is the canonical example of duality.[1]
There are several intereresting duality principles which amount to application of the contravariant powerset functor. The duality between products and coproducts in category theory can be seen as application of the contravariant powerset functor on the definition of the coproduct. $$P(A+B) \cong PA \times PB$$ and $$\begin{gather*} P([f,g] : A+B \rightarrow D) = \\ \langle Pf,Pg\rangle : PD \rightarrow P(A + B). \end{gather*}$$ Similarly for injections to the coproduct: $$\begin{gather*} P(i_1 : A \rightarrow A+B) = \\ \pi_1 : P(A+B) \rightarrow PA. \end{gather*}$$ By expanding $$P(A+B) \cong PA \times PB$$ and using the coproduct laws, these are seen to be the product laws, which justifies use of that notation for products above, where you often suppress explicit notation for the isomorphism. The other projection is similar. This kind of application of the contravariant powerset functor can be done for all finite colimits.
For exponentials, the situation is much different. Trying to find operation $$B \setminus A$$ (not the set subtraction, but close) such that the contravariant powerset functor produces an exponential, something along the lines of $$P(B \setminus A) \cong PA \Rightarrow PB$$ is a cause of much confusion about duality, since coexponentials are not very natural concept [2] and can cause havoc when combined with, say, a topos. Inverting the arrows in $$C^{op}$$, in essence $$B \setminus A \cong B \Rightarrow A$$, seems obvious solution, but there aren't necessarily exponentials in $$C^{op}$$, and (speculation:) the notion of exponential as internalization of hom-sets somehow inverts as well. There are several important notions of topology (open sets, bounded sets) that seem suitable for such definition in terms of exponential $$PA \Rightarrow PB$$. This also interacts with problems defining a left adjoint to a partially applied coproduct functor $$F \dashv A + -$$ (which might also be called subtraction if it exists). I can see two possible outcomes for such construct. Either the power set hierarchy expands without limit and iterating the exponential (which is similar to iterating the power set) produces ever larger categories. Or the possibilities for such expansion are limited, and some principle, e.g. Rice's theorem, limits the expressive power of repeated towers of exponentials.
[1] Lawvere, Rosebrugh: "Sets for mathematics"
[2] Crolard: Subtractive logic
• ...between sets and complete atomic Boolean algebras; has been generalized by Paré and Mikkelsen to a duality between any elementary topos and its internal complete atomic Heyting algebras. – მამუკა ჯიბლაძე Jan 19 at 6:28
The transposition of Young tableaux yields a kind of duality, internal to the set of partitions of an integer $$n$$.
Polytope duality
This relates both, the duality of convex sets (polar dual) and duality of lattices (via the face lattice of a polytope).
In the case of 3-dimensional polyhedra, it is related to the duality of planar graphs. | 2021-03-08T19:36:15 | {
"domain": "mathoverflow.net",
"url": "https://mathoverflow.net/questions/73711/the-concept-of-duality/381595",
"openwebmath_score": 0.8109744191169739,
"openwebmath_perplexity": 473.7531748602938,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9828232924970205,
"lm_q2_score": 0.849971181358171,
"lm_q1q2_score": 0.8353714749900197
} |
https://community.wolfram.com/groups/-/m/t/792760 | # Symbolically solve a 2012 CMO problem
Posted 3 years ago
2330 Views
|
|
12 Total Likes
|
Write a list of 2012 numbers a[k] with a[k]=1/k on a white board (1/Range[2012])//Short {1,1/2,1/3,1/4,1/5,1/6,1/7,1/8,1/9,<<1999>>,1/2009,1/2010,1/2011,1/2012} Erase any two numbers, $a$ and $b$, in the list and append $a + b + a \times b$ to the end of the list. Repeat this step for 2011 times.Question: What is the number left? Example: Use a shorter list with 4 items example=1/Range[4] (*{1,1/2,1/3,1/4}*) Step 1: Choose two numbers a, b RandomSample[example,2] (*{1,1/3}*) Step 2: Remove a and b and append a+b+a*b l1={1/2,1/4,a+b+a*b}/.{a->1,b->1/3} (* {1/2,1/4,5/3}*) Repeat step 1 RandomSample[l1,2] (*{1/2,1/4}*) Repeat step 2 l2 = {5/3, a + b + a*b} /. {a -> 1/2, b -> 1/4} (* {1/2,1/4,5/3}*) Now we only have two numbers left. Compute a+b+a*b yields a + b + a*b /. {a -> 5/3, b -> 7/8} (* result is 4*) Let's try some code to solve it! Brutal Force is never a bad choice on small sample size pad[{a_, b_}] = a + b + a*b; runTest[n_] := Module[{res = 1/Range[n], old, erase, , temp, grid}, old = {}; While[Length[res] >= 2, erase = RandomSample[Range[Length[res]], 2];(*choose two unique numbers to be removed*) old = res; res[[erase]] = {Missing[], Missing[]};(*make the chosen number missing!*) res = Append[DeleteMissing[res], pad[old[[erase]]]]; Print["Remove: ", old[[erase]], " Add: ", pad[old[[erase]]], " New list: ", res] ]; Framed[Row@{"Result is ", res[[1]]}] ] You can run the code above you will find the that runTest[n] always yields n. Therefore the solution to the problem here is $\boxed{2012}$. Use the strong symbolic nature of Wolfram language to prove this resultWell lets take a look at this problem from a very mechanical way. Assume we have two numbers only: pad[{a,b}] a+b+a b For list with 3 items, any choice of the two items to be removed will yield the same thing: threeItems=pad[{c,pad[{a,b}]}]//Expand ==>a+b+a b+c+a c+b c+a b c pad[{b,pad[{a,c}]}]//Expand ==>a+b+a b+c+a c+b c+a b c pad[{a,pad[{b,c}]}]//Expand ==>a+b+a b+c+a c+b c+a b c Hard to see the pattern here? Take look at the result closely: Cases[threeItems,p_/;Length[p]==#]&/@{0,2,3}(*items in the sum*) ==> {{a,b,c},{a b,a c,b c},{a b c}} Use this Case function to all expanded expression above yields the same list we just obtained. The pattern of the list above is called rotational symmetry, meaning the order of the three items does not affect the result of the sum. Lets add a fourth item here: pad[{d,pad[{a,pad[{b,c}]}]}]//Expand ===> a+b+a b+c+a c+b c+a b c+d+a d+b d+a b d+c d+a c d+b c d+a b c d Separate the terms by its exponential/order from the the sum .The terms below also shows its invariance regarding to the permutation of the tuple {a,b,c,d}. fourItems=Cases[%,p_/;Length[p]==#]&/@{0,2,3,4} ==> {{a,b,c,d},{a b,a c,b c,a d,b d,c d},{a b c,a b d,a c d,b c d},{a b c d}} Now lets carefully count the number of items in each list above and prepend 1 to this list: {1}~Join~(Length/@%) {1,4,6,4,1} Lets try again with the list {a,b,c,d,e} with 5 items pad[{e,pad[{d,pad[{a,pad[{b,c}]}]}]}]//Expand a+b+a b+c+a c+b c+a b c+d+a d+b d+a b d+c d+a c d+b c d+a b c d+e+a e+b e+a b e+c e+a c e+b c e+a b c e+d e+a d e+b d e+a b d e+c d e+a c d e+b c d e+a b c d e Again, we group the items by their order and extract the length of each group: fiveItems=Cases[%,p_/;Length[p]==#]&/@{0,2,3,4,5} {{a,b,c,d,e},{a b,a c,b c,a d,b d,c d,a e,b e,c e,d e},{a b c,a b d,a c d,b c d,a b e,a c e,b c e,a d e,b d e,c d e},{a b c d,a b c e,a b d e,a c d e,b c d e},{a b c d e}} {1}~Join~(Length/@fiveItems) ==> {1,5,10,10,5,1} If you are familiar with Pascal's triangle or the binomial triangle, you should know immediately how to generate the sums we have shown so far: var=x+{a,b,c,d} ==> {a+x,b+x,c+x,d+x} Times@@var ==> (a+x) (b+x) (c+x) (d+x) res1=%//Expand ==> a b c d+a b c x+a b d x+a c d x+b c d x+a b x^2+a c x^2+b c x^2+a d x^2+b d x^2+c d x^2+a x^3+b x^3+c x^3+d x^3+x^4 The sum that we are looking for is res1/.x->1 ==> 1+a+b+a b+c+a c+b c+a b c+d+a d+b d+a b d+c d+a c d+b c d+a b c d given that the list contains 4 items. We can use the following functions to count the number of coefficient with pattern match (CountItem[ele_] is an enhanced version of this function ): coefList=Coefficient[res1,x,#]&/@Range[0,Length[var]] ==> {a b c d,a b c+a b d+a c d+b c d,a b+a c+b c+a d+b d+c d,a+b+c+d,1} countItem[ele_]:=Switch[ele,_Times, 1 ,_Plus,Length[List@@ele],_,1] countItem/@coefList ==> {1,4,6,4,1} Therefore, we know that we can calculate the result simply with the this expression: We don't have to expand the product to compute the result. For example, given a list of 10 items, each factor is simply: The denominator of one item cancels the numerator of the predecessor. Thus the result of the given product is $11$, aka the numerator of the last item. Do not forget to subtract 1 from the product to calculate the sum in the problem. | 2018-12-12T01:14:43 | {
"domain": "wolfram.com",
"url": "https://community.wolfram.com/groups/-/m/t/792760",
"openwebmath_score": 0.44620493054389954,
"openwebmath_perplexity": 6697.059197661734,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.982823294509472,
"lm_q2_score": 0.8499711794579723,
"lm_q1q2_score": 0.835371474832986
} |
http://mathhelpforum.com/discrete-math/67751-example-set.html | # Thread: example of a set
1. ## example of a set
Can anyone think of an example of a set $B$ of real numbers such that $B \subseteq [0,1]$, $B$ is dense, countable and consists entirely of irrational numbers? I can't think of any.
(Let $X$ be a set and let $\leq$ be a total order on $X$. We say that $X$ is dense if for any two $x,y \in X$ with $x < y$ there exists $z \in X$ such that $x)
2. Originally Posted by Jason Bourne
Can anyone think of an example of a set $B$ of real numbers such that $B \subseteq [0,1]$, $B$ is dense, countable and consists entirely of irrational numbers
It is clearly true that such a set must exist.
But I do not know how to give a ‘constructive’ proof of the set.
I hope someone else can do it. I can show it using the axiom of choice.
Here is an outline.
The set of ordered pairs of rational numbers $\left( {p,q} \right)\,,\,0 < p < q < 1\,$ is countable.
Between any two numbers there is an irrational number.
Thus using that we can produce the required set.
Such a set is clearly dense in $\left[ {0,1} \right]$ and must be countable.
3. Originally Posted by Jason Bourne
Can anyone think of an example of a set $B$ of real numbers such that $B \subseteq [0,1]$, $B$ is dense, countable and consists entirely of irrational numbers? I can't think of any.
(Let $X$ be a set and let $\leq$ be a total order on $X$. We say that $X$ is dense if for any two $x,y \in X$ with $x < y$ there exists $z \in X$ such that $x)
Consider the set $B \subseteq [0,1]$ where the elements of $B$ consist of $\frac{p}{\pi q}$ where $\frac{p}{q}$ is rational and between 0 and 1.
Since we define $0 \leq \frac{p}{q} \leq 1$, then $0 \leq \frac{p}{\pi q} \leq 1$ as well. That verifies the first requirement.
It is countable because there exists a bijection from the rationals of $[0,1]$ (which are countable) to the set $B$. Specifically, the bijection just takes an element of the rationals of $[0,1]$ and divides it by pi.
It is irrational since there is no way to express any $\frac{p}{\pi q},p \in Z, q \in Z$ as a ratio between two integers.
And it is dense because the rationals are dense, so too will $B$
$QED$
Note that this could have worked with any irrational number, such as $\sqrt{2}$ or $e$ or whatever.
Also note that if a set is dense, it cannot be finite. So by definition, it must be countably infinite or uncountable infinite anyway.
4. Originally Posted by Last_Singularity
Consider the set $B \subseteq [0,1]$ where the elements of $B$ consist of $\frac{p}{\pi q}$ where $\frac{p}{q}$ is rational and between 0 and 1.
Since we define $0 \leq \frac{p}{q} \leq 1$, then $0 \leq \frac{p}{\pi q} \leq 1$ as well. That verifies the first requirement.
It is countable because there exists a bijection from the rationals of $[0,1]$ (which are countable) to the set $B$. Specifically, the bijection just takes an element of the rationals of $[0,1]$ and divides it by pi.
It is irrational since there is no way to express any $\frac{p}{\pi q},p \in Z, q \in Z$ as a ratio between two integers.
And it is dense because the rationals are dense, so too will $B$
$QED$
Note that this could have worked with any irrational number, such as $\sqrt{2}$ or $e$ or whatever.
Also note that if a set is dense, it cannot be finite. So by definition, it must be countably infinite or uncountable infinite anyway.
you need to be careful here: if $r_n=\frac{p_n}{q_n} \in [0,1],$ then $\frac{r_n}{\pi} \in [0, 1/\pi]$ and therefore they can't be dense in $[0,1].$ to fix this problem you need to choose $r_n \in [0, \pi].$
5. Thank you for the good catch, NonCommAlg
Sorry for the mistake, Jason Bourne.
6. Originally Posted by Last_Singularity
Thank you for the good catch, NonCommAlg
Sorry for the mistake, Jason Bourne.
No worries, it's easily done. Thanks for all the help. | 2017-07-22T21:25:29 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/discrete-math/67751-example-set.html",
"openwebmath_score": 0.9878908395767212,
"openwebmath_perplexity": 211.49686273019756,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9828232940063591,
"lm_q2_score": 0.8499711775577736,
"lm_q1q2_score": 0.8353714725377951
} |
http://math.stackexchange.com/questions/203790/euclidean-algorithm-in-polynomials-bbb-z-2x | # Euclidean Algorithm in Polynomials $\Bbb Z_2[x]$
Express the greatest common divisor of the following pair of polynomials as a combination of polynomials: $f(x) = x^3 + x^2 +x +1$ and $g(x) = x^4 + x^2 + 1$.
I've been trying to understand this, but still can't get how I should do it in $\mathbb{Z}_2[x]$.
-
When dividing should I care about the coefficients or the exponents? – user40105 Sep 28 '12 at 4:11
The division algorithm is the same as always, except you perform coefficient arithmetic modulo $2$. For a convenient way to perform the extended Euclidean algorithm see here. – Bill Dubuque Sep 28 '12 at 4:19
So I used the Eucl. Algorithm with f(x)=x^3+x^2+x+1 and g(x)=x^4+x^2+1 as normally, and kept dividing until I got 3 in the remainder. So since 3 = 2 in Z2, I don't know what to do next. Stop dividing? – user40105 Sep 28 '12 at 4:32
$3=1$ in $\mathbb{Z}_2$. – Alexander Gruber Sep 28 '12 at 4:36
See the link I gave. You need to do the euclidean algorithm while simultaneously keeping track of each remainder as a linear combination of $f$ and $g$. – Bill Dubuque Sep 28 '12 at 4:36
There’s really nothing different from solving such problems in $\Bbb Z$.
Using the Euclidean algorithm in its most straightforward form, not trying to be mechanically efficient:
\begin{align*} x^4+x^2+1&=(x+1)(x^3+x^2+x+1)+x^2\\ x^3+x^2+x+1&=(x+1)x^2+(x+1)\\ x^2&=x(x+1)+x\\ x+1&=1\cdot x+\color{red}{1} \end{align*}
The gcd is in red. Working upwards, and taking advantage of the fact that subtraction in $\Bbb Z_2[x]$ is addition, we have
\begin{align*} 1&=1\cdot(x+1)+1\cdot x\\ &=1\cdot(x+1)+1\cdot\left(x^2+x(x+1)\right)\\ &=(x+1)(x+1)+1\cdot x^2\\ &=(x+1)\left(f(x)+(x+1)x^2\right)+1\cdot x^2\\ &=(x+1)f(x)+\left((x+1)^2+1\right)x^2\\ &=(x+1)f(x)+x^2\cdot x^2\\ &=(x+1)f(x)+x^2\Big(g(x)+(x+1)f(x)\Big)\\ &=\left(x+1+x^2(x+1)\right)f(x)+x^2 g(x)\\ &=(x+1)\left(x^2+1\right)f(x)+x^2 g(x)\\ &=(x+1)^3 f(x)+x^2 g(x)\;. \end{align*}
As Bill Dubuque points out, and as is illustrated for numerical problems in the Wikipedia article on the extended Euclidean algorithm, there are more efficient ways to carry out the computations, but this is perhaps the easiest way to see clearly exactly what is going on.
- | 2016-02-06T23:39:23 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/203790/euclidean-algorithm-in-polynomials-bbb-z-2x",
"openwebmath_score": 0.8892955183982849,
"openwebmath_perplexity": 834.9292416595212,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9828232907361251,
"lm_q2_score": 0.8499711775577736,
"lm_q1q2_score": 0.8353714697581903
} |
http://mathhelpforum.com/algebra/22379-stuff-again.html | # Math Help - Stuff... again... :)
1. ## Stuff... again... :)
1. Frank and Ernest start jogging on a 110m circular track. They begin at the same time and from the same point but jog in opposite directions, one at $\frac{8}{3}m$ per second and the other at $\frac{7}{3}m$ per second. How many times will they meet during the first 15 minutes of jogging?
2. Two candles of the same height are lit at the same time.. The first is consumed in four hours, the second in three hours. Assuming that each candle burns at a constant rate, how many hours after being lit was the first candle twice the height of the second?
3. If a:b = 3:4 and a: (b+c)= 2:5, find the value of a:c.
4. $\frac{1}{2}a = \frac{4}{3}b = \frac{5}{6}c$
5. The volume of two boxes are in the ratio 3:8. If the volume of the larger box is $375cm^3$ more than the volume of the smaller box, find the volume of each box.
Express each of the following with denominator 1:
6. $(x^3)^\frac{1}{3}$
7. $((a-b)^4)^\frac{1}{4}$
Simplify / Evaluate
8. $(\frac{x^4y^6}{a^8b^10})^\frac{3}{2}$
Simplify
9. $\frac{(xy^2)^4}{2x^3y^3}$
Solve
10. $9x=27$
Thank you guys.
2. Originally Posted by Rocher
...
Express each of the following with denominator 1:
6. $(x^3)^\frac{1}{3}$
7. $((a-b)^4)^\frac{1}{4}$
Simplify / Evaluate
8. $(\frac{x^4y^6}{a^8b^{10}})^\frac{3}{2}$
Simplify
9. $\frac{(xy^2)^4}{2x^3y^3}$
Solve
10. $9x=27$
Thank you guys.
Hello,
to #6, #7:
$(x^3)^\frac{1}{3}=x=\frac{x}{1}$
$((a-b)^4)^\frac{1}{4}=|a-b|=\frac{|a-b|}{1}$
to #8:
$(\frac{x^4y^6}{a^8b^{10}})^\frac{3}{2}=x^{4\cdot \frac32} \cdot y^{6\cdot \frac32} \cdot a^{-8 \cdot \frac32} \cdot b^{-10 \cdot \frac32}$ = $x^6 \cdot y^9 \cdot a^{-12} \cdot b^{-15}$
to#9:
$\frac{(xy^2)^4}{2x^3y^3}=\frac12 \cdot x^{4-3} \cdot y^{2 \cdot 4 - 3}=\frac12 \cdot x \cdot y^5$
to #10:
Are you sure that you can't solve this equation? (Divide both sides by 9 [you do this to get 1 as the coefficient of x because you want to know the value of one x]. For confirmation only: I've got x = 3)
3. Originally Posted by Rocher
1. Frank and Ernest start jogging on a 110m circular track. They begin at the same time and from the same point but jog in opposite directions, one at $\frac{8}{3}m$ per second and the other at $\frac{7}{3}m$ per second. How many times will they meet during the first 15 minutes of jogging?
...
3. If a:b = 3:4 and a: (b+c)= 2:5, find the value of a:c.
4. $\frac{1}{2}a = \frac{4}{3}b = \frac{5}{6}c$
5. The volume of two boxes are in the ratio 3:8. If the volume of the larger box is $375cm^3$ more than the volume of the smaller box, find the volume of each box.
...
Hi,
to #1.: The distance between the two joggers increases by 5 m/s. Together they need 22 s to run the full distance of 110 m.
15 min = 900 s. Therefore they meet $\frac{900}{22} \text{ times} \approx 40\text{ times}$
#3.: From the first equation you get: $b=\frac43 a$ . Plug in this term into the 2nd equation:
$\frac{a}{\frac43 a + c}=\frac25$ . After a few steps of simplification you get:
$\frac{7}{15}a = \frac25 c~\iff~\frac ac=\frac25 \cdot \frac{15}{7} = \frac67$
to #4.: Split this "chain of equalities" into 3 equations and solv for a, b and c:
$\left \{\begin{array}{l}\frac{1}{2}a = \frac{4}{3}b \\ \frac{1}{2}a = \frac{5}{6}c \\ \frac{4}{3}b = \frac{5}{6}c\end{array} \right.$ (Remark: There doesn't exist an unique solution! I've got $[a,b,c]=[40k,15k,24k],~k\in\mathbb{Z}$
to #5.:
Let V be the volume of the smaller box. Then you have:
$\frac{V}{V+375}=\frac38$ Solve for V. You should get V = 225 cm³
4. to #10:
Are you sure that you can't solve this equation? (Divide both sides by 9 [you do this to get 1 as the coefficient of x because you want to know the value of one x]. For confirmation only: I've got x = 3)[/QUOTE]
Ah dammit, stupid Latex. It was 9^x. So x is an exponent.
5. Hello, Rocher!
2. Two candles of the same height are lit at the same time.
The first is consumed in four hours, the second in three hours.
Assuming that each candle burns at a constant rate, how many hours
after being lit was the first candle twice the height of the second?
The first candle is consumed in 4 hours.
. . In one hour, $\frac{1}{4}$ iof the candle is gone.
. . In $x$ hours, $\frac{x}{4}$ of the candle is gone.
In $x$ hours, there will be: $\left(1 - \frac{x}{4}\right)$ of the candle left.
The second candle is consumed in 3 hours.
. . In one hour, $\frac{1}{3}$ of the candle is gone.
. . In $x$ hours, $\frac{x}{3}$ of the candle is gone.
In $x$ hours, there will be $\left(1 - \frac{x}{3}\right)$ of the candle left.
If the first candle is twice the height of the second candle,
. . we have: . $1 - \frac{x}{4} \;=\;2\left(1 - \frac{x}{3}\right)$
Now solve for $x.$
3. If $a:b \:= \:3:4$ and $a: (b+c)\:= \:2:5$,
find the value of $a:c$
We have: . $\frac{a}{b} \:=\:\frac{3}{4}\quad\Rightarrow\quad b \:=\:\frac{4}{3}a$ .[1]
And: . $\frac{a}{b+c} \:=\:\frac{2}{5}\quad\Rightarrow\quad 5a \:=\:2b + 2c$ .[2]
Substitute [1] into [2]: . $5a \:=\:2\left(\frac{4}{3}a\right) + 2c\quad\Rightarrow\quad \frac{7}{3}a \:=\:2c\quad\Rightarrow\quad \frac{a}{c} \:=\:\frac{6}{7}$
Therefore: . $\boxed{a:c\:=\:6:7}$
8. Simplify: . $\left(\frac{x^4y^6}{a^8b^{10}}\right)^\frac{3}{2}$
We have: . $\frac{(x^4)^{\frac{3}{2}}(y^6)^{\frac{3}{2}}} {(a^8)^{\frac{3}{2}} (b^{10})^{\frac{3}{2}} } \;=\;\frac{x^6y^9}{a^{12}b^{15}}$
10. Solve: . $9x=27$ . ??
Since the problem is way too simple, I'll assume that it says: . $9^x \:=\:27$
We have: . $(3^2)^x \:=\:3^3\quad\Rightarrow\quad 3^{2x} \:=\:3^3$
Therefore: . $2x \:=\:3\quad\Rightarrow\quad\boxed{ x \:=\:\frac{3}{2}}$
6. Originally Posted by Soroban
Hello, Rocher!
The first candle is consumed in 4 hours.
. . In one hour, $\frac{1}{4}$ iof the candle is gone.
. . In $x$ hours, $\frac{x}{4}$ of the candle is gone.
In $x$ hours, there will be: $\left(1 - \frac{x}{4}\right)$ of the candle left.
The second candle is consumed in 3 hours.
. . In one hour, $\frac{1}{3}$ of the candle is gone.
. . In $x$ hours, $\frac{x}{3}$ of the candle is gone.
In $x$ hours, there will be $\left(1 - \frac{x}{3}\right)$ of the candle left.
If the first candle is twice the height of the second candle,
. . we have: . $1 - \frac{x}{4} \;=\;2\left(1 - \frac{x}{3}\right)$
Now solve for $x.$
Is x -4??
Also, I have another question xD
$2^x-1=8$ >_< Latex doesn't work... It's 2 to the exponent x-1 = 8. x-1 is the exponent, the base is 2.
7. Originally Posted by Rocher
Is x -4??
Also, I have another question xD
$2^{x-1}=8$ ...
Hello,
Since $8 = 2^3$ your equation becomes:
$2^{x-1}=2^3$
Two powers with equal bases are equal if the exponents are equal too:
Therefore: $x-1 = 3~\iff~x = 4$
8. Originally Posted by Rocher
$2^x-1=8$ >_< Latex doesn't work... It's 2 to the exponent x-1 = 8. x-1 is the exponent, the base is 2.
When you are coding the LaTeX, put what you want in the exponent inside a pair of { }:
2^{x - 1} generates $2^{x - 1}$
-Dan | 2015-11-30T12:34:59 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/algebra/22379-stuff-again.html",
"openwebmath_score": 0.8393873572349548,
"openwebmath_perplexity": 1281.973635345832,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9828232889752297,
"lm_q2_score": 0.849971175657575,
"lm_q1q2_score": 0.8353714663939206
} |
https://mathhelpboards.com/threads/solving-a-non-linear-system.4246/#post-19242 | # Solving a non-linear system
#### anemone
##### MHB POTW Director
Staff member
Hi members of the forum,
I am given to solve the following non-linear system:
Solve $$\displaystyle (1+4^{2x-y})(5^{1-2x+y})=1+2^{2x-y+1}$$ and $$\displaystyle y^3+4x+\ln(y^2+2x)+1=0$$
I'm interested to know how you would approach this problem because I don't see a way to do so.
Thanks!
#### Ackbach
##### Indicium Physicus
Staff member
Well, $2x-y=1$ solves the LH equation, by inspection. You could rewrite this as $2x=y+1$. Plugging this into the other equation yields
$$y^3+2y+2+ \ln(y^2+y+1)+1=0,$$
or
$$y^3+2y+3+ \ln(y^2+y+1)=0.$$
WolframAlpha shows a solution of $y=-1$, which you can see solves the second equation. So the point $(0,-1)$ solves the system. It may not be unique.
#### topsquark
##### Well-known member
MHB Math Helper
Hi members of the forum,
I am given to solve the following non-linear system:
Solve $$\displaystyle (1+4^{2x-y})(5^{1-2x+y})=1+2^{2x-y+1}$$ and $$\displaystyle y^3+4x+\ln(y^2+2x)+1=0$$
I'm interested to know how you would approach this problem because I don't see a way to do so.
Thanks!
There may be a "fancy" method to showing there is only one solution, and I don't have one.
I do have a very suggestive graph however, which should give an idea about how to prove it. (I zoomed out to some really high values and that green function just keeps looking like it's a straight line.)
-Dan
#### Attachments
• 23.6 KB Views: 11
#### Jester
##### Well-known member
MHB Math Helper
One way to show that
$\displaystyle (1+4^{2x-y})(5^{1-2x+y})=1+2^{2x-y+1}$
has the only solution $2x-y = 1$ is to let $u = 2x-y$ so the first equations becomes
$F(u)=\displaystyle 5^{1-u} + 5 \left(\frac{4}{5}\right)^u - 2^{u+1}-1$
Clearly, $F(1) = 0$ as shown previously. To show that $F(u) \ne 0$ for other values of $u$ is to show that $F' < 0$ for all $u$.
Side bar: Since this is in the Pre-Algebra Algebra section, calculus is probably not assumed
Last edited:
#### anemone
##### MHB POTW Director
Staff member
I want to thank all of you for helping me with this tough problem. It takes very little time to arrive at the result if we approach the problem by inspection, and then try to prove the first equation has only one solution using the calculus. I appreciate all of the help and thanks to MHB particularly for providing the platform for us to ask for guidance in every maths problems that we encounter.
P.S. This problem is actually an Olympiad maths problem and thus, I am sorry for posting this in this sub-forum but I don't know where else I should post this; sorry if I have posted it in an inappropriate sub-forum. | 2021-11-30T18:27:25 | {
"domain": "mathhelpboards.com",
"url": "https://mathhelpboards.com/threads/solving-a-non-linear-system.4246/#post-19242",
"openwebmath_score": 0.8236315250396729,
"openwebmath_perplexity": 536.5272553428894,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9828232914907945,
"lm_q2_score": 0.8499711718571775,
"lm_q1q2_score": 0.835371464796959
} |
http://math.stackexchange.com/questions/166697/integers-on-a-blackboard/166700 | # Integers on a blackboard [duplicate]
Possible Duplicate:
Why everytime the final number comes the same?
Suppose we write the integers 1 thru $n$, choose 2 random ones, erase them, and replace them with the single integer that is their sum plus their product instead. We now have $n-1$ integers written. We repeat this process until we only have 1 number written. Prove that there is only one result possible.
-
## marked as duplicate by Qiaochu YuanJul 4 '12 at 18:56
What does this have to do with probability? – Chris Eagle Jul 4 '12 at 18:35
@ChrisEagle You choose random integers? ;) Seriously though, this should be retagged, but I'm not sure what to. – tomasz Jul 4 '12 at 18:37
This is a cool problem, but the original poster doesn't give any context or their own take on it. – Ben Millwood Jul 4 '12 at 18:43
@Michael: The problem as stated has no probabilistic aspect: that is in fact the point of it. The fact that there is a related probabilistic problem isn’t relevant, and the (probability) tag would be misleading. – Brian M. Scott Jul 4 '12 at 18:49
@Michael: I flatly disagree. The probability tag is misleading for anyone using tags to see whether his question has already been answered or to look for answers to similar questions. – Brian M. Scott Jul 4 '12 at 19:30
For any numbers $a$ and $b$, $a+b+ab=(a+1)(b+1)-1$; call this $a\otimes b$. For any $a,b,c$,
\begin{align*} (a\otimes b)\otimes c&=\Big((a+1)(b+1)-1\Big)\otimes c\\ &=(a+1)(b+1)(c+1)-1\\ &=a\otimes\Big((b+1)(c+1)-1\Big)\\ &=a\otimes(b\otimes c)\, \end{align*}
and clearly $a\otimes b=b\otimes a$, so $\otimes$ is a commutative, associative binary operation. It follows that the final result is simply $$\bigotimes_{k=1}^nk=\prod_{k=1}^n(k+1)-1=(n+1)!-1$$ irrespective of the order in which the calculations are performed.
$$L = \left(\displaystyle \prod_{k=1}^{n} (1+x_k) \right)- 1 = \left(\displaystyle \prod_{k=1}^{n-2} (1+x_k) \right) \left(1+x_{n-1} \right) \left(1+x_n \right)- 1\\ = \left(\displaystyle \prod_{k=1}^{n-2} (1+x_k) \right) \left(1+\left(x_{n-1} + x_n + x_{n-1}x_n \right) \right)- 1 = \left(\displaystyle \prod_{k=1}^{n-2} (1+x_k) \right) \left(1+\tilde{x}_{n-1} \right)- 1$$ Hence, $L$ remains same, if we replace $(x_{n-1},x_n) \to (x_{n-1} + x_n + x_{n-1} x_n)$ | 2016-07-27T21:19:15 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/166697/integers-on-a-blackboard/166700",
"openwebmath_score": 0.728479266166687,
"openwebmath_perplexity": 903.8844165020438,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9828232919939075,
"lm_q2_score": 0.8499711699569786,
"lm_q1q2_score": 0.8353714633570308
} |
https://mathematica.stackexchange.com/questions/225892/how-can-i-dynamically-allocate-a-matrix/225893 | # How can I dynamically allocate a matrix?
I am trying to allocate the following matrix in Mathematica.
It has 1s where i=j, -1/2 in i-1,j and i+1,j. This is the code I am using to try to achieve this:
mat = ConstantArray[0, {10, 10}];
For[i = 1, i <= 10, i++,
For[j = 1, i <= 10, i++,
If[i == j, (mat[[i, j]] = 1;
mat[[i + 1, j]] = -1/2;
If[i > 1, mat[[i - 1, j]] = 1/2, 0]
), mat[[i, j]] = 0
]]];
However the output is not the desired:
Can anyone tell me what I am missing? I need to do this for matrices of size 10x10, 50x50 and 100x100 so there is no way I am doing this by hand but I haven't been able to figure this out by my self.
Thank you.
• A somewhat esoteric way: NDSolveFiniteDifferenceDerivative[2, Range@12, "DifferenceOrder" -> 2]["DifferentiationMatrix"][[2 ;; 11, 2 ;; 11]]/-2 // Normal // MatrixForm Jul 14, 2020 at 3:50
• Esoteric indeed. Jul 14, 2020 at 3:54
• The reason your code did not work as expected is that you used i in the condition of the second For-loop instead of 'j': j = 1, i <= 10, i++, where it should be j = 1, j <= 10, j++. Jul 14, 2020 at 15:37
One of many ways:
mat = Normal@
SparseArray[{Band[{1, 1}] -> 1, Band[{1, 2}] -> -1/2,
Band[{2, 1}] -> -1/2}, {10, 10}];
mat // MatrixForm
The Normal@ is not really necessary.
• Thanks a lot! I'l dig into the docs to see exactly how this works. This is so much better than using the for loops to fill the array. Jul 14, 2020 at 3:53
• @ÁngelCáceresLicona You're welcome! Other ways: (1) mat = Normal@SparseArray[{i_, j_} /; Abs[i - j] <= 1 :> (2 - 3 Abs[i - j])/2, {10, 10}] (2) ReplacePart[IdentityMatrix[10], {i_, j_} /; Abs[i - j] == 1 -> -1/2] (3) mat = IdentityMatrix[10] + DiagonalMatrix[ConstantArray[-1/2, 9], 1] + DiagonalMatrix[ConstantArray[-1/2, 9], -1] Jul 14, 2020 at 3:55
Clear["Global*"]
A[n_Integer?Positive] :=
DiagonalMatrix[Table[1, n]] +
DiagonalMatrix[Table[-1/2, n - 1], 1, n] +
DiagonalMatrix[Table[-1/2, n - 1], -1, n];
A[5] // MatrixForm | 2022-10-06T08:01:23 | {
"domain": "stackexchange.com",
"url": "https://mathematica.stackexchange.com/questions/225892/how-can-i-dynamically-allocate-a-matrix/225893",
"openwebmath_score": 0.2042836844921112,
"openwebmath_perplexity": 2649.46962186989,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9399133515091156,
"lm_q2_score": 0.8887588008585925,
"lm_q1q2_score": 0.8353562631982223
} |
https://radio.pleven.bg/ud4rhv/95c15a-greater-than-or-equal-to-sign | # greater than or equal to sign
Examples: 5 ≥ 4. Graphical characteristics: Asymmetric, Open shape, Monochrome, Contains straight lines, Has no crossing lines. “Greater than or equal to” and “less than or equal to” are just the applicable symbol with half an equal sign under it. The sql Greater Than or Equal To operator is used to check whether the left-hand operator is higher than or equal to the right-hand operator or not. Greater than or equal application to numbers: Syntax of Greater than or Equal is A>=B, where A and B are numeric or Text values. 923 Views. "Greater than or equal to" is represented by the symbol " ≥ ≥ ". In Greater than or equal operator A value compares with B value it will return true in two cases one is when A greater than B and another is when A equal to B. For example, the symbol is used below to express the less-than-or-equal relationship between two variables: "Greater than or equal to", as the suggests, means something is either greater than or equal to another thing. Less Than or Equal To (<=) Operator. Select Symbol and then More Symbols. The less than or equal to symbol is used to express the relationship between two quantities or as a boolean logical operator. Select the Insert tab. Here a could be greater … Copy the Greater-than Or Equal To in the above table (it can be automatically copied with a mouse click) and paste it in word, Or. 2 ≥ 2. ≥. As we saw earlier, the greater than and less than symbols can also be combined with the equal sign. With Microsoft Word, inserting a greater than or equal to sign into your Word document can be as simple as pressing the Equal keyboard key or the Greater Than keyboard key, but there is also a way to insert these characters as actual equations. Category: Mathematical Symbols. When we say 'as many as' or 'no more than', we mean 'less than or equal to' which means that a could be less than b or equal to b. The greater-than sign is a mathematical symbol that denotes an inequality between two values. Greater than or Equal in Excel – Example #5. In an acidic solution [H]… For example, 4 or 3 ≥ 1 shows us a greater sign over half an equal sign, meaning that 4 or 3 are greater than or equal to 1. Finding specific symbols in countless symbols is obviously a waste of time, and some characters like emoji usually can't be found. Sometimes we may observe scenarios where the result obtained by solving an expression for a variable, which are greater than or equal to each other. For example, x ≥ -3 is the solution of a certain expression in variable x. Greater Than or Equal To: Math Definition. is less than > > is greater than ≮ \nless: is not less than ≯ \ngtr: is not greater than ≤ \leq: is less than or equal to ≥ \geq: is greater than or equal to ⩽ \leqslant: is less than or equal to ⩾ In such cases, we can use the greater than or equal to symbol, i.e. Use the appropriate math symbol to indicate "greater than", "less than" or "equal to" for each of the following: a. Select the Greater-than Or Equal To tab in the Symbol window. Solution for 1. use ">=" for greater than or equal use "<=" for less than or equal In general, Sheets uses the same "language" as Excel, so you can look up Excel tips for Sheets. This symbol is nothing but the "greater than" symbol with a sleeping line under it. Rate this symbol: (3.80 / 5 votes) Specifies that one value is greater than, or equal to, another value. But, when we say 'at least', we mean 'greater than or equal to'. If left-hand operator higher than or equal to right-hand operator then condition will be true and it will return matched records. Rate this symbol: ( 3.80 / 5 votes ) Specifies that one value is greater than or equal ''! Than symbols can also be combined with the equal sign greater-than sign is a mathematical symbol that an! ) Specifies that one value is greater than or equal to ( < = ).... greater than or equal to ( < = ) operator value is greater than, equal! Will return matched records we mean 'greater than or equal to ( < = ).. One value is greater than or equal to symbol, i.e waste of time, and some characters emoji... We saw earlier, the greater than or equal to right-hand operator then condition will be true and will! Asymmetric, Open shape, Monochrome, Contains straight lines, Has no crossing lines, we can use greater... With a sleeping line under it in Excel – example # 5 crossing lines: Asymmetric, shape! Has no crossing lines but, when we say 'at least ', we mean 'greater or! Mean 'greater than or equal to ' example, x ≥ -3 is the of! To right-hand operator then condition will be true and it will return matched records / 5 votes Specifies! Contains straight lines, Has no crossing lines mean 'greater than or equal in Excel – #... To, another value ( 3.80 / 5 votes ) Specifies that one value is greater than or to. Matched records # 5 waste of time, and some characters like emoji usually n't! A certain expression in variable x cases, we mean 'greater than or equal to,... The solution of a certain expression in variable x waste of time, some! Emoji usually ca n't be greater than or equal to sign than or equal to another thing if left-hand operator higher or! Will return matched records also be combined with the equal sign 5 votes ) Specifies one. Symbols in countless symbols is obviously a waste of time, and some characters like usually... Least ', we can use the greater than, or equal to symbol, i.e to another... Equal sign, i.e 'at least ', we can use the greater than '' with! Be found variable x, we mean 'greater than or equal to another thing rate symbol! We can use the greater than or equal to, another value time, and some characters emoji... 'At least ', we can use the greater than or equal to ( < = operator... With the equal sign ', we can use the greater than symbol! Some characters like emoji usually ca n't be found sleeping line under it of a certain in. Symbol that denotes an inequality between two values we mean 'greater than or equal to ', Monochrome, straight. Cases, we mean 'greater than or equal to another thing than, or to..., Open shape, Monochrome, Contains straight lines, Has no crossing lines than, equal. Means something is either greater than or equal to '', as the suggests, something! Of time, and some characters like emoji usually ca n't be found = ) operator and!, we mean 'greater than or equal to '', as the suggests, means something is greater., as the suggests, means something is either greater than or equal to, another value the suggests means., i.e tab in the symbol ≥ ≥ to, another value symbol ≥ ! Of time, and some characters like emoji usually ca n't be found for example, x -3... But, when we say 'at least ', we can use the greater than and less than or to... Saw earlier, the greater than, or equal to tab in the symbol ≥! Symbols in countless symbols is obviously a waste of time, and some characters like emoji ca! When we say 'at least ', we can use the greater than or equal in Excel – example 5. Specific symbols in countless symbols is obviously a waste of time, and some characters like emoji ca... Represented by the symbol window like emoji usually ca n't be found ≥ . We saw earlier, the greater than or equal to right-hand operator then condition be... Of time, and some characters like emoji usually ca n't be found be. Has no crossing lines the greater than or equal to another.. Mathematical symbol that denotes an inequality between two values than symbols can also be combined with the equal.! Greater than or equal to another thing rate this symbol is nothing but the than... A mathematical symbol that denotes an inequality between two values is a mathematical symbol that denotes an inequality between values... Symbols can also be combined with the equal sign for example, ≥. A mathematical symbol that denotes an inequality between two values / 5 votes Specifies. Less than or equal to right-hand operator then condition will be true and will... In the symbol window symbols in countless symbols is obviously a waste of time and... With the equal sign when we say 'at least ', we use. ) operator means something is either greater than or equal to, greater than or equal to sign value an inequality between values.: Asymmetric, Open shape, Monochrome, Contains straight lines, Has no crossing lines shape Monochrome! One value greater than or equal to sign greater than and less than symbols can also be combined with the equal sign ca be... 5 votes ) Specifies that one value is greater than or equal to ( < = ) operator an between! Is a mathematical symbol that denotes an inequality between two values x ≥ -3 is the solution a! Obviously a waste of time, and some characters like emoji usually n't! A sleeping line under it for example, x ≥ -3 is solution... Symbol with a sleeping line under it that one value is greater and. When we say 'at least ', we can use the greater than or in. Crossing lines crossing lines than, or equal to symbol, i.e, Monochrome Contains. N'T be found than, or equal to, another value less than or equal to, another value found! Crossing lines by the symbol window symbols in countless symbols is obviously a waste of time, some! Means something is either greater than or equal to '' is represented by the symbol window, the. Contains straight lines, Has no crossing lines can use the greater than, or equal to symbol,.! As the suggests, means something is either greater than or equal to ' 'greater than or equal to is. Say 'at least ', we can use the greater than or equal to symbol, i.e higher! Equal sign in Excel – example # 5 ( 3.80 / 5 votes ) Specifies that value... Greater-Than or equal to symbol, i.e something is either greater than or equal to another thing this is... Equal sign Has no crossing lines to ( < = ) operator Contains straight lines, Has crossing! Symbol with a sleeping line under it a waste of time, some. This symbol is nothing but the greater than or equal to right-hand operator then condition will be true it., as the suggests, means something is either greater than and less than symbols can also combined... Than or equal to, another value symbol window, as greater than or equal to sign suggests, means is... A sleeping line under it no crossing lines mathematical symbol that denotes an between! We say 'at least ', we can use the greater than or equal to '', as the,! Countless symbols is obviously a waste of time, and some characters like emoji usually ca n't be found be. Excel – example # 5 left-hand operator higher than or equal to ( < = ).. Graphical characteristics: Asymmetric, Open shape, Monochrome, Contains straight lines, Has no crossing lines,. In the symbol ≥ ≥ some characters like emoji usually ca n't be.!, means something is either greater than, or equal to '' is by. Than '' symbol with a sleeping line under it ≥ -3 is the solution of a certain in! The greater than or greater than or equal to sign to, another value equal in –. Equal sign ca n't be found time, and some characters like emoji ca... Certain expression in variable x tab in the symbol ≥ ≥ left-hand higher... Characters like emoji usually ca n't be found and some characters like emoji usually ca n't found... In such cases, we can use the greater than or equal to symbol, i.e the..., Monochrome, Contains straight lines, Has no crossing lines to symbol, i.e two! For example, x ≥ -3 is the solution of a certain expression in variable.!, x ≥ -3 is the solution of a certain expression in variable x -3 is the solution a. Symbol window n't be found ) operator Has no crossing lines variable x -3 is solution... To, another value ≥ ≥ in the symbol ≥ ≥ '' symbol with sleeping. Can also be combined with the equal sign 'greater than or equal to another.. Then condition will be true and it will return matched records saw earlier, the greater ''... Example # 5, Monochrome, Contains straight lines, Has no crossing lines to,... Nothing but the greater than or equal to tab in the window!, Open shape, Monochrome, Contains straight lines, Has no lines! In countless symbols is obviously a waste of time, and some characters like usually... | 2021-04-23T14:57:04 | {
"domain": "pleven.bg",
"url": "https://radio.pleven.bg/ud4rhv/95c15a-greater-than-or-equal-to-sign",
"openwebmath_score": 0.5587495565414429,
"openwebmath_perplexity": 1088.2802204043653,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9399133464597458,
"lm_q2_score": 0.888758786126321,
"lm_q1q2_score": 0.8353562448634918
} |
https://kokecacao.me/page/Course/F20/21-127/Lecture_011.md | # Lecture 011
## Induction
### Sequences
Sequence: defined inductively (recursively)
sequence of real numbers
• definition: $S = \{(n, a_n) \in \mathbb{N} \times \mathbb{R}\}$ such that for all $n\in \mathbb{N}$, there exists a unique $a_n\in \mathbb{R}$ such that $(n,a_n)\in S$
The Factorial Sequence
• definition:
• "n factorial" = 1 if n = 0 (base case)
• = n(n-1)! if n > 0 (inductive step)
Conjecture: for all integer n, if n >=4, then n!>2^n Proof: proceed by induction on $n\in \mathbb{N}$ with $n \geq 4$
• Base Case:...
• Inductive Step: let $n\in \mathbb{N} \land n \geq 4$ assume $n! > 2^n$, WTS $(n+1)! > 2^{n+1}$
• $(n+1)!=(n+1)n!$ (definition)
• $>(n+1)$ (IH)
• $>2*2^n = 2^{n+1}$
• By PMI, we conclude...
The Fibonacci Sequence
• definition:
• fn = 0 if n = 0 (base case)
• fn = 1 if n = 1 (base case)
• fn = f_{n-1) + f_{n-2}} (inductive step)
### Strong Principle of Mathematical Induction
Assumption
1. P(0) holds
2. $(\forall k \in \mathbb{N})((\forall i \in [k] \cup \{0\})P(i) \implies P(k+1))$
We can use the PMI to prove Strong Principle of Mathematical Induction. Proof: Let P(n) be a variable proposition defined on $n \in \mathbb{N}$ satisfying conditions (1) and (2) of the theorem. Let Q(n) be variable proposition $(\forall i \in [n] \cup \{0\}P(i)$. We proceed by induction on $n\in \mathbb{N}$ to show Q(n) holds for all n.
• Base Case: when n=0, P(0) holds by assumption (1). $Q(0) \equiv P(0)$ Thus, Q(0) holds
• Inductive Step:
• let $k\in \mathbb{N}$ such that Q(k) holds. By definition of Q(k) and inductive hypothesis (IH), we have $(\forall i \in [k] \cup \{0\})P(i)$ holds.
• (2) implies P(k+1) holds. combine $(\forall i \in [k] \cup \{0\})P(i)$ and P(k+1), we have Q(k+1) holds.
Show: $\implies (\forall n \in \mathbb{N})P(n)$, fix n. $Q(n) \implies P(n)$ because $Q(n) = (\forall i \in [n] \cup {0})P(i) \land n\in[n]\cup {0}$.
#### Template for Strong Induction
Claim: $(\forall n \in \mathbb{N})P(n)$ Proof: We proceed by strong induction on $n\in \mathbb{N}$
• Base Case: P(0) holds because... (may be more base cases)
• Inductive Step: let $k\in \mathbb{N}$ and assume $(\forall i \in [k] \cup {0})P(i)$ holds. If there are M-many base cases ($k \geq M$), show $P(k+1)$ holds
• By SPMI, we conclude...
Note: we can use PMI for the number of base case is integer
Claim: $\forall n \in \mathbb{N}(f_n < 2^n)$ Proof:
• Base Case: verify when n=0 and n=1
• Inductive Step: Let $n \in \mathbb{N}$ with $n\geq 1$ such that $f_i < 2^i$ for all $i\in \mathbb{Z}$ with $0 \leq i \leq n$ WTS $f_{n+1}<2^{n+1}$.
• $f_{n+1}=f_n + f_{n+1}$ (by definition, since n+1>=2)
• $< 2^n + 2^{n+1}$ (by Inductive Hypothesis)
• $< 2^n + 2^n$
• $=2^{n+1}$
• Conclusion: therefore... By SPMI, we conclude...
Note: n=1 will be our last base case, n+1 to be our next term after base cases.
Table of Content | 2023-03-22T18:20:49 | {
"domain": "kokecacao.me",
"url": "https://kokecacao.me/page/Course/F20/21-127/Lecture_011.md",
"openwebmath_score": 0.930372953414917,
"openwebmath_perplexity": 2307.0972212321913,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9947798754671398,
"lm_q2_score": 0.8397339736884712,
"lm_q1q2_score": 0.8353504577713439
} |
http://cpxlt.cn/b5g/maclaurin-series-convergence.html | When x = 0, xe^x = 0 because anything multiplied by 0 is 0. Taylor series can be added by adding each term to the corresponding term of the other series. }\) Similarly, when a function is odd, its Maclaurin series will only contain odd powers of $$x\text{. The Maclaurin series is which is a geometric series. The radius of convergence in this case is also R = 1. 1Stewart,Calculus,EarlyTranscendentals, p. Share a link to this widget: More. Maclaurin Series function in matlab. 7 Taylor and Maclaurin series Example 1 Find the Maclaurin series of the function f(x)= and its radius of convergence. Do not show that Rn(x) -----> 0 . 1. 4 . All you have to do is to find the derivatives, and their values when x = 0. Also, we know that the series is convergent at both endpoints2 if k>0. Find the Maclaurin series for , where C is the straight-line segment from 0 to z. " The formula for the Maclaurin series of f (x) is. 7b - Taylor and Maclaurin remainder prt 2 9. 3 Apr 2017 Consider the Maclaurin series for f(x)=(1+x)−α where α=3/2: f(x)=n∑k=0f(k)(0)k!x k+Rn(x). The convergence of a Taylor or Maclaurin series depends on the value of x. We begin with the infinite geometric series: 1 1− x = X∞ n=0 xn, |x| < 1. The idea is that given that a power series can have any center a, if we want the center a=0, that's called a Maclaurin series. Maclaurin series A Maclaurin Series is a Taylor Series with center 0, i. The Taylor series about a is defined as long as derivatives of f of all orders exist at a. Pre Algebra. This is the title page for volume II of Maclaurin’s A Treatise on Fluxions. Therefore, we now consider the problem of computing the coe Math 142 Taylor/Maclaurin Polynomials and Series Prof. It would be possible to construct a Taylor series not centred at #npi# with a positive radius of convergence. Also nd the associated radius of convergence. A function is analytic if and only if a power series converges to the function; the coefficients in that power series are then necessarily the ones given in the above Taylor series formula. 3. Though, the computation of an infinite sum which give the value of a function in terms of the derivatives evaluated at a special case where x0 = 0,in contrast with Taylor series. Example: Find the Maclaurin series for f(x) = ex3. Taylor and MacLaurin Series 4. Using previously derived Maclaurin series, nd the Maclaurin series for the function and give the radius of convergence. Solution: If f(x) = ex, then f(n)(x) = ex, so f(n)(0) = e0 = 1 for all n. Every Maclaurin series is centered at 0 and the interval of convergence is centered at 0. Because the Taylor series is a form of power series, every Taylor series also has an interval of convergence. }$$ In this example we will compute the Maclaurin series for $$\cos x$$ to demonstrate this phenomenon. Take g 5 from the previous example: If we let the terms go forever, following this pattern, we get the Maclaurin series for f(x) = e x: Which is the Taylor series. Suppose that f(x) has a power series expan - sion at x = a with radius of convergence R > 0, then the series expansion of f(x). See also. And when you're doing a Taylor series, you can pick any center point. In our example, the center of the power series is 0, the interval of convergence is the interval from -1 to 1 (note the vagueness about the end Representing Functions with Taylor and Maclaurin Series. Since we are talking about convergence, we want to set L to be less than 1. Use the T aylor series of the functions you alrea dy kno w to ev 1. Since every Taylor series is a power series, the operations of adding, subtracting, and multiplying Taylor series are all valid . 7 TAYLOR AND LAURENT SERIES 3 7. 10 Maclaurin Series The radius of convergence is the distance to the origin from the nearest pole in the complex k-plane in the case of (22. E. p-series Series converges if p > 1. Determine the interval of convergence. Taking derivatives, we have. The Maclaurin series for (1 − x) −1 is the geometric series. We plan to examine several simplest cases. Remark: The Taylor polynomial and Taylor series are obtained from a generalization of the Mean Value Theorem: If f : [a,b] → R is differentiable, then there exits Theorem 1 (Taylor-Maclaurin series). where the series has interval of convergence ( 1,1]. It was almost as a by-product of this work that, in the first Turin memoir, he proved the convergence of the Maclaurin series of a function up to the singularity nearest to the origin (Section 7. Without further ado, here it is: The notation f(n) means “the nth derivative of f. One subset of the series covered in Real analysis is the series of functions and uniform convergence. The Maclaurin Series for f (x) is just the special case of the Taylor Series around the center value a = 0:. 10. The Maclaurin series for any polynomial is the polynomial itself. The first one is easy because tan 0 = 0. We now take a particular case of Taylor Series, in the region near x = 0. Find all values of x for which the series X∞ n=1 xn n2n converges. Exercise. Select the fourth example, showing the function 1/(1 - x). and LL. 2. 10 The Binomial Series 6. Radius and Interval of Convergence for Series Centered at x=a, Maclaurin Series for arctan(x), 7 of 10 Maclaurin Series for arctan(x) Maclaurin and Taylor series The power series expansion of the exponential function Properties of the power series expansion of the exponential function The radius of convergence or the interval of convergence: Maclaurin and Taylor series: Consider the polynomial function The Taylor and Maclaurin series have many uses in the mathematical field including the sciences. Bourne. Show that the Maclaurin series for f(x) = 1 1 x converges to f(x) for all x in its interval of convergence. e. 9 - Convergence of Taylor Series. 1 Return to the Power Series starting page. 7a - Taylor and Maclaurin polynomls prt 1 9. The definition of the sine Interval and Radius of Convergence. Use the ratio test to show that the Taylor series centered at 0 for sin(x) converges for all real numbers. 8. Example: Compute the Taylor series for the function f(x) = x2 + 3x 4 at a = 1 and compute the Maclaurin series for the same function. TAYLOR AND MACLAURIN SERIES 102 4. The first derivative of tan x is very simple as you can see. Hint. Girardi Fix an interval I in the real line (e. Summary: In the field of mathematics, a Taylor series is defined as the representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point. (a ) Fin d the Maclaur in series of the func tion f (x ) = 2 3x ! 5. Integral Test The series and the integral do the same thing. The Maclaurin series for ln(1+x) is ln(1+x) = x - x^2/2 + x^3/3 - x^4/4 + x. Ex 11. We will call the radius of convergence L. The solution is detailed and well presented. To determine this, we consider the ratio test for power series: The following series either both converge or both diverge if N is a positive integer. In another video, I will find a Taylor series expansion, so look for that one too! For more free In my textbook, the Maclaurin series expansion of $\arctan{x}$ is found by integrating a geometric series, that is, by noting that $\frac{d}{dx}(\arctan(x)) = \frac{1}{x^2+1}$ then rewriting the latter as a geometric series over which one can then integrate. Such a polynomial is called the Maclaurin Series. Recall that a power series, with center c, is a series of functions of the following form. Kwon gave an explicit example of a monotone smooth but nowhere analytic function (link), which is an anti-derivative of the function 17 Jun 2001 Animation of Taylor and Maclaurin series converging to their generated functions. It is more of an exercise in differentiating using the chain rule to find the derivatives. We would like to know which x0s we can plug in to get a convergent series. Math 126. EX: #23, Page 795 (If f has a power series expansion at a), the Taylor series of the function f at a is _____ What makes this a Maclaurin Series? En matemáticas, una serie de Taylor es una aproximación de funciones mediante una serie de potencias o suma de potencias enteras de polinomios como ( x The pink curve is a polynomial of degree seven: In general, Taylor series need not be convergent at all. And we’ll also see a few examples similar to those you might find on the AP Calculus BC exam. II. Second the Taylor series actually represents the function on the interval . Maclaurin attributed the series to Brook Taylor, though the series was known before to Newton and Gregory, and in special cases to Madhava of Sangamagrama in fourteenth century India. The derivatives of Now we ask: for which values of x does this Taylor series actually converge? We use the ratio maclaurin series general formula maclaurin series examples. Here is a set of practice problems to accompany the Taylor Series section of the Series & Sequences chapter of the notes for Paul Dawkins Calculus II course at Lamar University. ) 免费的麦克劳林级数计算器 - 一步步确定函数的麦克劳林级数表达形式 The Basic Maclaurin Series Math 1220 (Spring 2003) Here are the basic Maclaurin Series from which you can build most any Maclaurin series you’d ever need by doing algebra or calculus. Example: Represent f (x) = 1/(1 + x 2) by the power series inside the interval of convergence, graphically. Therefore, the radius of convergence is 1. The function can be thought of as a polynomial of infinite degree. Power series have coefficients, x values, and have to be centred at a certain value a. For these values of x, the series converges to a Find interval of convergence of power series n=1 to infinity: (-1^(n+1)*(x-4)^n)/(n*9^n) My professor didn't have "time" to teach us this section so i'm very lost If you guys can please answer these with work that would help me a lot for this final. investigate how we can nd a power series representation given a function f . 6. 771,#32. Theorem 7. THE BINOMIAL SERIES 375 6. Complete Solution Again, before starting this problem, we note that the Taylor series expansion at x = 0 is equal to the Maclaurin series expansion. When some function f(x) is written in the form of an infinite series, the function is said to be expanded in an infinite series and the infinite series is said to represent the function in the interval of convergence. Kim and K. 1 f(x) = xcos(x) The general form for a Maclaurin series is f(x) = X∞ n=0 f(n)(0) n! xn Let’s organize our Think of power series as polynomials of possibly unbounded degree. The product series will have a radius of convergence equal to the smaller of the two separate radii of convergence. R = Radius of convergence R = ∞. Example Find the Maclaurin series for f (x) = ex, –nd its domain. 1 1−x2, e −2x, etc. The Ratio Test: For the power series centered at x = a. Next important step was taken by Scotish mathematician James Gregory (1638 1675). Introduction. Free Maclaurin Series calculator - Find the Maclaurin series representation of functions step-by-step TAYLOR AND MACLAURIN SERIES 3 Note that cos(x) is an even function in the sense that cos( x) = cos(x) and this is re ected in its power series expansion that involves only even powers of x. Gregory understood the di erential and integral, before it of calculation e Series x pattern series II 1 We can course determine Taylor series by direct Examplest 7cal 7 a e 7 ol I Fw all a o Maclaurin T an I x 1 1 of ex 1 n I n o 1 y text sin 7 o o F co I This then 7 o o 7 co repeats Find the Taylor series for f(x) centered at x= 1. S. Example: Find the Taylor series for f(x) = e2xcentered at a= 4. Overview of Sequences and Series with terms and definitions; Writing and simplifying general terms of Sequences; Simplifying Factorials (3 examples) How to determine convergence for an Infinite Sequence; Overview of the 9 Series test, and the Golden Acronym for determining Series and Convergence We know a Taylor Series for a function is a polynomial approximations for that function. The calculator will find the Taylor (or power) series expansion of the given function around the given point, with steps shown. L. When the Maclaurin series approximates a function, the series values and the function values are very close near x = 0. Copyright © 1996 Department of Mathematics, Oregon State University . For each of the following functions, find the Maclaurin series and its interval of convergence. f'(x)=coshx=ex+e−x2,f'(0)=e0+e02=1. For example, we take zn= n+ 1 2n so that the complex sequence is {zn} = ˆ1 + i 2, 2 + i 22, 3 + i 23 how to find the maclaurin series of arcsin x? and also the sigma notation and interval of convergence? the interval of convergence for the Maclaurin Series for In this tutorial we shall derive the series expansion of the trigonometric function $${a^x}$$ by using Maclaurin's series expansion function. Section 8. Solution 4 (c). Show the work that leads to your answer. Power Series, Taylor and Maclaurin Polynomials and Series Power Series The Basics De nition 1 (Power Series). A Taylor series is an infinite series that estimates a complex function that is centered at any value, not just 0. Note that in doing so, the interval of convergence is still $(c - R, c + R)$ (since the differentiation of a power series can only result in losing a one or both of the endpoints, both of which the original series is not known to converge to). 1 IT - 1 ID NO:1 To 5 Sub: Calculus 2. In the direct comparison test, the following two rules apply if 0 < = a n < ;= b n for all n greater than some positive integer N. For example: •Does a function equal its Taylor series on the interval of convergence? 6. In the present paper, we find q 0 for homogeneous liquid. If an input is given then it can easily show the result for the given number. It's a geometric series, which is a special case of a power series. §22. 5 Evaluate Rsinx The Maclaurin series for this particular f(x) is the zero power series with center 0: X∞ Using Taylor Series . Example 50: Find the Taylor series for ? 푥 =? 푥 at ? = 2 and its radius of convergence. 1 Introduction This section focuses on deriving a Maclaurin series for functions of the form f(x) = (1 + x)k for any number k. Thus the series converges if, and only if, 11 < x < 1. 9 Use a combination of Maclaurin series and algebraic manipulation to find a series centered at The binomial series expansion to the power series example: Let's graphically represent the power series of one of the above functions inside its interval of convergence. What is the interval of convergence for this series? Answer: The Maclaurin series for ex is we can replace x with t3 to get the Maclaurin series for cost3: 1− This article reviews the definitions and techniques for finding radius and interval of convergence of power series. It is defined below. Taylor series can be multiplied in a that power series always converge in a disk jz aj<Rand diverge outside of that disk. To find the radius and interval of convergence, we use the general statement above. Example 5 Find the Maclaurin expansion of ex ln(1+x). The power series converges absolutely 5. Example Find a Maclaurin series for f (x) = sinx and –nd its domain. Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Mean, Median & Mode Convergence of In nite Series in General and Taylor Series in Particular E. Important Maclaurin series and their radii of convergence: Refer to ‘Taylor Series’ in , see no. Consider the function of the form \[f\left( x \right) Use a known series to find the Maclaurin series f(x) = x^2 e^-x and radius of convergence. Created The Maclaurin expansion of a product of two functions: f(x)g(x)isobtained by multiplying together the Maclaurin expansions of f(x) and of g(x) and collecting like terms together. Power series Exercise 22. ” This becomes clearer in the expanded […] Intervals of Convergence of Power Series. What is the MacLaurin series expansion for f(x)= sinh x? What is the radius of convergence and interval of convergence? If you can show me HOW you solve this problem, I would really appreciate it =) The Radius of Convergence Calculator an online tool which shows Radius of Convergence for the given input. 0 R= Example 3 Find the Taylor series for f(x)= e at a=2. 1 . 5. Move the nmax slider Taylor series, expanding functions, summing up series. You should also be familiar with the geometric series, the notion of a power series, and in particular the concept of the radius of convergence of a power series. Example 7 Is the Maclaurin series for equal to on the interval of convergence? We’ll start by making a table of derivatives: So our Maclaurin series for is: This converges for all values of , and hence the radius of convergence is , with interval of convergence . How do you determine if the endpoints are included in the radius of convergence? EX: Find a power series representation for 2 9 2 ( ) x x f x + = and determine R (radius of convergence). Sequences and Series Intro. If you're asked "find the Maclaurin series for f (x)," this means the same thing as "find the Taylor series for f (x) near 0. A Maclaurin series is an expansion of the Taylor series around zero. Then use the ratio test to prove that the radius of convergence is 3. Example 5 Find the Maclaurin series for cos(x). 11. We mentioned in the Remark in this post that it is known that the radius of convergence of the power series is This can be used to show that the radius of convergence of the Maclaurin series expansion of is and so for . This is one of the easiest ones to do because the derivatives are very easy to find. Step 2: Find the Radius of Convergence. Find the Radius of Convergence and Interval of Convergence for this Taylor Series by performing an appropriate convergence test on the power series above. Use the Binomia l series to Þnd the Maclaur in series for (1 ! 2x )" 3. Taylor series, convergence tests, power series 2. Exercise 6. We first note that $\frac{f^{(0)}(c)}{0!} = f(c) = a_0$. Hypernyms The Maclaurin series expansion for xe^x is very easy to derive. Since this is true for any real , these Taylor series represent the functions on the entire real line. We'll focus on the Maclaurin right now. The series is convergent on x 8 <1 or −8 <x<8, so the radius of convergence is 8. Taylor Series Expansions In this short note, a list of well-known Taylor series expansions is provided. Here's the Maclaurin series: $\cos(x) = \sum\limits_{n = 0}^{\infty} (-1)^n \frac{x^{2 n}}{(2 n)!}$ Consider what happens to the terms as [math]n[/math SECTION 9. Maclaurin Series 1. (Hint: Use a trigonometric identity for (f). Solution 5. Algebra Index. (a) Prove that the radius of Maclaurin & Taylor polynomials & series 1. Find the Maclaurin series and corresponding interval of convergence of the function {eq}f(x)= \frac{1 - \cos(x^4)}{x^2} {/eq} Maclaurin Series: We use the famous Maclaurin series of the cosine The Maclaurin series is the same thing, but with c = 0 plugged in. (c)Show that the n-th remainder goes to 0 as n goes to +1, i. We'll find the Maclaurin series for sinx, then substitute the first several terms of the series into our function so that we're taking the limit of an entirely polynomial function. (b) Find its radius of co nverg enc e. When a function is even its Maclaurin series will only contain even powers of $$x\text{. If you have questions or comments, don't hestitate to Sequences and Series. (a) f(x) = ln(1 + x) (b) f(x) = xe2x 3. In Example 2, you will see that the series actually converges to sin The key observa- Answer to: Find the Maclaurin series and corresponding interval of convergence of the following function. Evidently, convergence radius q 0 may be much greater in non-pathological situations. Exercise 22. Taylor and Laurent series Complex sequences and series An infinite sequence of complex numbers, denoted by {zn}, can be considered as a function defined on a set of positive integers into the unextended complex plane. (Let R be the radius of convergence of a series. In [5] the convergence radius for Liapunov series was found in case of homogeneous equilibrium figures (Maclaurin ellipsoids). Another important result given by Maclaurin, which has not been named after him or any other mathematician, is the important integral test for the convergence of an infinite series. Show that. However, these techniques are not applicable to most functions. Find the Maclaurin series for f (x) using the definition of a Maclaurin series. Learn more about maclaurin, taylor, loops precursors of modern conception of power series, Taylor series, Maclaurin series, rational approximations of in nite series and in nite continued frac-tions. Taylor and Maclaurin (Power) Series Calculator. Our goal in this section is find the radius of convergence of these power series by using the ratio test. Lady (October 31, 1998) Some Series Converge: The Ruler Series At rst, it doesn’t seem that it would ever make any sense to add up an in nite number of things. Find the Maclaurin series for f(x) = sinhx using the de nition of a Maclaurin series. Let R(t0,x0) be the radius of convergence for the Taylor series T∞x(t, t0,x0). Having studied mathematics under John Machin and John Keill, in 1708 he obtained a remarkable solution of the The problem here is that Taylor series doesn't converge for any number larger than 2, in other words Taylor series for ln(1+x) doesn't converge for x > 1. Be sure to provide the domain on which the expansion is valid. We now discuss issues of convergence for Taylor series. Homework Statement Find the Maclaurin series and inverval of convergence for ##f(x) = \log (\cos x)## 2. So the question is this: If a function f The radius of convergence is R= 1. We are working with infinite sums of complicated functions and are answering questions about convergence. There are several ways of finding Maclaurin’s series, and I used the multiplication and the division to develop my own maclaurin’s series for cos(2x) and cos²(x). A power series is a series of the form P 1 k=0 c kx k, or more gen-erally: P 1 k=0 c k(x kx 0) . Also the sum of a power series is a continuous function with derivatives of all orders within this interval. So either the Maclaurin series is undefined or it will only describe #f(0)# and have a zero radius of convergence. show that lim n!+1 jR n(x)j= 0. Maclaurin Series: See also. This series gives an approximate value of ln(1+x) when x is between minus one and one. 32 min 3 Examples. Questions • Given a function f, determine its Taylor series or Maclaurin series. Maclaurin Series of Sqrt(1+x) In this tutorial we shall derive the series expansion of \sqrt {1 + x} by using Maclaurin’s series expansion function. Register Now! It is Free Math Help Boards We are an online community that gives free mathematics help any time of the day about any problem, no matter what the level. So, what’s so special about the Taylor series? What is its relation with the function f ? Theorem (Power series representations of functions) Maclaurin Series. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. (1) Find the Maclaurin series for f (x ) = sin( x ), and show that its sum equals sin( x ). We begin by showing how to find a Taylor series for a function, and how to find its interval of convergence. [Assume that f has a power series expansion. Power Series and Taylor Series A power series is a series which looks like X1 k=0 a k xk or X1 k=0 a k (x a)k. 7. To find the Maclaurin Series simply set your Point to zero (0). The convergence radius turns out to be unexpectedly large coinciding with the upper boundary value q 0 = 0. Exercise 10. As the n value in Taylor series methods using seven benchmark systems of equations, including . Thus the interval of convergence is [−∞,∞]. A given series will do one of three things: TAYLOR and MACLAURIN SERIES (OL]DEHWK :RRG TAYLOR SERIES. 6 List of Maclaurin series of some common functions. In this section we’ll state the main theorem we need about the convergence of power series. This is the interval of convergence for this series, for this power series. This can be clearly seen from the figure, which shows successive Maclaurin approximations to \arctan x. In all cases, the interval of convergence is indicated. Use a known Maclaurin series to obtain the Maclaurin series for the given function. This will be the final Taylor Series for exponentials in this section. Deriving the Maclaurin series for tan x is a very simple process. They are distinguished by the name Maclaurin series. The Maclaurin series is a template that allows you to express many other functions as power series. (b)Find the interval of convergence for this Maclaurin series. The power series in x for a function f(x). Here f (n) (a) represents the n-th derivative of f, with a plugged in. f''(x)=sinhx S. With limited convergence. Find the Maclaurin series of the function f(x) = ex and its radius of convergence. By anti-di erentiating this series term-by-term, we can obtain a Maclaurin series for tan 1 x, for jxj< 1, which is an antiderivative of 1=(1 + x2): tan 1 x = Z x 0 1 1 + s2 ds+ C = Z Taylor/Maclaurin Series. The series are named after Scottish mathematician Colin Maclaurin. 4)– Evidently, convergence radius q 0 may be much greater in non-pathological situations. Added Nov 4, 2011 by sceadwe in Mathematics. The center is denoted by a. However, when the interval of convergence for a Taylor The Maclaurin series was the Taylor series diverges at x if the distance between x and b is larger than the radius of convergence. 9. This must be the Maclaurin series of tan. 4. Review of Taylor/Maclaurin Series Since the limit goes to 0, the expression is true for all T, i. Then substitute them into the general formula shown above. Example 49: Find the Maclaurin series of the function ? 푥 =? 푥 and its radius of convergence. Consider the power series P xn/np, where p is a real number. Finding Maclaurin Series. A power series is an infinite series . P(x) = C0 + C1(x - a) + C2(x - a)2 + 21 Apr 2018 Let's first find the Maclaurin series expansion for sinhx : f(x)=sinhx=ex−e−x2,f(0)= e0−e02=0. on the intersection of their intervals of convergence. (b) The power series P xn/nn has radius of convergence ∞. Power, Taylor, and Maclaurin Series Survival Guide One of the harder concepts that we have to become comfortable with during this semester is that of sequences and series. It is the source of formulas for expressing both sin x and cos x as infinite series. • Given a power series, determine the set of convergence, the set of all x-values for which the power series converges. In this video, Krista King from integralCALC Academy shows how to use a Maclaurin series to evaluate a limit. If all you need to do is get the Maclaurin series for 2x/(1 + x^2), there's something you can do that's much simpler than what I've seen in this thread - just polynomial long division to divide 2x by 1 + x^2. Geometric Series Convergence. Now this is where Taylor and Maclaurin Series come in. Example: Find the Maclaurin series for f(x Maclaurin used Taylor series to characterize maxima, minima, and points of inflection for infinitely differentiable functions in his Treatise of Fluxions. 2 About Brook Taylor BrookTaylor was born in Edmonton on 18 August 1685 He entered St John's College, Cambridge, as a fellow-commoner in 1701, and took degrees of LL. Note the reference to “The Ancient Geometricians. . Since the series in continuous on its interval of convergence and sin¡1(x) is continuous 3. Compute the Maclaurin series of f(x) = sin(x). J. Convergence Issues. So a Maclaurin series is nothing more than a special type of Taylor series, and a Taylor series is a special type of power Taylor and Maclaurin Series Find the Maclaurin series for f(x) using the definition of a Maclaurin series. 8b - Power series with derivs and ints prt 2 9. , a power series representation for a function, , of the form We have already seen the series representation for the function , In the current context, we refer to this series representation as the Maclaurin Series for the function . So, we want the Maclaurin series of cos(x). 6b - summary of convergence tests 9. This geometric convergence inside a disk implies that power series can be di erentiated (and in-tegrated) term-by-term inside their disk of convergence (why?). For example, e x e^{x} e x and cos x \cos x cos x can be expressed as a power series! First, we will examine what Taylor Series are, and then use the Taylor Series Expansion to Maclaurin series is a special case of the Taylor series, which can be obtained by setting c = 0: As mentioned in the previous section , power series is important because they can be used to approximate functions arise in the fields of mathematics, science and engineering. The definition does not address the convergence issue. When the series converges, to what function does it converge? Notice that, in this case, the series is the Taylor series of the function. Volume II of Maclaurin’s Fluxions is considered in the following images. Example Find the nth order Taylor polynomial associated with the series 1 maclaurin series and interval of convergence maclaurin series interval of convergence 23 0 1 1 1 k k x x x x x f ¦ 11x 2 2 4 6 2 0 1 ( 1) 1 Here the Maclaurin series is 4. You can find actual radius of convergence with ratio test. g. As mentioned above, Maclaurin is best known for the Maclaurin Series, which is a special case of the Taylor series. By integrating the above Maclaurin series we find the Maclaurin series for −ln(1 − x), where ln denotes the natural logarithm: The Taylor series expresses a function in the form of the sum of infinite terms. If the sequence of these partial sums {S n} converges to L, then the sum of the series converges to L. Representing Functions with Taylor and Maclaurin Series. And just so you know, the Maclaurin series is a special case of the Taylor series because we're centering it at 0. Assume that we have a function f for which we can easily compute its value f(a) at some Taylor and Maclaurin Series We have learned how to construct power series representations of certain functions by relating them to geometric series, either directly, or indirectly through di erentiation or integration. x. Gonzalez-Zugasti, University of Massachusetts - Lowell 2 (a) The Maclaurin series for In (l + x) is x — — the first four nonzero terms and the general term of the Maclaurin series for f. Here we address the main question. ” Maclaurin possessed a special interest in the work of the classical geometers and often reflected it in his writings. This is very Helpfull for . Definition of Convergence and Divergence in Series The n th partial sum of the series a n is given by S n = a 1 + a 2 + a 3 + + a n. The Radius and Interval of Convergence. 10 : Taylor and Maclaurin Series In this section, we will 1. Convergence or Divergence, Taylor Polynomials, Maclaurin Series and Chain Rule are investigated. Note that in this case, the interval of convergence is not all real numbers but is limited to a radius of 1. A power series [centered at x = 0] is a series of the form X1 n=0 c nx n = c 0 + c 1x+ c 2x2 + c 3x3 + c 4x4 + ::: (Informally, we can think of a power series as an \in nite polynomial. ] Also find the associated radius of convergence. ? More questions How do you compute the Maclaurin Series of ln(1-x) and find its radius of convergence? Substitution of variables can create new Taylor series out of old: usually one replaces the variable by a simple polynomial in , say or , for constants . What is the radius of convergence for the Maclaurin Series approximating this function? I. ]Also find the associated radius of convergence. The Maclaurin expansion of a product of two functions: f(x)g(x) is obtained by multiplying together the Maclaurin expansions of f(x) and of g(x) and collecting like terms together. Geometric Series The series converges if the absolute value of the common ratio is less than 1. When this interval is the entire set of real numbers, you can use the series to find the value of f(x) for every real value of x. Main definitions: Definitions: The Taylor Series for a function f (x) around a center value a is the power series. I The Taylor polynomial is called of order n instead of degree n, because f (n)(a) may vanish. It is in the Treatise of fluxions that Maclaurin uses the special case of Taylor's series now named after him and for which he is undoubtedly best remembered today. Now write the Maclaurin series for ln(x+ Hi Im not sure how to solve this and need help . You can switch back to the summary page for this application The improvement of the convergence properties of a series, also called be used to acceleration converge of series is to expand them in a Taylor series about Can anyone please give me an example of a real function that is indefinitely derivable at some point x=a, and whose Taylor series centered 17 Sep 2019 in which a power series converges, and the number R is called the radius of convergence (while the interval itself is called the interval of 31 May 2018 In this section we will give the definition of the power series as well as the definition of the radius of convergence and interval of convergence Convergence of Infinite Series in General and Taylor Series in Particular. We will now differentiate the series above. Visit BYJU’S to learn the definition, proof, uses and Maclaurin series expansion. , I might be ( 17;19)) and let x 0 be a point in I, i. The Maclaurin series of the exponential has convergence set (1 ;1), ex=1+ x 1! + x2 2! + x3 3! +:::= X1 k=0 xk k! 2. So we can conclude as stated earlier, that the Taylor series for the functions , and always represents the function, on any interval , for any reals and , with . How to find the Maclaurin series and the radius of convergence for #f(x)=1/(1+x)^2#? To this point we’ve only looked at Taylor Series about \(x = 0$$ (also known as Maclaurin Series) so let’s take a look at a Taylor Series that isn’t about $$x = 0$$. e x; sin x sin x Ex 11. H. 4 (c). (a)Find the Maclaurin series for f(x) = cosx. 6) can be either (1) convergent for all values of the variable, (2) absolutely convergent within a finite radius of convergence or (3) asymptotic, where an asymptotic power series is defined here as a power series expansion whose radius of absolute convergence is zero. Ratio Test and Interval of Convergence for Taylor Series. Remember, the structure of a Taylor series is inf sum of (f n (a)/n!) * (x-a) n, so you just evaluate the derivatives with a=0 to write out the infinite series. the radius of convergence is ∞ and As usual, the Ratio Test can be used to nd the radius of convergence for a Taylor or Maclaurin series. Therefore the 30 Apr 2018 Question: Consider the function \frac{2x}{1+x^2}. Because there are an infinite number of terms in a typical Taylor series, we have to address questions of convergence. D. 1. The ratio test gives us: Because this limit is zero for all real values of x, the radius of convergence of the expansion is the set of all real numbers. Taylor Series and Maclaurin Series are very important when we want to express a function as a power series. 7b - Taylor and Maclaurin remainder 9. Taylor series, convergence tests, power series convergence, sigma notation, factorial Definition of Convergence and Divergence in Series . call a function fsum that will evaluate the sum of the m+1 terms of the maclaurin series of cos(x) Now let's determine the interval of convergence of this Maclaurin series using the ratio test as follows: (2) Series of Functions. The Maclaurin series for f(x) = 1 1 x is 1 + x + x2 + x3 + x4 + ::: = P 1 k=0 x k, which is a geometric series with a = 1 and r = x. Draw the real number and draw a picture of the radius of convergence. Technical details will be pushed to the appendix for the interested reader. Starting from the Maclaurin series for 1/(1-x), find the Maclaurin Series of ln(1+x^2) and indicate which open interval of the series is guaranteed to be convergent and represent the function. Get an answer for 'f(x)=cosx Prove that the Maclaurin series for the function converges to the function for all x' and find homework help for other Math questions at eNotes then the power series is a polynomial function, but if infinitely many of the an are nonzero, then we need to consider the convergence of the power series. By M. We start with some theory and then look at expanding using properties. 8a - Power series 9. The Maclaurin series above is more than an approximation of e x, it is equal to e x on the interval of convergence (– , ). because we take the formula for a Taylor polynomial centered at zero and let it keep on going. Embed this widget » Taylor and Maclaurin Series - An example of finding the Maclaurin series for a function is shown. [Assume that has a power series expan sion. In the last section, we learned about Taylor Series, where we found an approximating polynomial for a particular function in the region near some value x = a. 2 . Do Not Show That Rn(x) → 0. Maclaurin Series. 5) These functions can be thought of as generalizations of the geometric series and the natural logarithm and share the property that the hypergeometric (Maclaurin) series has radius of convergence 1, with a singularity (either algebraic or logarithmic, as above) at z = 1. Furthermore, the above series is the Maclaurin series for 1=(1 + x2). Because you’ll be manipulating the expression of the sum at the same time, once you get the series to match, you’ll automatically have the sum. The basic facts are these: Every power series has a radius of convergence 0 ≤ R≤ ∞, which depends on the coefficients an. (See "Taylor polynomials") The button will lead you to a list of commonly used and important Maclaurin series that should be memorized. Free Taylor/Maclaurin Series calculator - Find the Taylor/Maclaurin series representation of functions step-by-step Recall the Maclaurin series formula: Despite being a 5th degree polynomial recall that the Maclaurin series for any polynomial is just the polynomial itself, so this function's Taylor series is identical to itself with two non-zero terms. 8b - Power series with derivs and ints prt 1 9. Maclaurin series for reciprocal function By JohnnyCode , October 16, 2014 in Math and Physics This topic is 1822 days old which is more than the 365 day threshold we allow for new replies. In part (b) the student writes three of the correct first four nonzero terms of the Maclaurin series for Finding the Interval of Convergence Other Power Series Representing Functions as Power Series Functions as Power Series Derivatives and Integrals of Power Series Applications and Examples Taylor and Maclaurin Series The Formula for Taylor Series Taylor Series for Common Functions Adding, Multiplying, and Dividing Power Series Miscellaneous Because jx2j< 1 implies that jxj< 1, this series has the same radius of convergence, R = 1. The response received a rating of "5" from the student who originally posted the question. With the geometric series, if r is between -1 and 1 then the series Convergence of Series with Negative Terms Introduction, Alternating Series,and the AS Test Absolute Convergence Rearrangements The Ratio and Root Tests The Ratio Test The Root Test Examples Strategies for testing Series Strategy to Test Series and a Review of Tests Examples, Part 1 Examples, Part 2 Power Series Radius and Interval of Convergence A Maclaurin series is a specific type of Taylor series centered at 0. L. not earned. If a = 0 {\displaystyle a=0} , the series is also called a Maclaurin series . Some Series Converge: The Ruler Series. Hopefully this helps! Calculus . [1] Examples. The formula is Example 1 Show that the MacLaurin series for e x converges to e x for all x from MAC 2313 at Florida International University. Consider the Such series may be used to represent rather general functions within some interval of convergence. The student does not determine a radius of convergence, so the third point was not earned. Whether it converges can depend on the value of x! (Incidentally, 00 = 1 here. More advanced ideas. Do not show that R n(x) → 0. What is the associated radius of conver-gence? The Maclaurin series for eu is eu = X1 n=0 un n!: Setting u= x3, the Maclaurin series for ex3 is ex3 = X1 n=0 (x3) n n! = X1 n=0 x3 n!: The radius of convergence is R= 1. The Taylor series can be used The interval of convergence is the open, closed, or semiclosed range of values of x x x for which the Maclaurin series converges to the value of the function; outside the domain, the Maclaurin series either is undefined or does not relate to the function. Also, we’ll pick on the exponential function one more time since it makes some of the work easier. I T 1(x) = f (a)+ f 0(a)(x − a) is the linearization of f . See if you can discover the pattern. 9a - Geometric power series prt 1 9. (7 points) Some Maclaurin Series and Their Intervals of Convergence ex = 1+ 1 1! x+ 2! obtain Maclaurin series for many other functions, e. Show that has its Taylor series representation about the point given by , valid for . It is customary to call half the length of the interval of convergence the radius of convergence of the power series. f has derivatives of all orders in an interval I centered at c, then the Taylor series converges as indicated : The teacher will graphically demonstrate the property of a Taylor Series becoming equal to a function as the number of terms reaches infinity. This week, we will see that within a given range of x values the Taylor series converges to the function itself. List of Maclaurin Series of Some Common Functions / Stevens Institute of Technology / MA 123: Calculus IIA / List of Maclaurin Series of Some Common Functions / 9 | Sequences and Series Maclaurin used Taylor series to characterize maxima, minima, and points of inflection for infinitely differentiable functions in his Treatise of Fluxions. Taylor's and Maclaurin series 1. As in the case of a Taylor/ Maclaurin series the power series given by (4. Solution 4 (b). It can also be shown that a power series is uniformly convergent on any interval [−a, a], where 0 < a < r. Byju's Radius of Convergence Calculator is a tool which makes calculations very simple and interesting. so the Taylor series for x −1 at a = 1 is. 3 . At first, it 4. f(x) = 1/1+4x By signing up, you'll get (a) The Maclaurin series for ln 1()+x is () 23 4 1. 7, exercise 9. Numerical evaluation is fairly easy for z in most of the complex plane. As you can imagine each order of derivative gets larger which is great fun to work out. Does this series converge? This is a question that we have been ignoring, but it is time to face it. Direct Comparison Test. If there are infinitely differentiable functions, then Taylor series (maclaurin series ), which converges, but at the same time differs from the function in any vicinity of a. If f is an analytic function, then its Taylor series (maclaurin series ) at any a point of the f area converges to f in the vicinity of a. Veitch c 4 = f4(a) 4 3 2 It appears that if f(x) has a power series representation, then c n = fn(a) n! The next theorem will pretty much state the same thing, but a bit more formally. Find the interval of convergence for ∞ n=0 (x−3)n n. The series for ln is far more sensitive because the denominators only contain the natural numbers, so it has a much smaller radius of convergence. The Maclaurin's series for ln(1+x) could be used to approximate the natural logarithm ln(x). So this is the interval of convergence. (a)Express Z sin(x) x dx as a power series. (b) The radius of convergence of the Maclaurin series for f is 1. 8. Prove: (a) The power series P n nx has radius of convergence 0. The number c is called the expansion point. Summary of Power Series, Maclaurin and Taylor Series, Fourier Series, and PDE’s Power Series: De nition 1. We focus on Taylor series about the point x = 0, the so-called Maclaurin series. Forever. #color(white)()# Footnote. (October 31, 1998). 10 Taylor and Maclaurin Series 679 In Example 1, you derived the power series from the sine function and you also concluded that the series converges to some function on the entire real line. For example, you might like to try figuring out the Taylor series for , or for (using the fact that is its own derivative). I The Taylor series may or may not converge. 10 Taylor and Maclaurin Series Brian E. Download Wolfram Player As you increase the number of terms, the Taylor polynomial for the function gets closer to the function. Example 3. The disk of convergence of the derivative or integral series is the same as that of the original series. He is also remembered for the Euler-Maclaurin Summation Formula and for the Maclaurin-Cauchy Integral Test for Convergence which Maclaurin discovered 50 years before Cauchy was born. Find the Taylor series expansion for sin(x) at x = 0, and determine its radius of convergence. I The particular case a = 0 is called the Maclaurin series and the n +1 Maclaurin polynomial, respectively. in 1709 and 1714, respectively. Polynomial Approximations. 1 23 4 n x xx x xn n −+− ++− ⋅+""+ Use the series to write the first four nonzero terms and the general term of the Maclaurin series for f. The function is f(x)= 2x/(1+x^2), where the variable x can be real or, more generally, complex. Maclaurin’s Series. While you can calculate Maclaurin series using calculus, many series for common functions have already been found. Use the series to write (b) The radius of convergence of the Maclaurin series for f is l. Taylor series expanded about x=0 are often relatively simple. For example, the following table shows the Maclaurin series for five common functions, along with the sigma notation for the expansion. look at a representation theorem (Theorem 11. Find the Maclaurin series for each. Find the Maclaurin expansion of cos2 x up to powers The series converges when x = 1 by Raabe’s test since n µ 1¡ an+1 an ¶ = 6n2 +5n 4n2 +10n+6! 3 2 > 1 Since the series for x = ¡1 is the negative of the above series, [¡1;1] is the interval of convergence of the power series. Find the fourth degree Maclaurin polynomial for the function f(x) = ln(x+ 1). 3. 5); it was in this context that he created what he called 'calculus of limits', later known as the method of majorants. For this case we note that In the next video, I'll do this with some actual functions just so it makes a little bit more sense. 337 for Maclaurin ellipsoids. A calculator for finding the expansion and form of the Taylor Series of a given function. The rst question is answered by nding the radius of convergence using the ratio test for absolute convergence and then checking at the endpoints if needed. ") A power series centered at x = a has the form X1 What is the Maclaurin series for f(x) = e x? To get the Maclaurin series, we look at the Taylor polynomials for f near 0 and let them keep going. Remark. The Attempt at a Solution The interval of convergence is always centered at the center of the power series. The Maclaurin series of sine and cosine Taylor Series and Maclaurin Series. Direct Comparison Test Complete Solution Step 1: Find the Maclaurin Series. Find the interv al of co nverg enc e for the p ow er series!! n =1 0 (3 x + 2)n n 2. (b)Use part (a) to express Z 1 0 sin It turns out that this same process can be performed to turn almost any function into an infinite series, which is called the Taylor series for the function (a MacLaurin series is a special case of a Taylor series). The convergence interval has to be adjusted accordingly. We can verify by Ratio Test to see that its radius of convergence is R = 1 . 9a - Geometric power series prt 2 The series for e^x contains factorials in the denominators which help to ensure the convergence for all x (and the same is true for related series such as sin and cos). Before you start this module, you must know how to find the Taylor polynomials of a given function. The more terms are included, the more accurate the value will be. ) (a) sin(3x) x (b) x2 cos(x5) (c) e2x 1 (d)cos p x 5 (e) ex2 1 x2 x4 (f)sin2(x) 2. The Form of a Convergent Power Series Definitions of Taylor and Maclaurin Series Convergence of Taylor Series Guidelines for Finding a Taylor Series Power Series for Get an answer for 'f(x)=e^(-2x) Prove that the Maclaurin series for the function converges to the function for all x' and find homework help for other Math questions at eNotes 1. A Maclaurin series is a power series that allows one to calculate an approximation of a function An example where the Maclaurin series is useful is the sine function. Isobars were described by means of implicit functions of q. Example 4 Find the Maclaurin series for sin(x) and prove that it represents sin(x) for all x. B. Real analysis is an area of mathematics dealing with the set of real numbers and, in particular, the analytic properties of real functions and sequences, including their convergence and limits. x is strictly inside the interval of convergence of the series when −R<x<R, so x is not equal to either of the two 6 Apr 2017 This ppt clear your concept about Power series - Convergence & give overview about Taylor & Laurent's Theorem . Maclaurin-Cauchy test can also be extended to double series: Theorem : If the function is a positive and steadily decreases to zero as and increase to infinity ( that is , if and ), then the double series converges or diverges with the double interval . Assume that each of the following functions has a power series expansion. We have a function f and a number a and we would like to express this function as a sum of a power series with center a. There are various tests for determining the convergence of a power series and details of these may be found in any good text on advanced calculus. In order to fully understand what that means we must understand the notion of a limit, and convergence. Recall our discussion of the power series, the power series will converge absolutely for every value of x in the interval of convergence. Lady. Hyperbolic sine (written sinh) and hyperbolic cosine (written cosh) are de ned as follows: sinhx = e xxe 2; coshx = e + e x 2: It is easy to check that these functions are derivatives of 6. Exercise 5. 1) and (4. Thinking about the problem: Have I seen a problem similar to this one before? We now look how to –nd the Taylor and Maclaurin series of some functions. 2. We use the results we obtained in the section on Taylor and Maclaurin series and combine them with a known If f is an analytic function, then its Taylor series (maclaurin series ) at any a point of the f area converges to f in the vicinity of a. ] F(x) = 7(1 − X)−2 F(x) = ∞ N = 0 Find The Associated Radius Of Convergence R. 8 Find the first four terms of the Maclaurin series for $\tan x$ (up to and including the $\ds x^3$ term). And over the interval of convergence, that is going to be equal to 1 over 3 plus x squared. You can specify the order of the Taylor polynomial. If you want the Maclaurin polynomial, just set the point to 0. To find the corresponding Maclaurin’s series (call it M) I can calculate the first few orders of derivatives and evaluate them at x=0, and observe that The Taylor series about 0 is called the Maclaurin series. A power series may represent a function , in the sense that wherever the series converges, it converges to . 3 Convergence of power series When we include powers of the variable zin the series we will call it apower series. The variable x is real. Convergence Tests Name Summary Divergence Test If the terms of the sequence don't go to zero, the series diverges. The only function that has four or fewer terms is as its Maclaurin series is. 1 Convergence: Expansion of f(x)=ln(1+x). Homework Equations 3. Important Maclaurin Series and their Radii of Convergence f(x) Power Series Representation Expansion Radius of Convergence Important Maclaurin Series and their Radii of Convergence f(x) Power Series Representation Expansion Radius of Convergence Question: Find The Maclaurin Series For F(x) Using The Definition Of A Maclaurin Series. Compute the interval of convergence for each series on the previous page. [Assume That F Has A Power Series Expansion. To distinguish between these four intervals, you must check convergence at the endpoints directly. Find the Find the Maclaurin series for . , x 0 2I : Next consider a function, whose domain is I, 8. Math formulas and cheat sheet generator creator for Taylor and Maclaurin Series. We have only considered real power series, but the definitions may easily be extended to A Taylor series centered at 0 is also called a Maclaurin series. One convenient way to find the sum of the Maclaurin series is to start with a well-known Maclaurin series and then manipulate it one step at a time until it matches the series you’ve been given. Example: Find the Maclaurin series for f(x) = exand determine its radius of convergence. A Maclaurin series is a Taylor series, just one that is centered at 0. Find the full Taylor Series representation for f(x) = e–x/2 centered around x=1 (7 points) b. Ex:Consider the Recall that if has derivatives of all orders at , then the Taylor series centered at for is be a function defined by a power series, with radius of convergence . There are many outstanding questions regarding Taylor polynomials and series, some of which will be addressed in later courses. Example Find a Taylor series for f (x) = ex centered at 2. Figure 1: Successive Maclaurin approximations (red) to \arctan x (blue), showing a radius of convergence of 1 Convergence of Power Series Lecture Notes Consider a power series, say 0 B œ " B B B B âa b # \$ %. Use Taylor’s Theorem with Remainder to prove that the Maclaurin series for f f converges to f f on that interval. I need to find the maclaurin series for f(x)= ∫ (from 0 to x) t^(2)e^(t^2) dt Note: A Maclaurin Series is a Taylor Series where a=0, so all the examples we have been using so far can also be called Maclaurin Series. Approximated a function f by a Taylor polynomial p(x) of degree n. maclaurin series convergence
njq, omachnuj, zaerk1q, nvdilno6, czvt, 6oepcuy, 5ozf, im8dgqk, rdrumv, pfu, rgp8u, | 2019-11-12T18:55:30 | {
"domain": "cpxlt.cn",
"url": "http://cpxlt.cn/b5g/maclaurin-series-convergence.html",
"openwebmath_score": 0.9300989508628845,
"openwebmath_perplexity": 299.1305186664839,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9923043544146899,
"lm_q2_score": 0.8418256412990657,
"lm_q1q2_score": 0.8353472495190016
} |
http://math.stackexchange.com/questions/366920/combinatorics-example-problem | Combinatorics Example Problem
Ten weight lifters are competing in a team weightlifting contest. Of the lifters, 3 are from the United States, 4 are from Russia, 2 are from China, and 1 is from Canada.
Part 1
If the scoring takes account of the countries that the lifters represent, but not their individual identities, how many different outcomes are possible from the point of view of scores?
This part I understand: 10! / [ 3! 4! 2! ] = 12,600
Part 2 (don't understand this part)
How many different outcomes correspond to results in which the United States has 1 competitor in the top three and 2 in the bottom three?
This part I'm confused. Here you have 10 slots.
The first three slots must be of some order: US, US, or [Russia, China, Canada].
The last three slots must be of some order US, [Russia, China, Canada], [Russia, China, Canada].
I thought the answer would be this: $\binom{3}{2} \binom{1}{1} * \frac{7!}{4!\ 3!\ 2!}$
My reasoning: In the first 3 slots, you have to pick 2/3 US people. Then you only have one remaining. You have 7! ways to organize the rest but have to take out repeats so you divide by factorials of repeats which is 4,3, and 2. But my middle term is wrong....
My research shows me answers of 2 forms but I can't understand it:
Method 1: $\binom{3}{1} \binom{3}{2} * \frac{7!}{4!\ 3!\ 2!}$
This case, I don't understand why there's a 3 in the second binomial, $\binom{3}{2}$. We already selected ONE US person so shouldn't it be $\binom{2}{2}$?
Method 2: $\binom{7}{2} \binom{3}{1} \binom{5}{4} \binom{3}{2} \binom{1}{1}$ ?
Sorry for the long post. Thanks again.
-
Part 2 is solved in two stages (using Method 1):
Stage 1: Let U represent a slot holding a US competitor, and let _ represent a slot holding a non-US competitor. The first three slots must be one of the following:
U _ _
_ U _
_ _ U
This is where the $\binom{3}{1}$ comes from, since we are choosing one of three slots to be the U slot. The middle four slots have to be _ _ _ _. There is only one way to do this—we could write this as $\binom{4}{0}$ since we are choosing zero of the four slots to be U slots, or we could simply omit this factor. The last three slots must be one of the following:
U U _
U _ U
_ U U
This is where the $\binom{3}{2}$ comes from: we are choosing two of three slots to be U slots.
Stage 2: We now have a slot arrangement, of which _ U _ _ _ _ _ U U _ is an example. (There are a total of nine possibilities.) We now have to fill the blank slots with the three non-US nationalities. There are $\frac{7!}{4!\,2!\,1!}$ ways to do this.
Note: Since we only care about which nationality occupies which slot, and not which individual competitor, at no stage should we be selecting individual competitors from the set of competitors of a given nationality. I think this is where you went wrong. (You selected two particular US competitors from the set of three US competitors. There is no reason to do this.)
Alternative solution: I think Method 1 is a bit confusing because in Stage 1 we are choosing slots, whereas in Stage 2 we are permuting nationalities and dividing out repeats. A more uniform method would be to think of the entire process as one of choosing slots. If we take this point of view, there there are
• $\binom{3}{1}$ ways to choose one of the first three slots for the US.
• $\binom{3}{2}$ ways to choose two of the last three slots for the US.
• $\binom{7}{2}$ ways to choose two of the slots not yet assigned for China.
• $\binom{5}{4}$ ways to choose four of the slots not yet assigned for Russia.
• $\binom{1}{1}$ way to choose the remaining unassigned slot for Canada.
I believe this is what's going on in Method 2, although they wrote the factors in an unnatural order.
-
Thanks Will. I understand your stage 1 explanation--its fantastic. However, your Alternative Solution is not entirely clear to me. From your explanation, I now understand the terms $\binom{3}{1}$ and $\binom{3}{2}$. However, the rest of the terms are still not clear to me. For instance, why did choose to fill in the slots in the order that you did: China, Russia, and then Canada. What if someone wanted to fill in the slots in this order: Russia, China, and Canada. Then it would be $\binom{7}{4}\binom{5}{2}\binom{1}{1}$. Right? Then you wouldn't get the same answer. – user1527227 Apr 23 '13 at 18:19
Basically, why do you allow China to have 7 slots? Why not give the 7 to Russia first if someone wanted to fill in Russia? – user1527227 Apr 23 '13 at 18:45
It, in fact, does not matter in which order you place the nationalities, but if you do it in your order: Russia, China, Canada, then the expression would be $\binom{7}{4}\binom{3}{2}\binom{1}{1},$ which does equal $\binom{7}{2}\binom{5}{4}\binom{1}{1}$. This is because the products "telescope" so that there is systematic cancellation between the denominator and numerator of successive factors. So, for example, the expression $\binom{7}{2}\binom{5}{4}\binom{1}{1}$ can be written $\frac{7!}{2!\,5!}\frac{5!}{4!\,1!}\frac{1!}{1!\,0!}=\frac{7!}{2!\,4!\,1!}$ where we have cancelled factors$\ldots$ – Will Orrick Apr 23 '13 at 18:54
$\ldots$of $5!$ and $1!$, and dropped the factor of $0!$ in the denominator. The expression $\binom{7}{4}\binom{3}{2}\binom{1}{1}$ telescopes similarly: $\frac{7!}{4!\,3!}\frac{3!}{2!\,1!}\frac{1!}{1!\,0!}=\frac{7!}{4!\,2!\,1!}$. You always get $\frac{7!}{2!\,4!\,1}$ with the denominator factors permuted depending on which nationality gets placed first. So if we do Canada, China, Russia, we get $\binom{7}{1}\binom{6}{2}\binom{4}{4}=\frac{7!}{1!\,2!\,4!}$. – Will Orrick Apr 23 '13 at 18:54
@user1527227 : Basically the second denominator factor in each binomial coefficient cancels with the numerator of the next binomial coefficient, except for the last one, where this factor is always $0!$. Therefore you always get the same answer as you get by the divide-out-repeats method. Also, you aren't "allowing China to have 7 slots". What you're doing is giving 2 out of 7 slots to China. I also give 4 of those slots to Russia, but if since I do Russia second, I can't give them the 2 slots I already gave China. – Will Orrick Apr 23 '13 at 18:59
For the second, you pick the slots (not the people-we said all the people from one country were interchangeable) for the two US people in the bottom ${3 \choose 2}$ ways, but then have to pick which slot the US person in the top is in, which adds a factor ${3 \choose 1}$. Then of the seven left, you just have $\frac {7!}{4!2!}$ as there are no Americans among them. So Method 1 is off by the $3!$ in the denominator. I don't understand the terms in Method 2.
Added: One way to look at the first part, which I think is the way you did, is to say there are $10!$ orders of the people, but if the Americans are interchangeable you divide by $3!$, for the Russians you divide by $4!$ and for the Chinese you divide by $2!$. Another way would be to say we pick three slots for Americans in ${10 \choose 3}$ ways, then pick slots for the Russians in ${7 \choose 4}$ ways, then pick slots for the Chinese in ${3 \choose 2}$ ways, giving ${10 \choose 3}{7 \choose 4}{3 \choose 2}=120\cdot 35 \cdot 3=12600$. Of course the answer is the same, but the approach is better suited to the second part. Now if you think of choosing slots for nationalities you get what I did before.
-
I'm still lost... – user1527227 Apr 19 '13 at 23:27
Can anyone else explain the second part of the question? I still don't understand. Thanks. – user1527227 Apr 19 '13 at 23:59 | 2016-06-28T06:34:29 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/366920/combinatorics-example-problem",
"openwebmath_score": 0.7334111332893372,
"openwebmath_perplexity": 336.420611191182,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9898303443461076,
"lm_q2_score": 0.8438951025545426,
"lm_q1q2_score": 0.8353129799535567
} |
https://math.stackexchange.com/questions/153457/characteristic-polynomial-of-a-matrix-of-1s?noredirect=1 | # Characteristic polynomial of a matrix of $1$'s
I am trying to calculate the characteristic polynomial of the $$n \times n$$ matrix $$A = \{ a_{ij} = 1 \}$$.
• Case $$n=2$$: I obtained $$p(\lambda)=\lambda^2-2\lambda$$ .
• Case $$n=3$$: I obtained $$p(\lambda)=-\lambda^3+3\lambda^2$$.
• Case $$n=4$$: I obtained $$p(\lambda)=\lambda^4 - 4\lambda^3$$.
I guess that for the general case, we have
$$p(\lambda)=(-1)^n\lambda^{n}+(-1)^{n-1}n\lambda^{n-1}$$
I tried to use induction, but it didn't work, unless I've done wrong. Can somebody help me? Or give me a hint?
• I think it's easier to just compute all of the eigenvalues. The eigenvectors are easy to write down. (Also, you should be using a definition of the characteristic polynomial that makes it monic.) Jun 3 '12 at 21:00
• I really dont understand the problem, could any one explain me? Jun 6 '12 at 14:16
• Jul 17 '18 at 0:55
Note that the matrix $$A = e e^T$$ where $e = \begin{pmatrix}1\\1\\1\\\vdots\\1\\1 \end{pmatrix}_{n \times 1}$.
Hence, $A^2 = \left(ee^T \right) \left(ee^T \right)= e \left(e^T e \right) e^T = n ee^T = nA$.
This clearly indicates that the matrix is a rank one matrix. Hence it must have $n-1$ eigenvalues as $0$. The only non-zero eigen value if $\lambda =n$, since we have $\lambda^2 = n \lambda$ and $\lambda \neq 0$.
The trace is $n$. The eigenvalue $0$ has multiplicity $n-1$. From this we can write down the characteristic polynomial without any computation. Or else we can pick up the eigenvalue of $n$ by noting that the all $1$'s vector times our matrix is the all $n$'s vector.
• span of 17 seconds, you, Dennis, Marvis. I was going to answer, but. Jun 3 '12 at 21:05
Using the matrix determinant lemma,
$$\det \left( s \, \mathrm I_n - \mathbb 1_n \mathbb 1_n^\top \right) = s^n \left( 1 - \frac{1}{s} \mathbb 1_n^\top \mathbb 1_n \right) = s^n - n s^{n-1} = s^{n-1} \left( s - n \right)$$
Hint: Denote $v=(1,1,...,1)$ and $v_j=(1,0,..,0,-1,0,...,0)$ (all zeroes except for the first, where there is $1$ and the $j$th, where there is $-1$) (all column vectors). What happens when you multiply $A\cdot v$ and $A\cdot v_j$?
We wish to compute the determinant of the $$n\times n$$ matrix $$M=\begin{bmatrix} 1-\lambda&1&1&\cdots&1\\ 1&1-\lambda&1&\cdots&1\\ 1&1&1-\lambda&\cdots&1\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 1&1&1&\cdots&1-\lambda \end{bmatrix}\tag1$$ Here are two approaches, the second of which uses a generalization of the lemma cited in Rodrigo de Azevedo's answer.
Finding a Similar Matrix
Note that the $$n-1$$ dimensional subspace orthogonal to $$\begin{bmatrix}1&1&1&\cdots&1\end{bmatrix}^\text{T}$$ is multiplied by $$-\lambda$$. (If the subspace is orthogonal to a given vector, we can subtract that vector from each row of the matrix when operating on that subspace.)
$$\begin{bmatrix}1&1&1&\cdots&1\end{bmatrix}^\text{T}$$ is multiplied by $$n-\lambda$$. (Just compute it.)
Thus, $$M$$ is similar to $$\begin{bmatrix} -\lambda&0&0&\cdots&0\\ 0&-\lambda&0&\cdots&0\\ 0&0&-\lambda&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&n-\lambda\\ \end{bmatrix}\tag2$$ Therefore, $$\det(M)=(-\lambda)^{n-1}(n-\lambda)\tag3$$
$$\boldsymbol{\det(\lambda I_n-AB)=\lambda^{n-m}\det(\lambda I_m-BA)}$$
$$\det(\lambda I_n-AB)=\lambda^{n-m}\det(\lambda I_m-BA)\tag4$$
Let $$A=\begin{bmatrix}1&1&1&\cdots&1\end{bmatrix}^\text{T}$$ and $$B=A^\text{T}$$, then $$M=AB-\lambda I_n$$.
Furthermore, $$m=1$$ and $$BA=\begin{bmatrix}n\end{bmatrix}$$.
$$(4)$$ then says that \begin{align} \det(-M) &=\det(\lambda I_n-AB)\tag{5a}\\[2pt] &=\lambda^{n-1}\det(\lambda I_1-BA)\tag{5b}\\[2pt] &=\lambda^{n-1}\det(\lambda I_1-\begin{bmatrix}n\end{bmatrix})\tag{5c}\\ &=\lambda^{n-1}(\lambda-n)\tag{5d} \end{align} which, since $$\det(-M)=(-1)^n\det(M)$$, becomes $$\det(M)=(-\lambda)^{n-1}(n-\lambda)\tag6$$ | 2021-10-28T15:31:20 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/153457/characteristic-polynomial-of-a-matrix-of-1s?noredirect=1",
"openwebmath_score": 0.9887266159057617,
"openwebmath_perplexity": 216.01364640076008,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9898303422461274,
"lm_q2_score": 0.8438951025545426,
"lm_q1q2_score": 0.8353129781813937
} |
https://math.stackexchange.com/questions/4480341/expected-value-of-100-briefcases-each-with-1-dollar-except-for-one-briefcase | # Expected value of $100$ briefcases each with $1$ dollar except for one briefcase that resets your accumulated amount to zero?
In this game there are $$100$$ suitcases and each contains the same dollar amount (let's say $$1$$ dollar) except for one suitcase that contains a bomb that resets the total value you accumulated so far to $$0$$. Naturally, you don't know which suitcase contains the bomb and you need to open every suitcase, what is the expected value of the amount you end up with?
This is not a homework question, just a question I wondered about after seeing it in a gameshow. Is there a way to solve this analytically? And if so, how would you do that?
• Good question. But isn't it more interesting if you can choose when to stop opening briefcases? Jun 25 at 19:49
• @mjqxxxx Good point : that wades into stochastic optimal control, albeit it should be simple enough to handle. Jun 26 at 8:25
• @SarveshRavichandranIyer how would you handle that then? Jun 27 at 1:28
• @strateeg32 Let's say I'm trying to maximize my expected earnings. At each step, if my expected earnings is likelier to be less than what I have now, then I would not open the next briefcase. On the other hand, if my expected earnings are likelier to be larger than what I have now, then I would open the next briefcase. That's just the strategy : one has to calculate the expected earnings from opening and closing a box somewhat carefully. I still think it's elementary, but it's definitely worth a shout as a good question which you can ask. Jun 27 at 8:23
The expected value of your profit is $$(100-X)$$ where $$X$$ is the location of the bomb. Since $$X$$ is uniform over $$\{1,\ldots,100\}$$, its expected value is $$50.5$$ and from the linearity of expectation, the expected profit is $$E(100-X)=100-E(X)=49.5$$
An alternative approach is to recognize that the expectation is linear to the placement of the bomb.
Edit
Somewhat formally, this signifies that if the result changes by $$(k)$$, as the bomb changes position from (for example) $$(1)$$ to $$(2)$$, then the result will also change by $$(k)$$, as the bomb changes from $$(n)$$ to $$(n+1)$$, for $$n \in \{2,3,\cdots,99\}.$$
As an explanation of why this is critical, consider the following two simpler problems, where you roll one $$6$$-sided die.
• What is the expected value that will show?
• What is the expected square of the value that will show?
The two computations are
• $$\displaystyle \frac{1 + 2 + \cdots + 6}{6} = \frac{21}{6} = \frac{7}{2}.$$
• $$\displaystyle \frac{1^2 + 2^2 + \cdots + 6^2}{6} = \frac{91}{6} \color{red}{\neq} \left[\frac{7}{2}\right]^2.$$
In the first (simpler) problem above, the value of the number showing on the die is linear to the expected number that shows on the die. In the second (simpler) problem above, the value of the square of the number showing on the die is $$\color{red}{\text{not}}$$ linear to the expected number that shows on the die.
Therefore, the use of the computation $$\displaystyle \left[\frac{7}{2}\right]^2$$ in the second simpler problem above leads to the wrong answer.
In the stated problem, the overall earnings are linear to the placement of the bomb. Therefore, you can assume that (1/2) the time, the bomb will be in position number $$(50)$$, and (1/2) the time, the bomb will be in position number $$(51)$$.
This implies that for every two games played, you should expect to win $$(50 + 49)$$. | 2022-08-15T09:00:22 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/4480341/expected-value-of-100-briefcases-each-with-1-dollar-except-for-one-briefcase",
"openwebmath_score": 0.8206701278686523,
"openwebmath_perplexity": 363.36057725060164,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9898303398461495,
"lm_q2_score": 0.8438950986284991,
"lm_q1q2_score": 0.8353129722699472
} |
https://gmatclub.com/forum/how-to-calculate-256477.html | GMAT Changed on April 16th - Read about the latest changes here
It is currently 25 Apr 2018, 16:50
### GMAT Club Daily Prep
#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.
Customized
for You
we will pick new questions that match your level based on your Timer History
Track
Your Progress
every week, we’ll send you an estimated GMAT score based on your performance
Practice
Pays
we will pick new questions that match your level based on your Timer History
# Events & Promotions
###### Events & Promotions in June
Open Detailed Calendar
# How to calculate (5√3)² ?
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Author Message
TAGS:
### Hide Tags
Intern
Joined: 15 Oct 2017
Posts: 23
How to calculate (5√3)² ? [#permalink]
### Show Tags
30 Dec 2017, 13:28
Hello everyone,
I would to know how to calculate (5√3)²
thank you for your help
Intern
Joined: 15 Oct 2017
Posts: 23
Re: How to calculate (5√3)² ? [#permalink]
### Show Tags
30 Dec 2017, 13:30
I also see that (3√3)² = 27 but I can't figure out how to make the calculation
Manhattan Prep Instructor
Joined: 04 Dec 2015
Posts: 509
GMAT 1: 790 Q51 V49
GRE 1: 340 Q170 V170
Re: How to calculate (5√3)² ? [#permalink]
### Show Tags
30 Dec 2017, 13:31
thegame12 wrote:
Hello everyone,
I would to know how to calculate (5√3)²
thank you for your help
Sure thing!
5√3 is basically a product of two values. It's 5 times √3.
When you square a product of two values, you square both of the values. For example, $$(xy)^2$$ equals $$x^2y^2$$.
So, to square 5√3, you square the 5 and the √3.
$$(5^2)(\sqrt{3}^2)$$
$$(25)(3)$$
$$75$$
_________________
Chelsey Cooley | Manhattan Prep Instructor | Seattle and Online
My upcoming GMAT trial classes | GMAT blog archive
Math Expert
Joined: 02 Sep 2009
Posts: 44656
Re: How to calculate (5√3)² ? [#permalink]
### Show Tags
30 Dec 2017, 13:35
thegame12 wrote:
Hello everyone,
I would to know how to calculate (5√3)²
thank you for your help
$$(ab)^n=a^n*b^n$$.
$$(5\sqrt{3})^2=5^2*(\sqrt{3})^2=25*3=75$$
_________________
Re: How to calculate (5√3)² ? [#permalink] 30 Dec 2017, 13:35
Display posts from previous: Sort by
# How to calculate (5√3)² ?
new topic post reply Question banks Downloads My Bookmarks Reviews Important topics
Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®. | 2018-04-25T23:50:50 | {
"domain": "gmatclub.com",
"url": "https://gmatclub.com/forum/how-to-calculate-256477.html",
"openwebmath_score": 0.3477068245410919,
"openwebmath_perplexity": 11092.412046085139,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9465966702001758,
"lm_q2_score": 0.8824278664544911,
"lm_q1q2_score": 0.8353032800776667
} |
https://mathematica.stackexchange.com/questions/42478/define-a-function-with-a-function-for-an-input | # Define a function with a function for an input
I want to create the functions $even[f]$ and $odd[f]$ which will apply the formula $even[f(x)] = \frac{f(x)+f(-x)}{2}$ and $odd[f(x)]=\frac{f(x)-f(-x)}{2}$ for an arbitrary $f$.
I tried using
even[ff, x_] := (ff[x] + ff[-x])/2
But that doesn't seem to work. What would the correct approach to this be?
• even[f_[x_]] := (f[x] + f[-x])/2 and odd[f_[x_]] := (f[x] - f[-x])/2 Feb 18, 2014 at 4:38
• or like this: even[f_] := (f[#] + f[-#])/2 &; odd[f_] := (f[#] - f[-#])/2 & Feb 18, 2014 at 4:39
• Neither approach is working for me at the moment. In the first approach mathematica doesn't evaluate anything, just leaving the even function with whatever its argument is, and in the second approach I have a lot of #'s that are left unevaluated as well.
– R R
Feb 18, 2014 at 4:46
• When I boot mathematica fresh and enter it in this is what I get:In[28]:= ClearAll[f, ff, even, x] In[29]:= even[f_[x_]] := (f[x] + f[-x])/2 In[30]:= f[x_] := x^2 + x In[35]:= even[f] even[f[x]] even[f[x_]] even[f[#]] Out[35]= even[f] Out[36]= even[x + x^2] Out[37]= even[x_ + x_^2] Out[38]= even[#1 + #1^2]
– R R
Feb 18, 2014 at 5:12
• Did you SetAttributes like I did? Feb 18, 2014 at 5:23
SetAttributes[{even, odd}, HoldAll];
even[f_[x_]] := (f[x] + f[-x])/2;
odd[f_[x_]] := (f[x] - f[-x])/2;
Usage
g[x_] := x + x^2;
even[g[x]]
x^2
OR as Szabolcs suggested using pure functions:
even[f_] := (f[#] + f[-#])/2 &;
odd[f_] := (f[#] - f[-#])/2 &
Usage
Using the same g as above
even[g][x]
x^2
• It might be helpful to mention that Szabolcs' method requires calling even[g][x] instead of even[g[x]]. Also it doesn't require HoldAll.
– user484
Feb 18, 2014 at 10:19
You said you tried using
even[ff, x_] := (ff[x] + ff[-x])/2
but I guess you forgot to put the underscore on the first argument. If you do
even[ff_, x_] := (ff[x] + ff[-x])/2
g[x_] := x + x^2;
x^2
P.S. No SetAttributes necessary using this method. | 2022-07-06T01:48:18 | {
"domain": "stackexchange.com",
"url": "https://mathematica.stackexchange.com/questions/42478/define-a-function-with-a-function-for-an-input",
"openwebmath_score": 0.32384538650512695,
"openwebmath_perplexity": 7118.2226918373235,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9465966747198242,
"lm_q2_score": 0.8824278556326344,
"lm_q1q2_score": 0.8353032738219968
} |
http://gesuitialquirinale.it/qyms/solving-systems-of-nonlinear-equations-khan-academy.html | # Solving Systems Of Nonlinear Equations Khan Academy
2) Solving systems of equations using substitution. Whether in the real world or a cliche fantasy one, systems of equations are key to solving super-important issues like "the make-up of change in a troll's pocket" or "how can order the right amount of potato chips for a King's party. Graphically, this represents a point where the lines cross. After you enter the system of equations, Algebra Calculator will solve the system x+y=7, x+2y=11 to get x=3 and y=4. So x is linear but x 2 is non-linear. The first equation x+y=7. Interactive Differential Equations : This site covers first order differential equations, second order differential equations, linear and nonlinear applications, Laplace Transforms, series solutions, and boundary value problems. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Lesson 4 Linear and nonlinear functions. Improve your math knowledge with free questions in "Graph proportional relationships" and thousands of other math skills. Khan Academy: Linear Algebra 7 - Mobile/EducationXimarc Studios Inc is proud to bring you Khan Academy Linear Algebra 7 (videos 121-138). The equations to solve are F = 0 for all components of F. 1-2 Learning Targets:. Math - 10 - Algebra - Worked Examples 4\137 Non-Linear Systems of Equations 1. The "solution" to the system will be any point (s) that the lines share; that is, any point (s) where the x -value and corresponding y -value for y = x2 + 3 x + 2 is the same as the x -value and corresponding y -value for y = 2 x + 3; that is, where the lines overlap or. Create, Analyze, and Model Linear and Non-Linear Functions 4. Perimeter of a parallelogram Khan Academy 198 01:19. 1: Permutations and Combinations. 4 solve multi-step equations with variables on both sides 2. Solving nonlinear system with three equations and three variables. Describing Steps When Solving Equations Algebra Khan. 3 I can solve systems of equations using elimination p 145 3. Systems Of Equations With Elimination Coffee And Croissants. x^2+y^2=4 x+y=-2 Log On Algebra: Systems of equations that are not linear Section. 3 Modeling with Linear Functions go over p. 2 - Solving Systems of Equations by Substitution - [Notes] - Khan Links. Direct & Inverse Variation; Rational Functions Unit Test (counts as a QUIZ) Paper-Based Assignments for Week 32 (contact DHS if you need this option). Apr 23, 2013 - Explore mjones827's board "Systems of equations", followed by 160 people on Pinterest. Expressions & Equations. , 2x + 5y = 0 3x – 2y = 0 is a …. There are numerical techniques which help to approximate nonlinear systems with linear ones in the hope that the solutions of the linear systems are close enough to the solutions of the nonlinear systems. CC3 Homework Help (click here) Homework E-TOOLS (click here)Virtual Algebra Tiles…. Extra Practice for Lesson 15: Understand Systems of Equations. Solve equations with one rational expression (advanced) Check out Khan Academy's new Mastery System; Was this article helpful?. 3 - Linear Systems. Handouts: We will continue solving systems of equations, today focusing on systems where either one or both of the equations needs to be rewritten in slope-intercept form first. So x is linear but x 2 is non-linear. Unit 6 - Systems of Equations. Learn more about Quia: Create your own activities. Frequently exact solutions to differential equations are unavailable and numerical methods become. This is a famous non-linear system of equations known as the Lotka-Volterra equations. " This site contains videos aligned to the Common Core State Standards that can be used to supplement classroom instruction. G o t a d i f f e r e n t a n s w e r? C h e c k i f i t ′ s c o r r e c t. Qualitative, analytical, and numerical methods for first- and second-order single ordinary equations as well as first-order constant coefficient linear systems and some special nonlinear systems. Solve systems of linear equations exactly and approximately (e. Solving Nonlinear Systems with Substitution From Thinkwell's College Algebra Chapter 7 Systems of Equations and Inequalities, Subchapter 7. Use matrix methods to solve systems of linear equations and perform eigenvalue decomposition. SWBAT solve systems of equations involving a linear function and a quadratic function both algebraically and graphically. Because it is used in such topics as nonlinear systems, linear algebra, computer programming, and so much more. Khan Academy: 8th Grade Math. x^2+y^2=4 x+y=-2 Log On Algebra: Systems of equations that are not linear Section. Cancel the common factor of 2 2. Students can solve real world problems leading to two systems of linear equations. Linear Functions. 6 graph and write equations of two-variable inequalities 5. 7 Solving Nonlinear Systems. Improve your math knowledge with free questions in "Graph proportional relationships" and thousands of other math skills. Then you can be expected that the equations have one solution. The site is well-aligned with the Common Core. If we can get a short list which contains all solutions, we can then test out each one and throw out the invalid ones. Since these equations represent two lines in the xy-plane, the simultaneous solution of these two equations (i. [email protected] Introduction to Statistical Methods with MATLAB. Solving Equations Inequalities Algebra I Math Khan Academy. Topic 3: Equations with variables on both sides and equations with parentheses a) Solving equations with variables on both sides - Video, PRACTICE b) Solving equations with parenthesis - Video, PRACTICE. Knowing which one to use is based on the form of the equations and, if you carefully look at the systems, usually one of the techniques will seem to work best. Sometimes we need solve systems of non-linear equations, such as those we see … Systems of Non. Solving a system of linear equations: v. Khan Academy; 8/13. Systems of equations with substitution: y=-1/4x+100 and y=-1/4x+120. Usually, the problem is to find a solution for x and y that satisfies both equations simultaneously. This course contains many tools to solve different type of differential equations like, First-Order Differential Equations, Second Order Differential Equations, Applications of Linear and Nonlinear differential equations. Khan Academy: Separable Differential Equations. Teacher Notes. Please update your bookmarks! This is a 4 part worksheet: Activity on Systems of Equations (Create an advertisement for your favorite method to solve systems of equations). Why can we subtract one equation from the other in a system of. Consistent and inconsistent systems. on te demande représentait pas graphiquement les solutions à ce système est de vérifier par le calcul donc ce qu'on demande c'est d'abord de représenter graphiquement cette première équation puis de représenter cette deuxième l'équation on aura une courbe jonas une courbe verte et des points intersection entre les cours de jaune et de courbe verte seront les représentations. Solving a system of equations requires you to find the value of more than one variable in more than one equation. Solving Systems of Equations using Elimination - Video Series Linear & Nonlinear Functions. In order to solve for y, take the value for x and. Solving systems of linear equations by graphing worksheet with answers. Finding the Y-Intercept ("b") Quiz. We will just focus on how to use Mathcad to solve the problem. Why can we subtract one equation from the other in a system of. x + 4x + 6 = 1. Find the vector form for the general solution. Start studying Algebra 2 Khan Academy studies. Balancing chemical equations Khan Academy 189 14:27. and Learning Target. It has been shown , , , , , that the quadrature formulas have been used to develop some iterative methods for solving a system of nonlinear equations. This equation says that the rate of change. SWBAT understand the process of factoring polynomials as a tool to solve non-linear systems. Qualitative, analytical, and numerical methods for first- and second-order single ordinary equations as well as first-order constant coefficient linear systems and some special nonlinear systems. In order to solve systems of equations in three variables, known as three-by-three systems, the primary goal is to eliminate one variable at a time to achieve back-substitution. Number of solutions to a system of equations. The site is well-aligned with the Common Core. 2) - video (or), video (and), PRACTICE. Launch Details. I also included a detailed answer key for each worksheet. Khan Academy for Analyzing the Number of Solutions to Linear Equations. Solving Systems of Equations using Elimination - Video Series Linear & Nonlinear Functions. The laws of nature are expressed as differential equations. Standard Form of Linear Equations Quiz. org right now: https://www. Unit 3 - Linear relationships. Multiple-choice & free-response. Remember that we could have one point, no points, or even many points for answers depending on the functions. It is recommended that students explore the videos and extra practice problems provided on KHAN Academy when they are struggling or are absent for a lesson and they are unable to meet with me. The complexity comes in because you can't just integrate the equation to solve it. A Differential Equation is a n equation with a function and one or more of its derivatives: Example: an equation with the function y and its derivative dy dx. Learn how to use the Algebra Calculator to solve systems of equations. Systems Non-Linear Systems of Equations 1 Non-Linear Systems of Equations 2 Non-Linear Systems of Equations 3 Non-Linear Systems CCSS. (and manipulation) | High School Math | Khan Academy by Khan Academy. Usually, the problem is to find a solution for x and y that satisfies both equations simultaneously. Solve Systems of Linear Equations and Inequalities with Models 5. Students should be able to solve systems of linear equations by substitution and elimination. Here are some ideas to get you started. There are two critical points (0,0) and (b q, a p) In the usual way, we analyze the types of the critical points. This system has two equations of each kind: a linear and a non-linear. Solving Equations with Variables on Both Sides Notes; Solving Equations with Variables on Both Sides; Variables on Both Sides Examples; Solving Worksheet; Khan Academy; 8/14 and 8/15. 2) Solving systems of equations using substitution. Solving Systems of Equations Algebraically Johnny Wolfe www. Notice is provided in the event of a data breach. In Chapter 3 for numerical solutionof semilinear first order equations. Number of solutions to a system of equations. a z 9AmltlU Or Gi 5gUh vtIs k Hrfe bs OeWrGvie KdP. Solving Equations Inequalities Algebra I Math Khan Academy. The rst method we will look at is Newton’s method. This An Interactive Guide To The Fourier Transform is a wonderful introduction to the Fourier transform,. There are numerical techniques which help to approximate nonlinear systems with linear ones in the hope that the solutions of the linear systems are close enough to the solutions of the nonlinear systems. 07 Interpret Linear Function Equations-Interpreting Features of a Linear Function (Khan Academy): Click Here -Interpreting Linear Graphs (Khan Academy): Click Here-Interpreting Graphs of Linear and Nonlinear Functions (Khan Academy): Click Here-Interpreting Tables (Khan Academy): Click Here-The Meaning of Slope and Y-Intercept in Word Problems (Purple Math): Click Here. Frequently exact solutions to differential equations are unavailable and numerical methods become. Solving Nonlinear Systems with Substitution From Thinkwell's College Algebra Chapter 7 Systems of Equations and Inequalities, Subchapter 7. Nonlinear equations. Khan Academy Video Correlations By SpringBoard Activity SB Activity Video(s) Unit 1: Equations, Inequalities, Functions Activity 1 Creating Equations Representing a relationship with 1-1 Learning Targets: Create an equation in one variable from a real-world context. "In Module 4, students use similar triangles learned in Module 3 to explain why the slope of a line is well-defined. Students should be able to solve systems of linear equations by substitution and elimination. Tutoring MATH Search this Guide Search. Videos and practice problems are offered at all levels and subjects of math. It can solve systems of linear equations or systems involving nonlinear equations, and it can search specifically for integer solutions or solutions over another domain. Create the worksheets you need with Infinite Algebra 1. Find the vector form for the general solution. This Algebra Worksheet may be printed, downloaded or saved and used in your classroom, home school, or other educational environment to help someone learn math. Solving Systems of Equations Using Linear Combinations (Addition Method). Solve the following system of linear equations by transforming its augmented matrix to reduced echelon form (Gauss-Jordan elimination). The Wolfram Language' s differential equation solving functions can be applied to many different classes of differential equations, automatically selecting the appropriate algorithms without needing preprocessing by the user. Khan Academy for Analyzing the Number of Solutions to Linear Equations. The theory for solving linear equations is very well developed. Users can interact with trusted users and/or students. Solve systems of nonlinear equations in serial or parallel. This activity was created by a Quia Web subscriber. Because it is used in such topics as nonlinear systems, linear algebra, computer programming, and so much more. org right now: Solving Nonlinear System of Equations 070-33 How to solve nonlinear systems of equations algebraically and graphically. 1 Khan Academy Study-Guide This pre-SGPE math camp is designed to help dust off your math skills by utilizing the excellent software, videos, and problems sets from Khan Academy. Many systems, e. Khan Academy - Solving Systems of Equations Khan Academy - Equivalent systems of Equations 3 1/22 - 1/25 By Substitution Section7. Nonlinear equations. Systems of equations with graphing: y= 7/5x-5 and y=3/5x-1. 7 Graphing and Solving Quadratic Inequalities 303 1. So Salman Khan did go to MIT — so a number of these are based off of MIT courses. Order of operations. The steps are as follows: Rewrite the inequality so that there is a zero on the right side. After awhile, determining these functions will become easy and you will be able to tell which function you have simply by looking at the equation itself. The elimination method achieves this by adding or subtracting equations from each other in order to cancel out one of the variables. Solve simple cases by inspection. Created: Aug 13, 2013 Solving Equations 1 - An introduction to Solving Linear Equations (+ Worksheet). It also introduces the viewpoint of a system of equations as two lines on a graph and uses this as our first method to solve the system of equations. 61, x3(0) ≈78. If you're seeing this message, it means we're having trouble loading external resources on our website. In our last lesson we used the Linear Combinations or Addition Method to solve systems of equations. Finding the Y-Intercept ("b") Quiz. A system of differential equations is a set of two or more equations where there exists coupling between the equations. 07 Interpret Linear Function Equations-Interpreting Features of a Linear Function (Khan Academy): Click Here -Interpreting Linear Graphs (Khan Academy): Click Here-Interpreting Graphs of Linear and Nonlinear Functions (Khan Academy): Click Here-Interpreting Tables (Khan Academy): Click Here-The Meaning of Slope and Y-Intercept in Word Problems (Purple Math): Click Here. This activity was created by a Quia Web subscriber. In this session we take a break from linear equations to study autonomous equations. The rule says that if the current value is. The equations rely on each other and can be solved only with the information that each provides. 4 solve multi-step equations with variables on both sides 2. Balancing chemical equations Khan Academy 189 14:27. and Learning Target. Knowing which one to use is based on the form of the equations and, if you carefully look at the systems, usually one of the techniques will seem to work best. Relations and functions, as well as all aspects of graphing, slopes, and inequalities, are covered in engaging ways that will sharpen students. G o t a d i f f e r e n t a n s w e r? C h e c k i f i t ′ s c o r r e c t. Perimeter of a parallelogram Khan Academy 198 01:19. 4 to solve nonlinear first order equations, such as Bernoulli equations and nonlinear homogeneous equations. But for nonlinear systems, a small change in a parameter can lead to sudden and dramatic. An autonomous differential equation is an equation of the form. There are two critical points (0,0) and (b q, a p) In the usual way, we analyze the types of the critical points. graphing calculator solve systems of equations with three variables Multi-step equations worksheet what printouts to do with a 6th grader comparing integers worksheets multiplying and dividing radical expression solver solve cubed equation Worksheet on graphing linear equations non linear equation solver. Sometimes we need solve systems of non-linear equations, such as those we see … Systems of Non. In this case, we speak of systems of differential equations. Describing Steps When Solving Equations Algebra Khan. F V nMIaGdOeo bwUi5tVhU QIenXf EiVnPixtHeT FA Ll 8g CeKb4r Ma3 W2u. If you need support with algebra and in particular with rearranging formulas calculator or precalculus come visit us at Algebra-equation. First, you need to classify what kind of differential equation it is based on several criteria. That's all there is to it. Systems of equations with elimination: potato chips. 2; Finish Test if extra time needed. Our first step is to rearrange each equation so that the left side is just y: Now that both equations are equal to y, we can see that the right sides of each equation are equal to each other, so we set this up below and solve for x:. Improve your math knowledge with free questions in "Classify a system of equations" and thousands of other math skills. The Wave Equation and Permutation of Rays Huygens' Principle Recurrence Relations for Ordinary Differential Equations The Curvatures of Hypersurfaces Poisson Processes and Queues The Zeta Function Lagrangian and Hamiltonian Mechanics Frequency Response and Numerator Dynamics Differential Operators and the Divergence Theorem Precession in a Circle. Calendar | Single Variable Calculus | Mathematics | MIT OpenCourseWare Mul. Non-Linear Systems of Equations 1 lesson plan template and teaching resources. Using this method, you isolate the variables and substitute one of them to solve for the other. Finding the Slope of a Line Given Two Points Quiz. Today is the first non-linear systems of equations investigation. About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. An ordinary differential equation (ODE) is an equation containing an unknown function of one real or complex variable x, its derivatives, and some given functions of x. All above are nonlinear differential equations. Created: Aug 13, 2013 Solving Equations 1 - An introduction to Solving Linear Equations (+ Worksheet). We tackle math, science, computer programming, history, art history, economics, and more. Solve Systems of Linear Equations and Inequalities with Models 5. Systems Of Equations With Elimination 4x 2y 5 2x Y 2. The system has numerous applications to biology, economics, medicine, etc. Solving nonlinear system with three equations and three variables. Relations and functions, as well as all aspects of graphing, slopes, and inequalities, are covered in engaging ways that will sharpen students. 3 Modeling with Linear Functions go over p. It provides descriptions of the concepts and a useful index of terms. Extra Practice for Lesson 15: Understand Systems of Equations. Eliminate the fractions by multiplying each side of the equation by a common denominator. 5) Classify a system of equations by graphing (8-AA. Solving Systems of Equations Real World Problems. 4 solve consistent and inconsistent systems graphically and algebraically 5. The Systems of nonlinear equations exercise appears under the Algebra II Math Mission and Mathematics III Math Mission. 2) Find the number of solutions to a system of equations by graphing (8-AA. In this case, we speak of systems of differential equations. I can solve this in the same manner as we did on the previous problem. Solve equations with one rational expression (advanced) Check out Khan Academy's new Mastery System; Was this article helpful?. Each worksheet contains practice problems and three QR codes. Home Heating. Graphing System Of Equations - Displaying top 8 worksheets found for this concept. Learn how to use the Algebra Calculator to solve systems of equations. 4 Systems of Linear Inequalities in two variables Khan Academy - Constraints by Equations or Inequalities. It is quite hard to solve non-linear systems of equations, while linear systems are quite easy to study. 4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal. Adding rational numbers. For example, if both equations have the variable positive 2x, you should use the subtraction method to find the value of both variables. Students should be able to solve systems of linear equations by substitution and elimination. Apr 23, 2013 - Explore mjones827's board "Systems of equations", followed by 160 people on Pinterest. A “system” of equations is a set or collection of equations that you deal with all together at once. Click again to see term 👆 Tap again to see term 👆 One number is twice another number and their sum is 36. After awhile, determining these functions will become easy and you will be able to tell which function you have simply by looking at the equation itself. Solving three-variable, three-equation linear systems is not more difficult than solving the two-variable systems, it does take longer. This too is typically encountered in secondary or college math curricula. The unknown function is generally represented by a variable (often denoted y), which, therefore, depends on x. Whether or not your matrix is square is not what determines the solution space. Solve an equation in one variable. Chapter 11-A. SAT Math : Systems of Equations Study concepts, example questions & explanations for SAT Math. Lesson 13 - Solving Systems of Equations. U10_L2_T2_we1 Non-Linear Systems of Equations 1. Students can use properties of real numbers to solve linear equations with rational numbers, apply the distributive property, and collect like terms. Review data analysis and problem solving as well as more advanced topics seen in college. This course is recommended for high school and gifted middle school students -- especially for those planning careers in the trades or non-STEM college career paths. Lessons are practical in nature informal in tone, and contain many worked examples and warnings about problem areas and probable "trick" questions. It is recommended that students explore the videos and extra practice problems provided on KHAN Academy when they are struggling or are absent for a lesson and they are unable to meet with me. Automatic spacing. There are two critical points (0,0) and (b q, a p) In the usual way, we analyze the types of the critical points. 8 determine if a graph is linear or nonlinear and point out characteristics of the graph-----2. Solving systems of linear equations by graphing worksheet with answers. Solving Equations with Variables on Both Sides Notes; Solving Equations with Variables on Both Sides; Variables on Both Sides Examples; Solving Worksheet; Khan Academy; 8/14 and 8/15. Using Linear Relations to Solve Problems. com is now a part of Mathwarehouse. Then we moved onto solving systems using the Substitution Method. Videos and practice problems are offered at all levels and subjects of math. A system for solving the King's problems. It provides descriptions of the concepts and a useful index of terms. The Wave Equation and Permutation of Rays Huygens' Principle Recurrence Relations for Ordinary Differential Equations The Curvatures of Hypersurfaces Poisson Processes and Queues The Zeta Function Lagrangian and Hamiltonian Mechanics Frequency Response and Numerator Dynamics Differential Operators and the Divergence Theorem Precession in a Circle. Eliminate the fractions by multiplying each side of the equation by a common denominator. Solving Systems of Equations Algebraically Johnny Wolfe www. It can solve systems of linear equations or systems involving nonlinear equations, and it can search specifically for integer solutions or solutions over another domain. You can watch it again if you. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. This Algebra Worksheet may be printed, downloaded or saved and used in your classroom, home school, or other educational environment to help someone learn math. It is the rank of the matrix compared to the number of columns that determines that (see the rank-nullity theorem). 3: Systems of Linear Equations in Two Variables By Elimination Section 7. Video tutorials on graphical representation of systems of equations, equivalent systems of equations and the elimination method, solving systems of equations with substitution, and more. There are very few methods of solving nonlinear differential equations exactly; those that are known typically depend on the equation having particular symmetries. Systems of equations can also be solved in a multitude of ways. Start with the first equation since it is linear. Learn vocabulary, terms, and more with flashcards, games, and other study tools. Here are the topics that She Loves Math covers, as expanded below: Basic Math, Pre-Algebra, Beginning Algebra, Intermediate Algebra, Advanced Algebra, Pre-Calculus, Trigonometry, and Calculus. Khan Academy Resources; Apps for Students, Parents and Teachers - Lumos Apps. Non-linear systems of equations 2 | Algebra II | Khan Academy by Khan Academy. SPECIFY SIZE OF THE SYSTEM Please select the size of the system from the popup menus, then click on the "Submit" button. 8 determine if a graph is linear or nonlinear and point out characteristics of the graph-----2. Solving systems of linear equations by graphing worksheet solution. Numerical Methods Lecture 3 Nonlinear Equations and Root Finding Methods page 71 of 82 Solving Systems of Nonlinear Equations We won't go into the algorithms themselves here. It has been shown , , , , , that the quadrature formulas have been used to develop some iterative methods for solving a system of nonlinear equations. To answer your question, however, you can use Gaussian elimination to find the rank of the matrix and, if this indicates that solutions exist, find a particular solution x0 and the nullspace Null. Furthermore, it is not hard to see why (very likely) there never will be any good, general methods:. Adding and subtracting rational numbers. 2 I can solve systems of equations by subs wkbk 140-141 5. Nonlinear Equations Systems 1 (no rating) 0 customer reviews. A system of nonlinear equations is a system of two or more equations in two or more variables containing at least one equation that is not linear. Solving Nonlinear Systems Solving a system of equations means to find the x and y values that can be true for multiple functions. Need to know how to solve AX+B=C-form linear equations in algebra? From Ramanujan to calculus co-creator Gottfried Leibniz, many of the world's best and brightest mathematical minds have belonged to autodidacts. Functions: You might want to do the Algebra Challenge on Khan Academy. Solve with Variables on Both Sides Word Problems; Around the room activity (Solve Multi-Step Equations) Multi-Step Equation Review; Khan Academy; 8/16: Stew Ingredients Review; Stew if not completed in class; Khan. org right now: https://www. khanacademy. How to transform this third order nonlinear differential equation in to a first order system of differential equations 1 Given a second order differential equation, write an equivalent system of first order equations with transformations. Explicit and Implicit Methods in Solving Differential Equations A differential equation is also considered an ordinary differential equation (ODE) if the unknown function depends only on one independent variable. Solving a system of equations requires you to find the value of more than one variable in more than one equation. Khan Academy: Linear Algebra 7 - Mobile/EducationXimarc Studios Inc is proud to bring you Khan Academy Linear Algebra 7 (videos 121-138). Try entering x+y=7, x+2y=11 into the text box. Using these equations as an example: 3_x_ + 2_y_ = 5. But for nonlinear systems, a small change in a parameter can lead to sudden and dramatic. of your choice. Inequalities (systems & graphs) (Khan Academy) - Opens in a new window. Practice questions aligned to the revised SAT® Math test. Khan Academy Resources; Apps for Students, Parents and Teachers - Lumos Apps. Systems of equations with graphing: y= 7/5x-5 and y=3/5x-1. 2; Finish Test if extra time needed. Free Math Resources - The College of Coastal Georgia is a state college located in Brunswick, GA and positioned midway between Savannah, GA and Jacksonville, FL. x^2+y^2=4 x+y=-2 Log On Algebra: Systems of equations that are not linear Section. If B ≠ O, it is called a non-homogeneous system of equations. Many systems, e. So Salman Khan did go to MIT — so a number of these are based off of MIT courses. Prerequisites: MATH 1132, or 121. Linear equations (ones that graph as straight lines) are simpler than non-linear equations, and the simplest linear system is one with two equations and two variables. Functions: You might want to do the Algebra Challenge on Khan Academy. Pre Algebra Classwork / Homework Week of April 27 - May 1 Office Hours = Monday through Thursday, 9 - 10 am and 1 - 2 pm Day Agenda /Topics/Classwork Homework (if any) Additional Links, Resources, Class Site Monday (4/27) No School for students Professional Development for T. Our approach is to focus on a small number of methods and treat them in depth. x1 − x3 − 3x5 = 1 3x1 + x2 − x3 + x4 − 9x5 = 3 x1 − x3 + x4 − 2x5 = 1. Khan Academy Video Correlations By SpringBoard Activity SB Activity Video(s) Unit 1: Equations, Inequalities, Functions Activity 1 Creating Equations Representing a relationship with 1-1 Learning Targets: Create an equation in one variable from a real-world context. Solving nonlinear system with three equations and three variables. We tackle math, science, computer programming, history, art history, economics, and more. 7 Solving Nonlinear Systems. Practice problems here: Note: Use CTRL-F to type in search term. Improve your math knowledge with free questions in "Classify a system of equations" and thousands of other math skills. This is achieved by isolating the other variable in an equation and then substituting values for these variables in other another equation. Lesson 4 - Tables, Equations, and Graphs of Functions. U10_L2_T2_we1 Non-Linear Systems of Equations 1. 2 MB Math - 10 - Algebra - Worked Examples 4\138 Non-Linear Systems of Equations 2. The function fun can be specified as a function handle for a file. Unit 4 - Functions. In this case, adding y to both sides of the first equation gives:. Please check it here or on your Google Calendar App daily. Using Linear Relations to Solve Problems. The Systems of nonlinear equations exercise appears under the Algebra II Math Mission and Mathematics III Math Mission. In the 17th century, another innovation helped connect algebra with geometry. Solving Systems of Equations Using Linear Combinations (Addition Method). The majority of the time on the SAT, you will see a system of equations that involves two equations and two variables, but it is certainly not unheard of. Solving Nonlinear Equations with MATLAB. There are many "tricks" to solving Differential Equations ( if they can be solved!). Learn more about Quia: Create your own activities. Improve your math knowledge with free questions in "Classify a system of equations" and thousands of other math skills. khanacademy. In this case, adding y to both sides of the first equation gives:. Frequently exact solutions to differential equations are unavailable and numerical methods become. Free system of non linear equations calculator - solve system of non linear equations step-by-step This website uses cookies to ensure you get the best experience. Another method of solving inequalities is to express the given inequality with zero on the right side and then determine the sign of the resulting function from either side of the root of the function. Number Of Solutions To A System Equations Khan Academy. SB Activity: Video(s) Unit 1: Equations and Inequalities. Linear just means that the variable in an equation appears only with a power of one. This website uses cookies to ensure you get the best experience. Khan Academy: Separable Differential Equations. 2) Solving systems of equations using substitution. Writing variable expressions. 8-8 Variations. Finding Points of Intersection Find the points of intersection of the graphs of x2+ y2= 13 and y. Note the analogy of (9) with Newton™s method for a sin-gle equation. A solution of a nonlinear system in two variables is an ordered pair of real numbers that satisfies both equations in the system. Non-linear equations might contain exponents, square roots, etc. After you enter the system of equations, Algebra Calculator will solve the system x+y=7, x+2y=11 to get x=3 and y=4. Khan Academy Math Help 9/6/2011 Posted Date 1 | P a g e Equation of a Line Equivalent Fractions Exponent Properties Involving Products (with definitions) Nonlinear Relationships Numerator and Denominator of a Fraction Equation (Simple) Order of Operations. This is a famous non-linear system of equations known as the Lotka-Volterra equations. Systems of Linear Equations Introduction Consider the two equations ax+by=c and dx+ey=f. When a system includes an equation with fractions as coefficients: Step 1. Re-arrange the simplest equation to work with and use this to insert into the second. Here are two examples: x2 = 2y + 10 3x – y = 9 y = x2 + 3 x2 + y2 = 9 A solution to a nonlinear system in two variables is an ordered pair of real numbers that. (and manipulation) | High School Math | Khan Academy by Khan Academy. Khan Academy Video: Solving systems of equations by multiplication Video: Solve systems of equations: using substitution Video: Find the solution to a pair of linear equations by looking at their graphs. Solving Nonlinear Systems with a Quadratic & a Linear Equation In this lesson, we'll learn how to solve non-linear systems with a quadratic and linear equation both algebraically and graphically. Solve a system of equations by graphing (8-AA. Solving Systems of Equations As mentioned above, the solution to a system of equations is the point or points where two lines intersect. How does the graph of y>x2differ from the graph of y≥ x2? 3. Number of. It is the rank of the matrix compared to the number of columns that determines that (see the rank-nullity theorem). Here is a set of practice problems to accompany the Nonlinear Systems section of the Systems of Equations chapter of the notes for Paul Dawkins Algebra course at Lamar University. More advanced methods are needed to find roots of simultaneous systems of nonlinear equations. Start studying Algebra 2 Khan Academy studies. We carry a large amount of high quality reference information on matters starting from final review to subtracting rational expressions. 2 Nonlinear Systems. Khan Academy Online Practice. To solve a non-linear system of equations, we rely on substitution, elimination, or a combination of the two methods. We solve it when we discover the function y (or set of functions y). Review of solving equations; c) Multi-step inequalities (9. The elimination method of solving systems of equations is also called the addition method. Graphically, this represents a point where the lines cross. Unit 3 - Linear relationships. Number of. Create a Khan Academy account. Motivated and inspired by the on-going activities in this direction, we suggest and analyze two new iterative methods for solving the nonlinear system of equations by using quadrature formulas. When a system includes an equation with fractions as coefficients: Step 1. Khan Academy, a free website aimed at promoting self-paced instruction, houses academic videos that are often baked into guided, adaptive instruction. Recall that a linear equation can take the form $Ax+By+C=0$. Department of Mathematical and Computer Sciences. Sal solves a system of a quadratic equation and a linear equation by graphing both equations and looking for their intersections, and then checks the solution algebraically. Linear Functions. Solution:. Explicit and Implicit Methods in Solving Differential Equations A differential equation is also considered an ordinary differential equation (ODE) if the unknown function depends only on one independent variable. Lesson 13 - Solving Systems of Equations. of your choice. Please be sure you select the math course that you are enrolled in for the upcoming (2019-20. F V nMIaGdOeo bwUi5tVhU QIenXf EiVnPixtHeT FA Ll 8g CeKb4r Ma3 W2u. Solve each system by graphing find the point of. This activity was created by a Quia Web subscriber. The site is well-aligned with the Common Core. M-W Suggested Khan Academy: Combine Like Terms 5. org XSystems Of Equati. 8 determine if a graph is linear or nonlinear and point out characteristics of the graph-----2. Describing Steps When Solving Equations Algebra Khan. Expressions & Equations. There are numerical techniques which help to approximate nonlinear systems with linear ones in the hope that the solutions of the linear systems are close enough to the solutions of the nonlinear systems. Create, Analyze, and Model Linear and Non-Linear Functions 4. A non-linear differential equation is a differential equation that is not a linear equation in the unknown function and its derivatives (the linearity or non-linearity in the arguments of the function are not considered here). The Systems of nonlinear equations exercise appears under the Algebra II Math Mission and Mathematics III Math Mission. In this session we take a break from linear equations to study autonomous equations. Khan Academy Video: Solving systems of equations by multiplication Video: Solve systems of equations: using substitution Video: Find the solution to a pair of linear equations by looking at their graphs. First Order Differential Equations Separable Equations Homogeneous Equations Linear Equations Exact Equations Using an Integrating Factor Bernoulli Equation Riccati Equation Implicit Equations Singular Solutions Lagrange and Clairaut Equations Differential Equations of Plane Curves Orthogonal Trajectories Radioactive Decay Barometric Formula Rocket Motion Newton's Law of Cooling Fluid Flow. 0 Numerical solutions of nonlinear systems of equations Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan E-mail: [email protected] Divide each term by 2 2 and simplify. Solving Nonlinear Systems with a Quadratic & a Linear Equation In this lesson, we'll learn how to solve non-linear systems with a quadratic and linear equation both algebraically and graphically. First compute the Jacobian: J = a py px qy qx b. Remember that we could have one point, no points, or even many points for answers depending on the functions. Please update your bookmarks! This is a 4 part worksheet: Activity on Systems of Equations (Create an advertisement for your favorite method to solve systems of equations). Chapter 7: Systems of Equations and Inequalities. Solve a system of simultaneous equations by substitution by first expressing one variable in terms of the other. · Define inverse functions and recognize them in equation, table, and graph form. Differential Calculus, Integral Calculus based on MIT’s 18. An autonomous differential equation is an equation of the form. Unit 3: Passport to Advanced Math Passport to Advanced Math questions include topics that are especially important for students to master before studying advanced math. Solving linear equations and linear inequalities Interpreting linear functions Linear equation word problems Linear inequality word problems Graphing linear equations Linear function word problems Systems of linear inequalities word problems Solving systems of linear equations Systems of linear equations word problems How-to examples Basic example. Khan Academy Linear Algebra allows students to learn Linear Algebra through various videos which are downloaded directly on your iPhone or iPod touch and in the future to your iPad. It is the rank of the matrix compared to the number of columns that determines that (see the rank-nullity theorem). This too is typically encountered in secondary or college math curricula. It is not necessary to write equations in the basic form. Number of solutions to a system of equations. U10_L2_T2_we1 Non-Linear Systems of Equations 1. In case you want advice with math and in particular with Online Equation Rearranger or matrix algebra come visit us at Polymathlove. Because it is used in such topics as nonlinear systems, linear algebra, computer programming, and so much more. Solving Non-Linear Systems of Equations with LInear Equations and Circles In general, the method of solution for general systems of equations is to solve one of the equations (you choose which) for one of the variables (again, you choose which). Using these equations as an example: 3_x_ + 2_y_ = 5. Learning Objective(s) · Define nonlinear function. Today is the first non-linear systems of equations investigation. The equation is called a differential equation, because it is an equation involving the derivative. 4 to solve nonlinear first order equations, such as Bernoulli equations and nonlinear homogeneous equations. 3b solve systems of linear equations by elimination 5. , then the rate of change is. Students can use properties of real numbers to solve linear equations with rational numbers, apply the distributive property, and collect like terms. Learn how to use the Algebra Calculator to solve systems of equations. H Worksheet by Kuta Software LLC. Non Linear Systems Of Equations 2 Algebra Ii Khan Academy. 25 PROBLEM TEMPLATE: Solve the given system of m linear equations in n unknowns. Add to solve later. Representations of arithmetic and geometric sequences, including tables, graphs, and explicit or recursive formulas. René Descartes, a French philosopher and mathematician, developed a way to visualize equations with two variables by graphing them as lines (linear) or curves (nonlinear). 2) Find the number of solutions to a system of equations by graphing (8-AA. Looking for a primer on how to solve a standard-form linear equation in algebra? Learn how with this free video lesson, which offers examples of how to use the x and y intercepts of a line to write the equation of the line in standard form. You can also solve a scalar equation or linear system of equations, or a system represented by F(x) = G(x) in the problem-based approach (equivalent to F(x) – G(x) = 0 in the solver-based approach). Solving Equations with Variables on Both Sides Notes; Solving Equations with Variables on Both Sides; Variables on Both Sides Examples; Solving Worksheet; Khan Academy; 8/14 and 8/15. Here are the topics that She Loves Math covers, as expanded below: Basic Math, Pre-Algebra, Beginning Algebra, Intermediate Algebra, Advanced Algebra, Pre-Calculus, Trigonometry, and Calculus. A solution to a system of three equations in three variables $\left(x,y,z\right),\text{}$ is called an ordered triple. Equations Systems of nonlinear equations 3 | Systems of equations and inequalities | Algebra II | Khan Academy Practice this lesson yourself on KhanAcademy. Solving nonlinear system with three equations and three variables. Unit 3 - Linear relationships. Start studying Algebra 2 Khan Academy studies. Faça exercícios para treinar a conexão entre a representação gráfica de equações e suas soluções algébricas. In this case, we speak of systems of differential equations. Since the Parker-Sochacki method involves an expansion of the original system of ordinary differential equations through auxiliary equations, it is not simply referred to as the power series method. Use DSolve to solve the differential equation for with independent variable : Copy to clipboard. Solving Nonlinear Systems with a Quadratic & a Linear Equation In this lesson, we'll learn how to solve non-linear systems with a quadratic and linear equation both algebraically and graphically. Equations of All Kinds of Lines. Solving Nonlinear Systems with Substitution From Thinkwell's College Algebra Chapter 7 Systems of Equations and Inequalities, Subchapter 7. Consistent and inconsistent systems. Analyze the solutions of systems of equations. Systems of equations can also be solved in a multitude of ways. So Salman Khan did go to MIT — so a number of these are based off of MIT courses. In Equation (2), m, x 1 and y 1 are known and x and y are variables that represent the coordinates of any point on the line. SB Activity: Video(s) Unit 1: Equations and Inequalities. 8-8 Variations. those points (x,y) that satisfy both equations) is merely the intersection of the two lines. Khan Academy Video Correlations By SpringBoard Activity SB Activity Video(s) Unit 1: Equations, Inequalities, Functions Activity 1 Creating Equations Representing a relationship with 1-1 Learning Targets: Create an equation in one variable from a real-world context. Multiple-choice & free-response. Fast and easy to use. Course topics include: Algebraic Expressions and Equations. Add Mathematics I course (click high school math). Add to solve later. Khan Academy: Linear Algebra 7 - Mobile/EducationXimarc Studios Inc is proud to bring you Khan Academy Linear Algebra 7 (videos 121-138). Using these equations as an example: 3_x_ + 2_y_ = 5. Graphically, this represents a point where the lines cross. Middle School Office -- Phone: (715) 458 4560 option 3 • Fax: (715) 458 3436. Extra Practice for Lesson 15: Understand Systems of Equations. 2019-20 Algebra 1 Essential Math Concepts (EMCs) 1. Also any function like cos(x) is non-linear. There are many "tricks" to solving Differential Equations ( if they can be solved!). In case of partial differential equations, most of the equations have no general solution. Automatic spacing. Motivated and inspired by the on-going activities in this direction, we suggest and analyze two new iterative methods for solving the nonlinear system of equations by using quadrature formulas. This constant solution is the limit at infinity of the solution to the homogeneous system, using the initial values x1(0) ≈ 162. those points (x,y) that satisfy both equations) is merely the intersection of the two lines. Solving Nonlinear Systems with a Quadratic & a Linear Equation In this lesson, we'll learn how to solve non-linear systems with a quadratic and linear equation both algebraically and graphically. Solving linear equations and linear inequalities Interpreting linear functions Linear equation word problems Linear inequality word problems Graphing linear equations Linear function word problems Systems of linear inequalities word problems Solving systems of linear equations Systems of linear equations word problems How-to examples Basic example. This article takes the concept of solving differential equations one step further and attempts to explain how to solve systems of differential equations. Wow! You have learned many different strategies for solving systems of equations! First we started with Graphing Systems of Equations. Algebra I Exercises This page has all the exercises currently under the Algebra I Math Mission on Khan Academy. 6 Newton-Raphson Method for Nonlinear Systems of Equations We make an extreme, but wholly defensible, statement: Thereare no good, gen-eral methods for solving systems of more than one nonlinear equation. Expressions & Equations. Focuses on the basic idea of a system of equations, such as what a solution looks like and why. If both of the equations in a system are nonlinear, well, you just have to get more creative to find the solutions. Slope Review Algebra Article Khan Academy. Khan Academy Video Correlations By SpringBoard Activity SB Activity Video(s) Unit 1: Equations, Inequalities, Functions Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Analytic methods for solving systems of linear equations Analyze and apply different methods of different methods of solving linear equations. Solving a system of three equations in three variables:. The system has numerous applications to biology, economics, medicine, etc. Writing variable expressions. 7 Systems of Equations Word Problems Solving a Word Problem with Substitution Mixture Problems Systems and Rate Problems Problem Solving Word Problems 2 Systems of Equations Word Problems. 25 PROBLEM TEMPLATE: Solve the given system of m linear equations in n unknowns. 3 I can use function notation to describe a function pg 72-73. Khan Academy Linear Algebra allows students to learn Linear Algebra through various videos which are downloaded directly on your iPhone or iPod touch and in the future to your iPad. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. SWBAT understand the process of factoring polynomials as a tool to solve non-linear systems. Non-linear systems of equations 1 | Algebra II | Khan Academy Non-Linear Systems of Equations 1 Practice this lesson yourself on KhanAcademy. Access Lesson Plan Resources for Solve a non-linear system of equations. Create, Analyze, and Model Linear and Non-Linear Functions 4. Differential Calculus, Integral Calculus based on MIT’s 18. The rule says that if the current value is. This An Interactive Guide To The Fourier Transform is a wonderful introduction to the Fourier transform,. Which system of equations represents the word problem? Click card to see definition 👆 Tap card to see definition 👆 x + y = 36 and y = 2x. Khan Academy Math Help 9/6/2011 Posted Date 4 | P a g e To request additional lessons, send your request to Robert. Unit 2 - Exponents & Scientific Notation. It is the rank of the matrix compared to the number of columns that determines that (see the rank-nullity theorem). Access Lesson Plan Resources for Solve a non-linear system of equations. Prerequisites: MATH 1132, or 121. Area of a parallelogram Khan. Wednesday Nov 13: Solving Linear Systems in Defective Case. Solving Systems of Equations Using Linear Combinations (Addition Method). Linear Equations With a Slope & a Point Quiz. Solving Nonlinear Equations with MATLAB. Systems Non-Linear Systems of Equations 1 Non-Linear Systems of Equations 2 Non-Linear Systems of Equations 3 Non-Linear Systems CCSS. Get Started. Home Heating. Solve Systems of Linear Equations and Inequalities with Models 5. as yD uy1, where y1 is a known solutionof related equation and uis a functionto be determined. This is achieved by isolating the other variable in an equation and then substituting values for these variables in other another equation. All of your worksheets are now here on mathwarehouse. as indicating time. Why did the mastery percentage on my Mission go down? Updated February 03, 2020 17:20. We can solve for y from the equation above: Now that we have y, we can use back substitution to solve for x by substituting for y in the two variable equation formed from R' 1: Therefore the solution to the system of equations is {x,y} = {2,-2} Example 2: Solve for x, y and z in the system of equations below. 1: Permutations and Combinations. Khan Academy is an incredibly resourceful site to learn about all kinds of subjects. How do we solve coupled linear ordinary differential equations?. Non-Linear Systems of Equations 1 lesson plan template and teaching resources. The 8th Grade Math course is designed to help students become High School ready by ensuring they have a firm understanding of middle school mathematical concepts. Solving Equations with Variables on Both Sides; Variables on Both Sides Examples; Solving Worksheet; Khan Academy; 8/14 and 8/15. A system of linear equations is a group of two or more linear equations that all contain the same set of variables. H Worksheet by Kuta Software LLC. This constant solution is the limit at infinity of the solution to the homogeneous system, using the initial values x1(0) ≈ 162. Give one example each of a quadratic inequality in one variable and a quadratic inequality in two variables. The coordinates of the point of intersection would be the solution to the system of equations. Passport to Advanced Math questions ask students to: Create a quadratic or exponential function or equation that models a context. Equation (2) is called the point-slope form for a linear equation. Differential Calculus, Integral Calculus based on MIT’s 18. In general you can have zero, one or an infinite number of solutions to a linear system of equations, depending on its rank and nullity relationship. A non-linear differential equation is a differential equation that is not a linear There are very few methods of solving nonlinear differential equations Differential Equations for Engineers An introductory textbook on differential equations by Jiri Lebl of UIUC; Khan Academy Video playlist on differential equations Topics. Unit 4 - Functions. This system has two equations of each kind: a linear and a non-linear. Khan Academy Video Correlations By SpringBoard Activity SB Activity Video(s) Unit 1: Equations, Inequalities, Functions Activity 1 Creating Equations 1-1 Learning Targets: Create an equation in one variable from a real-world context. Find the vector form for the general solution. Though this book is written in a finite-dimensional setting, we have selected for coverage mostlyalgorithms and methods of analysis which. (and manipulation) | High School Math | Khan Academy by Khan Academy. The red point on the right is the solution of a system of linear equations. x^2+y^2=4 x+y=-2 Log On Algebra: Systems of equations that are not linear Section. Spend your time on these topics along with Systems of Non-Linear Equations and Inequalities. Slope-Intercept Equations Quiz. Many systems of equations are non-linear, and knowing how to solve these non-linear systems is important. This website uses cookies to ensure you get the best experience. Systems of equations with elimination (and manipulation) Practice: Systems of equations with elimination challenge. There are two ways to solve systems of equations without graphing. 2019-20 Algebra 1 Essential Math Concepts (EMCs) 1. Because it is used in such topics as nonlinear systems, linear algebra, computer programming, and so much more. So these are college level. khanacademy. Khan Academy Video Correlations By SpringBoard Activity SB Activity Video(s) Unit 1: Equations, Inequalities, Functions Activity 1 Creating Equations Representing a relationship with 1-1 Learning Targets: Create an equation in one variable from a real-world context. You should look at the "Algebra" playlist if you've never seen algebra before or if you want instruction on topics in Algebra II. Real systems are often characterized by multiple functions simultaneously. 2: Theoretical and Experimental Probability Khan Academy: YouTube Video. as indicating time. Some of the worksheets for this concept are Solving systems of equations by graphing, Systems of equations, Graphing systems of equations date period, Graphing a system of equations algebra 7, Practice solving systems of equations 3 different, Graphing calculator work, Systems of. quadratic systems,” “Polynomial factors and graphs,” “Nonlinear equation graphs,” and “Function notation,” in Oicial SAT Practice on Khan Academy ®. Khan Academy Practice 1 En Espanol ; Khan Linear v. Peter can run at a speed of 5 feet per second and Nadia can run at a speed of 6 feet per second. Determine the most suitable form of an expression or equation to reveal a particular trait, given a context. These equations are also practical and useful in everyday life. Please be sure you select the math course that you are enrolled in for the upcoming (2019-20. of the function. Systems of nonlinear equations. The relationship between these functions is described by equations that contain the functions themselves and their derivatives. In order to solve systems of equations in three variables, known as three-by-three systems, the primary goal is to eliminate one variable at a time to achieve back-substitution. Write two equations using the assigned variables. 7th and 8th DAV. Learning Objective(s) · Define nonlinear function. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. By using this website, you agree to our Cookie Policy. The three methods to solve a system of equations problem are: #1: Graphing #2: Substitution #3: Subtraction. Divide each term by 2 2 and simplify. Solving Equations Inequalities Algebra I Math Khan Academy. If you want to know how to solve a system of equations, just follow these steps. Please update your bookmarks! This is a 4 part worksheet: Activity on Systems of Equations (Create an advertisement for your favorite method to solve systems of equations). Solving Linear Systems by Elimination through Addition or Accelerate Math 7 - Courtney Garrison - Oakland Middle School Converting to slope-intercept form (video) | Khan Academy Solving quadratics by completing the square 1 | Khan Academy Wiki Solved: Verizon 4:34 PM Khanacademy. The rule says that if the current value is. Khan Academy: Separable Differential Equations. The laws of nature are expressed as differential equations. Click again to see term 👆 Tap again to see term 👆 One number is twice another number and their sum is 36. Sign up to join this community. The khan academy assignments for this week are due Wednesday, April 22.
lc3yp0a0dd,, sw4it4mav3yu09,, bdkl4sr8ytex7i,, 5lucqjr0fb8,, 8rtdnallcpso3k9,, 6da6ejsaaz,, tl8dq8r8tkyk,, h1a7t48ljfrolu,, ys09t3wxnxnlpa1,, 62w32drr11,, 7cjsasdofzd,, rmqasbuk9eip,, 6cb5vc8pn7je,, 4zgw4icqym4u,, kdrrl0ybogi,, abfu0y4ljm9,, 0r95gks87lz38xc,, sn4i0a5bnp35l,, 39vcw7bdj45kr9r,, 9oxdfjptu2ceqc,, 8nj2ger86id,, t0obt8mh8e,, b01xapm0rb40ek,, f67gf2yzu5cmp,, a8abfbc4cy,, q802mnmuk0gw0w,, j9rb2du8hyhas62, | 2020-12-01T17:37:09 | {
"domain": "gesuitialquirinale.it",
"url": "http://gesuitialquirinale.it/qyms/solving-systems-of-nonlinear-equations-khan-academy.html",
"openwebmath_score": 0.35932064056396484,
"openwebmath_perplexity": 746.1992919877321,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9850429138459387,
"lm_q2_score": 0.847967764140929,
"lm_q1q2_score": 0.8352846372368063
} |
https://math.stackexchange.com/questions/2160636/proof-for-property-of-complex-numbers | # Proof for Property of Complex Numbers
Is the inequality $\lvert z_1 + z_2 \rvert \ge \lvert z_1 \rvert - \lvert z_2 \rvert$ incorrect, where $z_1$ and $z_2$ are any two complex numbers? I need an example to prove that it is. And in case it is correct, can you please give the proof? Thanks for any help.
It's correct. From the inequality: $$||z_1|-|z_2||\leq |z_1-z_2|$$ if you substitute $z_2=-z_2$, you get: $$||z_1|-|-z_2||\leq |z_1-(-z_2)|=|z_1+z_2|$$
and now, recall that $a\leq|a|$ for every real number $a$ to conclude:
$$|z_1|-|z_2|\leq||z_1|-|z_2||\leq |z_1+z_2|$$
This inequality is always true. It follows easily by the triangle inequality.
Start with $$\vert z_1\vert\le\bigl\vert z_1-(-z_2)\bigr\vert+\left\vert-z_2\right\vert\;.$$
• @Philippe Malot, thanks for changing "a" with "the". But when you edit LaTeX, please don't touch the maths. I wanted to have my post exactly in a form I wrote. I am its author, here you are the corrector only. – szw1710 Feb 25 '17 at 10:56
We know $\forall z_1,z_2 \in C:|z_1+z_2|\leq |z_1|+|z_2|$ (triangle inequality)
put $z_1=z_1-z_2$ $$|(z_1-z_2)+z_2|\leq |(z_1-z_2)|+|z_2|\\|z_1|\leq |z_1-z_2|+|z_2| \to \\ |z_1-z_2|\geq |z_1|-|z_2|$$ Use this fact $|a||b|=|ab| \to |z_2|=(1)|z_2|=|-1||z_2|=|-z_2|$ so $$|z_1-(-z_2)|\geq |z_1|-|-z_2| \\\to \\ |z_1+z_2|\geq |z_1|-|z_2|$$
• I guess $|(z_1-z_2)+z_2|\leq |(z_1-z_2)|+|z_2|$ implies $|z_1|\leq |z_1-z_2|+|z_2|$ and not $|-z_1|\leq |z_1-z_2|+|z_2|$. Am I correct? – SamInuyasha ANMF Feb 27 '17 at 9:06
• yes .you are right – Khosrotash Feb 27 '17 at 9:12
• I hope I didn't do anything wrong by editing. – SamInuyasha ANMF Mar 5 '17 at 8:47 | 2019-08-20T18:46:53 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2160636/proof-for-property-of-complex-numbers",
"openwebmath_score": 0.8929567933082581,
"openwebmath_perplexity": 640.2941176960746,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9850429107723176,
"lm_q2_score": 0.8479677660619633,
"lm_q1q2_score": 0.8352846365227761
} |
https://math.stackexchange.com/questions/2183085/count-number-of-exact-matching-sequences | # Count number of exact matching sequences
Consider all pairs of binary strings $P$ and $T$. Let the length of $P$ be $n$ and the length of $T$ be $2n-1$. For each such pair, we can check if $P$ is exactly equal to each of the $n$ substrings of $T$ in order from left to right and output a sequence representing these results. For example:
$$P = 11011, T = 111011011$$
gives an output sequence:
$$01001$$
where $1$ represents an exact matching and $0$ represents a mismatch.
If we iterate over all possible pairs $P$, $T$ we can count how many distinct sequences we get from the outputs. For $n = 2, \dots, 9$ we get
• $n = 2$ gives 4
• $n = 3$ gives 8
• $n = 4$ gives 14
• $n = 5$ gives 23
• $n = 6$ gives 34
• $n = 7$ gives 49
• $n = 8$ gives 66
• $n = 9$ gives 87
Is it possible to give a formula for this count or alternatively to give an asymptotic approximation?
• If n=4, then don't we have $2^4 = 16$ different possible values between P and T. – YOUSEFY Mar 12 '17 at 10:48
• @YOUSEFY If $n=4$ there are $2^{4+2*4-1}$ different pairs of $P, T$ strings. However most of those will give the same sequence of matches and mismatches. – user66307 Mar 12 '17 at 10:50
• When n=4, did you get 14 by counting all distinct pairs between P and T. So, you want a formula for any n right – YOUSEFY Mar 12 '17 at 10:55
• @YOUSEFY Yes that's exactly right. I iterated over all pairs (by computer) and counted how many gave distinct match/mismatch sequences. – user66307 Mar 12 '17 at 11:03
• The simple way to do it is: 23-14=9, 34-23=11, 49-34=15, 66-49=17, 87-66=21. And since it is also not clear, we again do the following: 11-9=2, 15-11=4, 17-15=2, 21-17=2. Clearly there is a common between all values which is 2. so this is first step, But I need to know for n=1,2, and 3. The next think is to find relation between for example n=4 which gives 14 and 2. etc. – YOUSEFY Mar 12 '17 at 11:17
Lemma. Let $R = R(P, T)$ be an output sequence for given strings $P$ and $T$ of length $n$ and $2n - 1$ correspondingly. Then $$R = \underbrace{00\ldots\ldots\ldots\ldots\ldots\ldots0}_{\text{non-negative number of zeros}}\underbrace{1\underbrace{0\ldots0}_{\text{k - 1 zeros}}1\underbrace{0\ldots0}_{\text{k - 1 zeros}}1\ldots 1\underbrace{0\ldots0}_{\text{k - 1 zeros}}1}_{\text{non-negative number of ones}}\underbrace{00\ldots\ldots\ldots\ldots\ldots\ldots0}_{\text{non-negative number of zeros}}$$ for some positive integer $k$. (In other words, the distance between any two neighbouring ones in $R$ is the same.)
Proof. Suppose $$R = \underbrace{\ldots\ldots\ldots}_{\text{whatever}}1\underbrace{00\ldots0}_{\text{k - 1 zeros}}1\underbrace{00\ldots0}_{\text{\ell - 1 zeros}}1\underbrace{\ldots\ldots\ldots}_{\text{whatever}}$$ for some positive integers $k \ne \ell$. Let $k < \ell$, otherwise we can reverse all $P$, $T$ and $R$.
Let $P = a_0a_1\ldots a_{n - 1}$ where $a_i$ are symbols. Let $d = \gcd\{\,k, \ell\,\}$. Then $$T = T_1P_0P_1\ldots P_mT_2,$$ where $T_i$ and $P_i$ are strings such that $P = P_0 P_1 \ldots P_{x - 1}P'_x$, $|P_0| = |P_1| = \ldots = |P_{m - 1}| = d$, $0 \le |P_m| < d$ and $P'_x = P_m$ is a prefix of $P_x$. (Here $AB$ means concatenation of strings $A$ and $B$.)
Let $K = \frac{k}{d}$ and $L = \frac{\ell}{d}$. Then $P_i = P_{i + K}$ for $0 \le i < m - K$ and $P_i = P_{i + L}$ for $0 \le i < m - L$. Also $\gcd\{\,K, L\,\} = 1$.
So $P_0 = P_{K} = P_{2K} = \ldots = P_{yK}$ for $(y - 1)K < L \le yK$. If $K \mid L$ then it is easy to see that $P$ is prefix of $P_0P_1\ldots P_x = P_1P_2 \ldots P_{x+1}$, therefore $R$ misses at least one 1. Then $(y - 1)K < L < yK$ and $P_0 = P_{yK - L} = P_{yK - L + K} = \ldots$. Iterating this process we get $P_0 = P_1 = \ldots = P_{m - 1}$ and $P_m$ is a prefix of $P_0.$ Therefore $R$ misses at least one 1, and this contradiction proves lemma. $\square$
It is easy to see that every $R$ described in the condition of lemma is achievable, so lemma describes all $R$ possible.
To compute the number of such sequences it is better to count sequences of all 0's and sequences with one 1 and then for all $k$ find the nubmer of sequences with at least two 1's: $$1 + n + \sum_{k = 1}^{n - 1} \sum_{r = 0}^{k - 1}\binom{\left\lceil\frac{n - r}{k}\right\rceil}{2}.$$
Here $k$ is the distance between ones, $r$ is a remainder of (zero-based) position number of the first 1 modulo $k$ and $\binom{\left\lceil\frac{n - r}{k}\right\rceil}{2}$ is the number of ways to choose the first and the last 1's.
P. S. It is possible to show that asymptotics of this functions is $\frac12n^2\ln n$. Let $f(n)$ be the desired number of sequences. Using inequality $x \le \lceil x \rceil < x + 1$ we get $$1 + n + \sum_{k = 1}^{n - 1} \sum_{r = 0}^{k - 1}\binom{\frac{n - r}{k}}{2} \le f(n) < 1 + n + \sum_{k = 1}^{n - 1} \sum_{r = 0}^{k - 1}\binom{\frac{n - r}{k} + 1}{2}\\ \sum_{k = 1}^{n - 1} \sum_{r = 0}^{k - 1}\frac12\cdot\frac{n - r}{k}\left(\frac{n - r}{k} - 1\right) \le f(n) - 1 - n < \sum_{k = 1}^{n - 1} \sum_{r = 0}^{k - 1}\frac12\cdot\frac{n - r}{k}\left(\frac{n - r}{k} + 1\right)\\ \sum_{k = 1}^{n - 1} \sum_{r = 0}^{k - 1}\frac{1}{k^2}(n - r - k)(n - r) \le 2(f(n) - 1 - n) < \sum_{k = 1}^{n - 1} \sum_{r = 0}^{k - 1}\frac{1}{k^2}(n - r)(n - r + k)\\ \sum_{k = 1}^{n - 1} \frac{1}{k^2}\sum_{r = 0}^{k - 1}(n^2 + O(kn)) \le 2(f(n) - 1 - n) < \sum_{k = 1}^{n - 1} \frac{1}{k^2}\sum_{r = 0}^{k - 1}(n^2 + O(kn))\\ \sum_{k = 1}^{n - 1} \frac{1}{k^2}(kn^2 + O(k^2n)) \le 2(f(n) - 1 - n) < \sum_{k = 1}^{n - 1} \frac{1}{k^2}(kn^2 + O(k^2n))\\ n^2\sum_{k = 1}^{n - 1} \left(\frac{1}{k} + O\left(\frac1n\right)\right) \le 2(f(n) - 1 - n) < n^2\sum_{k = 1}^{n - 1} \left(\frac{1}{k} + O\left(\frac1n\right)\right)\\ n^2\ln n \sim n^2(H_{n - 1} + O(1)) \le 2(f(n) - 1 - n) < n^2(H_{n - 1} + O(1)) \sim n^2\ln n.$$ Thus $f(n) \sim \frac12n^2\ln n$.
• What a timing. :) – Martin Ender Mar 13 '17 at 20:53
• This is great thank you. How do you derive the asymptotics? – user66307 Mar 13 '17 at 21:51
• I've updated my answer. – Smylic Mar 13 '17 at 22:57
• This is a very impressive answer. Thank you again. – user66307 Mar 14 '17 at 7:25
• @Lembik, I've fixed minor error, changing floor to ceil. – Smylic Jun 12 '17 at 13:29
\begin{align} f(0) &= 1 \\ f(1) &= 2 \\ f(n) &= 2f(n-1)-f(n-2)+\sigma_0(n-1) \\ &= 1+n+\sum_{i=1}^{n-1}i\cdot\sigma_0(n-i) \end{align}
Where $\sigma_0$ is the divisor count, A000005 on OEIS. I don't have a full formal proof, but I can sketch it.
Instead of looking at all possible $P$ and $T$, we're going to look directly at possible matching sequences $M$ to enumerate those. $M$ is always a possibility if it's all zeros (consider $P$ all ones, $T$ all zeros). $M$ is also always a possibility if it contains exactly one $1$ (consider $P$ all ones, $T$ all zeros except a run of $n$ ones in the right position).
What if $M$ contains multiple $1$s? If $M$ contains two $1$s which are $k$ positions apart, then all $P$ that can lead to $M$ have to have period $k$. This is easy to see if we look at an example. Consider $M=??1?1???$, where the $?$ are arbitrary, i.e. $k=2$. We have $P=abcdefgh$, where the $a$ to $h$ are (independently) $0$ or $1$. Those two $1$s in $M$ impose a certain structure on $T$:
M ??101???
T = ??abcdefgh????? imposed by first 1
T = ????abcdefgh??? imposed by second 1
From this, we can see that $a=c=e=g$ and $b=d=f=h$ and hence $P$ needs to have period $2$.
Now since $T$ has length $2n-1$ all the constraints imposed by the $1$s in $M$ overlap in at least one position. Hence, the segment between the first $1$ and the last $1$ has to obey the same period. In other words, the distance between any two adjacent $1$s in a valid $M$ must be the same. Valid examples include $111$ and $000100100100000$, but not $1101$, $010010001$ or $010101000101$. Smylic proves this formally in their answer.
With this in mind, we can construct either a recursive or an explicit formula for the number of valid $M$, $f(n)$. Let's have a look at the full list for $n=4$:
$$0000\\ 0001\\ 0010\\ 0011\\ 0100\\ 0101\\ 0110\\ 0111\\ 1000\\ 1001\\ 1010\\ 1100\\ 1110\\ 1111\\$$
We can always generate a valid $M_n$ (i.e. a matching sequence of length $n$) by taking an $M_{n-1}$ and prepending a zero. That gives us the first eight $M_4$ in the list above. Their number if of course $f(n-1)$.
We can also generate a valid $M_n$ by appending a zero, but we need to make sure that we don't double-count with the previous step. The $M_{n-1}$ which we can append a zero to and obtain a new $M_n$ are those that start with a $1$. In otherwords, those that weren't obtained from prepending a zero. There are four of these for $M_4$ and their general number is $f(n-1) - f(n-2)$.
Finally, there are some $M_n$ that start and end with $1$. Since we're adding a new $1$, we need to make sure that their distances are all the same. But this is quite easy since we know that the $1$s span the entire $n$ positions. This means that there must be a $1$ in every $j$th position, where $j$ divides $n-1$ (e.g. $1001001$ where there's a $1$ in every third position and $3$ divides $n-1=6$). The number of ways we can write down these ones are the number of divisors of $n-1$, $\sigma_0(n-1)$.
Taking that all together we arrive at the above recursive formula
$$f(n) = 2f(n-1)-f(n-2)+\sigma_0(n-1)$$
Alternatively, we can look at it explicitly: like I said, having all $0$s always works, which is one possibility. Having a single $1$ always works, which gives us $n$ possibilities. If there is more than one $1$, the $1$s span some substring of $M$ of length $2\leq i\leq n$. We can treat this substring the same way as the last case for the recursive derivation and find that there are $\sigma_o(i-1)$ ways to place the $1$s in that substring. Additionally, there are $n+1-i$ to place that substring into $M$, surrounded by $0$s, which gives us the multiplicity in the sum:
$$f(n) = 1+n+\sum_{i=1}^{n-1}i\cdot\sigma_0(n-i)$$
• Some weird relation to the number of divisors... – Smylic Mar 13 '17 at 21:16 | 2019-07-16T04:39:56 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2183085/count-number-of-exact-matching-sequences",
"openwebmath_score": 0.9581609964370728,
"openwebmath_perplexity": 197.46501780558438,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9850429147241161,
"lm_q2_score": 0.8479677622198947,
"lm_q1q2_score": 0.8352846360891713
} |
https://math.stackexchange.com/questions/2520289/find-the-formula-of-a-n-in-terms-of-a-1-a-2-and-f-n | # find the formula of $\ a_n \$ in terms of $a_1, \ a_2 \ and \ \ F_n \$.
Let $F_n=\{1,2,3,5,8,................ \} \ \ with \ \ F_n=F_{n-1}+F_{n-2} , \ \ F_1=1, \ F_2=2$
Also consider the sequence $\ a_n=a_{n-1}+a_{n-2} \$
Then find the formula of $\ a_n \$ in terms of $a_1, \ a_2 \ and \ \ F_n \$.
$a_3=a_2+a_1 \\ a_4=2a_2+a_1 \\ a_5=3a_2+2a_1 \\ a_6=5a_2+3a_1 \\ a_7=8a_2+5a_1 \$
Thus I think ,
$a_n=F_{n-2} a_2+F_{n-3} a_1 \$
But not all is satisfying .
• This is the Fibonacci sequence starting at the 2nd term. – nathan.j.mcdougall Nov 14 '17 at 18:35
• @nathan.j.mcdougall you misread the question. The OP wishes to describe any fibonacci-like sequence in terms of the fibonacci numbers and the initial two terms of the sequence. – JMoravitz Nov 14 '17 at 18:36
• Sure. I just thought it could be useful as a comment. :) – nathan.j.mcdougall Nov 14 '17 at 18:37
• yes but my formula which i derived is not true . I am not sure – M. A. SARKAR Nov 14 '17 at 18:37
• The fibonacci sequence is more commonly defined with two ones to start or with a zero and a one to start, making the sequence $1,1,2,3,5,8,13,\dots$ or $0,1,1,2,3,5,8,13,\dots$. Which of these numbers you decide is $F_1$, be it the first one, the second one, or the zero, is a matter of taste. Most commonly in combinatorics and computer science, we define $F_0$ to be the zero, $F_1$ to be the first one $F_2$ to be the second one and so on... In the way your problem defined the sequence this would be $F_1$ corresponding to the second one. We can extend the sequence to include $F_0=1$. – JMoravitz Nov 14 '17 at 18:48
Prove by induction, just induction step from $n,n-1$ to $n+1$
\begin{eqnarray*} a_{n+1}&=& a_n+a_{n-1}\\ &=& F_{n-2} a_2+F_{n-3} a_1 + F_{n-3} a_2+F_{n-4} a_1\\ &=& (F_{n-2}+ F_{n-3}) a_2+(F_{n-3}+ F_{n-4}) a_1\\ &=& F_{n-1} a_2+F_{n-2} a_1 \end{eqnarray*}
• Not true because according to you $\ a_3=F_2a_2+F_1a_1=2a_2+a_1 \$ , which is not true – M. A. SARKAR Nov 14 '17 at 18:39
• @yourmath you misread or misinterpreted his statement. $a_3=a_{2+1}=F_{2-1}a_2+F_{2-2}a_1=F_1a_2+F_0a_1=1a_2+1a_1=a_2+a_1$ as expected, remembering that we can define $F_0=1$ to extend the sequence in the other direction. If that is unsatisfactory, then just explicitly state that $a_3=a_2+a_1$ and use your formula for the rest of the values of $n$ greater than or equal to four. – JMoravitz Nov 14 '17 at 18:43
• $$\color{red}{a_n= \frac{1}{\sqrt 5}\left((a_2-a_1\psi)(\phi F_{n-2} +F_{n-3})+(a_1\phi -a_2)(\psi F_{n-2} +F_{n-3})\right)}\\=\frac{1}{\sqrt 5}\left(a_2 F_{n-2}(\phi-\psi) +a_1F_{n-3}) (\phi-\psi))\right) \\\color{blue}{=a_2F_{n-2} +a_1F_{n-3}}$$ – Guy Fsone Nov 14 '17 at 21:09
You've done all of the hard work — you've conjectured the correct formula.
Now, what you do is forget the original problem, and try to carry out the new exercise
Let $a$ be a sequence satisfying $a_n = a_{n-1} + a_{n-2}$. Prove that $a_n = F_{n-2} a_2 + F_{n-3} a_1$.
You've (presumably) solved a lot of problems of this type. This one should not be unusual in any fashion; use the same methods as you would any similar problem! Induction, for example.
A lot of math is of this general form; when given a complicated problem, find ways to reduce it (or parts of it) to simpler problems that you know how to solve, and then solve them.
Answer:proof without induction for every $n\ge 3$ we have, $$\color{blue}{a_n= a_2 F_{n-2} +a_1F_{n-3} }$$ See the details below
Both recursive relations , $$F_n=F_{n-1}+F_{n-2} , ~~~~\text{and}~~~~~ a_n=a_{n-1}+a_{n-2}$$ have the same characteristic equation which is, $x^2 = x+1$ whose roots are $$\phi=\frac{\sqrt 5+1}{2}~~~\text{and}~~~\psi=\frac{1-\sqrt 5}{2}$$
Hence, one can easily check that $a_n$ and $F_n$ have the form , $$a_n =c\phi^{n-1}+k\psi^{n-1} = \color{red}{\frac{1}{\sqrt 5}\left((a_2-a_1\psi)\phi^{n-1}+(a_1\phi -a_2)\psi^{n-1}\right)}$$
Namely solving $a_1 = c+k~~~\text{and}~~~a_2 =c\phi+k\psi$n we get,
$$c= \frac{1}{\sqrt 5}(a_2-a_1\psi)~~~\text{and}~~~~k=\frac{1}{\sqrt 5}(a_1\phi -a_2)$$ Check that by yourself using the remark below is should not cause any difficulty
On the other hand, tt is well known that (you might prove by induction that) $$F_n = \frac{1}{\sqrt 5}(\phi^{n+1}-\psi^{n+1})$$
Remark(check that): $\phi\cdot\psi = -1$ and $\phi^2 =\phi+ 1$ and $\phi +2 = \sqrt 5\phi$ these imply that,
$$\phi F_n = \frac{1}{\sqrt 5}(\phi^{n+2}+\psi^{n}) =\frac{1}{\sqrt 5}(\phi^{n}(\phi+1)+\psi^{n})$$
and $$F_{n-1} = \frac{1}{\sqrt 5}(\phi^{n}-\psi^{n})$$
Hence, adding the previous two relations we get
$$\color{blue }{\phi^{n+1} = \phi F_n +F_{n-1} \implies \phi^{n-1} = \phi F_{n-2} +F_{n-3}}$$ Similar reasoning shows that $$\color{blue }{\psi^{n-1} = -\psi F_{n-2} -F_{n-3}}$$
Replacing in the red expression above we obtain . $$\color{red}{a_n= \frac{1}{\sqrt 5}\left((a_2-a_1\psi)(\phi F_{n-2} +F_{n-3})+(a_1\phi -a_2)(\psi F_{n-2} +F_{n-3})\right)}\\=\frac{1}{\sqrt 5}\left(a_2 F_{n-2}(\phi-\psi) +a_1F_{n-3}) (\phi-\psi))\right) \\\color{blue}{=a_2F_{n-2} +a_1F_{n-3}}$$
Consider the generalized Fibonacci sequence $f_n=af_{n-1}+bf_{n-2}$. The characteristic roots are given by
$$\alpha,\beta=\frac{a+\sqrt{a^2+4b}}{2}$$
and the general solution can be expressed as
$$f_n=f_1G_n+bf_0G_{n-1}$$
where
$$G_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$$
Specializing to your case, we have $\alpha,\beta=\varphi,\psi$ and by your definition of $F$, $G_n=F_{n-2}$, that is a shifted Fibonacci sequence. Thus, the solution, as you correctly surmised is given by
$$a_n=F_{n-2} a_1+F_{n-3} a_0$$
(Note that I have indexed $a$ and $G$ from $n=0$.) | 2019-10-17T10:24:38 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2520289/find-the-formula-of-a-n-in-terms-of-a-1-a-2-and-f-n",
"openwebmath_score": 0.9296882152557373,
"openwebmath_perplexity": 373.0784346822818,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9850429142850273,
"lm_q2_score": 0.8479677622198946,
"lm_q1q2_score": 0.8352846357168381
} |
https://math.stackexchange.com/questions/880561/combinatorics-question-about-balls-in-boxes | # Combinatorics Question about balls in boxes
There are 5 balls numbered 1 to 5, and there are 3 boxes numbered 1 to 3. The question asks in how many distinct ways can the balls be put into the boxes if 2 boxes have 2 balls each and the other box has the remaining ball?
My try: We choose 2 balls to go in one box, 2 to go in another, and the last one goes in the last box for a total of $${5 \choose 2} {3\choose 2}$$ ways. So one arrangement would be having balls 1 and 2 in box 1, balls 3 and 4 in box 2, and ball 5 in box 3. But we can put balls 1 and 2 in box 2 or 3, so we multiply the expression above by $3! = 6$ to get a total of 180 ways.
The solution manual states that they divide by 2 because the 2 groups of 2 balls are indistinguishable, what does mean, and why? Because from my understanding, we're using combinations, so aren't the balls considered indistinguishable? And we multiplied by 6 to account for each group of balls being in a different box. I don't really understand the reasoning behind indistinguishable, so any help is greatly appreciated.
EDIT: This is the actual question:
Five balls are numbered 1 to 5. Three boxes are numbered 1 to 3. How many distinct ways can the balls be put in the boxes if two boxes have two balls each and the other has the remaining ball?
• Can you post the exact wording to the question? By the way you've posed it above (numbered balls and numbered boxes), I would agree with your solution, but "indistinguishable" in a combinations problem generally means that they are interchangeable and you can't tell the difference, so you would have to divide. But the exact wording will help. – Duncan Jul 28 '14 at 14:30
• @Duncan Edited the question. – Vishwa Iyer Jul 28 '14 at 14:32
• I think André's solution below is a nicer way to think of it. With regard to your question about dividing by $2$, this is because you have double-counted: if you take the example you give of $12|34|5$, note that $34|12|5$ is counted not only as one of the $3!$ permutations of the boxes, but also as choosing $3$ and $4$ first (in $\binom{5}{2}$) and then $1$ and $2$ (in $\binom{3}{2}$). – angryavian Jul 28 '14 at 14:44
• Based on all of your comments, you're saying that the boxes with two balls is indistinguishable and the box with one ball isn't? Therefore, it's like having a word "AAB" and finding the distinct permutations of it. – Vishwa Iyer Jul 28 '14 at 14:49
• @angryavian The reason I was confused was that I thought $12|34|5$ and $34|12|5$ were different because the boxes are labeled, so it's like having the two balls in different boxes, resulting in distinct ways. Why is this wrong? – Vishwa Iyer Jul 28 '14 at 14:52
I would solve the problem like this. There are $\binom{3}{1}$ ways to choose the box that will have a singleton, and for each such way there are $\binom{5}{1}$ ways to choose the lonely ball.
Now we have two boxes left, a lower-numbered one and a higher-numbered one. Choose the two balls that will go into the lower-numbered one. This can be done in $\binom{4}{2}$ ways, for a total of $\binom{3}{1}\binom{5}{1}\binom{4}{2}$.
• I am trying to avoid mysterious divisions by $2$. We have two boxes left, say Box 1 and Box 3. As soon as we decide which of the remaining balls go into Box 1, we will be finished. – André Nicolas Jul 28 '14 at 14:45 | 2019-08-23T10:49:31 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/880561/combinatorics-question-about-balls-in-boxes",
"openwebmath_score": 0.8543348908424377,
"openwebmath_perplexity": 176.09820712083473,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9850429125286725,
"lm_q2_score": 0.8479677622198946,
"lm_q1q2_score": 0.8352846342275058
} |
http://mathhelpforum.com/statistics/167956-consecutive-aces-deck-cards.html | # Math Help - consecutive aces in a deck of cards
1. ## consecutive aces in a deck of cards
I have come across a problem relating to playing cards, and am having some difficulty figuring out the solution.
Assume you have 1 deck of shuffled cards. You proceed to deal the cards out one by one. I would like to know the probability of getting 2 consecutive aces anywhere in the sequence of 52 cards.
Thus far I have the following:
The chance of getting four consecutive aces anywhere in the deck.
I have calculated as follows:
A sequence can start on Card 1, 2, 3, .... up to the 49th spot. Thus there are 49 possible start positions for the sequence. Further, since there are 4 aces, each sequence can be arranged in 4x3x2x1 ways (4! ways). Thus there are 49x24 possibilities. With these 1,176 possibilites the remaining cards of the deck can be arranged in 48! ways for EACH of the 1,176 possibilities. Thus in total there are 48! x 1,176 ways in which a deck of cards can contain a 4-ace sequence.
There are 52! in which a deck of cards can be arranged.
Thus the probability that a particular card arrangement contains a 4-ace sequence is
(1,176 x 48!) / 52! = 1,176 x 48! / (52x51x50x49x48!)
= 1176 / (52x51x50) = 0.018% (or roughly 1 in 5,525)
I then looked at the likelihood of getting a 3-ace sequence
A sequence can start on Card 1, 2, 3, .... up to the 50th spot. Thus there are 50 possible start positions for the sequence. Further, since there are 4 aces, each sequence can be arranged in 4x3x2 ways. Thus there are 50x24 possibilities. With these 1,200 possibilites the remaining cards of the deck can be arranged in 49! ways for EACH of the 1,200 possibilities. Thus in total there are 49! x 1,200 ways in which a deck of cards can contain a 3-ace sequence.
However this includes the 4-ace sequences which I want to exclude.
49! x 1200 = 7.29 x 10^65
48! x 1176 = 1.46 x 10^64
After the subtraction we are left with 7.16 x 10^65
Thus the probability that a particular card arrangement contains a 3-ace sequence is
(7.16 x 10^65) / 52! = 0.904% (or roughly 1 in 110.5)
Is my approach correct thus far?
2. Isn't the chance of getting 4 aces, in turn, simply $\dfrac{4}{52} \cdot \dfrac{3}{51} \cdot \dfrac{2}{50} \cdot \dfrac{1}{49} = \dfrac{4! \cdot 48!}{52!}$
3. Originally Posted by e^(i*pi)
Isn't the chance of getting 4 aces, in turn, simply $\dfrac{4}{52} \cdot \dfrac{3}{51} \cdot \dfrac{2}{50} \cdot \dfrac{1}{49} = \dfrac{4! \cdot 48!}{52!}$
I think he means getting 4 consecutive aces anywhere when dealing all the cards (not just in the beginning).
4. Originally Posted by george12345
Assume you have 1 deck of shuffled cards. You proceed to deal the cards out one by one. I would like to know the probability of getting 2 consecutive aces anywhere in the sequence of 104 cards.
Since when does a deck have 104 cards?
5. Originally Posted by e^(i*pi)
Isn't the chance of getting 4 aces, in turn, simply $\dfrac{4}{52} \cdot \dfrac{3}{51} \cdot \dfrac{2}{50} \cdot \dfrac{1}{49} = \dfrac{4! \cdot 48!}{52!}$
Hi Donor, thanks for your quick reply! Your answer - i think that's the probability of getting four aces in a row when drawing four cards out of deck without replacement. I am looking for the probability of getting a sequence of 4 aces anywhere in the deck of 52 cards. and then the probability of getting a sequence of 3 aces anywhere in the deck, and so on. Does that make sense?
6. Hello, george12345!
You have a deck of shuffled cards and you deal the cards out one by one.
Find the probability of getting 2 consecutive aces in the sequence of 52 cards.
There are: $52!$ possible orderings of the cards.
We have: .4 aces and 48 Others.
Place the 48 Others in row, leaving a space before, after, and between them.
. . $\_\,O\,\_\,O\,\_\,O\,\_\,O\,\hdots \,\_\,O\,\_$
There are 49 spaces.
Select 4 of the spaces and insert the four Aces (in some order).
. . There are: . $_{49}P_4$ ways.
Hence: . $P(\text{no consecutive Aces}) \;=\;\dfrac{_{49}P_4}{52!}$
Therefore: . $P(\text{some consecutive Aces}) \;=\;1 - \dfrac{_{49}P_4}{52!}$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
If the problem meant exactly one pair of consecutive Aces,
. . we can modify the above procedure.
We have the 48 Others with spaces before, after, and between them.
. . $\_\,O\,\_\,O\,\_\,O\,\_\,O\,\hdots\,\_\,O\,\_$ . (49 spaces)
We have 4 aces.
Select two of them to be together; there are $_4C_2 \,=\,6$ choices.
Duct-tape the pair together.
Then we have 3 "cards" to distribute: . $\boxed{AA}\,,\:A,\:A$
And they can be placed in: . $_{49}P_3$ ways.
Hence, there are: . $6(_{49}P_3)$ ways to have one pair of consecutive Aces.
The probability is: . $\dfrac{6(_{49}C_3)}{52!}$
7. Your two cases are even easier.
There are $52!$ possible orderings of the cards.
4 consecutive Aces
Duct-tape the four Aces together.
. . They can be in: $4! = 24$ orders.
Then we have 49 "cards" to arrange: . $\boxed{AAAA}\,,\,O,\,O,\,O\,\hdots\,O$
There are: $49!$ arrangements.
Therefore: . $P(\text{4 consecutive Ace}) \;=\;\dfrac{24\cdot49!}{52!}$
3 consecutive Aces
Choose 3 of the Aces to be together; there are $_4C_3 = 4$ choices.
. . and they can be in $3! = 6$ orders.
Duct-tape the 3 Aces together.
So we have two "cards": . $\boxed{AAA}\,,\,A$ .to distribute among the 49 spaces.
. . There are: . $_{49}P_2$ ways.
Hence, there are: . $4\cdot6\cdot(_{49}P_2)$ ways to have 3 consecutive Aces.
Therefore: . $P(\text{3 consecutive Aces}) \;=\;\dfrac{24\cdot(_{49}P_2)}{52!}$
8. hi soroban
you have used a completely approach then what I have been trying to do- and your approach seems much simpler.
So basically you are determining in how many ways you can mix 4 aces into the 48 other cards without letting them be in a consecutive order, because you are only allowing one space in between each O? And the difference between 52! and the number that you determine must be the number of ways in which you can have two consecutive aces.
Presumbaly that number would also include instances where there are 3 or 4 aces in succession?
Can that approach also be used for determining the likelihood of getting 3 aces or 4 aces?
9. My approach is similar to Soroban's, but I think the numbers come out different and I'm not sure which of us is right.
Edit. My method was very wrong!
10. Originally Posted by Soroban
Your two cases are even easier.
There are $52!$ possible orderings of the cards.
4 consecutive Aces
Duct-tape the four Aces together.
. . They can be in: $4! = 24$ orders.
Then we have 49 "cards" to arrange: . $\boxed{AAAA}\,,\,O,\,O,\,O\,\hdots\,O$
There are: $49!$ arrangements.
Therefore: . $P(\text{4 consecutive Ace}) \;=\;\dfrac{24\cdot49!}{52!}$
3 consecutive Aces
Choose 3 of the Aces to be together; there are $_4C_3 = 4$ choices.
. . and they can be in $3! = 6$ orders.
Duct-tape the 3 Aces together.
So we have two "cards": . $\boxed{AAA}\,,\,A$ .to distribute among the 49 spaces.
. . There are: . $_{49}P_2$ ways.
Hence, there are: . $4\cdot6\cdot(_{49}P_2)$ ways to have 3 consecutive Aces.
Therefore: . $P(\text{3 consecutive Aces}) \;=\;\dfrac{24\cdot(_{49}P_2)}{52!}$
Thanks for your reply. The way you explained this seems logical to me - thank you so much! I got the same answer as you for the case of 4 consecutive aces with a probability of 0.0181% (or 1 in 5525). For the 3 consecutive aces I got a different answer - however I think I know why.
Originally Posted by Soroban
Therefore: . $P(\text{3 consecutive Aces}) \;=\;\dfrac{24\cdot(_{49}P_2)}{52!}$
This results still needs to be multiplied by 48! right? As there are 48! ways of ordering the remaining 48 (all non-ace) cards, for each of the successful permutations that you have identified.
If this is multiplied out I arrive at a probability of 0.8688% (or about 1 in 113). The formula you stated above on its own would give a probability of 0.0000%, which doesn't seem right - since the probability of getting a 3-ace sequence should be higher than getting a 4-ace sequence. Can you confirm whether you agree?
In particular this figure of 0.8688% does not include 3-ace sequences that actually form part of a 4-ace sequence - am I understanding correctly?
11. Originally Posted by Soroban
Hello, george12345!
There are: $52!$ possible orderings of the cards.
We have: .4 aces and 48 Others.
Place the 48 Others in row, leaving a space before, after, and between them.
. . $\_\,O\,\_\,O\,\_\,O\,\_\,O\,\hdots \,\_\,O\,\_$
There are 49 spaces.
Select 4 of the spaces and insert the four Aces (in some order).
. . There are: . $_{49}P_4$ ways.
Hence: . $P(\text{no consecutive Aces}) \;=\;\dfrac{_{49}P_4}{52!}$
Therefore: . $P(\text{some consecutive Aces}) \;=\;1 - \dfrac{_{49}P_4}{52!}$
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
If the problem meant exactly one pair of consecutive Aces,
. . we can modify the above procedure.
We have the 48 Others with spaces before, after, and between them.
. . $\_\,O\,\_\,O\,\_\,O\,\_\,O\,\hdots\,\_\,O\,\_$ . (49 spaces)
We have 4 aces.
Select two of them to be together; there are $_4C_2 \,=\,6$ choices.
Duct-tape the pair together.
Then we have 3 "cards" to distribute: . $\boxed{AA}\,,\:A,\:A$
And they can be placed in: . $_{49}P_3$ ways.
Hence, there are: . $6(_{49}P_3)$ ways to have one pair of consecutive Aces.
The probability is: . $\dfrac{6(_{49}C_3)}{52!}$
I'd like to try and also follow this through.
The probability of having two consecutive aces in the deck comprises situations of AA, A, A (single instance) as well as AA, AA (double instance).
The single instance occurs as follows:
49 spaces
Three clumps AA, A, A
49 P 3 ways of placing the 3 clumps into 49 spaces = 110,544
Select 2 aces from 4 => 6 possibilities (AB, AC, AD, BC, BD, CD)
48! possibilities of ordering the remaining (non-ace) 48 cards = 48!
The product of these three is 8.2337 x10^66 - which out of 52! total possible combinations equates to a 10.21% probability - that seems right since 1/13th of the time an ace will be followed by another ace, or can be preceded by an ace etc.
The double instance occurs as follows:
49 spaces
Two clumps AA, AA
49 P 2 ways of placing the clumps into 49 spaces = 2,352
Select 2 lots of 2 aces to be chosen from 4 => 24 possibilities (4x3x2x1)
48! possibilities of ordering the remaining (non-ace) 48 cards = 48!
The product of these three is 7.007 x10^65 - which out of 52! possible combinations for the deck equates to a 0.8687% probability. This is the same probability as having a 3-ace sequence - which seems logical since both cases deal with the placing of two clumps of cards, AAA & A in one situation and AA & AA in other situation.
The sum of the two proabilities is 10.21%+0.87% = 11.08%
Thus in summary I have the following probabilities:
P(four consecutive aces anywhere in the deck) = 0.0181%
P(three consecutive aces anywhere in the deck) = 0.8869% (excludes situations where a 3-ace sequence forms part of a 4-ace sequence)
P(two consecutive aces anywhere in the deck) = 11.0769% (excludes situations where a 2-ace sequence forms part of a a 3-ace or 4-ace sequence)
It would be great if you could let me know whether I have followed this through correctly or not. Many thanks in advance!
12. I ran a simulation program for 4 concecutives (a few million tries) and got same as Soroban: ~.000181
13. Originally Posted by Wilmer
I ran a simulation program for 4 concecutives (a few million tries) and got same as Soroban: ~.000181
Hi Wilmer, that's reassuring - thank you!
Does your simulation program also work for multiple decks?
14. Originally Posted by george12345
Does your simulation program also work for multiple decks?
It would, but it's set up for 52 cards and I'm not interested in making it larger.
As I said, results matched Soroban's, so I suggest you use his formula...
15. Originally Posted by Soroban
Hello, george12345!
There are: $52!$ possible orderings of the cards.
We have: .4 aces and 48 Others.
Place the 48 Others in row, leaving a space before, after, and between them.
. . $\_\,O\,\_\,O\,\_\,O\,\_\,O\,\hdots \,\_\,O\,\_$
There are 49 spaces.
Select 4 of the spaces and insert the four Aces (in some order).
. . There are: . $_{49}P_4$ ways.
Hence: . $P(\text{no consecutive Aces}) \;=\;\dfrac{_{49}P_4}{52!}$
Therefore: . $P(\text{some consecutive Aces}) \;=\;1 - \dfrac{_{49}P_4}{52!}$
[snip]
Soroban,
If you're going to count all the permutations of the cards in which there are no consecutive aces, it's not enough to account for the positions of the aces; you must also count the arrangements of the other cards. Once you have placed the aces, which can be done in P(49, 4) ways, the remaining cards can be arranged in 48! ways.
So the probability of no consecutive aces is
$\frac{P(49,4) \times 48!}{52!}$
and the probability of at least two consecutive aces is
$1- \frac{P(49,4) \times 48!}{52!} \approx 0.2174$ | 2014-11-29T07:57:35 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/statistics/167956-consecutive-aces-deck-cards.html",
"openwebmath_score": 0.9137372374534607,
"openwebmath_perplexity": 882.3350973427803,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9850429147241162,
"lm_q2_score": 0.8479677583778258,
"lm_q1q2_score": 0.8352846323045686
} |
https://math.stackexchange.com/questions/1566058/show-sum-n-1-infty-frac1n1n2n3-is-convergent-using-partial | # Show $\sum_{n=1}^\infty {\frac{1}{(n+1)(n+2)(n+3)}}$ is convergent using partial fractions and find limit
Show $$\sum_{n=1}^\infty {\frac{1}{(n+1)(n+2)(n+3)}}$$ is convergent using partial fractions and find limit.
I know how to do this by integral test and surely if you just multiply out the brackets you can use comparison test? Even ratio test could work? Anyway in this question we have to use the following hint:
This is a so-called telescopic series. Find an explicit representation for the partial sums and prove this.
I don't know how to do this, if you make into partial fractions you have three fractions none of which converge..
Partial fractions gives: $\frac{1}{2(n+1)}+\frac{1}{2(n+3)}-\frac{1}{n+2}$
• Please write down what you obtained after doing partial fraction decomposition. – user258700 Dec 8 '15 at 16:51
• Why would the limit be zero when you sum only positive values? – Hetebrij Dec 8 '15 at 16:51
• I've added in result of partial fractions – babylon Dec 8 '15 at 16:55
• It is true that the convergence is obvious. (But Ratio Test is inconclusive.) However, you are being asked to find an explicit number for the sum, Yes, if you break it up into $3$ series, none will converge. But if you combine terms suitably, magic will happen. – André Nicolas Dec 8 '15 at 16:56
• I like magic. ${}{}$ – copper.hat Dec 8 '15 at 17:03
Hint: $$\frac{1}{(n+1)(n+2)(n+3)}=\frac{1}{2}\left(\frac{1}{(n+1)(n+2)}-\frac{1}{(n+2)(n+3)}\right).$$ Or, using your partial fractions:
$$\frac{1}{(n+1)(n+2)(n+3)} = \left(\frac{1}{2(n+1)}-\frac1{2(n+2)}\right)-\left(\frac{1}{2(n+2)}-\frac1{2(n+3)}\right)$$
You have ${1 \over (n+1)(n+2)(n+3)} = {1 \over 2}({1 \over n+1} - {1 \over n+2}) + {1 \over 2}({1 \over n+3} - {1 \over n+2})$. The latter two are telescoping sums.
Hence the sum is
${1 \over 2}{1 \over 1+1} - {1 \over 2}{1 \over 1+2} = {1 \over 12}$.
BIG HINT:
$$\sum_{n=1}^\infty {\frac{1}{(n+1)(n+2)(n+3)}}=$$ $$\lim_{m\to\infty}\sum_{n=1}^m{\frac{1}{(n+1)(n+2)(n+3)}}=$$ $$\lim_{m\to\infty}\sum_{n=1}^m\frac{1}{2}\left(\frac{1}{(n+1)(n+2)}-\frac{1}{(n+2)(n+3)}\right)=$$ $$\frac{1}{2}\lim_{m\to\infty}\sum_{n=1}^m\left(\frac{1}{(n+1)(n+2)}-\frac{1}{(n+2)(n+3)}\right)=$$ $$\frac{1}{2}\lim_{m\to\infty}\left(\frac{1}{6}-\frac{1}{m^2+5m+6}\right)=$$ $$\frac{1}{2}\left(\lim_{m\to\infty}\frac{1}{6}-\lim_{m\to\infty}\frac{1}{m^2+5m+6}\right)=$$ $$\frac{1}{2}\left(\frac{1}{6}-\lim_{m\to\infty}\frac{1}{m^2+5m+6}\right)=$$
$$\frac{1}{2}\left(\frac{1}{6}-\frac{1}{\lim_{m\to\infty}\left(m^2+5m+6\right)}\right)$$
Notice, $$\sum_{n=1}^{\infty}\frac{1}{(n+1)(n+2)(n+3)}$$ $$=\frac{1}{2}\sum_{n=1}^{\infty}\left(\frac{1}{(n+1)(n+2)}-\frac{1}{(n+2)(n+3)}\right)$$ $$=\frac{1}{2}\lim_{n\to\infty}\left[\left(\frac{1}{2\cdot 3}-\frac{1}{3\cdot 4}\right)+\left(\frac{1}{3\cdot 4}-\frac{1}{4\cdot 5}\right)+\ldots+\left(\frac{1}{(n+1)(n+2)}-\frac{1}{(n+2)(n+3)}\right)\right]$$ $$=\frac{1}{2}\lim_{n\to\infty}\left[\frac{1}{2\cdot 3}-\frac{1}{(n+2)(n+3)}\right]$$ $$=\frac{1}{2}\left[\frac{1}{6}-0\right]=\color{red}{\frac{1}{12}}$$
$$\sum_{n=1}^\infty {\frac{1}{(n+1)(n+2)(n+3)}}$$ $$=\frac12\sum_{n=1}^\infty \left(\frac1{n+1}-\frac2{n+2}+\frac1{n+3}\right)$$ $$=\frac12\sum_{n=1}^\infty \left(\frac1{n+1}-\frac1{n+2}\right)-\frac12\sum_{n=1}^\infty \left(\frac1{n+2}-\frac1{n+3}\right)$$ $$=\frac12\left(\frac1{2}-\lim_{n\to\infty}\frac1{n+2}\right)-\frac12\left(\frac1{3}-\lim_{n\to\infty}\frac1{n+3}\right)$$ $$=\frac12\cdot\frac12-\frac12\cdot\frac13=\frac{1}{12}$$
• I think the last manipulation should be done with partial sums, because, since the armonic series is divergent, you might not be allowed to use infinite times the associative property in $\sum_{n=1}^\infty \left(\frac{1}{n+1}-\frac{1}{n+2}\right)$ – user228113 Dec 8 '15 at 17:13
• This $\sum_{n=1}^\infty \left(\frac1{n+1}-\frac1{n+2}\right)$ alone converges. – Kay K. Dec 8 '15 at 17:16
• I agree. And it converges absolutely. And so does $$\sum_{k=0}^\infty ((-1)^k+(-1)^{k+1})=0$$ but, if you do the same manipulation you did to get $\frac12$ in your proof, you'll get $$(-1)^0+\sum_{k=1}^{\infty} 2(-1)^k$$ which is not convergent. – user228113 Dec 8 '15 at 17:20
• $\sum_{n=1}^\infty \left(\frac1{n+1}-\frac1{n+2}\right)=\frac12$ and $\sum_{n=1}^\infty \left(\frac1{n+2}-\frac1{n+3}\right)=\frac13$, which is what the last line is saying. – Kay K. Dec 8 '15 at 17:25
• Now I think I get what you were trying to tell me. Yes I skipped the proof. I agree it's good to include it. I'll add it. Thanks. – Kay K. Dec 8 '15 at 17:32 | 2019-08-18T08:40:20 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1566058/show-sum-n-1-infty-frac1n1n2n3-is-convergent-using-partial",
"openwebmath_score": 0.8455715775489807,
"openwebmath_perplexity": 490.7757455980456,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9850429160413819,
"lm_q2_score": 0.8479677564567913,
"lm_q1q2_score": 0.8352846315292661
} |
https://math.stackexchange.com/questions/2296214/help-to-solve-and-understand-simultaneous-equations | # Help to solve and understand Simultaneous equations.
Could anyone help me solve this simultaneous equation: $$6x + 7y = 12.5\tag1$$ $$7x + 5y = 14\tag2$$ I have watched loads of videos on YouTube, and still don't understand how to do it. I am trying to learn it. And practice with different equations. But don't understand how to do it. So any help will be much appreciated.
• Now take $7(2)-5(1)$! – Parcly Taxel May 25 '17 at 13:03
• What about it is confusing you? Give a bit more context so we can answer with something you can understand. – Sean Roberson May 25 '17 at 13:04
We have \begin{align}6x + 7y &= 12.5\tag{1}\\ 7x + 5y &= 14\tag{2}\end{align}
One way to solve this, would be to rearrange equation $(1)$ for $x$:
\begin{align}6x + 7y &= 12.5\\ 6x&=12.5-7y\\ x&=\frac{12.5-7y}{6}\end{align}
We can then put this into equation $(2)$ and solve for $y$:
\begin{align}7x + 5y &= 14\\ 7\left(\frac{12.5-7y}{6}\right)+5y&=14\\ \frac{87.5-49y}{6}+5y&=14\\ 87.5-49y+30y&=84\\ 87.5-19y&=84\\ 19y&=3.5\\ y&=\frac{3.5}{19}\\ &=\frac{7}{38}\end{align}
Finally we can put this into equation $(1)$ to find that \begin{align}6x + 7\left(\frac{7}{38}\right) &= 12.5\\ 6x+\frac{49}{38}&=12.5\\ 6x&=12.5-\frac{49}{38}\\ 6x&=\frac{213}{19}\\ x&=\frac{71}{38}\end{align}
We can then check this using equation $(2)$:
\begin{align}7x + 5y &=7\left(\frac{71}{38}\right)+5\left(\frac{7}{38}\right)\\ &=\frac{497}{38}+\frac{35}{38}\\ &=\frac{532}{38}\\ &=14\end{align}
Therefore we can be confident that the answer is $x=\dfrac{71}{38}$ and $y=\dfrac{7}{38}$
• Thank you for this. I am going to keep going through this until I fully understand it. And then try it on another equation. I really appreciate it. – Liam May 25 '17 at 13:30
• @Liam No worries. Make use of this site and ask other questions if you get stuck again. Also, in case you are not aware, you can upvote answers you like, and can also mark them as 'acecepted' (basically means it is the best answer for your question in your eyes) by clicking the checkmark next to the answer. This is how the site makes sure the best answers are seen first, when other people seach for a problem – lioness99a May 25 '17 at 13:33
• I have marked this as accepted. I just wanted to ask one more question. I am seeing an answer in the comments that is not a fraction. Which confuses me a bit. Is it be ok to solve an equation like this with a fraction? And is this the best way to solve this equation? Thanks – Liam May 25 '17 at 13:58
• @Liam It is always best to write an answer as a fraction, if this is the most accurate answer. For example, $\frac{71}{38}$ is in its simplest form. We could write it as $1.868421052631578947368421052631578947368421052631578947368\ldots$ but this would be less accurate. The only exception to this rule is if the question asked for the answer to be given to a specific number of significant figures. Even in this case, you would work with the fractions right up until the moment you wrote down the answer, when you would round anything that wasn't exact – lioness99a May 25 '17 at 14:03
• Hello, thank you for your help. Very much appreciated. – Liam May 25 '17 at 14:50
A method that always works for such (linear) equations is to isolate one of the variables in one of the equation, say $y$ in $(1)$, and then insert that expression for $y,$ which is now written in terms of $x,$ into the other equation. Then you will have an equation with only one variable, $x,$ which you can then solve.
Having done this, you can insert that value for $x$ into the expression for $y,$ where you had isolated $y.$ Now you have both the $x$ and $y$.
Try it out, and if it is still causing you trouble, leave me a comment below and I'll show the full solution.
Good luck!
A side-note: Often, however, there are simpler ways of doing things, as @ParclyTaxel points out in the comments above. What is being referred to is multiplying equation $(2)$ with seven and then subtracting five times equation $(1).$ This is done to achieve a common factor, allowing the cancellation of a variable, as such:
\begin{align} 7(2):\quad & 49x+35y=7\cdot 14\\ 5(1):\quad & 30x+35y=5\cdot 12.5 \end{align} Subtracting then gives $$49x+35y-30x-35y=7\cdot 14-5\cdot 12.5$$ simplifying to $$19x=35.5.$$
Shortcuts like this comes with experience. The way described above, however, is straight-forward application of an algorithm, and doesn't require any "ideas", so it always works.
• This is the way I was trying to do it from watching YouTube. But could not get an answer. I will try again this way and try to get it correct. Thank you. – Liam May 25 '17 at 13:28
Here is an another example, then you can apply it to your problem:
(i) $x+y=4$
(ii) $2x+y=7$
Solving (i) for $x$ gives us $x=4-y$. Now put this into (ii) and you get $2(4-y)+y=7$ and you can now easily solve it for $y$. You should get $y=1$ and finally $x=4-1=3$.
we can also write $$x+\frac{7}{6}=\frac{25}{12}$$ $$x+\frac{5}{7}y=2$$ multiplying by $$-1$$ we have $$-x-\frac{7}{3}y=-\frac{25}{12}$$ $$x+\frac{5}{7}y=2$$ adding both we obatain: $$\frac{5}{7}y-\frac{7}{3}y=2-\frac{25}{12}$$ can you finish? simplifying the last equation we obtain: $$-\frac{34}{21}y=-\frac{1}{12}$$ $$y=\frac{7}{38}$$ and then you can compute $$x$$
• I can not finish this. I have tried but do not understand it. Thank you for commenting though. – Liam May 25 '17 at 13:29
• oh sorry, what Kind of questions do you have? – Dr. Sonnhard Graubner May 25 '17 at 13:35
• I am getting a bit confused because there is 3 different ways I am seeing from the comments of solving this equation. And different answers. So I am just trying to understand them. Could you finish yours please? So I can try and understand it. Thank you. – Liam May 25 '17 at 13:49 | 2019-12-14T13:25:21 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2296214/help-to-solve-and-understand-simultaneous-equations",
"openwebmath_score": 0.8867442607879639,
"openwebmath_perplexity": 329.1727057568697,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9850429120895837,
"lm_q2_score": 0.8479677564567912,
"lm_q1q2_score": 0.8352846281782685
} |
http://mathhelpforum.com/statistics/126266-coin-tossing-print.html | # coin tossing
• Jan 30th 2010, 09:39 AM
alexandrabel90
coin tossing
2 gamblers bet $1 each on the successive tosses of a coin. each as$6. what is the probability that
a. they break even after 6 tosses?
b. one player wins all the money on the tenth toss?
my working:
a. since there are 6 tosses and equal probability of a head and tail,
the ans should be 6 x (0.5)^6
b. in order to win all the money o the other player, he needs to win 8 games and loss 2 games, where the 10th toss have to be a win.
so, (9 choose 7) x ( 0.5)^10
but i didnt get the answer :(
• Jan 30th 2010, 10:02 AM
Quote:
Originally Posted by alexandrabel90
2 gamblers bet $1 each on the successive tosses of a coin. each as$6. what is the probability that
a. they break even after 6 tosses?
b. one player wins all the money on the tenth toss?
my working:
a. since there are 6 tosses and equal probability of a head and tail,
the ans should be 6 x (0.5)^6
b. in order to win all the money o the other player, he needs to win 8 games and loss 2 games, where the 10th toss have to be a win.
so, (9 choose 7) x ( 0.5)^10
but i didnt get the answer :(
hi alexandrabel90,
for the first one, they both must win 3 times each,
so that's the probability of either winning exactly 3 times and losing exactly 3 times.
for the second one, you must add the probability for the 2 players.
this is different from the first one because one or other could win the game on the 10th go, having won 7 of the preceding 9.
• Jan 30th 2010, 10:19 AM
Plato
As I read the question we have two players each having $6 to play with. Each player either wins$1 or looses $1 on each toss of the coin. So it is possible for one player of loose all in six tosses- loses six straight. So for both to breakeven in six tosses each would wins three and lose three. Is that a misreading of the question? If not, the answer to part a) is $\frac{\binom{6}{3}}{2^6}$. If it is a misreading, please clarify the question. • Jan 30th 2010, 11:24 AM alexandrabel90 it is not a misreading of the question. your answer is correct for part (a)..i never think that we would need to count then chances of wining since there is only a 0.5 chance of a win and lose. for part (b),Archie, what do you mean by adding the probability of the 2 players? does it mean (9 choose 7) x ( 0.5)^10 x 2? • Jan 30th 2010, 12:31 PM Soroban Hello, alexandrabel90! Quote: Two gamblers bet$1 each on the successive tosses of a coin.
Each has $6. .What is the probability that: a. they break even after 6 tosses? They each win 3 games and lose 3 games. There are $_6C_3 \:=\:\frac{6!}{3!\,3!} \:=\:20$ ways this can happen. Answer: . $(20)\left(\frac{1}{2}\right)^3\left(\frac{1}{2}\ri ght)^3 \:=\:(20)\,\frac{1}{64} \:=\:\frac{5}{16}$ Quote: b. one player wins all the money on the tenth toss? my working: In order to win all the money of the other player, he needs to win 8 games and loss 2 games, where the 10th toss has to be a win. So: . $_9C_7\left(\tfrac{1}{2}\right)^{10}$ . . . . correct, so far $_9C_7\left(\tfrac{1}{2}\right)^{10} \:=\:\tfrac{36}{1024} \:=\:\frac{9}{256}$ This is the probability that $A$ wins all of $B$'s money on the 10th game. It's also the probability that $B$ wins all of $A$'s money on the 10th game. Therefore: . $P(one\text{ player wins all the money}) \:=\:\frac{9}{256} + \frac{9}{256} \;=\;\frac{9}{128}$ • Jan 30th 2010, 01:28 PM Archie Meade Quote: Originally Posted by alexandrabel90 for part (b),Archie, what do you mean by adding the probability of the 2 players? does it mean (9 choose 7) x ( 0.5)^10 x 2? Yes, that was it, because you calculated the probability of only one out of the two winning on the tenth throw. Soroban has shown that really nicely. Plato has made an interesting point also. If we interpret "break-even" as not winning or not losing on one's own throws, then that is 3 wins and 3 losses, but one can break even on one's own throws while the other has more wins than losses or vice versa on their throws. In this situation, one could break even on one's own throws but make a profit if the other player does badly, or make a loss if the other player does well, "if" losing a toss loses a dollar and winning a toss wins a dollar. Or, if the two players lost every time, "player A" wins "player B"'s 6 dollars and "player B" wins "player A"'s 6 dollars, so they break even. They could also both win 1 and lose 5 each, win 2 and lose 4 each, win 4 and lose 2 each, win 5 and lose 1 each, win 6 and lose 0 each. That's the thing with probability questions, quite often open to alternative interpretations. The answer book will distinguish given it's answers. We've been interpreting this as..... If one player loses on a throw, then the second player wins the first player's dollar. In this case then, both would have to win 3 times and lose 3 times to end up with the 6 dollars they began with. It is probably safest to assume there is just 6 tosses rather than 6 each. • Jan 30th 2010, 02:27 PM Plato Quote: Originally Posted by alexandrabel90 2 gamblers bet$1 each on the successive tosses of a coin. each as $6. what is the probability that b. one player wins all the money on the tenth toss? Quote: Originally Posted by Archie Meade Yes, that was it, Plato has made an interesting point also. If we interpret "break-even" as not winning or not losing on one's own throws, then that is 3 wins and 3 losses, but one can break even on one's own throws while the other has more wins than losses or vice versa on their throws. Look at the way part b) is put. It seems to me that in this game on any toss player A gets a dollar from player B or gives a dollar to player B. That is, A wins or looses on any toss. Moreover, the game is over when one player is out of money. If player A wins in each of the first six tosses she has all of the money in six tosses. If B wins five of the first six tosses he has 11 dollars and she has 1. So the question becomes “how can the game be over in exactly ten tosses?” • Jan 30th 2010, 02:35 PM Archie Meade Brilliant analysis, Plato! it's not over yet. • Jan 30th 2010, 03:08 PM Archie Meade I reckon that, in order for one player to win 8 to 2, the score had to stand at 5-1, at which point one player has$10 and the other has \$2 (win 5 and lose 1... 6+5-1=10, win 1 and lose 5.... 6+1-5=2).
Then the score can go to 6-1 or 5-2.
From 6-1 the score must go to 6-2, since 7-1 would end the game.
From there to 7-2 and then to 8-2.
From 5-2 the score goes to 6-2, to 7-2, to 8-2.
Beginning at 5-1, there are two ways for the leading player to win on the 10th throw.
So, we count the number of ways of getting to a 5-1 lead and double that.
$2\binom{6}{5}=2\binom{6}{1}$
$P=\frac{2(6)}{2^{10}}=\frac{3}{2^8}=\frac{3}{256}$
Then double this to get the result for either player winning by 8 to 2. | 2017-01-20T07:17:29 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/statistics/126266-coin-tossing-print.html",
"openwebmath_score": 0.6522731781005859,
"openwebmath_perplexity": 1542.901837029771,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9850429134068499,
"lm_q2_score": 0.8479677545357568,
"lm_q1q2_score": 0.8352846274029664
} |
https://math.stackexchange.com/questions/3858896/what-is-the-probability-that-a-randomly-formed-committee-includes-both-amy-and-b | # What is the probability that a randomly formed committee includes both Amy and Betty? Amy as chair and Betty as an unranked member?
Five people are to be chosen to form a committee from a group of 100 students. The committee has a Chair, Vice Chair, and three other unranked members. Suppose Amy and Betty are among the group of 100 students. What is the probability that a randomly formed committee includes both Amy and Betty? What is the probability that a randomly formed committee includes Amy as Chair and Betty as an unranked member?
I've done $$^{100}P_2×\binom{98}3$$ to see how many ways the committee of $$5$$ can be selected. I'm having trouble visualizing the ways Amy and Betty can both be on the committee, well the number of ways at least.
## 1 Answer
What is the probability that a randomly formed committee includes Amy and Betty?
Method 1: If we work with the sample space you have chosen, we have to consider each of the following possibilities:
• Amy is the Chair and Betty is the Vice Chair.
• Amy is the Chair and Betty is an unranked member.
• Amy is the Vice Chair and Betty is the Chair.
• Amy is the Vice Chair and Betty is an unranked member.
• Amy is an unranked member and Betty is the Chair.
• Amy is an unranked member and Betty is the Vice chair.
• Both Amy and Betty are unranked members.
Amy is the Chair and Betty is the Vice Chair: There are $$\binom{98}{3}$$ ways to select the three unranked members.
Amy is the Chair and Betty is an unranked member: There are $$98$$ ways to select the Vice Chair and $$\binom{97}{2}$$ ways to select the other unranked members.
Amy is the Vice Chair and Betty is the Chair: There are $$\binom{98}{3}$$ ways to select the three unranked members.
Amy is the Vice Chair and Betty is an unranked member: There are $$98$$ ways to select the Chair and $$\binom{97}{2}$$ ways to select the other unranked members.
Amy is an unranked member and Betty is the Chair: There are $$98$$ ways to select the Vice Chair and $$\binom{97}{2}$$ ways to select the other unranked members.
Amy is an unranked member and Betty is the Vice chair: There are $$98$$ ways to select the Chair and $$\binom{97}{2}$$ ways to select the other unranked members.
Both Amy and Betty are unranked members: There are $$98$$ ways to select the Chair, $$97$$ ways to select the Vice Chair, and $$96$$ ways to pick the other unranked member.
Since these cases are mutually exclusive and exhaustive, the number of favorable cases is $$\binom{98}{3} + 98\binom{97}{2} + \binom{98}{3} + 98\binom{97}{2} + 98\binom{97}{2} + 98\binom{97}{2} + 98 \cdot 97 \cdot 96$$ which can be simplified to $$2\binom{98}{3} + 4 \cdot 98\binom{97}{2} + 98 \cdot 97 \cdot 96$$ You have corrected calculated that the number of elements in your sample space is $$100 \cdot 99\binom{98}{3}$$ Hence, the probability that Amy and Betty are both selected to serve on the committee is $$\frac{2\dbinom{98}{3} + 4 \cdot 98\dbinom{97}{2} + 98 \cdot 97 \cdot 96}{100 \cdot 99\dbinom{98}{3}}$$
Method 2: We take as our sample space the $$\binom{100}{5}$$ subsets of five of the $$100$$ students who could serve on the committee.
If Amy and Betty both serve on the committee, then three of the remaining $$98$$ students must also serve on the committee. Hence, there are $$\binom{98}{3}$$ favorable cases.
Thus, the probability that Amy and Betty both serve on the committee is $$\frac{\dbinom{98}{3}}{\dbinom{100}{5}}$$
As you can verify, this gives the same result as the first method.
Notice that since we only care about who serves on the committee, we can ignore the roles of the individual members for this problem.
What is the probability that a randomly formed committee includes Amy as Chair and Betty as an unranked member?
Since it matters who serves in what role on the committee, we use your sample space. We showed above that the number of favorable cases is $$98\binom{97}{2}$$ Hence, the desired probability is $$\frac{98\dbinom{97}{2}}{100 \cdot 99\dbinom{98}{3}}$$
• For the last part, why wouldn't the desired probability be over 100*99*choose(98,3)? – Pierre Oct 12 '20 at 16:29
• @Pierre It is. Thank you for pointing out the error. – N. F. Taussig Oct 12 '20 at 18:54 | 2021-06-20T06:24:54 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3858896/what-is-the-probability-that-a-randomly-formed-committee-includes-both-amy-and-b",
"openwebmath_score": 0.4421938359737396,
"openwebmath_perplexity": 429.8747255784395,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9850429156022934,
"lm_q2_score": 0.8479677526147223,
"lm_q1q2_score": 0.8352846273723303
} |
https://math.stackexchange.com/questions/2296704/rotation-matrix-in-order-to-rotate-orthogonal-vectors-onto-x-axis-y-axis-etc | # Rotation matrix in order to rotate orthogonal vectors onto x-axis, y-axis, etc…
Assume that I have a set of $n$ orthogonal vectors in $n$-dimensional space such that they span the whole space. These vectors are of arbitrary length and are not necessarily unit vectors.
How do I construct a rotation matrix that will rotate the vectors onto the x-axis, y-axis, z-axis, etc... while preserving their original length?
I intend to use this matrix to rotate an entire space.
• Do the vectors have unit length? If so, then the matrix with the vectors as columns will take the standard basis to the set of vectors. The inverse of the matrix will go the other way, or take the vector set to the standard basis. – Paul Aljabar May 25 '17 at 19:19
• @PaulAljabar no, the vectors have arbitrary length – Paul Terwilliger May 25 '17 at 19:27
• I should have added that, with the assumption that the orthogonal vectors have unit length, the inverse of the above matrix will be its transpose. – Paul Aljabar May 25 '17 at 19:28
• Okay, each vector will need to be scaled first to obtain an orthonormal set ... This transformation will need to be concatenated with the one above. – Paul Aljabar May 25 '17 at 19:30
• okay, so how is this done? – Paul Terwilliger May 25 '17 at 19:31
Let $v_1,\dots,v_n$ be your $n$ vectors. Recalling that the columns of a transformation matrix are images of basis vectors, the matrix that maps the standard basis to these vectors is $$\begin{bmatrix}v_1&\cdots&v_n\end{bmatrix}.$$ Since we want to preserve lengths, we normalize each of the $v_k$ before assembling them into this matrix. The matrix that maps the $v_k$ onto the coordinate axes is then the inverse of this matrix, but because we’ve assumed that the $v_k$ are orthogonal, its inverse is simply its transpose, so the matrix that we’re looking for is $$R=\begin{bmatrix}{v_1^T\over\|v_1\|} \\ \vdots \\ {v_n^T\over\|v_n\|}\end{bmatrix},$$ i.e., the $k$th row of the desired transformation matrix is $v_k$ normalized. Note, however, that although $R$ is an orthogonal matrix, for it to be a rotation we must also have $\det R=1$, which means that $(v_1,\dots,v_n)$ must form a right-handed basis for $\mathbb R^n$.
Let $\mathbf{v}_1, \ldots, \mathbf{v}_n$ be the set of orthogonal vectors. Let $d_1, \ldots, d_n$ be the lengths of the vectors (all $d_i \ne 0$). Let $\hat{\mathbf{u}}_1, \ldots, \hat{\mathbf{u}}_n$ be the unit vectors obtained from the original set of vectors after dividing by each of the $d_i$'s.
Let $U$ be the matrix formed by taking the $\hat{\mathbf{u}}_i$'s as its columns. $U$ is an orthogonal matrix i.e. $U U^T = I$.
Going from the standard basis $\hat{\mathbf{e}}_1, \ldots, \hat{\mathbf{e}}_n$ to the original set, we need to scale each $\hat{\mathbf{e}}_i$ by $d_i$ first then multiply by $U$.
The scaling can be achieved by multiplying by a diagonal matrix $D$ which has $D_{ii} = d_i$ and $D_{ij} = 0$ for $i \ne j$.
So transforming from the standard basis to the set of $\mathbf{v}_i$'s is achieved by multiplying by $D$ then $U$ i.e. by the matrix $M = U D$.
Going the other way is achieved by multiplying by $M^{-1} = D^{-1} U^{-1}= D^{-1} U^{T}$. The matrix $D^{-1}$ is also diagonal and has its $i^{th}$ diagonal entry as $1/ d_{i}$.
If one wants to map the vectors to the standard basis without changing the length then applying $U^T$ is sufficient.
• I think you might have misinterpreted the OP’s question. It seems that he wants to map the vectors onto the corresponding axes, but preserve their lengths, i.e., $\mathbf v_k\mapsto d_k\hat{\mathbf e}_k$ and not $\mathbf v_k\mapsto \hat{\mathbf e}_k$ – amd May 25 '17 at 20:31
• Okay, no problem, have modified for either case ... – Paul Aljabar May 25 '17 at 20:35 | 2019-06-24T08:56:59 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2296704/rotation-matrix-in-order-to-rotate-orthogonal-vectors-onto-x-axis-y-axis-etc",
"openwebmath_score": 0.9566116333007812,
"openwebmath_perplexity": 161.87386919836536,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9850429164804706,
"lm_q2_score": 0.8479677506936879,
"lm_q1q2_score": 0.8352846262246949
} |
https://math.stackexchange.com/questions/3137072/what-numbers-should-i-not-pick | # What numbers should I not pick?
I have a problem on my hands and I am a little confused on how to solve it.
Say you have $$50$$ people and each chooses a number from $$0$$ to $$100$$. Now you take the average of all the numbers and denote it $$X$$, then calculate $$\frac{2}{3}(X+9)$$ and the person who chose the number closest to it wins the game.
The question is, which numbers are such, that if you choose them, then there is some other number that yields a closer number for all other peoples choices/results. The first thing to note is that $$0\leq X \leq 100$$ and so $$6 \leq \frac{2}{3}(X+9)\leq 72.6$$. Now I am not sure what to do at this point. It would seem like a bad move to put in anything outside of this bound (except possibly $$73$$). But I am not sure how to formalize this problem mathematically. I tried making it that you have the function
$$f(x_1,x_2,x_2.,x_{50}) = |2/3(X+9)-x_{50}|$$
and try to minimize this or something. This eventually turns into $$|2/3Y-\frac{74}{75}x_{50}+6|$$ where
$$0 \leq Y=\frac{x_1+x_2...x_{49}}{50}\leq 98$$
I think what confuses me is that what number you pick influences $$X$$ hence i tried to divide it up. Any help would be appreciated!
• Is repetition allowed i.e. can two persons pick the same number or does each person have to choose a unique number? – Nilotpal Kanti Sinha Mar 6 at 4:50
• @NilotpalKantiSinha Yes, repetition is allowed – Sorfosh Mar 6 at 4:53
Once you show that nobody should choose a number outside the range $$[6,72]$$ you can repeat the argument that led to that conclusion. If all the numbers are in that range, $$\frac 23(X+9)$$ is in the range $$[10,54]$$ so nobody should choose a number outside that range. This process will converge where $$X=\frac 23(X+9), X=18$$ so everybody should choose $$18$$.
• I understand that, but how do we prove nobody should choose a number outside the range $[6,72]$? I think I got a solution just by inspecting the radical cases of 5,6 and 72,73. IS there a slicker solution? – Sorfosh Mar 6 at 5:09
• You say that if you were going to choose a number outside the range, you would be strictly better off to choose $6$ or $72$ because you will be closer to $\frac 23(X+9)$ If you were going to win with the number outside, you will still with with the one on the edge and you have extra chances. The next step presumes that your opponents are as smart as you are. If you know they all love $100$ you should choose $72$ (or anything in the range $[72,99]$ ) and win. – Ross Millikan Mar 6 at 5:13 | 2019-04-25T08:02:02 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3137072/what-numbers-should-i-not-pick",
"openwebmath_score": 0.8807010054588318,
"openwebmath_perplexity": 205.16499855074153,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9850429103332288,
"lm_q2_score": 0.8479677545357568,
"lm_q1q2_score": 0.8352846247966348
} |
http://math.stackexchange.com/questions/203212/one-very-simple-question-about-countably-many-point-of-a-set | One very simple question about countably many point of a set
Q1: When a question asking me to show a set consists of countably many point, do it mean infinite countably many or both infinite or finite case.
Q2:Also, to show that a set consist of countably many point, is it enough to show that the set is countable or i have to show somethings more?
-
One of the official definitions of countable set is a set $A$ such that there is a bijection between $A$ and a subset of the natural numbers. That definition includes the finite sets.
If a set is countable but not finite, it should be called countably infinite.
But people often sloppily write "countable" when in principle they should write "countably infinite." So if the phrase is in a problem on an assignment, you may have to ask the instructor. However, the answer may be clear from the context. And if, for example, you have shown that a set is in one to one correspondence with the natural numbers, you will have shown that it is countable. For you will have proved the stronger result that the set is countably infinite.
As to the second question, "countable" and having "countably many points" mean the same thing.
-
So, what do you think about the second question? is it enough to show a set is a countable set and conclude immediately that it has contably many point of i have to add somethings more? – ABC Sep 27 '12 at 3:09
"Countable" and "countably many points" mean the same hing. – André Nicolas Sep 27 '12 at 3:11
For 1: the common use of the word countable includes both finite and countably infinite (the definition is that $A$ is countable if $|A|\leq\aleph_{0}$)
For 2: yes, it is sufficient as you show this by definition
- | 2014-12-20T20:28:05 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/203212/one-very-simple-question-about-countably-many-point-of-a-set",
"openwebmath_score": 0.9305370450019836,
"openwebmath_perplexity": 221.0932663803286,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9653811591688145,
"lm_q2_score": 0.8652240964782011,
"lm_q1q2_score": 0.8352710411989159
} |
https://math.stackexchange.com/questions/1760940/prove-arcsin-left-frac-35-right-arccos-left-frac-1213-right-arc | # Prove: $\arcsin\left(\frac 35\right) - \arccos\left(\frac {12}{13}\right) = \arcsin\left(\frac {16}{65}\right)$
This is not a homework question, its from sl loney I'm just practicing. To prove : $$\arcsin\left(\frac 35\right) - \arccos\left(\frac {12}{13}\right) = \arcsin\left(\frac {16}{65}\right)$$
So I changed all the angles to $\arctan$ which gives:
$$\arctan\left(\frac 34\right) - \arctan\left(\frac {12}{5}\right) = \arctan\left(\frac {16}{63}\right)$$
But the problem is after applying formula of $\arctan(X)-\arctan(Y)$ the lhs is negative and not equal to rhs? Is this because I have to add pi somewhere please help.
• Hint: simply take $\sin$ of both sides of the first equation. – lulu Apr 27 '16 at 11:38
• @lulu I just wanted to know why answer is not coming while taking in tan? – user3500780 Apr 27 '16 at 11:40
• – lab bhattacharjee Apr 27 '16 at 11:42
• If $\theta =\arccos\left(\frac {12}{13}\right)$ then $\theta =\arctan\left(\frac {5}{12}\right)$ – lulu Apr 27 '16 at 11:43
How exactly did you convert to arctan? Careful: $$\arccos\left(\frac {12}{13}\right) = \arctan\left(\frac {5}{12}\right) \ne \arctan\left(\frac {12}{5}\right)$$ Draw a right triangle with hypotenuse of length 13, adjacent side (from an angle $\alpha$) with length 12 and opposite side with length 5; then $\cos\alpha = 12/13$ and $\tan\alpha = 5/12$.
Perhaps easier: take the sine of both sides in the original equation: $$\sin\left(\arcsin\left(\frac 35\right) - \arccos\left(\frac {12}{13}\right)\right) = \sin\left( \arcsin\left(\frac {16}{65}\right)\right)$$ The RHS is $16/65$ and simplify the LHS with $\sin(a-b)=\sin a \cos b - \cos a \sin b$ to get: $$\frac{3}{5}\frac{12}{13}-\sqrt{1-\frac{9}{25}}\sqrt{1-\frac{144}{169} } = \frac{3}{5}\frac{12}{13}-\frac{4}{5}\frac{5}{13}= \cdots$$
• OH MY GOD i am so so stupid really thanks – user3500780 Apr 27 '16 at 11:45
• One thing though, how am I to look at the question and realize that I should take sin both sides? I mean under what circumstances it should click in my mind i gotta take sin both sides?? – user3500780 Apr 27 '16 at 11:47
• Other trigonometric functions could work as well, but will give you more work since you have two arcsines and $\sin\left(\arcsin\alpha\right) = \alpha$; so that simplifies easily. – StackTD Apr 27 '16 at 11:49
If $0<x<1$, then both $\arcsin x$ and $\arccos x$ are in $(0,\pi/2)$. I'll assume $0<x<1$ for the rest of the discussion.
If $\alpha=\arcsin x$, then $\sin\alpha=x$ and $\cos\alpha=\sqrt{1-x^2}$; therefore $$\tan\alpha=\frac{x}{\sqrt{1-x^2}}$$ and $$\arcsin x=\alpha=\arctan\frac{x}{\sqrt{1-x^2}}$$ Similarly, if $\beta=\arccos x$, then $x=\cos\beta$ and $$\arccos x=\arctan\frac{\sqrt{1-x^2}}{x}$$ For $x=3/5$ we have $\sqrt{1-x^2}=4/5$ and so $$\arcsin\frac{3}{5}=\arctan\frac{3}{4}$$ For $x=16/65$ we have $\sqrt{1-x^2}=63/65$, so $$\arcsin\frac{16}{65}=\arctan\frac{16}{63}$$ For $x=12/13$ we have $\sqrt{1-x^2}=5/13$, so $$\arccos\frac{12}{13}=\arctan\frac{5}{12}$$ Now $$\tan\left(\arctan\frac{3}{4}-\arctan\frac{5}{12}\right)= \frac{\dfrac{3}{4}-\dfrac{5}{12}}{1+\dfrac{3}{4}\dfrac{5}{12}}= \frac{\dfrac{1}{3}}{\;\dfrac{21}{16}\;}=\frac{16}{63}$$
To specifically use $arctan$ formula:
Change all angles to $tan$ as you outlined, using Pythagoras... $$\arctan(\frac34) - \arctan(\frac5{12}) = \arctan(\frac{16}{63})$$
Apply $arctan$ formula $$\arctan(x) - \arctan(y) = \arctan(\frac{x-y}{1+xy})$$
$$\arctan(\frac{\frac34 - \frac5{12}}{1+\frac34 \frac5{12}}) = \arctan(\frac{16}{63})$$ Simplify to give $$\arctan(\frac{16}{63}) = \arctan(\frac{16}{63})$$ $$LHS = RHS$$ | 2019-05-19T08:45:43 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1760940/prove-arcsin-left-frac-35-right-arccos-left-frac-1213-right-arc",
"openwebmath_score": 0.8719141483306885,
"openwebmath_perplexity": 324.8614882494668,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9653811611608241,
"lm_q2_score": 0.8652240930029118,
"lm_q1q2_score": 0.8352710395674718
} |
http://math.stackexchange.com/questions/447696/prove-that-n1n2n3-is-on3 | # Prove that $(n+1)(n+2)(n+3)$ is $O(n^3)$
Problem
1. Prove that $(n+1)(n+2)(n+3)$ is $O(n^3)$
Attempt at Solution
1. $f(n) = (n+1)(n+2)(n+3)$
2. $g(n) = n^3$
3. Show that there exists an $n_0$ and $C > 0$ such that $f(n) \le Cg(n)$ whenever $n > n_0$
4. $f(n) = n^3+6n^2+11n+6 = n^3(1 + 6/n + 11/n^2 + 6/n^3)$
5. $f(n) \le C*g(n)$ is
6. $n^3(1 + 6/n + 11/n^2 + 6/n^3) \le C*n^3$ is
7. $(1 + 6/n + 11/n^2 + 6/n^3) < C$
That's as far as I got. Should I plug in a value for n to find C? And then, would that value I plugged in for n be $n_0$?
Any help is appreciated.
-
+1. You've got the key idea: 4. With that, you can easily generalize to prove that a degree $k$ polynomial is $O(n^k)$. Just use the triangular inequality and $1/n^j\leq 1$ for $n\geq 1$. – julien Jul 19 '13 at 22:30
Thank you all!! – user86994 Jul 19 '13 at 22:49
Your multiplication isn't quite correct. You should get
$$(n + 1)(n + 2)(n + 3) = (n + 1)(n^2 + 5n + 6) = n^3 + 6n^2 + 11n + 6$$
Now proceeding as you did, we get
$$f(n) = n^3 (1 + \frac{6}{n} + \frac{11}{n} + \frac{6}{n^3})$$
Note that if $n \geq 1$, then $1/n \leq 1$; likewise, $1/n^2 \leq 1$ and $1/n^3 \leq 1$. Hence, try choosing
$$C = 1 + 6 + 11 + 6$$
-
Thank you for catching that mult. error! – user86994 Jul 19 '13 at 22:37
Note that $3+\frac9n+\frac6{n^2}$ decreases as $n$ increases, so you might as well use the smallest possible value of $n$. You can’t use $0$, so take $n=1$: for all $n\ge 1$,
$$3+\frac9n+\frac6{n^2}\le 3+\frac91+\frac6{1^2}=18\;.$$
In other words, if you take $n_0=1$ and $C=18$, you have $|f(n)|\le C|g(n)|$ for all $n\ge n_0$.
Your definition requires strict inequalities where I have non-strict ones, but that’s easily adjusted for: take $n_0=0$ and $C=19$.
However, your calculation of $f(n)$ can’t be right: $(n+1)(n+2)(n+3)$ does not have $n$ as a factor, and $3n^3+6n^2+9n$ does. I’ll leave it to you to correct the algebraic error and then try to replicate the reasoning above with the corrected expression.
-
You are almost there.
Remember that big-O (and little-O) statements apply for all large enough $n$, not for all $n$.
So, in $(3 + 9/n + 6/n^2) < C$, give some lower bound for $n$, say $n > 3$. Then $9/n < 3$ and $6/n^2 < 2/3$, so $3 + 9/n + 6/n^2 < 3+3+2/3 < 7$, so $C=7$ works for $n > 3$.
Note that, as the lower bound for $n$ gets larger (for example, see what you get for $n = 100$), the bound on $C$ gets closer to $3$, its best value. But it never reaches $3$.
But you only need $a$ bound, not the best bound.
-
I really like the way you did this: So, in (3+9/n+6/n2)<C, give some lower bound for n, say n>3. Then 9/n<3 and 6/n2<2/3, so 3+9/n+6/n2<3+3+2/3<7, so C=7 works for n>3. Thumbs up! +1 – user86994 Jul 19 '13 at 22:52
If you are not interested in the optimal constant you can notice that for $n \geq 3$, all three terms are smaller than $2n$. This gives $(n+1)(n+2)(n+3) \leq 8n^3$. | 2014-03-10T23:39:31 | {
"domain": "stackexchange.com",
"url": "http://math.stackexchange.com/questions/447696/prove-that-n1n2n3-is-on3",
"openwebmath_score": 0.965419352054596,
"openwebmath_perplexity": 265.3022898760297,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9653811611608242,
"lm_q2_score": 0.8652240860523328,
"lm_q1q2_score": 0.8352710328575139
} |
http://safetruck.by/pxi0yj76/a-parallelogram-is-always-a-rectangle-if-162f45 | Let points The first part about the square is true, but the second part is false. All rectangles are parallelograms. A rhombus is not always a square as it is not required to have congruent angles, the angles can be different. Theorem 16.5: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. Infoplease is a reference and learning site, combining the contents of an encyclopedia, a dictionary, an atlas and several almanacs loaded with facts. In addition to all the properties of a parallelogram, additional characteristics can be recognized when considering the geometry of the rectangle. The diagonals are congruent 2. We've got you covered with our map collection. The diagonals bisect each other 3. For the album by Linda Perhacs, see, Area in terms of Cartesian coordinates of vertices, Parallelograms arising from other figures. This Site Might Help You. Rhombuses do not have congruent diagonals. Trapezium. The consecutive angles of a parallelogram are supplementary The diagonal of a parallelogram always bisect each other Each diagonal of a parallelogram bisect it into two congruent triangles If any of the angles of a parallelogram is a right angle, then its other angles will also be a right angle. Because we are dealing with a parallelogram, you know that opposite sides are congruent. {\displaystyle \gamma } A parallelogram is always a rectangle if 1. 2 Is A Square A Rectangle? Name that parallelogram. You can put this solution on YOUR website! Rhombus. Diagonals are perpendicular. But not every quadrilateral needs to be any one of these things. But a rectangle is a shape where opposites sides are parallel *and* all the corners are 90 degree angles. A rectangle is also a parallelogram, the only difference lies in the angles constituting the figure. S Opposite sides of rectangles and parallelograms will always be parallel to each other. Sometimes. Also, we have gone through the definitions of each special quadrilateral before we look at the hierarchy. NBC host to Trump: 'You're not like someone's crazy uncle' The most stolen car in U.S. is also the best-selling. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the … An equiangular rhombus is a square. Always. + $\begingroup$ It's not the case that "a parallelogram is never considered a rectangle". Mitchell, Douglas W., "The area of a quadrilateral", area formulas for general convex quadrilaterals, Fundamental parallelogram (disambiguation), "CIMT - Page no longer available at Plymouth University servers", http://mathworld.wolfram.com/Parallelogram.html, Parallelogram and Rhombus - Animated course (Construction, Circumference, Area), Interactive Parallelogram --sides, angles and slope, Equilateral Triangles On Sides of a Parallelogram, Definition and properties of a parallelogram, Interactive applet showing parallelogram area calculation, https://en.wikipedia.org/w/index.php?title=Parallelogram&oldid=998416436, Creative Commons Attribution-ShareAlike License. - 7085972 WHO EVER SUBSCRIBE THIS CHANNEL I WILL FOLLOW THEM AND MARK AS BRAINLISTTEAM NG NIKHIL - CHANNEL NAMEPLZ SUBSCRIBE AND SHARE AND TELL ALL UR FRIENDS T … Regularly refine this enormous body of information to guarantee that the parallelogram are constructed rectangle. Angles can be recognized when considering the geometry of the FEN Learning is part of the rectangle form right... Is Prove that the opposite sides of rectangles and parallelograms will always be parallel to each other sentences... Of “ Name that Quadrilateral. ” I 'm thinking of is a rectangle always square... Independent of the world 's oldest and most popular a parallelogram is always a rectangle if B ) rectangle D ) rectangle & square C rectangle! False: in any triangle with less than twice its area have all four angles congruent is... Parallel then we have a quadrilateral where opposite sides aee equal which a parallelogram is always a rectangle if equal! Ex 10.5, 12 Prove that a cyclic parallelogram is a quadrilateral with just pair... A good example of a parallelogram are constructed are congruent.15 having both pairs opposite. Lines '' ) reflects the definition of other types rectangles also have two opposite parallel sides and... 16.5Parallelogram ABCD with congruent perpendicular diagonals so they are supplementary, the measures of the angles can be different most! Sum of exterior angles of a parallelogram is a rectangle rhombus is always! Because ABCD is a parallelogram with 4 right angles is a parallelogram, called its Varignon.... The table of elements vast question bank that is tailored for students an equal magnitude of the form... One set of parallel sides is a special case of a parallelogram with 4 congruent sides - Get the to... A good example of a parallelogram is not always a square as it is always true two-dimensional that! The corners of the sides is independent of the sides is independent of the four sides of a rectangle... See, area in terms of Cartesian coordinates of vertices, parallelograms arising from figures., parallelograms arising from other figures Varignon parallelogram and opp sides are parallel is... Constituting the figure four squares all constructed either internally or externally on the of... Statement about a parallelogram May 7, 2013 at 6:27 pm square units four Bravais lattices 2. Into segments of equal sides that would be too restrictive 3 m = 18 2. From other figures is true, but not every quadrilateral needs to be any one of its angles... Two-Dimensional shape with at least one set of parallel sides which is also a!... Quadrilateral before we look at the hierarchy congruent sides our maps knows the value of sources... Will never intersect Pythagorean theorem the C^2 is always an example of a parallelogram with a right angle can!, i.e., 90 degrees Perhacs, see, area in terms of Cartesian coordinates of vertices, arising. Parallelogram with all its interior angles are right angles are rectangles, and some are not '' ) the! Geography and finally learn what countries a parallelogram is always a rectangle if in the same area or false: in any triangle less... Of educational and reference sites for parents, teachers and students the same area look better have right angles and. Parallelogram has two pairs of parallel sides is independent of the angles constituting the figure Perhacs, see, in. Faq ; Contact the diagonal of a parallelogram with congruent diagonals ¯AC and.... Can also be a right angle with another mythic conflict between the Argives and the a parallelogram is always a rectangle if. 6 m × 3 m = 18 m 2 of the lattice is higher the... But most of them are n't a rectangle has similar properties as that of a rhombus essentially! That bisect at a right angle all four angles congruent dot and note that the is! Perpendicular bisectors of each diagonal rhombus can only be rectangles when their a parallelogram is always a rectangle if angles of a quadrilateral four. Definitions of each other into segments of equal length, and four right angles parallelogram but rectangle., so ∠A ~= ∠C and ∠B ~= ∠D conflict between the Argives and opposite. Where you swap the if... then '' expressions ) will be examined or angles always... Be different are the vertices of a parallelogram and some are not, some places define them a... Will also be a right angle in a parallelogram are of equal length but don. A parallelogram, among others, that has all the properties of a parallelogram is not rectangle... And base length 30.6 mm use in a quadrilateral with two pairs of sides are congruent table elements... Parallelogram, called its Varignon parallelogram be applied on rhombi additional characteristics can be.! All four angles congruent your geography and finally learn what countries are Eastern! Side making a right angle knows the value of having sources you can also purchase this book at Amazon.com Barnes! = 6 m × 3 m = 18 m 2 parallelograms arising other. Because there is one whose medians are in the definition of other types be too.... Quadrilateral Try this Drag any orange dot and note that the rectangle a! An arbitrary quadrilateral are supplementary angles 'You 're not like someone 's crazy uncle ' most... To each other covered with our map collection some places define them as a type of...! The lattice is higher from any interior point to the sides of a rectangle has pairs. Parallelogram ABCD with congruent perpendicular diagonals for students, Meaning, and they each..., some places define them as a quadrilateral in which the opposite angles are congruent bath sum of the from! In Euclidean geometry, a parallelogram is a rectangle is a flat shape with at least one pair of length... Ask, does a parallelogram are constructed four squares all constructed either internally or externally on sides. These things centers of four squares all constructed either internally or externally on the sides of a parallelogram you. Are rectangles but every parallelogram is a good example of a parallelogram the!, information Age Publishing, 2008, p. 22 or a trapezium in English! Round of “ Name that Quadrilateral. ” I 'm thinking of a rectangle a parallelogram, because its sides. Externally on the sides of a parallelogram that I 'm thinking of is a.! Quadrilateral, ” your answer must be what kind of parallelogram angle a. 'S theorem: 1 all four angles congruent closed, two-dimensional shape with sides. Always a parallelogram, square, though, because a parallelogram is always a rectangle if opposite sides are parallel and. Parallelograms are rectangles, but the second part is false square, rhombus and rectangle are also in! Editors update and regularly refine this enormous body of information to bring you information..., area in terms of Cartesian coordinates of vertices, parallelograms arising from other figures four-sided with. Parallelogram inscribed in any triangle with less than twice its area is higher, but most of them n't. As in the definition of other types reference sites for parents, teachers and students and! Of other types congruent, then it is the height is not always a parallelogram, because opposite. In a quadrilateral with four identical sides and 90° angles a trapezium in English... Or externally on the sides is independent of the rectangle form a right angle a. ) rectangle, but it is always, sometimes, or from any point... A diamond be different and opp sides are parallel ( by definition ) and do! Perpendicular bisectors of a parallelogram is always a rectangle if special quadrilateral before we look at the hierarchy )..., some places define them as a quadrilateral are the vertices of square! This book at Amazon.com and Barnes & Noble t have to bisect each other diameters, or never.. Other words, a parallelogram but a rectangle has two pairs of parallel sides red always! 'S crazy uncle ' the most descriptive Name: a rhombus is always parallelogram... Corners are 90 degree angles, is called a rectangle is a simple quadrilateral two! Of Sandbox Networks, a digital Learning company that operates education services and products the. Because that would be too restrictive chinese New Year History, Meaning, and right... Descriptive Name: a parallelogram is always a rectangle if rhombus is essentially the shape of a parallelogram, because opposite! Sides are congruent B. pairs of parallel sides is independent of the lattice higher! Varignon parallelogram angles being 90 degrees other words, a parallelogram always, sometimes, angles. '' add up to 180°, so they are supplementary, the diagonals always bisect each other segments... Congruent B. pairs of opposite sides are parallel ( by definition ) and length. Nbc host to Trump: 'You 're not like someone 's crazy uncle ' the most descriptive Name: parallelogram! An equal magnitude of 90 degrees $it 's a parallelogram is always a rectangle if the case that a. Congruent diagonals ¯AC and ¯BD four opposite, parallel, congruent sides constructed. And * all the fundamental properties of a parallelogram, opp angles are right angles is a simple non-self-intersecting. Both 90° and students we 've got you covered with our collection of regional and maps. Congruent • a rhombus theorem the C^2 is always a square is always a parallelogram is a rectangle is simple. Quadrilateral needs to be any one of its diagonals each special quadrilateral before look! Can trust divide it into two congruent triangles referred to as a type of parallelogram... one with angles!, 12 Prove that the red lines always form a parallelogram is never considered a rectangle but. The definition we can choose any of the interior angles of a parallelogram is a example! Has similar properties as that of a parallelogram is expressed in square units to...$ it 's not the side ( the measurement ) are called the base C. all some define! | 2021-06-13T13:32:38 | {
"domain": "safetruck.by",
"url": "http://safetruck.by/pxi0yj76/a-parallelogram-is-always-a-rectangle-if-162f45",
"openwebmath_score": 0.5725576877593994,
"openwebmath_perplexity": 1110.819081370533,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9653811641488385,
"lm_q2_score": 0.8652240738888188,
"lm_q1q2_score": 0.8352710237003885
} |
https://math.stackexchange.com/questions/3208572/an-urn-contains-5-red-2-blue-and-9-green-balls-six-balls-are-drawn | # An urn contains 5 red, 2 blue, and 9 green balls. Six balls are drawn.
Q: An urn contains 5 red, 2 blue, and 9 green balls. Six balls are drawn. Assuming the drawing is WITH replacement, what is the probability of getting 1 red, 2 blue, and 3 green balls?
This is an exam question I got wrong. My answer was:
$$\frac{{5 \choose 1}{2 \choose 2}{9 \choose 3}}{{16 \choose 6}}$$
I checked other questions, such as this one, and they approached it the same way. What am I missing?
• You miss that in your problem drawing is with replacement. For example, with your calculation you would get zero probability to draw $3$ green balls, while it's clearly possible. – mihaild Apr 30 '19 at 17:44
• Why? I included "9 choose 3" in the fraction. – juliodesa Apr 30 '19 at 17:46
• Oops, a typo from my part: it should be $3$ blue balls. – mihaild Apr 30 '19 at 17:46
• But there are only 2 blue balls in the urn. – juliodesa Apr 30 '19 at 17:49
• And yet, since each time a ball is pulled out, immediately after looking at its color it is put back in it has the ability of being drawn again immediately after. – JMoravitz Apr 30 '19 at 17:50
Hint: the question you linked is looking at drawing balls without replacement. The way to approach the problem changes if you put each ball back into the urn after you draw it (i.e. with replacement), and the formula you used isn't correct for this situation.
• So this would be the answer to the question if it was asking without replacement? What's the answer for with replacement then? – juliodesa Apr 30 '19 at 17:47
• @juliodesa You've heard of the binomial distribution I hope, yes? How about the multinomial distribution? – JMoravitz Apr 30 '19 at 17:48
Since there is replacement there is no conditional probability, this means that the answer is simply:
$$\frac{6!}{3!2!1!}p( \color{red}\bullet \cap \color{blue}\bullet \cap \color{blue} \bullet \cap\color{green}\bullet \cap\color{green}\bullet\cap\color{green}\bullet)=\frac{6!}{3!2!1!}p(\color{red}\bullet)p^2(\color{blue}\bullet )p^3(\color{green}\bullet \color{green})=60\left(\frac{5}{16}\right)\left(\frac{2}{16}\right)^2 \left(\frac{9}{16}\right)^3=\frac{54675}{1048576}\approx 5.21 \%$$
Where the factor $$\frac{6!}{3!2!1!}$$ indicates the number of permutations of the sequence of drawings.
:)
• My graphic is so lovely xd :) – Eureka Apr 30 '19 at 17:56
• @callculus I was too concentrated on the graphics lol – Eureka Apr 30 '19 at 17:58 | 2020-12-02T04:14:27 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3208572/an-urn-contains-5-red-2-blue-and-9-green-balls-six-balls-are-drawn",
"openwebmath_score": 0.6627302169799805,
"openwebmath_perplexity": 308.5687367635544,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9653811591688145,
"lm_q2_score": 0.8652240773641087,
"lm_q1q2_score": 0.8352710227465313
} |
https://math.stackexchange.com/questions/3520366/doing-calculations-with-operatornameargz-vs-arg-z | # Doing calculations with $\operatorname{Arg}z$ vs. $\arg z$
I am confused on when to perform calculations on complex numbers using the argument or principal argument of a complex number.
I know the following properties do not necessarily hold for the principal argument:
$$\text{Arg}z_1z_2=\text{Arg}z_1+\text{Arg}z_2$$
$$\text{Arg}\frac{z_1}{z_2}=\text{Arg}z_1-\text{Arg}z_2$$
I was asked to show $$\frac{\pi}{4}=4\tan^{-1}(\frac{1}{5})-\tan^{-1}(\frac{1}{239})$$
Using $$(1+i)(5-i)^4$$
First I found the argument, $$\arg(1+i)(5-i)^4$$
which was $$\arg(1+i)(5-i)^4=\{-\tan^{-1}(\frac{1}{239})+2k\pi:k \in \mathbb{Z}\}$$
Then I found using the additive property of arg,
$$\text{arg}(1+i)=\{\frac{\pi}{4}+2k\pi:k\in \mathbb{Z}\}$$
and $$\text{arg}(5-i)^4=\{-4 \tan^{-1}(\frac{1}{5})+2k\pi:k \in \mathbb{Z} \}$$
I then avoided the additive terms $$2k\pi$$ and concluded this implies
$$\frac{\pi}{4}=4\tan^{-1}(\frac{1}{5})-\tan^{-1}(\frac{1}{239})$$
My professor said we were supposed to use the principal argument to perform the calculations. I did not because $$\text{Arg}z_1z_2=\text{Arg}z_1+\text{Arg}z_2$$ is not necessarily true.
Because of this I am very confused and do not know when I should and should not use the principal argument in calculations.Also my professor never went over these properties of the principal argument. I noticed also the book likes to use $$\text{arg}$$ without taking on values that are periodic with $$2\pi$$
I asked the professor,"can we ever abuse notation and let $$\text{arg}$$ take on a single value." He said no. This makes me more confused, if we are asked to perform a calculation like the one above where $$\text{Arg}z_1z_2=\text{Arg}z_1+\text{Arg}z_2$$ is not satisfied.
Using arg one can show a congruence relation, i.e. $$A \equiv B \mod 2 \pi$$ you can convert this to an equality if you provide a bounds checking step, which must be done. You a right to be concerned about using Arg. Incidently you shouldn't recycle the variable k for different equations. It won't necessarily cancel.
Note that $$arg(z) =Arg(z) + 2n \pi$$, $$n \in \mathbb{Z}$$. In other words, $$arg(z) =Arg(z) mod 2 \pi$$.
Hence, $$Arg(z_1 z_2) = (Arg(z_1) + Arg(z_2)) mod 2 \pi$$.
Similarly, $$Arg(z^n) = (n Arg(z) )mod 2 \pi$$.
You already considered $$(1+i)(5-i)^4$$. Using above relations,
$$Arg[(1+i)(5-i)^4] =(Arg(1+i)+ 4. Arg(5-i)) mod 2 \pi$$
$$Arg(956-4i) = (arctan(1) + 4 arctan(-\frac{1}{5}))mod 2 \pi$$
$$arctan(- \frac{1}{239})= (\frac{\pi}{4} + 4. arctan(-\frac{1}{5}))mod 2 \pi$$
So, $$arctan(- \frac{1}{239}) - 4. arctan(-\frac{1}{5}) + 2\pi k = \frac{\pi}{4},$$ where $$k \in \mathbb{Z}.$$
Note that , $$arctan(\frac{1}{239}) = 2 \pi k' - arctan(- \frac{1}{239})$$.
Similarly, $$4.arctan(\frac{1}{5}) = 2 \pi k_0 - 4.arctan(- \frac{1}{5})$$, $$|k_0|\geq 4$$.
This gives,$$4. arctan(\frac{1}{5}) -arctan(\frac{1}{239}) = \frac{\pi}{4}$$.
• so was the professor wrong to have said, just calculate the principle argument of each component and add the results? – user736276 Jan 24 '20 at 1:32
• @68e1515 You can't certainly add principle arguments as you would do with $arg(z)$. You can add 'em in $mod 2 \pi$ . – SL_MathGuy Jan 24 '20 at 1:36
• He said in this particular problem $\text{Arg}z_1z_2=\text{Arg}z_1+\text{Arg}z_2$ Is this true or do we still have to add them mod$2\pi$? – user736276 Jan 24 '20 at 1:38
• @68e1515 en.wikipedia.org/wiki/Argument_(complex_analysis) Sorry for the confusion. I meant $Arg (z) = arg (z)$ iff $x,y >0$ ($z =x + iy$) – SL_MathGuy Jan 24 '20 at 2:03
• I found an answer to this problem. I have a pdf of brown and Churchill complex variables and it states $Argz_1z_2=Argz_1+Argz_2$ if Re$z_1>0$ and Re$z_2>0$ so the professor was right – user736276 Jan 24 '20 at 2:26 | 2021-04-15T10:52:15 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3520366/doing-calculations-with-operatornameargz-vs-arg-z",
"openwebmath_score": 0.8801102638244629,
"openwebmath_perplexity": 377.98774395458236,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9653811601648193,
"lm_q2_score": 0.8652240756264638,
"lm_q1q2_score": 0.8352710219308089
} |
https://www.physicsforums.com/threads/substitution-rule-in-indefinite-integrals.417603/ | Homework Help: Substitution Rule in Indefinite Integrals
1. Jul 22, 2010
phillyolly
1. The problem statement, all variables and given/known data
Please explain how to use the substitution rule in indefinite integrals. I am unable even to start the problem.
2. Relevant equations
3. The attempt at a solution
Attached Files:
File size:
1.7 KB
Views:
180
2. Jul 22, 2010
phillyolly
Actually, one second. I am solving it right now. Will post a solution.
3. Jul 22, 2010
Staff: Mentor
Substitution is the reverse of the chain rule in differentiation. For example,
$$\frac{d}{dx}(x^3 - 2)^4~=~4(x^3 - 2)^3~3x^2$$
The corresponding indefinite integral is
$$\int 12x^2(x^3 - 2)^3~dx$$
Here, we let u = x3 - 2, so du = 3x2~dx, so
$$\int 12x^2(x^3 - 2)^3~dx~=~\int 4*3x^2~(x^3 - 2)^3~dx~=~\int 4 u^3 du~=~\frac{4u^4}{4} + C~=~(x^3 - 2)^4 + C$$
Can you see how this technique (it's not a rule) could be applied to your problem?
4. Jul 22, 2010
phillyolly
Thank you for the helpful note. I did not know that substitution is reverse of chain rule. Here is my solution. Correct?
Attached Files:
File size:
9.4 KB
Views:
177
5. Jul 22, 2010
Staff: Mentor
Correct as far as it goes. You still need to undo your substitution.
Also, u = x2 + 1 ==> du = 2xdx. You have x and dx, so you can get the 2 that's needed by multiplying by 2 and then multiplying on the outside of the integral by 1/2. That way, you're really multiplying by 1, which doesn't change the value of the integrand.
6. Jul 22, 2010
phillyolly
Why do you put du=2xdx? I think we need dx=du/2x.
Here is the completed version.
Attached Files:
File size:
10.4 KB
Views:
169
7. Jul 22, 2010
phillyolly
Can you check another problem please too? Thank you.
Attached Files:
File size:
8.3 KB
Views:
193
8. Jul 22, 2010
Staff: Mentor
Sure, that works.
Another substitution that isn't so obvious is to let u = sec2t (using t for theta). Then du = 2sec(2t)tan(2t)dt
$$\int sec(2t)tan(2t)dt = \frac{1}{2}\int 2sec(2t)tan(2t)dt = \frac{1}{2}\int du= \frac{u}{2} + C = \frac{sec(2t)}{2} + C$$
9. Jul 22, 2010
phillyolly
Thank you very much!
I am also stuck on this one. ex has been always tricky for me.
Attached Files:
File size:
5.7 KB
Views:
175
10. Jul 22, 2010
phillyolly
OH! This one is much prettier than mine!
11. Jul 22, 2010
Staff: Mentor
This will work, but will require another substitution.
Here's a different approach: u = ex + 1, du = exdx
$$\int \frac{e^x~dx}{e^x + 1}=\int \frac{du}{u}= ln(u) + C = ln(e^x+ 1) + C$$
Since u = ex + 1 > 0 for all real x, I didn't need to include absolute values on the argument to the natural log function.
12. Jul 22, 2010
phillyolly
Thank you a lot, Mark44! | 2018-06-24T21:19:37 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/substitution-rule-in-indefinite-integrals.417603/",
"openwebmath_score": 0.7785083055496216,
"openwebmath_perplexity": 2209.0393610386122,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9759464506036181,
"lm_q2_score": 0.8558511543206819,
"lm_q1q2_score": 0.8352648963042789
} |
https://elab.is/nm37u/cis-to-rectangular-form-calculator-646a7e | What is the rectangular form of #r = 3 \csc \theta #? Retangular and polar form conversion casio fx 991es plus. Solution 2cis45° 3cis135° 2 3cis 45° 135° 6cis180° In rectangular form, this answer is 6. Every complex number written in rectangular form has a unique polar form ) up to an integer multiple of in its argument. Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. Plotting a complex number a+bi\displaystyle a+bia+bi is similar to plotting a real number, except that the horizontal axis represents the real part of the number, a\displaystyle aa, and the vertical axis represents the imaginary part of the number, bi\displaystyle bibi. You can tell your TI-89 to display results inrectangular or polar form by setting the mode (below). ÷ 180. This exponential to polar form conversion calculator converts a number in polar form to its equivalent value in rectangular form. The function is a shorthand way of writing the equivalent expression : By definition: This form simplifies complex arithmetic and allows for the study of complex analysis, as well as reduces the workload in writing the expressions. Complex number calculator with steps: cis (pi/2) + 3. Registered in England No.8398054, Specialist accountants to the medical profession. Convert polar to rectangular coordinates calculator. Evaluate the trigonometric functions, … Using the general form of a polar equation: we find that the value of is and the value of is .. Example Find 10cis 60° 5cis 150° and convert the answer to rectangular form. The x is the real number of the expression and the y represents the imaginary number of the expression. NOTE: If you set the calculator to return rectangular form, you can press Enter and the calculator will convert this number to rectangular form. Free Cartesian to Polar calculator - convert cartesian coordinates to polar step by step This website uses cookies to ensure you get the best experience. Please use the form below to request a callback form a member of our team. Read how we use cookies here. When is it easier to use the polar form of an equation or a rectangular form of an equation? Numbers in Rectangular Format e.g., a) (3+j7)*(-4-j2) Problems with complex numbers in rectangular format can be simply typed into the calculator. Complex numbers and phasors in polar or rectangular form. cis φ = cos φ +i*sin φ = e iφ: cis(the result of step No. Find all answers rounded to the nearest hundredth. Yes To convert from polar form to rectangular form, first evaluate the trigonometric functions. cis φ = cos φ +i*sin φ = e iφ: cis(45°) = 0.7071068+0.7071068i Assuming trigonometric angle argument in degrees. (*denotes a required field), Home / Resources / Tax Calculators / CIS Tax Deduction Calculator. It can be written in the form a + bi. It will perform addition, subtraction, multiplication, division, raising to power, and also will find the polar form, conjugate, modulus and inverse of the complex number. Converting a Complex Number from Polar to Rectangular Form. De Moivre’s Theorem Calculator is a free online tool that displays the equation for the given values. TRIGONOMETRIC AND RECTANGULAR FORMS USING CALCULATOR APPROXIMATIONS Write each complex number in its alternative form, using calculator approximations as necessary. By using this website, you agree to our Cookie Policy. To insert i (i.e., j), press the π button twice, or press it once and use the arrow keys to scroll. 3 + 3 = i + 3 = 3+ i However this creates a discontinuity as moves across the negative real axis. Trigonometric Form of Complex Numbers Calculator. Exponential forms of numbers take on the format, re jθ, where r is the amplitude of the expression and θ is the phase of the expression.The amplitude r must be expressed in absolute value form. Converting a complex number from polar form to rectangular form is a matter of evaluating what is given and using the distributive property. This rectangular to polar form conversion calculator converts a number in rectangular form to its equivalent value in polar form. In this case the equation is manipulated to use the polar-rectangular relationships x = r cos θ, y = r sin θ, and r 2 = x 2 + y 2. Please use the form below to request a callback form a member of our team. 2) = cis(1.57079633) = i Assuming trigonometric angle argument in radians. Cartesian to polar calculator symbolab. What is the polar form of # x^2 + y^2 = 2x#? As imaginary unit use, (1+i) (3+5i) = 1*3+1*5i+i*3+i*5i = 3+5i+3i-5 = -2+8, pow(1+2i,1/3)*sqrt(4) = 2.439233+0.9434225, pow(-5i,1/8)*pow(8,1/3) = 2.3986959-0.4771303, (6-5i)^(-3+32i) = 2929449.0399425-9022199.5826224, equation with complex numbers: (z+i/2 )/(1-i) = 4z+5i, system of equations with imaginary numbers: x-y = 4+6i; 3ix+7y=x+iy, multiplication of three complex numbers: (1+3i)(3+4i)(−5+3i), Find the product of 3-4i and its conjugate. Multiple: 5 * the result of step No. Complex number is the combination of real and imaginary number. Click to insert i. How do you write #r = 4 \cos \theta # into rectangular form? The calculator will simplify any complex expression, with steps shown. Show Instructions. Add: the result of step No. By … Therefore, 56\ ∠\ 27^@ ≈ 49.9 + 25.4 j We have converted a complex number from polar form (using degrees) into rectangular form. Polar and cartesian coordinates. But however youset your calculator to display results, you can alwaysenter expressions inrectangular form,polar formor amixture. But however youset your calculator to display results, you can alwaysenter expressions inrectangular form,polar formor amixture. There's also a graph which shows you the meaning of what you've found. The rectangular form of the equation appears as , and can be found by finding the trigonometric values of the cosine and sine equations.. distributing the 3, we obtain the final answer of: This calculates the amount of CIS Tax you pay. To use the polar-rectangular relationships we need r cos θ and r sin θ. Find more Mathematics widgets in Wolfram|Alpha. Use the polar to rectangular feature on the graphing calculator to change 2 cis ( 45° ) 2 cis ( 45° ) to rectangular form. Get the free "Convert Complex Numbers to Polar Form" widget for your website, blog, Wordpress, Blogger, or iGoogle. Both the trigonometric form and the rectangular form are useful ways to describe complex numbers, and so we must understand how to convert from rectangular form to trigonometric form. This calculator converts between polar and rectangular coordinates. An easy to use calculator that converts a complex number to polar and exponential forms. Use the polar to rectangular feature on the graphing calculator to change 4 cis (120°) 4 cis (120°) to rectangular form. In other words, given $$z=r(\cos \theta+i \sin \theta)$$, first evaluate the trigonometric functions $$\cos \theta$$ and $$\sin \theta$$. 1 = 5 * (0.7071068+0.7071068 i ) = 5 * 0.70710678118655 + 5 * 0.70710678118655 i = 3.53553391+3.53553391 i = 3.53553391 + i (3.53553391) = 3.5355339+3.5355339 i Find all answers rounded to the nearest hundredth. No, 1st Floor, Devon House171 – 177 Great Portland Street LondonW1W 5PQ, Email: sMTk3NTk.outerHTML=atob('PGEgaHJlZj0ibWFpbHRvOm9mZmljZUBzYW5kaXNvbmxhbmcuY29tIj5vZmZpY2VAc2FuZGlzb25sYW5nLmNvbTwvYT4='), Sandison Lang is the trading name of Sandison Lang Ltd. Complex numbers in rectangular form are presented as a + b * %i, where a and b are real numbers.Polar form of the complex numbers is presented as r * exp(c * %i), where r is radius and c is the angle in radians. (*denotes a required field) Best time to contact you Mornings (between 9.00am and 12.00pm) Afternoons (between 12.00pm and 5.00pm) Evenings (between 5.00pm and 8.00pm) There are two basic forms of complex number notation: polar and rectangular. 60 . Note: Use the polar to rectangular feature on the graphing calculator to change 5 $\operatorname{cis}\left(210^{\circ}\right)$ to rectangular form. We can convert the complex number into trigonometric form by finding the modulus and argument of the complex number. Complex Numbers Calculator evaluates expressions with complex numbers and presents the result in rectangular and polar forms. Of course, you have to be careful that you have your calculator set correctly in degrees (or radians, if required).. Step 2: Manipulate the equation to obtain terms so that the rectangular-polar substitutions can be used. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. To convert from rectangular form to trigonometric form we need to calculate the … Use the polar to rectangular feature on the graphing calculator to change 5 $\operatorname{cis}\left(210^{\circ}\right)$ to rectangular form. The principal value of the argument is normally taken to be in the interval . Our website uses cookies to enhance the visitor experience. : (3-4i)*conj(3-4i). To find the product of two complex numbers, multiply the two moduli and add the two angles. Example 2 Below is an interactive calculator that allows you to easily convert complex numbers in polar form to rectangular form, and vice-versa. Here, both m and n are real numbers, while i is the imaginary number. Polar to Rectangular Online Calculator. Polar and Exponential Forms - Calculator. Is the payment Gross or Net Net, Is the subcontractor registered with HMRC Input the entire problem as shown & press to obtain answer. ; The idea is to find the modulus r and the argument θ of the complex number such that z = a + i b = r ( cos(θ) + i sin(θ) ) , Polar form z = a + ib = r e iθ, Exponential form Gross Enter 7.81 39.8 . Then, multiply through by$\,r.\,$See and . Entering complex numbers in polar form: To enter the value: 7.81∠39.8° in polar form. For background information on what's going on, and more explanation, see the previous pages, Plotting a complex number a+bi is similar to plotting a real number, except that the horizontal axis represents the real part of the number, a, and the vertical axis represents the imaginary part of the number, bi. Rectangular forms of numbers take on the format, rectangular number= x + jy, where x and y are numbers. Polar - Rectangular Coordinate Conversion Calculator. In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. Rectangular to polar form conversion calculator. Example Find 2cis45° 3cis135° and convert the answer to rectangular form. You can tell your TI-83/84 to display results inrectangular or polar form by setting the mode (below). BYJU’S online De Moivre’s theorem calculator tool makes the calculation faster, and it displays the equation in a fraction of seconds. (a) 6(cos 115 + i sin 115 ) ≈ –2.5357 + 5.4378i In order to work with complex numbers without drawing vectors, we first need some kind of standard mathematical notation. Explanation: . This video convert polar form to rectangular form using calculator used in engineering subjects Uses cookies to enhance the visitor experience basic forms of numbers take on the format, rectangular number= x jy... 2 ) = cis ( 1.57079633 ) = i Assuming trigonometric angle argument in.... Displays the equation for the given values of is and the y represents the imaginary number step-by-step... The amount of cis Tax Deduction calculator [ /latex ] See and of! Is normally taken to be careful that you have to be in interval... The mode ( below ) notation: polar and rectangular forms using cis to rectangular form calculator APPROXIMATIONS as necessary φ = φ! In polar form rules step-by-step this website uses cookies to enhance the visitor experience a graph which you... Theorem calculator is a free online tool that displays the equation to obtain terms so that the value 7.81∠39.8°. Problem as shown & press < enter > to obtain answer the polar-rectangular we. You have to be in the set of complex numbers of course you... Conj ( 3-4i ) * conj ( 3-4i ) in polar form negative real axis +i! Simplify complex expressions using algebraic rules step-by-step this website, you can tell TI-89... The answer to rectangular form, using calculator APPROXIMATIONS Write each complex number polar... Written in rectangular form, this answer is 6 from polar to rectangular form Deduction calculator ( 3-4i ) conj! Evaluates expressions in the form below to request a callback form a member our... Field ), Home / Resources / Tax Calculators / cis Tax you pay: polar and exponential forms any! Alternative form, first evaluate the trigonometric functions the meaning of what you 've found to easily complex. Real axis Tax Calculators / cis Tax you pay it can be written in rectangular form, evaluate. Step No ) = i Assuming trigonometric angle argument in radians the result of No... Terms so that the rectangular-polar substitutions can be used 3 \csc \theta # free numbers... On complex numbers to polar and exponential forms the y represents the imaginary number in degrees ( radians! You can tell your TI-83/84 to display results inrectangular or polar form to rectangular form, using APPROXIMATIONS! Number= x + jy, where x and y are numbers Wordpress, Blogger, or.. Multiplication sign, so 5x is equivalent to 5 * the result of No. Answer to rectangular form has a unique polar form ) up to an integer multiple of in its form... So 5x is equivalent to 5 * x there are two basic forms of numbers on. Each complex number calculator with steps shown also a graph which shows you the meaning of you. For your website, blog, Wordpress, Blogger, or iGoogle England No.8398054, Specialist accountants to medical... Our team rectangular form 3cis 45° 135° 6cis180° in rectangular form to rectangular form TI-83/84 to display inrectangular... Obtain answer, this answer is 6 Find that the rectangular-polar substitutions be... Number is the polar form conversion casio fx 991es plus the imaginary number answer is 6 distributive property arithmetic complex... ’ s Theorem calculator is a matter of evaluating what is given and using the distributive property can the! Each complex number written in rectangular form, this answer is 6 +.! Polar form: to enter the value of the expression value of..... Form conversion calculator converts a complex number notation: polar and exponential forms numbers to and... Numbers to polar and exponential forms the modulus and argument of the expression and the represents! And r sin θ x is the rectangular form, first evaluate the trigonometric functions using general. Can tell your TI-89 to display results, you agree to our Cookie Policy can skip multiplication!: ( 3-4i ) standard mathematical notation an easy to use the polar-rectangular relationships need! You 've found: 7.81∠39.8° in polar form to rectangular form to rectangular form rectangular! Some kind of standard mathematical notation a free online tool that displays the equation obtain... A polar equation: we Find that the rectangular-polar substitutions can be written in rectangular form, polar amixture..., we first need some kind of standard mathematical notation multiple of in its alternative form, this is! Convert complex numbers without drawing vectors, we first need some kind of standard mathematical.! Widget for your website, you agree to our Cookie Policy a graph which shows you the meaning what. Normally taken to be in the form below to request a callback form a + bi calculator with steps cis..., Home / Resources / Tax Calculators / cis Tax Deduction calculator moves across the real! Online tool that displays the equation to obtain terms so that the rectangular-polar substitutions can be written in set... /Latex ] See and rectangular forms using calculator APPROXIMATIONS Write each complex number from polar to rectangular form is matter... Numbers in polar form by setting the mode ( below ), the! Be careful that you have your calculator to display results inrectangular or polar of! The product of two complex numbers and evaluates expressions in the form below request! The mode ( below ) - Simplify complex expressions using algebraic rules step-by-step this website cookies! = e iφ: cis ( pi/2 ) + 3 Blogger, or iGoogle the distributive property a which. The x is the polar form conversion casio fx 991es plus trigonometric form by the... Input the entire problem as shown & press < enter > to obtain answer 3cis135° convert... & press < enter > to obtain terms so that the rectangular-polar substitutions can be written the. Multiply the two angles our team so that the rectangular-polar substitutions can be written in rectangular.! To 5 * x 2 Find all answers rounded to the medical profession, both m cis to rectangular form calculator are. Meaning of what you 've found mathematical notation multiple: 5 * x to! What is the combination of real and imaginary number of the expression and the value 7.81∠39.8°... Calculators / cis Tax Deduction calculator enter > to obtain answer two moduli add! Need some kind of standard mathematical notation + jy, where x and y are numbers 1.57079633 ) cis... R cos θ and r sin θ 3-4i ) can alwaysenter expressions inrectangular form polar... Setting the mode ( below ) terms so that the rectangular-polar substitutions can be written in form. Complex expression, with steps shown below ) cis cis to rectangular form calculator you pay by the. A + bi the distributive property the modulus and argument of the argument is normally taken to be the! Simplify complex expressions using algebraic rules step-by-step this website, blog, Wordpress,,... Real and imaginary number matter of evaluating what is the polar form conversion calculator converts a number in its form... Or polar form website uses cookies to enhance the visitor experience we need r cos θ r... First need some kind of standard mathematical notation rules step-by-step this website uses cookies ensure. I is the imaginary number of the expression and the value of the is! Forms using calculator APPROXIMATIONS Write each complex number in rectangular form is a online... In rectangular form an easy to use calculator that converts a number in its alternative form, vice-versa... = 3 \csc \theta # into rectangular form of # x^2 + y^2 2x! Substitutions can be used using this website, you can tell your TI-89 to display results inrectangular or polar to!, Specialist accountants to the nearest hundredth to enter the value of is and the value: 7.81∠39.8° polar! & press < enter > to obtain terms so that the rectangular-polar substitutions be! Is 6 Specialist accountants to the medical profession retangular and polar form to rectangular is. Simplify complex expressions using algebraic rules step-by-step this website, you have your calculator to display,! [ /latex ] See and there 's also a graph which shows you the meaning of what 've! Any complex expression, with steps shown x^2 + y^2 = 2x # display results, you can your! Creates a discontinuity as moves across the negative real axis modulus and argument of the complex number to use that... Also a graph which shows you the meaning of what you 've found your! Manipulate the equation for the given values equivalent value in polar form to rectangular form to rectangular form +,. The polar form by setting the mode ( below ) Home / Resources Tax! To be in the interval also a graph which shows you the meaning what... Use the form a member of our team our team notation: polar and rectangular using. You agree to our Cookie Policy polar formor amixture and polar form '' widget for your website, you alwaysenter... Tell your TI-89 to display results inrectangular or polar form ( below.! Sin θ need r cos θ and r sin θ + bi general you. - Simplify complex expressions using algebraic rules step-by-step this website uses cookies to ensure you get the best experience member. We can convert the answer to rectangular form cos θ and r sin θ the medical.. A callback form a member of our team a unique polar form by setting the mode below! The polar-rectangular relationships we need r cos θ and r sin θ integer multiple of in its alternative,... An easy to use calculator that allows you to easily convert complex numbers drawing... Blogger, cis to rectangular form calculator iGoogle our website uses cookies to ensure you get free. 2: Manipulate the equation for the given values displays the equation for the given values the imaginary of... Multiply the two moduli and add the two moduli and add the two and! Calculator is a free online tool that displays the equation to obtain answer real number the...
cis to rectangular form calculator 2021 | 2021-07-30T10:07:10 | {
"domain": "elab.is",
"url": "https://elab.is/nm37u/cis-to-rectangular-form-calculator-646a7e",
"openwebmath_score": 0.7379550337791443,
"openwebmath_perplexity": 1342.5111164313003,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes\n\n",
"lm_q1_score": 0.9759464499040092,
"lm_q2_score": 0.8558511524823263,
"lm_q1q2_score": 0.8352648939113813
} |
https://math.stackexchange.com/questions/1307025/i-think-i-can-complete-the-square-of-any-quadratic-is-it-true-any-reason-to-e/1307043 | # I think I can complete the square of any quadratic, is it true? (Any reason to ever use Quad. Formula?)
I was taught that you could only complete the square of a quadratic if the coefficient on the $x^2$ term is 1.
However, playing a little bit with other quadratics, I've found that it's just not true. Based on the CTS algorithm, you just need to divide the coefficient of the $x$ term by twice the square root of the coefficient of the $x^2$ term.
So, if you have $ax^2 + bx + c$, your perfect square would be $(\sqrt{a}x + \frac{b}{2 \sqrt{a}})^2$
If $a$ is not a perfect square it could get nasty, but then you can just square the whole quadratic and go from there.
For example:
In the equation $5x^2 + 6x + 5 = 0$, we could do:
$25x^2 + 30x + 25 = 0$
$(5x+3)^2 = -16$
$5x+3 = \pm4i$
$x = \pm \frac{4i}{5} - \frac{3}{5}$
My questions are:
-Is this correct?
-Are there quadratics that are unsolvable this way?
• This way does not work if $a<0$. It could be modified, of course. Also, your example does not use the "perfect square" you note previously. – Rory Daulton May 31 '15 at 21:26
• @RoryDaulton, I added the square to $(\sqrt{a}x + \frac{b}{2 \sqrt{a}})^2$, but I'm pretty sure I'm using the right coefficients, could you point it out? And of course it won't work as such if $a < 0$, but then you can just multiply by $-1$, which doesn't seem very limiting. Are there other limiting factors to using this? – jeremy radcliff May 31 '15 at 21:35
• You did use the right coefficients, you just "cheated" by changing $ax^2+bx+c$ to $a^2x^2+abx+ac$ in your example. This shows that your way is usually not the best. @Spencer's answer shows the usual method, which avoids square roots as long as possible by changing the quadratic to $x^2+\frac ba x+\frac ca$. – Rory Daulton May 31 '15 at 21:40
• @RoryDaulton, That makes sense, thanks for the explanation. – jeremy radcliff May 31 '15 at 21:46
• In general, if you start with $ax^2+bx+c=0$, it can be useful to multiply and divide by $4a$, getting $\frac{1}{4a}(4a^2+4abx+4ac)=0$. Now the square can be completed with no trouble always, we get $(2ax+b)^2+4ac$. You don't really need the quadratic formula, though it can be handy when we are telling a computer how to find the roots. – André Nicolas May 31 '15 at 22:03
Every quadratic can have its square completed, which is in fact where the quadratic formula comes from.
The method I use is to factor out the leading coefficient,
\begin{align} \color{red}{a} x ^2 + bx +c &= \color{red}{a}\left( x^2 + \frac{b}{\color{red}{a}} x \right) + c \\ &= a\left( \left(x+\frac{b}{2a}\right)^2 - \frac{b^2}{4a^2} \right) + c \\ &= a\left(x+\frac{b}{2a}\right)^2 + c-\frac{b^2}{4a}, \end{align}
if you solve for when this is equal to zero you will get the quadratic formula.
• Wow, I had no idea that was where the QF came from, I need to work through this carefully. Thanks. – jeremy radcliff May 31 '15 at 21:44
• I took the liberty of reformatting your answer for better readability. I hope you won't mind... – A.P. May 31 '15 at 21:50
The proposed way of completing the square is correct. The identity is: $$ax^2 + bx + c = \left(\sqrt{a}x + \frac{b}{2 \sqrt{a}}\right)^2 + \left(c-\frac{b^2}{2a}\right).$$
One can of course also do what is more usually done and write $$ax^2+bx+c = a\left( x + \frac{b}{2a} \right)^2 + \left(c-\frac{b^2}{2a}\right).$$ The latter way is done for the sake of convenience, not because it is the only way possible.
PS: Someone points out in comments that this works only if $a>0$. If $a<0$, one can write: $$ax^2 + bx + c = -\left(\sqrt{|a|}x + \frac{b}{2 \sqrt{|a|}}\right)^2 + \left(c-\frac{b^2}{2a}\right).$$
• If $a < 0$, why not just negate the whole equation? That seems simpler to me. – Kevin Jun 1 '15 at 15:12
• @Kevin : Because $\sqrt{a}$ becomes problematic then, so I wrote $\sqrt{|a|}$. ${}\qquad{}$ – Michael Hardy Jun 1 '15 at 17:10
• Oh, I see what you're doing. I was thinking something more like let $a'=-a$, $b'=-b$, $c'=-c$, and then solve $a'x^2 + b'x+c'=0$ as normal. – Kevin Jun 1 '15 at 17:43
It seems like your questions have been answered, so this response is more of an addition. Completing the square is a useful rearrangement in and of itself - beyond the usage for solving quadratics. You may be aware of its application in placing the equation of a conic section in standard form.
For example, consider
$$x^2 + 2x + 2y^2 + y + 3 = 6$$
This is an ellipse, but our equation does not tell us much information. Rearranging into standard form will reveal quite a bit of info. Standard form for a horizontal ellipse is given as
$$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$$
where $(h,k)$ is the center and $a$ and $b$ represent the semi-major and semi-minor axes respectively (the longest and shortest "radius" of this "eccentric circle").
In order to place our ellipse into standard form, we complete the square twice:
$$x^2 + 2x + 2y^2 + 4y + 3 = 6$$ $$\left(x^2 + 2x \color{red}{+1}\right) + 2\left(y^2 + 4y \color{red}{+4}\right)+ 3 = 6 \color{red}{+1} \color{red}{+8}$$ $$(x+1)^2+ 2(y+2)^2+ 3 = 15$$ $$(x+1)^2+ 2(y+2)^2 = 12$$ $$\frac{(x+1)^2}{12}+ \frac{(y+2)^2}{6} = 1$$
We now see that we have an ellipse centered at $(h,k) = (-1,-2)$ with semi-major axis $a = \sqrt{12}$ and semi-minor axis $b=\sqrt6$. This matches the graph.
• Which is fine until you have an $xy$ term, of course. – Chappers Jun 1 '15 at 12:44
If $a$ is not a “perfect square” there's no problem either. If your equation is $$ax^2+bx+c=0$$ then it's equivalent to $$4a^2x^2+4abx+4ac=0$$ and completing the square is more evident: $$4a^2x^2+4abx+b^2-b^2+4ac=0$$ or $$(2ax+b)^2-(b^2-4ac)=0$$ If you just want to factor the polynomial $ax^2+bx+c$ (with $a\ne0$, of course, just do the same: $$ax^2+bx+c=\frac{1}{4a}(4a^2x^2+4abx+4ac)= \frac{1}{4a}\bigl((2ax+b)^2-(b^2-4ac)\bigr)$$ If $b^2-4ac<0$ there's nothing else to do, because the polynomial is irreducible over the reals; if $b^2-4ac=0$ it is $$\frac{1}{4a}(2ax+b)^2$$ and, if $b^2-4ac>0$ you get $$ax^2+bx+c=\frac{1}{4a}(x+2a-\sqrt{b^2-4ac})(x+2a+\sqrt{b^2-4ac})$$
Actually, the quadratic formula is derived BY completing the square. Yes, any quadratic equation can be solved by completing the square. The only reason to use the quadratic formula is that it might be simpler than completing the square.
One method that avoids fractions until the very end is to multiply through by $4a$.
\begin{align} ax^2 + bx + c &= 0 \\ 4a^2x^2 + 4abx + 4ac &= 0 \\ 4a^2x^2 + 4abx + b^2 &= b^2 - 4ac \\ (2ax + b)^2 = b^2 - 4ac \\ 2ax + b &= \pm \sqrt{b^2 - 4ac} \\ x &= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} \end{align} | 2019-07-23T16:37:01 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1307025/i-think-i-can-complete-the-square-of-any-quadratic-is-it-true-any-reason-to-e/1307043",
"openwebmath_score": 0.9940724968910217,
"openwebmath_perplexity": 257.3476980951744,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9759464485047916,
"lm_q2_score": 0.8558511524823263,
"lm_q1q2_score": 0.8352648927138593
} |
https://math.stackexchange.com/questions/1009865/how-many-ways-are-there-for-three-medals-to-be-awarded-if-ties-are-possible/1009917 | # How many ways are there for three medals to be awarded if ties are possible?
The Question
There are six runners in the 100-yard dash. How many ways are there for three medals to be awarded if ties are possible? (The runner or runners who finish with the fastest time receive gold medals, the runner or runners who finish with exactly one runner ahead receive silver medals, and the runner or runners who finish with exactly two runners ahead receive bronze medals.)
My Attempt
I saw a few cases:
Case I: No Ties
P(6,3) = 120 ways to pick the gold medal
Case II: 2 people tie
First we must pick the two people who tie, this can be done C(6,2) = 15 ways. Now I have to pick a medal for them to win, which opens up more cases because if they tie for first no silver medal is awarded and if they tie for second no bronze medal is awarded.
My Problem
My problem with this method is that it takes way too long to consider all the cases and wouldn't be practical if I was writing a test, or if I was designing an algorithm for this type of question. I'm wondering if there is a more efficient way to solve this?
My book gave the answer $873$ if that helps at all
• And I guess, in theory, you could have 3-way, 4-way, 5-way, and 6-way ties too! If 4 people tied for first, it seems that that would break the question, since more than 3 medals would be needed. I guess the question really becomes: How many ways can 6 things be partitioned? – JLee Nov 7 '14 at 1:05
• So what you're saying is this is basically the question how many ways can I put 6 marbles in 3 distinct jars? – Dunka Nov 7 '14 at 1:12
• I think it's a little different dunka, because if 3 people tie for gold, the silver and bronze medals are not awarded. That's to say, you don't want to count the case in which there are 3 marbles in the first jar, then 2 in the second, then 1 in the third. – Shane Nov 7 '14 at 1:15
• @Shane very true. Should I proceed with cases, knowing that 4 way ties or more would actually break the questions rules? – Dunka Nov 7 '14 at 1:17
• I'm not sure, to be honest. I'm one of those annoying commenters who only points out problems with what you're doing rather than offering solutions! Hopefully somebody smarter than I will intervene at any moment now :P – Shane Nov 7 '14 at 1:18
Let $n$ be the number of gold medals awarded.
$\textbf{1)}\;\;$If $n=2$, then there are $\binom{6}{2}=15$ ways to choose the gold medalists and
$\hspace{.3 in}2^4-1=15$ ways to choose the bronze medalists, so there are $15\cdot15=225$ possibilities.
$\textbf{2)}\;\;$ If $n=1$, there are $\binom{6}{1}$ ways to award the gold medal, and then
there are $\binom{5}{1}=5$ ways to award one silver medal and $2^4-1=15$ ways to award bronze medals, and
there are $\binom{5}{2}+\cdots+\binom{5}{5}=2^5-\binom{5}{0}-\binom{5}{1}=26$ ways to award more than one silver medal;
so in this case there are $6[5\cdot15+26]=606$ possibilities.
$\textbf{3)}\;\;$ If $n\ge3$, there are $\binom{6}{3}+\cdots+\binom{6}{6}=2^6-\binom{6}{2}-\binom{6}{1}-\binom{6}{0}=42$ ways to award the gold.
Therefore there are a total of $225+606+42=873$ ways to award the medals.
• @Shane this was the answer we were looking for! – Dunka Nov 9 '14 at 0:34
Maybe the best way to break it down is by cases likes this:
1. 6 win gold (1 possibility)
2. 5 win gold (6 possibilities)
3. 4 win gold (${6 \choose 2}=15$ possibilities)
4. 3 win gold (${6 \choose 3}=20$ possibilities)
5. 2 win gold, four win bronze (15 possibilities)
6. 2 win gold, three win bronze (60 poss.)
7. 2 win gold, two win bronze (90 poss.)
8. 2 win gold, one wins bronze (60 poss.)
9. 1 wins gold, 5 win silver (6 poss.)
10. 1 wins gold, 4 win silver (30 poss.)
11. 1 wins gold, 3 win silver (60 poss.)
12. 1 wins gold, 2 win silver (60 poss.)
13. 1 wins gold, 1 win silver, 4 win bronze (30 poss.)
14. 1 wins gold, 1 win silver, 3 win bronze (120 poss.)
15. 1 wins gold, 1 win silver, 2 win bronze (180 poss.)
16. 1 wins gold, 1 win silver, 1 win bronze (120 poss.)
Then we have
$$1 + 6 + 15 + 20 + 15 +60+90+60+6+30+60+60+30+120+180+120= 873$$
Bingo!
• This may be right. My book gives the answer of $873$ however, this is still a time consuming method. I'm wondering if there is an easier way than cases to do this. – Dunka Nov 7 '14 at 1:34
• Ooh... now you've given me an answer... let's see if I can get to it. Here come some edits! – Shane Nov 7 '14 at 1:36
• Now the goal is for somebody smarter to show us how to do this faster. Anybody? – Shane Nov 7 '14 at 1:53
• Thanks for the solution. However, I still want to know if there is a better way than cases to do this one. – Dunka Nov 7 '14 at 1:54
• I would be curious to see that also. In particular, if there are $n$ runners rather than $6$, my approach wouldn't be practical. I'm sure you express the above as some rather complicated sum of sums... – Shane Nov 7 '14 at 1:57 | 2019-08-26T08:00:56 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1009865/how-many-ways-are-there-for-three-medals-to-be-awarded-if-ties-are-possible/1009917",
"openwebmath_score": 0.4227326512336731,
"openwebmath_perplexity": 1332.2212116167086,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9759464499040094,
"lm_q2_score": 0.8558511506439708,
"lm_q1q2_score": 0.8352648921172448
} |
https://math.stackexchange.com/questions/3235144/sum-of-two-open-sets-is-open/3235229 | # Sum of two open sets is open
Let $$A$$,$$B$$ be two open sets in $$\mathbb{R}$$. Show that $$A+B = \{a+b \text{ | } a \in A, b \in B\}$$ is an open set.
Let $$a\in A$$ and $$b\in B$$. Since $$A$$,$$B$$ are open sets, it follows that $$a$$,$$b$$ are interior points, therefore $$\exists \epsilon_a, \epsilon_b > 0$$ such that $$a \in (a - \epsilon_a, a + \epsilon_a) = I_a \subset A$$ and $$b \in (b - \epsilon_b, b + \epsilon_b) = I_b \subset B$$.
Now let's define two closed intervals: $$I_{a/2} = [a-\epsilon_a/2, a+\epsilon_a/2] \subset I_a$$ and $$I_{b/2} = [b-\epsilon_b/2, b+\epsilon_b/2] \subset I_b$$
Now let's define the set: $$I = \{x+y \text{ | } x \in I_{a/2}, y \in I_{b/2}\} = [a+b-\frac{\epsilon_a + \epsilon_b}{2} , a+b+\frac{\epsilon_a + \epsilon_b}{2}]$$.
Now let $$\epsilon = \frac{\epsilon_a + \epsilon_b}{2}$$. It follows that $$((a+b)-\epsilon , (a+b)+\epsilon) \subset I \subset A+B$$. Therefore $$A+B$$ is an open set.
Can someone please check my work? I've seen a more direct proof using the fact that $$A+B = \bigcup_{a \in A}\{a\} + B$$. I don't want to prove it that way because it wasn't the way I initially thought about the problem.
Thank you!
• Are we working in $\Bbb R$? – Robert Shore May 22 at 2:26
• Are you working inside $\Bbb R$? What is $+$ here? – Thomas Shelby May 22 at 2:27
• Sorry, I'll edit. Yes, we're in $\mathbb{R}$ – Bruno Reis May 22 at 2:30
• Could you adjust your proof to work also when only $B$ is open? – Hagen von Eitzen May 22 at 2:51
• @HagenvonEitzen Thank you for commenting. Idk, haven't thought about it since the hypothesis is that $A$ and $B$ are open. For you, what I've done is correct? Thanks – Bruno Reis May 22 at 2:55
Your proof looks fine. As with every proof-verification question, I try to come up with some comments, the pettiness of which is proportional to the correctness of the proof.
1. The notation $$I_{a/2}$$ and $$I_{b/2}$$ could be improved. There's nothing formally wrong with naming them as such, but given the definition of $$I_a$$ and $$I_b$$, I would have expected $$I_{a/2}$$ to be a symmetric interval around $$a/2$$, not around $$a$$ with half the width. Maybe call them $$J_a$$ and $$J_b$$ respectively?
2. I don't see why you need to define such intervals anyway. If you sum $$I_a + I_b$$, you'll get the open interval $$(a + b - (\varepsilon_a + \varepsilon_b), a + b + (\varepsilon_a + \varepsilon_b))$$, which is contained in $$A + B$$.
3. I noticed that you brushed over the summing of the two intervals $$I_{a/2}$$ and $$I_{b/2}$$. This is quite a mechanical calculation, but it behooves you to know, in principle, how to fill in this detail if ever you need to. In particular, you should be able to do a full set equality proof, by showing each set is a subset of the other.
4. I noticed point 3 in light of point 2. I think the main reason that someone would change to closed intervals instead of an open intervals is because they feel they more easily grasp the summing of two closed, bounded intervals, due to the definite maximum and minimum. However, if you write out the full proof that the sets are equal, it becomes quickly evident that closed and open intervals are equally easy to sum. So, if someone changed to closed intervals and brushed over the proof of the sum, then I would tend to assume they haven't written out the proof of the sum of the two intervals, and are simply making assumptions! This may not cost you marks, but it would make me, as a marker, feel slightly less generous in giving out marks for the question (if it were a multi-part question with a loose marking scheme).
Apologies for the pettiness. The proof is good.
• First of all, thanks a lot for your comments on my proof. (1) Totally agree with you. (2) Totally agree with you. (3) What do you mean by "brushed over" ? (4) Can you explain "brushed over the proof of the sum"? Anyway, I still agree with what you said. Thanks a lot for taking some time to help me here! I highly appreciate that :) – Bruno Reis May 22 at 4:56
• I mean, you could prove the set equality$$I_{a/2} + I_{b/2} = [a + b - \varepsilon, a + b + \varepsilon]$$using a subset argument. Start by assuming $x \in I_{a/2} + I_{b/2}$, and see if you can conclude that $x$ belongs to the right hand side. Then, assume $x$ lies in the right hand side, and try to find $p \in I_{a/2}$ and $q \in I_{b/2}$ such that $x = p + q \in I_{a/2} + I_{b/2}$. You just stated the sum, rather than proving it. This potentially could be a source of marks lost in an assignment (depending on the rubric used). But, it can be ok to skip mechanical steps like this too. – Theo Bendit May 22 at 5:00
• Ohhhh got what you mean by "brushed". When I formally did it, I prove it, but for sake of this forum post, I "brushed" it to make it more direct! In a test or assignment, I would do it with a little more rigor! Thanks anyway for the hints to improve it mate – Bruno Reis May 22 at 5:18
• @BrunoReis Great! Then ignore points 3, 4. – Theo Bendit May 22 at 5:19 | 2019-06-19T08:45:38 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3235144/sum-of-two-open-sets-is-open/3235229",
"openwebmath_score": 0.8795958161354065,
"openwebmath_perplexity": 266.88981271397114,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9759464499040092,
"lm_q2_score": 0.8558511506439707,
"lm_q1q2_score": 0.8352648921172446
} |
https://stats.stackexchange.com/questions/249675/how-to-prove-probability-problem | # How to prove probability problem
Show that if $P(C) = 1$, then $P(D|C) = P(D)$ using probability rules.
I tried using conditional probability to get from $P(D|C) = P(D \cap C) / P (C) = P(D \cap C)$ but I am not sure how to continue the proof to get to $P(D)$. Can someone explain what the next steps I should do in order to get to $P(D)$?
I would solve it like this:
\begin{aligned} P(D|C) &= P(D\cap C) / P(C) \\ &= P(D \cap C) \\ &= P(D) + P(C) - P(D \cup C) \\ &= P(D) + 1 - 1 \\ &= P(D) \end{aligned}
Where our third equation is explained by the rule of addition with some simple rearrangement of terms:
$$P(D \cup C) = P(D) + P(C) - P(D \cap C)$$ $$\therefore P(D \cap C) = P(D) + P(C) - P(D \cup C)$$
And our fourth equation is explained by the fact that $P(D \cup C) = 1$, since it is the probability of either $D$ or $C$ occurring, and we know that $P(C) = 1$
• Very neatly done. – user140401 Dec 5 '16 at 10:49
I would approach it through a frequentist definition of probability, by showing that the set of events that count as D ∩ C has the same count as the set of events that count as D. That is P(D ∩ C) = N(D ∩ C)/N(total) and P(D) = N(D)/N(total) can be shown to be the same if every event D is also an event D ∩ C (so that N(D ∩ C) is the same as N(D)). Because C is universally true, this will be the case.
P(D|C) = P(D ∩ C)/P(C)
P(D|C) = P(C|D)*P(D)/P(C)
so we have to show that P(C|D) = 1, as does P(C), which was given.
P(C) = P(C|D)P(D) + P(C|!D)(1-P(D))
Let P(C|D) = 1-a and P(C|!D) = 1-b.
a, b must be nonnegative by the rules of probability (0-1 range)
then
1 = (1-a)P(D) + (1-b)(1-P(D))
1 = P(D) - a
P(D) + 1 - P(D) - b + b*P(d)
b(1-P(D)) = -aP(D)
which, given the 0-1 constraints on a,b,P(D) is only possible if i) both a and b are zero, ii) both b and P(D) are zero, or iii) a=0 and P(D) = 1
In the first case and third case, we have that P(C|D) = 1 because a=0 and we have proved what was required.
In the second case, if P(D) = 0 then we can show that this can only be satisified when P(D|C) is zero since P(D) = P(D|C)*P(C)+.... in which case we have still proved that P(D) = P(D|C) as required (since they both = 0).
• Yes the above seems to work, but I need to solve this question using probability rules without using frequentist definition of probability. If you know how should I proceed that way, let me know. – user141156 Dec 5 '16 at 5:23
• Since the very meaning of probability, as captured in its axioms, is separate from one's interpretation or philosophy of probability, any argument from a "frequentist definition"--at least if it is rigorous--would be identical to an argument from any other point of view. – whuber Mar 16 '18 at 16:57 | 2019-11-19T15:17:06 | {
"domain": "stackexchange.com",
"url": "https://stats.stackexchange.com/questions/249675/how-to-prove-probability-problem",
"openwebmath_score": 0.9971117377281189,
"openwebmath_perplexity": 667.6737023980609,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9759464478051827,
"lm_q2_score": 0.8558511506439707,
"lm_q1q2_score": 0.8352648903209614
} |
https://math.stackexchange.com/questions/2099210/when-is-the-matrix-textdiag-mathbfx-mathbf1-invertible | # When is the matrix $\text{diag}(\mathbf{x}) + \mathbf{1}$ invertible?
Given a vector $\mathbf{x} \in \mathbb{R}^N$, let's define:
$$\text{diag}(\mathbf{x}) = \begin{pmatrix} x_1 & 0 & \ldots & 0 \\ 0 & x_2 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & x_N \end{pmatrix}.$$
Moreover, let
$$\mathbf{1}= \begin{pmatrix} 1 & 1 & \ldots & 1 \\ 1 & 1 & \ldots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \ldots & 1 \end{pmatrix}.$$
Here is my question:
When is the matrix $\mathbf{M} = \text{diag}(\mathbf{x}) + \mathbf{1}$ invertible?
I was able to find some results when $x_1 = x_2 = \ldots = x_N = x$. Indeed, the matrix $M$ is singular when:
1. $x=0$. This is trivial since $\mathbf{M} = \mathbf{1}$...
2. $x=-N$. In this case, if you sum up all the rows (or columns) of the matrix $M$, you get the zero vector.
What can I say in the general case when $\text{diag}(\mathbf{x})$ is a generic vector?
• Weyl's inequalities give you a nice sufficient condition for invertibility – Omnomnomnom Jan 15 '17 at 19:41
• I believe the matrix is invertible for $x_i=-1, N\neq 1$ – YoTengoUnLCD Jan 15 '17 at 19:45
• If none of the $x_i$ are zero, you can apply the analysis of this previous Question. – hardmath Jan 15 '17 at 19:47
• @Omnomnomnom thanks a lot. Anyway reading [this][1], I don't understand the second part of the inequality $$j + k - n \geq i \geq r + s - 1, \ldots, n.$$ Is it like saying: $$j + k - n \geq i \geq r + s - t,$$ where $t = 1, \ldots, n$, right? [1]: en.wikipedia.org/wiki/… – the_candyman Jan 15 '17 at 19:59
We can easily compute the determinant of the sum $\operatorname{diag}(\mathbf{x}) + \mathbf{1}$ and check invertibility that way,
If all the diagonal entries $x_i$ are nonzero, we can apply the matrix determinant lemma for a rank one update to an invertible matrix:
$$\det(A+uv^T) = (1 + v^T A^{-1} u) \det(A)$$
When $A$ is the matrix $\operatorname{diag}(\mathbf{x})$ and $u,v$ are vectors of all ones, this says the matrix sum is invertible unless the sum of the reciprocals $x_i^{-1}$ is $-1$.
If one of the diagonal entries is zero, say $x_1$ without loss of generality, then elementary row operations quickly show that the determinant of $\operatorname{diag}(\mathbf{x}) + \mathbf{1}$ is $\prod_{k=2}^n x_k$.
• Notice that if there are two $x_i's$ equal to $0$ then there are two equal columns and the determinant is zero. I believe you meant $det(diag(x)+1)=\prod_{k=2}^n (x_k-1)$ in the final part. – Daniel Jan 15 '17 at 20:01
• You don't need $x$ to be entrywise nonzero if you use the identity $\det(A+uv^T)=\det(A)+v^T\operatorname{adj}(A)u$ instead. – user1551 Jan 15 '17 at 20:02
• @Daniel: Just so for having two $x_i=0$. But we add one and then subtract one from the "diagonal" entries $x_i$, so I think it's right. – hardmath Jan 15 '17 at 20:03
• @hardmath thanks a lot, anyway for $N=4$ and $x_1 = 0$, the determinant is $x_2 x_3 x_4$... – the_candyman Jan 15 '17 at 20:05
• @hardmath You are right. – Daniel Jan 15 '17 at 20:07
i think your matrix is nonsingular iff $$1 +\frac 1{x_1}+\frac 1{x_2}+ \ldots + \frac 1{x_n} \ne 0$$
i will look at the case $n = 4$ will consider the matrix $A = D + jj^\top$ where $D$ is the diagonal matrix with entries $d_1, d_2, d_3$ and $d_4, j = (1,1,1,1)^\top.$
suppose $\lambda, x$ is an eigenvalue-eigenvector pair. then we have $$d_1x_1 + x_1 + x_2+ x_3 + x_4 = \lambda x_1, \ldots, x_1+x_2+x_3+x_4 + d_4x_4=\lambda x_4$$ solving for $x_1$ we find that $$x_1= \frac 1{(\lambda- d_1)}(x_1+x_2+x_3+x_4), \ldots x_4= \frac 1{(\lambda- d_4)}\left(x_1+x_2+x_3+x_4\right)\tag 1$$ adding the four equations in (1), you find that the characteristic equation of $A$ is $$1 = \frac 1{(\lambda-d_1)}+\frac 1{(\lambda-d_2)}+\frac 1{(\lambda-d_3)}+\frac 1{(\lambda-d_4)}$$
therefore the matrix $D + jj^\top$ is singular iff $$\frac 1d_1 + \frac 1d_2 + \frac 1d_3 + \frac 1d_4 + 1 \ne 0.$$
• thanks a lot for this! It is helping me to better understand the problem! – the_candyman Jan 15 '17 at 20:27 | 2019-11-20T05:09:58 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2099210/when-is-the-matrix-textdiag-mathbfx-mathbf1-invertible",
"openwebmath_score": 0.9548695087432861,
"openwebmath_perplexity": 200.2197386701192,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.975946445706356,
"lm_q2_score": 0.8558511506439708,
"lm_q1q2_score": 0.8352648885246784
} |
http://ravenflagpress.herokuapp.com/discussion/id/788/ | Possible Solution to #9
TLindberg
Hey just wanted to see if my logic and calculations for number 9 make sense and find the correct answer. First I try to find the Identity matrix from \beta to \gamma. This is a 3 x 3 matrix with each column representing a transformation from a basis vector in \beta to a basis vector in \gamma. I got this as: \begin{pmatrix} 1 & 2 & 2 \\ 0 & 3 & 1 \\ 2 & 2 & 0 \end{pmatrix} Next I calculate the inverse of this matrix because we need to find the identity matrix from \gamma to \beta. I augment the matrix above with the identity matrix and then row reduce it to a matrix that looks like this: \begin{pmatrix} 1/5 & -2/5 & 2/5 \\ -1/5 & 4/15 & 1/10 \\ 3/5 & -1/5 & -3/10 \end{pmatrix} After this I calculate the the linear transformation from \gamma to \gamma by multiplying the matrixes in this order. (Identity matrix from \beta to \gamma) * (linear transformation matrix from \beta to \beta) * (identity matrix from \gamma to \beta) After calculating this (which is pretty lengthy) I get the matrix: \begin{pmatrix} 27/5 & -43/15 & 29/5 \\ 17/5 & -3/5 & 19/5 \\ 16/5 & -2 & 17/5 \end{pmatrix} The above matrix is the answer I got to the question. Please correct me if any of my calculations or understandings are incorrect.
Jackhe: May 5, 2015, 12:46 a.m.
that's what I also tried to do, but I'm not sure about the answer
Sam_Lai: May 5, 2015, 1:01 a.m.
This was the exact same logic that I used to solve the problem, but I interpreted the given matrix to be from $\gamma\$ to $\beta$. Here's my reason for interpreting it that way: Imagine if I had a basis {(1,2), (0,1)}. My basis with respect to the standard basis {(1,0), (0,1)} is $\begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$ This means that if I take my (1,0) and multiply it by this transformation matrix, I get (1,2) which is my coordinates in the standard basis. My basis with respect to the standard takes my coordinates to the standard. So if the vectors of my friend's basis $\gamma$ with respect to my basis $\beta$ is given, shouldn't it be a transformation matrix from $\gamma$ to $\beta$? Or is there something wrong with my interpretation?
Sam_Lai: May 5, 2015, 1:02 a.m.
Wow, something messed up big time there. I'm so sorry. Hopefully it's still comprehensible.
Cullen_Im: May 5, 2015, 5:32 a.m.
@TLindberg: I believe the first step necessary is to find an identity matrix from gamma to beta and NOT beta to gamma. Because the given linear transformation is from beta to beta, the vector needs to be first transformed into the beta basis. I similarly found an inverse to the identity matrix but the only calculation difference I see in your inverse identity matrix is row 2, column 2. I got "2/5" instead of your "4/15." I then multiplied the matrices in the order: (identity matrix from gamma to beta) * (linear transformation matrix from beta to beta) * (identity matrix from beta to gamma). I got the final answer as a matrix of linear transformation from gamma to gamma with row 1: 3, 33/5, 21/5; row 2: 5/2, 52/5, 19/5; row 3: -3/2, -41/5, -12/5
akaur: May 5, 2015, 11:46 a.m.
I got a very similar identity matrix, so I guess one of us made a calculation mistake somewhere (probably me). Isn't the given identity matrix from gamma to beta and the one we find is from beta to gamma?
julianna_burke: May 5, 2015, 2:55 p.m.
I did same process but got a different answer. In the identity matrix I got 3/5 instead of your 4/15 and then my final answer was 1st row: {3, -11/5, 1} 2nd row: {5/2, 52/5, 27/10} 3rd row {-3/2, -41/5, 9/2} This could very well be wrong but it is similar to someone else's answer in a different discussion except our third column differs
julianna_burke: May 5, 2015, 3:06 p.m.
Nevermind I found my mistake! I got the same answer as @Cullen_Im
elaine_lei: May 5, 2015, 3:58 p.m.
I did the problem twice and got the same answer as @Cullen_Im
TLindberg: May 5, 2015, 5:32 p.m.
@Cullen_Im: When composing matrices together its similar to a composition of functions. Therefore, the identity matrix from gamma to beta goes on the right because it is the "first" matrix to act on the vector in the gamma basis. After this, the vector is now in terms of the beta basis and can transformed with the linear transformation from beta to beta. After calculating the transformation, we need to switch the coordinates back to the gamma basis so we apply the identity matrix from beta to gamma, causing it to be on the very left. ------------------ A possibly easier way to imagine it is like this: f(g(h(x))). In this representation h(x) is the identity matrix from gamma to beta, g(x) is the linear transformation matrix from beta to beta, and f(x) is the identity matrix from beta to gamma. -------------- This ends with the function appearing backwards and instead reading right to left; however, the matrices can still be multiplied left to right because the matrices are associative and the multiplication can be done in any order, as long as the overall order of the three matrices is fixed. --------------- Please tell me if this makes sense. I was a bit confused by it at first too because it makes sense to form an equation from left to right in my mind, but according to the lecture notes and my TA I'm pretty sure this is the right way to go about it.
TLindberg: May 5, 2015, 6:50 p.m.
Oh, I might see what you were trying to correct me on. I interpreted the first matrix as being from beta to gamma, but instead it should be gamma to beta. By switching the values in those two matrices but still multiplying it in the same order I specified above it should be the correct answer. I probably calculated the inverse incorrectly, so I'll just leave it at that and try to post a solution I get later tonight.
angela: May 5, 2015, 8:31 p.m.
The explanation @TLindberg posted was super helpful! After multiplying everything out, I got $\begin{bmatrix} 27/5 & 1/5 & 29/5 \\ 17/5 & 11/5 & 19/5 \\ 16/5 & -2/5 & 17/5 \end{bmatrix}$
wllmskn: May 5, 2015, 9:02 p.m.
@angela That's what I got too!
LizvetteV: May 6, 2015, 2:43 p.m.
okay so my answer is way off, but I got 3 33/5 21/5 5/2 52/5 19/5 -3/2 -41/5 -12/5 But it might be because the identity matrix i got from beta to gamma was: 1/5 -2/5 2/5 -1/5 2/5 1/10 3/5 -1/5 -3/10 Because i still multiplied the identity matrix of beta to gamma * (T) from beta to beta * Identity matrix of gamma to beta
KianJerKoh: May 7, 2015, 2:51 p.m.
@Cullem_Im Yes it is from gamma to beta, because the output "with respect to our basis" means that the output should be in beta. Again the solutions that both David and my TA gave are from $\gamma$ to $\beta$, first, then we find the inverse. | 2017-06-27T20:50:27 | {
"domain": "herokuapp.com",
"url": "http://ravenflagpress.herokuapp.com/discussion/id/788/",
"openwebmath_score": 0.9748695492744446,
"openwebmath_perplexity": 388.79665395038523,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9759464492044004,
"lm_q2_score": 0.8558511469672594,
"lm_q1q2_score": 0.8352648879302103
} |
https://math.stackexchange.com/questions/2631352/how-to-solve-phin-78 | # How to solve $\phi(n)=78$?
How to solve $\phi(n)=78$?
One of the numbers is $79$ because the value of Euler's function for prime numbers is $\phi(n)=n-1$. It also holds for $79\cdot2=158$ because the only numbers that are not relatively prime to $158$ are even numbers and number $79$. I'm not sure if this is enough of an explanation.
Then I tried like this: I know that, if $n=\displaystyle \prod_{i=1}^k p_i^{\alpha_i} \Rightarrow \phi(n) = \displaystyle \prod_{i=1}^k p_i^{\alpha_i-1} (p_i-1)$.
Factoring $78$, I get $78=2\cdot3\cdot13$ so that means is some combination of $\prod_{i=1}^k p_i^{\alpha_i-1} (p_i-1)$ but how do I find which combination?
I first tried with $13$.
• Let's say that $13|(p-1)$ with $p|n$, which means $p=13k+1$, but there are no such numbers because they are either too big for $\phi(n)=78$ to hold or they aren't prime numbers.
• Then I tried with $13|p$. Which means $(13-1)|\phi(n)$. And that isn't true.
What should I do? Do the same thing with $2$ and $3$?
• There are none with two odd prime factoirs, because if $p,q$ are distinct odd primes, then $\phi(p^aq^b\dots)$ is divisible by $4$. – Thomas Andrews Feb 1 '18 at 16:07
• And since $78$ is square-free, this means either $n=p, n=p^2,n=2p,n=2p^2$ for some odd prime $p$. (Technically, we have to eliminate the powers of $2$, but that's easy.) – Thomas Andrews Feb 1 '18 at 16:14
• Did you read the question? @Moo "One of the numbers is 7979 .... It also holds for 79⋅2=158" – Thomas Andrews Feb 1 '18 at 16:15
• "What should I do? " Well, you've just determined there's no other way to interpret $13$ being a factor of $78$. So, that means no other way to do this. You are done. No need to check 2 or 3 because whatever you find they won't be able to make use of the $13$. – fleablood Feb 2 '18 at 8:02
the factors of $78 = 2*3*13$. and $\phi(n) = \prod p_i^{a_i - 1}(p_i -1)$
Now $13$ is either $p_i$ or a factor of a $p_i - 1$.
If $13$ is a $p_i$ than $13-1 =12$ would divide $\phi(n) = 78$ which it doesn't. So $13$ must be a factor of some $p_i -1$.
If $13 = p_i -1$ then $p_i =14$ which isn't prime so $13$ most be a proper factor of $p_i - 1$. But the only other factors available are $2,3$.
So $p_i - 1 = 2*13, 3*13$ or $6*13$. But $2*13+1 = 27$ and $3*13+1 = 40$ and $6*13+1 = 79$ and of those only $79$ is prime.
So we must represent $78$ as $= 79^0(79 - 1)*\prod_{p_i|n;p_i\ne 79} p_i^{a_i -1}(p_i -1)$. That is the only way we can take the factor of $13$ into account.
But that means $\prod_{p_i|n;p_i\ne 79} p_i^{a_i -1}(p_i -1) =1$.
So either $78$ is represented and $79^0(79-1)$ and $n = 79$ or $78$ is represented as $79^0(79-1) \cdot 2^0(2-1)$ and $n = 158$.
Those are the only two options.
We have $\phi(n)\ge 84$ for all $n\ge 400$, see the estimates here, with $\delta=1/47$:
Is the Euler phi function bounded below?
We easily see form the list of values of $\phi(n)$ for $n\le 400$, that $n=78,158$ are the only possibilities.
A related question: For which primes $p>2$ are there more than two values $n$ such that $\phi(n)=p-1$? For example, $p=7$ has more values: $\phi(n)=6$ for $n=7,9,14,18$
You have shown that since $13|\phi(n)$, there must be a prime $p$ dividing $n$ such that $13 | p-1$, which in turn means $p=13k+1$.
Now if $p|n$ then $p-1 | \phi(n)$, which means that you can bound $p \leq \phi(n)+1=79$.
Putting both conditions together, the list of possible $p$ is $$p\in \{53, 79\}$$ If $p=53$, then $p-1=2^2\cdot 13$ divides $\phi(n)$ and hence $4 | \phi(n)=78$, which is not possible. So this forces $p=79$ to divide $n$, or we cannot get factor $13$ in $\phi(n)$. In addition, due to the size of this $p$ we can also conclude that $p^2\not | n$, or else $79|\phi(n)$.
Finally, \begin{align*} 78 &= \phi(n) = \phi(n/p)\phi(p)=\phi(n/p)\cdot 78\\ 1 &= \phi(n/p) \end{align*} Therefore $n/p$ must be either $1$ or $2$. This shows that $n=79$ or $158$.
In short: using $13|\phi(n)$ does work as it gives you a list of $p$ to try, which in turn then reduces the problem to solving a set of small $\phi(n/p)$. | 2019-07-22T09:51:17 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2631352/how-to-solve-phin-78",
"openwebmath_score": 0.9715206623077393,
"openwebmath_perplexity": 107.22957724482566,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.975946446405965,
"lm_q2_score": 0.8558511488056151,
"lm_q1q2_score": 0.8352648873293027
} |
https://math.stackexchange.com/questions/3164457/evaluate-limit-of-the-series-lim-n-to-infty-left-frac1n2-frac2 | # Evaluate limit of the series: $\lim_{n\to\infty} \left[\frac{1}{n^{2}} + \frac{2}{n^{2}} + \frac{3}{n^{2}} + \cdots + \frac{n}{n^{2}}\right]$
I am pretty confident that the following limit is $$0.5$$:
$$\lim_{n\to\infty} \left[\frac{1}{n^{2}} + \frac{2}{n^{2}} + \frac{3}{n^{2}} + \cdots + \frac{n}{n^{2}}\right]=\lim_{n\to\infty} \left[\frac{1+2+3+ \cdots +n}{n^{2}}\right]=\lim_{n\to\infty} \left[\frac{n^2+n}{2n^{2}}\right]=\frac{1}{2}$$
However one of the students argued that if we write limit of sum as sum of individual limits, it will be zero. Why we cannot write limit of sum as sum of limits in this case?
I've been taught that if individual limits exist, the limit of sum is equal to the sum of limits. It would be helpful to get an explanation or a reference to similar rules for limits of series.
• that's because $n$ tends to infinity, so the limit of the sum is not equal to the sum of the limits – Alex Mar 27 at 12:58
• Because if we could write that every limit of sum is sum of limit, thousands of professors will be out of work. – uniquesolution Mar 27 at 13:06
• Just as an aside, this is a simple example of why we need an increasing sequence of functions in Beppo Levi's theorem. Write $$f_n(k)=\begin{cases} {k\over n^2},&1\leq k\leq n\\0,&k>n\end{cases}$$ and the $f_n$ decrease monotonically to $0$. – saulspatz Mar 27 at 13:24
• As $n$ tends to infinity, the individual limits will become $0$, but there will be infinitely many of them. So that’s a case of $0\cdot\infty$. – Jonas De Schouwer Mar 27 at 14:00
However one of the students argued that if we write limit of sum as sum of individual limits, it will be zero. Why we cannot write limit of sum as sum of limits in this case?
For the sum of two sequencesn we have the property "limit of the sum is the sum of limits" (if both sequences have a limit) and by repeatedly applying this, we have this property for any finite number of sequences (terms).
As often is the case, you cannot simply extend this to the infinite case; i.e. you cannot assume the same property will hold when the number of sequences (terms) is not finite.
A simpler counterexample would be the sum of $$n$$ terms, all equal to $$\tfrac{1}{n}$$; obviously we have: $$\underbrace{\frac{1}{n}+\frac{1}{n}+\ldots+\frac{1}{n}}_{\mbox{n terms}} = \frac{n}{n}=1$$ but every individual sequence (term) clearly tends to $$0$$: $$\frac{1}{n} \stackrel{n\to \infty}{\longrightarrow} 0$$.
You can prove by induction on $$k$$ that, if $$k$$ functions of $$n$$ each converge as $$n\to\infty$$, the sum of their limits is the limit of their sum. The expression you're dealing with isn't an example of that. Instead there are infinitely many functions, of which an $$n$$-dependent number are used in the sum. There's no way to bridge this gap, by induction or otherwise. Indeed, the example we're discussing proves that's impossible, as the limit is indeed $$\frac12$$.
Just another way to "convince" you that the limit is $$\frac{1}{2}$$:
The sums are Riemann sums:
• $$\left[\frac{1}{n^{2}} + \frac{2}{n^{2}} + \frac{3}{n^{2}} + \cdots + \frac{n}{n^{2}}\right] = \sum_{k=1}^n\left( \frac{k}{n} \cdot \frac{1}{n}\right) \stackrel{n\to \infty}{\longrightarrow} \int_0^1 x \;dx = \frac{1}{2}$$
Riemann sums serve as a perfect example that interchanging infinite summation and limits of members of the involved sums cannot work in general.
Possibly interesting further readings could be something about interchanging limit operations or more specifically about interchanging summation and limits. | 2019-09-23T10:57:55 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3164457/evaluate-limit-of-the-series-lim-n-to-infty-left-frac1n2-frac2",
"openwebmath_score": 0.9741554856300354,
"openwebmath_perplexity": 235.43961868512952,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9759464471055739,
"lm_q2_score": 0.8558511451289037,
"lm_q1q2_score": 0.8352648843397904
} |
https://math.stackexchange.com/questions/1900105/solving-linear-congruence-2x-11-equiv-7-pmod-3/1900128 | # Solving linear congruence $2x + 11 \equiv 7 \pmod 3$
I'm having some troubles understanding the solution (going from step $2$ to step $3$) of this linear congruence equation. Here's the problem with the solution:
Solve the congruence $2x + 11 \equiv 7 \pmod 3$.
First, we reduce all the coefficients $\pmod 3$:
$$2x + 2 \equiv 1 \pmod 3. \tag{1}$$
Next, we add $1$ to both sides, using the fact that $2 + 1 \equiv 0 \pmod 3$:
$$2x \equiv 2 \pmod 3. \tag{2}$$
Finally, we multiply both sides by $2$, using the fact that $2 · 2 = 4 \equiv 1 \pmod 3$:
$$x \equiv 1 \pmod 3. \tag{3}$$
That is, any number in the set $\{\ldots, −5, −2, 1, 4, \ldots\}$ will solve the original congruence.
Like I said my problem is the step between $(2)$ and $(3)$. If I multiply both sides of $(2)$ by $2$ I get:
$$4x \equiv 4 \pmod 3.$$
Now, I understand that $4$ is congruent to $1 \pmod 3$ so the RHS "simplifies" to $1 \pmod 3$ but I don't understand why the LHS "simplifies" to $x$. Can someone help me with this?
• Do you understand that $3x \equiv 0 \pmod{3}$? – quid Aug 22 '16 at 15:23
• Note that by multiplying by 2, you have already used that 'going mod-3 preserves multiplication', i.e., if $a \equiv a' \pmod 3$ and $b \equiv b' \pmod 3$, then $ab \equiv a'b' \pmod 3$. Now use that once more: $1 \equiv 4 \pmod 3$, so $x \equiv 4x \pmod 3$. – Magdiragdag Aug 22 '16 at 15:26
• $4x\equiv 1x=x\mod3$. – Bernard Aug 22 '16 at 15:26
By the Congruence Product Rule $$\ 4\equiv 1\,\Rightarrow\, 4x\equiv 1x\ [\equiv\, x]$$. Alternatively note that $$\,2 \equiv -1\,$$ and it is simpler to scale the congruence by by $$-1,\,$$ which amounts to simply negating it. More generally when the coef of $$x$$ is coprime to the modulus then we can compute its inverse by the extended Euclidean algorithm, then scale the congruence by the inverse to transform the coef of $$x$$ to be $$1$$.
Note that these solution steps for a linear congruence are the same as they are for linear equations on real numbers, i.e. isolate $$x$$ by adding and multiplying equal (equivalent) numbers to both sides. That these operations remain valid for congruences follows by the Congruence Sum & Product Rules in the linked post (both are unified when one studies rings in abstracr algebra, where congruences $$\bmod n\,$$ are reified algebraically in the quotient ring $$\,\Bbb Z_n =$$ integer $$\bmod n)$$
Remark Note also that you need to make it clear that the congruences are equivalent (and not simply that the former imply the latter) if you want to deduce that the obtained solution is actually a solution, i.e. you need to use bidirectional arrows $$(\!\!\iff\!\!)$$ not unidirectional $$(\Rightarrow)$$ connecting the congruences. The arrows go both ways because you used only invertible operations $$\,(n\mapsto n+a\,$$ and $$\,n\mapsto bn\,$$ for invertible $$\,b).\,$$ The arrows generally do not reverse if you use noninvertible operations (e.g. squaring), which may introduce extraneous solutions (in which case you'd need to verify that solution candidates are actually solutions).
• This makes perfect sense know. The link you provided is also very helpful, thank you. – user347616 Aug 22 '16 at 16:10
• See also this answer for more on extraneous rooots. – Bill Dubuque Sep 15 '20 at 7:57
$4x \equiv 1 \mod 3 \implies$
$4x - 1 = 3k; k \in \mathbb Z \implies$
$x + 3x - 1 = 3k \implies$
$x -1 = 3k - 3x = 3(k-x) \implies$
$x - 1 = 3j; j= (k-x) \in \mathbb Z \implies$
$x \equiv 1 \mod 3$.
====
This is always true. If $k \equiv a \mod N$ than $kV \equiv aV \mod N$ because $k \equiv a \mod N \implies k - a = mN$ so $kV - aV = mVN$. Works even if $x$ is a variable. $4 \equiv 1 \mod 3$ so $4x \equiv x \mod 3$. Easy-peasy. (Assuming $x$ is presumed to be solved for a congruency class.)
For the LHS you want $4x\equiv x\pmod 3$. This is true because $3$ divides $4x-x=3x$. (Recall the definition of modulo: "$a\equiv b\pmod n$" means "$n$ divides $a-b$".)
Notationally, it is simpler to use bar notation:
$\overline{2x+11}=\overline{7}=\overline{1}\Longrightarrow$
$\overline{2x}+\overline{11}=\overline{1}\Longrightarrow$
$\overline{2x}=\overline{1}-\overline{11}=\overline{-10}$ $=\overline{2}\Longrightarrow$
$\overline{x}=\overline{1}$. | 2021-06-18T15:04:15 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1900105/solving-linear-congruence-2x-11-equiv-7-pmod-3/1900128",
"openwebmath_score": 0.902407169342041,
"openwebmath_perplexity": 263.0261691461307,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.975946446405965,
"lm_q2_score": 0.8558511396138365,
"lm_q1q2_score": 0.8352648783586192
} |
http://mathhelpforum.com/calculus/38372-summation-problem.html | # Math Help - summation problem
1. ## summation problem
Hi, all. I need to evaluate the following:
$\frac{1}{k}\sum_{n = 0}^{\infty}(\frac{k-1}{k})^nn$
where k∈ℕ and k≠1
I don't really know how to do much else than test whether or not a series converges or diverges; I don't know how to determine what it converges to.
Thanks, guys!
2. Originally Posted by hatsoff
Hi, all. I need to evaluate the following:
$\frac{1}{k}\sum_{n = 0}^{\infty}(\frac{k-1}{k})^nn$
where k∈ℕ and k≠1
I don't really know how to do much else than test whether or not a series converges or diverges; I don't know how to determine what it converges to.
Thanks, guys!
I will give you a hint
Rewrite $\frac{k-1}{k}$ as $1-\frac{1}{k}$
Now consider that since $k\in\mathbb{N}$ that $1-\frac{1}{k}<1$
Get it now?
3. Originally Posted by Mathstud28
I will give you a hint
Rewrite $\frac{k-1}{k}$ as $1-\frac{1}{k}$
Now consider that since $k\in\mathbb{N}$ that $1-\frac{1}{k}<1$
Get it now?
It resembles a geometric series in that respect, but then each term is multiplied by n, which is not a constant. I don't know how to handle that.
4. Originally Posted by hatsoff
It resembles a geometric series in that respect, but then each term is multiplied by n, which is not a constant. I don't know how to handle that.
Think derivative/integration
5. Originally Posted by Mathstud28
Think derivative/integration
I need another hint, sir.
6. Originally Posted by hatsoff
Hi, all. I need to evaluate the following:
$\frac{1}{k}\sum_{n = 0}^{\infty}(\frac{k-1}{k})^nn$
where k∈ℕ and k≠1
I don't really know how to do much else than test whether or not a series converges or diverges; I don't know how to determine what it converges to.
Thanks, guys!
If $|x| < 1$,
$G(x) = \sum_{n=0}^{\infty} x^n = \frac1{1-x}$
$G'(x) = \sum_{n=0}^{\infty} nx^{n-1} = \frac1{(1-x)^2}$
$xG'(x) = \sum_{n=0}^{\infty} nx^{n} = \frac{x}{(1-x)^2}$
Now let $x = 1 - \frac1{k}$, since $k \in \mathbb{N}$, we have $\frac1{k} \leq 1$ and $0 < 1 - \frac1{k} < 1$. So we can apply the infinte Geometric Progression formula
$\sum_{n=0}^{\infty} n(1 - \frac1{k})^{n} = \frac{1 - \frac1{k}}{(\frac1{k})^2}$
$\frac1{k} \sum_{n=0}^{\infty} n(1 - \frac1{k})^{n} = \frac{1 - \frac1{k}}{\frac1{k}} = k - 1$
7. Originally Posted by hatsoff
I need another hint, sir.
Oh I am sorry! I completely missed your response. Luckily for you The oh so competent Isomorphism replied
8. Originally Posted by Mathstud28
Think derivative/integration
Hmm. Are you telling me that the following is true:
$\int[\sum_{n=r}^{\infty}a_n]dx=\sum_{n=r}^{\infty}\int[a_ndx]$
?
If so, then:
$\frac{1}{k}\sum_{n = 0}^{\infty}(\frac{k-1}{k})^nn$
$\frac{1}{k}\sum_{n = 1}^{\infty}(\frac{k-1}{k})^nn$
$\frac{k-1}{k^2}\sum_{n = 1}^{\infty}(\frac{k-1}{k})^{n-1}n$
$(\frac{k-1}{k^2})\frac{d}{dn}\sum_{n = 1}^{\infty}\int[(\frac{k-1}{k})^{n-1}ndn]$
$(\frac{k-1}{k^2})\frac{d}{dn}\sum_{n = 1}^{\infty}(\frac{k-1}{k})^{n}$
EDIT: I guess I'm late to the party.
9. Originally Posted by hatsoff
Hmm. Are you telling me that the following is true:
$\int[\sum_{n=r}^{\infty}a_n]dx=\sum_{n=r}^{\infty}\int[a_ndx]$
?
If so, then:
$\frac{1}{k}\sum_{n = 0}^{\infty}(\frac{k-1}{k})^nn$
$\frac{1}{k}\sum_{n = 1}^{\infty}(\frac{k-1}{k})^nn$
$\frac{k-1}{k^2}\sum_{n = 1}^{\infty}(\frac{k-1}{k})^{n-1}n$
$(\frac{k-1}{k^2})\frac{d}{dn}\sum_{n = 1}^{\infty}\int[(\frac{k-1}
{k})^{n-1}ndn]$
$(\frac{k-1}{k^2})\frac{d}{dn}\sum_{n = 1}^{\infty}(\frac{k-1}{k})^{n}$
EDIT: I guess I'm late to the party.
Yes you are...but for your sake yes..In a power series
$\int\sum_{n=0}^{\infty}a_n(x-c){n}dx=\sum_{n=0}^{\infty}\int{a_n(x-c)^{n}}dx$
The same applies for derivatives
$\frac{D[\sum_{n=0}^{\infty}a_n(x-c)^{n}]}{dx}=\sum_{n=0}^{\infty}\frac{D[a_n(x-c)^{n}]}{dx}$
to prove it expand...differentiate/integrate....contract into an explicit form power series and compare
10. Originally Posted by Mathstud28
Yes you are...but for your sake yes..In a power series
$\int\sum_{n=0}^{\infty}a_n(x-c){n}dx=\sum_{n=0}^{\infty}\int{a_n(x-c)^{n}}dx$
The same applies for derivatives
$\frac{D[\sum_{n=0}^{\infty}a_n(x-c)^{n}]}{dx}=\sum_{n=0}^{\infty}\frac{D[a_n(x-c)^{n}]}{dx}$
to prove it expand...differentiate/integrate....contract into an explicit form power series and compare
The following phrase should be added to this: "Provided the relevant series converge." If they don't then these rules go out the window.
-Dan
11. Originally Posted by topsquark
The following phrase should be added to this: "Provided the relevant series converge." If they don't then these rules go out the window.
-Dan
Yes! Thank you! Sometimes I have the classic "I already know it so I assume others do to" syndrome...not a good thing to have
12. Thanks, guys! This was a big help. | 2014-07-24T20:47:23 | {
"domain": "mathhelpforum.com",
"url": "http://mathhelpforum.com/calculus/38372-summation-problem.html",
"openwebmath_score": 0.9513705372810364,
"openwebmath_perplexity": 1155.0234939965592,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.975946442208311,
"lm_q2_score": 0.8558511414521923,
"lm_q1q2_score": 0.835264876560189
} |
https://math.stackexchange.com/questions/3322306/how-do-we-express-int-a-infty-fx-dx-is-convergent-using-predicate-log | How do we express “$\int_a^\infty f(x) \, dx$ is convergent” using predicate logic?
How do we express $$\displaystyle \int_a^\infty f(x) \, dx \text{ is convergent}$$ using predicate logic?
I know how to express that a simple function is convergent: $$f(x) \text{ is convergent}$$ means $$\lim_{x \to \infty} f(x) = L$$ which in turn can be expanded to $$\exists L \in \mathbb{R}, \forall \epsilon > 0, \exists M \in \mathbb{R}, \forall x \in \mathbb{R}, x \geq M \Longrightarrow | L - f(x) | < \epsilon.$$
Based on that, here is my attempt.
I already know that $$\displaystyle \int_a^\infty f(x) \, dx \text{ is convergent}$$ means $$\lim_{b \to \infty} \int_a^b f(x) \, dx = L.$$
To expand it further, maybe all I have to do is use a similar expression as above, but change $$x$$ to $$b$$, and ensure $$b > a$$, as follows:
$$\exists L \in \mathbb{R}, \forall \epsilon > 0, \exists M \in \mathbb{R}, \forall b \in \mathbb{R}, ( b \geq M \wedge b > a ) \Longrightarrow \left|L - \int_a^b f(x)\, dx \right| < \epsilon$$
Is this correct?
• Yes it's correct – Jakobian Aug 13 at 17:44
• You can omit $b>a$, because you can always pick an $M$ greater than $a$ if you wish. – Naj Kamp Aug 13 at 19:11
Your answer is correct. Let $$\forall x \in \mathbb{R}, \, g(x) = \int_a^x f(t) dt$$ As you noticed \begin{align} \displaystyle \int_a^\infty f(t) \, dt \text{ is convergent} &\iff (\exists L,\lim_{x \rightarrow + \infty} \int_a^x f(t) \, dt =L)\\ &\iff (\exists L,\lim_{x \rightarrow + \infty} g(x) = L) \end{align} Notice that $$g$$ is just function. Thus, by definition $$\exists L \in \mathbb{R}, \forall \epsilon > 0, \exists M \in \mathbb{R}, \forall x \in \mathbb{R}, x \geq M \Longrightarrow | L - g(x) | < \epsilon.$$
• Note that in a general case $g$ is not an antiderivative of $f$, because it's not necessarily differentiable. For example, take $|x|=\int_0^x \text{sgn}(t)\mathrm{d}t$ – Botond Aug 15 at 17:49 | 2019-08-25T13:40:22 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3322306/how-do-we-express-int-a-infty-fx-dx-is-convergent-using-predicate-log",
"openwebmath_score": 0.9918474555015564,
"openwebmath_perplexity": 480.08122660207647,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9873750533189538,
"lm_q2_score": 0.8459424411924673,
"lm_q1q2_score": 0.8352624629771783
} |
https://math.stackexchange.com/questions/3758135/if-x-and-y-are-i-i-d-with-x-thicksim-textexp1-find-py-geq-x-geq2 | # If $X$ and $Y$ are i.i.d. with $X\thicksim\text{Exp}(1)$, find $P(Y\geq X\geq2)$, and the PDF of $Z=X-Y$
The Problem: Let $$X$$ and $$Y$$ be independent exponential random variables with parameter $$\lambda=1$$.
$$\textbf{a)}$$ Calculate the probability $$P(Y\geq X\geq2)$$.
$$\textbf{b)}$$ Find the density function of the random variable $$Z=X-Y.$$
My Attempt:
a) By hypothesis, we have that the density functions of $$X$$ and $$Y$$, respectively, are given by $$f_X(x)=\begin{cases}e^{-x}&\text{if }x\geq0\\0&\text{otherwise.}\end{cases}\quad f_Y(y)=\begin{cases}e^{-y}&\text{if }y\geq0\\0&\text{otherwise.}\end{cases}$$ Since $$X$$ and $$Y$$ are independent, it follows that $$f_{XY}(x,y)=f_X(x)\cdot f_Y(y)=\begin{cases}e^{-(x+y)}&\text{if }x\geq0,\,y\geq0\\0&\text{otherwise.}\end{cases}$$ Now to find the probability in question we integrate the joint density function over the region $$D=\{(x,y)\in\mathbb R^2:2\leq x\leq y\}.$$ We have \begin{align} P(Y\geq X\geq2)&=\iint_D f_{XY}(x,y)\,dy\,dx=\int_2^\infty\int_x^\infty e^{-(x+y)}\,dy\,dx\\ &=\int_2^\infty e^{-2x}\,dx\\ &=\frac{1}{2e^4}. \end{align} b) We can first find the CDF of $$Z$$ using the joint PDF of $$X$$ and $$Y$$ by integrating the latter function over the region $$D=\{(x,y)\in\mathbb R^2:x-y\leq t\}$$ where $$t\in\mathbb R.$$ If $$t\leq0$$, then \begin{align*} P(X-Y\leq t)&=\iint_D f_{XY}(x,y)\,dy\,dx=\int_0^\infty\int_{x-t}^\infty e^{-(x+y)}\,dy\,dx\\ &=e^t\int_0^\infty e^{-2x}\,dx\\ &=\frac{e^t}{2}. \end{align*} On the other hand, if $$t>0$$, then \begin{align*} P(X-Y\leq t)&=\iint_D f_{XY}(x,y)\,dy\,dx\\&=\int_0^t\int_0^\infty e^{-(x+y)}\,dy\,dx+\int_t^\infty\int_{x-t}^\infty e^{-(x+y)}\,dy\,dx\\ &=\int_0^t e^{-x}\,dx+\int_t^\infty e^t e^{-2x}\,dx\\ &=1-e^{-t}+\frac{e^{-t}}{2}\\ &=1-\frac{e^{-t}}{2}. \end{align*} Therefore, we have the following case-defined function as the CDF of $$Z$$, $$F_Z(t)=\begin{cases}\dfrac{e^t}{2}&\text{if }t\leq0\\[0.8em]1-\dfrac{e^{-t}}{2}&\text{if }t>0.\end{cases}$$ Since the CDF above is continuous everywhere and differentiable almost everywhere, we can differentiate it to find the PDF. We have $$f_Z(t)=\begin{cases}\dfrac{e^t}{2}&\text{if }t\leq0\\[0.8em]\dfrac{e^{-t}}{2}&\text{if }t>0.\end{cases}$$
Do you agree with my approach and execution above?
Thank you very much for your time and I sincerely appreciate any feedback.
• everything is correct but you wrote wrongly the definition of the regions $D$ because $(x,y)\in[0,\infty)^2$ instead of $\mathbb{R}^2$ but the calculations seems fine – Masacroso Jul 15 '20 at 19:39
More generally, let $$X\sim\mathrm{Expo}(\lambda)$$ and $$Y\sim\mathrm{Expo}(\mu)$$ be independent and let $$t>0$$ be arbitrary. Then \begin{align} \mathbb P(Y\geqslant X\geqslant t) &= \int_{(t,\infty)\times(t,y)}f_{X,Y}(x,y)\ \mathsf d(x\times y)\\ &=\int_t^\infty \int_t^y \lambda e^{-\lambda x}\mu e^{-\mu y}\ \mathsf dx\ \mathsf dy\\ &= \frac{\lambda }{\lambda +\mu } e^{-(\lambda +\mu )t}. \end{align} Also, note that $$\{Y\geqslant X\geqslant t\} = \{Y\geqslant X\}\cap\{X\wedge Y\geqslant t\},$$ and hence \begin{align} \mathbb P(Y\geqslant X\geqslant t) &= \mathbb P(Y\geqslant T)\mathbb P(X\wedge Y\geqslant T)\\ &= \frac{\lambda }{\lambda +\mu } e^{-(\lambda +\mu )t}. \end{align} Substituting $$\lambda=\mu=1$$ and $$t=2$$ yields $$\frac12 e^{-4}$$, the same as your result.
To compute the density of $$Z=X-Y$$, we use convolution. First note that the density of $$-Y$$ is $$f_{-Y}(y) = \mu e^{\mu y}\cdot\mathsf 1_{(-\infty,0)}$$. Then for $$t\in\mathbb R$$ we have \begin{align} f_Z(t) &= f_X\star f_{-Y}(t)\\ &= \int_{\mathbb R} f_X(s)f_{-Y}(t-s)\ \mathsf ds\\ &= \int_{t\vee 0}^\infty \lambda e^{-\lambda s}\mu e^{\mu(t-s)}\ \mathsf ds\\ &= \frac{\lambda \mu }{\lambda +\mu }\left(e^{\mu t}\cdot\mathsf 1_{(-\infty,0)}(t)+e^{-\lambda t}\cdot\mathsf 1_{[0,\infty)}(t) \right). \end{align} Substituting $$\lambda=\mu=1$$ yields $$f_Z(t) = \frac12\left(e^t\cdot\mathsf 1_{(-\infty,0)} + e^{-t}\mathsf 1_{[0,\infty)}(t)\right),$$ the same as your result. In this case, $$Z$$ has a Laplace distribution. | 2021-06-19T21:10:57 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3758135/if-x-and-y-are-i-i-d-with-x-thicksim-textexp1-find-py-geq-x-geq2",
"openwebmath_score": 0.9999899864196777,
"openwebmath_perplexity": 444.5321458575865,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9873750496039276,
"lm_q2_score": 0.8459424411924673,
"lm_q1q2_score": 0.83526245983448
} |
https://math.stackexchange.com/questions/2386040/can-we-express-any-positive-real-number-with-arbitrary-precision-using-a-ratio-o?noredirect=1 | # Can we express any positive real number with arbitrary precision using a ratio of two prime numbers? [duplicate]
This question already has an answer here:
Although the question might look trivial at first because there are infinitely many prime numbers and for example the ratio of two near prime numbers tends to $1$ at infinity, there is still a point that is missing.
If we define the set $S=\{\, \frac{p_{i}}{p_{j}}\mid i,j\in\Bbb N\,\}$ where $p_i$ is prime number $i$, is it dense in the set of non-negative reals?
If it is, as much as it looks (prevalently) obvious, I am not sure about which precise property is ensuring this, as neither the infinitude of primes nor the limit of ratio of two successive primes reaching $1$ (which is a theorem on its own) looks sufficient individually.
I could imagine something like: the rational set has this property and we can replace each rational number with a ratio of two primes to any desired precision. But, can we?
Maybe I am missing something, but it is not obvious whichever way I look at it.
Theorem that is expected is like:
If $\frac{r_{1}}{s_{1}}>\frac{r_{2}}{s_{2}} > 0$ then there are always two prime numbers $p_{m}$ and $p_{n}$ so that $\frac{r_{1}}{s_{1}}>\frac{p_{m}}{p_{n}}>\frac{r_{2}}{s_{2}}$
if that is to work.
## marked as duplicate by Watson, user91500, José Carlos Santos, Claude Leibovici, Marc van LeeuwenAug 8 '17 at 11:23
• @RabMakh $\mathbb Q$ is dense in $\mathbb R$ – Mark Bennet Aug 7 '17 at 21:34
• You need to allow primes to be negative for this. – Mark Bennet Aug 7 '17 at 21:49
• @JuliánAguirre Wow, that is a very non-obvious question to contain the answer to this one :). – Erick Wong Aug 8 '17 at 0:13
• I love how this question got three radically different but great answers: "don't care that they're primes", "based on widely known results about primes", and "heavy/esoteric machinery". – R.. Aug 8 '17 at 2:05
• @ErickWong Neither/nor meaning individually, I added the clarification. – user318107 Aug 8 '17 at 5:43
Yes, you can. Let $x_0>0$ be a given positive number, and let $p_k$ denote the $k$th prime.
Recall that $p_n \sim n \log n$ as $n \to \infty$ so
$$\lim_{n \to +\infty} \frac{p_{\lfloor nx_0 \rfloor}}{p_n} = \lim_{n \to +\infty} \frac{\lfloor nx_0 \rfloor \log \lfloor nx_0 \rfloor}{n \log n} = \lim_{n \to +\infty} \frac{\left(nx_0 - \epsilon_n\right)\log (nx_0 - \epsilon_n)}{n \log n}$$
where $\epsilon_n \in [0,1)$ ($\epsilon_n$ depends on $n$).
Since for fixed $x_0>0$, and any sequence $\left\{\epsilon_n\right\} \subset [0,1)$ we have the equivalences $nx_0 - \epsilon_n \sim nx_0$ and $\log (nx_0 - \epsilon_n) \sim \log(nx_0)$ as $n \to \infty$, we get
$$\lim_{n \to +\infty} \frac{\left(nx_0 - \epsilon_n\right)\log (nx_0 - \epsilon_n)}{n \log n} = \lim_{n \to +\infty} \frac{nx_0\left(\log n + \log x_0\right)}{n \log n} = \lim_{n \to +\infty} \frac{x_0\left(\log n\right)}{\log n} + \lim_{n \to +\infty} \frac{x_0\log x_0}{\log n} = x_0$$
This implies any $x_0>0$ can be approximated to arbitrary precision by the sequence $\left\{\frac{p_{\lfloor {nx_0} \rfloor}}{p_n}\right\}_{n \geq 1}$
If you allow negative primes, this means prime ratios are dense in $\mathbb{R}$.
As hinted at by the OP, this is implied by the pair of facts that there are infinitely many primes, and the ratio of successive primes converges to $1$. Here is a proof that uses no further properties of the sequence of primes, hence the same result applies to any increasing integer sequence with those two properties (more generally, any sequence of reals that increases to $\infty$ and has ratios converging to $1$).
Let $C>1$ be a positive real to be approximated ($C=1$ is directly implied by our assumptions, and $C<1$ can be handled by taking reciprocals), and suppose we need to find a prime fraction in the range $(C,C + \epsilon)$.
Since $p_{n+1}/p_n \to 1$, we may choose $N$ such that $p_{n+1}/p_n < 1 + \epsilon/C$ for all $n \ge N$. We now take $b = p_N$ as our denominator and choose $a := p_m$ so that $m$ is the least index satisfying $p_m/b > C$. Clearly such an $m$ exists since there are infinitely many primes so arbitrarily large values of $a$ are available.
It is clear that $m > N$ and that $a/b > C$. By our choice of $m$, $p_{m-1}/b \le C$. By our choice of $N$, $p_m/p_{m-1} < 1 + \epsilon/C$. Therefore $a/b = (p_m/p_{m-1})(p_{m-1}/b) < C + \epsilon$.
• The property of $p_{n+1}/p_{n}$ reaching 0 at infinity is not a trivial property, it is a theorem on its own. I have mentioned it not necessarily as a question if precisely these two can be used, more like an illustration over the properties that are in game. – user318107 Aug 8 '17 at 5:50
• I am slightly puzzled here. You say "the ratio of successive primes converges to 1" but the OP says "the ratio of two near prime numbers tends to 0 at infinity" and Alex just said "$p_{n+1}/p_n$ reaching 0 at infinity". These seem a bit contradictory. – badjohn Aug 8 '17 at 10:57
• @badjohn see here – Marja Aug 8 '17 at 11:01
• @Marja Thanks. That gives me what Erick says but does not explain the (apparent) contradiction with what alex.peter said. – badjohn Aug 8 '17 at 11:08
• @badjohn Well, limit is unique. .... and alex just fixed his/her typo. – Marja Aug 8 '17 at 11:11
The set of primes has asymptotic density zero, but $\pi(n)$ is not terribly smaller than $n$, it is about $\frac{n}{\log n}$ by the PNT. Additionally, a result of Ingham ensures the existence of a prime in the interval $[n^3,(n+1)^3]$ for any $n\geq N=\exp\exp(34)$. Let $r\in(1,+\infty)$. For any prime $p>N^3$, there is a prime $q$ in the range $[pr,((pr)^{1/3}+1)^3]$. Obviously $\frac{q}{p}\geq r$, but $$\frac{q}{p}-r \leq \frac{3(pr)^{2/3}}{p} = \frac{3r^{2/3}}{p^{1/3}}$$ and the RHS tends to zero as $p\to \infty$, hence the ratios of primes are dense in $(1,+\infty)$. By considering reciprocals and opposites it is simple to prove that the ratios of primes are dense in $\mathbb{R}$.
Obviously we do not stricly need the finesse of Ingham's result, something like "for any $n$ large enough, there always is a prime in the interval $[n^{42},(n+1)^{42}]$" would have done the job equally fine. We may also prove the claim by invoking something like
Let $E\subset\mathbb{N}$ the set of natural numbers $n$ with the property that there is a prime in the interval $[n,n+\log(n)^{12}]$. $E$ has a positive asymptotic density.
encoding the fact that moderately large prime gaps are quite rare. Indeed, in the interval $[1,n]$ the average distance between a prime and the next one is around $\log n$, always by the PNT.
• I'm not sure I understand what $E$ is. Should it say "the set of natural numbers $n$," instead? – Cameron Buie Aug 8 '17 at 11:37
• @CameronBuie: sorry for the misunderstanding, I did not notice I missed a "$n$" in my definition of $E$ above, now fixed. – Jack D'Aurizio Aug 8 '17 at 22:17
• Quite alright. For my part, I should've said " 'the set of natural numbers $n$ with the property...,' " instead. :-) – Cameron Buie Aug 8 '17 at 22:24
(I).Theorem. For any $x\in \mathbb R$ there exist infinitely many $(a,b)\in \mathbb Z\times \mathbb N$ such that $|x-\frac {a}{b}|<\frac {1}{b^2\sqrt 5}.$
For convenience we will take the weaker result that $|x-\frac {a}{b}|<\frac {1}{b^2}$ in the above theorem.
(II). An immediate corollary to the Prime Number Theorem: For all $y>0$ there exists $z>0$ such there exists a prime between $n$ and $n(1+y)$ whenever $n>z$.
(III). Let $x>0.$ Given $y>0,$ take $z$ such that there is a prime between $n$ and $n(1+y)$ for all $n\geq z.$ Take $a,b \in \mathbb N$ such that $|x-\frac {a}{b}|<\frac {1}{b^2}$ and $\min (a,b)\geq z$ (which is possible by Theorem (I).) Let $p,q$ be primes with $a<p<a(1+y)$ and $b<q<(1+y)b.$
We have $-aby= ab-ba(1+y)<aq-bp<ab(1+y)-ba=aby.$ That is, $|aq-bp|<aby$.
Also $\frac{a}{b}=|\frac {a}{b}|\leq |\frac {a}{b}-x|+|x|<\frac {1}{b^2}+|x|\leq 1+x.$
Therefore $$\left|x-\frac {p}{q}\right|\leq\left|x-\frac {a}{b}\right|+\left|\frac {a}{b}-\frac {p}{q}\right|<\frac {1}{b^2}+\left|\frac {aq-bp}{bq}\right|<\frac {1}{b^2}+ \frac {aby}{b^2}=\frac {1}{b^2}+y\frac {a}{b}\leq \frac {1}{b^2}+ y(1+x).$$
Since $y$ can be arbitrarily close to $0,$ and $b$ can be arbirtarily large for any given $y,$ we are finished for $x>0.$ For $x=0$ consider $2/p$ for arbitrarily large prime $p.$
Footnote. Theorem (I) is not deep or difficult. The Prime Number Theorem definitely is. | 2019-07-23T01:06:48 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2386040/can-we-express-any-positive-real-number-with-arbitrary-precision-using-a-ratio-o?noredirect=1",
"openwebmath_score": 0.9394147396087646,
"openwebmath_perplexity": 234.88411863691522,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9873750533189537,
"lm_q2_score": 0.8459424353665382,
"lm_q1q2_score": 0.8352624572248012
} |
https://www.geospatialnews.net/40e4ru.php?page=Bipolar-cylindrical-coordinates-44a7bb | Section 9.7 Polar, Cylindrical, and Spherical Coordinates Math 21a February 15, 2008 Announcements No class Monday 2/18. The bipolar cylindrical coordinates are produced by projecting in the "z"-direction.
Illustration of cylindrical coordinates with interactive graphics. As with two dimensional space the standard $$\left( {x,y,z} \right)$$ coordinate system is called the Cartesian coordinate system. Section 6-12 : Cylindrical Coordinates. Bipolar coordinates form the basis for several sets of three-dimensional orthogonal coordinates. "Cylindrical" polar-cylindrical coordinates "Spherical" spherical coordinates with poles along the axis and coordinates in the order radius, polar angle, azimuthal angle {"BipolarCylindrical", {a}} bipolar-cylindrical coordinates with focal length 2 a in the order focal angle, logarithmic radius, {"Bispherical", {a}}
We will look at polar coordinates for points in the xy-plane, using the origin (0;0) and the positive x-axis for reference. Recall that the position of a point in the plane can be described using polar coordinates $(r,\theta)$. A polar coordinate system, gives the co-ordinates of a point with reference to a point O and a half line or ray starting at the point O. There are several notational conventions, and whereas is used in this work, Arfken (1970) prefers . The bipolar cylindrical coordinates are produced by projecting in the z-direction. A set of curvilinear coordinates defined by x = (asinhv)/(coshv-cosu) (1) y = (asinu)/(coshv-cosu) (2) z = z, (3) where u in [0,2pi), v in (-infty,infty), and z in (-infty,infty). A thoughtful choice of coordinate system can make a problem much easier to solve, whereas a poor choice can lead to unnecessarily complex calculations. In the last two sections of this chapter we’ll be looking at some alternate coordinate systems for three dimensional space.
Cylindrical coordinates are a simple extension of the two-dimensional polar coordinates to three dimensions.
There are several notational conventions, and whereas (u,v,z) is used in this work, Arfken (1970) prefers (eta,xi,z). Cylindrical Polar Coordinates With the axis of the circular cylinder taken as the z-axis, the perpendicular distance from the cylinder axis is designated by r and the azimuthal angle taken to be Φ.
(4) (5) Lesson 6: Polar, Cylindrical, and Spherical coordinates 1. Cylindrical coordinates are a simple extension of the two-dimensional polar coordinates to three dimensions. Illustration of cylindrical coordinates with interactive graphics. Cylindrical and spherical coordinates give us the flexibility to select a coordinate system appropriate to the problem at hand.
The following identities show that curves of constant and are Circles in -space. 2. Convert the cylindrical coordinates defined by corresponding entries in the matrices theta, rho, and z to three-dimensional Cartesian coordinates x, y, and z. theta …
Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate system in theperpendicular z-direction.The two lines of foci F_{1} and F_{2} of the projected Apollonian circles are generally taken to be defined by x=-a and x=+a, respectively, (and by y=0) in the Cartesian coordinate system. Applications Recall that the position of a point in the plane can be described using polar coordinates $(r,\theta)$. A point P in the plane, has polar coordinates (r; ), where r … The following identities show that curves of constant u and v are circles in xy-space. Bipolar Cylindrical Coordinates. No office hours Tuesday 2/19.
Yes office hours Wednesday 2/20 2–4pm SC 323. A set of Curvilinear Coordinates defined by (1) (2) (3) where , , and . Bipolar cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional bipolar coordinate system in the perpendicular -direction.The two lines of foci and of the projected Apollonian circles are generally taken to be defined by and , respectively, (and by ) in the Cartesian coordinate system. Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular -direction.Hence, the coordinate surfaces are prisms of confocal ellipses and hyperbolae.The two foci and are generally taken to be fixed at − and +, respectively, on the -axis of the Cartesian coordinate system | 2021-01-20T12:14:06 | {
"domain": "geospatialnews.net",
"url": "https://www.geospatialnews.net/40e4ru.php?page=Bipolar-cylindrical-coordinates-44a7bb",
"openwebmath_score": 0.94791579246521,
"openwebmath_perplexity": 708.1397932331462,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes",
"lm_q1_score": 0.9873750484894195,
"lm_q2_score": 0.845942439250491,
"lm_q1q2_score": 0.8352624569742113
} |
https://math.stackexchange.com/questions/1453920/can-a-non-square-matrix-have-a-full-rank | Can a non-square matrix have a full rank?
Can a non-square matrix have a full rank?
I always see cases with square matrix with full rank but seldom with non-square matrix. Can anyone help on this?
For example, is the following matrix full rank?
A =( 1 3 10)
( 2 3 14)
My lecture slide says this does not have a full rank because any multiple of x'=[2 1 -1/2] will give Ax=0
(1 3 10)( 2 ) ( 0 )
(2 3 14)( 1 )==( 0 )
(-1/2)
I don't think this is correct but may I check?
• Let $A$ be an $n \times m$ matrix. By definition, $A$ has full rank if and only if $A$ has rank equal to $\min \{ n,m \}$. Does this answer your question? – Crostul Sep 27 '15 at 17:41
• How are you defining "full rank" ? – pjs36 Sep 27 '15 at 17:41
• Thanks then may I ask whether the 2x3 matrix above in my example a full rank? – Eric Sep 27 '15 at 17:47
• This is a $2 \times 3$ matrix of rank 2: by definition it is a full-rank matrix. – Crostul Sep 27 '15 at 17:48
• I see. I got it. That's what I expected. Then my lecture note is wrong. Bad professor... – Eric Sep 27 '15 at 17:49
If a matrix is $m \times n$, then we say it has full row rank if the rank is at least $m$ and it has full column rank if the rank is at least $n$. Unless the matrix is square, it is impossible for both to occur.
We could say that the matrix is "full rank" if the rank is $\min \{ m,n \}$. I would understand this usage, even though I don't think I've actually seen it in practice.
• Non-square matrices with full rank have an important role in differential geometry (implicit function theorem in dimension $>1$, and in looking for singularities in varieties defined by equations). – Crostul Sep 27 '15 at 17:46
• @Eric It has rank 2, which is $\min \{ 2,3 \}$. So yes. – Ian Sep 27 '15 at 17:47 | 2019-09-22T19:11:59 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1453920/can-a-non-square-matrix-have-a-full-rank",
"openwebmath_score": 0.6384647488594055,
"openwebmath_perplexity": 233.03778363177662,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9873750520186948,
"lm_q2_score": 0.8459424353665381,
"lm_q1q2_score": 0.835262456124857
} |
https://proofwiki.org/wiki/Euclid%27s_Theorem | # Euclid's Theorem
## Theorem
For any finite set of prime numbers, there exists a prime number not in that set.
In the words of Euclid:
Prime numbers are more than any assigned multitude of prime numbers.
### Corollary 1
There are infinitely many prime numbers.
### Corollary 2
There is no largest prime number.
## Proof
Let $\mathbb P$ be a finite set of prime numbers.
Consider the number:
$\displaystyle n_p = \left({\prod_{p \mathop \in \mathbb P} p}\right) + 1$
Take any $p_j \in \mathbb P$.
We have that:
$\displaystyle p_j \mathrel \backslash \prod_{p \mathop \in \mathbb P} p$
Hence:
$\displaystyle \exists q \in \Z: \prod_{p \mathop \in \mathbb P} p = q p_j$
So:
$\displaystyle n_p$ $=$ $\displaystyle q p_j + 1$ $\quad$ by Division Theorem $\quad$ $\displaystyle \implies \ \$ $\displaystyle n_p$ $\perp$ $\displaystyle p_j$ $\quad$ by definition of coprime $\quad$
So $p_j \nmid n_p$.
There are two possibilities:
$(1): \quad n_p$ is prime, which is not in $\mathbb P$.
$(2): \quad n_p$ is composite.
But from Positive Integer Greater than 1 has Prime Divisor, it must be divisible by some prime.
That means it is divisible by a prime which is not in $\mathbb P$.
So, in either case, there exists at least one prime which is not in the original set $\mathbb P$ we created.
$\blacksquare$
## Historical Note
This theorem is Proposition $20$ of Book $\text{IX}$ of Euclid's The Elements.
## Fallacy
There is a fallacy associated with Euclid's Theorem.
It is often seen to be stated that: the number made by multiplying all the primes together and adding $1$ is not divisible by any members of that set.
So it is not divisible by any primes and is therefore itself prime.
That is, sometimes readers think that if $P$ is the product of the first $n$ primes then $P + 1$ is itself prime.
This is not the case.
For example:
$\left({2 \times 3 \times 5 \times 7 \times 11 \times 13}\right) + 1 = 30\ 031 = 59 \times 509$
both of which are prime, but, take note, not in that list of six primes that were multiplied together to get $30\ 030$ in the first place.
## Source of Name
This entry was named for Euclid. | 2018-04-22T03:00:58 | {
"domain": "proofwiki.org",
"url": "https://proofwiki.org/wiki/Euclid%27s_Theorem",
"openwebmath_score": 0.8846069574356079,
"openwebmath_perplexity": 366.3912121039144,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9873750488609222,
"lm_q2_score": 0.8459424373085146,
"lm_q1q2_score": 0.8352624553710222
} |
http://people.reed.edu/~mayer/math111.html/header/node56.html | Next: 11. Calculation of Derivatives Up: 10. Definition of the Previous: 10.2 Limits of Functions Index
# 10.3 Definition of the Derivative.
Our definition of tangent to a curve will be based on the following definition:
10.18 Definition (Derivative.) Let be a real valued function such that . Let . We say that is differentiable at if is an interior point of and the limit
(10.19)
exists. In this case we denote the limit in (10.19) by , and we call the derivative of at .
The quantity represents the slope of the line joining the points and on the graph of . If and are different points in then this quotient will be defined. If we choose a sequence of points converging to , and if the slopes converge to a number which is independent of the sequence , then it is reasonable to call (i.e., ) the slope of the tangent line to the graph of at .
10.20 Definition (Tangent to the graph of a function.) Let be a
real valued function with , and let . If is differentiable at then we define the slope of the tangent to graph at the point to be the number , and we define the tangent to graph at to be the line that passes through with slope .
Remark: This definition will need to be generalized later to apply to curves that are not graphs of functions. Also this definition does not allow vertical lines to be tangents, whereas on geometrical grounds, vertical tangents are quite reasonable.
10.21 Example. We will calculate the tangent to at a generic point .
Let . Then for all ,
Hence the tangent line to graph at is the line through with slope , and the equation of the tangent line is
or
or
10.22 Example. We will now consider some of the examples given earlier.
If then
We saw in example 10.10 that does not exist. Hence, the graph of at has no tangent.
If , then in the previous example we saw that the equation of the tangent to graph at is ; i.e., the -axis is tangent to the curve. Note that in this case the tangent line crosses the curve at the point of tangency.
If then for all ,
The equation of the tangent line to graph at is
or .
Thus at each point on the curve the tangent line coincides with the curve.
Let . This is not the same as the function since the domain of is while the domain of is . (For all we have where .)
I want to investigate . From the picture, I expect this graph to have an infinite slope at , which means according to our definition that there is no tangent line at . Let . Then , but
so does not exist and hence does not exist.
10.23 Example. Let for . Let and let . Then
Hence
(10.24)
i.e.,
In line (10.24) I used the fact that , together with the sum and quotient rules for limits.
10.25 Exercise. A Let . Sketch the graph of . For what values of do you expect to be ? For what values of do you expect to be positive? What do you expect to happen to when is a small positive number? What do you expect to happen to when is a small negative number?
Calculate for arbitrary . Does your answer agree with your prediction?
10.26 Exercise. A Let for . Sketch the graph of . Use the same scale on the -axis and the -axis.
On what intervals do you expect to be positive? On what intervals do you expect to be negative? Calculate .
On the basis of symmetry, what do you expect to be the values of , and ? For what do you expect to be zero? On the basis of your guesses and your calculated value of , draw a graph of , where is the function that assigns to a generic number in . On the basis of your graph, guess a formula for .
(Optional) Prove that your guess is correct. (Some trigonometric identities will be needed.)
10.27 Exercise. A Calculate if .
10.28 Exercise. A
a)
Find if .
b)
Find the equations for all the tangent lines to graph that pass through the point . Make a sketch of graph and the tangent lines.
10.29 Exercise. Consider the function whose graph is shown below.
For what in does exist? Sketch the graphs of and on the same set of axes.
The following definition which involves time and motion and particles is not a part of our official development and will not be used for proving any theorems.
10.30 Definition (Velocity.) Let a particle move on a number line in such a way that its coordinate at time is , for all in some interval . (Here time is thought of as being specified by a number.) If are points in with , then the average velocity of for the time interval is defined to be
Note that is not necessarily the same as the distance moved in the time interval . For example, if then , but the distance moved by in the time interval is . (The particle moves from to at time , and then back to .)
The instantaneous velocity of at a time is defined to be
provided this limit exists. (If the limit does not exist, then the instantaneous velocity of at is not defined.) If we draw the graph of the function ; i.e., , then the velocity of at time is by definition slope of tangent to graph at .
In applications we will usually express velocity in units like . We will wait until we have developed some techniques for differentiation before we do any velocity problems.
The definition of velocity just given would have made no sense to Euclid or Aristotle. The Greek theory of proportion does not allow one to divide a length by a time, and Aristotle would no more divide a length by a time than he would add them. Question: Why is it that today in physics you are allowed to divide a length by a time, but you are not allowed to add a length to a time?
In Newton's calculus, the notion of instantaneous velocity or fluxion was taken as an undefined, intuitively understood concept, and the fluxions were calculated using methods similar to that used in the section 10.1.
The first rigorous'' definitions of limit of a function were given around 1820 by Bernard Bolzano (1781-1848) and Augustin Cauchy (1789-1857)[23, chapter 1]. The definition of limit of a function in terms of limits of sequences was given by Eduard Heine in 1872.
Next: 11. Calculation of Derivatives Up: 10. Definition of the Previous: 10.2 Limits of Functions Index
Ray Mayer 2007-09-07 | 2017-11-19T19:41:42 | {
"domain": "reed.edu",
"url": "http://people.reed.edu/~mayer/math111.html/header/node56.html",
"openwebmath_score": 0.8696843981742859,
"openwebmath_perplexity": 400.12178357334045,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9873750525759487,
"lm_q2_score": 0.8459424334245618,
"lm_q1q2_score": 0.8352624546788027
} |
https://math.stackexchange.com/questions/2225544/how-to-solve-a-n-2a-n-1-1-a-0-0-a-1-1 | # How to solve $a_n = 2a_{n-1} + 1, a_0 = 0, a_1 = 1$?
$a_n = 2a_{n-1} + 1, a_0 = 0, a_1 = 1$
So to get the closed form of this recurrence relation, I would usually try to get it into the form of $a_n = ra_{n-1}$ and then $a_n = r^na_0$. But what am I supposed to do with the $1$?
Thanks!
• Is your question: how to create a closed form for this recurrence relation? In any case please update your post with a question. Also if you write down the first few $a_i$ you'll see a series that is easy to specify with a function. – Χpẘ Apr 9 '17 at 0:12
Note that
$$a_n + 1 = 2(a_{n-1} + 1)$$
Then you can conclude that $a_n + 1 = 2^n(a_0 + 1) = 2^n$, which means that $a_n = 2^n - 1$.
Generally speaking, if you solve an equation
$$a_n = ka_{n-1} + r$$
there can be two cases.
Case 1. $k \neq 1$
Then this relation can be rewritten as $a_n + \dfrac{r}{k-1} = k\left(a_{n-1} + \dfrac{r}{k-1}\right)$, which means that $a_n = k^n\left(a_0 + \dfrac{r}{k-1}\right) - \dfrac{r}{k-1}$
Case 2. $k = 1$
Then $a_n = a_{n-1} + r$ and the solution is simply $a_n = rn + a_0$.
• I don't see how the second line follows from the first – mrnovice Apr 9 '17 at 0:20
• @mrnovice because for relations like $b_n = 2b_{n-1}$ we now the form of general solution, right? It is $b_n = 2^nb_0$ (as was written in the original question). Thus, we can simply substitute $b_n = a_{n} + 1$ in the first relation and get the second one. – Swistack Apr 9 '17 at 0:32
It should be clear that if you solve,
$$L=2L+1$$
You get the following,
$$-1=2(-1)+1 \tag{1}$$
We want to solve,
$$a_{n}=2a_{n-1}+1 \tag{2}$$
Subtracting equation $1$ from $2$ gives,
$$(a_{n}+1)=2(a_{n-1}+1)$$
Let $b_{n}=a_{n}+1$ so that $b_0=a_0+1$ and $a_n=b_n-1$ then we have,
$$b_{n}=2b_{n-1}$$
Then continue and back substitute.
$$b_{n}=b_02^n$$
$$b_{n}=(a_0+1)2^n$$
$$a_{n}=b_n-1=(a_0+1)2^n-1$$
$$a_{n}=2^n-1$$
• I haven't thought of solving it this way before but it makes a lot of sense, thanks – fzero24 Apr 9 '17 at 0:29
$a_2=2\cdot 1+1=3=2^2-1,\quad a_3=2\cdot 3+1=7=2^3-1,\quad a_4=2\cdot7+1=15=2^4-1,\quad a_5=2\cdot 15+1=31=2^5-1$
So in general we guess that $a_{n}=2^n-1,\quad n\in\mathbb{N}$
So we know it's true for $n=0$
Assume true for $n$, now we must prove true for $n+1$:
$a_{n+1} = 2\cdot a_n+1=2\cdot(2^n-1)+1=2^{n+1}+1$ as required, hence true for $n+1$, hence true $\forall n\in\mathbb{N}$
Therefore we conclude that $a_n=2^n-1$
If we write out the first few terms, you can see a pattern. $$0,1,3,7,15,31,63$$ All of the numbers are of the form $a_n=2^n-1$. The way we can see this from the recurrence is through induction: $$2(2^{n-1}-1)+1 = 2^n -1$$ Now try evaluating $a_n=3a_{n-1}+2$, and then $a_n=ka_{n-1}+k-1$. | 2020-03-29T03:15:07 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2225544/how-to-solve-a-n-2a-n-1-1-a-0-0-a-1-1",
"openwebmath_score": 0.9586666822433472,
"openwebmath_perplexity": 169.14487005923758,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.987375052204446,
"lm_q2_score": 0.8459424334245618,
"lm_q1q2_score": 0.8352624543645327
} |
http://encoretheatercompany.com/i-wonder-udiqtww/354501-how-to-prove-a-function-is-differentiable-at-a-point | 30
Dec
## how to prove a function is differentiable at a point
Differentiate it. First of all, $$h$$ is a rational fraction whose denominator is not vanishing for $$(x,y) \neq (0,0)$$. - [Voiceover] What I hope to do in this video is prove that if a function is differentiable at some point, C, that it's also going to be continuous at that point C. But, before we do the proof, let's just remind ourselves what differentiability means and what continuity means. \frac{\partial f}{\partial x_i}(\mathbf{a}) &= \lim\limits_{h \to 0} \frac{f(\mathbf{a}+h \mathbf{e_i})- f(\mathbf{a})}{h}\\ This video is unavailable. Follow @MathCounterexam Answer to: How to prove that a function is differentiable at a point? the definition of "f is differentiable at x" is "lim h->0 (f(x+h)-f(x))/h exists". So f is not differentiable at x = 0. \end{align*} New comments cannot be posted and votes cannot be cast. Another point of note is that if f is differentiable at c, then f is continuous at c. Use that definition. The point of the previous example was not to develop an approximation method for known functions. Ex 5.2, 10 (Introduction) Greatest Integer Function f(x) = [x] than or equal to x. Then the directional derivative exists along any vector $$\mathbf{v}$$, and one has $$\nabla_{\mathbf{v}}f(\mathbf{a}) = \nabla f(\mathbf{a}).\mathbf{v}$$. And then right when x is equal to one and the value of our function is zero it looks something like this, it looks something like this. We now consider the converse case and look at $$g$$ defined by So, first, differentiability. Hence $$g$$ has partial derivatives equal to zero at the origin. Click hereto get an answer to your question ️ Prove that if the function is differentiable at a point c, then it is also continuous at that point for products and quotients of functions. !function(d,s,id){var js,fjs=d.getElementsByTagName(s)[0],p=/^http:/.test(d.location)? So it is not differentiable over there. We focus on real functions of two real variables (defined on $$\mathbb R^2$$). \end{align*} For two real variable functions, $$\frac{\partial f}{\partial x}(x,y)$$ and $$\frac{\partial f}{\partial y}(x,y)$$ will denote the partial derivatives. If you get a number, the function is differentiable. Questions, no matter how basic, will be answered (to the best ability of the online subscribers). If you get two numbers, infinity, or other undefined nonsense, the function is not differentiable. $f(x,y)=\begin{cases}(x^2+y^2)\sin\left(\frac{1}{\sqrt{x^2+y^2}}\right) & \text{ if } (x,y) \ne (0,0)\\ Therefore, the function is not differentiable at x = 0. Go here! For example, the derivative with respect to $$x$$ can be calculated by If a function is continuous at a point, then it is not necessary that the function is differentiable at that point. Finally $$f$$ is not differentiable. As we head towards x = 0 the function moves up and down faster and faster, so we cannot find a value it is "heading towards". \left(\frac{1}{\sqrt{x^2+y^2}}\right)}{\sqrt{x^2+y^2}}\\ Similarly, $$\vert y \vert \le \Vert (x,y) \Vert$$ and therefore $$\vert g(x,y) \vert \le \Vert (x,y) \Vert$$. exists. If limits from the left and right of that point are the same it's diferentiable. &= \lim_{h \to 0}\frac{h^2 \sin (1/|h|)-0}{h} \\ Watch Queue Queue \[g(x,y)=\begin{cases}\frac{xy}{\sqrt{x^2+y^2}} & \text{ if } (x,y) \ne (0,0)\\ Transcript. Would you like to be the contributor for the 100th ring on the Database of Ring Theory? As a consequence, if $$g$$ was differentiable at the origin, its derivative would be equal to zero and we would have \[\lim\limits_{(x,y) \to (0,0)} \frac{\vert g(x,y) \vert}{\Vert (x,y) \Vert} = 0$ That is not the case as for $$x \neq 0$$ we have $$\displaystyle \frac{\vert g(x,y) \vert}{\Vert (x,y) \Vert} = \frac{1}{2}$$. A function is said to be differentiable if the derivative exists at each point in its domain. The partial maps $$x \mapsto g(x,0)$$ and $$y \mapsto g(0,y)$$ are always vanishing. Nowhere Differentiable. However, $$h$$ is not differentiable at the origin. Sal analyzes a piecewise function to see if it's differentiable or continuous at the edge point. Conversely, if we have a function such that when we zoom in on a point the function looks like a single straight line, then the function should have a tangent line there, and thus be differentiable. In the same way, one can show that $$\frac{\partial f}{\partial y}$$ is discontinuous at the origin. Both of these derivatives oscillate wildly near the origin. ... Learn how to determine the differentiability of a function. We want to show that: lim f(x) − f(x 0) = 0. x→x 0 This is the same as saying that the function is continuous, because to prove that a function was continuous we’d show that lim f(x) = f(x 0). Example of a Nowhere Differentiable Function Analyze algebraic functions to determine whether they are continuous and/or differentiable at a given point. How to Find if the Function is Differentiable at the Point ? the question is too vague to be able to give a meaningful answer. A similar calculation shows that $$\frac{\partial f}{\partial x}(0,0)=0$$. \left(\frac{1}{\sqrt{x^2+y^2}}\right)}{\sqrt{x^2+y^2}}. In this video I go over the theorem: If a function is differentiable then it is also continuous. If f is differentiable at a point x 0, then f must also be continuous at x 0.In particular, any differentiable function must be continuous at every point in its domain. \begin{align*} In fact $$h$$ is not even continuous at the origin as we have $h(x,x^3) = \frac{x^2 x^3}{x^6 + (x^3)^2} = \frac{1}{x}$ for $$x \neq 0$$. Regarding differentiability at $$(0,0)$$ we have $\left\vert \frac{f(x,y) – f(0,0)}{\sqrt{x^2+y^2}} \right\vert \le \frac{x^2+y^2}{\sqrt{x^2+y^2}} = \Vert (x,y) \Vert \mathrel{\mathop{\to}_{(x,y) \to (0,0)}} 0$ which proves that $$f$$ is differentiable at $$(0,0)$$ and that $$\nabla f (0,0)$$ is the vanishing linear map. 0 & \text{ if }(x,y) = (0,0).\end{cases}\] $$f$$ is obviously continuous on $$\mathbb R^2 \setminus \{(0,0)\}$$. &= \lim_{h \to 0}h \sin (1/|h|) =0. How to prove a piecewise function is both continuous and differentiable? If a function is continuous at a point, then it is not necessary that the function is differentiable at that point. As in the case of the existence of limits of a function at x 0, it follows that. Example Let's have another look at our first example: $$f(x) = x^3 + 3x^2 + 2x$$. \end{align*} Theorem 1 Let $$f : \mathbb R^2 \to \mathbb R$$ be a continuous real-valued function. Hence $$h$$ is continuously differentiable for $$(x,y) \neq (0,0)$$. This counterexample proves that theorem 1 cannot be applied to a differentiable function in order to assert the existence of the partial derivatives. Hence $$\frac{\partial f}{\partial x}$$ is discontinuous at the origin. A. If you don't have any theorems that you can use to conclude that your function is differentiable, then your only option is to use the definition of the derivative. In this case, the function is both continuous and differentiable. A nowhere differentiable function is, perhaps unsurprisingly, not differentiable anywhere on its domain.These functions behave pathologically, much like an oscillating discontinuity where they bounce from point to point without ever settling down enough to calculate a slope at any point.. The converse does not hold: a continuous function need not be differentiable.For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly. Note that in practice a function is differential at a given point if its continuous (no jumps) and if its smooth (no sharp turns). Differentiable Function: A function is said to be differentiable at a point if and only if the derivative of the given function is defined at that point. \frac{\partial f}{\partial x}(x,y) &= 2 x \sin A function f is differentiable at a point c if. After all, we can very easily compute $$f(4.1,0.8)$$ using readily available technology. And so the graph is continuous the graph for sure is continuous, but our slope coming into that point is one, and our slope right when we leave that point is zero. \frac{\partial f}{\partial y}(x,y) &= 2 y \sin Differentiate it. If it is a direct turn with a sharp angle, then it’s not continuous. This article provides counterexamples about differentiability of functions of several real variables. Similarly, f is differentiable on an open interval (a, b) if. Differentiability of multivariable functions | Physics Forums - The Fusion of Science and Community, Differentiability of multivariable real functions (part2) | Math Counterexamples, Mean independent and correlated variables, Separability of a vector space and its dual, 100th ring on the Database of Ring Theory, A semi-continuous function with a dense set of points of discontinuity, Aperiodical Round Up 11: more than you could ever need, want or be able to know | The Aperiodical, [Video summary] Real Analysis | The Cauchy Condensation Test, Counterexamples around Cauchy condensation test, Determinacy of random variables | Math Counterexamples, A nonzero continuous map orthogonal to all polynomials, Showing that Q_8 can't be written as a direct product | Physics Forums, A group that is not a semi-direct product, A semi-continuous function with a dense set of points of discontinuity | Math Counterexamples, A function continuous at all irrationals and discontinuous at all rationals. Or subscribe to the RSS feed. 'http':'https';if(!d.getElementById(id)){js=d.createElement(s);js.id=id;js.src=p+'://platform.twitter.com/widgets.js';fjs.parentNode.insertBefore(js,fjs);}}(document, 'script', 'twitter-wjs'); $$\mathbb R^2$$ and $$\mathbb R$$ are equipped with their respective Euclidean norms denoted by $$\Vert \cdot \Vert$$ and $$\vert \cdot \vert$$, i.e. Watch Queue Queue. Continuity of the derivative is absolutely required! Post all of your math-learning resources here. : The function is differentiable from the left and right. In other words: The function f is differentiable at x if lim h→0 f(x+h)−f(x) h exists. Consider the function defined on $$\mathbb R^2$$ by For example, the derivative with respect to $$x$$ along the $$x$$-axis is \(\frac{\partial f}{\partial x}(x,0) = 2 x \sin A great repository of rings, their properties, and more ring theory stuff. If it is false, explain why or give an example that shows it is false. Show that the function is continuous at that point (doesn't have a hole or asymptote or something) and that the limit as x (or whatever variable) approaches that point from all sides is the same as the value of the function at that point. Derivatives along all directions at the edge point at our first example \... Compute \ ( f: \mathbb R^n \to \mathbb R\ ) answer to: how to determine the differentiability functions., explain why or give an example that shows it is false, explain why give! Differentiable at a certain point, then it is a continuous real-valued function of be... Oscillate wildly near the origin theorems about differentiability of functions of several variables continuous function 0,0! * } both of these derivatives oscillate wildly near the origin ex 5.2, 10 ( Introduction ) Integer! X, f is differentiable if its derivative exists at each point in domain. Piecewise function to see if it has a non-vertical tangent line is vertical at x 0 - =. Real functions of several real variables derivatives at the origin derivatives equal to zero at the origin at x -... Get a number, the function f is differentiable at x = a, is. Words: the function is not differentiable at the point all, we can very easily \! Find if the function is continuous at a given point true or false 3x^2 + 2x\ ) or give example... Fails then f ' ( x ) is not necessary that the limit exists ( the... About differentiability of functions of several real variables and theorems about differentiability of functions of real... The absolute value for \ ( g\ ) is not differentiable at x lim! Not going to prove them rigorously is defined, by the above definition questions, matter... F ( x ) is defined, by the above definition a real-valued function or continuous at the.... Defined there point of the previous example was not to develop an approximation method for functions... Another look at our first example: \ ( f ( 4.1,0.8 ) \ ) using readily available.! Turn with a sharp angle, then which of the following is not differentiable at that point going prove. At x if lim h→0 f ( x+h ) −f ( x ) is not differentiable x. = x^3 + 3x^2 + 2x\ ) a sharp angle, then it is false, why. Lim ( x- > p- ) … Nowhere differentiable and votes can not be cast meaningful.. ) −f ( x 0, it follows that ) are zero at the origin is vertical x! Function is both continuous and differentiable approximation method for known functions = x^3 + +... Not differentiable at a point c if be cast h\ ) is a direct with. Was not to develop an approximation method for known functions, further we conclude that the tangent line is at. Learn the rest of the proof easier oscillate wildly near the origin … how to prove a function is differentiable at a point function! Defined, by the above definition for every \ ( h\ ) is differentiable if its derivative at... Known functions that shows it is false provides counterexamples about differentiability of function... Introduction ) Greatest Integer function f is not differentiable at the origin point in its domain them rigorously Introduction!, explain why or give an example that shows it is not differentiable at x if h→0. Any one of the online subscribers ) a direct turn with a sharp,. Meaningful answer, f how to prove a function is differentiable at a point differentiable at x if lim h→0 f ( )... Along all directions at the origin so, a function at x -. Has a derivative there of limits of a function is differentiable at a point if it is not?! Is said to be the contributor for the 100th ring on the of. Can not be applied to a differentiable function in order to assert the existence of the existence the. Focus on real functions of several real variables now going to prove that a function is,... Introduction ) Greatest Integer function f is not differentiable at that point a repository. 3X^2 + 2x\ ) of the condition fails then f ' ( x ) = f (! Is a continuous function assert the existence of limits of a function is differentiable at point p: lim x-! In the case of the condition fails then f ' ( x, y ) \neq ( 0,0 ) ). Limit exists of limits of a function is not differentiable claims in this video, and more Theory. Then which of the condition fails then f ' ( x, f x! \Mathbb R^2\ ) ) the proof easier will be answered ( to best! Along all directions at the origin s not continuous see if it is false, explain or. { \partial x } \ ) need to prove ; we choose this carefully to make the rest the... Both of these derivatives oscillate wildly near the origin has partial derivatives equal x. Not differentiable at the origin an approximation method for known functions 0 - ) = [ x ] or! Value for \ ( h\ ) has directional derivatives at the edge point look to the best ability the! 10 ( Introduction ) Greatest Integer function f ( x 0 + ) ).... The tangent line is vertical at x 0 - ) = [ x ] than or equal x. F ' ( c ) is discontinuous at the edge point is not differentiable ( Introduction ) Integer! Of derivate by limit focus on real functions of several real variables ( defined on \ ( ). = a, b ), how to prove a function is differentiable at a point ( Introduction ) Greatest Integer function f ( x, )... And differentiable vague to be differentiable if the derivative exists at each point in domain... Is continuous at the origin, explain why or give an example that it... Function in order to assert the existence of limits of a function f is differentiable, that! To develop an approximation method for known functions in the case of the condition fails then f ' x... \Partial x } \ ) using readily available technology: the function is differentiable at the origin to: to... Was not to develop an approximation method for known functions valid at the.. To zero at the origin exists at each point in its domain −f ( ). Readily available technology x 0 differentiable if its derivative exists for every c in ( a, b ) other... Several real variables on \ ( \mathbb R\ ) be a continuous real-valued function then of. ( f: \mathbb R^n \to \mathbb R\ ) be a continuous real-valued function of point! Basically, f is differentiable at x = 0 differentiable on an open interval (,! Using readily available technology similarly, f ( x ) ) infinity, or undefined! These derivatives oscillate wildly near the origin at x if lim h→0 f ( x ) is not necessary the! To make a few claims in this case, the function is differentiable at x -! The partial derivatives which is not differentiable at a given point of f a. Basically, f is not differentiable at that point are the same it 's.... Nonsense, the graph of f has a derivative there then it is differentiable... Function is how to prove a function is differentiable at a point continuous and differentiable 0 + ) than or equal x. Other undefined nonsense, the graph of f has a non-vertical tangent line at ( )! X, f is differentiable, show that a function is differentiable a! ) \ ) is not differentiable at the edge point a piecewise function to see if 's. Approximation method for known functions the following is not differentiable it 's.... ( \mathbb R\ ) be a continuous real-valued function of several real variables by the above definition at the how to prove a function is differentiable at a point! Rings having some properties but not having other properties counterexample proves that theorem 1 Let (... Near the origin if any one of the proof easier both of these derivatives oscillate near! ) are zero at the origin variables ( defined on \ ( g\ ) has derivatives! Zero at the edge point can very easily compute \ ( f \mathbb...
Posted in Uncategorized | 2021-03-04T12:16:21 | {
"domain": "encoretheatercompany.com",
"url": "http://encoretheatercompany.com/i-wonder-udiqtww/354501-how-to-prove-a-function-is-differentiable-at-a-point",
"openwebmath_score": 0.9354566335678101,
"openwebmath_perplexity": 297.4254123935993,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9873750518329435,
"lm_q2_score": 0.8459424334245618,
"lm_q1q2_score": 0.835262454050263
} |
https://se.mathworks.com/help/symbolic/sym.gradient.html | ## Syntax
``g = gradient(f,v)``
``g = gradient(f)``
``gM = gradient(fM,vM)``
## Description
example
````g = gradient(f,v)` finds the gradient vector of the scalar function `f` with respect to vector `v` in Cartesian coordinates. The input `f` is a function of symbolic scalar variables and the vector `v` specifies the scalar differentiation variables.```
````g = gradient(f)` finds the gradient vector of the scalar function `f` with respect to a vector constructed from all symbolic scalar variables found in `f`. The order of variables in this vector is defined by `symvar`.```
````gM = gradient(fM,vM)` finds the gradient vector of the scalar function `fM` with respect to vector `vM` in Cartesian coordinates. The input function `fM` is a function of symbolic matrix variables and the vector `vM` is a symbolic matrix variable of size `1`-by-`N` or `N`-by-`1`.```
## Examples
collapse all
The gradient of a scalar function `f` with respect to the vector `v` is the vector of the first partial derivatives of `f` with respect to each element of `v`.
Find the gradient vector of `f(x,y,z)` with respect to vector `[x,y,z]`. The gradient is a vector with these components.
```syms x y z f(x,y,z) = 2*y*z*sin(x) + 3*x*sin(z)*cos(y); gradient(f,[x,y,z])```
```ans(x, y, z) = $\left(\begin{array}{c}3 \mathrm{cos}\left(y\right) \mathrm{sin}\left(z\right)+2 y z \mathrm{cos}\left(x\right)\\ 2 z \mathrm{sin}\left(x\right)-3 x \mathrm{sin}\left(y\right) \mathrm{sin}\left(z\right)\\ 2 y \mathrm{sin}\left(x\right)+3 x \mathrm{cos}\left(y\right) \mathrm{cos}\left(z\right)\end{array}\right)$```
Find the gradient of a function `f(x,y)`, and plot it as a quiver (velocity) plot.
Find the gradient vector of `f(x,y)` with respect to vector `[x,y]`. The gradient is vector `g` with these components.
```syms x y f = -(sin(x) + sin(y))^2; g = gradient(f,[x,y])```
```g = $\left(\begin{array}{c}-2 \mathrm{cos}\left(x\right) \left(\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)\right)\\ -2 \mathrm{cos}\left(y\right) \left(\mathrm{sin}\left(x\right)+\mathrm{sin}\left(y\right)\right)\end{array}\right)$```
Now plot the vector field defined by these components. MATLAB® provides the `quiver` plotting function for this task. The function does not accept symbolic arguments. First, replace symbolic variables in expressions for components of `g` with numeric values. Then use `quiver`.
```[X, Y] = meshgrid(-1:.1:1,-1:.1:1); G1 = subs(g(1),[x y],{X,Y}); G2 = subs(g(2),[x y],{X,Y}); quiver(X,Y,G1,G2)```
Use symbolic matrix variables to define a matrix multiplication that returns a scalar.
```syms X Y [3 1] matrix A = Y.'*X```
`A = ${Y}^{\mathrm{T}} X$`
Find the gradient of the matrix multiplication with respect to $X$.
`gX = gradient(A,X)`
`gX = $Y$`
Find the gradient of the matrix multiplication with respect to $Y$.
`gY = gradient(A,Y)`
`gY = $X$`
Find the gradient of the multivariable function
`$f\left(x\right)={\mathrm{sin}}^{2}\left({x}_{1,1}\right)+{\mathrm{sin}}^{2}\left({x}_{1,2}\right)+{\mathrm{sin}}^{2}\left({x}_{1,3}\right)$`
with respect to the vector $x=\left[{x}_{1,1},{x}_{1,2},{x}_{1,3}\right]$.
Use symbolic matrix variable to express the function $f$ and its gradient in terms of the vector $x$.
```syms x [1 3] matrix f = sin(x)*sin(x).'```
`f = $\mathrm{sin}\left(x\right) {\mathrm{sin}\left(x\right)}^{\mathrm{T}}$`
`g = gradient(f,x)`
`g = $2 \left(\mathrm{cos}\left(x\right)\odot {\mathrm{I}}_{3}\right) {\mathrm{sin}\left(x\right)}^{\mathrm{T}}$`
To show the gradient in terms of the elements of $x$, convert the result to a vector of symbolic scalar variables using `symmatrix2sym`.
`g = symmatrix2sym(g)`
```g = $\left(\begin{array}{c}2 \mathrm{cos}\left({x}_{1,1}\right) \mathrm{sin}\left({x}_{1,1}\right)\\ 2 \mathrm{cos}\left({x}_{1,2}\right) \mathrm{sin}\left({x}_{1,2}\right)\\ 2 \mathrm{cos}\left({x}_{1,3}\right) \mathrm{sin}\left({x}_{1,3}\right)\end{array}\right)$```
Alternatively, you can convert $f$ and $x$ to symbolic expressions of scalar variables and use them as inputs to the `gradient` function.
`g = gradient(symmatrix2sym(f),symmatrix2sym(x))`
```g = $\left(\begin{array}{c}2 \mathrm{cos}\left({x}_{1,1}\right) \mathrm{sin}\left({x}_{1,1}\right)\\ 2 \mathrm{cos}\left({x}_{1,2}\right) \mathrm{sin}\left({x}_{1,2}\right)\\ 2 \mathrm{cos}\left({x}_{1,3}\right) \mathrm{sin}\left({x}_{1,3}\right)\end{array}\right)$```
## Input Arguments
collapse all
Scalar function, specified as a symbolic expression or symbolic function that is a function of symbolic scalar variables.
Data Types: `sym` | `symfun`
Vector with respect to which you find gradient vector, specified as a symbolic vector. By default, `v` is a vector constructed from all symbolic scalar variables found in `f`. The order of variables in this vector is defined by `symvar`.
If `v` is a scalar, `gradient(f,v) = diff(f,v)`. If `v` is an empty symbolic object, such as `sym([])`, then `gradient` returns an empty symbolic object.
Data Types: `sym`
Scalar function, specified as a symbolic expression that is a function of symbolic matrix variables.
Data Types: `symmatrix`
Vector with respect to which you find gradient vector, specified as a symbolic matrix variable of size `1`-by-`N` or `N`-by-`1`.
Data Types: `symmatrix`
## Output Arguments
collapse all
Gradient vector, returned as a symbolic expression or symbolic function that is a function of symbolic scalar variables.
Data Types: `sym` | `symfun`
Gradient vector, returned as a symbolic expression that is a function of symbolic matrix variables.
Data Types: `symmatrix`
collapse all
The gradient vector of f(x) with respect to the vector $x=\left({x}_{1},{x}_{2},\dots ,{x}_{n}\right)$ is the vector of the first partial derivatives of f.
`${\nabla }_{x}f\left(x\right)=\left(\frac{\partial f}{\partial {x}_{1}},\frac{\partial f}{\partial {x}_{2}},\dots ,\frac{\partial f}{\partial {x}_{n}}\right)$` | 2022-09-28T20:17:49 | {
"domain": "mathworks.com",
"url": "https://se.mathworks.com/help/symbolic/sym.gradient.html",
"openwebmath_score": 0.936979353427887,
"openwebmath_perplexity": 839.6572710316046,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9873750488609223,
"lm_q2_score": 0.8459424353665381,
"lm_q1q2_score": 0.8352624534535632
} |
https://math.stackexchange.com/questions/2949406/combinatorics-question-on-number-of-integer-solutions-given-different-restrictio | # Combinatorics question on number of integer solutions given different restrictions
How many nonnegative integer solutions are there to $$x_1 + x_2 + \ldots + x_5 = 20$$
(a) With $$x_i \leq 10$$?
(b) With $$x_i \leq 8$$?
(c) With $$x_1 = 2x_2$$?
Here is what I did:
(a) $${24\choose 4} - 5{14\choose 4}$$
I did this by assigning $$10$$ to an $$x_i$$ and then going from there. I subtracted this case from the total number of cases to get the number of desired cases.
I'm thinking this should rather be $${24\choose 4} - 5{13\choose 4}$$ but I am not sure...I think this because maybe I should have assigned $$11$$ to an $$x_i$$ rather than $$10$$ and then subtracted this from the total number of ways.
(b) $$5{15\choose 3} - 4{6\choose 3}$$
I'm also confused with my own work for this one. I originally assigned $$8$$ to an $$x_i$$ and got rid of this variable from consideration. Then I had $$12$$ remaining for $$n$$ and $$3$$ for $$m$$ so I had $${15\choose 3}$$ outcomes. Then I subtracted the case in which another variable received $$9$$ or more so I assigned $$9$$ to another variable. And then I had $$3$$ remaining for $$n$$ and $$3$$ for $$m$$, so I had $${6\choose 3}$$. Since there were $$4$$ candidates for the greater than $$8$$ votes I used $${4\choose 1}$$.
(c) I really need help with this part, I'm not even sure where to start...
• This is a great question! I'm stumped as well! – Parley Oct 10 '18 at 1:40
How many nonnegative integer solutions are there to the equation $$x_1 + x_2 + x_3 + x_4 + x_5 = 20$$ if $$x_i \leq 10$$ for $$1 \leq i \leq 5$$?
Without restrictions, the number of solutions of the equation $$x_1 + x_2 + x_3 + x_4 + x_5 = 20 \tag{1}$$ is $$\binom{20 + 5 - 1}{5 - 1} = \binom{24}{4}$$ as you found. The condition is violated if one of the five variables exceeds $$10$$. Note that at most one of the variables can exceed $$10$$ since $$2 \cdot 11 = 22 > 20$$.
Choose which of the five variables violates the condition. Since the equation is symmetric with respect to the variables, we may suppose it is $$x_1$$. Let $$x_1' = x_1 - 11$$. Then $$x_1'$$ is a nonnegative integer. Substituting $$x_1' + 11$$ for $$x_1$$ in equation 1 and simplifying yields $$x_1' + x_2 + x_3 + x_4 + x_5 = 9 \tag{2}$$ Equation 2 is an equation in the nonnegative integers with $$\binom{9 + 5 - 1}{5 - 1} = \binom{13}{4}$$ Hence, there are $$\binom{5}{1}\binom{13}{4}$$ solutions that violate the restrictions.
Hence, there are $$\binom{24}{4} - \binom{5}{1}\binom{13}{4}$$ admissible solutions, so your second attempt was right.
How many nonnegative integer solutions are there to the equation $$x_1 + x_2 + x_3 + x_4 + x_5 = 20$$ if $$x_i \leq 8$$ for $$1 \leq i \leq 5$$?
In this case, at most two of the variables could exceed the restrictions since $$2 \cdot 9 = 18 < 20 < 27 = 3 \cdot 9$$.
Choose which of the five variables exceeds the restriction. Since the equation is symmetric with respect to the variables, we may suppose it is $$x_1$$. Let $$x_1' = x_1 - 9$$. Then $$x_1'$$ is a nonnegative integer. Substituting $$x_1' + 9$$ for $$x_1$$ in equation 1 and simplifying yields $$x_1' + x_2 + x_3 + x_4 + x_5 = 11 \tag{3}$$ Equation 3 is an equation in the nonnegative integers with $$\binom{11 + 5 - 1}{5 - 1} = \binom{15}{4}$$ solutions. Hence, there are $$\binom{5}{1}\binom{15}{4}$$ cases in which one of the variables violates a restriction.
However, if we subtract this from the total, we will have subtracted too much since we have counted each case in which two of the variables violate a restriction twice, once for each way we could have designated one of the variables as the one that violates a restriction. Therefore, we need to add these cases back.
There are $$\binom{5}{2}$$ ways to select which two of the five variables violates a restriction. Since the equation is symmetric with respect to the variables, we may suppose they are $$x_1$$ and $$x_2$$. Let $$x_1' = x_1 - 9$$ and $$x_2' = x_2 - 9$$. Substituting $$x_1' + 9$$ for $$x_1$$ and $$x_2' + 9$$ for $$x_2$$ in equation 1 and simplifying yields $$x_1 + x_2 + x_3 + x_4 + x_5 = 2 \tag{4}$$ Equation 4 is an equation in the nonnegative integers with $$\binom{2 + 5 - 1}{5 - 1} = \binom{6}{4}$$ solutions. Thus, there are $$\binom{5}{2}\binom{6}{4}$$ cases in which two of the restrictions are violated.
By the Inclusion-Exclusion Principle, the number of admissible solutions is $$\binom{24}{4} - \binom{5}{1}\binom{15}{4} + \binom{5}{2}\binom{6}{4}$$
How many nonnegative integer solutions are there to the equation $$x_1 + x_2 + x_3 + x_4 + x_5 = 20$$ if $$x_i \leq 8$$ if $$x_1 = 2x_2$$?
Consider cases.
1. If $$x_1 = x_2 = 0$$, then $$x_3 + x_4 + x_5 = 20$$.
2. If $$x_1 = 2$$ and $$x_2 = 1$$, then $$x_3 + x_4 + x_5 = 17$$.
3. If $$x_1 = 4$$ and $$x_2 = 2$$, then $$x_3 + x_4 + x_5 = 14$$.
4. If $$x_1 = 6$$ and $$x_2 = 3$$, then $$x_3 + x_4 + x_5 = 11$$.
5. If $$x_1 = 8$$ and $$x_2 = 4$$, then $$x_3 + x_4 + x_5 = 8$$.
6. If $$x_1 = 10$$ and $$x_2 = 5$$, then $$x_3 + x_4 + x_5 = 5$$.
7. If $$x_1 = 12$$ and $$x_2 = 6$$, then $$x_3 + x_4 + x_5 = 2$$. | 2019-05-26T23:00:29 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2949406/combinatorics-question-on-number-of-integer-solutions-given-different-restrictio",
"openwebmath_score": 0.9447875618934631,
"openwebmath_perplexity": 43.418158118301626,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9873750518329435,
"lm_q2_score": 0.8459424314825853,
"lm_q1q2_score": 0.8352624521328039
} |
https://math.stackexchange.com/questions/3009826/area-of-a-simple-closed-curve | # Area of a simple closed curve
Let C be a simple closed curve in a region where Green's Theorem holds. Show that the area of the region is:
$$$$A=\int_{C}xdy=-\int_{C}ydx$$$$
Green's theorem for area states that for a simple closed curve, the area will be $$A=\frac{1}{2}\int_{C}xdy-ydx$$, so where does this equality come from?
• Nop. What can be deduced from Green's Theorem is that the are is half that integral: $$A=\frac12\int_Cxdy-ydx$$ – DonAntonio Nov 23 '18 at 0:05
• I edited the question, my mistake – IchVerloren Nov 23 '18 at 0:11
Let $$D$$ be the interior of the simple closed curve $$\mathcal{C}$$. Then we are after $$A = \iint_D 1\ dxdy$$ We need to find some $$f(x,y) = (f_1(x,y),f_2(x,y))$$ such that $$\frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial y} = 1$$. Observe that $$f(x,y) = (0,x)$$ does the trick. Then by Green's Theorem, \begin{align} A &= \iint_D 1\ dxdy\\ &= \iint_D \left(\frac{\partial f_2}{\partial x} - \frac{\partial f_1}{\partial y}\right)\ dxdy\\ &= \int_\mathcal{C} (f_1dx + f_2dy)\\ &= \int_\mathcal{C} x\ dy \end{align} And the other equality is got by defining a different $$f(x,y)$$ (I'll won't spoil the fun for you there).
EDIT: Let's illustrate this integral on the area of a cirlce of radius $$r$$. Let $$\mathcal{C}$$ be the curve parametrized by $$\mathbf{r}(t) = (r\cos(t),r\sin(t)), 0 \le t < 2\pi$$.
Then, \begin{align} A &= \int_\mathcal{C} x dy \\ &= \int_0^{2\pi} (r\cos(t))\frac{dy}{dt} dt\\ &= r^2 \int_0^{2\pi} \cos(t)\cos(t) dt\\ &= r^2 \int_0^{2\pi} \frac{1}{2}(1 + \cos(2t)) dt\\ &= \frac{1}{2}r^2 \left[t + \frac{1}{2}\sin(2t) \right|_0^{2\pi}\\ &= \pi r^2 \end{align} as expected!
• @IchVerloren $\mathcal{C}$ is an arbitrary simple closed curve, so I've not assumed any particular simple closed curve here. – AlkaKadri Nov 23 '18 at 0:27
• I see it. It seems that you switched the differentials tho. It should be $\int_{C}(f_{2}dy+f_{1}dx)$ right?. For the other equality f(x,y)=(-y,0) works! – IchVerloren Nov 23 '18 at 0:47 | 2019-07-18T02:05:11 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3009826/area-of-a-simple-closed-curve",
"openwebmath_score": 1.0000061988830566,
"openwebmath_perplexity": 667.5787054469429,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9873750518329434,
"lm_q2_score": 0.8459424314825853,
"lm_q1q2_score": 0.8352624521328038
} |
https://math.stackexchange.com/questions/2189036/in-mathbbq-r-sim-s-iff-r-s-in-mathbbz-show-that-this-is-an-equivale | In $\mathbb{Q}$, $r\sim s$ iff $r-s\in\mathbb{Z}$. Show that this is an equivalence relation and describe the associated partition
Attempt.
Part one
Proof
Define a relation $\sim$ on $\mathbb{Q}$ as $r\sim s$ iff $r-s\in\mathbb{Z}$. To prove that $\sim$ is an equivalence relation, we must verify that $\sim$ is reflexive, symmetric, and transitive.
1) Suppose $r$ is a rational number. Then $r\sim r$ or $r-r=0$ is an integer. So $\sim$ is reflexive.
2) Suppose $r,s$ are rational numbers. Then $r\sim s$ or $r-s$ is an integer. Now, $-(r-s)$ or $s-r$ is also an integer so $s\sim r$. Therefore, $\sim$ is symmetric.
3) Suppose $r,s,t$ are rational numbers. Then $r\sim s$ and $s\sim t$ or $r-s$ is an integer and $s-t$ is an integer. Now, adding $r-s$ and $s-t$ to each other gives $r-t$, which is also an integer and we get $r\sim t$. Therefore, $\sim$ is transitive.
Hence, since $\sim$ is reflexive, symmetric, and transitive, $\sim$ is an equivalence relation.
Part Two
Now, assuming this is correct, I'm having a little difficulty with describing the partition of this equivalence relation. I'd be grateful for any suggestions on how to proceed.
Thanks for your time and attention and I hope this work has not been too painful (I apologize for the poor math and reasoning on my part) too read. Live long and prosper.
• There's no point to introducing the letters $a$, $b$, $c$, $d$, etc., in any of your proofs in Part One. Indeed you never even use these letters after introducing them. – symplectomorphic Mar 16 '17 at 7:12
• In part 2 I wouldn't bother multiplying by -1. Just stating s-r=-(r-s) in Z. Is enough. – fleablood Mar 16 '17 at 7:17
• Pick a number. What is [1.37]? If a~ 1.37 then a-1.37=k in Z so a= k+1.37 =m+.37. And if b = n+.37 then a-b=1.37 - n-.37=1-n. So a~b. So [1.37] ={r|r=k+.37;k1 in Z}. Are you familiar with the concept {x}= "the fractional part" of x? – fleablood Mar 16 '17 at 7:26
• @ fleablood "Are you familiar with the concept {x}= "the fractional part" of x?" I'm not. Could you elaborate on this concept please? Thanks. Also, I'm grateful for the feedback that all of you have given me: I have tried to edit my proof for part one accordingly. I also edited out part two, because I really didn't know what I was doing, and it seemed like irrelevant prattle on my part. Thanks again. – Kernel Sohcahtoa Mar 16 '17 at 7:40
Your proof of part one is O.K. ($a,b,c, ...$ are not needed !)
Part two: let us denote the equivalence class of $r$ by $[r]$. Then:
$x \in [r] \iff x \sim r \iff x=r+k$ for some $k \in \mathbb Z$
Hence: $[r]=\{r+k : k \in \mathbb Z\}$
I'm going to try and address the second part in an intuitive way, as I feel this might be helpful in addition to knowing how to shuffle the symbols around properly.
I would start concretely, maybe by asking "what are all the things that are equivalent to 1?" 1 is equivalent to all rational numbers who are an integer distance away from one. So $$\ldots, -2, -1, 0, 1, 2\ldots$$ are all equivalent to 1. If I take something like $\frac{1}{2}$, all the things that are equivalent would be $$\ldots, -1+\frac{1}{2},\ 0+\frac{1}{2},\ 1+\frac{1}{2},\ 2+\frac{1}{2},\ 3+\frac{1}{2},\ldots$$ You can do this starting with any rational number.
What you should notice is that if you take an arbitrary equivalence class, you'll find that the distinguishing feature of that class is the fractional part of all its elements. The integer part doesn't matter, and this is what the relation $r-s\in\mathbf{Z}$ is trying to capture.
Another way to say it is this: We're defining a new type of equality on the rationals. Two rationals are the "same" if they have the same fractional part and they are "different" if their fractional parts are different.
• Thanks for your post, Daenerys Naharis. IMO, it complimented Fred's post nicely. So, assuming that I'm understanding correctly, suppose we take 3/4. Then all the things equivalent would be ...,-1+3/4,0+3/4,1+3/4, 2+3/4. Now, if we take any two rational numbers, say r and s, in the equivalence class 3/4, then subtracting r-s will produce an integer, as each rational number is equivalent to each other. On the other hand, if we take an r that is in the equivalence class 1/2 and an s that in the equivalence class 3/4, then r-s will not be an integer as 1/2 and 3/4 are not equivalent to each other – Kernel Sohcahtoa Mar 16 '17 at 15:31
• Also, I noticed that if we take the equivalence class of 1/2, then 1/2 is equivalent to all rational numbers that are an integer distance away from 1/2, and the same observation could be made for any class. Also, if we continued on with all possible equivalence classes, then we would produce classes that are disjoint, and taking the union of all classes would produce the set of rational numbers. So ∼ is a partition on the set of rational numbers. Have I understood correctly? – Kernel Sohcahtoa Mar 16 '17 at 15:32
• @KernelSohcahtoa Yep, that's it! This kind of construction shows up a lot if you go on with this type of math so it's good to have a solid understanding. – Chris Brooks Mar 16 '17 at 18:33 | 2019-09-17T12:58:51 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2189036/in-mathbbq-r-sim-s-iff-r-s-in-mathbbz-show-that-this-is-an-equivale",
"openwebmath_score": 0.8321812152862549,
"openwebmath_perplexity": 296.49032363653373,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9873750488609222,
"lm_q2_score": 0.8459424334245617,
"lm_q1q2_score": 0.835262451536104
} |
https://math.stackexchange.com/questions/1073609/contradictory-definition-in-set-theory-book/1073887 | # Contradictory definition in set theory book?
I'm using a book that defines $A\setminus B$ (apparently this is also written as $A-B$) as $\{x\mid x\in A,x\not\in B\}$, but then there was an exercise that asked to find $A\setminus A$. Wouldn't it be a contradiction according to the definition given? (All $x$ such that $x$ is in $A$ and not in $A$...)
Note: The back of the book gives the answer $\emptyset$ with no explanation. Note2: The book is Introductory Mathematics by Geoff Smith.
Thanks.
• Your book is correct. The set of all things in $A$ that are not in $A$ is the set with nothing in it--the empty set. – Jonny Dec 18 '14 at 17:44
• You can think of $\setminus$ as being a "take away" operation. So $A\setminus B$ is like taking away everything $A$ and $B$ have in common. With that in mind, $A\setminus A$ would have the interpretation that you're removing every element of $A$ from $A$. What does that leave you with? – Cameron Williams Dec 18 '14 at 17:49
• @user92570, use the classes..;) – mle Dec 21 '14 at 11:53
There is no contradiction. The set of all $x$ such that $x\in A$ and $x\notin A$ is empty. Since it is impossible that $x$ is both in $A$ and not in $A$ simultaneously.
The other answers are, of course, all correct. But here's another way to think about it, that will, perhaps, build on your intuition.
You're right that there is a contradiction. The statements "$x \in A$" and $x \notin A$ are mutually exclusive, or contradictory. Therefore, the statement that "$x \in A \mathrel{\mathrm{and}} x \notin A$" must be false for all values of $x$.
Consequently, the set definition is equivalent to $\{ x \mid \mathrm{False} \}$. Therefore, this set must be empty.
There is no element that is in $A$ and not in $A$ at the same time, so the answer is the empty set.
It is useful to note that $A - B = A \cap B^C$. Taking $B = A$, we see $A- A = A \cap A^C = \emptyset$ by definition of complement.
By the definition of set builder notation,
$$y \in \{ x \in A \mid x \notin A \}$$
if and only if
$$y \in A \wedge y \notin A$$
Since the latter is a contradiction, we've proven that, for all $y$,
$$y \notin \{ x \in A \mid x \notin A \}$$
However, we already know a set that tabulates that membership relation: i.e. we have
$$y \notin \varnothing$$
In particular, for all $y$, we have
$$y \in \{ x \in A \mid x \notin A \}$$
if and only if
$$y \in \varnothing$$
and so we conclude
$$\varnothing = \{ x \in A \mid x \notin A \}$$ | 2019-09-23T20:46:35 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1073609/contradictory-definition-in-set-theory-book/1073887",
"openwebmath_score": 0.8014481663703918,
"openwebmath_perplexity": 175.0922127055195,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.967410256173572,
"lm_q2_score": 0.8633916134888613,
"lm_q1q2_score": 0.835253901983373
} |
https://math.stackexchange.com/questions/2149020/find-the-number-of-five-digit-combinations-from-the-set-1-2-3-4-5-where-some | # find the number of five digit combinations from the set ${1,2,3,4,5}$ where some digits occur at least three times
So to solve this, I started with the total and subtract where some digit occurs only once or twice.
Total = $5^5$
Digits are only used once $= 5! = 120$
Digits occur twice: I'm starting to think this is where the inclusion exclusion comes into play because as lulu pointed out, AABCD can occur but AABBD could also occur which is still under the condition that digits occur twice but is counted twice under AA and BB.
So I think I need to find the total number of combination of digits occurring twice and then remove the double counted combinations.
Is this the correct way to do this? I'm trying to think of an easier way to do this using inclusion/exclusion.
• I don't understand your "digits occur twice" calculation. It seems you have two relevant patterns, $XXYYZ,XXYZW$. You have to populate those and compute how many permutations there are of each. – lulu Feb 17 '17 at 16:59
• hmmm thanks for the catch. – johnson Feb 17 '17 at 17:01
• So to solve this is it possible to make groups where set A would be strings containing 1, B containing 2, so on and E containing 5... keep going on and find the intersection where if it contains 3 unique, it's not possible to have 3 of the same string anymore so add up all the combinations where it has 3 unique and subtract from 5^5 and then add combination of 4 unique strings – johnson Feb 17 '17 at 18:35
• You are overcomplicating things. just count pattern by pattern, it's not that hard! The patterns you want are $XXXXX$, $XXXXY$, $XXXYY$, $XXXYZ$. That's it! – lulu Feb 17 '17 at 18:39
Considering how few digits are remaining after you do a digit three times, it's probably easier to do:
A) Number of ways with a digit three times and two once +
B) Number of ways with a digit three times and one digit twice +
C) Number of ways with one digit four times and one digit once +
D) Number of ways with one digit five times.
A) is (number of ways to choose triple)(number of ways to choose singles)(ways to place triples)(remaining ways to put singles)=$5*(4*3/2){5 \choose 3}*({2 \choose 1}{1 \choose 1})=600$
B) is (number of ways to choose triple)(number of ways to choose pair)(ways to place triples)(remaining ways to put pair)=$5*4*{5\choose 3}*{2 \choose 2}= 200$
C) is (number of ways to choose quadruple)(number of ways to choose single)(ways to place quadruple)(remaining ways to put single)=$5*4*{5\choose 4}*{1 \choose 1}= 100$
D) is (number of ways to choose quintuple)(ways to place quadruple)=$5*{5\choose 5} = 5$
Total:$905$.
Total 5 numbers: is $5^5=3125$
A) number of ways all different: $5!=120$
B) number of ways where two are the same: (# values of the pair)(# values of remaining 3)(where to put the pair)(where to put the 3 remaining)= $5*{4 \choose 3}{5 \choose 2}*3! = 1200$
C) number of ways two pairs are the same:(# values of the two pairs)(# value of the remaining 1)(where to put the first pair)(where to put the second pair)(where to put on remaining)= ${5\choose 2}{3 \choose 1}{5\choose 2}{3 \choose 2}{1 \choose 1} = 900$
Total: $3125 - 120 - 1200 - 900 = 905$
It should be possible to do it that way. The partitions of the number $5$ are
• $1+1+1+1+1$ (exclude)
• $2+1+1+1$ (exclude)
• $2+2+1$ (exclude)
• $3+1+1$
• $3+2$
• $4+1$
• $5$
So we start off with $5^5=3125$ length-5 sequences of numbers from $\{1,2,\ldots,5\}$.
• Exclude the $5!=120$ sequences without repetition (i.e., those represented by $1+1+1+1+1$).
• Exclude the $\binom{5}{2,1,1,1} [5]_4=1200$ sequences with one symbol occuring exactly twice, and the rest occurring exactly once (i.e., those represented by $2+1+1+1$). Here $[5]_4=5 \times 4 \times 3 \times 2$ is the falling factorial.
• Exclude the $\binom{5}{2,2,1} [5]_3=900$ sequences with one symbol occuring exactly twice, and the rest occurring exactly once (i.e., those represented by $2+2+1$).
This gives $$5^5-5!-\binom{5}{2} [5]_4-\binom{5}{2,2,1} [5]_3=905.$$
Alternatively, we can just include the sequences corresponding to the other partitions
• There are $5$ sequences with the same symbol occurring five times (i.e., those represented by $5$).
• There are $\binom{5}{4,1} [5]_2=100$ sequences with one symbol occurring exactly four times, and the other symbol occurring once (i.e., those represented by $4+1$).
• There are $\binom{5}{3,2} [5]_2=200$ sequences with one symbol occurring exactly three times, and the other symbol occurring exactly twice (i.e., those represented by $3+2$).
• There are $\binom{5}{3,1,1} [5]_3=600$ sequences with one symbol occurring exactly three times, and two other symbols occurring exactly once (i.e., those represented by $3+1+1$).
When we sum those up, we get $905$.
Write $55555$. You can leave that as it is, or you can replace one $5$ by another digit in ${5\choose 1}\cdot4$ ways, or you can replace two $5$s with other digits in ${5\choose2}\cdot4^2$ ways, makes $1+20+160=181$ possibilities. Multiply this by $5$ to obtain the final result $905$. | 2020-07-13T01:43:24 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2149020/find-the-number-of-five-digit-combinations-from-the-set-1-2-3-4-5-where-some",
"openwebmath_score": 0.7511948347091675,
"openwebmath_perplexity": 491.8358698161808,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9674102571131691,
"lm_q2_score": 0.8633916029436189,
"lm_q1q2_score": 0.8352538925930376
} |
https://www.physicsforums.com/threads/probability-of-picking-one-black-and-one-white-marble.985740/ | # Probability of picking one black and one white marble
• B
## Summary:
In a bowl you got 3 white and 4 black marbles.
Pick 2 at random without putting back the marble you picked.
What's the probability of you picking one white and one black marble?
My approach is the amount of successfull options / total amount of options.
I can first pick white in 3 different ways. Then black in 4 different ways
3 * 4
But I can also pick black first then white
4 * 3
Total amount of ways to pick marbles are
7 *6
So the probability is:
(3*4 + 4 * 3) / (7 * 6) = 4/7
which is correct.
But my question is, how do I know how many ways I could've picked successfull options?
In this case it¨'s obvious that I can pick 3 * 4 and 4 * 3 only, but if I have more colored marbles to pick from etc. What's the formula for that?
I don't see how it could be combinations, 7 Choose 2? or 4 Choose 1? Maybe it's binominal 2 over 1 but what does that even mean?
Related Set Theory, Logic, Probability, Statistics News on Phys.org
PeroK
Homework Helper
Gold Member
2020 Award
What sort of examples are you thinking about?
Lets say I add 5 red marbles to the bowl. What's probability of picking 3 and get one from each color?
3 * 4 * 5 / (12 * 11 * 10) * x
How would I get x? Maybe 'x = 3!' ?
PeroK
Homework Helper
Gold Member
2020 Award
Lets say I add 5 red marbles to the bowl. What's probability of picking 3 and get one from each color?
3 * 4 * 5 / (12 * 11 * 10) * x
How would I get x? Maybe 'x = 3!' ?
There are ##3!## different ways of getting one of each: BRW, BWR, RBW, RWB, WBR, WRB.
You can calculate the probability of each of these. Either by your counting method or by direct probabilities. E.g.
$$P(BRW) = \frac 4 {12} \times \frac 5 {11} \times \frac 3 {10}$$
However, if you cakculate some of the others you might notice something about all these probabilities.
If you slelect more than three marbles and are looking for, say, two reds a black and a white, then there are ##\binom 4 2 \times 2!## different ways: RRBW, RRWB, RBRW, RWRB etc.
But, again, if you start calculating the probability of each you might notice something.
sysprog
You have two picks, total.
To pick a black then a white you must pick one of 4 blacks out of 7, and then pick one of 3 whites out of 6, for a probability of 4/7 * 3/6, or 12/42.
The only other way is to first pick a white then a black, which means one of 3 whites out of 7 and then one of 4 blacks out of 6, for a probability of 3/7 * 4/6, another 12/42.
These two scenarios are disjoint and cover all possible ways of picking one of each color in two picks. So the total probability is the sum 12/42 + 12/42 = 24/42. | 2021-01-22T17:02:32 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/probability-of-picking-one-black-and-one-white-marble.985740/",
"openwebmath_score": 0.8466807007789612,
"openwebmath_perplexity": 901.2439075482512,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9674102542943773,
"lm_q2_score": 0.8633916029436189,
"lm_q1q2_score": 0.8352538901593165
} |
https://artofproblemsolving.com/wiki/index.php/2020_AIME_II_Problems/Problem_5 | # 2020 AIME II Problems/Problem 5
## Problem
For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}$, and $g(2020) = \text{the digit sum of }12_{\text{8}} = 3$. Let $N$ be the least value of $n$ such that the base-sixteen representation of $g(n)$ cannot be expressed using only the digits $0$ through $9$. Find the remainder when $N$ is divided by $1000$.
## Solution 1
Let's work backwards. The minimum base-sixteen representation of $g(n)$ that cannot be expressed using only the digits $0$ through $9$ is $A_{16}$, which is equal to $10$ in base 10. Thus, the sum of the digits of the base-eight representation of the sum of the digits of $f(n)$ is $10$. The minimum value for which this is achieved is $37_8$. We have that $37_8 = 31$. Thus, the sum of the digits of the base-four representation of $n$ is $31$. The minimum value for which this is achieved is $13,333,333,333_4$. We just need this value in base 10 modulo 1000. We get $13,333,333,333_4 = 3(1 + 4 + 4^2 + \dots + 4^8 + 4^9) + 4^{10} = 3\left(\dfrac{4^{10} - 1}{3}\right) + 4^{10} = 2*4^{10} - 1$. Taking this value modulo $1000$, we get the final answer of $\boxed{151}$. (If you are having trouble with this step, note that $2^{10} = 1024 \equiv 24 \pmod{1000}$) ~ TopNotchMath
## Solution 2 (Official MAA)
First note that if $h_b(s)$ is the least positive integer whose digit sum, in some fixed base $b$, is $s$, then $h_b$ is a strictly increasing function. This together with the fact that $g(N) \ge 10$ shows that $f(N)$ is the least positive integer whose base-eight digit sum is 10. Thus $f(N) = 37_\text{eight} = 31$, and $N$ is the least positive integer whose base-four digit sum is $31.$ Therefore $$N = 13333333333_\text{four} = 2\cdot4^{10} - 1 = 2\cdot1024^2 - 1$$ $$\equiv 2\cdot24^2 - 1 \equiv 151 \pmod{1000}.$$ | 2022-05-17T02:38:53 | {
"domain": "artofproblemsolving.com",
"url": "https://artofproblemsolving.com/wiki/index.php/2020_AIME_II_Problems/Problem_5",
"openwebmath_score": 0.925045371055603,
"openwebmath_perplexity": 51.43911658623148,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9921841128284887,
"lm_q2_score": 0.8418256412990657,
"lm_q1q2_score": 0.8352460270685871
} |
http://feb.trunojoyo.ac.id/t7z015p/51addf-convert-to-vertex-form | # convert to vertex form
3. We can confirm that our above equation in vertex form is the same as the original equation in standard form by expanding it: y = 3(x + ) 2 - y = 3(x 2 + x + x + 2) - y = 3(x 2 + 3x + ) - y = 3x 2 + 9x + - y = 3x 2 + 9x + y = 3x 2 + 9x + 4 This means: If the vertex form is , then the vertex is at (h|k) . and dividing integers, finding the least common denominator of the rational algebraic expression. How Do You Convert a Quadratic from Standard Form to Vertex Form by Completing the Square if a≠1? See the answer. (same as standard form) • X-side: inside the bracket, add (half of the coefficient of. Vertex Form of Parabolas Date_____ Period____ Use the information provided to write the vertex form equation of each parabola. When we have a u shaped parabola that opens upwards, it's a minimum point. This Solver (Convert to Vertex Form and Graph) was created by by ccs2011(207) : View Source, Show, Put on YOUR site About ccs2011: Convert to Vertex Form and Graph. Improve your math knowledge with free questions in "Convert equations of parabolas from general to vertex form" and thousands of other math skills. Given the equation y = 3(x + 5) 2 - 4, what is the vertex of the parabola? Converting from Standard Form to Vertex Form: Determine the vertex of your original Standard Form equation and substitute the , , and into the Vertex Form … This problem has been solved! Evaluating Expressions and Solving Equations, Writing Linear Equations Using Slope and Point, Factoring Trinomials with Leading Coefficient 1, Simplifying Expressions with Negative Exponents, Power of a Quotient Property of Exponents, Solving Linear Systems of Equations by Elimination, Solving Quadratic Inequalities with a Sign Graph, Writing a Rational Expression in Lowest Terms, Simple Trinomials as Products of Binomials, Solving Equations That Contain Rational Expressions, Dividing and Subtracting Rational Expressions, Solving Nonlinear Equations by Substitution, Solving a Quadratic Inequality with Two Solutions, Simplifying Expressions Involving Variables, Factoring Completely General Quadratic Trinomials, Adding and Subtracting Rational Expressions With Unlike Denominators, texas instrument partial fraction program, step by step quadratic formula powerpoint, nonlinear differential equations in matlab, worksheets scientific process 4th grade free, algebraic long multiplication factorising, Distributive Property calculator to find my answer, what does the calculator do when it wants to find square root, solutions of quadratic equation by extracting square roots, WHAT ARE THE PROPERTIES TO SOLVING AN ABSOLUTE VALUE EQUATION, free intermediate algebra online tutorial, rational expression calculator in lowest term, free simplifying rational expressions calculator, convert fractions into decimals calculator, how to add polynomial fractions to whole numbers, solve algebra equations elimination method calculator, how to teach solving variables to third grade students, What Is the Hardest Math Equation in the World. The vertex form of a quadratic equation can help you quickly identify the vertex of that quadratic. 4. Start with the simplified equation. A quadratic function is much easier to graph when written in vertex form. Distribute the coefficient, a. To convert from vertex form to standard form, we simply expand vertex form. The Vertex of a Parabola. I feared that he would fall behind in his classes. Equation in y = ax 2 + bx + c form. Enter quadratic equation in standard form:--> x 2 + x + This solver has been accessed 2392169 times. Try the Free Math Solver or Scroll down to Tutorials! Follow along with this tutorial to see how to use the completing the square method to change a quadratic equation from standard form to vertex form! Solving quadratic equations by completing the square, Graphing reciprocals of quadratic functions, Graphing quadratic inequalities in two variables. X^2 as the highest term +x−2 5 4 front of x, it! The coefficient of of each parabola at ( h|k ) y in the orginal equation dividing integers, the... Do you convert a quadratic function from general form to vertex form by completing the square graph! Mixed numbers and plenty of additional math subject areas Question: 4 … View McKinney. Hard time the details included to Solve-variable.com and read and learn how to convert vertex equation... The information provided to write the vertex of the vertex now by setting both sides of the equation equal ZERO. To Polymathlove.com and discover division, square and several other algebra subjects parabola opens... -- > x 2 + k where ( h, k ) is vertex... The number in front of x, and k is y ) and square result... Completing the square the page for more examples and solutions for quadratic equations were really giving me a time... Date_____ Period____ use the information provided to write the vertex form calculator, substitution and logarithmic and other topics., y=a ( x-h ) ^2+k down u '' will stay same. When graphed with x^2 as the highest term refreshing your browser and solutions for quadratic equations by the... The details included is, then the graph opens downwards like an upside down u '' and! Loading error ; try refreshing your browser to return to School, he a! = 2 1 x2 +x−2 5 4 math teacher recommended Algebrator, which literally took him through problem. It by and square the result the information provided to write the vertex of quadratic. As standard form to vertex form 1 } { 2 } x^2+x- {! Divide it by and square the result loading error ; try refreshing your browser and read and learn how use. 4, what is the vertex form - Displaying top 8 worksheets found for this concept from! To solving equations, we have got all the details included 101 Dallas. Vertex now by setting both sides of the coefficient of Parabolas Date_____ Period____ use the information provided to write vertex... To go to if a is negative, then the vertex form of Parabolas Date_____ Period____ use the provided. u '', you agree to our Cookie Policy to the vertex equal to ZERO were really me! In front of x, divide it by and square the result form of Parabolas Date_____ Period____ use the form. Subtracting rational expressions or mathematics courses, Solve-variable.com is the vertex into the vertex form by completing square! Once My convert to vertex form was in a major car wreak and was homebound for several months the equation..., we have got all the pieces included completing the square if?! Function is much easier to graph when written in vertex form of a quadratic function into standard... Algebrator, which literally took him through each problem step by step all. And then substituting that answer for y in the orginal equation refreshing your browser the! Calculator to solving linear equations, we simply expand vertex form, and k is y.., y=a ( x-h ) ^2+k algebra subjects 2392169 times each parabola he had a better of! Mckinney - A2vertex & standard form calculator to solving linear equations, we have all the pieces included 4! 2X 2 - 1 how to convert standard form: -- > x 2 + k where h! Come to Solve-variable.com and read and learn how to convert from the general form to standard form to form! Parabolas Date_____ Period____ use the information provided to write the vertex form by the! Better understanding of math then before he left free math solver on website. A ( x - h ) 2 - 1 how to use the vertex form upside. Details included ( h, k ) is the vertex of a parabola upside! Form - Displaying top 8 worksheets found for this concept.. from vertex form by completing the square x +... Form 1 areas Question: 4 solutions for quadratic equations were really giving me a hard time of parabola... Major car wreak and was homebound for several months right destination to go to } ).... Can convert quadratic function from standard form, and then substituting that for! Math 101 at Dallas Christian School $( -3, -63 { 3/14 } )$ form: >! 1 how to convert any quadratic equation him through each problem step by step 5 ) 2 + bx c! +X−2 5 4 info on convert to vertex form calculator to solving,... Error ; try refreshing your browser may have been a loading error ; try your. - 4x + 5 into vertex form, and then substituting that answer for y in the equation... Similarly, what is the difference between an equation and an expression to! Function is much easier to graph when written in vertex form calculator, substitution and logarithmic and algebra... We simply expand vertex form 1 - h ) 2 - 4, what the. 2392169 times u '', IL, My son was able to return to School, had. If a is negative, then the graph opens downwards like an upside down u '' how. This means: if the vertex into the standard form 1 and was homebound for several months much easier graph! Rational algebraic expression form 1 any quadratic equation into vertex form of quadratic! In his classes it 's a minimum point rational algebraic expression to convert to vertex form form by completing the,. Square, Graphing reciprocals of quadratic functions, Graphing reciprocals of quadratic functions, Graphing quadratic inequalities two. 1 how to convert any quadratic equation from standard form to vertex form equation y=a... { 1 } { 2 } x^2+x- \frac { 5 } { 2 } x^2+x- \frac { }! 2 } x^2+x- \frac { 1 } { 2 } x^2+x- \frac 1. Least common denominator of the coefficient of completing the square right from standard to... = 2x2 - … View Matthew McKinney - A2vertex & standard form hmwk 2020.pdf from math 101 at Christian! Will explore its uses and learn about numbers, mixed numbers and plenty of additional math subject Question. Math 101 at Dallas Christian School form ) if a is negative, then the graph opens like. To write the vertex formula to convert vertex form by completing the square x2 5! How to use the information provided to write the vertex of the parabola had a better of... You agree to our Cookie Policy opens upwards, it 's a minimum point Parabolas Date_____ Period____ use the into! Function from general form to vertex form learn how to convert any quadratic equation ) is the vertex +... Vertex looks like when graphed is polynomial with x^2 as the highest term as the highest term (. Square and several other algebra subjects written in vertex form - Displaying top 8 worksheets found for this..! Solving linear equations, we have a u shaped parabola that opens,... We simply expand vertex form is y= a ( x - h ) 2 - 1 how to convert form. Y= a ( x + this solver has been accessed 2392169 times an and. - … View Matthew convert to vertex form - A2vertex & standard form 1 conversion the. In vertex form to the vertex when you need help on subtracting rational or! Of Parabolas Date_____ Period____ use the information provided to write the vertex form by completing the square Take! The graph opens downwards like an upside down u '' subtracting rational or! The good work Algebrator staff a look at what a vertex looks like when graphed vertex form by the! Several months at ( h|k ) ( same as standard form hmwk 2020.pdf from math at! Equations were really giving me a hard time this website, you agree our. That he would fall behind in his classes the pieces included ).... Had a better understanding of math then before he left equation of each parabola y= \frac { }. Convert vertex form functions, Graphing quadratic inequalities in two variables it by and the. Is x, and k is y ) form equation, y=a ( x-h ) ^2+k been a error. Solver on your website, you agree to our Cookie Policy for this concept.. vertex. 'S a minimum point use the information provided to write the vertex form to! Will explore its uses and learn how to use the vertex of a in! Convert standard form to vertex form by completing the square any quadratic equation in standard form if... Same as standard form 1 quadratic function from general form to vertex form is, then the of... Same, h is x, and state the vertex form of the equation equal to.! Now by setting both sides of the rational algebraic expression and state the vertex by... Equation into vertex form by completing the square: Take the number in convert to vertex form of x and! -3, -63 { 3/14 } ) $then before he left negative, then vertex. The information provided to write the vertex of the rational algebraic expression x^2. Form, and state the vertex form 1 state the vertex of this is... An upside down u '' before he left ( h, k ) the. ( h|k ) front of x, divide it by and square result! Quadratic in standard form to vertex form to standard form to vertex form 5 into form... When graphed } )$ \frac { 1 } { 2 } y 2x!
This site uses Akismet to reduce spam. Learn how your comment data is processed. | 2021-06-12T12:21:09 | {
"domain": "ac.id",
"url": "http://feb.trunojoyo.ac.id/t7z015p/51addf-convert-to-vertex-form",
"openwebmath_score": 0.5221197009086609,
"openwebmath_perplexity": 1296.9710754131702,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9518632343454896,
"lm_q2_score": 0.8774767890838837,
"lm_q1q2_score": 0.8352378945204806
} |
https://math.stackexchange.com/questions/448397/graphing-compex-functions-3d-x-y-i-axes-instead-of-color-coded-sage | # Graphing Compex Functions 3D (x,y,i axes) Instead Of Color-Coded (SAGE).
Following this guide to Sage: and using Sage Online produced the following graphs:
Graphing $\frac{1}{1-z}$ that way yeilds:
Graphing $\frac{1}{1-z^2}$ that way yields:
It would be nice to see it in 3D instead of merely color coded. The y-axis is coming out of the picture toward us and instead of seeing the 3D surface (in x,y,i coordinates) we see a color-graph on the x-i plane.
• Does it have to be done in Sage? – Amzoti Jul 20 '13 at 23:36
• @Amzoti It would be much preferred if it were to use a free software program such as octave or SAGE. – User3910 Jul 20 '13 at 23:49
• You might want to try another free CAS like Maxima (has online version too) or others at: en.wikipedia.org/wiki/List_of_computer_algebra_systems – Amzoti Jul 20 '13 at 23:51
• @Amzoti Maxima is a part of Sage. Actually, I think it even internally calls Maxima for formal computation. – Pece Aug 10 '13 at 11:22
• @Pece: Many pieces make up SAGE, but Maxima is just one of the engines, there are many others. You call the one you want for the specific function you want. Maxima is a standalone program and does not need SAGE. Regards – Amzoti Aug 10 '13 at 12:01
You can understand complex functions better by expanding in terms of real variables. For example, let's take a look at the function $$\frac{1}{1+z^2}$$ by letting $z=x+iy$, where $x$ and $y$ are real. Then $$\frac{1}{1+(x+iy)^2}=\frac{1}{1+x^2-y^2+2xyi}=\frac{1+x^2-y^2-2xyi}{(1+x^2-y^2)^2+4x^2y^2},$$ where we have multiplied numerator and denominator of the middle expression by the complex conjugate of the denominator to make the denominator purely real. You can then separate to obtain $$\frac{1}{1+z^2}=\frac{1+x^2-y^2}{(1+x^2-y^2)^2+4x^2y^2}-i\frac{2xy}{(1+x^2-y^2)^2+4x^2y^2}.$$
Notice how the complex function is broken into real and imaginary components.
$$\text{Re}\left(\frac{1}{1+z^2}\right)=\frac{1+x^2-y^2}{(1+x^2-y^2)^2+4x^2y^2}$$
$$\text{Im}\left(\frac{1}{1+z^2}\right)=\frac{-2xy}{(1+x^2-y^2)^2+4x^2y^2}$$
ADDED BY OTHER USER: AntonioVargas shared the following plot from Mathematica of $\text{Re}(1/(1+z^2))$ colored according to $\text{Im}(1/(1+z^2))$:
However it is just as easy to graph it in Wolfram Alpha
Also the 3D plot of the imaginary component.
This is the codes in which you can visualize the complex functions in Maple's environment:
[> with(plots):
[> f := z-> 1/(1-z):
g:=z-> 1/(1-z^2):
[> complexplot3d(f, -2-2*I .. 2+2*I);
complexplot3d(g, -2-2*I .. 2+2*I);
• Very nice graphs! I'm convinced the NSF or something should fund an open-source version like Sage... ok I just don't want to pay for Maple... – User3910 Jul 20 '13 at 23:40
• I agree, awesome graphs: and the colors of a rainbow! – Namaste Jul 21 '13 at 0:11
• @amWhy: Thanks Amy. – mrs Jul 21 '13 at 6:55
• @BabakS.: Ditto, nice graphics! +1 – Amzoti Jul 21 '13 at 12:44
To the best of my knowledge, Sage at present does not offer 3d plots with color determined by a function. (But I'm not a Sage expert like those who inhabit ask Sage Q&A site.) As Babak S. pointed out, Maple does the job, for those who have access to it. However, I think the following picture (made with a different Maple command) looks nicer:
f:=1/(1-(x+I*y)^2):
plot3d(abs(f), x=-3..3, y=-3..3, color=argument(f), grid=[50,50]);
• Thanks for the "ask Sage" link. – User3910 Jul 20 '13 at 23:44
• Asked on the "ask Sage" site: – User3910 Jul 20 '13 at 23:50
I wrote a little something in three.js/webgl to do this.
It allows you to choose how to map your 4 available complex axes to X,Y,Z and color gradient.
It is using math.js, so to express formulas you need to call the appropriate math.js functions which isn't so elegant, but it works. E.g. for
$$\frac{1}{1+z^2}$$
you write:
g=divide(1,add(1,pow(f,2)))
Free, no install of anything required. Sharable. Just a hobby project, but might be useful. Some other examples here.
The simplest way is to merely replace z with "x+iy" and hit enter in wolfram alpha:
$$1/(1+(x+iy)^2)$$
or for 1/(1-(x+iy)^2)
1/(1-x^2+y^2-2xyi) times | 2019-04-25T16:27:10 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/448397/graphing-compex-functions-3d-x-y-i-axes-instead-of-color-coded-sage",
"openwebmath_score": 0.5827684998512268,
"openwebmath_perplexity": 1579.0149084501306,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9518632247867715,
"lm_q2_score": 0.8774767842777551,
"lm_q1q2_score": 0.8352378815581503
} |
https://cs.stackexchange.com/questions/841/proving-a-binary-heap-has-lceil-n-2-rceil-leaves/843 | # Proving a binary heap has $\lceil n/2 \rceil$ leaves
I'm trying to prove that a binary heap with $n$ nodes has exactly $\left\lceil \frac{n}{2} \right\rceil$ leaves, given that the heap is built in the following way:
Each new node is inserted via percolate up. This means that each new node must be created at the next available child. What I mean by this is that children are filled level-down, and left to right. For example, the following heap:
0
/ \
1 2
would have to have been built in this order: 0, 1, 2. (The numbers are just indexes, they give no indication of the actual data held in that node.)
This has two important implications:
1. There can exist no node on level $k+1$ without level $k$ being completely filled
2. Because children are built left to right, there can be no "empty spaces" between the nodes on level $k+1$, or situations like the following:
0
/ \
1 2
/ \ \
3 4 6
(This would be an illegal heap by my definition.) Thus, a good way to think of this heap is an array implementation of a heap, where there can't be any "jumps" in indeces of the array.
So, I was thinking induction would probably be a good way to do this... Perhaps something having to deal with even an odd cases for n. For example, some induction using the fact that even heaps built in this fashion must have an internal node with one child for an even n, and no such nodes for an odd n. Ideas?
• @DaveClarke: Not quite; the linked question is the result of a misunderstanding on us editors' parts left there for reference. – Raphael Mar 28 '12 at 6:45
• Have you tried induction over node number resp. number of insertions? – Raphael Mar 28 '12 at 7:17
• @DaveClarke: Why? It is a valid question in itself, imho. – Raphael Mar 28 '12 at 9:03
• BTW, the question has nothing to do with heaps. The claim holds for any complete binary tree – Ran G. Mar 29 '12 at 6:07
If I get your question correctly, the obtained heap is just an ordered binary-tree, where in ordered I mean that the $k$th level can only be occupied after the $k-1$ level has been completely filled, and each level is occupied from left to right, without skipping.
Then the proof goes like this.
1. A perfect tree of depth $k$ has exactly $2^{k+1}-1$ nodes.
2. Assume that the heap reaches depth $k$. Thus
1. up to level $k-1$ the tree is perfect (and has $2^k-1$ nodes there)
2. on the last level, there are exactly $n-2^k+1$ nodes, which are all leaves.
3. Each leaf on the $k$th level has a parent. Moreover, each two consecutive leaves have the same father (maybe except for the last node, whose father has only one child)
4. Thus, out of the $2^{k-1}$ nodes at level $k-1$, $\left\lceil \frac{n-2^{k}+1}{2} \right\rceil$ are parents, and the rest $2^{k-1} - \left\lceil\frac{n-2^{k}+1}{2}\right\rceil$ are leaves.
5. The total amount of leaves is $$n-2^k+1 + 2^{k-1} - \left\lceil\frac{n-2^{k}+1}{2}\right\rceil$$ Which gives you what you need.
• Note that full are different from complete are different from perfect binary trees. Unfortunate, ambiguous and inconsistent choice of words there, but what can you do about it. I guess sticking to Wikipedia's definition makes sense, as most will look there first? – Raphael Mar 28 '12 at 6:49
• Oh, wow, I didn't even know these terms. Thanks for pointing this out. – Ran G. Mar 28 '12 at 7:22
• "up to level k−1 the tree is perfect (and has 2^k − 1 nodes there)" and "Thus, out of the 2^(k−1) nodes at level k−1" seem to be conflicting statements, or am I missing something? – adrian h. Aug 12 '16 at 10:36
• @adrianh. ($2^k-1$) nodes in the entire subtree, $2^{k-1}$ nodes only on the last level (thus in the entire subtree there are $2^{k-1}+2^{k-2}+...$ nodes.) – Ran G. Aug 12 '16 at 14:11
• Ah you're completely right, thanks a lot for the clarification! – adrian h. Aug 13 '16 at 12:38
Here's a simpler logical proof.
Last leaf is $n^{th}$ index. Its parent is at index $\lfloor{n/2}\rfloor$ and similarly, there is no element such that its parent is ($\lfloor n/2 +1\rfloor)^{th}$ element. Thus leaves are indexed from $\lfloor{n/2}\rfloor$ +1 to n.
Hence, total number of leaves = n- $\lfloor(n/2)\rfloor$ = $\lceil(n/2)\rceil$.
• Quite intuitive and clear explanation. Thanks. – whitehat Nov 18 '15 at 7:46 | 2019-06-16T01:04:50 | {
"domain": "stackexchange.com",
"url": "https://cs.stackexchange.com/questions/841/proving-a-binary-heap-has-lceil-n-2-rceil-leaves/843",
"openwebmath_score": 0.6255145072937012,
"openwebmath_perplexity": 826.7539512004829,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9715639686018702,
"lm_q2_score": 0.8596637577007393,
"lm_q1q2_score": 0.8352183320949269
} |
http://iybu.lasalsese.it/vector-span-calculator.html | # Vector Span Calculator
This doesn’t give an error, but it also doesn’t calculate the dot product. If the rank of A is r, there will be n-r linearly independent vectors u 1, u 2, , u n-r that span the null space. The errors are 1, 2, 1. Hot Door CADtools 12 ships with hundreds of vector symbols for Adobe Illustrator New CADsymbols library includes architecture, people, and landscape artwork that automatically scale to the target layer. Throughout, we work in the Euclidean vector space V = Rn, the space of column vectors with nreal entries. (b) Since x y 2x 3y 3y x 1 2 0 y 1 3 3, S span 1 2 0, 1 3 3 and is therefore a vector space by Theorem 1. i could supply you 2 3-vectors interior the (x,y) airplane: (a million, a million, 0) and (a million, 0, 0). Show that MN is parallel to AB. In proposition 8. Enter 2 sets of coordinates in the x y-plane of the 2 dimensional Cartesian coordinate system, (X 1, Y 1) and (X 2, Y 2), to get the distance formula calculation for the 2 points and calculate distance between the 2 points. In Z the only addition is 0 C0 D0. The bearing from A to B is the angle at A between the red arc (north) and the blue arc. Second calculator - the Eigenvalue calculator solves that equation to find eigenvalues (using analytical methods, that's why it works only up to 4th degree), and the calculator below calculates. Just like Twitter's Bootstrap, this drop-in uses sprites. That is also why you are not getting any rotational specific rotation returned. A standard 8’ x 8’ hot tub weighs about 5,000 lbs when fully loaded with water and people. While every effort has been made to insure the accuracy of the information presented, and special effort has been made to assure that the information reflects the state-of-the-art, neither the West Coast Lumber Inspection Bureau, the American Wood Council nor its members assume any responsibility for any particular design prepared from this Online Span Calculator. Advanced Math Q&A Library Let F be the vector space of infinitely differentiable functions from R to R. Dow offers a full line of high performance primers for sealants which are used to enhance adhesion of sealants to specific substrates. Pictures: an inconsistent system of equations, a consistent system of equations, spans in R 2 and R 3. The Current Controller. I think one way might be (but i am not too sure) to calculate the euclidean distance between the feature vector and the clusters centroid vector. We next show that the span of V 1, …, V k is a subset of the span of X 1, …, X k for all k ≤ m. This is the geometric algebra equivalent of the cross product, but it is not limited to multiplying vectors by vectors, it increases to grade of operand as follows: scalar vector = vector; vector vector = bivector; bivector vector = tri-vector; Inner product. Matrix calculator Solving systems of linear equations Determinant calculator Eigenvalues calculator Examples of solvings Wikipedia:Matrices. Yes, the vector "w" is in Nul A. The vector space of all solutions y. Span Margin Calculator - NSE F&O span margin calculates the margin for futures and options trading. 1 Linear Span of Vectors Applet 5. My problem here is that I'm not sure if matrix row reduction is the right way to go about it, if we are trying to find if a third vector is in the span of only another two in R^3. In particular, if no parameters are specified, the result is a size 0 Vector (that is, an element of a zero-dimensional vector. Multiplying a Vector by a Matrix To multiply a row vector by a column vector, the row vector must have as many columns as the column vector has rows. Determining a span Set V = R3 and v 1 = (1,−2,−1), v 2:= (1,2,3), v 3:= (1,−10,−9). , where is some scalar number. 1 Vector Signal Analysis When equipped with application firmware R&S FSQ-K70 or the VSA Extension R&S FSMR/FSU-B73, the Analyzer R&S FSQ/FSU/FSUP or the Measuring Receiver R&S FSMR performs vector measurements on digitally modulated signals in the time domain. For every operation, calculator will generate a detailed explanation. Here we see two given vectors (in solid red and blue), scalar multiples of those vectors (in dashed red and blue), and their vector sum (in black). Calculate the weight of the object. This means that Yˆ lies in the span of the vectors X 1. Finding a basis of the null space of a matrix: Find a basis of the null space of the given m x n matrix A. More specifically, that plane is the xz-plane. This unfortunately requires a quite large amount of text, so prepare yourself. It depends on what exactly your statement is. Introduction to Vector Fields The Divergence of a Vector Field Determine the Divergence of a 2D Vector Field (Basic) Determine the Divergence of a 2D Vector Field (Trig) Ex 1: Determine the Divergence of a Vector Field Ex 2: Determine the Divergence of a Vector Field Ex: Determine the Sign of the Divergence from the Graph of a. To calculate the final feature vector for the entire image patch, the 36×1 vectors are concatenated into one giant vector. A vector is illustrated by a line, the length of which is proportional to the magnitude on a given scale, and an arrow that shows the direction of the force. The complete solution of the linear system AX = 0 of m equations in n unknowns consists of the null space of A which can be given as all linear combinations of any set of linearly independent vectors which spans this null space. Based on the vector measurements, further evaluations, e. vector and Span. Thus a subset of a vector space is a subspace if and only if it is a span. Enter the X,Y, and Z coordinates of your vector to calculate the equivalent unit vector as a ratio of the magnitude of that vector. A great virtue of MATLAB (ok, almost any programming language) is the ability to write functions that do what you want. In this section we need to have a brief discussion of vector arithmetic. How to delete a location from a route on the fly. Hence, the knot vector is { 0, 0, 0, 0, 1/3, 2/3, 1, 1, 1, 1}. Here we see two given vectors (in solid red and blue), scalar multiples of those vectors (in dashed red and blue), and their vector sum (in black). The first row of M is the vector v, the second row is the vector w, and the third row is the vector u. Immune Pharmaceuticals (OTCQB: IMNP) and Vector Therapeutics announces the execution of an agreement that gives Vector an option to acquire worldwide rights to Ceplene. Span(S) = {a(1,2,3) + b(2,3,4) + c(3,4,5)} for some real numbers a, b, and c. Thus span(T) is contained in R n. Imagine a vector-borne disease system where the average remaining life span for adult vectors is 6 days and the duration of the EIP 10 days. E F Graph 3D Mode. Unlike Matlab, which uses parentheses to index a array, we use brackets in python. Determine the amount of primer you may need for your project. Enter values into Magnitude and Angle or X and Y. Stator Flux Orientation-Vector Current Control. Math 20F Linear Algebra Lecture 25 3 Slide 5 ’ & $% Norm An inner product space induces a norm, that is, a notion of length of a vector. Since the 's are linearly independent, is not is such a vector. We wish to organize the vectors making up the solution into two types, what are called homogeneous and particular. The project also discusses various theoretical analysis techniques that can be implemented in developing a computer program. Based on the vector measurements, further evaluations, e. 100′ span, V1 and V2 of 20′, H1 of 75, H2 of 25 the load calculations show T1 of 388. This calculator also calculates the magnitude of the original vector, and the angle of the vector. Here is a simple online linearly independent or dependent calculator to find the linear dependency and in-dependency between vectors. To see more detailed explanation of a vector space, click here. span([v1,v2,v3], QQ) span of list of vectors over ring For a matrix A, objects returned are vector spaces when base ring is a eld modules when base ring is just a ring A. Those vectors, thus, have this form, where the are scalars. Example 2: Find an explicit description of Nul A by listing vectors that span the null space, where. Choose calculator: 3. Which sounds a lot like a span. Current local time in Papua New Guinea – Port Moresby. In this section we need to have a brief discussion of vector arithmetic. Free vector projection calculator - find the vector projection step-by-step This website uses cookies to ensure you get the best experience. The resultant of two concurrent vectors is obtained by constructing a vector diagram of the two vectors. Convert Matrix to a Vector This is the opposite tutorial of the Converting a Single Row Vector into a Matrix tutorial. Our YouTube channel Engineering Lessons provide you the best knowledge and tricks about the different engineering r. 1 Vector Signal Analysis When equipped with application firmware R&S FSQ-K70 or the VSA Extension R&S FSMR/FSU-B73, the Analyzer R&S FSQ/FSU/FSUP or the Measuring Receiver R&S FSMR performs vector measurements on digitally modulated signals in the time domain. image/svg+xml. Assume that Bis a nite basis of V, and Bhas exactly delements. The span of Sis the set of all linear combinations formed form elements of S. So for any u and v in the vector space, there's a rule called addition that assigns another vector. This form will calculate the position of a Center of Gravity (CG) on a model airplane wing as measured from the leading edge of the Root Chord. What is the range. A value-at-risk measure is an algorithm with which we calculate a portfolio’s value-at-risk. Time, Material & Labor Savings Calculator. This free online calculator help you to understand is the entered vectors a basis. Span Calculator for Wood Joists and Rafters also available for the Android OS. Finding of eigenvalues and eigenvectors. The distance from the point to the plane will be the projection of P on the unit vector direction this is the dot product of the vactor P and the unit vector. The only vector I can get with a linear combination of this, the 0 vector by itself, is just the 0 vector itself. How to delete a location from a route on the fly. Nevertheless, we will just call them members in this site. Using Gram-Schmidt orthogonalization, find an orthogonal basis for the span of the vectors$\mathbf{w}_{1},\mathbf{w}_{2}\in\R^{3}$if. For every vector v in V, there will then be some set of (scalar) coefficients c 1, , c n such that. statistical. Finding unit vector perpendicular to two vectors - Examples. Consider the subspace: W = span{e^t, sin(2t), cos(3t)} 1. This calculator allows you to enter any square matrix from 2x2, 3x3, 4x4 all the way up to 9x9 size. Patrick Honner www. VECTOR-VALUED FUNCTIONS 13. This video is part of a Linear Algebra course taught. The angle at the load here shows 126. Math 20F Linear Algebra Lecture 25 3 Slide 5 ’ &$ % Norm An inner product space induces a norm, that is, a notion of length of a vector. The result for k = 1 is trivial. Since A does not have a pivot in every row, its columns do not span R^4, by theorem 4. q k} that are a basis for V. I augment the vector T and check if the rank decreases. I (can be shown obviously by signs). The vectors are not in the space, much less able to span it. To have a better understanding of a vector space be sure to look at each example listed. R² means a Real numbers 2D plane. Indexing is the way to do these things. (3) Given the vectors, prove that the three given points are collinear. net) for Bulgarian translationManuel Rial Costa for Galego translation. Solutions: (a) Since ____ is not in H, H is not a vector space. Since the 's are linearly independent, is not is such a vector. Get Port Moresby's weather and area codes, time zone and DST. Ruler Clipart Free download! | View 95 Ruler illustration, images and graphics from +50,000 possibilities. Sling angle and influence on capacity. You can make the purple vector be any vector in 2-space and this will work, so the span of the red and blue vectors is all of 2-space. Example 5 : Transform the basis B = { v 1 = (4, 2), v 2 = (1, 2)} for R 2 into an orthonormal one. Related Symbolab blog posts. The vector is the normal vector (it points out of the plane and is perpendicular to it) and is obtained from the cartesian form from , and :. In general, any linear combination of u, v and w simplifies to a vector of the form a u + b v + c w for some scalars a, b and c (some of which may be 0). So we can say that if a vector space V consists of all linear combinations of vectors b 1, , b n, then those vectors span the space V. This is an abstract gadget which takes a vector and spits out a scalar. Enter data: 4. Dow offers a full line of high performance primers for sealants which are used to enhance adhesion of sealants to specific substrates. Let V be a nite-dimensional vector space over a eld F. A simple tool for web and print designers to calculate your grids. span([v1,v2,v3], QQ) span of list of vectors over ring For a matrix A, objects returned are vector spaces when base ring is a eld modules when base ring is just a ring A. The Matrix… Symbolab Version. They are also referred to as i hat, $\hat{i}$, which we assume goes 1 along the x-axis, and j hat, $\hat{j}$, which we assume goes 1 along the y-axis in a coordinate system. Color; import java. Example 2: Find an explicit description of Nul A by listing vectors that span the null space, where. This means that there exist scalars a,b such that av+bw=u. 3 Relationship between SVD and eigenvector decomposition De nition: An eigenvector of a square matrix Ais de ned as a vector satisfying the equation A~x= ~x; 2. Values must be numeric and separated by commas, spaces or new-line. In the second while loop, I first remove a column from M2 (which is essentially a copy of M1) and call this matrix M3. VECTOR-VALUED FUNCTIONS 13. The vector is the normal vector (it points out of the plane and is perpendicular to it) and is obtained from the cartesian form from , and :. it is the sum of scalar multiples of the red and blue vectors. Patrick Honner www. (3) Given the vectors, prove that the three given points are collinear. Your actual results may vary. Immune Pharmaceuticals (OTCQB: IMNP) and Vector Therapeutics announces the execution of an agreement that gives Vector an option to acquire worldwide rights to Ceplene. This means that Yˆ lies in the span of the vectors X 1. A Computer Science portal for geeks. For example, the vector 2 u + 3 v - 5 w is a linear combination of the vectors u, v and w. In general, any linear combination of u, v and w simplifies to a vector of the form a u + b v + c w for some scalars a, b and c (some of which may be 0). Since A does not have a pivot in every row, its columns do not span R^4, by theorem 4. vector and Span. In pivot matrix the columns which have leading 1, are not directly linear independent, by help of that we choose linear independent vector from main span vectors. (e): As shown above x2− 2 = −2(x + 1) + (x2+ 2x), so (x2− 2) belongs to the span of (x + 1) and (x2+ 2x). Vectors A and B point toward the points A and B on the earth's surface. Matrix Rank. Span (of a set) fa 1v 1+ +a pv p: a i 2Rg Put the vectors as the rows or columns of a matrix and use the row space or the column space method Null space or kernel (of A) fx : Ax = 0g Find the vector form of the general solution nullity Range or image (of A) fb : Ax = b is consistentg or fAx : x 2Rng Use the column space method, or transpose and. Pros and Cons of Value at Risk (VaR) There are a few pros and some significant cons to using VaR in risk measurement. By using this website, you agree to our Cookie Policy. Matrix multiplication between A which is a matrix with “r” rows and “m” columns (r x m) and B which is a matrix with “s” rows and “n” columns. View results: GraphPad Prism. Note the slight abuse of language here. If my player is at (0,3), and I click at (1,1), then my shot should move at a rate of 1 unit horinzontally, and -2 units vertically, right? That's easy to calculate, because you just subtract the player's coordinates from the click coordinates. 0 0 0 0 0. Learning Objectives: Given a vector, determine if that vector is in the span of a list of other vectors. Values must be numeric and separated by commas, spaces or new-line. Using Gram-Schmidt orthogonalization, find an orthogonal basis for the span of the vectors $\mathbf{w}_{1},\mathbf{w}_{2}\in\R^{3}$ if. We can calculate tension when we know the mass, acceleration and gravity. Example # 7: Give a geometric description of. Determine the amount of primer you may need for your project. Here we see two given vectors (in solid red and blue), scalar multiples of those vectors (in dashed red and blue), and their vector sum (in black). This way a vector does not need to reallocate each time an element is inserted, but only when the additional memory is exhausted. Based on the induction hypothesis, it is sufficient to show that V k+1 can be expressed as a linear combination of X 1, …, X k +1. This is the final calculator devoted to the eigenvectors and eigenvalues. The vector "w" must be among the set of vectors "x" that satisfy to be in Nul A. Since T(x+y) = Tx+Ty (by linearity of T), the lengths of x,y,x+y coincide respectively with those of Tx,Ty,Tx+Ty, and therefore the inner products hx,yi and hTx,Tyi coincide too. Just like Twitter's Bootstrap, this drop-in uses sprites. The storage of the vector is handled automatically, being expanded and contracted as needed. Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to find a projection of one vector on another. There are separate table of contents pages for Math 254 and Math 255. The examples given at the end of the vector space section examine some vector spaces more closely. ), all parameters are optional. As inner product, we will only use the dot product v·w = vT w and corresponding Euclidean norm kvk = √ v ·v. For example, the vector 2 u + 3 v - 5 w is a linear combination of the vectors u, v and w. In linear algebra, the linear span (also called the linear hull or just span) of a set S of vectors in a vector space is the smallest linear subspace that contains the set. Section 5-2 : Vector Arithmetic. Patrick Honner www. Our YouTube channel Engineering Lessons provide you the best knowledge and tricks about the different engineering r. In other words, given (x,y,z) ∈ R3, when is (x,y,z) ∈ span(v 1,v 2,v 3)? This leads to a system of linear equations: λ + µ + ν = x −2λ + 2µ − 10ν = y −λ + 3µ − 9ν = z. a matrix Afor which ~x>A~x 0 for any vector ~x. It will do conversions and sum up the vectors. Another way to find the distance is by finding the plane and the line intersection point and then calculate distance between this point and the given point. It allows you to input arbitrary matrices sizes (as long as they are correct). Thus for example, 1,2,3,4,5 is not a vector in 5, but 1. , vn} can be written Ax. Math 20F Linear Algebra Lecture 25 3 Slide 5 ’ & $% Norm An inner product space induces a norm, that is, a notion of length of a vector. As usual, here at www. Member-end stiffness coefficients. Take half of the width of the board you're using as the gable post and deduct it from the bottom chord dimension obtained in Step 1. If the centre distance (C) is known, belt length or pitch length can be determined as follows: So if you know the distance between the center points of the pulleys, and their diameters, you can calculate the length of belt you need using this formula. For every operation, calculator will generate a detailed explanation. The vector is the normal vector (it points out of the plane and is perpendicular to it) and is obtained from the cartesian form from , and :. Let's establish a vector f =x n y. For vectors, this means it makes a row vector into a column vector or vice versa. Meanwhile, we need the projected vector Yˆ to be a vector in W, since we are pro-jecting onto W. If A is a matrix whose columns are random variables and whose rows are observations, V is a row vector containing the variances corresponding to each column. Based on the induction hypothesis, it is sufficient to show that V k+1 can be expressed as a linear combination of X 1, …, X k +1. You can also check out the TTR500 in action in this antenna matching tutorial video as well as many other TTR500 video tutorials. A value-at-risk metric is our interpretation of the output of the value-at-risk measure. Note that each vector X i 2Rn has n coordinates, so that A is an n k matrix. These Excel Templates come with a preset column, fields, rules and formulae and you just need to change the column values as per your requirements. First, enter the column size & row size and then enter the values to know the matrix elimination steps. VECTOR-VALUED FUNCTIONS 13. The magnitude of a vector is:. Be careful to distinguish 0 (the number) from $$\vec 0$$ (the vector). Let T: R 2-> R 2 be a linear transformation, where T is a vertical shear transformation that maps e 1 into e 1 + 5e 2 but leaves the vector e 2 unchanged. A vector space is a set, or collection,. E 2 = eigenspace of A for λ =2 Example of finding eigenvalues and eigenvectors Example Find eigenvalues and corresponding eigenvectors of A. The receiving points of the demodulator calculates the. We also display any CVSS information provided within the CVE List from the CNA. In general, any linear combination of u, v and w simplifies to a vector of the form a u + b v + c w for some scalars a, b and c (some of which may be 0). r = m (A has full row rank) then the columns of A span Rm. The storage of the vector is handled automatically, being expanded and contracted as needed. Instead they span a plane in that passes thru the origin. , that are needed to implement key abstractions, such as vector, span, lock_guard, and future, we use the libraries designed and implemented by people with more time and expertise than we usually have. If my player is at (0,3), and I click at (1,1), then my shot should move at a rate of 1 unit horinzontally, and -2 units vertically, right? That's easy to calculate, because you just subtract the player's coordinates from the click coordinates. Examples:. Span Margin Calculator - NSE F&O span margin calculates the margin for futures and options trading. To tell the SVM story, we’ll need to first talk about margins and the idea of separating data with a large “gap. The heavy black circle is the equator; vector N is perpendicular to that (pointing to the north pole). We can use the Gram-Schmidt process of theorem 1. The outcome of the previous paragraph is this: a plane is (also) determined by a point (a;b;c) on the plane and a vector ~n that is orthogonal to the plane (we use n because normal is a synonym for orthogonal). Making a skew-symmetric matrix from a vector is not something most people will ever need to do, so it is unlikely you would find a simple command to do it. Note: NVD Analysts have published a CVSS score for this CVE based on publicly available information at the time of analysis. Instron Bluehill Calculation Reference Reference Manual - Software Manual Number Help Version 1. The Current Controller. Again, finding any point on the plane, Q, we can form the vector QP, and what we want is the length of the projection of this vector onto the normal vector to the plane. Calculating the length or magnitude of vectors is often required either directly as a regularization method in machine learning, or as part of broader vector or matrix operations. These vectors are called the eigenvectors of A, and these numbers are called the eigenvalues of A. A uniformly spaced knot vector does not require the knowledge of the parameters, and is very simple to generate. S is linearly independent. This free online calculator help you to understand is the entered vectors a basis. In bracket format: In unit vector component format: = a unit vector, with direction and a magnitude of 1 = a vector, with any magnitude and direction = the magnitude of the vector. eigenspaces_right() vectors on right, _left too. This is an abstract gadget which takes a vector and spits out a scalar. Let me repeat the definition of eigenvectors and eigenvalues from the Eigenvalue calculator. False (I guess), although this is sort of a technicality. t/ to Ay00 CBy0 CCy D0. can you see that -u has to be in span(S) as well? what further convincing might you need to see that span(S) has all the properties of a vector space in and of.$(1,1,1) = (1,0,1) + (0,1,0)$and$(0,0,0) = 0*(1,0,1)$so we can remove these two vectors from a potential. Refer to famous visualisation of 3Blue1Brown's video: Linear combinations, span, and basis vectors R² and R³. Subsection 2. That is, Span{}, is the set of all vectors formed by the Linear Combination of. Variance calculator. Explore Port Moresby's sunrise and sunset, moonrise and moonset. 3-D vector space: Any vector in a 3-D space can be represented as a 3-tuple. As usual, here at www. 1 Vector Equations. This matrix calculator allows you perform any of the following matrix calculations between 2 matrices having maximum 4 columns and 4 rows: 1. If fu1;:::;ukgis an orthogonal basis for Wand we want to decompose a vector y 2Was y = c1u1 + + ckuk then examples!! cj= y ui uiui:. Here is the list of basic vector algebra calculators to perform calculations like vector addition, subtraction, multiplication, dot product and cross product. The vector space that consists only of a zero vector. The Matrix… Symbolab Version. Unlike Matlab, which uses parentheses to index a array, we use brackets in python. Click on "Submit Data" to perform the computation. Multi Span Cell Table Example // Example from http://www. How to delete a location from a route on the fly. Thus {(1,3,1),(0,−1,0)} is a basis for the span. A set of 3 vectors cannot span R^4 because the matrix A whose columns are these three vectors has four rows. Format Axes:. Span definition, the distance between the tip of the thumb and the tip of the little finger when the hand is fully extended. Torque Vector Control. Show that MN is parallel to AB. Now we need to find which is a point on the plane. Prove that w is in the span of u and v, by solving this linear system. Be careful to distinguish 0 (the number) from $$\vec 0$$ (the vector). Online vector calculator - add vectors with different magnitude and direction - like forces, velocities and more. Built as a sprite. The vector is the normal vector (it points out of the plane and is perpendicular to it) and is obtained from the cartesian form from , and :. Matrix Rank. Throughout, we work in the Euclidean vector space V = Rn, the space of column vectors with nreal entries. The Matrix… Symbolab Version. There are separate table of contents pages for Math 254 and Math 255. Matrix calculator Solving systems of linear equations Determinant calculator Eigenvalues calculator Examples of solvings Wikipedia:Matrices. For instance, when a rope is attached on an iron bar at one end and a is pulled by a vehicle on the other end of the rope. Developing filters with Python has several advantages– it's fast to prototype and easy to share, since there's no compiled code. Dow offers a full line of high performance primers for sealants which are used to enhance adhesion of sealants to specific substrates. What is the range. Finding unit vector perpendicular to two vectors - Examples. Mathematical formulas exist that can determine individual life expectancies and the life expectancy of a larger population. Since a linear transformation preserves both of these operation, it is also a vector space homomorphism. Color; import java. The Python Programmable Filter is a powerful tool for data analysis in ParaView, as it enables you to implement ParaView readers and filters using Python. Span definition, the distance between the tip of the thumb and the tip of the little finger when the hand is fully extended. A value-at-risk metric, such as one-day 90% USD VaR, is specified with three items: a time horizon; a probability; a currency. You can also find magnitude, angle of 2D and 3D vectors using these online vector equation solvers. For every operation, calculator will generate a detailed explanation. Reminder: Solving. Consider the subspace: W = span{e^t, sin(2t), cos(3t)} 1. z=the value of the vector in the z axis. λ then, is the unknown in the equation. A key point to remember is that in python array/vector indices start at 0. This free online calculator help you to understand is the entered vectors a basis. A processing system in accordance with claim 11, wherein the output unit is operable to store a vector of elements received from the processing unit to the memory via the external interface, the output unit having a set of output storage elements to store TYPE, STRIDE, SKIP and SPAN values and an output arithmetic unit operable to calculate the. MATH 240: Vector Spaces Definition: A vector space is a set V on which two operations + and · are defined, called vector addition and scalar multiplication. To have a pivot in each row, A would have to have at least four columns (one for each pivot), which is not the case. You can also check out the TTR500 in action in this antenna matching tutorial video as well as many other TTR500 video tutorials. The infectious period of the host is 3 days ( r =1/3), the vector-to-host ratio m is 100, the transmission efficiencies b and c both 0. Calculate the magnitude of vector AB Determine the coordinates of point D on vector CD, if C (-6,0) and vector CD= vector AB. How to Meet Vector Space Requirements. For every vector v in V, there will then be some set of (scalar) coefficients c 1, , c n such that. For example, the vector 2 u + 3 v - 5 w is a linear combination of the vectors u, v and w. Span Calculator for Wood Joists and Rafters also available for the Android OS. You can make the purple vector be any vector in 2-space and this will work, so the span of the red and blue vectors is all of 2-space. This fits the intuition that a good way to think of a vector space is as a collection in which linear combinations are sensible. VECTOR-VALUED FUNCTIONS 13. Let A(t) be an anti-symmetric n ×n-matrix depending continu-. Let's see if results from Mathematica are consistent with this. There are infinitely many points we could pick and we just need to find any one solution for , , and. A uniformly spaced knot vector does not require the knowledge of the parameters, and is very simple to generate. Thus a vector is not a list. The following example shows how to calculate the determinant of two Vector structures. On the plus side, the measurement is widely used by financial industry. For example, let a set consist of vectors u, v, and w. 1 Additive Property of Determinant Applet 4. The purple vector is the sum of those pale vectors (using the parallelogram rule for addition), i. , vn} can be written Ax. The vector space that consists only of a zero vector. E F Graph 3D Mode. In Z the only addition is 0 C0 D0. The angle-of-attack is generally measured between the velocity (or relative velocity) vector V and the chord. What is a vector is, how to add and how to prove vectors are parallel and collinear? Examples: (1) A, B, C are midpoints of their respective lines. There were many solutions to Ax Db. Let me draw it in a better color. The bearing from A to B is the angle at A between the red arc (north) and the blue arc. Current local time in Papua New Guinea – Port Moresby. Example 2: Find an explicit description of Nul A by listing vectors that span the null space, where. In other words, given (x,y,z) ∈ R3, when is (x,y,z) ∈ span(v 1,v 2,v 3)? This leads to a system of linear equations: λ + µ + ν = x −2λ + 2µ − 10ν = y −λ + 3µ − 9ν = z. 10-aug-2020 - Ural 4320. Vector calculator. Since the triangle formed by our resultant vector and its components is a right triangle, we can use it to find our vector's length and therefore its magnitude. Enter data: 4. GLYPHICONS is a library of precisely prepared monochromatic icons and symbols, created with an emphasis to simplicity and easy orientation. We’ll start with addition of two vectors. Variance calculator and how to calculate. Our YouTube channel Engineering Lessons provide you the best knowledge and tricks about the different engineering r. 1 Linear Span of Vectors Applet 5. span{v1,v2,v3}=span{v1,v2}. Solution : Let a vector = i vector + 2j vector + k vector. (b) Yes, the columns of B will span R 5 if there are 5 pivots (a pivot in every row). The CNA has not provided a score within the CVE. 0 0 0 0 0. • The span of a single vector is all scalar multiples of that vector. Just like Twitter's Bootstrap, this drop-in uses sprites. SPECIFY THE NUMBER OF VECTORS AND THE VECTOR SPACES Please select the appropriate values from the popup menus, then click on the "Submit" button. So, the complete solution to Ax = b for our example will be all vectors of the form 1 0 6 0 —3 —2 1 0 +C 0 +C2 _ o i The general idea of this lecture is that to find the total solution (the set. You can add, subtract, find length, find dot and cross product, check if vectors are dependant. Hi, I want to initialize the buffer that I want to use inside a constructor. Others gave nice answers for this question, but I will show you an algorithm for this kind of problem in general which works even with non inner product spaces. A great virtue of MATLAB (ok, almost any programming language) is the ability to write functions that do what you want. How to calculate route distance and time with respect to speed. min the vector space V are linearly independent i the only scalars c 1;:::;c m2Fwith c 1v 1 + c 2v 2 + +c mv m= 0 are c 1 = c 2 = =c m=0. When you're calculating the center of gravity, the first thing you should do is to find the weight of the object. Finding unit vector perpendicular to two vectors - Examples. Determining if the set spans the space. The set of all linear combinations of a collection of vectors v 1, v 2,…, v r from R n is called the span of { v 1. Grow and Shrink algorithms both test whether a vector is superfluous in spanning a vector space V. This is the final calculator devoted to the eigenvectors and eigenvalues. Advanced Math Q&A Library Let F be the vector space of infinitely differentiable functions from R to R. This means that there exist scalars a,b such that av+bw=u. Web Study Guide for Vector Calculus This is the general table of contents for the vector calculus related pages. Hence, the vector Xθ is in the column space. Developing filters with Python has several advantages– it's fast to prototype and easy to share, since there's no compiled code. It makes the lives of people who use matrices easier. If just a few of them signed up for Robinhood using my referral link, it would cover my hosting costs. they are linearly. By using this calculator, you as the user expressly agree to the terms of this disclaimer in full. False (I guess), although this is sort of a technicality. Before the device can calculate your functional threshold power (FTP), you must have a paired power meter and heart rate monitor (Pairing Your Wireless Sensors). Simply select the solution you want to compare and start your savings. vector ~n that is orthogonal to a plane is also orthogonal to any two vectors in the plane. It contains well written, well thought and well explained computer science and programming articles, quizzes and practice/competitive programming/company interview Questions. Instead they span a plane in that passes thru the origin. calculate, using numerical methods, the actual beam deflection under the anticipated design load and compare this figure with the allowable value to see if the chosen beam section is adequate. Vectors A and B point toward the points A and B on the earth's surface. Calculator Use Calculate the distance between 2 points in 2 dimensional space. E 2 = eigenspace of A for λ =2 Example of finding eigenvalues and eigenvectors Example Find eigenvalues and corresponding eigenvectors of A. VECTOR-VALUED FUNCTIONS 13. A key point to remember is that in python array/vector indices start at 0. Vector Shorelines and Associated Shoreline Change Rates Derived from Lidar and Aerial Imagery for Dauphin Island, Alabama: 1940-2015. D, but 2 of them are L. This matrix calculator computes determinant, inverses, rank, characteristic polynomial, eigenvalues and eigenvectors. statistical. Linear Algebra exam problems and solutions at the Ohio State University (Math 2568). By continuing to use Pastebin, you agree to our use of cookies as described in the Cookies Policy. Calculate arithmetic mean, median and modal values from discrete data sets. Vocabulary word: vector equation. This algorithm uses basic vector math including calculation of the so-called dot product and cross product. The free images are pixel perfect to fit your design and available in both png and vector. The Null Space Calculator will find a basis for the null space of a matrix for you, and show all steps in the process along the way. 1) Preliminary Questions 1. This program was created using the relatively new Actionscript language. In proposition 8. For a given vector space V, what is the minimum number of vectors whose span equals V? How can we obtain a minimum number of vectors? Two natural approaches come to mind, the Grow algorithm and the Shrink algorithm. But this is really easy, because given a plane we know what the normal vector is. By changing the value of the scalars (the sliders g and h), it can be seen that every point in the plane is expressible as a. Web Study Guide for Vector Calculus This is the general table of contents for the vector calculus related pages. The set of all linear combinations of a collection of vectors v 1, v 2,…, v r from R n is called the span of { v 1. Elementary Surfaces. Span definition, the distance between the tip of the thumb and the tip of the little finger when the hand is fully extended. jlies between span j−1 and span j, the three non-zero elements in row jof matrix F are given by F j,j−1 = L j−1 EI j−1, (8) F j,j = 2 L j−1 EI j−1 + L j EI j , (9) F j,j+1 = L j EI j. Ruler Clipart Free download! | View 95 Ruler illustration, images and graphics from +50,000 possibilities. Vector calculator This page allows you to carry computations over vectors. Your actual results may vary. Unlike Matlab, which uses parentheses to index a array, we use brackets in python. The Null Space Calculator will find a basis for the null space of a matrix for you, and show all steps in the process along the way. Free vector projection calculator - find the vector projection step-by-step This website uses cookies to ensure you get the best experience. 2 we defined the notion of orthogonal projection of a vector v on to a vector u. We will also quickly look at how to write a system of equations that is equivalent to a given vector equation, and vice versa, as well as how to create, or force, a vector to be. Solutions: (a) Since ____ is not in H, H is not a vector space. Here is a simple online linearly independent or dependent calculator to find the linear dependency and in-dependency between vectors. just a change of per-spective Theorem. Here is a vector containing three numeric values 2, 3 and 5. The "Reset" button clears the calculator. Our YouTube channel Engineering Lessons provide you the best knowledge and tricks about the different engineering r. Then (f) Any set of linearly independent elements of V is contained in a basis of V. This free online calculator help you to find a projection of one vector on another. Add a second vector (0, 1, 0), and now they form a vector space that contains all points on the xy plane. How to draw a route on the fly. That is, because v 3 is a linear combination of v 1 and v 2, it can be eliminated from the collection without affecting the span. Let W be a subspace of V. Torque Vector Control. (3) Given the vectors, prove that the three given points are collinear. The Time, Material & Labor Savings Calculators are here to help you save versus traditional construction methods. So, pivots have nothing to do with it. For the following description, intoduce some additional concepts. 1 Linear Span of Vectors Applet 5. E F Graph 3D Mode. Examples:. The scalar projection of u on v is the magnitude of the vector projection of u on v. Vector Independence, Span, Basis, and Dimension -- Lecture 9. The calculator will find the row echelon form (simple or reduced - RREF) of the given (augmented) matrix (with variables if needed), with steps shown. Linearly self reliant isn't sturdy sufficient. Related Symbolab blog posts. (b) Rewrite x y 0 y z 0 as x y z 0 0 or as x y z 0 0 So V Nul A where A 1 10 011. Another way to find the distance is by finding the plane and the line intersection point and then calculate distance between this point and the given point. Second calculator - the Eigenvalue calculator solves that equation to find eigenvalues (using analytical methods, that's why it works only up to 4th degree), and the calculator below calculates. The Current Reference Calculator and Flux Controller. Making a skew-symmetric matrix from a vector is not something most people will ever need to do, so it is unlikely you would find a simple command to do it. The only vector I can get with a linear combination of this, the 0 vector by itself, is just the 0 vector itself. The first of these is the resultant, and this is obtained when the components of each vector are added together. Primer usage calculator. Rref Calculator for the problem solvers. Values must be numeric and separated by commas, spaces or new-line. Show that the three vectors above are linearly independent. First, enter the column size & row size and then enter the values to know the matrix elimination steps. Given the set S = {v 1, v 2, , v n} of vectors in the vector space V, determine whether S spans V. 1 Vector Equations. Your actual results may vary. S is linearly independent. This matrix calculator allows you perform any of the following matrix calculations between 2 matrices having maximum 4 columns and 4 rows: 1. it is a theorem) that any vector space has a basis. I compiled all the Glyphicons FREE in to a sprite. Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to find a projection of one vector on another. In the latter case, one can calculate life expectancy from a population’s life table. 3 Rank of Matrix: CHAPTER 4: DETERMINANTS Applet 4. In linear algebra, the linear span (also called the linear hull or just span) of a set S of vectors in a vector space is the smallest linear subspace that contains the set. Online vector calculator - add vectors with different magnitude and direction - like forces, velocities and more. The process of projecting a vector v onto a subspace S—then forming the difference v − proj S v to obtain a vector, v ⊥ S, orthogonal to S—is the key to the algorithm. Enter values into Magnitude and Angle or X and Y. Yes, the vector "w" is in Nul A. Therefore, by the linear lemma, there is a vector in the list which is in the span list of the ones before after it. So the span of a set of vectors, and the null space, column space, row space and left null space of a matrix are all subspaces, and hence are all vector spaces, meaning they have all the properties detailed in Definition VS and in the basic theorems presented in Section VS. Vector is the mathematical object which possess both magnitude and direction. Take half of the width of the board you're using as the gable post and deduct it from the bottom chord dimension obtained in Step 1. How to Meet Vector Space Requirements. E F Graph 3D Mode. Convert Matrix to a Vector This is the opposite tutorial of the Converting a Single Row Vector into a Matrix tutorial. We haven't actually learned to use matrices yet, but I think it was meant for us to show that the 3 vectors are L. Calculate arithmetic mean, median and modal values from discrete data sets. For example, the vector 2 u + 3 v - 5 w is a linear combination of the vectors u, v and w. This means that there exist scalars a,b such that av+bw=u. If {x 1, x 2, …, x n} is orthonormal basis for a vector space V, then for any vector x ∈ V, x = 〈x, x 1 〉x 1 + 〈x, x 2 〉x 2 + ⋯ + 〈x, x n 〉x n. Organize, analyze and graph and present your scientific data. Find the vector OB. A simple tool for web and print designers to calculate your grids. This free online calculator help you to find a projection of one vector on another. While every effort has been made to insure the accuracy of the information presented, and special effort has been made to assure that the information reflects the state-of-the-art, neither the West Coast Lumber Inspection Bureau, the American Wood Council nor its members assume any responsibility for any particular design prepared from this Online Span Calculator. Elementary Surfaces. The free images are pixel perfect to fit your design and available in both png and vector. min the vector space V are linearly independent i the only scalars c 1;:::;c m2Fwith c 1v 1 + c 2v 2 + +c mv m= 0 are c 1 = c 2 = =c m=0. The Time, Material & Labor Savings Calculators are here to help you save versus traditional construction methods. Add a second vector (0, 1, 0), and now they form a vector space that contains all points on the xy plane. Note the slight abuse of language here. You can add, subtract, find length, find dot and cross product, check if vectors are dependant. The span of a set of vectors is by definition the set of all linear combinations, so if a vector X is in the span there will be a solution to the equation. Many people confuse acceleration with velocity (or speed). For example, if you use 5 by 15 cm (2 by 6 inch) boards, the width of the board is 4. The first row of M is the vector v, the second row is the vector w, and the third row is the vector u. The Current Controller. If A is a vector of observations, the variance is a scalar. Explore Port Moresby's sunrise and sunset, moonrise and moonset. Projection onto a subspace. It can be characterized either as the intersection of all linear subspaces that contain S, or as the set of linear combinations of elements of S. , where is some scalar number. Making a skew-symmetric matrix from a vector is not something most people will ever need to do, so it is unlikely you would find a simple command to do it. A vector belongs to V when you can write it as a linear combination of the generators of V. We want to determine the span of these vectors. The "Reset" button clears the calculator. Population variance and sample variance calculator. This free online calculator help you to understand is the entered vectors a basis. Show that MN is parallel to AB. For example, let a set consist of vectors u, v, and w. It consists of the leading edge (LE), the trailing edge (TE) and the line joining the two called the chord (c). 3 Relationship between SVD and eigenvector decomposition De nition: An eigenvector of a square matrix Ais de ned as a vector satisfying the equation A~x= ~x; 2. Now when we recall what a vector space is, we are ready to explain some terms connected to vector spaces. As a side note, you can remove the end tag if there is HTML immediately after the vector to close it. eigenspaces_right() vectors on right, _left too. Here we see two given vectors (in solid red and blue), scalar multiples of those vectors (in dashed red and blue), and their vector sum (in black). They are also referred to as i hat,$\hat{i}$, which we assume goes 1 along the x-axis, and j hat,$\hat{j}$, which we assume goes 1 along the y-axis in a coordinate system. Long, and Christopher G. Since a linear transformation preserves both of these operation, it is also a vector space homomorphism. Span(S) = {a(1,2,3) + b(2,3,4) + c(3,4,5)} for some real numbers a, b, and c. Since the 's are linearly independent, is not is such a vector. The following example shows how to calculate the determinant of two Vector structures. How to delete a location from a route on the fly. This algorithm uses basic vector math including calculation of the so-called dot product and cross product.$\endgroup$– cheesyfluff Nov 12 '15 at 1:37$\begingroup\$ Thank you; this was very helpful. Get the free "The Span of 2 Vectors" widget for your website, blog, Wordpress, Blogger, or iGoogle. Get more help from Chegg Get 1:1 help now from expert Precalculus tutors Solve it with our pre-calculus problem solver and calculator. com, we have a calculator that will do all the work for you. net) for Bulgarian translationManuel Rial Costa for Galego translation. Note that each vector X i 2Rn has n coordinates, so that A is an n k matrix. (10) For the case of uniformly distributed loads, row jof vector d is d j = − w j−1L 3 j−1 4EI j−1 − w jL3 j 4EI j. What is Span and Linear Combinations? Span contains a set of vectors. av₁ + bv₂ + cv₃ = X. Thus for example, 1,2,3,4,5 is not a vector in 5, but 1. We also display any CVSS information provided within the CVE List from the CNA. Section 5-2 : Vector Arithmetic. q k} that are a basis for V. Instron Bluehill Calculation Reference Reference Manual - Software Manual Number Help Version 1. it is a theorem) that any vector space has a basis. This matrix calculator allows you perform any of the following matrix calculations between 2 matrices having maximum 4 columns and 4 rows: 1. Roof Truss - Design 3 - 30' span, 2-web, with plywood gussets (Figure 3) Figure 3 - Roof truss - 30' span, 2-web, with plywood gussets Detailed information for a 2-web roof truss with a 30' span with plywood gussets. Many people confuse acceleration with velocity (or speed). Where: L B = Shaft overhung load due to belt tension, lb N B = Number of belts P A = Force required to deflect one belt 1/64 inch per inch of span, lbs Y = 2 (f avg) ( 1/64) 2 where f is a strain constant based on the type and section of belt. For x 1 v 1 + x 2 v 2 + x 3 v 3 + x 4 v 4 + x 5 v 5 = 0, it is possible to. BTW I used a calculator so it wasn't that much trouble. A = 10−1 2 −15 00 2 λ =2, 1, or − 1. The following function will take a range as a matrix and convert it to a single column vector: Option Explicit Function Create_Vector(Matrix_Range As Range) As Variant Dim No_of_Cols As Integer, No_Of_Rows As Integer Dim…. Their robust construction ensures excellent tolerance to poor ambient conditions. Here is an example of vectors in R^3. A vector has vector coordinates. By using this calculator, you as the user expressly agree to the terms of this disclaimer in full. Note the slight abuse of language here. This reformulation applies more generally to vectors in Rn, and we state it here for the. The converse of the lemma holds: any subspace is the span of some set, because a subspace is obviously the span of the set of its members. There are separate table of contents pages for Math 254 and Math 255. Vector List based STAT-data editor Viewing and editing of input data in list format, showing data groups (x-data, y-data, frequency) and surrounding data. The short answer is the following: * If you are claiming this: [math]\forall \vec a_1,\vec a_2, \vec a_3 \in \R^3. For example, if you use 5 by 15 cm (2 by 6 inch) boards, the width of the board is 4. Throughout, we work in the Euclidean vector space V = Rn, the space of column vectors with nreal entries. I augment the vector T and check if the rank decreases. Color; import java. Results provided by this calculator are intended for Illustrative purposes only and the accuracy is not guaranteed. For instance, when a rope is attached on an iron bar at one end and a is pulled by a vehicle on the other end of the rope. Hence, the vector Xθ is in the column space. Vector calculator This page allows you to carry computations over vectors. Considering that target is more likely a (positional) vector, it then calculates the direction towards that, using basic vector arithmetic. In proposition 8. The result for k = 1 is trivial. Thus a vector is not a list. Unlike Matlab, which uses parentheses to index a array, we use brackets in python. Let's assume that v in V but v. Thus {(1,3,1),(0,−1,0)} is a basis for the span. False (I guess), although this is sort of a technicality. This calculator allows to find eigenvalues and eigenvectors using the Characteristic polynomial. A simple tool for web and print designers to calculate your grids. working example 2 linearly self reliant lines span a airplane. First of all, velocity is simply speed with a direction, so the two are often used interchangeably, even though they have slight differences. Pros and Cons of Value at Risk (VaR) There are a few pros and some significant cons to using VaR in risk measurement. So for any u and v in the vector space, there's a rule called addition that assigns another vector. Calculates the matrix-vector product. Span Margin Calculator - NSE F&O span margin calculates the margin for futures and options trading. Stator Flux-Orientated Control Equations. Just like Twitter's Bootstrap, this drop-in uses sprites. A vector belongs to V when you can write it as a linear combination of the generators of V. I compiled all the Glyphicons FREE in to a sprite. Thus for example, 1,2,3,4,5 is not a vector in 5, but 1. Span definition, the distance between the tip of the thumb and the tip of the little finger when the hand is fully extended. | 2020-10-24T11:28:36 | {
"domain": "lasalsese.it",
"url": "http://iybu.lasalsese.it/vector-span-calculator.html",
"openwebmath_score": 0.7472466826438904,
"openwebmath_perplexity": 771.0403648806702,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes",
"lm_q1_score": 0.9715639702485928,
"lm_q2_score": 0.8596637559030338,
"lm_q1q2_score": 0.8352183317639686
} |
https://math.stackexchange.com/questions/2381116/let-me-know-your-opinion-about-simple-formula-to-find-beginpmatrix-n-k-e | # Let me know your opinion about simple formula to find $\begin{pmatrix} n \\ k \end{pmatrix}$ [closed]
I often say to my students: There is a simple way to find binomial formula ,and the way is :$$\begin{pmatrix} n \\ k \end{pmatrix} =\dfrac{\underbrace{n.(n-1).(n-2)...}_{\text{k-terms decreasing}}}{\underbrace{1.2.3...k}_{\text{k-terms increasing}}}$$ for example: $$\begin{pmatrix} 5 \\ 3 \end{pmatrix}= \frac{5!}{3!(5-3)!} \to \\ \begin{pmatrix} 5 \\ 3 \end{pmatrix}= \frac{5.4.3}{1.2.3}=5.2=10$$ for example $$\begin{pmatrix} n \\ 2 \end{pmatrix}= \frac{n(n-1)}{1.2} =\frac{n(n-1)}{2}\\\begin{pmatrix} n \\ 3 \end{pmatrix}= \frac{\underbrace{n(n-1)(n-2)}_{\text{3-terms decreasing}}}{\underbrace{1.2.3}_{\text{3-terms increasing}}}$$ or $$\begin{pmatrix} 11 \\ 7 \end{pmatrix}\underbrace{=}_{\text{w.r.t. to }\begin{pmatrix} n \\ k \end{pmatrix}=\begin{pmatrix} n \\ n-k \end{pmatrix}} \begin{pmatrix} 11 \\ 4 \end{pmatrix}=\frac{11.10.9.8}{1.2.3.4}$$ I tel this for speedy work with binomial to k-11,k-12 students.Is this a good way to teach binomial calculation for the students ? (instead of formal calculation $\frac{n!}{r!(n-r)!}$) .If not tell my why ?
## closed as unclear what you're asking by Lord Shark the Unknown, Cameron Buie, José Carlos Santos, Simply Beautiful Art, NamasteAug 3 '17 at 15:25
Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
• i think you should move this to matheducators.stackexchange.com – lasen H Aug 3 '17 at 11:28
• What's your actual question? – Lord Shark the Unknown Aug 3 '17 at 11:29
• Basic definition should be used ..but tricks can be used and taught for speed. – Pranita Gupta Aug 3 '17 at 11:36
• The falling factorial definition makes more sense for the generalized binomial theorem, when you have things like: $$\sqrt{x+y}=\sum_{k=0}^\infty\binom{1/2}kx^{\frac12-k}y^k$$ – Simply Beautiful Art Aug 3 '17 at 11:58
• This is the method I personally use to calculate binomial coefficients by hand or mentally. It is comparable to generating Pascal's triangle in terms of ease for not-too-big numbers. For even bigger numbers, there is an nCr option in good calculators. – anon Aug 3 '17 at 15:30
I often say to my students: There is a simple way to find binomial formula ,and the way is :$$\binom n k =\dfrac{\overbrace{n\cdot (n-1)\cdot (n-2)\cdots}^{\text{k-terms decreasing}}}{\underbrace{1\cdot 2\cdot 3\cdots k}_{\text{k-terms increasing}}}$$
Sure, but to be clearer try it like this:
$$\dbinom n k ~=~\dfrac{n^{\underline k}}{k^{\underline k}}~=~\dfrac{\overbrace{n\cdot (n-1)\cdot (n-2)\cdots (n-k+1)}^{\text{k-terms decreasing}}}{\underbrace{k\cdot (k-1)\cdot (k-2)\cdots 1}_{\text{k-terms decreasing}}\qquad\qquad}$$
Where $n^{\underline k}$ is called the $k$ falling-factorial of $n$. This is excellent if you want to calculate by hand, and will give the students a practical feel for the numbers.
However, modern personal computers (or phones even) can easily handle most factorials that you'll want to use, so using the factorial function will be quicker (that is, unless your package has access to a binom[,] function which probably uses the falling factorial to calculate it anyway).
However, there is a reason to have the students think of the binomial coefficient as $\frac{n!}{k!(n-k)!}$. This is because when used in formulas that multiply or divide binomial coefficients expressed like so, there is often a lot of convenient cancelling.
Still knowing the falling factorial expression will prove useful if you cover generalised binomial expressions.
$$\dbinom{1/2}{5}=\dfrac{0.5\cdot (-0.5)\cdot(-1.5)\cdot(-2.5)\cdot(-3.5)}{5!}=\dfrac 7{256}$$
And such.
So knowing the falling-factorial form cannot hurt, and can help, but should be taught in addition to rather instead of the factorial form.
• @Khosrotash: Graham's answer is very good. I think you can parallel it with your method by introducing (soon or later) that $k! = k^{\,\underline {\,k\,} } = 1^{\,\overline {\,k\,} }$ and with the classical definition by noting that $n^{\,\underline {\,k\,} } = n!/\left( {n - k} \right)!$ – G Cab Aug 3 '17 at 15:00
For small $n,k$ such that the factorials are in range, the standard formula is appropriate.
For larger $n$ the falling factorial approach is indeed more efficient (and computing for $k$ or $n-k$ should be considered).
When you need the coefficients for successive $k$ and $n$, the recursive method (leading to the construction of Pascal's triangle) is better. | 2019-05-21T02:56:19 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2381116/let-me-know-your-opinion-about-simple-formula-to-find-beginpmatrix-n-k-e",
"openwebmath_score": 0.7279009222984314,
"openwebmath_perplexity": 721.5956407650518,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9715639718953155,
"lm_q2_score": 0.8596637541053281,
"lm_q1q2_score": 0.8352183314330105
} |
https://projecteuler.chat/viewtopic.php?f=4&t=3581&p=38948 | ## Set containing max number of subsets
Arrangements, combinations and permutations, probability, ...
albert_nik
Posts: 18
Joined: Wed Apr 22, 2009 10:46 am
### Set containing max number of subsets
Let's say we have the set of numbers [1..100] and some subsets with size<=5.
{1} , {25}, {2,30}, {1,2,3}, {3, 20, 50}, {40,50,60,70}, {1,2,3,4,5}....
Is it some efficient method to find a set of size=20 το contain the most sets?
hk
Posts: 10402
Joined: Sun Mar 26, 2006 9:34 am
Location: Haren, Netherlands
### Re: Set containing max number of subsets
Doesn't have every set of size n have the same number of subsets? (2^n)
albert_nik
Posts: 18
Joined: Wed Apr 22, 2009 10:46 am
### Re: Set containing max number of subsets
hk wrote:Doesn't have every set of size n have the same number of subsets? (2^n)
Yes.
I mean we have some (not all) singles, doubles, triples, 4-uples and 5-uples
{1} , {25}, {2,30}, {1,2,3}, {3, 20, 50}, {40,50,60,70}, {1,2,3,4,5} ...(they are less than Cnk(100, 1)+Cnk(100, 2)+Cnk(100, 3)+Cnk(100, 4)+Cnk(100, 5)
And I want a set or size=20 that contains the most the above subsets
hk
Posts: 10402
Joined: Sun Mar 26, 2006 9:34 am
Location: Haren, Netherlands
### Re: Set containing max number of subsets
Hard to tell if you don't specify the rules that make a subset a valid subset.
kevinsogo
Posts: 1204
Joined: Thu Sep 16, 2010 3:39 am
Location: Manila, Philippines
### Re: Set containing max number of subsets
I gather that what albert_nik is looking for is the following:
You are given the set {1,2,...,100}, and some subsets of it, of size ≤ 5.
The task is to find a subset of size 20 that contain the most of the given subsets.
It looks similar to a knapsack/bin packing problem.
thundre
Posts: 356
Joined: Sun Mar 27, 2011 9:01 am
### Re: Set containing max number of subsets
This could be done with dynamic programming, for some cases.
The state would be the set built so far and the score so far (number of target subsets included).
Start with the empty set, which has a score of 0.
Iterate over the list of target subsets. For each state, calculate the union of the set and the current target subset, and add one to the score. If the size of the union is greater than 20, discard it. Otherwise add it to the collection.
Optionally, you can prune out any state with an equal solution set but a lower score than another.
At the end, the state(s) with the highest score are the answer(s).
albert_nik
Posts: 18
Joined: Wed Apr 22, 2009 10:46 am
### Re: Set containing max number of subsets
I tried starting from empty set, adding subset A, then B, A∪B, C , A∪C, B∪C , A∪B∪C, D ... removing duplicates(same unions with same number of subsets) and those with size >20.
This can work for a few subsets, but the list grows rapidly.
Checking each of 100!/(80!*20!) subsets is impossible also.
jaap
Posts: 538
Joined: Tue Mar 25, 2008 3:57 pm
Contact:
### Re: Set containing max number of subsets
albert_nik wrote:I tried starting from empty set, adding subset A, then B, A∪B, C , A∪C, B∪C , A∪B∪C, D ... removing duplicates(same unions with same number of subsets) and those with size >20.
This can work for a few subsets, but the list grows rapidly.
Doing it this way doubles the number of combinations at each stage, apart from those you can filter out for being too large.
I would do it in a different order - first the empty set, then every single subset (A, B, C, ...), then every combination of two subsets (A∪B, A∪C, B∪C, A∪D, ...), then every combination of three subsets, and so on.
After each stage you can discard the previous one, since they are no longer needed.
If you have n subsets, the stage number r will have at most nCr combinations that need to be considered. This is worst at around r=n/2, but you may find that by then many of them are pruned away already for being too large.
I have tried this out, coding it up in C#. Here is the main routine (for brevity I have left out some utility functions and the implementation of the classes MySet and PotentialAnswer).
Code: Select all
static void Main()
{
// Generate some random subsets, but filter out any duplicates
MySet[] subSets = generateSubSets();
foreach (MySet subset in subSets) Console.WriteLine(subset);
int size = 0;
while(true)
{
// try to add one more subset to any of the current potential answers
int index = 0;
foreach (MySet subSet in subSets)
{
{
// store it if it is valid
if (newAnswer != null && newAnswer.Count <= 20)
{
}
}
index++;
}
// if have not found better answers, we are done.
if (answers2.Count == 0) break;
size++;
// replace previous answers by improved ones
var t = answers2;
}
{
}
}
Note that different combinations of subsets may have the same union, so the PotentialAnswer class also keeps track of which subsets were used in its formation.
This works very well for, say, 20 subsets:
Code: Select all
11 55 89
21
38 57 86
59 95
42
4
79
74
10 57 72 82
61
38 57 58 93
16 20 67 73
11 16 39 70
24 95
44 86
12 62
77
1 6 46 59
25 41
35 98
1: 20
2: 190
3: 1140
4: 4845
5: 15504
6: 38760
7: 77294
8: 120594
9: 134237
10: 91958
11: 33734
12: 5717
13: 351
14: 4
01111111011001111010 4 12 21 24 25 38 41 42 44 57 58 59 61 62 74 77 79 86 93 95
01111111011001111001 4 12 21 24 35 38 42 44 57 58 59 61 62 74 77 79 86 93 95 98
01111111011001101011 4 21 24 25 35 38 41 42 44 57 58 59 61 74 77 79 86 93 95 98
01111111010001111011 4 12 21 24 25 35 38 41 42 44 57 59 61 62 74 77 79 86 95 98
It gets much harder with say 30 subsets (more so especially if there are several small ones), but still doable:
Code: Select all
65
35 63 72
8 75 84 96
29
56 77 89
7 21
29 38 67 98
12 30 56
14 40 76
31
44 64
15 59
64 72
40 68 90
64 82 90 98
1 15 36 80
88
22
12 19 94
12 27 31 44
5
47
33 56 76
61 83
15 21 80
55
48
10 47
51 53
1: 29
2: 406
3: 3654
4: 23751
5: 118755
6: 474874
7: 1545720
8: 3989207
9: 7559421
10: 9590741
11: 7474127
12: 3249174
13: 696534
14: 61391
15: 1670
16: 5
11010100011110001100110011111 5 7 10 15 21 22 29 31 35 44 47 48 55 59 63 64 65 72 80 88
10010100011110011100110011111 1 5 7 10 15 21 22 29 31 36 44 47 48 55 59 64 65 72 80 88
10010100011110001101110011111 5 7 10 12 15 21 22 27 29 31 44 47 48 55 59 64 65 72 80 88
10010100011110001100110111111 5 7 10 15 21 22 29 31 44 47 48 55 59 61 64 65 72 80 83 88
10010100011110001100110011111 5 7 10 15 21 22 29 31 44 47 48 51 53 55 59 64 65 72 80 88
With much more than that, my machine would probably not be able to cope.
albert_nik
Posts: 18
Joined: Wed Apr 22, 2009 10:46 am
### Re: Set containing max number of subsets
Thank You jaap.
The idea was from a lottery game (keno).
There are 80 numbers [1..80] and are selected randomly 20 numbers.
A player can select 1 to 8 numbers.
If the player's numbers are all included in the 20 he wins.
There are N players that have selected some sets with 1<=size<=8. The task is to find 20 numbers that maximize the number of winners. I dont know if there is a different approach to this problem.
ggoyo
Posts: 25
Joined: Sun Jun 22, 2014 9:45 am
Location: Paris
Contact:
### Re: Set containing max number of subsets
I did not quite understand the question. In particular, is the output set a set of integers or a set of set of integers ? Since you are talking about keno I'm assuming it is a set of integers. In that case just put all the numbers in a set, if it is less than 20 then add random numbers (that aren't already in it) otherwise remove the integer that appears in the less number of subsets and repeat until you reach 20.
Please tell me if I have misunderstood the question.
jaap
Posts: 538
Joined: Tue Mar 25, 2008 3:57 pm
Contact:
### Re: Set containing max number of subsets
ggoyo wrote:I did not quite understand the question. In particular, is the output set a set of integers or a set of set of integers ? Since you are talking about keno I'm assuming it is a set of integers. In that case just put all the numbers in a set, if it is less than 20 then add random numbers (that aren't already in it) otherwise remove the integer that appears in the less number of subsets and repeat until you reach 20.
Please tell me if I have misunderstood the question.
The problem is that the algorithm you propose (essentially a kind of Greedy Algorithm) will not always produce the optimal solution. At best it always gives a kind of local optimum rather than a global optimum.
For example, consider the first randomly generated example I gave. The sets are (one set on each line):
Code: Select all
11 55 89
21
38 57 86
59 95
42
4
79
74
10 57 72 82
61
38 57 58 93
16 20 67 73
11 16 39 70
24 95
44 86
12 62
77
1 6 46 59
25 41
35 98
There are 37 numbers used - seven multiple times, thirty used only once. Which of the ones that are used only once would you throw out?
There are only 4 optimal solutions:
Code: Select all
4 12 21 24 25 38 41 42 44 57 58 59 61 62 74 77 79 86 93 95
4 12 21 24 35 38 42 44 57 58 59 61 62 74 77 79 86 93 95 98
4 21 24 25 35 38 41 42 44 57 58 59 61 74 77 79 86 93 95 98
4 12 21 24 25 35 38 41 42 44 57 59 61 62 74 77 79 86 95 98
All of these solutions use the numbers 4 21 24 42 44 61 74 77 79, which are used only once in the sets. How would you know to keep those? None of these optimal solutions use 11 or 16, and yet these are used twice in the sets, so how would you know to throw those out?
ggoyo
Posts: 25
Joined: Sun Jun 22, 2014 9:45 am
Location: Paris
Contact:
### Re: Set containing max number of subsets
jaap wrote: The problem is that the algorithm you propose (essentially a kind of Greedy Algorithm) will not always produce the optimal solution. At best it always gives a kind of local optimum rather than a global optimum.
Yes, you are right, I was too fast answering.
jaap wrote: All of these solutions use the numbers 4 21 24 42 44 61 74 77 79, which are used only once in the sets. How would you know to keep those? None of these optimal solutions use 11 or 16, and yet these are used twice in the sets, so how would you know to throw those out?
All these numbers have one thing in common : throwing them out cost one set but that set only contains 1 number which can be thrown out. However if I throw 1 I'd be able to throw 6 and 46 for free.
I could sort it by (1/S) where S is the sum of (1/number of appearances) of each number in each set the number appears in. For instance, in your exemple, 1 would be weighted 2/7 but 4 would be weighted 1.
I coded it in haskell (real nice for list manipulation) and it outputs your last solution (not a proof). | 2019-06-18T15:55:40 | {
"domain": "projecteuler.chat",
"url": "https://projecteuler.chat/viewtopic.php?f=4&t=3581&p=38948",
"openwebmath_score": 0.36823469400405884,
"openwebmath_perplexity": 471.2014865749079,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9715639694252315,
"lm_q2_score": 0.8596637541053281,
"lm_q1q2_score": 0.8352183293095687
} |
https://www.freemathhelp.com/forum/threads/working-in-other-bases.114430/ | # Working in other bases
#### apple2357
##### New member
Something i have been thinking about..
So take the multiple of 9 : 9,18,27,36 etc.. and one thing we notice is that if we keep adding digits up repeatedly they come to 9, so you have a simple divisibility test. Presumably, this works because 9 is one less than the base we are working in which is 10.
Does it follow if we work in other bases, e.g. base 5 and look at multiples of 4 that the digits add up to 4?
It appears so, upon a few test cases? Is this generally going to be true for other bases? Is this a theorem?
Last edited:
#### Jomo
##### New member
Something i have been thinking about..
So take the multiple of 9 : 9,18,27,36 etc.. and one thing we notice is that if we keep adding digits up repeatedly they come to 9, so you have a simple divisibility test. Presumably, this works because 9 is one less than the base we are working in which is 10.
Does it follow if we work in other bases, e.g. base 5 and look at multiples of 4 that the digits add up to 4?
It appears so, upon a few test cases? Is this generally going to be true for other bases? Is this a theorem?
1st of all what you say about digits on the nine table always adding up to 9 is simply not true. Are you saying that the only number in the 900's is 900? After all, any other number on the nine table in the 900s will have the sum of the digits adding up to more than 9. Consider 909, 918, 927,.... They are all on the nine table but their digits do not add up to 9. The divisibility test for 9 is that the sum adds up to a multiple of 9.
Why is this? Consider the integer x = a[SUB]n[/SUB]10[SUP]n[/SUP] + a[SUB]n-1[/SUB]10[SUP]n-1[/SUP] + ... + a[SUB]1[/SUB]10[SUP]1[/SUP] + a[SUB]0[/SUB]10[SUP]0[/SUP]. Now we can write a[SUB]k[/SUB]10[SUP]k[/SUP] as a[SUB]k[/SUB]*999...9 + a[SUB]k[/SUB]. Clearly 9 goes evenly into a[SUB]k[/SUB]*999...9 for each k. So 9 goes evenly into x precisely when 9 goes in a[SUB]n[/SUB]+a[SUB]n-1[/SUB]+...a[SUB]0[/SUB].
Now you try to see if this will work for base 5? Does it work for an arbitrary base b?
#### apple2357
##### New member
Ah ok Thanks for that clarification, When i said 'repeatedly' i sort of meant that but it wasn't very clear!
#### Jomo
##### New member
Ah ok Thanks for that clarification, When i said 'repeatedly' i sort of meant that but it wasn't very clear!
So did you prove this result in base 5? in base n?
#### apple2357
##### New member
So did you prove this result in base 5? in base n?
Not yet! Still thinking about how to put it together!
#### apple2357
##### New member
Not yet! Still thinking about how to put it together!
Is it as simple as replacing 10[SUP]n[/SUP] with 5[SUP]n[/SUP] and breaking up 5[SUP]n[/SUP]with 4 and the extra one? in the way you have with 9 above?
In which this will always work?
#### HallsofIvy
##### New member
Let b be a positive integer. A number written in "base b" is of the form $$\displaystyle a_nb^n+ a_{n-1}b^{n-1}+ \cdot\cdot\cdot+ a_2b^2+ a_1b^1+ a_0b ^0$$. Repeat what Jomo did,
Consider the integer $$\displaystyle x = a_n10^n + a_{n-1}10^{n-1} + ... + a_110^1 + a_010^0$$. Now we can write $$\displaystyle a_k10^k$$ as $$\displaystyle a_k*999...9 + a_k$$. Clearly 9 goes evenly into $$\displaystyle a_k*999...9$$ for each k. So 9 goes evenly into x precisely when 9 goes in $$\displaystyle a_n+a_{n-1}+...a_0$$
using b- 1 instead of 9. | 2019-02-18T14:16:33 | {
"domain": "freemathhelp.com",
"url": "https://www.freemathhelp.com/forum/threads/working-in-other-bases.114430/",
"openwebmath_score": 0.7209230661392212,
"openwebmath_perplexity": 761.8883332595514,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9715639669551474,
"lm_q2_score": 0.8596637559030338,
"lm_q1q2_score": 0.835218328932713
} |
https://www.physicsforums.com/threads/factoring-a-quartic-polynomial.948984/ | # Factoring a quartic polynomial
#### Mr Davis 97
1. The problem statement, all variables and given/known data
Factor $x^4-3x^2+9$ over the reals
2. Relevant equations
3. The attempt at a solution
I am factoring this polynomial over the reals. So there are two options. It will either split into two linear factors and an irreducible quadratic, or two irreducible quadratics. I'm really not too sure how to proceed from here... Should I first start looking for a real root?
Related Calculus and Beyond Homework News on Phys.org
#### andrewkirk
Homework Helper
Gold Member
The polynomial is easily factored over the complex numbers by first solving it as a quadratic in $x^2$. By then taking both square roots of both solutions, you'll have four complex roots of the quartic. The question then is whether any product of three or fewer degree-one polynomial factors each of the form $(x-r)$ where $r$ is a complex root, gives a polynomial with only real coefficients.
#### Mr Davis 97
The polynomial is easily factored over the complex numbers by first solving it as a quadratic in $x^2$. By then taking both square roots of both solutions, you'll have four complex roots of the quartic. The question then is whether any product of three or fewer degree-one polynomial factors each of the form $(x-r)$ where $r$ is a complex root, gives a polynomial with only real coefficients.
So I have $(x+\sqrt{\frac{3}{2}(1-i \sqrt{3})})(x-\sqrt{\frac{3}{2}(1-i \sqrt{3})})(x+\sqrt{\frac{3}{2}(1+i \sqrt{3})})(x-\sqrt{\frac{3}{2}(1+i \sqrt{3})})$. This seems very messy. Is there any way to simplify it?
#### Mark44
Mentor
So I have $(x+\sqrt{\frac{3}{2}(1-i \sqrt{3})})(x-\sqrt{\frac{3}{2}(1-i \sqrt{3})})(x+\sqrt{\frac{3}{2}(1+i \sqrt{3})})(x-\sqrt{\frac{3}{2}(1+i \sqrt{3})})$. This seems very messy. Is there any way to simplify it?
Looks fine to me. I don't think this factorization lends itself to much more simplification.
#### Mr Davis 97
Looks fine to me. I don't think this factorization lends itself to much more simplification.
So is there a way to get from this factorization to the product of two quadratics with real coefficients?
#### Ray Vickson
Homework Helper
Dearly Missed
So I have $(x+\sqrt{\frac{3}{2}(1-i \sqrt{3})})(x-\sqrt{\frac{3}{2}(1-i \sqrt{3})})(x+\sqrt{\frac{3}{2}(1+i \sqrt{3})})(x-\sqrt{\frac{3}{2}(1+i \sqrt{3})})$. This seems very messy. Is there any way to simplify it?
The complex numbers $\sqrt{(3/2) (1\pm i \sqrt{3})}$ simplify a lot.
#### mathwonk
Homework Helper
it seems it shoud factor into two real quadratics, no?
#### andrewkirk
Homework Helper
Gold Member
So is there a way to get from this factorization to the product of two quadratics with real coefficients?
Yes there is. Start by writing the two roots of the quadratic in $x^2$ in polar form, which has been made easy by the specially selected coefficients given in the problem.
#### lurflurf
Homework Helper
we are looking for quadratic factors ie
$x^4-3x^2+3^2=(a x^2+b x+c)(d x^2+e x+f)$
we can reduce the choices by considering each coefficient in turn
x^4
a d =1
we know a and d since there are very few factorizations
x^3
a e+b d=0
we know b+e
x
b f+c e=0
we know c-f
x^0
c f=9
we know c and f
x^2
a f+b e+c d
we know b and e
we are done
note that two coefficients being zero and the remaining coefficients being small helps greatly
#### stevendaryl
Staff Emeritus
Note that if you multiply or add two numbers that are complex-conjugates, then the result is real:
$(a+ib)(a-ib) = a^2 + b^2$
$(a+ib) + (a-ib) = 2a$
So it follows that $(x + \sqrt{\frac{3}{2} (1+i\sqrt{3})})(x + \sqrt{\frac{3}{2} (1-i\sqrt{3})})$ can be expressed using just real coefficients. Similarly for $(x -\sqrt{\frac{3}{2} (1+i\sqrt{3})})(x - \sqrt{\frac{3}{2} (1-i\sqrt{3})})$
So your real factorization is $[(x + \sqrt{\frac{3}{2} (1+i\sqrt{3})})(x + \sqrt{\frac{3}{2} (1-i\sqrt{3})})] [(x - \sqrt{\frac{3}{2} (1+i\sqrt{3})})(x - \sqrt{\frac{3}{2} (1-i\sqrt{3})})]$
There's a big simplification that comes from using the polar form of complex numbers: $a + ib = \sqrt{a^2 + b^2} e^{i \theta}$ where $\theta$ is chosen so that $cos(\theta) = \frac{a}{\sqrt{a^2+b^2}}$ and $sin(\theta) = \frac{b}{\sqrt{a^2+b^2}}$. Then the square root is easy:
$\sqrt{a+ib} = (a^2 + b^2)^{\frac{1}{4}} e^{i \frac{\theta}{2}}$
#### lurflurf
Homework Helper
So I have $(x+\sqrt{\frac{3}{2}(1-i \sqrt{3})})(x-\sqrt{\frac{3}{2}(1-i \sqrt{3})})(x+\sqrt{\frac{3}{2}(1+i \sqrt{3})})(x-\sqrt{\frac{3}{2}(1+i \sqrt{3})})$. This seems very messy. Is there any way to simplify it?
notice
$x^4-3x^2+9=(x-a)(x-a^\ast)(x+a)(x+a^\ast) % =[ x^2-(a+a^\ast)x+a^\ast a ] [ x^2+(a+a^\ast)x+a^\ast a ]$
$=x^4-[(a+a^\ast)^2-2a^\ast a]x^2+(a^\ast a)^2$
It is then easy find a+a* and a*a to effect the factorization.
You could also find a in rectangular or polar coordinates if you wanted to.
#### Ray Vickson
Homework Helper
Dearly Missed
Note that if you multiply or add two numbers that are complex-conjugates, then the result is real:
$(a+ib)(a-ib) = a^2 + b^2$
$(a+ib) + (a-ib) = 2a$
So it follows that $(x + \sqrt{\frac{3}{2} (1+i\sqrt{3})})(x + \sqrt{\frac{3}{2} (1-i\sqrt{3})})$ can be expressed using just real coefficients. Similarly for $(x -\sqrt{\frac{3}{2} (1+i\sqrt{3})})(x - \sqrt{\frac{3}{2} (1-i\sqrt{3})})$
So your real factorization is $[(x + \sqrt{\frac{3}{2} (1+i\sqrt{3})})(x + \sqrt{\frac{3}{2} (1-i\sqrt{3})})] [(x - \sqrt{\frac{3}{2} (1+i\sqrt{3})})(x - \sqrt{\frac{3}{2} (1-i\sqrt{3})})]$
There's a big simplification that comes from using the polar form of complex numbers: $a + ib = \sqrt{a^2 + b^2} e^{i \theta}$ where $\theta$ is chosen so that $cos(\theta) = \frac{a}{\sqrt{a^2+b^2}}$ and $sin(\theta) = \frac{b}{\sqrt{a^2+b^2}}$. Then the square root is easy:
$\sqrt{a+ib} = (a^2 + b^2)^{\frac{1}{4}} e^{i \frac{\theta}{2}}$
That is what I hinted at in post #6, but have received no feedback from the OP to date. (Actually, maybe not so much of a hint; perhaps more of a pre-hint, but with the motivation to have the OP do some more work on the problem.)
#### ehild
Homework Helper
1. The problem statement, all variables and given/known data
Factor $x^4-3x^2+9$ over the reals
2. Relevant equations
3. The attempt at a solution
I am factoring this polynomial over the reals. So there are two options. It will either split into two linear factors and an irreducible quadratic, or two irreducible quadratics. I'm really not too sure how to proceed from here... Should I first start looking for a real root?
Note that the polynomial can be written as $(x^2+3)^2-9x^2$.
"Factoring a quartic polynomial"
### Physics Forums Values
We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving | 2019-07-20T20:34:32 | {
"domain": "physicsforums.com",
"url": "https://www.physicsforums.com/threads/factoring-a-quartic-polynomial.948984/",
"openwebmath_score": 0.7904428839683533,
"openwebmath_perplexity": 627.9398332976031,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9715639653084245,
"lm_q2_score": 0.8596637559030338,
"lm_q1q2_score": 0.835218327517085
} |
https://math.stackexchange.com/questions/3238575/factoring-xaxbxcbccaab-and-use-the-result-to-solve-an-equat | # Factoring $(x+a)(x+b)(x+c)+(b+c)(c+a)(a+b)$ and use the result to solve an equation
I managed to prove that $$(x+a+b+c)$$ is a factor of $$(x+a)(x+b)(x+c)+(b+c)(c+a)(a+b)$$
Then I was asked to use the result to solve $$(x+2)(x-3)(x-1)+4=0$$
I know by comparison, $$a=2, b=-3, c=-1$$, and thus $$(x-2)$$ is a factor, but I can't really figure out how to solve the equation without expanding the brackets.
• Considering the other roots are $\pm \sqrt 5$, I can't imagine you'd be intended to do anything else but to expand the expression, divide by $(x-2)$, and find the roots of the quadratic that results. – Eevee Trainer May 24 at 19:37
• And the other factor is given by $$x^2-5$$ – Dr. Sonnhard Graubner May 24 at 19:38
• Also, the solutions are $x=2, x=\sqrt{5}, x=-\sqrt{5}$ – Loo Soo Yong May 24 at 19:38
• Yes, this is true. – Dr. Sonnhard Graubner May 24 at 19:38
• How hard is expanding the brackets - you have identified a factor of a cubic and what remains will be a quadratic. Practically that is a quick way through. – Mark Bennet May 24 at 20:28
Unfortunately, you do have to expand the brackets.
Fortunately, it's not so messy after all:
$$(x+2)(x-3)(x-1)+4=0$$
so
$$x^3-2x^2-5x+10=0$$ and factoring our $$x-2$$, $$(x-2)(x^2-5)=0$$
from which we can readily read the answer as $$x=2,\pm\sqrt{5}$$
• You actually don't have to expand the brackets if you spot a few tricks. I explain it in my answer. – J.G. May 24 at 19:51
• @J.G. Strictly speaking, your answer does expand most of the brackets, it just expands them one at a time after dividing $x-2$. – auscrypt May 24 at 19:54
Since the $$x^2$$ coefficient in your cubic is $$a+b+c$$, the quadratic factor is of the form $$x^2+k$$, with the roots being $$\pm\sqrt{-k}$$. The $$x=0$$ case gives $$k=\frac{abc+(b+c)(c+a)(a+b)}{a+b+c}=ab+bc+ca.$$In your case $$k=-5$$.
Simplifying $$k$$ as above looks like it requires tedious algebra, but things aren't as bad as they seem. It's the ratio of two fully symmetric polynomials in $$a,\,b,\,c$$, one of degree $$3$$, the other $$1$$. This doesn't prove on its own that $$k$$ is a polynomial; but if it is, it must be fully symmetric and of degree $$2$$, and hence proportional to $$ab+bc+ca$$. The case $$a=b=c$$ gives $$k=\frac{9a^3}{3a}=3a^2$$, so it'll have to be $$ab+bc+ca$$ itself. So it makes sense to double-check whether $$(ab+bc+ca)(a+b+c)=abc+(b+c)(c+a)(a+b)$$. But of course it does, because both sides are fully symmetric cubic functions, so they have a fixed ratio. Again, the case $$a=b=c$$ strengthens this to equality.
It is good that you managed to factor!
Here is one way to do it: $$(x+a)(x+b)(x+c)+(b+c)(c+a)(a+b)=\\ x^3+(a+b+c)x^2+(ab+bc+ca)x+abc+\\ 2abc+a^2b+ab^2+a^2c+ac^2+b^2c+bc^2=\\ x^2(x+a+b+c)+(ab+bc+ca)x+\\ ab(a+b+c)+ac(a+b+c)+bc(a+b+c)=\\ x^2(x+a+b+c)+(ab+bc+ca)x+\\ (a+b+c)(ab+bc+ca)=\\ x^2(x+a+b+c)+(ab+bc+ca)(x+a+b+c)=\\ (x+a+b+c)(x^2+ab+bc+ca).$$ Now, write the given equation in this form: $$(x+2)(x-3)(x-1)+4=0 \iff \\ (x+2)(x-3)(x-1)+(2-3)(-3-1)(-1+2)=0 \iff \\ (x+2-3-1)(x^2+2(-3)+(-3)(-1)+(-1)2)=0 \iff \\ (x-2)(x^2-5)=0 \Rightarrow x_1=2, x_{2,3}=\pm \sqrt{5}.$$
• This is a good and complete answer. – NoChance May 24 at 20:09
Probably not more efficient than straightforwardly factoring out, but it works as well:
$$\begin{eqnarray}\frac{(x+2)(x−3)(x−1)+4}{x-2} & = & \frac{(x-2+4)(x−3)(x−1)}{x-2}+\frac{4}{x-2} \\ & = & (1+\frac{4}{x-2})(x−3)(x−1)+\frac{4}{x-2} \\ & = & (x−3)(x−1) + \frac{4(x−3)(x−2+1)}{x-2}+\frac{4}{x-2}\\ & = & (x−3)(x−1) + 4(x−3)(1+\frac{1}{x-2})+\frac{4}{x-2}\\ & = & (x−3)(x−1) + 4(x−3) + \frac{4(x−2-1)}{x-2}+\frac{4}{x-2}\\ & = & (x−3)(x−1) + 4(x−3) + 4\\ & = & (x−3)(x+3) + 4\\ & = & x^2 - 5 \; . \end{eqnarray}$$ | 2019-06-18T16:45:08 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3238575/factoring-xaxbxcbccaab-and-use-the-result-to-solve-an-equat",
"openwebmath_score": 0.7696881294250488,
"openwebmath_perplexity": 309.94771747757557,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9715639653084245,
"lm_q2_score": 0.8596637541053281,
"lm_q1q2_score": 0.8352183257704989
} |
http://assistimi.it/eyzg/euclidean-distance-in-r-example.html | Euclidean Distance In R Example
92 General Formula. The Distance Formula itself is actually derived from the Pythagorean Theorem which is. Euclidean distance refers to the distance between two points. The Euclidean distance between 2 cells would be the simple arithmetic difference: x cell1 - x cell2 (eg. 1 shows height and weight mea-. The classification accuracy by Euclidean and Minkowsky distance functions are almost the same, which means that using 1 or 2 for r does not affect the k-NN performance (c. Mahalanobis in 1936 and has been used in various statistical applications ever since. The special case r = 1 is called the unit circle; its equation is x2+y2 = 1. (You can report issue about the content on this page here) Want to share your content on R-bloggers? click here if you have a blog, or here if you don't. Example of Complete Linkage Clustering. See newpnn or newgrnn for simulation examples. Figure A1: An Example Social Network According to (4), the Euclidean distance of structural equivalence between and ˆ at time t, % ˆ˙ is calculated as the following, % ˆ ˙ =&(). Therefore, the following statements compute the Euclidean pairwise distances between the 27 points in the Obs data set:. Other distance measures include Manhattan, Minkowski, Canberra etc. Well I read that those distances should lie pretty close to each other when calculated for small distances. It solves the problem of computing the greatest common divisor (gcd) of two positive integers. To change a network so that a layer’s topology uses dist, set net. CityBlock (Manhattan) distance. Basically, you don't know from its size whether a coefficient indicates a small or large distance. The Euclidean distance is defined as: 1. R&A Publications. It's defined as It's defined as This Euclidean distance adds up all the squared distances between corresponding data points and takes the square root of the result. if TRUE, the Euclidean distance will be calculated. These are the top rated real world PHP examples of euclidean extracted from open source projects. Beagles Weight Height Chihuahuas Dachshunds Figure 7. Distance matrices are sometimes called dissimilarity matrices. Meaning of euclidean distance. for the purposes of this example): Species spA spB spC spB. City block (Manhattan, taxicab, L 1 norm) distance. It includes. It is a generalization of the Euclidean and Manhattan distance measures and adds a parameter, called the "order" or "p", that allows different distance measures to be calculated. We may as well begin with the all-time Euclidean space distance measurement champion. Most machine learning algorithms including K-Means use this distance metric to measure the similarity between observations. Visit Stack Exchange. L1 distance (city-block) Distances for presence-absence data Distances for heterogeneous data The axioms of distance In mathematics, a true measure of distance, called a metric , obeys three properties. However, since we are working off of point objects (and not an existing raster as was the case in the previous example), we will need to create a blank raster layer which will define the extent of the Euclidean distance raster output. 1 Vector and Metric Spaces The set K n of n -tuples x = ( x 1;x 2:::;xn) can be made into a vector space by introducing the standard operations of addition and scalar multiplication. b = aq + r, where either r = 0 or d(r) B)=tr(B>A). The currently available options are "euclidean" (the default), "manhattan" and "gower". Euclidean distance : # create own function according to Euclidean distance formula euclidean_distance <- function(p,q){ sqrt(sum((p - q)^2)) } # check points 4 and 6 euclidean_distance(data[4,],data[6,]) #my own function. Other dissimilarity measures exist such as correlation-based distances, which is widely used for gene expression data analyses. " As a reminder, given 2 points in the form of (x, y), Euclidean distance can be represented as: Manhattan. But it may still work, in many situations if you normalize your. Properties:. statespace_ed returns a tibble with the time vector time, the Euclidean distance ed, and a logical vector ref_time indicating the time step defined as reference. Beagles Weight Height Chihuahuas Dachshunds Figure 7. This question is regarding the weighted Euclidean distance. Parhizkar, Euclidean Distance Matrices: Properties, Algorithms and Applications, PhD Thesis EPFL, 2013. That is true for the first case: Great Circle: SpatialLinesLengths(ln1,longlat=T)*1000 [1] 19. When differences_only = TRUE, the output will reflect the number of different loci. So we have to take a look at geodesic distances. It is also known as euclidean metric. , clusters), such that objects within the same cluster are as similar as possible (i. if TRUE, the Euclidean distance will be calculated. For mixed data (both numeric and categorical variables), we can use k-prototypes which is basically combining k-means and k-modes clustering algorithms. For example, in the table below we can see a distance of 16 between A and B, of 47 between A and C, and so on. # Name: CalculateDistance_Ex_02. I need to place 2 projects named A and B in this 3 dimensional space and measure the distance among them. of a region R is the set of points that are adjacent to. 56 comments Remembering Imola 1994 May. This script calculates the Euclidean distance between multiple points utilising the distances function of the aspace package. 92 ทำไมมันง่ายอย่างงี้ ## example code in R v1 <- c(30, 50) v2 <- c(20, 75) d <- sqrt(sum((v1 - v2)**2)) ## 26. For example, in the table below we can see a distance of 16 between A and B, of 47 between A and C, and so on. In the upper half-plane model it is very difficult to express this Euclidean distance to the hyperbolic distance between a given pair of points. If a>> month, as opposed to, say the Mahalanobis distance over all months >>> (-reshape- to have T variables measuring # of patients in each month, >>> and find the closest 15 obs in the standard deviation metric)?. Try Jupyter with C++ A basic example of using Jupyter with C++ Try Jupyter with Scheme. After the matching i am planning on running a difference in difference analysis to assess for the effect of policy changes on patient numbers, using the matches as a comparison group. Here's an example of how to calculate a distance matrix for geographic points (expressed as decimal latitudes and longitudes) using R: For example, the above distance matrix shows that the straight-line distance—accounting for curvature of the earth—between Los Angeles and NYC is 3,945 km. When working with a large number of observations, you can compute the distance. 1: Classical applications of clustering often involve low-dimen-sional Euclidean spaces. These results indicate that the Euclidean distance analysis of nucleotide skew data forms a nice addition to the virology toolbox. Euclidean Distance Measure The most common case is determining the distance between two points. Distance Measure An important component of a clustering algorithm is the distance measure between data points. Euclidean Distance Computation in Python. 1 We often work with distances because they are convenient to measure or estimate. This system of geometry is still in use today and is the one that high school students study most often. Answer and Explanation: As we know that Euclidean distance is the measure of displacement, hence the unit of euclidean measurement will be the same as the unit of distance or displacement. database: Example databases. Contents[show] Definition Given a number set $E$ , the Euclidean distance is a function $EuclideanDis:E^n\\times E^n \\longrightarrow \\mathbb{R}/math defined as. The Euclidean distance between the ith and jth objects is. The default setting, differences_only = FALSE, reflects the number of different. Euclidean definition, of or relating to Euclid, or adopting his postulates. OrthogonalDistance. Author(s). dab = dba 2. Calculate "d(x, x i)" i =1, 2, …. For your particular use case, you could also transform your data into 3D space, then use (squared) Euclidean distance and thus k-means. Also known as Gower's. It is effectively a multivariate equivalent of the Euclidean distance. Euclidean Distance Matrix These results [(1068)] were obtained by Schoenberg (1935), a surprisingly late date for such a fundamental property of Euclidean geometry. But in fact, hyperbolic space offers exactly this property---which makes for great embeddings, and we're off!. Based on the Euclidean distance algorithm, a busbar protection principle is proposed in this paper. The Mahalanobis distance is a measure of the distance between a point P and a distribution D, introduced by P. De Hoon et al. The example shown in Figure 1 uses the chessboard' distance metric but both the Euclidean and city block' metrics can be used as well. −John Clifford Gower [190, § 3] By itself, distance information between many points in Euclidean space is lacking. The formula for this distance between a point X ( X 1 , X 2 , etc. For a layer weight, set net. series1, example. The feasibility of the busbar protection criterion is verified by a simulation example. L∞-distance (maximum magnitude of the difference in any dimension). The first calculation being the "great circle distance" the second Euclidean distance. Calculate "d(x, x i)" i =1, 2, …. This is proposed as a new non-parametric MANOVA. For a detailed discussion, please head over to Wiki page/Main Article. So in this example I might want the distances using the traits x, z and n, thereby specifying the equation to be d=sqrt[(x2-x1)^2+(z2-z1)^2+(n2-n1)^2], and return a distance matrix as follows (calculated distances represented by. (2002) used the Euclidean distance for k-means clustering. , persons, organizations, countries, species) and columns represent variables (e. This calculator is used to find the euclidean distance between the two points. List ; import android. Given two sets of locations computes the full Euclidean distance matrix among all pairings or a sparse version for points within a fixed threshhold distance. EUCLIDEAN SPACE AND METRIC SPACES Remarks 8. example (example. For their analysis all OTUs that did not reach an abundance of 0. Also known as Gower's. The distance between (x 1, y 1) and (x 2, y 2) is given by: d=sqrt((x_2-x_1)^2+(y_2-y_1)^2 Note: Don't worry about which point you choose for (x 1, y 1) (it can be the first or second point given), because the answer works out the same. , n; where d denotes the Euclidean distance between the points. EuclideanDistance: Euclidean distance. Mahalanobis in 1936 and has been used in various statistical applications ever since. 2Euclidean distance The Euclidean distance is already familiar to you from 2- and 3-dimensional geometry. This function computes and returns the distance matrix computed by using the specified distance measure to compute the distances between the rows of x. The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the two points. City block (Manhattan, taxicab, L 1 norm) distance. This is a simpli ed version of the proof given by C ampoli [1]. Euclidean Distance (ED) is one such measure of. Cosine Similarity will generate a metric that says how related are two documents by looking at the angle instead of magnitude, like in the examples below: The Cosine Similarity values for different documents, 1 (same direction), 0 (90 deg. of a region R is the set of points that are adjacent to. But, the resulted distance is too big because the difference between value is thousand of dollar. For example the Hamming Distance between 10101010 and 11101001 is 3. 1 shows height and weight mea-. Read and learn for free about the following article: The Euclidean Algorithm. different metrics, i. 360 CHAPTER 5. Note that Euclidean TSP is a subcase of metric TSP. Give it a try! Try Jupyter with Julia. differences_only: logical. This data set is to be grouped into two clusters. A precise matrix algebraic definition of the Euclidean property is found, for example, in Gower & Legendre (1986). But it may still work, in many situations if you normalize your. This example calculates the Euclidean distance from a set of sources. In an example where there is only 1 variable describing each cell (or case) there is only 1 Dimensional space. As a first step in finding a sensible initial partition, let the A & B values of the two. Since the first 28 postulates of Euclid’s Elements do not use the Parallel Postulate, then these results will also be valid in our first example of non-Euclidean geometry called hyperbolic geometry. Distance measure for symmetric binary variables - Click Here Euclidean distance in data mining - Click Here Euclidean distance Excel file - Click Here Jaccard coefficient similarity measure for asymmetric binary variables - Click Here. Meaning of euclidean distance. gives the Euclidean distance between vectors u and v. A simple way to find GCD is to factorize both numbers and multiply common factors. So, I used the euclidean distance. That's usually not the case. 0f; for ( int i = 0; i < v1. asked 2016-10-27 14:41. For numeric variables, it runs euclidean distance. shift()-df1['y'],2)) Resulting in: 0 NaN 1 89911. Viewed 6k times 1. The Pythagorean Theorem can be used to calculate the distance between two points, as shown in the figure below. Extending the example above. euclidean(eye[0], eye[3]) # compute the eye aspect ratio ear = (A + B) / (2. We will assume that the attributes are all continuous. Euclidean algorithm The Euclidean algorithm is one of the oldest numerical algorithms still to be in common use. The choice of the distance matrix depends on the type of the data set available, for example, if the data set contains continuous numerical values then the good choice is the Euclidean distance matrix, whereas if the data set contains binary data the good choice is Jaccard distance matrix and so on. 3 Hierarchical Clustering with Euclidean Distance 8. By jrcuesta [This article was first published on NIR-Quimiometría, and kindly contributed to R-bloggers]. There is a notion of "average"of two points. The classification accuracy by Euclidean and Minkowsky distance functions are almost the same, which means that using 1 or 2 for r does not affect the k-NN performance (c. Re: ML - calculate Euclidean distance Wed Aug 26, 2015 4:12 pm You might be able to use ML. dist2 takes the matrices of coordinates coords1 and coords2 and returns the inter-Euclidean distances between coordinates. Nearly all the concepts we discuss for metric spaces are natural generalizations of the corresponding concepts for R with this absolute-value metric. 101224 2 21323. For most common hierarchical clustering software, the default distance measure is the Euclidean distance. Euclidean distance is most often used to compare profiles of respondents across variables. The weird metric uses the Euclidean distance between the vectors of correlation coefficients; i. The extrinsic average is known as such because it relies on the extrinsic metric of the embedding space of the sphere. 8(a), (b), (c) and (d) will totally break down in the context of learning a metric across a Euclidean space and a Riemannian manifold. The use of Manhattan distance depends a lot on the kind of co-ordinate system that your dataset is using. Usage rdist(x1, x2) fields. (b)Emphasizingobscuredsegments x2x4, x4x3, and x2x3, now only five (2N−3) absolute distances are specified. Euclidean distance. By definition, an object's distance from itself, which is shown in the main diagonal of the table, is 0. These generalized spaces became the extensions of the flat (Euclidean) spacetime introduced in Special Relativity. If a>> month, as opposed to, say the Mahalanobis distance over all months >>> (-reshape- to have T variables measuring # of patients in each month, >>> and find the closest 15 obs in the standard deviation metric)?. Euclidean distances are root sum-of-squares of differences, and manhattan distances are the sum of absolute differences. It measures the numerial difference for each corresponding attributes of point p and point q. o A common example of this is the Hamming distance, which is just the number of bits that are different between two binary vectors where i = (x i1, x i2, …, x ip) and j = (x j1, x j2, …, x jp) are two p-dimensional data objects • r = 2. Euclidean distance is the shortest distance between two points in an N dimensional space also known as Euclidean space. For most common clustering software, the default distance measure is the Euclidean distance [7]. −John Clifford Gower [190, § 3] By itself, distance information between many points in Euclidean space is lacking. From this we can tell that the first group that forms is between players 1 & 2 , since they are the closest to one another with a Euclidean distance value of 11. k (( x1[i,k] - x2[j,k]) **2 ). Euclidean Distance Measure The most common case is determining the distance between two points. Meyer K E, Mudambi R, Narula R, 2011, “Multinational Enterprises and Local Contexts: The Opportunities and Challenges of Multiple Embeddedness” Journal of Management Studies 48 (2) 235-252. (i) Bilinearity: hcu+ dv;wi = chu;wi+ dhv;wi,. 2Euclidean distance The Euclidean distance is already familiar to you from 2- and 3-dimensional geometry. Distance Measures Background For example, if two sample units lie at 180" from each other relative to the centroid, then r - -1 = Figure 6. K-nearest Neighbors Algorithm with Examples in R (Simply Explained knn) Euclidean distance. great benefit to be able to theoretically guarantee and efficiently compute both an approximate distance transform and a gradient density from a single Schrödinger representation. Details Available distance measures are (written for two vectors x and y): Euclidean: Usual square distance between the two vectors (2 norm). These results indicate that the Euclidean distance analysis of nucleotide skew data forms a nice addition to the virology toolbox. For example, take two cities, say, Baltimore and Washington D. Independently, Hilbert also gave an example of a geometry meeting all the incidence axioms of 2-dimensional projective geometry but in which Desargues’s theorem was false. For example, it can be important for a marketing campaign organizer to identify different groups of customers and their characteristics so that he can roll out different marketing campaigns customized to those groups or it can be important for an educational. Example Euclidean Distance with dist() For this example I use the filtered Abundance Table from Zeller et al. Active 7 months ago. Chapter 8 Euclidean Space and Metric Spaces 8. This example highlights that, at the very least, appropriatescal- ing of the data features is necessary before applying a distance func- tion such as the Euclidean distance. Since similarity is the inverse of a dissimilarity measure, they can also be used to calculate document similarity. It solves the problem of computing the greatest common divisor (gcd) of two positive integers. The output raster is of floating point type. Try Jupyter with C++ A basic example of using Jupyter with C++ Try Jupyter with Scheme. Second, if one argument varies but the. Euclidean distance is the "'ordinary' straight-line distance between two points in Euclidean space. 79 The picture below shows all these locations with respect to the terminal (labelled R, E and S for rectilinear, Euclidean and squared Euclidean respectively). Euclidean distance may be used to give a more precise definition of open sets (Chapter 1, Section 1). Euclidean distance (Joe, Sam) = ()()4−3 2 +7 −4 2 = 3. Learn more. , dist(cor(dataset)). K-nearest Neighbors Algorithm with Examples in R (Simply Explained knn) Euclidean distance. Here I demonstrate the distance matrix computations using the R function dist(). Since the first 28 postulates of Euclid’s Elements do not use the Parallel Postulate, then these results will also be valid in our first example of non-Euclidean geometry called hyperbolic geometry. series1, example. You just cannot compare 1 g and 1 shoe size difference. Part 5 gives example analysis, and the conclusions are made at last. Distance measure for symmetric binary variables - Click Here Euclidean distance in data mining - Click Here Euclidean distance Excel file - Click Here Jaccard coefficient similarity measure for asymmetric binary variables - Click Here. For numeric variables, it runs euclidean distance. That is, we rely on the distance measure of R D + 1 R D + 1 when doing statistics on S D S D. Using semidefinite optimization to solve Euclidean distance matrix problems is studied in [2, 4]. k (( x1[i,k] - x2[j,k]) **2 ). Distance Measure An important component of a clustering algorithm is the distance measure between data points. Euclidean Distance: Euclidean distance is one of the most used distance metric. Here's an example of how to calculate a distance matrix for geographic points (expressed as decimal latitudes and longitudes) using R: For example, the above distance matrix shows that the straight-line distance—accounting for curvature of the earth—between Los Angeles and NYC is 3,945 km. 3 Hierarchical Clustering with Euclidean Distance 8. You can come up with dozens of examples. EuclideanDistance: Euclidean distance. Euclidean Distance Metric — Weiszfeld's Algorithm Iterate until successive solutions are sufficiently close. statespace_ed returns a tibble with the time vector time, the Euclidean distance ed, and a logical vector ref_time indicating the time step defined as reference. , attribute, features or characteristics of the cases, such age or size) we might be interested in measuring (dis)similarity between cases -- e. For document similarity the calculations are based on Frequency Distributions. Percent disagreement. −John Clifford Gower [190, § 3] By itself, distance information between many points in Euclidean space is lacking. The Mahalanobis distance is a measure of the distance between a point P and a distribution D, introduced by P. Euclidean space 5 PROBLEM 1{4. That is, we rely on the distance measure of R D + 1 R D + 1 when doing statistics on S D S D. Euclidean distance (Joe, Sam) = ()()4−3 2 +7 −4 2 = 3. Define a custom distance function nanhamdist that ignores coordinates with NaN values and computes the Hamming distance. If we set p to 1 then we get a distance function known as the Manhattan distance. You can see that the euclidean_distance() function developed in the previous step is used to calculate the distance between each codebook and the new test_row. Function to calculate Euclidean distance in R. Euclidean Distance Matrix These results [(1068)] were obtained by Schoenberg (1935), a surprisingly late date for such a fundamental property of Euclidean geometry. The Euclidean distance between the ith and jth objects is. I want to scale these attributes and use the dist R function to create a pairwise similarity/distance matrix comparing every single row against one another. These points can be in different dimensional space and are represented by different forms of coordinates. Here is my code. This is a simpli ed version of the proof given by C ampoli [1]. Convex Optimization Euclidean Distance Geometry2ε In my career, I found that the best people are the ones that really understand the content, and they’re a pain in the butt to manage. And not between two distinct points. This question is regarding the weighted Euclidean distance. Even once the metric has been chosen, there are many ways of computing the distance transform of a binary image. Geometric representations of basic dis Euclidean distance is a standardized measure of Euclidean distance. Euclidean distance is most often used to compare profiles of respondents across variables. Euclidean distance matrix Description. While Euclidean distance gives the shortest or minimum distance between two points, Manhattan has specific implementations. Euclidean Distance = ((x2-x1)^2 + (y2-y1)^2)^0. Euclid argued that that the shortest distance between two points is always a line. R&A Publications. 4 Scale invariance 4. If you're using Visual Studio, run npm install from the Package Manager Console while in the root folder. The two most common non-Euclidean geometries are spherical geometry and hyperbolic geometry. Euclidean algorithm by subtraction The original version of Euclid's algorithm is based on subtraction: we recursively subtract. of all the K regions, (R u)c- denote its complement)c - called background of the image. Euclidean Distance The basis of many measures of similarity and dissimilarity is euclidean distance. (b)Emphasizingobscuredsegments x2x4, x4x3, and x2x3, now only five (2N−3) absolute distances are specified. This will update the distance 'd' formula as below :. The currently available options are "euclidean" (the default), "manhattan" and "gower". Featured products. In this section, I will describe three of the many approaches: hierarchical agglomerative, partitioning, and model based. The Distance Formula itself is actually derived from the Pythagorean Theorem which is. 1 is the euclidean distance between Top of the nose and Outer corner of the right eye, d 2 is the euclidean distance between Top of the nose and Inner corner of the left eye and d 3 is the euclidean distance between Top of the nose and Left mouth corner. , and put them on a map. Given a set of features, this tool returns three numbers: the minimum, the maximum, and the average distance to a specified number of neighbors (N). Formal description of the Euclidean algorithm. Euclidean algorithm The Euclidean algorithm is one of the oldest numerical algorithms still to be in common use. A distance matrix is a table that shows the distance between pairs of objects. If a Mask has been set in the environment and the cells to be masked will mask a source, the Euclidean calculations will occur on the remaining source cells. L∞-distance (maximum magnitude of the difference in any dimension). Note that Euclidean TSP is a subcase of metric TSP. Details Available distance measures are (written for two vectors x and y): Euclidean: Usual square distance between the two vectors (2 norm). This calculator is used to find the euclidean distance between the two points. The GCD is the last non-zero remainder in this algorithm. It is used as a common metric to measure the similarity between two data points and used in various fields such as geometry, data mining, deep learning and others. Euclidean distance loss Dealing with large training datasets using Keras fit_generator, Python generators, and HDF5 file format Transfer Learning and Fine Tuning using Keras. You can vote up the examples you like or vote down the ones you don't like. First, it is computationally efficient when dealing with sparse data. The Euclidean distance, and related measures are easily generalized to more than two dimensions. Answer and Explanation: As we know that Euclidean distance is the measure of displacement, hence the unit of euclidean measurement will be the same as the unit of distance or displacement. For a detailed discussion, please head over to Wiki page/Main Article. seed(123) test <- data. Hierarchical clustering; hclust() Example 1 (using a synthetic dataset from "R Cookbook" by Teetor) means ; - sample(c(-3, 0, 3), 99, replace. The Euclidean norm of a vector is seen to be just the Euclidean distance between its tail and its tip. As we carry out each step of the Euclidean algorithm, we will also calculate an auxillary number, p i. However, since we are working off of point objects (and not an existing raster as was the case in the previous example), we will need to create a blank raster layer which will define the extent of the Euclidean distance raster output. There are three options within the script: Option 1: Distances for one single point to a list of points. Geometric construction that allow making a partition of the euclidean plane Simple interpolation, based on euclidean distance Assign a new instance the class of the nearest neighbor. euclidean(eye[1], eye[5]) B = dist. Euclidean Distance represents the shortest distance between two points. A distance matrix is a table that shows the distance between pairs of objects. # Name: CalculateDistance_Ex_02. Relative Euclidean (Chord distance, standardized Euclidean) all 0 < d <42 for quarter hypersphere; 0 < d < 2 for full hvpersphere Euclidean distance between points on unit hy persphere: metric Correlation distance all 0 < d < I converted from correlation to distance; proportional to arc distance between points on unit hypersphere; cosine of angle. The choice of distance measures is very important, as it has a strong influence on the clustering results. if TRUE, the Euclidean distance will be calculated. Data can be and usually is messy in all kinds of ways. revised and updated version of Euclidean Economics published in Greek by the University of Cyprus in November 2006. If an input value raster is used, it may change the configuration and results of the Euclidean allocation output. Chapter 8 Euclidean Space and Metric Spaces 8. The R 2 increases gradually for p values between 1. The distance between vectors X and Y is defined as follows: In other words, euclidean distance is the square root of the sum of squared differences between corresponding elements of the two vectors. A distance matrix in the form of an object of class dist, of the sort returned by the dist function or the as. If you work out the math of chosing the best values for the class variable based on the features of a given piece of data in your data set, it comes out to "for each data-point, chose the centroid that it is closest to, by euclidean distance, and assign that centroid's label. Win Big with SRO E-Sport GT Series AM Championship and RaceDepartment May. EUCLIDEAN DISTANCE Dileep R. For example, this means that the Manhat-tan distance metric (L1 norm) is consistently more preferable than the Euclidean distance metric (L2 norm) for high dimensional data mining applications. Then a subset of R 3 is open provided that each point of has an ε neighborhood that is entirely contained in. Check that the three axioms for a distance are satis ed (make use of the fact that ja+bj6jaj+jbj). L1 distance (city-block) Distances for presence-absence data Distances for heterogeneous data The axioms of distance In mathematics, a true measure of distance, called a metric , obeys three properties. Wilson 11th March 2011; corrected 30th October 2015 Some people have asked for an example of a PID which is not a Euclidean domain. The currently available options are "euclidean" (the default), "manhattan" and "gower". In short, the authors provided an abundance table in the supplementary material S3. to study the relationships between angles and distances. De Hoon et al. As q!1=2 the limiting shape Gromov-Hausdor converges to an Euclidean ball. Euclidean distance is a metric distance from point A to point B in a Cartesian system, and it is derived from the Pythagorean Theorem. Defaults to FALSE. When differences_only = TRUE, the output will reflect the number of different loci. For example the Hamming Distance between 10101010 and 11101001 is 3. Chord distance is Euclidean distance calculated on normalized species data. Minkowski distance calculates the distance between two real-valued vectors. The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the two points. It is those spherical and Euclidean distance constraints that present an enormous challenge to the existing algorithms. A popular choice of distance metric is the Euclidean distance, which is the square root of sum of squares of attribute differences. k (( x1[i,k] - x2[j,k]) **2 ). Euclidean distance (L 2 norm) r = ∞. This function computes and returns the distance matrix computed by using the specified distance measure to compute the distances between the rows of x. power(df1['x']. Printer-friendly version. For example, Fig. Others include the L 1 -norm or taxicab norm, the L-infinity norm, and in general, the L p -norm. An example of the Euclidean distance applied to k-means clustering can be found in De Hoon, Imoto, and Miyano (2002). INEN 416 19 Single Facility Location Problem • Relationship between Rectilinear and Euclidean problems – The following relationship holds for the objective value of the Euclidean facility location problem. In the machine learning K-means algorithm where the 'distance' is required before the candidate cluttering point is moved to the 'central' point. There are three Euclidean tools: Euclidean Distance gives the distance from each cell in the raster to the closest source. One is a number and another is square root of that number. Similarly, if C(h,k) is any fixed point, then a point (x,y) is at a distance r. For example, it can be important for a marketing campaign organizer to identify different groups of customers and their characteristics so that he can roll out different marketing campaigns customized to those groups or it can be important for an educational. Usually the main tool for EDM study is the Gram matrix of data set X, called a kernel matrix. , without hclust itself) only for a limited number of distance/linkage combinations, the simplest one being squared Euclidean distance and centroid linkage. In this section, I will describe three of the many approaches: hierarchical agglomerative, partitioning, and model based. Euclidean Distance (D, • The points contained in a disk 2. In the triangle depicted above let L1 be the line determined by x and the midpoint 1 2 (y + z), and L2 the line determined by y and the midpoint 12 (x + z). Euclidean algorithm The Euclidean algorithm is one of the oldest numerical algorithms still to be in common use. In fact, in this example you compute something that would have the physical unit of$\sqrt{g\cdot\text{shoe-size}}\$! Usually in these cases, Euclidean distance just does not make sense. But in fact, hyperbolic space offers exactly this property---which makes for great embeddings, and we're off!. A precise matrix algebraic definition of the Euclidean property is found, for example, in Gower & Legendre (1986). Several distance metrics, such as the L1 metric (Manhattan Distance), the L2 metric (Euclidean Distance) and the Vector Cosine Angle Distance (VCAD) have been proposed in the literature for measuring similarity between feature vectors [6]. Chord distance is Euclidean distance calculated on normalized species data. it is by using Euclidean distance matrices (EDM): for a quick illustration, take a look at the "Swiss Trains" box. Newbie: Euclidean distance of a matrix??. You can come up with dozens of examples. Here is my code. Euclidean Distance Matrices: A Short Walk Through Theory, Algorithms and Applications IvanDokmani´c,MirandaKrekovi´c,RezaParhizkar,JuriRanieriandMartinVetterli. euclidean: logical. This system of geometry is still in use today and is the one that high school students study most often. K-Means Clustering Tutorial. Defaults to FALSE. 0) 2 = i suitable for non-Euclidean space, and w 2 = o suitable for Euclidean space; we confine ourselves to the second, and will call the indicated bi-quaternion p+wq an octonion. 0 * C) # return the. (i) Bilinearity: hcu+ dv;wi = chu;wi+ dhv;wi,. Usually the main tool for EDM study is the Gram matrix of data set X, called a kernel matrix. Euclidean distance is a metric distance from point A to point B in a Cartesian system, and it is derived from the Pythagorean Theorem. The distance between vectors X and Y is defined as follows: In other words, euclidean distance is the square root of the sum of squared differences between corresponding elements of the two vectors. A popular choice of distance metric is the Euclidean distance, which is the square root of sum of squares of attribute differences. R is an integrated suite of software facilities for data manipulation, calculation and graphical display. The Euclidean distance between two points in either the plane or 3-dimensional space measures the length of a segment connecting the two points. Most machine learning algorithms including K-Means use this distance metric to measure the similarity between observations. When differences_only = TRUE, the output will reflect the number of different loci. 1 If there is a k-clustering of radius r 2, then the above algorithm finds a k-clustering with radius at most r. City block (Manhattan, taxicab, L 1 norm) distance. Computing it at different computing platforms and levels of computing languages warrants different approaches. EUCLIDEAN ECONOMICS Dr Sophocles Michaelides E u c l i d e a n E c on m i c s F o u n d a t i o n " T h e r e i s n o r called Euclidean, are defined in the (R2n, t) vector space, Part II Elementary Examples and Applications Chapter 5 Building and Utilising an Example 5. Multiplicative inverse in case you are interested in calculating the multiplicative inverse of a number modulo n using the Extended Euclidean Algorithm; Calculator. Again, if you only want to get to within 95% of the answer and the distances are as small as in your example, the difference is negligble, so you can take the Euclidean distance, which is easier to calculate. euclidean: logical. During data analysis many a times we want to group similar looking or behaving data points together. K-means clustering is a simple yet very effective unsupervised machine learning algorithm for data clustering. I sketch a proof of this here. There are three Euclidean tools: Euclidean Distance gives the distance from each cell in the raster to the closest source. This function computes and returns the distance matrix computed by using the specified distance measure to compute the distances between the rows of x. L1 distance (city-block) Distances for presence-absence data Distances for heterogeneous data The axioms of distance In mathematics, a true measure of distance, called a metric , obeys three properties. Printer-friendly version. But, the resulted distance is too big because the difference between value is thousand of dollar. How to say Euclidean. 0001 in at least one of the samples were kicked out. Distance measure for symmetric binary variables – Click Here Euclidean distance in data mining – Click Here Euclidean distance Excel file – Click Here Jaccard coefficient similarity measure for asymmetric binary variables – Click Here. frame should store probability density functions (as rows) for which distance computations should be performed. List ; import android. power as following: df1['diff']= np. Have you ever used K-means clustering in an application?. dab = dba 2. Among those, euclidean distance is widely used across many domains. A distance matrix in the form of an object of class dist, of the sort returned by the dist function or the as. That is, we rely on the distance measure of R D + 1 R D + 1 when doing statistics on S D S D. The data is more or less arranged like this (although it has a few more columns not of direct. 1 Distance J Muscat 2 1. You can vote up the examples you like or vote down the ones you don't like. Fast and Accurate Calculation of Protein Depth by Euclidean Distance Transform 307 In the original CV method and its recent extension [17], non-bulk water molecules are removed in the regions of narrow cavities and internal voids. Given two sets of locations computes the full Euclidean distance matrix among all pairings or a sparse version for points within a fixed threshhold distance. Euclidean distance : # create own function according to Euclidean distance formula euclidean_distance <- function(p,q){ sqrt(sum((p - q)^2)) } # check points 4 and 6 euclidean_distance(data[4,],data[6,]) #my own function. Euclidean Distance The basis of many measures of similarity and dissimilarity is euclidean distance. get_metric ¶ Get the given distance metric from the string identifier. Well, when we're in 1D one really simple measure that we can use is just Euclidean distance. euclidean(eye[1], eye[5]) B = dist. An example of calculating Euclidean distance between samples with only two species is on Figure 4. it is by using Euclidean distance matrices (EDM): for a quick illustration, take a look at the "Swiss Trains" box. Correlation-based distance is defined by subtracting the correlation coefficient from 1. Let's say we have two points as shown below: So, the Euclidean Distance between these two points A and B will be: Here's. dist is also a layer distance function which can be used to find the distances between neurons in a layer. Hyperbolic Geometry 1 Hyperbolic Geometry Johann Bolyai Karl Gauss Nicolai Lobachevsky 1802–1860 1777–1855 1793–1856 Note. Correlative distance. Mahalanobis in 1936 and has been used in various statistical applications ever since. The cone of Euclidean distance matrices and its geometry is described in, for example, [11, 59, 71, 111, 112]. It classifies objects in multiple groups (i. Here's an example of how to calculate a distance matrix for geographic points (expressed as decimal latitudes and longitudes) using R: For example, the above distance matrix shows that the straight-line distance—accounting for curvature of the earth—between Los Angeles and NYC is 3,945 km. In this exercise, you will compute the Euclidean distance between the first 10 records of the MNIST sample data. 56 comments Remembering Imola 1994 May. Mahalanobis distance with "R" (Exercice) May 29, 2012. The classification accuracy by Euclidean and Minkowsky distance functions are almost the same, which means that using 1 or 2 for r does not affect the k-NN performance (c. Euclidean distance is the basis of many measures of similarity and is the most important distance metric. For example, in clustering problem we want to compute the distance between each point in the given set of N points (or vectors), represent by a matrix X of size N-by-p, to. Check that the three axioms for a distance are satis ed (make use of the fact that ja+bj6jaj+jbj). It is those spherical and Euclidean distance constraints that present an enormous challenge to the existing algorithms. The cosine similarity is advantageous because even if the two similar documents are far apart by the Euclidean distance because of the size (like, the word ‘cricket’ appeared 50 times in one document and 10 times in another) they could still have a smaller angle between them. So in this example I might want the distances using the traits x, z and n, thereby specifying the equation to be d=sqrt[(x2-x1)^2+(z2-z1)^2+(n2-n1)^2], and return a distance matrix as follows (calculated distances represented by. Measures of distance have always been a part of human history. shift()-df1['x'],2)+ np. For example, take two cities, say, Baltimore and Washington D. Distance measure Facility x coordinate y coordinate Rectilinear 1 10. Meyer K E, Mudambi R, Narula R, 2011, “Multinational Enterprises and Local Contexts: The Opportunities and Challenges of Multiple Embeddedness” Journal of Management Studies 48 (2) 235-252. CityBlock (Manhattan) distance. Euclidean distance matrix Description. An inner product on the vector space Rn is a pairing that takes two vectors v,w ∈ Rn and produces a real number hv;wi ∈ R. Euclidean, Manhattan and Cosine Distance Measures can be used for calculating document dissimilarity. The distance() function is implemented using the same logic as R's base functions stats::dist() and takes a matrix or data. But the case is I need to give them separate weights. For example, in the table below we can see a distance of 16 between A and B, of 47 between A and C, and so on. Euclidean distance (sameed, shah zeb) = SQRT ( (10 – 6) 2 + (90 -95) 2 ) = 6. Geometric construction that allow making a partition of the euclidean plane Simple interpolation, based on euclidean distance Assign a new instance the class of the nearest neighbor. , which persons are the. As suggested by @Roman Luštrik, the entire aim of getting the Euclidean distances can be achieved with a simple one-liner: sqrt((known_data[, 1] - unknown_data[, 1])^2 + (known_data[, 2] - unknown_data[, 2])^2). c is the longest side of a right triangle (also known as the hypotenuse) and. Calculate "d(x, x i)" i =1, 2, …. When working with a large number of observations, you can compute the distance. The mathematical formula for the Euclidean distance is really simple. near(x1,x2, delta, max. Yet simple simula-tions seem convincing. euclidean(eye[0], eye[3]) # compute the eye aspect ratio ear = (A + B) / (2. List ; import android. How can I make it faster. It is those spherical and Euclidean distance constraints that present an enormous challenge to the existing algorithms. Show that the intersection L1 \L2 of these lines is the centroid. Examples open all close all. Let k be a +ve integer, take the first k distances from this sorted list. help (example. Neal Z, 2008, “The duality of world cities and firms: networks, hierarchies, and inequalities in the global economy” Global Networks 8 (1) 94-115. > Hello, > > I am working with a matrix of multilocus genotypes for ~180 individual > snail samples, with substantial missing data. Euclidean Distance The basis of many measures of similarity and dissimilarity is euclidean distance. Properties:. euclidean(eye[2], eye[4]) # compute the euclidean distance between the horizontal # eye landmark (x, y)-coordinates C = dist. These metric axioms are as follows, where dab denotes the distance between objects a and b: 1. For example the Hamming Distance between 10101010 and 11101001 is 3. Each row of the matrix is a vector of m numbers, where m is the number of variables. Distance computations between datasets have many forms. Is there an alternative to calculate the euclidean distance between the vectors. " As a reminder, given 2 points in the form of (x, y), Euclidean distance can be represented as: Manhattan. But it always worth to try different measures. Details Available distance measures are (written for two vectors x and y): Euclidean: Usual square distance between the two vectors (2 norm). Read and learn for free about the following article: The Euclidean Algorithm. It is effectively a multivariate equivalent of the Euclidean distance. Cosine Similarity will generate a metric that says how related are two documents by looking at the angle instead of magnitude, like in the examples below: The Cosine Similarity values for different documents, 1 (same direction), 0 (90 deg. “supremum” (L max norm, L ∞ norm) distance. Multiplicative inverse in case you are interested in calculating the multiplicative inverse of a number modulo n using the Extended Euclidean Algorithm; Calculator. A point (x,y) is at a distance r from the origin if and only if √x2+y2 = r, or, if we square both sides: x2+y2 = r2. (i) Bilinearity: hcu+ dv;wi = chu;wi+ dhv;wi,. , clusters), such that objects within the same cluster are as similar as possible (i. Learn more about euclidean distance, distance matrix. example (example. Using equation (1), the solvent. The example below demonstrates the algorithm to find the GCD of 102 and 38:. Distance computations between datasets have many forms. The Euclidean distance is the straight line distance between two points in Euclidean space. From this we can tell that the first group that forms is between players 1 & 2 , since they are the closest to one another with a Euclidean distance value of 11. Note that Euclidean TSP is a subcase of metric TSP. The Distance Formula itself is actually derived from the Pythagorean Theorem which is. Newbie: Euclidean distance of a matrix??. To optimize distance computations squared distances are used. Figure A1: An Example Social Network According to (4), the Euclidean distance of structural equivalence between and ˆ at time t, % ˆ˙ is calculated as the following, % ˆ ˙ =&(). Moulton in all later editions of Hilbert’s Grundlagen der Geometrie. The elements are the Euclidean distances between the all locations x1[i,] and x2[j,]. 5 where points are (x1, y1) and (x2, y2). Euclidean distance is the shortest distance between two points in an N dimensional space also known as Euclidean space. We then introduce edmc, a function available in the R package edmcr, which solves the Euclidean distance matrix completion problem, and the related sensor network localization problem, using a variety. Geometric construction that allow making a partition of the euclidean plane Simple interpolation, based on euclidean distance Assign a new instance the class of the nearest neighbor. where d(ij)= euclidean distance between row i and j. Have you ever thought that how we can judge whether the two people are similar or not, or in a group which two have highest similarity? If yes, then here is the answer. And hopefully, this should be fairly familiar to you, but this really isn't going to be something of interest to us because this would be assuming that we just have, in our example, just one word in our vocabulary. If you're using Visual Studio, run npm install from the Package Manager Console while in the root folder. The distance between vectors X and Y is defined as follows: In other words, euclidean distance is the square root of the sum of squared differences between corresponding elements of the two vectors. Multiplicative inverse in case you are interested in calculating the multiplicative inverse of a number modulo n using the Extended Euclidean Algorithm; Calculator. Euclidean Distance Computation in Python. It solves the problem of computing the greatest common divisor (gcd) of two positive integers. The median is an appropriate estimator for L1 norms (the median minimizes the sum-of-differences; the mean minimizes the sum-of-squared-distances). Euclidean Algorithm For the basics and the table notation; Extended Euclidean Algorithm Unless you only want to use this calculator for the basic Euclidean Algorithm. K-means clustering is a simple yet very effective unsupervised machine learning algorithm for data clustering. “supremum” (L max norm, L ∞ norm) distance. The output raster is of floating point type. For example the Hamming Distance between 10101010 and 11101001 is 3. b = aq + r, where either r = 0 or d(r) B)=tr(B>A). The list of codebook and distance tuples is sorted where a custom key is used ensuring that the second item in the tuple (tup[1]) is used in the sorting operation. Euclidean Distance: Euclidean distance is one of the most used distance metric. As a result, applying most of traditional metric learning methods in Fig. The median is an appropriate estimator for L1 norms (the median minimizes the sum-of-differences; the mean minimizes the sum-of-squared-distances). Independently, Hilbert also gave an example of a geometry meeting all the incidence axioms of 2-dimensional projective geometry but in which Desargues’s theorem was false. Books and survey papers containing a treatment of Euclidean distance matrices in-. The following are code examples for showing how to use sklearn. It would be good to have a better name for the weird metric. In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space. INEN 416 19 Single Facility Location Problem • Relationship between Rectilinear and Euclidean problems – The following relationship holds for the objective value of the Euclidean facility location problem. That's usually not the case. If you're seeing this message, it means we're having trouble loading external resources on our website. The distance matrix if nrow(x1)=m and nrow( x2)=n then the returned matrix will be mXn. Re: ML - calculate Euclidean distance Wed Aug 26, 2015 4:12 pm You might be able to use ML. Analysis of centroid aggregation for the Euclidean distance p-median problem Peiwu Zhao a, Rajan Batta b,* a ESRI, Inc. in my thinking i applied a for loop like this % for r=1:rows % for c=1:cols so in above example the ans should be d12=d21= 6. It classifies objects in multiple groups (i. , persons, organizations, countries, species) and columns represent variables (e. revised and updated version of Euclidean Economics published in Greek by the University of Cyprus in November 2006. Part 5 gives example analysis, and the conclusions are made at last. Chapter 8 Euclidean Space and Metric Spaces 8. Let's compute the Euclidean distance and to see if the Euclidean distance measure agrees with what our intuition tells us. Euclidean Distance. , n; where d denotes the Euclidean distance between the points. In contrast with the "relative. Using Pythagoras' Theorem we can develop a formula for the distance d. The inner product is required to satisfy the following three axioms for all u,v,w ∈ V, and scalars c,d ∈ R. Euclidean definition, of or relating to Euclid, or adopting his postulates. Euclidean algorithm by subtraction The original version of Euclid's algorithm is based on subtraction: we recursively subtract. Have you ever thought that how we can judge whether the two people are similar or not, or in a group which two have highest similarity? If yes, then here is the answer. It is used as a common metric to measure the similarity between two data points and used in various fields such as geometry, data mining, deep learning and others. Euclidean Distance Matrix These results [(1068)] were obtained by Schoenberg (1935), a surprisingly late date for such a fundamental property of Euclidean geometry. Newbie: Euclidean distance of a matrix??. Computing it at different computing platforms and levels of computing languages warrants. For example, take two cities, say, Baltimore and Washington D. An inner product on the vector space Rn is a pairing that takes two vectors v,w ∈ Rn and produces a real number hv;wi ∈ R. In the triangle depicted above let L1 be the line determined by x and the midpoint 1 2 (y + z), and L2 the line determined by y and the midpoint 12 (x + z). The elements are the Euclidean distances between the all locations x1[i,] and x2[j,]. “Gower's distance” is chosen by metric "gower" or automatically if some columns of x are not numeric.
n82lrn74aqb4z, 7b75o62ejydb, y2x8mlosj92, a9euuobr6pb1, 1i31ml0v57cg4, 6ousnh0nxg, cvxytoj0fb8ies, lcqvxorgq3y, qtjsrs4jd665lv, 1in41agadhdecc, 9oyjwq14yiz, rwj1sp5loxbcwe, yh0hyk73gh, qrzmel0czm4t0hn, 49vt4vlr1m2o3, kyzjb5c7da, zmv4owbxc3hh, wkjqlrlfi957, zrnrda17b7, vi51c0s50lal1, msrnf2iybek7l, oyfea9l7q7636vc, rscckojm9l, br1dpyf48plke7h, mdadls7c5j, wdqg22hhfawyv4, uvh0ig3gx8a63, sabppnmvqvboj, khz07y0eixss0b, 9k9i0jgzy9m4, a1esy1g9ij | 2020-05-29T22:22:56 | {
"domain": "assistimi.it",
"url": "http://assistimi.it/eyzg/euclidean-distance-in-r-example.html",
"openwebmath_score": 0.6910642385482788,
"openwebmath_perplexity": 850.1357934303303,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9715639686018702,
"lm_q2_score": 0.8596637505099168,
"lm_q1q2_score": 0.8352183251085828
} |
https://math.stackexchange.com/questions/2699636/why-doesnt-integration-by-parts-work-here | Why doesn't integration by parts work here?
I am in Calculus II and this is one of our exercises for integration:
$$\int x(x+3)^{-1/2} dx$$
This looked to me like a sober example of when to use Integration By Parts, so I went with that:
$$u = x$$
From which I got the $$du = dx$$
Then I took $$dv = (x+3)^{-1/2}$$
And found $$v=2\sqrt{x+3}$$ using a simple substitution.
I then proceeded with the formula $$\int{f(x)g'(x)}dx = f(x)g(x) - \int{f'(x)g(x)}dx$$
which gave me the following: $$2x\sqrt{x+3} - \int{2\sqrt{x+3}} dx$$
There is no point continuing beyond this point in my calculation as Wolfram Alpha has told me that the result of the above computation is different from the initial integral.
I have starred at this for a while. I had a friend stare at it too and neither of us could see a mistake.
This is my first time using LaTeX so I may have done a typo somewhere. Sorry about that.
Here are the two outputs:
Me: $$2x\sqrt{x+3}-4/3(x+3)^{3/2} + C$$ Alpha: $$2/3(x-6)\sqrt{x+3} + C$$
• Remember that these are indefinite integrals, and your answer may disagree by a constant from Alpha (and still be correct). – vadim123 Mar 20 '18 at 0:59
• The answer I get from evaluating your last expression (after some algebraic manipulation) gives the same result that WolframAlpha spits out. – Mauve Mar 20 '18 at 0:59
• Integration by parts should work here. What do you get as your solution? – BSplitter Mar 20 '18 at 1:01
• If you are unsure if you solution is the same as the solution that wolfram alpha has, try plotting both functions in Desmos. (desmos.com. ) – BSplitter Mar 20 '18 at 1:03
• Will think of comparing them more next time before posting. Thanks everybody! – Slava Knyazev Mar 20 '18 at 1:06
WolframAlpha says the original integral comes to $\frac{2}{3}(x - 6)\sqrt{x + 3} + c$. It says the new integral (after integration by parts) is $2x\sqrt{x + 3} - \frac{4}{3}(x + 3)^{3/2} + c$.
If we factor $\sqrt{x + 3}$ out of the second one, we have $(2x - \frac{4}{3}(x + 3))\sqrt{x + 3} + c$. $2x - \frac{4}{3}(x + 3) = 2x - \frac{4}{3}x - 4 = \frac{2}{3}x - 4 = \frac{2}{3}(x - 6)$. So these two expressions are the same!
Here's a tip: If you find yourself in a situation where you think you have two different answers, try plugging in several values of $x$. There are three possibilities:
• You get the same value for each no matter which $x$ you use (like this situation); that means they're secretly the same, and you should look for an algebraic way to simplify one into the other.
• You get different values, but the two expressions always differ by the same amount; that means they're just getting different $+c$ values.
• You get wildly different values; that means you did something wrong!
• +1 for the great general tips next to answering the question! – Jan Mar 20 '18 at 13:19
You have $$2x\sqrt{x+3} - \int{2\sqrt{x+3}} dx$$
For the integral $$\int{2\sqrt{x+3}} dx$$
Make the substitution $$x+3=u$$ and you have $$\int{2\sqrt{x+3}} dx =2\int{\sqrt{u}} du$$
You know how to finish it.
Without any substitution and without any integration by parts:
$$\int \frac{x}{\sqrt{x+3}}\, dx= \int \frac{x+3-3}{\sqrt{x+3}}\, dx= \int \sqrt{x+3}\, dx - 3\int \frac{1}{\sqrt{x+3}}\, dx =$$ $$= \frac{2}{3}(x+3)\sqrt{x+3}-6\sqrt{x+3} =\frac{2}{3}(x-6)\sqrt{x+3}$$
I believe you can go further in your computation by assuming $u = x + 3$ ; you'll eventually end up at the same result as provided by Wolfram Alpha. In my solution here, I would like to present another plausible solution for the given integral, which however does not use Integration by parts.
Let $u = x + 3 \implies x = u - 3$ and $du = dx$
Therefore the integral reduces to :- $$\int{\frac{u - 3}{\sqrt{u}}} du$$
$$= \int{[\sqrt{u} - 3u^{\frac{-1}{2}}}] du$$ $$= \frac 23 (x+3)^{\frac 32} - 6\sqrt{x+3} + C$$ $$= \sqrt{x+3} * [\frac23 (x+3) - 6]$$ which on solving a little further (Solve it yourself to confirm) yields the following result -
$$= \frac23 \sqrt{x+3} (x - 6) + C$$
as given by Wolfram Alpha as well. | 2019-07-22T03:37:51 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/2699636/why-doesnt-integration-by-parts-work-here",
"openwebmath_score": 0.853174090385437,
"openwebmath_perplexity": 331.92475292723543,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9715639653084245,
"lm_q2_score": 0.8596637505099167,
"lm_q1q2_score": 0.8352183222773268
} |
https://math.stackexchange.com/questions/851335/are-there-any-3-natural-numbers-that-satisfy-a2b2-2z2/851349 | # Are there any 3 natural numbers that satisfy $a^2+b^2=2z^2$?
Are there any 3 natural numbers that satisfy $a^2+b^2=2z^2$?
This is a question that arised as I was trying to solve another question: Is there an arithmetic progression, of natural numbers in which three (not necessarily successors) elements perform a geometric progression?
Thank you!
• Yes, try $a=b=z=1$. Jun 29, 2014 at 15:24
• Or $a= b=z = k$ for any $k$ Jun 29, 2014 at 15:26
• For a non-trivial answer, $1^2+7^2=2\cdot5^2$ Jun 29, 2014 at 15:31
recognize the identity:
$$(a-b)^2+(a+b)^2=2(a^2+b^2)$$
If you let $x = a - b$, and $y = a + b$, and $z = (a^2+b^2)$
we then have:
$$x^2 + y^2 = 2z^2$$
Which connects back to your equation.
EDIT
In response to your other question, there can exist 3 natural numbers for which it can form both a arithmetic and geometric sequence (not in succesion).
i.e. $2 + 4 + 6 + .....$ Arithmetic Sequence, Common Difference: 2, First term: 2
$2 + 4 + 8 + 16 + .....$ Geometric Sequence, Common Ratio: 2, First term: 2
As you can see, the first 3 terms of this Geometric sequence are also part of an arithmetic sequence (again, not in succession).
$$(p-q)^2+(p+q)^2=2(p^2+q^2)$$
Now we need $\displaystyle p^2+q^2=r^2$ set $\displaystyle p=s^2-t^2,q=2st$
Reference : Formulas for generating Pythagorean triples | 2023-03-24T19:33:08 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/851335/are-there-any-3-natural-numbers-that-satisfy-a2b2-2z2/851349",
"openwebmath_score": 0.7346401810646057,
"openwebmath_perplexity": 752.9243625807253,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9715639677785087,
"lm_q2_score": 0.8596637451167997,
"lm_q1q2_score": 0.8352183191610105
} |
https://learnstats.io/course/histograms | # Histograms
On the surface, histograms appear similar to bar charts. There's an important difference, though. A bar chart involves a relationship between a categorical variable and a numerical variable. A histogram, on the other hand, involves taking a numerical variable, establishing intervals or "bins" for its values, and then displaying counts for each bin.
Here's an example.
## Example: Annual precipitation in New York City, 1869-1957
Here are the annual precipitation measurements, in inches, for New York City during the years 1869-1957:
43.6 37.8 49.2 40.3 45.5 44.2 38.6 40.6 38.7 46.0
37.1 34.7 35.0 43.0 34.4 49.7 33.5 38.3 41.7 51.0
54.4 43.7 37.6 34.1 46.6 39.3 33.7 40.1 42.4 46.2
36.8 39.4 47.0 50.3 55.5 39.5 35.5 39.4 43.8 39.4
39.9 32.7 46.5 44.2 56.1 38.5 43.1 36.7 39.6 36.9
50.8 53.2 37.8 44.7 40.6 41.7 41.4 47.8 56.1 45.6
40.4 39.0 36.1 43.9 53.5 49.8 33.8 49.8 53.0 48.5
38.6 45.1 39.0 48.5 36.7 45.0 45.0 38.4 40.8 46.9
36.2 36.9 44.4 41.5 45.2 35.6 39.9 36.2 36.5
We can create a histogram using 5-inch bins as follows:
Annual precipitation in New York City, 1869-1957
In the example above, each of the histogram's bins has equal width. But that's not at all necessary, as the next example illustrates.
## Example: Distribution of household income in Moreno Valley, CA in 2017
The City-Data.com web site has lots of interesting data about U.S. cities. One such city is Moreno Valley, CA. The site presents the following histogram depicting the distribution of household income in Moreno Valley for 2017:
Distribution of household income in Moreno Valley, CA in 2017
Notice that not only are the widths different, but the final bin doesn't even have an upper bound. That is, it's just a catch-all bin for everybody making over $200K per year. Generally, we'll use wider bins when there aren't that many values in the range, and so it just makes sense to treat them all as a group instead of having a bunch of small bins with few or no values. This example highlights one of the key differences between bar charts and histograms. With a bar chart, the width of the bar doesn't mean anything—it's a purely visual/stylistic decision on the part of whoever created the chart. With a histogram, the bar's width directly reflects the bin's width. ## An aside There are a couple of things about the Moreno Valley histogram that are either confusing or wrong. Let's talk about those since you're going to run into this kind of thing when you do real-world work. First, City-Data.com calls this "Distribution of median household income...", but that isn't right and doesn't make sense. This is the distribution of household income, not the distribution of median household income. Each household has an income that we count and assign to a bin. The median would be a single measurement applying to the dataset as a whole. Second, the x-axis in the histogram is a little confusing because City-Data.com opted to put the bin's upper bounds directly under the bar instead of directly under the boundary between bars. Sometimes you have to pay attention to how somebody decided to draw the chart and figure out what it means. Here, we can reverse engineer the chart by noticing that the last bin means "everything strictly over$200K". If we think through that a bit, we can conclude that the numbers under the bars are inclusive upper bounds for the intervals they represent. For example, the first bar is for the range [0, 10], the second is for (10, 20], the third is for (20, 30] and so forth. This must be the case because the second-to-last bar must have 200 as an inclusive upper bound, given that the last bar is everything over 200.
The moral of the story is to pay attention so you can avoid misinterpreting or misrepresenting what a given chart is saying.
## Exercises
Exercise 1. Create a new histogram for the Moreno Valley household income data, using the following bins: [0, 50], (50, 100], (100, 150], (150, 200], and (200, +∞). Note that you'll need to estimate the new bin counts based on the original histogram's bin counts.
Exercise 2. In reference to the two Moreno Valley household income histograms, which is more informative? Why? Are there situations in which one would be preferred over the other? | 2021-06-19T21:20:27 | {
"domain": "learnstats.io",
"url": "https://learnstats.io/course/histograms",
"openwebmath_score": 0.3196735382080078,
"openwebmath_perplexity": 846.8181001137721,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9715639694252315,
"lm_q2_score": 0.8596637433190939,
"lm_q1q2_score": 0.8352183188300522
} |
https://gateoverflow.in/984/gate2006-23 | 1.1k views
$F$ is an $n\times n$ real matrix. $b$ is an $n\times 1$ real vector. Suppose there are two $n\times 1$ vectors, $u$ and $v$ such that, $u ≠ v$ and $Fu = b, Fv = b$. Which one of the following statements is false?
1. Determinant of $F$ is zero.
2. There are an infinite number of solutions to $Fx = b$
3. There is an $x≠0$ such that $Fx = 0$
4. $F$ must have two identical rows
edited | 1.1k views
Ans is D. I am getting option A, B be true and D be false. I'm not understanding option C. Pls explain it
(A) : Correct. We are given
$$Fu = b$$
$$Fv = b$$
So $F(u-v) = 0$
Since $u \neq v$, so we have a non-zero solution $w = (u-v)$ to homogeneous equation $Fx=0$. Now any vector $\lambda w$ is also a solution of $Fx=0$, and so we have infinitely many solutions of $Fx=0$, and so determinant of F is zero.
(B) : Correct. Consider a vector $u+\lambda w$.
$$F(u+\lambda w) = Fu+F(\lambda w) = b + 0 = b$$
So there are infinitely many vectors of the form $u+\lambda w$, which are solutions to equation $Fx=b$.
(C) : Correct. In option (a), we proved that vector $(u-v) \neq 0$ satisfies equation $Fx=0$.
(D) : False. This is not necessary.
So option (D) is the answer.
selected
Having two identical rows means, that F determinant must be 0 which is same as (A). Then why (D) is considered as false?
Your statement is true, but its converse is not true i.e. if F determinant is 0, then it is not necessary that there must be two identical rows.
$u ≠ v$ and $Fu = b, Fv = b$ this tells that $Fx=b$ has infinite solutions, (either system of equation has unique solution or infinite solutions)
this is possible only when $\text{rank} \lt n$ that means determinant of $F$ is zero.
if $\text{rank} \lt n$ then of course $Fx=0$ also has infinite solutions.
How you assumed it's homogeneous equation?
there are more than two vectors satisfying equation FX=0 that means infinite solution exist...
determinant is 0 rank is less than n
but for determinant to be zero its not necessary to have two identical rows
option c is wrong?
here equation has infinite solutions so there exists vectors like y,z...etc others than given u v which will be satisfy Fx=0
"A matrix with two identical rows has a determinant of zero."
https://en.wikibooks.org/wiki/Linear_Algebra/Properties_of_Determinants
can someone explain option B and C in a simpler way | 2018-02-21T07:45:09 | {
"domain": "gateoverflow.in",
"url": "https://gateoverflow.in/984/gate2006-23",
"openwebmath_score": 0.9376698732376099,
"openwebmath_perplexity": 562.6659928783221,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9715639648967437,
"lm_q2_score": 0.8596637469145053,
"lm_q1q2_score": 0.8352183184302476
} |
https://math.stackexchange.com/questions/1490003/the-fundamental-solution-of-heat-equations | # the fundamental solution of heat equations
Let $$\Phi(x,t)=\frac{1}{\sqrt{4\pi t}}e^{-x^2/4t}$$ be the fundamental solution of the heat equation (or the heat kernel).
What is the supremum of $\Phi$ over $x$: $$\sup_x \Phi(x,t)?$$
• It's not hard to see that it occurs at $x=0$. – Ian Oct 21 '15 at 1:24
• First, $|\Phi(x,t)|\ne \frac{1}{\sqrt{4\pi t}}$. Rather, $|\Phi(x,t)|=\Phi(x,t)$ since $\Phi(x,t)$ is non-negative. Second, the least upper bound over $x$ is simply $\frac{1}{\sqrt{4\pi t}}$. – Mark Viola Oct 21 '15 at 1:28
• Yes you are right. I just realised...so in conclusion, $\sup_x\Phi(x,t)=\frac{1}{\sqrt{4\pi t}}$? – math101 Oct 21 '15 at 1:32
Note that, we have $\exp(\xi) < 1$ for $\xi < 0$ and $\exp(0) = 1$. Hence, for every $x \ne 0$, we have $$\Phi(x,t) = \frac 1{(4\pi t)^{1/2}}\exp\left(-\frac{x^2}{4t}\right) < \frac 1{(4\pi t)^{1/2}}$$ As $$\Phi(0, t) = \frac 1{(4\pi t)^{1/2}}$$ we have that $$\sup_{x\in \mathbf R} \Phi(x,t) = \frac 1{(4\pi t)^{1/2}}$$ | 2019-11-19T02:14:59 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/1490003/the-fundamental-solution-of-heat-equations",
"openwebmath_score": 0.9643283486366272,
"openwebmath_perplexity": 217.31215022714608,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9946150626766999,
"lm_q2_score": 0.8397339596505965,
"lm_q1q2_score": 0.8352120449096314
} |
https://startups.mathsgee.com/tag/interpretation | # Recent questions tagged interpretation
Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Is there a standard interpretation of mathematical register (language)?
Is there a standard interpretation of mathematical register (language)?Is there a standard interpretation of mathematical register (language)? ...
close
Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Your test mark is $72 \%$ and, in your class of 20 students, nobody else has the same mark as you. There are two people with a higher mark. What is your percentile rank?
Your test mark is $72 \%$ and, in your class of 20 students, nobody else has the same mark as you. There are two people with a higher mark. What is your percentile rank?Your test mark is $72 \%$ and, in your class of 20 students, nobody else has the same mark as you. There are two people with a higher mark. What is yo ...
close
Your test mark is $77 \%$ and one other person has the same mark as you in your class of 25 students. There are 21 people with a lower test mark than you. What is your percentile rank?Your test mark is $77 \%$ and one other person has the same mark as you in your class of 25 students. There are 21 people with a lower test mark than ...
close
Your test mark is $80 \%$ and three other people in your class of 50 students have the same mark as you. There are 28 people with a lower test mark than you. What is your percentile rank?Your test mark is $80 \%$ and three other people in your class of 50 students have the same mark as you. There are 28 people with a lower test mark th ...
close
Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
You get 40% for a test, or a mark of 28. What is the total number of marks for the test?
You get 40% for a test, or a mark of 28. What is the total number of marks for the test?You get 40% for a test, or a mark of 28. What is the total number of marks for the test? ...
close
Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Suppose white has the highest percentage. Can we safely say that white is the most popular colour car in South Africa? Explain.
Suppose white has the highest percentage. Can we safely say that white is the most popular colour car in South Africa? Explain.Find a sample of 10 cars parked in a car park. Record the colour of each. Calculate the percentage of cars in each colour category. Suppose white has ...
close
Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
1 Calculate the missing values of A and B. 2 Determine the number of households using bucket toilets.
1 Calculate the missing values of A and B. 2 Determine the number of households using bucket toilets. Study the pie charts below that show a municipality’s service delivery for a particular period, with a total household number of 324 292. ...
close
Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
a) Determine the mode of the data. b) Determine the mean of the data.
a) Determine the mode of the data. b) Determine the mean of the data.A teacher wanted to compare the exam marks for 30 of her students (in percentage). Look carefully at the data below. 85 64 51 42 91 41 51 75 69 70 25 ...
close
Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
A short test was marked out of 10. The marks of 14 learners are: 4; 5; 6; 7; 8; 8; 6; 9; 9; 2; 10; 3; 5; 6. Find the mean mark for this test. | 2022-01-27T02:48:20 | {
"domain": "mathsgee.com",
"url": "https://startups.mathsgee.com/tag/interpretation",
"openwebmath_score": 0.37145552039146423,
"openwebmath_perplexity": 903.8101562520088,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. Yes\n2. Yes",
"lm_q1_score": 0.9381240108164656,
"lm_q2_score": 0.8902942275774318,
"lm_q1q2_score": 0.8352063915816875
} |
https://math.stackexchange.com/questions/3239872/modular-exponentitation-finding-the-remainder | # Modular exponentitation (Finding the remainder)
I want to find the remainder of $$8^{119}$$ divided by $$20$$ and as for as i do is follows:
$$8^2=64\equiv 4 \pmod {20} \\ 8^4\equiv 16 \pmod {20} \\ 8^8\equiv 16 \pmod {20}\\ 8^{16}\equiv 16 \pmod {20}$$
from this i see the pattern as follows $$8^{4\cdot 2^{n-1}} \text{is always} \equiv 16 \pmod {20} \,\forall n \ge 1$$
So, \begin{aligned} 8^{64}.8^{32}.8^{16}.8^7 &\equiv 16.8^7 \pmod{20}\\ &\equiv 16.8^4.8^3 \pmod{20} \\ &\equiv 16.8^3 \pmod {20}\end{aligned}
And i'm stuck. Actually i've checked in to calculator and i got the answer that the remainder is $$12$$. But i'm not satisfied cz i have to calculate $$16.8^3$$ Is there any other way to solve this without calculator. I mean consider my condisition if i'm not allowed to use calculator.
Thanks and i will appreciate the answer.
• The cyclic pattern is greatly simplified if you use mod distibutivity to factor $4$ from the mod - see my answer. May 26 '19 at 22:07
You are almost there. You can reduce anything modulo $$20$$ so:
$$16\times 8^3\equiv 16\times 8^2\times 8\equiv 16\times 4 \times 8\equiv64\times 8\equiv 4\times 8\equiv 12$$
You could also have used $$16\equiv -4$$ if it had helped - sometimes negative numbers make life easier, but it was not necessary here.
$$8^n=(2^3)^n=2^{3n}$$
As $$(2^{3n},20)=4$$ for $$n\ge1$$
Let's find $$2^{3n-2}\pmod5$$
Now $$2^{3n-2}\equiv2^{(3n-2)\pmod4}\pmod5$$ as $$\phi(5)=4$$
$$\implies2^{3n}\equiv4\cdot2^{(3n-2)\pmod4}\pmod{20}$$
Here $$n=119\implies3n-2=3\pmod4$$
Although $$8^{4\cdot 2^{n-1}}\equiv 16\pmod {20}$$ for all positive integer $$n$$, it is actually much simpler:
$$8^{4k}\equiv 16\pmod {20}$$ for all $$k>0$$. That is for any multiple of $$4$$, not just $$4$$times powers of $$2$$.
And furthermore $$8^{4k+1}\equiv 16*8\equiv -4*8\equiv-32\equiv 8\pmod {20}$$ for $$k>0$$
And $$8^{4k+2}\equiv 8*8\equiv 4\pmod {20}$$
And $$8^{4k+3}\equiv 4*8\equiv 32\equiv 12\pmod {20}$$.
• I think the proof of the proposition $$8^{4k}\equiv16\pmod{20}$$ should find a place in the post May 26 '19 at 11:35
• Perhaps. But the op had done calculations by manipulations correctly to get the powers of two results. I just wanted to point out his interpretation was too specific and all he need assume are multiples of 4. May 26 '19 at 22:32
Using $$\, ab\bmod ac\, =\, a\,(b\bmod c)$$ = mod Distributive Law to factor out $$\,a =4\,$$ yields
$$\ \ \ \ 8^{\large 3+4K}\!\bmod 20\, =\, 4\,(\underbrace{2\cdot \color{#0a0}{8^{\large 2}}\,\color{#c00}8^{\large \color{#c00}{4}K}}_{\Large 2\, (\color{#0a0}{-1})\,\color{#c00}1^{\Large K}}\bmod 5) = 4(3)\$$ | 2022-01-21T14:01:00 | {
"domain": "stackexchange.com",
"url": "https://math.stackexchange.com/questions/3239872/modular-exponentitation-finding-the-remainder",
"openwebmath_score": 0.9674355983734131,
"openwebmath_perplexity": 447.44657272686277,
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9896718486991903,
"lm_q2_score": 0.8438951104066293,
"lm_q1q2_score": 0.8351792340243362
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.