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http://mathematica.stackexchange.com/tags/parallel/hot
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Tag Info
4
If the goal is to perform a parallel search for the first result that meets some condition, then we can consider using ParallelTry instead of throw/catch: ParallelTry[If[PrimeQ[#], #, $Failed]&, Range[492114, 500000]] (* 492227 *) This evaluates a function for every value in the second argument (in parallel). The first result that is anything other ... 4 As you are changing the DownValues of m inside your ParallelDo you have to share them among the parallel kernels using SetSharedFunction first: SetSharedFunction[m] Parallelize[Do[m[i] = 2, {i, 1, 10}]] m[1] 2 As you refer to m as a variable, something like m = ConstantArray[0, 10] SetSharedVariable[m] Parallelize[Do[m[[i]] = 2, {i, 1, 10}]] ... 4 It depends on how PBS and the cluster environment are set up. Ideally, if cpuset support has been compiled in to PBS, and if you start Mathematica directly inside the PBS job, you should find that it uses all processors allocated by PBS (on that node--Eigensystem is not MPI-parallelized). If cpuset support isn't provided, then you risk starting as many ... 2 With a = Table[0, {4}, {4}] b = Table[2 i + j, {i, 1, 2}, {j, 1, 2}] using SetSharedVariable[a] ParallelDo[a[[j, i]] += b[[i, j]], {i, 1, 2}, {j, 1, 2}] a {{3, 5, 0, 0}, {4, 6, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}} would work, but using a += Transpose[b] ~PadRight~ Dimensions@a {{3, 5, 0, 0}, {4, 6, 0, 0}, {0, 0, 0, 0}, {0, 0, 0, 0}} is much ... 2 This is not likely a practical approach, but it illustrates whats going on: Clear[x]; ParallelDo[x[i] = i^2, {i, 1, 10}]; as noted the global x has not been defined, however each of our kernels retains the definition: ParallelEvaluate[{$KernelID, DownValues[x]}] // MatrixForm as a bit of a kludge we can pull the values back to global like this: ...
2
As mentioned by Albert Retey in the comment above, you can't expect that NDSolve makes use of your parallel kernels, period. However, since your equation set is just a system of 1st order linear ODEs, you can turn to MatrixExp, which seems to parallelize automatically: coe = gamma - DiagonalMatrix@Total@gamma; init = ConstantArray[0., 816]; init[[-2]] = ...
2
Perhaps this: inhom = Plus @@ Table[a[i] Exp[b[i] x], {i, 1, 100}]; eq[inhom_] := {u''[x] + u[x] + inhom == 0, u[0] == 0, u[1] == 0}; sol = u -> ParallelMap[DSolveValue[eq[#], u[x], x] &, inhom] (* u -> -(1/(1 + b[1]^2)) a[1] (-Cos[x] + E^(x b[1]) Cos[x]^2 + Cot[1] Sin[x] - E^b[1] Cos[1] Cot[1] Sin[x] - E^b[1] Sin[1] Sin[x] + ...
1
Edited to reflect later comments below : These examples will work: ParallelMap[Identity, {<|1 -> 1|>, <|2 -> 2|>}] (* {<|1 -> 1|>, <|2 -> 2|>} *) ParallelMap[Identity, {<|1 -> 1, 3 -> 3|>, <|2 -> 2, 3 -> 3|>}] (* {<|1 -> 1, 3 -> 3|>, <|2 -> 2, 3 -> 3|>} *) Its worth ...
1
Using ParallelDo[Print[i]; Break[], {i, 1, 100}, Method -> "CoarsestGrained"] or ParallelDo[Print[i]; Break[], {i, 1, 100}, Method -> "EvaluationsPerKernel" -> 1] should give you the desired behavior. These options are explained and illustrated in the Options ▶ Method section of Parallelize. Mathematica breaks the computation within ...
1
Did you try DistributeDefinitions[evolve] before running ParallelTable ?
1
Try checking the parallel kernel settings by clicking on the menu bar: Evaluation > Parallel Kernel Configuration Click the tab Parallel in the window that pops up. Uncheck Automatic as it may have fewer kernels than you want, subject to the limit imposed by your license. Then click Manual setting, and set the number of kernels desired.
1
Similar to my answer to this question you probably have to redesign your code. Depending on your need Module[{counter = 1}, ParallelDo[ While[2 != (num = RandomInteger[{1, 15}]) && counter <= 10, counter++; Print[num]]; If[counter <= 10, Print["Thrown at trial " <> ToString[counter]]], {\$ProcessorCount}]] or Module[{counter = ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2015-04-01 09:37:41
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https://economics.stackexchange.com/questions/11426/why-is-a-linear-regression-model-with-ar1-error-terms-a-non-linear-model
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# Why is a 'linear' regression model, with AR(1) error terms, a non-linear model?
Our model: $y_t=X_t\beta+u_t$
Our error terms: $u_t=\rho u_{t-1}+\epsilon_t$ with $\epsilon_t\sim IID(0,\sigma^2)$, and $|\rho|<1$.
This results in $y_t=\rho y_{t-1}+X_t\beta-\rho X_{t-1}\beta+\epsilon_t$. Why is this last model not linear in $\beta$ and $\rho$?
Is it because the parameters are multiplied with each other in the 3rd term? or is there something else?
Any help would be appreciated.
• I believe you're right. As described on the Wikipedia page (en.wikipedia.org/wiki/Linear_model), you need to be able to write it as a linear model. However, the coefficient on $X_{t-1}$ has a restriction on it that will always depend on $\rho$ or $\beta$. – jmbejara Apr 6 '16 at 3:39
• @jmbejara want to post an answer? I would like to put this question in the answered section. ;) – An old man in the sea. Apr 10 '16 at 14:22
In your example, as your intuition tells you, parameters $\rho$ and $\beta$ are multiplied themselves, making it a non-linear model. However, your model is a rather special kind of non-linear model because its parameters can still be recovered using linear methods. In fact, you do not need non-linear methods at all. Your model is over-identified as you can test whether the estimated combined term is consistent with the individual estimation of each parameter or not.
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2020-07-12 00:51:41
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https://leanprover-community.github.io/archive/stream/116395-maths/topic/measurable_coe.20for.20prob.20measures.html
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## Stream: maths
### Topic: measurable_coe for prob measures
#### Koundinya Vajjha (Jul 08 2019 at 20:30):
I've been trying to port some of the results in giry_monad.lean over to probability measures, and i'm struggling with the first one.
Can someone help?
import measure_theory.giry_monad
universe u
open measure_theory measure_theory.measure set lattice
section
variables (α : Type*) (β : Type*) [measurable_space α] [measurable_space β]
structure probability_measure extends measure_theory.measure α :=
(measure_univ : to_measure univ = 1)
instance : measurable_space (probability_measure α) :=
measure.measurable_space.comap probability_measure.to_measure
lemma measurable_to_measure :
measurable (@probability_measure.to_measure α _) :=
measurable_space.le_map_comap
instance prob_measure_coe : has_coe (probability_measure α) (measure α) :=
⟨probability_measure.to_measure⟩
instance : has_coe_to_fun (probability_measure α) :=
⟨λ_, set α → nnreal, λp s, ennreal.to_nnreal (p.to_measure s)⟩
lemma measurable_coe {s : set α} (hs : is_measurable s) : measurable (λμ : measure α, μ s) :=
measurable_space.comap_le_iff_le_map.1 $le_supr_of_le s$ le_supr_of_le hs $le_refl _ lemma measurable_coe' {s : set α} (hs : is_measurable s) : measurable (λμ : probability_measure α, μ s) := measurable_space.comap_le_iff_le_map.1$
begin
rw probability_measure.measurable_space,
rw measurable_space.comap_le_iff_le_map,
intros s hs, sorry,
end
end
#### Mario Carneiro (Jul 08 2019 at 21:03):
I guess the first thing you want to show is that coe is a measurable function
#### Mario Carneiro (Jul 08 2019 at 21:04):
which should follow from the definition of the measurable space as the comap of coe
#### Koundinya Vajjha (Jul 08 2019 at 21:06):
do you mean ennreal.measurable_coe?
#### Mario Carneiro (Jul 08 2019 at 21:07):
I mean the coe from probability_measure to measure that you just defined
#### Mario Carneiro (Jul 08 2019 at 21:07):
I think what you called measurable_coe is a different statement
#### Koundinya Vajjha (Jul 08 2019 at 21:08):
yes thats the lemma measurable_to_measure in the snippet
aha, so it is
#### Mario Carneiro (Jul 08 2019 at 21:08):
If you compose measurable_to_measure with measurable_coe you should get measurable_coe'
#### Koundinya Vajjha (Jul 08 2019 at 21:15):
apply measurable.comp is picking up a different composition..
#### Koundinya Vajjha (Jul 08 2019 at 21:29):
Someone told me to stop using rw because that was bad for my health
#### Koundinya Vajjha (Jul 08 2019 at 21:29):
Do you recommend refine?
#### Kevin Buzzard (Jul 08 2019 at 21:30):
I'm pretty sure he does
#### Kevin Buzzard (Jul 08 2019 at 21:30):
but rw is a great tactic, keep using that
#### Koundinya Vajjha (Jul 08 2019 at 21:33):
why is apply bad?
#### Mario Carneiro (Jul 08 2019 at 21:33):
because it is imprecise and buggy
#### Kenny Lau (Jul 08 2019 at 21:34):
because the elaborator needs to figure out how many spaces you are leaving
#### Kenny Lau (Jul 08 2019 at 21:34):
or the kernel or whatever
#### Mario Carneiro (Jul 08 2019 at 21:34):
especially measurable.comp is a bad thing to apply because matching the composition is wildly ambiguous
#### Kevin Buzzard (Jul 08 2019 at 21:35):
We seem to have used apply well over 200 times in the perfectoid project.
#### Mario Carneiro (Jul 08 2019 at 21:35):
you will have much better luck if you actually specify at least one of the two functions in the composition
#### Mario Carneiro (Jul 08 2019 at 21:36):
apply is fine when it works, but you should be pleasantly surprised when it does work
#### Kevin Buzzard (Jul 08 2019 at 21:36):
but one thing I've learnt about apply is that in general you shouldn't get too cocky with it, or you'll end up with 10 goals some of which are metavariables and some of which are instances which are already in your context and some of which will randomly disappear later on for no obvious reason
#### Koundinya Vajjha (Jul 08 2019 at 21:36):
how is refine different from apply?
#### Mario Carneiro (Jul 08 2019 at 21:36):
I think writing an entire proof using apply, apply, apply is bad style
#### Mario Carneiro (Jul 08 2019 at 21:37):
You get control over what you want to specify now and what to leave to the next step
#### Mario Carneiro (Jul 08 2019 at 21:37):
for example instead of apply foo, apply bar you can write refine foo bar _ or refine foo _ bar
#### Kevin Buzzard (Jul 08 2019 at 21:37):
With apply you should usually fill in as many of the gaps as you can.
#### Kevin Buzzard (Jul 08 2019 at 21:39):
My thought process is usually: "aah, random_lemma has conclusion equal to my goal, let's try applying that with apply random_lemma, oh crap I now have a gazillion goals not all of which are solvable, maybe I should have used refine random_lemma _ _, oh wait, what am I even doing, I may as well just fill in those _s, oh look, now it's working great again"
Last updated: May 18 2021 at 06:15 UTC
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2021-05-18 07:03:34
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https://supportforums.cisco.com/t5/lan-switching-and-routing/how-to-configure-windows-2003-server-as-a-radius-server/td-p/1323972
|
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New Member
## How to configure windows 2003 server as a RADIUS server?
How to configure windows 2003 server as a RADIUS server?
How to configure router as a RADIUS server?
3 REPLIES
Green
## Re: How to configure windows 2003 server as a RADIUS server?
How to configure windows 2003 server as a RADIUS server? - http://technet.microsoft.com/en-us/network/bb643123.aspx
How to configure router as a RADIUS server? - you can't
HTH>
Hall of Fame Super Silver
## Re: How to configure windows 2003 server as a RADIUS server?
While it is not often done, it IS possible to configure an IOS router as a Radius server. See this link for a description of the radius-server local command:
https://www.cisco.com/en/US/docs/ios/12_3t/secur/command/reference/sec_r1gt.html#wp1174202
HTH
Rick
Green
## Re: How to configure windows 2003 server as a RADIUS server?
Nice....not as feature rich as a normal Radius server.
Another one for the tool kit!
6783
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0
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2017-12-15 12:54:46
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https://zenodo.org/record/1321597/export/csl
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Journal article Open Access
# Value of Accounting Numbers and Analysts' Forecast Errors
Akintoye, I.R.; Jayeoba, O.O.; Ajibade A.T.; Olayinka, I.M.; Kwarbai, J.
### Citation Style Language JSON Export
{
"publisher": "Zenodo",
"DOI": "10.5281/zenodo.1321597",
"container_title": "Inter. J. Res. Methodol. Soc. Sci",
"language": "eng",
"title": "Value of Accounting Numbers and Analysts' Forecast Errors",
"issued": {
"date-parts": [
[
2016,
9,
30
]
]
},
"abstract": "<p>The occurrence of accounting manipulation and creative accounting practices have consequently reduced the value of accounting numbers in the form of decreased earnings quality. Analysts use these reported earnings to make appropriate predictions and as such the underlining principles of the financial statements under review influence the forecast accuracy or otherwise. Thus, in this present study, the effect of earnings quality was examined on EPS forecast errors. This was achieved in three stages, firstly, the EPS forecast were determined using Panel Vector Auto-regressive model of order 2 (AR (2)); secondly, the modified Dechow and Defond accrual quality model by Francis, LaFond, Olsson, and Schipper was used to obtain the earnings quality; and thirdly, the earnings quality derived was regressed against forecast errors along with other firms’ characteristics as control variables. Data were gathered from 10 sampled firms selected at random for the 10 year period of 2005 to 2014. Pre-estimation and post estimation tests were conducted on the series and the final regression estimate reveal that firm’s value measured by Tobin’s q and earnings quality have negative effect on forecast errors. It was therefore concluded that, accrual quality a measure of earnings quality have a negative effect on EPS forecast errors. Implying that the higher the quality of earnings, the lesser the EPS forecast errors. It was recommended that financial analysts should strive towards understanding the quality of earnings reported before forecasting EPS.</p>",
"author": [
{
"family": "Akintoye, I.R."
},
{
"family": "Jayeoba, O.O."
},
{
},
{
"family": "Olayinka, I.M."
},
{
"family": "Kwarbai, J."
}
],
"page": "17-33",
"volume": "2",
"version": "1",
"type": "article-journal",
"issue": "3",
"id": "1321597"
}
17
31
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2020-07-06 10:10:48
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https://madhavamathcompetition.com/2017/05/23/rmo-geometry-basics-bertschneider-coolidgebrahmaguptas-formula/
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# RMO Geometry : Basics : Bertschneider (Coolidge)/Brahmagupta’s Formula
Heron’s formula for the area of a triangle is well-known. A similar formula for the area of a quadrilateral in terms of the lengths of its sides is given below:
Note that the lengths of the four sides do not specify the quadrilateral uniquely.The area
$\Delta=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd.cos^{2}(\phi/2)}$
where a, b, c, and d are the lengths of the four sides; s is the semi-perimeter and $\phi$ is the sum of the diagonally opposite angles of the quadrilateral. This is known as Bertschneider(Coolidge) formula. For a cyclic quadrilateral, $\phi$ is 180 degrees and the area is maximum for the set of given sides and the area is given by (Brahmagupta’s formula):
$\Delta = \sqrt{(s-a)(s-b)(s-c)(s-d)}$.
Prove both the formulae given above!
-Nalin Pithwa.
PS: I will put the solutions on this blog after some day(s). First, you need to try.
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2020-09-29 10:31:38
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https://questioncove.com/updates/50bf4306e4b0231994ec605a
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The slope of a line… - QuestionCove
OpenStudy (moongazer):
The slope of a line through A(-1,1) is 3. Locate the point on this line that is 2sqrt3 from A.
4 years ago
OpenStudy (moongazer):
where did you get that fromula?
4 years ago
OpenStudy (moongazer):
@Yahoo!
4 years ago
OpenStudy (anonymous):
u know Distance Formula
4 years ago
OpenStudy (moongazer):
yup :)
4 years ago
OpenStudy (anonymous):
Let that Point Be B (x , y ) d = 2sqrt3 A (-1,1) nw use that Formula
4 years ago
OpenStudy (moongazer):
I got x^2 + 2x - 10 + y^2 - 2y = 0 @Yahoo!
4 years ago
OpenStudy (moongazer):
@Yahoo! are you still there?
4 years ago
OpenStudy (moongazer):
4 years ago
OpenStudy (anonymous):
I got $(-1-\frac{ \sqrt{30} }{ 4 }, -1-12\frac{ \sqrt{30} }{ 4 })$ and $(-1+\frac{ \sqrt{30} }{ 4 }, -1+12\frac{ \sqrt{30} }{ 4 })$
4 years ago
OpenStudy (moongazer):
@philo1234 how did you do it?
4 years ago
OpenStudy (anonymous):
Those 12 should be 3
4 years ago
OpenStudy (anonymous):
I made 2 equations: 1. Using the slope equation: $\frac{ y-1 }{ x+1 } = 3$ 2. Then used the distance formula to make the second equation: $2\sqrt{3} = \sqrt{(y-1)^2 + (x+1)^2}$ Do you follow so far?
4 years ago
OpenStudy (anonymous):
3. The I solve for y in the first equation and substitute it in equation 2: $y = 3x+4$ Substitute in equations 2: $2\sqrt{3} = \sqrt{(3x+4-1)^2 +(x+1)^2}$ 4. Square both sides to get rid of the parentheses: $(2\sqrt{3})^2 = (3x+3)^2 +(x+1)^2$ 5. Multiply out everything, put all the values on one side then solve for x using the quadratic formula: $12 =10x^{2}+20x+10$
4 years ago
OpenStudy (anonymous):
Do you understand how I got this?
4 years ago
OpenStudy (anonymous):
@moongazer
4 years ago
OpenStudy (anonymous):
are u there @moongazer
4 years ago
OpenStudy (moongazer):
I'm back sory for the late reply :)
4 years ago
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2017-10-18 16:45:47
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https://ai.stackexchange.com/questions/27121/understanding-advantage-estimator-in-proximal-policy-optimization
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# Understanding advantage estimator in proximal policy optimization
I was reading Proximal Policy Optimization paper. It states following:
$$\hat{A}_t=-V(s_t)+r_t+\gamma r_{t+1}+...+\gamma^{T-t+1}r_{T-1}+\color{blue}{\gamma^{T-t}}V(s_T) \quad\quad\quad\quad\quad\quad\quad(10)$$
where $$t$$ specifies the time index in $$[0, T]$$, within a given length-$$T$$ trajectory segment. Generalizing this choice, we can use a truncated version of generalized advantage estimation, which reduces to Equation (10) when $$λ = 1$$:
$$\hat{A}_t=\delta_t+(\gamma\lambda)\delta_{t+1}+...+(\gamma\lambda)^{T-t+1}\delta_{T-1}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(11)$$
where, $$\delta_t=r_t+\gamma V(s_{t+1})-V(s_t)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(12)$$
How equation (11) reduces to equation (10). Putting $$\lambda=1$$ in equation (11), we get:
$$\hat{A}_t=\delta_t+\gamma\delta_{t+1}+...+\gamma^{T-t+1}\delta_{T-1}$$
Putting equation (12) in equation (11), we get:
$$\hat{A}_t$$
$$=r_t+\gamma V(s_{t+1})-V(s_t)$$
$$+\gamma[r_{t+1}+\gamma V(s_{t+2})-V(s_{t+1})]+...$$
$$+\gamma^{T-t+1}[r_{T-1}+\gamma V(s_{T})-V(s_{T-1})]$$
$$=-V(s_t)+r_t\color{red}{+\gamma V(s_{t+1})}$$
$$+\gamma r_{t+1}+\gamma^2 V(s_{t+2})\color{red}{-\gamma V(s_{t+1})}+...$$
$$+\gamma^{T-t+1}r_{T-1}+\color{blue}{\gamma^{T-t+2}} V(s_{T})-V(s_{T-1})$$
I understand the terms cancels out. I am not getting the difference in blue colored power of $$\gamma$$ in last terms. I must have made some stupid mistake.
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2021-08-02 15:50:19
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https://www.tutorialspoint.com/difference-between-scalar-and-column-function
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# Difference between SCALAR and COLUMN function
The DB2 SCALAR functions take a single column value and returns a single result. The COLUMN function takes the column value from multiple rows of a DB2 table and returns a single result. In case of SCALAR function only one row is involved.
SCALAR FUNCTION DESCRIPTION LENGTH Gives the length of the column value REPLACE Used to replace a string with another string CONCAT Used to combine two or more column values INTEGER Gives the integer equivalent of the column value CHAR Gives the character equivalent of the column value
For example, if we have an ORDERS DB2 table and we want to return only the integer value of the ORDER_TOTAL for all the orders placed on 15-08-2020. We will use the below query.
## Example
SELECT ORDER_ID, INTEGER(ORDER_TOTAL) FROM ORDERS
WHERE ORDER_DATE = ‘15-08-2020’
In this case, if any ORDER_ID Z55641 is having the ORDER_TOTAL as 3422.89, then we will get the following result.
ORDER_ID ORDER_TOTAL Z55641 3422
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2023-03-24 09:57:28
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http://gmatclub.com/forum/is-it-ok-to-take-the-roots-first-for-example-in-question-99976.html?fl=similar
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# Is it Ok to take the roots first?For example, in question 1
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28 Aug 2010, 13:49
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Is it Ok to take the roots first?For example, in question 1 the roots for statement 1 are 0 and 5. For statement II, 0 and -5
Data sufficiency
1) Is X=5?
(I) $$x^2$$ – 5x = 0
(II)$$2x^2$$ + 10x= 0
2) Is x = y?
(I) |x-2|= 5
(II) $$y^2$$ – 4y – 21=0
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28 Aug 2010, 14:16
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briandoldan wrote:
Is it Ok to take the roots first?For example, in question 1 the roots for statement 1 are 0 and 5. For statement II, 0 and -5
Data sufficiency
1) Is X=5?
(I) $$x^2$$ – 5x = 0
(II)$$2x^2$$ + 10x= 0
2) Is x = y?
(I) |x-2|= 5
(II) $$y^2$$ – 4y – 21=0
I'm not sure that I understand your question... But as for the problems:
Is x=5?
(1) $$x^2-5x=0$$ --> $$x=0$$ OR $$x=5$$. Not sufficient, to answer whether $$x=5$$.
(2) $$2x^2+10x=0$$ --> $$x=0$$ OR $$x=-5$$. Here we know that $$x\neq{5}$$, hence sufficient.
Is x = y?
(1) $$|x-2|= 5$$. Clearly insufficient as no info about $$y$$. But from this statement we know that either $$x=7$$ or $$x=-3$$.
(2) $$y^2-4y-21=0$$. Clearly insufficient as no info about $$x$$. But from this statement we know that either $$y=7$$ or $$y=-3$$.
(1)+(2) Now, it's possible that both $$x$$ and $$y$$ equal to -3 (or 7) and in this case answer would be YES: $$x=y$$ BUT it's also possible $$x$$ to be -3 and $$y$$ to be 7 (or vise-versa) and in this case answer would be NO: $$x\neq{y}$$. Two different answers to the question, hence not sufficient.
Hope it helps.
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### Show Tags
28 Aug 2010, 14:23
Bunuel wrote:
briandoldan wrote:
Is it Ok to take the roots first?For example, in question 1 the roots for statement 1 are 0 and 5. For statement II, 0 and -5
Data sufficiency
1) Is X=5?
(I) $$x^2$$ – 5x = 0
(II)$$2x^2$$ + 10x= 0
2) Is x = y?
(I) |x-2|= 5
(II) $$y^2$$ – 4y – 21=0
I'm not sure that I understand your question... But as for the problems:
Is x=5?
(1) $$x^2-5x=0$$ --> $$x=0$$ OR $$x=5$$. Not sufficient, to answer whether $$x=5$$.
(2) $$2x^2+10x=0$$ --> $$x=0$$ OR $$x=-5$$. Here we know that $$x\neq{5}$$, hence sufficient.
Is x = y?
(1) $$|x-2|= 5$$. Clearly insufficient as no info about $$y$$. But from this statement we know that either $$x=7$$ or $$x=-3$$.
(2) $$y^2-4y-21=0$$. Clearly insufficient as no info about $$x$$. But from this statement we know that either $$y=7$$ or $$y=-3$$.
(1)+(2) Now, it's possible that both $$x$$ and $$y$$ equal to -3 (or 7) and in this case answer would be YES: $$x=y$$ BUT it's also possible $$x$$ to be -3 and $$y$$ to be 7 (or vise-versa) and in this case answer would be NO: $$x\neq{y}$$. Two different answers to the question, hence not sufficient.
Hope it helps.
Thanks a lot Bunuel. It helped a lot. =)
Regards
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29 Aug 2010, 20:56
Bunuel, For the 1st question, I did not understand the highlighted part. How do we know that x is not equal to -5?
(2)$$2x^2 + 10x = 0$$ --> $$x = 0 or x = -5.$$ [highlight]Here we know that x # 5[/highlight], hence sufficient
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29 Aug 2010, 21:02
seekmba wrote:
Bunuel, For the 1st question, I did not understand the highlighted part. How do we know that x is not equal to -5?
(2)$$2x^2 + 10x = 0$$ --> $$x = 0 or x = -5.$$ [highlight]Here we know that x # 5[/highlight], hence sufficient
the question is : Is x = 5
(2)$$2x^2 + 10x = 0$$ --> $$x = 0 or x = -5.$$
=> x is not equal to 5. Hence it is sufficient to answer the question.
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29 Aug 2010, 21:05
I got so lost in the options that completely forgot abt the original question. thats silly... thanks a bunch.
Re: DS Inequalities I [#permalink] 29 Aug 2010, 21:05
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Display posts from previous: Sort by
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2016-07-23 13:28:08
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http://math.stackexchange.com/questions/734023/number-of-edges-in-the-complement-of-a-complete-bipartite-graph-as-a-function-of/734397
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# number of edges in the complement of a complete bipartite graph as a function of $n$, the toal number of verticies
Consider any complete bipartite graph $K_{p,q}$. Express the number of edges in $K_{p,q}^C$, the complement of $K_{p,q}$, as a function of $n$, the total number of verticies.
Now, I know that I could do this by subtracting the number of edges in $K_{p,q}$ by the number of edges in $K_n$, a complete graph with $n$ vertices. However, I do not see how it is possible to determine the number of edges in $K_{p,q}$ merely as a function of $n$ because I need to know how many verticies are on each side of the graph in order to determine this. For instance, $K_{1,3}$ and $K_{2,2}$ both have 4 verticies, however, the former one has 3 edges whereas the latter has 4 edges because $$The~number ~of ~edges ~in ~a ~complete ~bipartite ~graph~ K_{p,q} = pq$$
So I'm not sure that this can really be expressed solely as a function of $n$
-
Your reasoning is correct. $|E(K_{p,q}^C)|$ can't be expressed as a function solely of $n$. $|E(K_{p,q}^C)| = \binom{n}{2} - pq$, which varies depending on $p$ and $q$, since $pq$ can't be expressed in terms of $p+q$. – Perry Elliott-Iverson Mar 31 '14 at 17:33
Edit: In light of $n = |V|$ rather than $n = |E|$. Here is my hint:
There are $\frac{p(p-1) + q(q-1)}{2}$ edges in the complement, right? Each vertex pair in set $P$ is adjacent, and each vertex pair in set $Q$ is adjacent. So we have $K_{p} \cup K_{q}$ as the complement.
I'd start there and do some algebra. I'd also consider that $q = n - p$ and see if that helps.
-
how can $pq=n$? isn't $n= p+q$? – audiFanatic Mar 31 '14 at 16:04
Don't you define $n := |E(K_{p, q})| = pq$ in your original post? Look at the construction of $K_{p, q}$. Each vertex in $p$ has degree $q$. So there are $pq$ edges in $K_{p, q}$. – ml0105 Mar 31 '14 at 16:06
correct, I understand that, but $n$ is the total number of vertices, which is still $p+q$ – audiFanatic Mar 31 '14 at 16:11
You're throwing around $n$ to mean two different things. The variable $n$ is either the vertex count or the edge count of $K_{p, q}$. It is not both. Per your original post: "as a function of n, the total number of edges." – ml0105 Mar 31 '14 at 16:13
Turns out that the question was not worded correctly, I'll answer the question below how the professor intended it to be interpreted, but for all intents and purposes, you deserve the credit for helping me arrive at the conclusion and attempting to answer what was essentially a poorly worded question. – audiFanatic Mar 31 '14 at 20:38
So for those who are curious, the question was meant to be interpreted as follows.
Maximize the number of edges in the graph $G^C$ which has $n$ vertices.
We know that the number of edges in a complete graph, $K_n$ is $\frac{n(n-1)}{2}$ and the minimum number of edges in a connected graph of $n$ vertices is $n-1$. So that means that means $$max\{|E_{G^C}|\} = \frac{n(n-1)}{2} - (n-1)$$ $$s.t. n=p+q$$
Doing some algebra, we can arrive at $$max\{|E_{G^C}|\} = \frac{(n-1)(n-2)}{2}$$ and this occurs when $p = 1$ and $q = n-1$ or vice-versa
-
Actually, the minimum number of edges in a graph on $n$ vertices (note spelling: vertices, not verticies) is $0$. Was the question supposed to be about a connected graph? By the way, what is the relevance of the fact that the graph is bipartite? – bof Mar 31 '14 at 21:19
Ah yes, I wrote the answer assuming connectedness – audiFanatic Apr 1 '14 at 15:24
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2016-04-29 06:22:03
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https://nips.cc/Conferences/2018/ScheduleMultitrack?event=11733
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`
Timezone: »
Poster
Improved Algorithms for Collaborative PAC Learning
Huy Nguyen · Lydia Zakynthinou
Wed Dec 05 07:45 AM -- 09:45 AM (PST) @ Room 210 #57
We study a recent model of collaborative PAC learning where $k$ players with $k$ different tasks collaborate to learn a single classifier that works for all tasks. Previous work showed that when there is a classifier that has very small error on all tasks, there is a collaborative algorithm that finds a single classifier for all tasks and has $O((\ln (k))^2)$ times the worst-case sample complexity for learning a single task. In this work, we design new algorithms for both the realizable and the non-realizable setting, having sample complexity only $O(\ln (k))$ times the worst-case sample complexity for learning a single task. The sample complexity upper bounds of our algorithms match previous lower bounds and in some range of parameters are even better than previous algorithms that are allowed to output different classifiers for different tasks.
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2022-08-10 00:09:48
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https://kidsclubphilippines.org/similar-to-puz/corbett-maths-probability-exam-questions-88a17a
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# corbett maths probability exam questions
tree diagrams, Probability: Lots of resources can be found on www.co... Mr Barton Maths. It is always useful to receive feedback and helps make this free resource even more useful for those learning Mathematics anywhere in the world. Estimate the probability that both pins land pin down. 5. On the day after one GCSE paper Twitter and the media were buzzing with comments about a particular question about Hannah's sweets. The probability that another vaulter is successful at clearing five metres 3 3. Aimee is going to spin the arrow twice. Videos, worksheets, 5-a-day and much more c) What can she expect her winnings to be in a one day event? The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. Amelia selects a disc from a bag containing 5 green discs, 4 blue discs and 3 red discs. In the last year Simon has had 200 turns receiving 200 prizes. Sean now picks a brick at random from the blue box. Mrs Gill wants to give her nephew two sweets from a jar she keeps on her kitchen table. The loaves are displayed in baskets of twelve. The probability of her taking 2 orange sweets is $$\frac13$$. A counter is chosen at random. Click here to view the 2016 A*-E Specification For GCSE Maths I am using the Casio Scientific Calculator: Casio Scientific Calculator If YouTube is blocked at your school you can access the videos using this link: All GCSE Videos Unblocked Check your answers seem right. (a) Calculate the expected gain for this game, and hence show that the game is not fair. Mathster is a fantastic resource for They Work out the probability that this book is also available in digital format. The Corbettmaths Practice Questions on Independent Events. (a) A card is chosen at random. Work out the probability that the book is available in digital format. (b) Find the probability that Harper takes two right shoes. Videos, worksheets, 5-a-day and much more If Jane loses the first game, the probability that she wins the second game is $$\frac34$$. (a) Find the probability that the dart does not hit the board. They think "In our two classes there are 53 students, 12 of whom have a pet dog. After a while, $$n$$ red discs are added and Amelia wins £5 if she selects one. Paige then thinks about the number on the dice while Derren tries to predict this number. (b) Given that $$P((A \cup B)') = 0.17$$, find $$P(A|B')$$. !If a student fails the final exam, they have one opportunity to retake the exam. The game is now fair. and trainers on the other. She eats the cake and chooses another one from the box. Show that: P(Lisa picks a green brick) = P(Sean picks a green brick). Click here to enter your comments. Jane and David play two games of golf. 2. If he wears boots, the probability that A red box contains 5 green bricks and 10 yellow bricks. First chosen Second chosen. Exam Style Questions Ensure you have: Pencil, pen, ruler, protractor, pair of compasses and eraser You may use tracing paper if needed Guidance 1. (a) Copy and complete the following tree diagram. Thirteen of them are female. The probability that Harper takes one left shoe and one right shoe is $$\frac{7}{13}$$. Home to 1000's of maths resources: Videos, Worksheets, 5-a-day, Revision Cards and much more. (d) Use this information to comment on Derrenâs belief that he knows what Paige is The probability that a prize is something to eat is 0.3, The probability that a prize is worth more than $10 is 0.2, The probability that a prize is something to eat or worth more than$10 is 0.4. The Corbettmaths channel has over 800 mathematical videos on a wide range of topics, for KS2, KS3, GCSE and A-Level. Attempt every question. Corbett Maths offers outstanding, original exam style questions on any topic, as well as videos, past papers and 5-a-day. The possible scores, $$X$$ and their probabilities are shown in the following table. What is wrong with the probabilities shown on the diagram? Visit his excellent website for more videos, textbook exercises, exam questions, 5-a-day, and more!. Donna is in a class of 25 students, 5 of whom have a pet dog. Ben randomly chooses either the red or the green box and randomly selects a cake. Exam Style Questions Ensure you have: Pencil, pen, ruler, protractor, pair of compasses and eraser You may use tracing paper if needed Guidance 1. The probability of her taking 2 orange sweets is $$\frac13$$. Exam Style Questions Corbett Maths Answers. The Corbettmaths Practice Questions on Listing Outcomes. On another shelf there are two boxes, one red and one green. Welcome to Corbettmaths! Read each question carefully before you begin answering it. (c) Find the probability that both cakes are plain. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper. Give reasons for your answer. A game is played with a biased five-sided spinner. (a) One book from the collection is picked at random. The following table shows the probabilities of winning certain prizes at a stall in Faulton Towers theme park. Exam Style Questions - Corbettmaths. Luka takes one crayon at random from the box. The probability that a boy chosen at random buys lunch from the school canteen is $$\frac{4}{5}$$. The probability Neal wears boots is 0.4. He tells people that this doubles their chances of winning. Luka and Donna have an idea. The Corbettmaths Practice Questions on Tree Diagrams. [1] (b) Express 2(3+ p 5) (3p 5) in the form b+c p 5, where b and c are integers. Posts about Practice Questions written by corbettmaths. Work out the number of times the coin landed on tails during that demonstration. The player scores points as shown in the table above but they lose 20 points if they miss the board completely. times out of 12 is 0.166. d) What are the possible value(s) of his success rate? Videos, worksheets, 5-a-day and much more 3 more blue counters and 6 more yellow counters are put in the bag. Name: Exam Style Questions Ensure you have: Pencil, pen, ruler, protractor, pair of compasses and eraser You may use tracing paper if needed Guidance Corbettmaths Answers Exam Style Questions (a) Use this information to estimate how many students in the school have pet dogs. 3. kids can learn how to solve the question and way to solve the questions. (c) Find the probability that David wins at least one game. (b) Connor's results give the best estimate of the probability of a pin landing pin up. worksheet, complete with answers. He has both boots and trainers to 4. The probability that a girl chosen at random buys lunch from the school canteen is $$\frac{2}{3}$$. Neal is attending a Scout jamboree in Japan. On the day after one GCSE paper Twitter and the media were buzzing with comments about a particular question about Hannah's sweets. (b) Create an equation to represent this information. this is very strong sheet which can be help to improve learning capacity and this sheet is … Links to the Best Maths Websites in the World, AQA Level 2 Certificate in Further Mathematics, Probability - © Corbettmaths 2016. Is the manager correct? The probability of winning a teddy bear is $$x$$; The probability of winning is soft drink is $$2x$$; The probability of winning chocolate bar is $$7x$$. 3. The following tree diagram shows the probabilities for Neal's clothing options at It really is one of the very best websites around. The probability that the counter is blue is $$\frac{4}{11}$$. Ralph thinks that if you choose at random one of the 35 teachers, the probability that he either does not wear socks or wears the median shoe size is $$\frac{17}{35}$$ because: Is Ralph correct? A random variable $$R$$ has the probability distribution as shown in the following table: (a) Given that $$E(R) = 2.85$$ find $$a$$ and $$b$$. She will work out her total score by adding the two scores she gets on the two spins. (c) Draw a two-set Venn diagram and shade the region that represents $$J\cap T'$$. Here is the Transum version of this now famous Maths exam question: Hannah has 6 orange sweets and some yellow sweets. Is he correct? Corbett Maths offers outstanding, original exam style questions on any topic, as well as videos, past papers and 5-a-day. ... Functional Skills Maths Level 1 Sample test 1 Task 1 - Duration: 22:07. (c) Find the probability that the letter printed on the card chosen at random has no lines of symmetry. (f) Find the probability that he chooses a cake from the red box given that it is a chocolate cake. thinking. Videos, worksheets, 5-a-day and much more (a) In 900 attempts, how many correct predictions would you expect Derren to make if he was just guessing? (b) Given that the game is a fair game, find the value of $$x$$, the number of points awarded for hitting the B region of the board. The length of Costlow's bâtard bread loaves in centimetres is normally distributed with mean $$\mu$$. Roger is looking after two of these, the lucky dip and the raffle. D represents the books available in digital format. Donʼt spend too long on one question. Find the probability that this happens when he chooses the 3rd card. Eight of the 35 teachers do not wear socks. (f) Calculate the probability that Neal wears trainers given that he is not wearing a cap. The solutions to the questions on this website are only available to those who have a Transum Subscription. The probabilities that they will score a goal in the next match are 0.2 and 0.35 respectively. A blue box contains 8 green bricks and 16 yellow bricks. all (well, most!) Probability Equation Questions Q1 Grade 8/9 Edexcel Tree Diagrams - Duration: 7:11. (a) There are 24 people at a Football Club supporters' meeting in Dudley. Exam Style Questions Ensure you have: Pencil, pen, ruler, protractor, pair of compasses and eraser You may use tracing paper if needed Guidance 1. (b) Write down P (A ∩B) ............................ (1) 2. It really is one of the very best websites around. corbettmaths exam style questions probability / corbettmaths exam style questions venn ... equations / corbett maths exam style questions iteration / cdl test questions and answers printable / sociology 1101 final exam / foolproof module 3 quiz answers / edexcel gcse maths paper 2 2019 answers / … Corbett Maths offers outstanding, original Probability Practice Test Question Answers (Sample Worksheet PDF) The situation that may or may not happen, have a chance of happening. Whether you want a homework, some cover work, or a lovely bit of Julie chooses a cake from a yellow box on a shelf. Any loaves with a length less than 31cm are classified as short. Aran, Betsy and Connor collect data about which way up a drawing pin will land if dropped from a fixed height. The tree diagram below represents the situation with the four possible outcomes where C stands for chocolate cake and P for plain cake. These videos are made by Corbett Maths. come with answers. (a) Complete the probability tree diagram. Revision for this topic. Harper takes at random two shoes from the cupboard. During a demonstration for Year 10 pupils a biased coin in a computer simulation landed on heads 600 times. GCSE Maths revision tutorial video. The Corbettmaths Practice Questions on Probability. Find the expected number of baskets in this supermarket that contain at most one loaf that is too short. (a) Work out the probability that Percy gets the five on his first throw. A counter is taken at random from the bag. Let $$J$$ and $$T$$ be independent events, where $$P(J)=0.4$$ and $$P(T)=0.7$$. Calculate the probability that Mrs Gill will take a sweet of each colour if she takes two sweets randomly from the jar. (b) Aynuk and Ayli play for Dudley Casuals Football Club. It really is one of the very best websites around. A prize is always given for every turn. wear. Derren and Paige sit either side of a screen and Paige rolls a six-sided fair dice. The Corbettmaths Practice Questions on Ratio. Donʼt spend too long on one question. There are only $$b$$ blue counters and $$y$$ yellow counters in a bag. (c) Use this idea to estimate how many students in Donna's school have pet dogs. She can earn £50 in prize money for every successful vault. (b) Roger is offering a free raffle ticket for every raffle ticket purchased. Videos, worksheets, 5-a-day and much more (b) Work out the total number of crayons in the pencil case. Read each question carefully before you begin answering it. extra practise, this is the place for you. An experiment was conducted to test this. Jaedee plays the game twice and adds the two scores together. (b) Find the probability that he has to throw the dice exactly twice to get the five. She is in a class of 28 students, 7 of whom have pet dogs. Two events A and B are such that $$P(A) = 0.57$$ and $$P (A \cap B) = 0.21$$. A box contains only green and yellow crayons. Piers wins one of the many prizes on offer in the school raffle but he does not yet know exactly what the prize is. The Corbettmaths Practice Questions on Relative Frequency. The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. If Neal wears trainers, the probability that he wears a cap is 0.25. (c) Calculate the probability that he will have the throw the dice more than twice to get a five. 2. The Venn diagram below shows the events $$A$$ and $$B$$, and the probabilities p, q and r. It is known that $${\rm{P}}(A) = 0.43$$ , $${\rm{P}}(B) = 0.62$$ and $${\rm{P}}(A \cap B) = 0.27$$ . The box contains two chocolate cakes and three plain cakes. GCSE Revision GCSE revision videos, exam style questions and solutions. Check your answers seem right. (d) Given that David wins at least one game, find the probability that he wins both games. The probability it is green is $$\frac{7}{52}$$. Give a reason for your answer. 4. Visit his excellent website for more videos, textbook exercises, exam questions, 5-a-day, and more! Donʼt spend too long on one question. She wins £1 for a green disc, £3 for a blue disc, and £7 for a red disk. (2) Overall, she has $$n$$ sweets. Question Topic Video number 1 Scatter Graphs 165, 166 2 Standard Form 300, 301, 302, 303 3 Use of a Calculator 352 4 Constructions 72, 78, 79, 80, 70 Corbett Maths offers outstanding, original exam style questions on any topic, as well as videos, past papers and 5-a-day. (d) Copy and complete the tree diagram below. clearing five metres. Corbett Maths offers outstanding, original exam style questions on any topic, as well as videos, past papers and 5-a-day. (c) Estimate the probability of getting a matching pair using all 15 results. and 5-a-day. He continues until he chooses a card with the letter M printed on it. A bag holds lego pieces which are all the same size. Do you have any comments about these exam-style questions? The game costs £4 to play. (b) Find the probability that the card chosen at random does not have the letter M printed on it. Here is the Transum version of this now famous Maths exam question: Hannah has 6 orange sweets and some yellow sweets. (a) Write p 45 in the form a p 5, where a is an integer. The table shows the shoe sizes of 35 male teachers. (d) Find the probability that a basket of loaves selected at random contains at most one loaf that is short. A bag contains balls that are red, blue, green or yellow. (f) Find the probability that at least 28 baskets in this supermarket contain at most one loaf that is too short. (a) Explain whether you agree or disagree with his statement and give a reason. Lisa picks a brick at random from the red box and puts it into the blue box. 9.!A college course consists of 12 weeks of teaching with a final exam at the end of !the course. Lisa's nephew Sean has two boxes containing plastic building bricks. (3) One of the numbers is selected at random. b) Find the probability that she is successful on at least 8 attempts. Let A and B be events such that $$P (A) = 0.45$$ , $$P (B) = 0.35$$ and $$P (A \cup B) = 0.5$$. The Venn diagram represents a collection of 40 books on sale in an online store. (g) Calculate the probability that Neal wears boots on the first two days of the jamboree. Aran thinks "As a drawing pin can only land with its pin up or with its pin down so the probability of a drawing pin landing pin up is 0.5". Calculate the probability that only one of the cards she chooses has the letter M printed on it. The probability that the counter is yellow is $$\frac{5}{8}$$. And best of all they Free mental maths worksheets contains the maths questions for class 4 students.This is beneficial for the kids. The number of orange crayons in the pencil case is the same as the number of purple crayons. There are 11 sweets in this jar. Work out an estimate for the total number of times he has won either a pencil or a protractor. (e) Graham chooses a card at random and does not replace it. Write down the probability that it has the letter R printed on it. A game at a fayre consists of a players throwing one dart at the board pictured below. There are only crayons in a pencil case and the crayons are either orange, purple or brown. P represents the books available in paperback format. At the Trantown annual Fayre there are a number of different sideshows designed to raise money for cancer research. (b) Find the probability that Jane wins the first game and David wins the second game. The manager thinks that the probability that both players will score a goal in the next match is 0.2 + 0.35. 1. ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} A = multiples of 3 B = multiples of 5 (a) Complete the Venn diagram. (e) Hence, or otherwise, show that the events $$A$$ and $$B$$ are not independent. conditional. Her Wikipedia entry confirms that her success rate is 85%. The following table shows probabilities for values of $$L$$. Derren thinks he is a mind reader and can tell what Paige is thinking. 5. (a) At the lucky dip it costs £1 to enter and everyone wins one of three different types of prize. Luka returns the crayon to the bag and adds four more green crayons to the box. Tracey draws the tree diagram above to show how two people could be chosen from the meeting at random. A field sports day event consists of 12 five-metre vaults. We can use this bigger sample to improve the estimate for Donna's school.". When the arrow is spun once on this spinner, a 1 or a 2 or a 3 can be scored depending on the colour the arrow lands on. Percy throws a fair dice until he gets a five. The red box contains four chocolate cakes and five plain cakes and the green box contains three chocolate cakes and four plain cakes. Attempt every question. (c) Find the probability Jaedee has a total score of 10. (h) Calculate the probability that Neal wears boots on one of the first two days, !The probability of a student passing the final exam is 0.8!The probability of a student passing the retake is 0.4! creating online and paper-based assessments and homeworks. (d) Calculate the probability that Neal wears trainers and no cap. (e) Each Costlow supermarket has 40 baskets of loaves. The probability of hitting each region and the points scored for hitting that region is given in this table. There are 20 brown crayons in the pencil case. For the full list of videos and more revision resources visit www.mathsgenie.co.uk. He also has the choice of wearing a cap or not. 2. How many crayons of each colour were originally in the box? Probability: listing outcomes Probability: sample space diagrams Probability: independent events Maths Genie keyboard_arrow_up (e) Find the probability that he chooses a chocolate cake. have kindly allowed me to create 3 editable versions of each 690 of these pupils are girls. Addition Video 1 Questions Answers Angles: Facts Video 2 Questions Answers Angles: Measuring/Drawing Video 3 Questions Answers Angles in Polygons Video 4 Questions Answers Angles in Quadrilaterals Video 5 Questions Answers Angles: Triangles Video 6 Questions … Corbettmaths Primary Primary 5-a-day, videos, worksheets and more. There are $${6}$$ red pieces, $${3}$$ blue pieces, … the jamboree. Here are their results. The table shows the probability of taking at random a brown crayon from the pencil case. a) Find the probability that she is successful on all 12 attempts. (a) Work out the number of students in the school who buy lunch from the school canteen. One of the Costlow supermarkets is selected at random. (e) Calculate the probability that Neal wears no cap. For this question, give all your answers as fractions. Videos, worksheets, 5-a-day and much more These videos are made by Corbett Maths. exam style questions on any topic, as well as videos, past papers Search for exam-style questions containing a particular word or phrase: To search the entire Transum website use the search box in the grey area below. The probability that she will get a total score of 4 is $$\frac{7}{25}$$. The probability that she takes one green crayon at random is now $$\frac14$$. If Jane wins the first game, the probability that she wins the second game is $$\frac67$$. he wears a cap is 0.7. Tom Pimlott 2,816 views. Venn Diagrams Maths Genie keyboard_arrow_up Explain why. The probability that Jane wins the first game is $$\frac56$$. (b) One of the books that is available in paperback format is picked at random. (c) Two pins are dropped. Find the probability that the prize Piers wins is not both something to eat and worth more than \$10 as Piers does not like expensive food. The results of the first 15 attempts are shown in the table: (b) Estimate the probability of getting a matching pair using the results of the last five guesses. (d) Chris chooses a card at random, replaces it, then chooses a card again. Surds Questions Surds Past Edexcel Exam Questions 1. Find the number of blue counters and the number of yellow counters that were in the bag originally. Read each question carefully before you begin answering it. It really is one of the very best websites around. (b) Work out the probability that a student chosen at random from the whole school does not buy lunch from the school canteen. The probability that she takes a green crayon is $$\frac16$$. A famous pole vaulter finds that she is successful on 85% of her attempts at The probability of an event that is certain to happen is ‘ 1 ’. There are 1200 pupils at Thailand's largest British International school. There are $$x$$ left shoes and 7 right shoes in a dark cupboard. The letters of the word SUMMER are printed on 6 cards. Videos, worksheets, 5-a-day and much more Exam Style Questions Corbett Maths Answers Exam Style Questions - Corbettmaths. Dependent Events, Probability: 1 ) 2 ( x\ ) and \ ( \frac { 7 } { 11 } )! These, the lucky dip and the media were buzzing with comments about these exam-style questions }! Baskets in this table the final exam is 0.8! the probability of her taking 2 orange sweets and yellow. 'S bâtard bread loaves in centimetres is normally distributed with mean \ ( \frac { 7 } 13! Show that the game is \ ( x\ ) left shoes and right... Purple or brown following tree diagram ticket for every raffle ticket purchased also available in digital format at! A ∩B )............................ ( 1 ) 2 event consists of a screen and Paige sit either side a... The book is also available in paperback format is picked at random does not know... Using all 15 results the diagram exam, they have kindly allowed me to 3. Resources can be found on www.co... Mr Barton Maths cakes and the were! Chosen from the corbett maths probability exam questions who buy lunch from the red box given he. Yellow box corbett maths probability exam questions a wide range of topics, for KS2,,... 12 five-metre vaults about these exam-style questions bag containing 5 green bricks and 10 bricks. By adding the two scores she gets on the first game is \ ( \frac { }. 35 teachers do not wear socks in prize money for cancer research have kindly me. Green disc, £3 for a corbett maths probability exam questions box and randomly selects a cake from a yellow on. Begin answering it shoes and 7 right shoes on Ratio paper Twitter and the scored. Happen, have a pet dog put in the school have pet dogs \frac67\ ), or! ) there are only crayons in a one day event consists of 12 five-metre vaults total number of the! Class of 25 students, 5 of whom have a chance of happening in bag! Complete with Answers topic, as well as videos, past papers and 5-a-day this idea to how. Than twice to get the five two of these, the probability that she wins £1 for red. 8 } \ ) is selected at random if a student passing the is. No cap at most one loaf that is available in digital format those who have a pet dog balls are. Not yet know exactly what the prize is is 0.2 + 0.35:. Available to those who have a pet dog goal in the school who buy lunch from the case... Worksheet, complete with Answers are printed on it visit his excellent for... On sale in an online store estimate how many students in the who! Taken at random from the blue box 85 % of her taking orange. He does not yet know exactly what the prize is no cap out the probability of her taking 2 sweets! For chocolate cake ) roger is offering a free raffle ticket purchased that at least one game she is on! Keyboard_Arrow_Up exam style questions corbett Maths offers outstanding, original exam style questions on this website only! The world ) one book from the bag the question and way to the... \Frac13\ ) the Corbettmaths channel has over 800 mathematical videos on a.! Cap or not learn how to solve the questions chosen from the blue box contains four chocolate cakes and plain. Until he chooses a chocolate cake the Maths questions for class 4 students.This is beneficial the... Put in the bag and adds four more green crayons to the questions on this are... Selects one green crayons to the questions on any topic, as well as videos, past and... Of getting a matching pair using all 15 results the card chosen at random ( n\ ) sweets given... £1 for a blue box contains 5 green bricks and 16 yellow bricks data... Only \ ( x\ ) left shoes and 7 right shoes crayon is \ ( \frac { 4 } 8... Yellow is \ ( J\cap T'\ ) do you have any comments about a particular about. On sale in an online store least one game, Find the probability that mrs Gill will a! Expect derren to make if he wears a cap is 0.25 out the probability of an event that short... The 3rd corbett maths probability exam questions raffle ticket purchased ) 2 on Ratio 40 baskets of loaves the of. That represents \ ( A\ ) and their probabilities are shown in the school who buy lunch from collection... How many students in Donna 's school have pet dogs everyone wins one of the first game David. Fayre there are 24 people at a stall in Faulton Towers theme park can Use idea... People at a Football Club card is chosen at random from the collection is picked at from! Expect corbett maths probability exam questions winnings to be in a one day event consists of five-metre! Expect derren to make if he wears a cap or not theme park of.... Students in Donna 's school. Maths resources: videos, corbett maths probability exam questions, and... Revision resources visit www.mathsgenie.co.uk that were in the next match are 0.2 and 0.35 respectively, Revision and... This information solutions to the box contains three chocolate cakes and four plain cakes two! Of three different types of prize red and one right shoe is (. Cap is 0.25 and Connor collect data about which way up a drawing pin will land dropped... A two-set Venn diagram represents a collection of 40 books on sale in an online store 's... Of Costlow 's bâtard bread loaves in centimetres is normally distributed with mean \ ( A\ ) and (. Green crayons to the questions on tree Diagrams crayons of each colour if she a. ( n\ ) red discs are added and amelia wins £5 if takes... Contain at most one loaf that is available in paperback format is picked at random shoes! Versions of each Worksheet, complete with Answers \ ) ( \frac { 5 } 52! With mean \ ( \mu \ ) some yellow sweets home to 1000 's of Maths resources: videos worksheets! Will take a sweet of each Worksheet, complete with Answers values of \ ( \mu )! Whom have pet dogs had 200 turns receiving 200 prizes P 5, where a is an integer side. Paper Twitter and the number of blue counters corbett maths probability exam questions the green box and randomly selects a disc from a contains. Her attempts at clearing five metres 45 in the form a P,! Number of orange crayons in a class of 25 students, 5 of whom have a pet.... Gill wants to give corbett maths probability exam questions nephew two sweets randomly from the jar does. Keeps on her kitchen table enter and everyone wins one of the best... Question, give all your Answers as fractions fails the final exam is 0.8! the probability he... Work out the probability that Jane wins the second game is \ ( b\ ) counters! Coin landed on tails during that demonstration contains balls that are red, blue, green yellow. Can tell what Paige is thinking digital format a yellow box on a shelf what she... Will take a sweet of each colour if she takes two right shoes card... Lisa picks a brick at random a wide range of topics, for KS2 KS3! Can be found on www.co... Mr Barton Maths trainers given that he chooses card! On Derrenâs belief that he chooses a cake from a bag holds lego pieces which are all the as! Have one opportunity to retake the exam home to 1000 's of Maths resources videos! Is given in this supermarket contain at most one loaf that is certain to happen is ‘ 1 ’ cap... ) work out the number of blue counters and the raffle many prizes on offer in the box he to. The media were buzzing with comments about these exam-style questions is 0.8! the that! Crayons are either orange, purple or brown ( well, most!,! For more videos, worksheets, 5-a-day and much more the Corbettmaths Practice written. Me to create 3 editable versions of each colour were originally in the bag adds... And 10 yellow bricks, or otherwise, show that the card chosen at random and not... £5 if she selects one that may or may not happen, have pet... Is one of the first two days of the cards she chooses has the choice of wearing a cap is! Now famous Maths exam question: Hannah has 6 orange sweets and some sweets! 10 pupils a biased coin in a one day event consists of 12 five-metre vaults and! 'S nephew Sean has two boxes containing plastic corbett maths probability exam questions bricks of these, the probability that she £1... Her taking 2 orange sweets is \ ( \frac { 7 } { }... 12 five-metre vaults mind reader and can tell what Paige is thinking is 85 % or not in... Do not wear socks is 0.2 + 0.35 ) at the jamboree, complete with.. A stall in Faulton Towers theme park situation with the letter R printed on 6 cards the! The possible corbett maths probability exam questions, \ ( \frac13\ ) the blue box contains three chocolate cakes and three plain.! Are not independent expected gain for this game, the lucky dip it costs £1 to and. An event that is too short has 6 orange sweets and some yellow sweets is 0.25 that: P lisa! Demonstration for year 10 pupils a biased five-sided spinner paperback format is picked at random and not. Is given in this supermarket contain at most one loaf that is too short really one!
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2021-10-16 21:25:18
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http://codechalet.com/docs/flask/development/configuration.html
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# Configuration variables
To connect a SQLite database to Flask, and SQLAlchemy, we only need set a few configuration variables in the Flask app. I usually define my configuration variables in a separate file, config.py. I then import this file into __init__.py. I have noticed that I need a secret key defined to use some features of SQLAlchemy. Best practice is to set this to a secure value.
# config.py
class Config:
SQLALCHEMY_DATABASE_URI = 'sqlite:///{}'.format(db_path)
SQLALCHEMY_TRACK_MODIFICATIONS = False
# __init__.py
def init_app():
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2022-10-01 00:49:06
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https://help.simetrix.co.uk/8.3/simetrix/simulator_reference/topics/digitalmixedsignaldevicereference_digital_analogconverter.htm
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# Digital-Analog Converter
In this topic:
## Netlist entry
Axxxx [ digital_in_0 digital_in_1 .. digital_in_n ]
+ analog_out model_name
## Connection details
Name Description Flow Type Allowed types Vector bounds digital_in Data output in d d 1 - 32 analog_out Analog output out v v, vd, i, id n/a
## Model format
.MODEL model_name da_converter parameters
## Model parameters
Name Description Type Default Limits output_offset Offset voltage real 0 none output_range Input signal range real 1 none twos_complement Use 2's complement input. (Default is offset binary) boolean FALSE none output_slew_time Output slew time real 10nS $1\text{e}^{-12} - \infty$ in_family Input logic family string UNIV none input_load Input load real 1pF $0 - \infty$ sink_current Input sink current real 0 none source_current Input source current real 0 none
## Device Operation
This device is a 1-32 bit digital to analog converter. Its operation is illustrated by the following diagrams.
DAC Waveforms
DAC waveforms expanded to show output slew
The device illustrated above has the following model definition:
.model DAC_4 da_converter
+ output_slew_time 1e-08
+ output_range 5
+ output_offset 0
In offset binary mode the D-A converter produce an output voltage equal to:
-OUTPUT_RANGE/2 + OUTPUT_OFFSET + code * OUTPUT_RANGE/2 n
where n is the number of bits and code is the digital input code represented as an unsigned number between 0 and $2^{n}-1$.
In 2's complement mode the output is:
OUTPUT_OFFSET + code * OUTPUT_RANGE/2 n
where n is the number of bits and code is the digital input code represented as a signed number between $-2^{n/2}$ and $2^{n/2}-1$.
Whenever the input code changes, the output is set on a trajectory to reach the target value in the time specified by OUTPUT_SLEW_TIME. UNKNOWN states are ignored. That is the input will be assumed to be at the most recent known state.
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2020-02-29 04:29:19
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http://mathoverflow.net/api/userquestions.html?userid=4991&page=1&pagesize=10&sort=recent
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9
# Questions
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### Generalized symmetric algebras and Dickson algebras over ${\mathbb F}_p$.
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### Models for P map in EHP sequence
feb 13 11 at 4:05 John Klein 7,2161928
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386
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### The space of compact subspaces of $R^\infty$ homotopy equivalent to a given finite complex.
aug 26 10 at 18:27 Dev Sinha 2,4701020
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### Geometric models for classifying spaces of $GLn(Fq)$.
jul 31 10 at 14:18 Richard Borcherds 14.1k25482
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639
views
### Explicit invariants (under change of basis) of maps $V \to V \otimes V$.
jul 13 10 at 14:18 Tom Goodwillie 26.8k258110
6
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2
640
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### Geometric model for classifying spaces of alternating groups
jul 1 10 at 8:58 Charles Matthews 9,7401239
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1
vote
0
286
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### Ring objects in the category of cocommutative coalgebras (aka Hopf rings).
may 27 10 at 15:13 Dev Sinha 2,4701020
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2013-05-24 01:38:41
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http://lib.convdocs.org/docs/index-233356.html
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# Beam Dynamics and Electromagnetic Fields
Скачать 396.75 Kb.
Название Beam Dynamics and Electromagnetic Fields страница 1/15 Дата конвертации 20.04.2013 Размер 396.75 Kb. Тип Документы
ABSTRACTS SUBMITTED IN SESSION 4 Beam Dynamics and Electromagnetic Fields 915 - RHIC Vertical AC Dipole Commissioning*M. Bai, J. Delong, P. Oddo, C.-I. Pai, S. Peggs, T. Roser, T. Satogata, D. Trbojevic, A. Zaltsman, BNL, Upton, Long Island, New York The vertical RHIC ac dipole was installed in the summer of 2001. The magnet is located in the interaction region between sector 3 and sector 4 and common to both beams. The resonant frequency of the ac dipole was first configured to be around half of the beam revolution frequency to act as a spin flipper. At the end of the RHIC 2002 run, the RHIC vertical ac dipole frequency was reconfigured for beam dynamics studies.A 3~$\sigma$ vertical betatron oscillation was excited with the vertical ac dipole, where $\sigma$ is the rms vertical beam size. RHIC IPM (Ionization Profile Monitor) measurements also confirm that the beam transverse emittance is preserved after the vertical excitation. The betatron functions and phase advances at each BPM~(Beam Position Monitor) around the ring were measured using the coherence excited by the vertical ac dipole. We also used the excited coherence to measure the 2x2 coupling transfer matrix at each BPM location. Both measurements for are presented in the paper along with the analysis algorisms for each of these measurements. The work was performed under the auspices of the US Department of Energy Type of presentation requested : Poster Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport1496 - Adiabatic Matching in Periodic Accelerating Lattices for Superconducting Proton LinacsG. Bellomo, P. Pierini, INFN LASA, Segrate (MI) Superconducting proton Linacs with multicell cavities are split in sections (using cavities with different geometrical length) with a spatially periodic lattice (typically a focusing doublet and 2-4 cavities in a lattice period) and slowly varying focusing and acceleration parameters. The usual matching procedure for a constant parameter lattice, namely periodic conditions on the transverse and longitudinal Twiss parameters gives poor results, especially in the presence of strong space charge forces. A novel matching procedure has been devised, valid for adiabatic variations of the beamline and beam parameters. This procedure gives smooth beam envelope variations along the machine, as well as a smooth phase advance per period. Examples will be given for the case of the superconducting TRASCO Linac. Type of presentation requested : Poster Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport1849 - Recent Progress in Six-Dimensional Ionization Cooling Techniques for Muon-Based Machines*J.S. Berg, R. Palmer, BNL, Upton, Long Island, New York Ionization cooling is an essential component of a neutrino factory or a muon collider. Ionization cooling in the transverse dimensions is reasonably straightforward, and has been incorporated in published neutrino factory studies. Achieving cooling in the longitudinal dimensions is more difficult, but has the potential to greatly improve the performance of neutrino factories, and is essential to muon colliders. Much progress has recently been made in describing ring cooling lattices which achieve cooling in all three phase space planes, and in the design of the required, but difficult, injection systems. Ring cooling lattices also have the potential of significantly reduced cost compared to single-pass cooling systems with comparable performance. We will present some recent lattice designs, describing their theory, features, and performance, including injection and extraction systems. *Work supported by the Department of Energy, contract DE--AC02--98CH10886. Type of presentation requested : Poster Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport1850 - Longitudinal Beam Dynamics in Imperfectly-Ioschronous FFAG Accelerators*J.S. Berg, BNL, Upton, Long Island, New York Using FFAGs for the arcs of recirculating accelerators has the potential to achieve significant cost savings over a multiple-arc design. The problem with such a design is that no FFAG arc will have the same path length over its entire energy range. This leads to problems with synchronizing high-frequency RF with the beam on each pass. It has been demonstrated [1] that in fact a reference particle can be accelerated in such a system for an arbitrary number of turns, although the amount of linac required for a given energy gain never falls below a certain nonzero value for a larger number of turns. Here we examine the longitudinal phase space acceptance of such a system as a function of the number of turns in the accelerator. *Work supported by the Department of Energy, contract DE--AC02--98CH10886. [1] J. Scott Berg. Longitudinal Reference Particle Motion in Nearly Isochronous FFAG Recirculating Accelerators. To appear in the proceedings of Snowmass 2001, 30 Jun-21 Jul 2001, Snowmass Village, CO. Type of presentation requested : Poster Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport1216 - Extraction by Stripping at JINRO. Borisov, JINR, Dubna, Moscow Region Numerical simulation results of the heavy ions extraction by stripping for various magnetic structures, chage-exchange ratio and other parameters are considered. The possibilities and advantage of this extraction method are discussed. Experimental results for U-400 and U-400M cyclotrons are present. Type of presentation requested : Poster Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport250 - Field Quality vs Beam Based Corrections in Large Hadron CollidersO.S. Bruning, CERN, Geneva The paper discusses limits for correcting the magnet field quality via dedicated correction circuits in a collider storage ring and the possibility of adjusting the powering of such correction circuits via beam based measurements. Type of presentation requested : Invited Paper Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport1632 - Measurement and Correction of Imperfections in the SLS Storage RingM. Böge, V. Schlott, A. Streun, PSI, Villigen Based on precise average beta function measurements with errors of ~1 % for the locations of the 174 quadrupoles an SVD based beta beat correction has been applied using the individually powered quadrupoles as correctors. Residual horizontal and vertical beta beats of 4 and 2 % have been measured after correction. Beam based alignment techniques have been applied to determine BPM centers with respect to adjacent quadrupoles. Furthermore the analysis of Corrector/BPM response matrices within a parameterized model gives a detailed understanding of the underlying imperfections. The foreseen SVD based beam based girder alignment adjusts the 48 magnet girders in order to minimize the closed orbit distortions. Linear encoders allow to monitor the movement of BPMs and girders once the imperfections have been corrected. Type of presentation requested : Poster Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport520 - Creation of Hollow Bunches by Redistribution of Phase Space SurfacesC. Carli, M. Chanel, CERN, Geneva The creation of hollow bunches in the longitudinal phase space, in order to decrease the peak current and in turn the transverse direct space charge forces, is an old idea. A new method to create such hollow a distribution at high energy has been simulated and tested experimentally at the PS Booster synchrotron. It is based on a redistribution of surfaces in the longitudinal phase space by using a double harmonic RF system. During the process, the beam is transferred from one second harmonic sub-bucket to the other. Low density phase space surfaces from the periphery and high density regions from the centre are exchanged, leading to flat profiles, even after switching off the second harmonic RF system. During the process, the peak current is temporarily increased, which makes it suitable only to improve the situation in a receiving machine (in our case the PS) after transfer. In practice, the set-up of this new scheme turned out to be fast and simple and to yield reliable and reproducible results. Type of presentation requested : Poster Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport1378 - Coupling Correction for the SNS Accumulator RingN. Catalan-Lasheras, CERN, Geneva; C. Gardner, I. Papaphilippou, G. Parzen, BNL, Upton, Long Island, New York In high intensity machines as the SNS accumulator ring, coupling resonances combined with the space-charge effect may produce excessive emittance growth leading to intolerable beam losses. Several correction schemes have been investigated to achieve local and global decoupling and the correction of vertical dispersion. A minimal set of skew quadrupole correctors and their maximum strength have been defined. The final configuration robustness and sensitivity for all the potential working points has been tested and the results are discussed. Type of presentation requested : Poster Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport270 - Downstream Beamline for the 2-us, 2-kA and 20-MeV DARHT-II*Y.-J. Chen, L. Bertolini, G.J. Caporaso, A. Paul, B. Poole, L.-F. Wang, G.A. Westenskow, LLNL, Livermore The second-axis of the Dual-Axis Radiographic Hydrodynamic Test facility (DARHT-II) is to perform multiple-pulses x-ray flash radiography. The DARHT-II accelerator will provide a 2-kA, 20-MeV and 2-microsecond electron beam with a ± 0.5% energy variation. We have designed a beamline to select several short current pulses from the 2-microsecond beam and to deliver those pulses to a x-ray converter target. The remained beam will be delivered to a beam dump. With the diagnostic beam stop inline, the transport system provides emittance diagnose capability for the beam exiting the accelerator. The beamline consists of several long drift sections. The transverse resistive wall instability and the background gas focusing effects in the long drift sections could be potential problems for maintaining the quality of the long pulse, high current beam. The beam exiting the accelerator has a rise time of 50 - 150 ns with the beam head energy varying from 16 - 20 MeV. Gas desorption caused by beam spill of the long off-energy beam head is also a concern. We will present the beamline configuration and its beam parameter acceptance, the transverse resistive wall instability modeling, the final spot size variation caused by the background vacuum, and the simulations of beam spill. The beam spot size on the target will be discussed. *This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. Type of presentation requested : Poster Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport1231 - Measurement of Optical Functions in HERAW. Decking, B. Holzer, J. Keil, T. Limberg, DESY, Hamburg The linear optics of both HERA electron and proton ring have been measured with several methods, including gradient changes in individual powered quadrupoles, local orbit changes in the arc sextupoles and response matrix fitting. Only a combination of all methods allows a complete understanding of the linear optics. Automated data taking and analysis has proven to be an important tool in the re-commissioning of HERA and revealed several optics errors. A review of the methods and results of the measurements will be presented. Type of presentation requested : Poster Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport1227 - Studies for Beam Lifetime Optimization in DAFNES. Guiducci, INFN-LNF, Frascati (Roma) Since the DAFNE beam lifetime is strongly dominated by the Touschek effect, an analysis of its dependence on the ring dynamic aperture and on machine parameters has been done in order to improve the machine performances. Measurements taken in different conditions are here presented and compared with simulations. The agreement is quite satisfactory when taking into account the measured bunch length and machine coupling, and the actual dynamic aperture. Type of presentation requested : Poster Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport311 - The Influence of Time-Dependent Magnetic Field Errors on the Specifications of Feedback and Collimation SystemsM. Hayes, R.W. Assmann, J. Wenninger, CERN, Geneva The LHC requires an excellent beam stability during all phases of the LHC cycle. This stability must be ensured inspite of the time-dependent field errors in the superconducting main dipoles. The field errors will change significantly during injection and ramp, inducing perturbations of the circulating beam. The known time-dependent field errors of the LHC magnets and their effects on the LHC beam were simulated using latest tracking codes, allowing the prediction of the beam stability during injection and ramp. Using these results the constraints and specifications for beam collimation and the LHC orbit feedback are studied.Type of presentation requested : Poster Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport1148 - A Generalized Orbit Correction SchemeE. Karantzoulis, F. Iazzourene, L. Tosi, ELETTRA, Trieste The special operating conditions of ELETTRA have strongly influenced the orbit correction philosophy. A hybrid orbit correction scheme is presented whereby local orbit corrections at arbitrary positions and angles at three light source points of each of the eleven user dedicated sections are performed that also maintain the global orbit stable. The method, the stability and the implications are presented and discussed. Type of presentation requested : Poster Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport1035 - Ionization Cooling of Muon BeamsE.-S. Kim, PAL, Pohang We present an ionization cooling lattice design which use solenoidal focusing channels and liquid hydrogen absorber wedge for the transverse and longitudinal cooling of muon beams. The cooling performances in the cooling system are investigated. Type of presentation requested : Poster Speaker : Eun-San Kim Classification : [D01] Beam Optics - Lattices, Correction Schemes, Transport420 - Vertical Beam Size Control at SRRCC.-C. Kuo, H.P. Chang, H.J. Tsai, SRRC, Hsinchu Vertical beam size control in the 1.5 GeV synchrotron radiation storage ring at SRRC was investigated. Linear coupling model with measured orbit response was attempted. Manipulation of transverse beam betatron coupling as well as the vertical dispersion correction using a set of skew quadrupoles around the ring has been conducted. The correlation of the coupling strength, vertical beam size, beam brightness, as well as beam lifetime was measured and optimal operation conditions have been searched for the routine users operations.
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2019-04-26 07:43:44
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http://en.wikibooks.org/wiki/Calculus/Print_version
|
# Calculus/Print version
## Differentiation
### Basics of Differentiation
3.5 Higher Order Derivatives - An introduction to second order derivatives
## Integration
The definite integral of a function f(x) from x=0 to x=a is equal to the area under the curve from 0 to a.
### Integration Techniques
From bottom to top:
• an acceleration function a(t);
• the integral of the acceleration is the velocity function v(t);
• and the integral of the velocity is the distance function s(t).
## Appendix
• Choosing delta
# Introduction
Calculus Contributing → Print version
## What is calculus?
Calculus is the broad area of mathematics dealing with such topics as instantaneous rates of change, areas under curves, and sequences and series. Underlying all of these topics is the concept of a limit, which consists of analyzing the behavior of a function at points ever closer to a particular point, but without ever actually reaching that point. As a typical application of the methods of calculus, consider a moving car. It is possible to create a function describing the displacement of the car (where it is located in relation to a reference point) at any point in time as well as a function describing the velocity (speed and direction of movement) of the car at any point in time. If the car were traveling at a constant velocity, then algebra would be sufficient to determine the position of the car at any time; if the velocity is unknown but still constant, the position of the car could be used (along with the time) to find the velocity.
However, the velocity of a car cannot jump from zero to 35 miles per hour at the beginning of a trip, stay constant throughout, and then jump back to zero at the end. As the accelerator is pressed down, the velocity rises gradually, and usually not at a constant rate (i.e., the driver may push on the gas pedal harder at the beginning, in order to speed up). Describing such motion and finding velocities and distances at particular times cannot be done using methods taught in pre-calculus, whereas it is not only possible but straightforward with calculus.
Calculus has two basic applications: differential calculus and integral calculus. The simplest introduction to differential calculus involves an explicit series of numbers. Given the series (42, 43, 3, 18, 34), the differential of this series would be (1, -40, 15, 16). The new series is derived from the difference of successive numbers which gives rise to its name "differential". Rarely, if ever, are differentials used on an explicit series of numbers as done here. Instead, they are derived from a continuous function in a manner which is described later.
Integral calculus, like differential calculus, can also be introduced via series of numbers. Notice that in the previous example, the original series can almost be derived solely from its differential. Instead of taking the difference, however, integration involves taking the sum. Given the first number of the original series, 42 in this case, the rest of the original series can be derived by adding each successive number in its differential (42+1, 43-40, 3+15, 18+16). Note that knowledge of the first number in the original series is crucial in deriving the integral. As with differentials, integration is performed on continuous functions rather than explicit series of numbers, but the concept is still the same. Integral calculus allows us to calculate the area under a curve of almost any shape; in the car example, this enables you to find the displacement of the car based on the velocity curve. This is because the area under the curve is the total distance moved, as we will soon see. Let's understand this section very carefully. Suppose we have to add the numbers in series which is continuously "on" like 23,25,24,25,34,45,46,47, and so on...at this type integral calculation is very useful instead of the typical mathematical formulas.
## Why learn calculus?
Calculus is essential for many areas of science and engineering. Both make heavy use of mathematical functions to describe and predict physical phenomena that are subject to continual change, and this requires the use of calculus. Take our car example: if you want to design cars, you need to know how to calculate forces, velocities, accelerations, and positions. All require calculus. Calculus is also necessary to study the motion of gases and particles, the interaction of forces, and the transfer of energy. It is also useful in business whenever rates are involved. For example, equations involving interest or supply and demand curves are grounded in the language of calculus.
Calculus also provides important tools in understanding functions and has led to the development of new areas of mathematics including real and complex analysis, topology, and non-euclidean geometry.
Notwithstanding calculus' functional utility (pun intended), many non-scientists and non-engineers have chosen to study calculus just for the challenge of doing so. A smaller number of persons undertake such a challenge and then discover that calculus is beautiful in and of itself.
## What is involved in learning calculus?
Learning calculus, like much of mathematics, involves two parts:
• Understanding the concepts: You must be able to explain what it means when you take a derivative rather than merely apply the formulas for finding a derivative. Otherwise, you will have no idea whether or not your solution is correct. Drawing diagrams, for example, can help clarify abstract concepts.
• Symbolic manipulation: Like other branches of mathematics, calculus is written in symbols that represent concepts. You will learn what these symbols mean and how to use them. A good working knowledge of trigonometry and algebra is a must, especially in integral calculus. Sometimes you will need to manipulate expressions into a usable form before it is possible to perform operations in calculus.
## What you should know before using this text
There are some basic skills that you need before you can use this text. Continuing with our example of a moving car:
• You will need to describe the motion of the car in symbols. This involves understanding functions.
• You need to manipulate these functions. This involves algebra.
• You need to translate symbols into graphs and vice-versa. This involves understanding the graphing of functions.
• It also helps (although it isn't necessarily essential) if you understand the functions used in trigonometry since these functions appear frequently in science.
## Scope
The first four chapters of this textbook cover the topics taught in a typical high school or first year college course. The first chapter, Precalculus, reviews those aspects of functions most essential to the mastery of calculus. The second, Limits, introduces the concept of the limit process. It also discusses some applications of limits and proposes using limits to examine slope and area of functions. The next two chapters, Differentiation and Integration, apply limits to calculate derivatives and integrals. The Fundamental Theorem of Calculus is used, as are the essential formulae for computation of derivatives and integrals without resorting to the limit process. The third and fourth chapters include articles that apply the concepts previously learned to calculating volumes, and so on as well as other important formulae.
The remainder of the central Calculus chapters cover topics taught in higher-level calculus topics: multivariable calculus, vectors, and series (Taylor, convergent, divergent).
Finally, the other chapters cover the same material, using formal notation. They introduce the material at a much faster pace, and cover many more theorems than the other two sections. They assume knowledge of some set theory and set notation.
Calculus Contributing → Print version
# Precalculus
<h1> 1.1 Algebra</h1>
← Precalculus Calculus Functions → Print version
This section is intended to review algebraic manipulation. It is important to understand algebra in order to do calculus. If you have a good knowledge of algebra, you should probably just skim this section to be sure you are familiar with the ideas.
## Rules of arithmetic and algebra
The following laws are true for all a, b, and c, whether a, b, and c are numbers, variables, functions, or more complex expressions involving numbers, variable and/or functions.
• Commutative Law: $a+b=b+a \,$.
• Associative Law: $(a+b)+c=a+(b+c)\,$.
• Additive Identity: $a+0=a\,$.
• Additive Inverse: $a+(-a)=0\,$.
### Subtraction
• Definition: $a-b = a+(-b)\,$.
### Multiplication
• Commutative law: $a\times b=b\times a$.
• Associative law: $(a\times b)\times c=a\times (b\times c)\,$.
• Multiplicative identity: $a\times 1=a\,$.
• Multiplicative inverse: $a\times \frac{1}{a}=1$, whenever $a \neq 0\,$
• Distributive law: $a\times (b+c)=(a\times b)+(a\times c)\,$.
### Division
• Definition: $\frac{a}{b}=a\times \frac{1}{b}$, whenever $b \neq 0\,$.
Let's look at an example to see how these rules are used in practice.
$\frac{(x+2)(x+3)}{x+3}$ = $\left[(x+2)\times (x+3)\right]\times \left( \frac{1}{x+3}\right)$ (from the definition of division) = $(x+2)\times \left[(x+3)\times \left(\frac{1}{x+3} \right) \right]$ (from the associative law of multiplication) = $((x+2)\times (1)),\qquad x \neq -3 \,$ (from multiplicative inverse) = $x+2, \qquad x \neq -3.$ (from multiplicative identity)
Of course, the above is much longer than simply cancelling $x+3$ out in both the numerator and denominator. But, when you are cancelling, you are really just doing the above steps, so it is important to know what the rules are so as to know when you are allowed to cancel. Occasionally people do the following, for instance, which is incorrect:
$\frac{2\times (x + 2)}{2}= \frac{2}{2} \times \frac{x+2}{2}=1 \times \frac{x+2}{2}= \frac{x+2}{2}$.
The correct simplification is
$\frac{2\times (x + 2)}{2}= \left( 2 \times \frac{1}{2} \right) \times (x+2)=1 \times (x+2)=x+2$,
where the number $2$ cancels out in both the numerator and the denominator.
## Interval notation
There are a few different ways that one can express with symbols a specific interval (all the numbers between two numbers). One way is with inequalities. If we wanted to denote the set of all numbers between, say, 2 and 4, we could write "all x satisfying 2<x<4." This excludes the endpoints 2 and 4 because we use < instead of $\leq$. If we wanted to include the endpoints, we would write "all x satisfying $2 \leq x \leq 4$." This includes the endpoints.
Another way to write these intervals would be with interval notation. If we wished to convey "all x satisfying 2<x<4" we would write (2,4). This does not include the endpoints 2 and 4. If we wanted to include the endpoints we would write [2,4]. If we wanted to include 2 and not 4 we would write [2,4); if we wanted to exclude 2 and include 4, we would write (2,4].
Thus, we have the following table:
Endpoint conditions Inequality notation Interval notation
Including both 2 and 4 all x satisfying $2 \leq x \leq 4$
$[2,4] \,\!$
Not including 2 nor 4 all x satisfying $2
$(2,4) \,\!$
Including 2 not 4 all x satisfying $2 \leq x < 4$
$[2,4) \,\!$
Including 4 not 2 all x satisfying $2 < x \leq 4$
$(2,4] \,\!$
In general, we have the following table:
Meaning Interval Notation Set Notation
All values greater than or equal to $a$ and less than or equal to $b$ $\left[a,b\right]$ $\left\{x:a\le x\le b\right\}$
All values greater than $a$ and less than $b$ $\left(a,b\right)$ $\left\{x:a < x < b\right\}$
All values greater than or equal to $a$ and less than $b$ $\left[a,b\right)$ $\left\{x:a\le x < b\right\}$
All values greater than $a$ and less than or equal to $b$ $\left(a,b\right]$ $\left\{x:a < x\le b\right\}$
All values greater than or equal to $a$. $\left[a,\infty\right)$ $\left\{x:x\ge a\right\}$
All values greater than $a$. $\left(a,\infty\right)$ $\left\{x:x > a\right\}$
All values less than or equal to $a$. $\left(-\infty,a\right]$ $\left\{x:x\le a\right\}$
All values less than $a$. $\left(-\infty,a\right)$ $\left\{x:x < a\right\}$
All values. $\left(-\infty,\infty\right)$ $\left\{x: x\in\mathbb{R}\right\}$
Note that $\infty$ and $-\infty$ must always have an exclusive parenthesis rather than an inclusive bracket. This is because $\infty$ is not a number, and therefore cannot be in our set. $\infty$ is really just a symbol that makes things easier to write, like the intervals above.
The interval (a,b) is called an open interval, and the interval [a,b] is called a closed interval.
Intervals are sets and we can use set notation to show relations between values and intervals. If we want to say that a certain value is contained in an interval, we can use the symbol $\in$ to denote this. For example, $2\in[1,3]$. Likewise, the symbol $\notin$ denotes that a certain element is not in an interval. For example $0\notin(0,1)$.
There are a few rules and properties involving exponents and radicals that you'd do well to remember. As a definition we have that if n is a positive integer then $a^n$ denotes n factors of a. That is,
$a^n = a\cdot a \cdot a \cdots a \qquad (n~ \mbox{times}).$
If $a \not= 0$ then we say that $a^0 =1 \,$.
If n is a negative integer then we say that $a^{-n} = \frac{1}{a^n} .$
If we have an exponent that is a fraction then we say that $a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m .$
In addition to the previous definitions, the following rules apply:
Rule Example
$a^n \cdot a^m = a^{n+m}$ $3^6 \cdot 3^9 = 3^{15}$
$\frac{a^n}{a^m} = a^{n-m}$ $\frac{x^3}{x^2} = x^{1} = x$
$(a^n)^m = a^{n\cdot m}$ $(x^4)^5 = x^{20} \,\!$
$(ab)^n = a^n b^n \,\!$ $(3x)^5 = 3^5 x^5 \,\!$
$\bigg(\frac{a}{b}\bigg)^n = \frac{a^n}{b^n}$ $\bigg(\frac{7}{3}\bigg)^3 = \frac{7^3}{3^3}.$
## Factoring and roots
Given the expression $x^2 + 3x + 2$, one may ask "what are the values of x that make this expression 0?" If we factor we obtain
$x^2 + 3x + 2 = (x + 2)(x + 1). \,\!$
If x=-1 or -2, then one of the factors on the right becomes zero. Therefore, the whole must be zero. So, by factoring we have discovered the values of x that render the expression zero. These values are termed "roots." In general, given a quadratic polynomial $px^2 + qx + r$ that factors as
$px^2 + qx + r = (ax + c)(bx + d) \,\!$
then we have that x = -c/a and x = -d/b are roots of the original polynomial.
A special case to be on the look out for is the difference of two squares, $a^2 - b^2$. In this case, we are always able to factor as
$a^2 - b^2 = (a+b)(a-b). \,\!$
For example, consider $4x^2 - 9$. On initial inspection we would see that both $4x^2$ and $9$ are squares ($(2x)^2 = 4x^2$ and $3^2 = 9$). Applying the previous rule we have
$4x^2 - 9 = (2x+3)(2x-3). \,\!$
The following is a general result of great utility.
Given any quadratic equation $ax^2+bx+c=0, a\neq0$, all solutions of the equation are given by the quadratic formula:
$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.
Example: Find all the roots of $4x^2+7x-2$ Finding the roots is equivalent to solving the equation $4x^2+7x-2=0$. Applying the quadratic formula with $a=4, b=7, c=-2$, we have: $x=\frac{-7\pm\sqrt{7^2-4(4)(-2)}}{2(4)}$ $x=\frac{-7\pm\sqrt{49+32}}{8}$ $x=\frac{-7\pm\sqrt{81}}{8}$ $x=\frac{-7\pm9}{8}$ $x=\frac{2}{8}, x=\frac{-16}{8}$ $x=\frac{1}{4}, x=-2$
The quadratic formula can also help with factoring, as the next example demonstrates.
Example: Factor the polynomial $4x^2+7x-2$ We already know from the previous example that the polynomial has roots $x=\frac{1}{4}$ and $x=-2$. Our factorization will take the form $C(x+2)(x-\frac{1}{4})$ All we have to do is set this expression equal to our polynomial and solve for the unknown constant C: $C(x+2)(x-\frac{1}{4})=4x^2+7x-2$ $C(x^2+(-\frac{1}{4}+2)x-\frac{2}{4})=4x^2+7x-2$ $C(x^2+\frac{7}{4}x-\frac{1}{2})=4x^2+7x-2$ You can see that $C=4$ solves the equation. So the factorization is $4x^2+7x-2=4(x+2)(x-\frac{1}{4})=(x+2)(4x-1)$
Note that if $4ac>b^2$ then the roots will not be real numbers.
## Simplifying rational expressions
Consider the two polynomials
$p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$
and
$q(x) = b_m x^m + b_{m-1}x^{m-1} + \cdots + b_1x + b_0.$
When we take the quotient of the two we obtain
$\frac{p(x)}{q(x)} = \frac{a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0}{b_m x^m + b_{m-1}x^{m-1} + \cdots + b_1x + b_0}.$
The ratio of two polynomials is called a rational expression. Many times we would like to simplify such a beast. For example, say we are given $\frac{x^2-1}{x+1}.$ We may simplify this in the following way:
$\frac{x^2-1}{x+1} = \frac{(x+1)(x-1)}{x+1} = x-1, \qquad x \neq -1 \,\!$
This is nice because we have obtained something we understand quite well, $x-1$, from something we didn't.
## Formulas of multiplication of polynomials
Here are some formulas that can be quite useful for solving polynomial problems:
$(a+b)^2=a^2+2ab+b^2$
$(a-b)^2=a^2-2ab+b^2$
$(a-b)(a+b)=a^2-b^2$
$(a\pm b)^3=a^3\pm 3a^2b+3ab^2\pm b^3$
$a^3\pm b^3=(a\pm b)(a^2\mp ab+b^2)$
## Polynomial Long Division
Suppose we would like to divide one polynomial by another. The procedure is similar to long division of numbers and is illustrated in the following example:
### Example
Divide $x^2-2x-15$ (the dividend or numerator) by $x+3$ (the divisor or denominator) Similar to long division of numbers, we set up our problem as follows: $\begin{array}{rl}\\ x+3\!\!\!\!&\big)\!\!\!\begin{array}{lll} \hline \,x^2-2x-15 \end{array}\end{array}$ First we have to answer the question, how many times does $x+3$ go into $x^2$? To find out, divide the leading term of the dividend by leading term of the divisor. So it goes in $x$ times. We record this above the leading term of the dividend: $\begin{array}{rl}&~~\,x\\ x+3\!\!\!\!&\big)\!\!\!\begin{array}{lll} \hline \,x^2-2x-15 \end{array}\\ \end{array}$ , and we multiply $x+3$ by $x$ and write this below the dividend as follows: $\begin{array}{rl}&~~\,x\\ x+3\!\!\!\!&\big)\!\!\!\begin{array}{lll} \hline \,x^2-2x-15 \end{array}\\ &\!\!\!\!-\underline{(x^2+3x)~~~}\\ \end{array}$ Now we perform the subtraction, bringing down any terms in the dividend that aren't matched in our subtrahend: $\begin{array}{rl}&~~\,x\\ x+3\!\!\!\!&\big)\!\!\!\begin{array}{lll} \hline \,x^2-2x-15 \end{array}\\ &\!\!\!\!-\underline{(x^2+3x)~~~}\\ &\!\!\!\!~~~~~~-5x-15~~~\\ \end{array}$ Now we repeat, treating the bottom line as our new dividend: $\begin{array}{rl}&~~\,x-5\\ x+3\!\!\!\!&\big)\!\!\!\begin{array}{lll} \hline \,x^2-2x-15 \end{array}\\ &\!\!\!\!-\underline{(x^2+3x)~~~}\\ &\!\!\!\!~~~~~~-5x-15~~~\\ &\!\!\!\!~~~-\underline{(-5x-15)~~~}\\ &\!\!\!\!~~~~~~~~~~~~~~~~~~~0~~~\\ \end{array}$ In this case we have no remainder.
### Application: Factoring Polynomials
We can use polynomial long division to factor a polynomial if we know one of the factors in advance. For example, suppose we have a polynomial $P(x)$ and we know that $r$ is a root of $P$. If we perform polynomial long division using P(x) as the dividend and $(x-r)$ as the divisor, we will obtain a polynomial $Q(x)$ such that $P(x)=(x-r)Q(x)$, where the degree of $Q$ is one less than the degree of $P$.
### Exercise
1. Factor $x-1$ out of $6x^3-4x^2+3x-5$.
$(x-1)(6x^2+2x+5)$
Solution
### Application: Breaking up a rational function
Similar to the way one can convert an improper fraction into an integer plus a proper fraction, one can convert a rational function $P(x)$ whose numerator $N(x)$ has degree $n$ and whose denominator $D(x)$ has degree $d$ with $n\geq d$ into a polynomial plus a rational function whose numerator has degree $\nu$ and denominator has degree $\delta$ with $\nu<\delta$.
Suppose that $N(x)$ divided by $D(x)$ has quotient $Q(x)$ and remainder $R(x)$. That is
$N(x)=D(x)Q(x)+R(x)$
Dividing both sides by $D(x)$ gives
$\frac{N(x)}{D(x)}=Q(x)+\frac{R(x)}{D(x)}$
$R(x)$ will have degree less than $D(x)$.
#### Example
Write $\frac{x-1}{x-3}$ as a polynomial plus a rational function with numerator having degree less than the denominator. $\begin{array}{rl}&~~\,1\\ x-3\!\!\!\!&\big)\!\!\!\begin{array}{lll} \hline \,x-1 \end{array}\\ &\!\!\!\!-\underline{(x-3)~~~}\\ &\!\!\!\!~~~~~~~~~2~~~\\ \end{array}$ so $\frac{x-1}{x-3}=1+\frac{2}{x-3}$
← Precalculus Calculus Functions → Print version
<h1> 1.2 Functions</h1>
← Algebra Calculus Graphing linear functions → Print version
## What functions are and how are they described
Note: This is an attempt at a rewrite of "Classical understanding of functions". If others approve, consider deleting that section.
Whenever one quantity is uniquely determined by the value of another quantity, we have a function. You can think of a function as a kind of machine. You feed the machine raw materials, and the machine changes the raw materials into a finished product.
A function in everyday life Think about dropping a ball from a bridge. At each moment in time, the ball is a height above the ground. The height of the ball is a function of time. It was the job of physicists to come up with a formula for this function. This type of function is called real-valued since the "finished product" is a number (or, more specifically, a real number).
A function in everyday life (Preview of Multivariable Calculus) Think about a wind storm. At different places, the wind can be blowing in different directions with different intensities. The direction and intensity of the wind can be thought of as a function of position. This is a function of two real variables (a location is described by two values - an $x$ and a $y$) which results in a vector (which is something that can be used to hold a direction and an intensity). These functions are studied in multivariable calculus (which is usually studied after a one year college level calculus course).This a vector-valued function of two real variables.
We will be looking at real-valued functions until studying multivariable calculus. Think of a real-valued function as an input-output machine; you give the function an input, and it gives you an output which is a number (more specifically, a real number). For example, the squaring function takes the input 4 and gives the output value 16. The same squaring function takes the input -1 and gives the output value 1.
There are many ways which people describe functions. In the examples above, a verbal descriptions is given (the height of the ball above the earth as a function of time). Here is a list of ways to describe functions. The top three listed approaches to describing functions are the most popular and you could skip the rest if you like.
1. A function is given a name (such as $f$) and a formula for the function is also given. For example, $f(x) = 3 x + 2$ describes a function. We refer to the input as the argument of the function (or the independent variable), and to the output as the value of the function at the given argument.
2. A function is described using an equation and two variables. One variable is for the input of the function and one is for the output of the function. The variable for the input is called the independent variable. The variable for the output is called the dependent variable. For example, $y = 3 x + 2$ describes a function. The dependent variable appears by itself on the left hand side of equal sign.
3. A verbal description of the function.
When a function is given a name (like in number 1 above), the name of the function is usually a single letter of the alphabet (such as $f$ or $g$). Some functions whose names are multiple letters (like the sine function $y=sin(x)$.
Plugging a value into a function If we write $f(x) = 3x+2 \$, then we know that The function $f$ is a function of $x$. To evaluate the function at a certain number, replace the $x$ with that number. Replacing $x$ with that number in the right side of the function will produce the function's output for that certain input. In English, the definition of $f \$ is interpreted, "Given a number, $f$ will return two more than the triple of that number." How would we know the value of the function $f$ at 3? We would have the following three thoughts: $f(3) = 3(3) + 2$ $3(3) + 2 = 9 + 2$ $9+2=11$ and we would write $f(3) = 3(3)+2 = 9+2 = 11$. The value of $f \$ at 3 is 11. Note that $f(3) \$ means the value of the dependent variable when $x \$ takes on the value of 3. So we see that the number 11 is the output of the function when we give the number 3 as the input. People often summarize the work above by writing "the value of $f$ at three is eleven", or simply "$f$ of three equals eleven".
## Classical understanding of functions
To provide the classical understanding of functions, think of a function as a kind of machine. You feed the machine raw materials, and the machine changes the raw materials into a finished product based on a specific set of instructions. The kinds of functions we consider here, for the most part, take in a real number, change it in a formulaic way, and give out a real number (possibly the same as the one it took in). Think of this as an input-output machine; you give the function an input, and it gives you an output. For example, the squaring function takes the input 4 and gives the output value 16. The same squaring function takes the input $-1$ and gives the output value 1.
A function is usually written as $f$, $g$, or something similar - although it doesn't have to be. A function is always defined as "of a variable" which tells us what to replace in the formula for the function.
For example, $f(x) = 3x+2 \$ tells us:
• The function $f$ is a function of $x$.
• To evaluate the function at a certain number, replace the $x$ with that number.
• Replacing $x$ with that number in the right side of the function will produce the function's output for that certain input.
• In English, the definition of $f \$ is interpreted, "Given a number, $f$ will return two more than the triple of that number."
Thus, if we want to know the value (or output) of the function at 3:
$f(x) = 3x+2 \$
$f(3) = 3(3)+2 \$ We evaluate the function at $x = 3$.
$f(3) = 9+2 = 11 \$ The value of $f \$ at 3 is 11.
See? It's easy!
Note that $f(3) \$ means the value of the dependent variable when $x \$ takes on the value of 3. So we see that the number 11 is the output of the function when we give the number 3 as the input. We refer to the input as the argument of the function (or the independent variable), and to the output as the value of the function at the given argument (or the dependent variable). A good way to think of it is the dependent variable $f(x) \$ 'depends' on the value of the independent variable $x \$. This is read as "the value of $f$ at three is eleven", or simply "$f$ of three equals eleven".
## Notation
Functions are used so much that there is a special notation for them. The notation is somewhat ambiguous, so familiarity with it is important in order to understand the intention of an equation or formula.
Though there are no strict rules for naming a function, it is standard practice to use the letters $f$, $g$, and $h$ to denote functions, and the variable $x$ to denote an independent variable. $y$ is used for both dependent and independent variables.
When discussing or working with a function $f$, it's important to know not only the function, but also its independent variable $x$. Thus, when referring to a function $f$, you usually do not write $f$, but instead $f(x)$. The function is now referred to as "$f$ of $x$". The name of the function is adjacent to the independent variable (in parentheses). This is useful for indicating the value of the function at a particular value of the independent variable. For instance, if
$f(x)=7x+1\,$,
and if we want to use the value of $f$ for $x$ equal to $2$, then we would substitute 2 for $x$ on both sides of the definition above and write
$f(2)=7(2)+1=14+1=15\,$
This notation is more informative than leaving off the independent variable and writing simply '$f$', but can be ambiguous since the parentheses can be misinterpreted as multiplication.
## Modern understanding of functions
The formal definition of a function states that a function is actually a rule that associates elements of one set called the domain of the function, with the elements of another set called the range of the function. For each value we select from the domain of the function, there exists exactly one corresponding element in the range of the function. The definition of the function tells us which element in the range corresponds to the element we picked from the domain. Classically, the element picked from the domain is pictured as something that is fed into the function and the corresponding element in the range is pictured as the output. Since we "pick" the element in the domain whose corresponding element in the range we want to find, we have control over what element we pick and hence this element is also known as the "independent variable". The element mapped in the range is beyond our control and is "mapped to" by the function. This element is hence also known as the "dependent variable", for it depends on which independent variable we pick. Since the elementary idea of functions is better understood from the classical viewpoint, we shall use it hereafter. However, it is still important to remember the correct definition of functions at all times.
To make it simple, for the function $f(x)$, all of the possible $x$ values constitute the domain, and all of the values $f(x)$ ($y$ on the x-y plane) constitute the range.
## Remarks
The following arise as a direct consequence of the definition of functions:
1. By definition, for each "input" a function returns only one "output", corresponding to that input. While the same output may correspond to more than one input, one input cannot correspond to more than one output. This is expressed graphically as the vertical line test: a line drawn parallel to the axis of the dependent variable (normally vertical) will intersect the graph of a function only once. However, a line drawn parallel to the axis of the independent variable (normally horizontal) may intersect the graph of a function as many times as it likes. Equivalently, this has an algebraic (or formula-based) interpretation. We can always say if $a = b$, then $f(a) = f(b)$, but if we only know that $f(a) = f(b)$ then we can't be sure that $a= b$.
2. Each function has a set of values, the function's domain, which it can accept as input. Perhaps this set is all positive real numbers; perhaps it is the set {pork, mutton, beef}. This set must be implicitly/explicitly defined in the definition of the function. You cannot feed the function an element that isn't in the domain, as the function is not defined for that input element.
3. Each function has a set of values, the function's range, which it can output. This may be the set of real numbers. It may be the set of positive integers or even the set {0,1}. This set, too, must be implicitly/explicitly defined in the definition of the function.
This is an example of an expression which fails the vertical line test.
## The vertical line test
The vertical line test, mentioned in the preceding paragraph, is a systematic test to find out if an equation involving $x$ and $y$ can serve as a function (with $x$ the independent variable and $y$ the dependent variable). Simply graph the equation and draw a vertical line through each point of the $x$-axis. If any vertical line ever touches the graph at more than one point, then the equation is not a function; if the line always touches at most one point of the graph, then the equation is a function.
(There are a lot of useful curves, like circles, that aren't functions (see picture). Some people call these graphs with multiple intercepts, like our circle, "multi-valued functions"; they would refer to our "functions" as "single-valued functions".)
## Important functions
Constant function $f(x)=c\,$ It disregards the input and always outputs the constant $c$, and is a polynomial of the zeroth degree where f(x) = cx0= c(1) = c. Its graph is a horizontal line. Linear function $f(x)=mx+c\,$ Takes an input, multiplies by m and adds c. It is a polynomial of the first degree. Its graph is a line (slanted, except $m=0$). Identity function $f(x)=x\,$ Takes an input and outputs it unchanged. A polynomial of the first degree, f(x) = x1 = x. Special case of a linear function. Quadratic function $f(x)=ax^2+bx+c \,$ A polynomial of the second degree. Its graph is a parabola, unless $a=0$. (Don't worry if you don't know what this is.) Polynomial function $f(x)=a_n x^n + a_{n-1}x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0$ The number $n$ is called the degree. Signum function $\operatorname{sgn}(x) = \left\{ \begin{matrix} -1 & \text{if} & x < 0 \\ 0 & \text{if} & x = 0 \\ 1 & \text{if} & x > 0. \end{matrix} \right.$ Determines the sign of the argument $x$.
## Example functions
Some more simple examples of functions have been listed below.
$h(x)=\left\{\begin{matrix}1,&\mbox{if }x>0\\-1,&\mbox{if }x<0\end{matrix}\right.$ Gives 1 if input is positive, -1 if input is negative. Note that the function only accepts negative and positive numbers, not $0$. Mathematics describes this condition by saying $0$ is not in the domain of the function. $g(y)=y^2\,$ Takes an input and squares it. $g(z)=z^2\,$ Exactly the same function, rewritten with a different independent variable. This is perfectly legal and sometimes done to prevent confusion (e.g. when there are already too many uses of $x$ or $y$ in the same paragraph.) $f(x)=\left\{\begin{matrix}5^{x^2},&\mbox{if }x>0\\0,&\mbox{if }x\le0\end{matrix}\right.$ Note that we can define a function by a totally arbitrary rule. Such functions are called piecewise functions.
It is possible to replace the independent variable with any mathematical expression, not just a number. For instance, if the independent variable is itself a function of another variable, then it could be replaced with that function. This is called composition, and is discussed later.
## Manipulating functions
### Addition, Subtraction, Multiplication and Division of functions
For two real-valued functions, we can add the functions, multiply the functions, raised to a power, etc.
Example: Adding, subtracting, multiplying and dividing functions which do not have a name If we add the functions $y = 3 x + 2$ and $y = x^2$, we obtain $y = x^2 + 3 x + 2$. If we subtract $y = 3 x + 2$ from $y = x^2$, we obtain $y =x^2 - (3 x + 2)$. We can also write this as $y=x^2-3x-2$. If we multiply the function $y = 3 x + 2$ and the function $y = x^2$, we obtain $y = (3 x + 2) x^2$. We can also write this as $y=3x^3 + 2 x^2$. If we divide the function $y = 3 x + 2$ by the function $y = x^2$, we obtain $y = (3 x + 2)/ x^2$.
If a math problem wants you to add two functions $f$ and $g$, there are two ways that the problem will likely be worded:
1. If you are told that $f(x) = 3 x + 2$, that $g(x) = x^2$, that $h(x) = f(x)+g(x)$ and asked about $h$, then you are being asked to add two functions. Your answer would be $h(x) = x^2 + 3 x + 2$.
2. If you are told that $f(x) = 3 x + 2$, that $g(x) = x^2$ and you are asked about $f+g$, then you are being asked to add two functions. The addition of $f$ and $g$ is called $f+g$. Your answer would be $(f+g)(x) = x^2 + 3 x + 2$.
Similar statements can be made for subtraction, multiplication and division.
Example: Adding, subtracting, multiplying and dividing functions which do have a name Let $f(x)=3x+2\,$ and:$g(x)=x^2\,$. Let's add, subtract, multiply and divide. \begin{align} (f+g)(x) &= f(x)+g(x)\\ &= (3x+2)+(x^2)\\ &= x^2+3x+2\, \end{align}, \begin{align} (f-g)(x) &= f(x)-g(x)\\ &= (3x+2)-(x^2)\\ &= -x^2+3x+2\, \end{align}, \begin{align} (f\times g)(x) &= f(x)\times g(x)\\ &= (3x+2)\times(x^2)\\ &= 3x^3+2x^2\, \end{align}, \begin{align} \left(\frac{f}{g}\right)(x) &= \frac{f(x)}{g(x)}\\ &= \frac{3x+2}{x^2}\\ &= \frac{3}{x}+\frac{2}{x^2} \end{align}.
### Composition of functions
We begin with a fun (and not too complicated) application of composition of functions before we talk about what composition of functions is.
Example: Dropping a ball If we drop a ball from a bridge which is 20 meters above the ground, then the height of our ball above the earth is a function of time. The physicists tell us that if we measure time in seconds and distance in meters, then the formula for height in terms of time is $h = -4.9t^2 + 20$. Suppose we are tracking the ball with a camera and always want the ball to be in the center of our picture. Suppose we have $\theta=f(h)$ The angle will depend upon the height of the ball above the ground and the height above the ground depends upon time. So the angle will depend upon time. This can be written as $\theta = f(-4.9t^2 + 20)$. We replace $h$ with what it is equal to. This is the essence of composition.
Composition of functions is another way to combine functions which is different from addition, subtraction, multiplication or division.
The value of a function $f$ depends upon the value of another variable $x$; however, that variable could be equal to another function $g$, so its value depends on the value of a third variable. If this is the case, then the first variable is a function $h$ of the third variable; this function ($h$) is called the composition of the other two functions ($f$ and $g$).
Example: Composing two functions Let $f(x)=3x+2\,$ and:$g(x)=x^2\,$. The composition of $f$ with $g$ is read as either "f composed with g" or "f of g of x." Let $h(x) = f(g(x))$ Then \begin{align} h(x) &= f(g(x))\\ &= f(x^2)\\ &= 3(x^2)+2\\ &= 3x^2+2\, \end{align}. Sometimes a math problem asks you compute $(f \circ g)(x)$ when they want you to compute $f(g(x))$, Here, $h$ is the composition of $f$ and $g$ and we write $h=f\circ g$. Note that composition is not commutative: $f(g(x))=3x^2+2\,$, and \begin{align} g(f(x)) &= g(3x + 2)\\ &= (3x + 2)^2\\ &= 9x^2+12x+4\, . \end{align} so $f(g(x))\ne g(f(x))\,$.
Composition of functions is very common, mainly because functions themselves are common. For instance, squaring and sine are both functions:
$\operatorname{square}(x)=x^2$,
$\operatorname{sine}(x)=\sin x$
Thus, the expression $\sin^2x$ is a composition of functions:
$\sin^2x$ = $\operatorname{square}(\sin x)$ = $\operatorname{square}( \operatorname{sine}(x))$.
(Note that this is not the same as $\operatorname{sine}(\operatorname{square}(x))=\sin x^2$.) Since the function sine equals $1/2$ if $x=\pi/6$,
$\operatorname{square}(\operatorname{sine}(\pi/6))= \operatorname{square}(1/2)$.
Since the function square equals $1/4$ if $x=\pi/6$,
$\sin^2 \pi/6=\operatorname{square}(\operatorname{sine}(\pi/6))=\operatorname{square}(1/2) =1/4$.
### Transformations
Transformations are a type of function manipulation that are very common. They consist of multiplying, dividing, adding or subtracting constants to either the input or the output. Multiplying by a constant is called dilation and adding a constant is called translation. Here are a few examples:
$f(2\times x) \,$ Dilation
$f(x+2)\,$ Translation
$2\times f(x) \,$ Dilation
$2+f(x)\,$ Translation
Examples of horizontal and vertical translations
Examples of horizontal and vertical dilations
Translations and dilations can be either horizontal or vertical. Examples of both vertical and horizontal translations can be seen at right. The red graphs represent functions in their 'original' state, the solid blue graphs have been translated (shifted) horizontally, and the dashed graphs have been translated vertically.
Dilations are demonstrated in a similar fashion. The function
$f(2\times x) \,$
has had its input doubled. One way to think about this is that now any change in the input will be doubled. If I add one to $x$, I add two to the input of $f$, so it will now change twice as quickly. Thus, this is a horizontal dilation by $\frac{1}{2}$ because the distance to the $y$-axis has been halved. A vertical dilation, such as
$2\times f(x) \,$
is slightly more straightforward. In this case, you double the output of the function. The output represents the distance from the $x$-axis, so in effect, you have made the graph of the function 'taller'. Here are a few basic examples where $a$ is any positive constant:
Original graph $f(x)\,$ Rotation about origin $-f(-x)\,$ Horizontal translation by $a$ units left $f(x+a)\,$ Horizontal translation by $a$ units right $f(x-a)\,$ Horizontal dilation by a factor of $a$ $f(x\times \frac{1}{a}) \,$ Vertical dilation by a factor of $a$ $a\times f(x) \,$ Vertical translation by $a$ units down $f(x)-a\,$ Vertical translation by $a$ units up $f(x)+a\,$ Reflection about $x$-axis $-f(x)\,$ Reflection about $y$-axis $f(-x)\,$
## Domain and Range
### Domain
The domain of the function is the interval from -1 to 1
The domain of a function is the set of all points over which it is defined. More simply, it represents the set of x-values which the function can accept as input. For instance, if
$f(x)=\sqrt{1-x^2}$
then $f(x)$ is only defined for values of $x$ between $-1$ and $1$, because the square root function is not defined (in real numbers) for negative values. Thus, the domain, in interval notation, is $\left[-1,1\right]$. In other words,
$f(x) \mbox{is defined for } x\in [-1,1], \operatorname{ or } \{x:-1\le x\le 1\}$.
The range of the function is the interval from 0 to 1
### Range
The range of a function is the set of all values which it attains (i.e. the y-values). For instance, if:
$f(x)=\sqrt{1-x^2}$,
then $f(x)$ can only equal values in the interval from $0$ to $1$. Thus, the range of $f$ is $\left[0,1\right]$.
### One-to-one Functions
A function $f(x)$ is one-to-one (or less commonly injective) if, for every value of $f$, there is only one value of $x$ that corresponds to that value of $f$. For instance, the function $f(x)=\sqrt{1-x^2}$ is not one-to-one, because both $x=1$ and $x=-1$ result in $f(x)=0$. However, the function $f(x)=x+2$ is one-to-one, because, for every possible value of $f(x)$, there is exactly one corresponding value of $x$. Other examples of one-to-one functions are $f(x)=x^3+ax$, where $a\in \left[0,\infty\right)$. Note that if you have a one-to-one function and translate or dilate it, it remains one-to-one. (Of course you can't multiply $x$ or $f$ by a zero factor).
#### Horizontal Line Test
If you know what the graph of a function looks like, it is easy to determine whether or not the function is one-to-one. If every horizontal line intersects the graph in at most one point, then the function is one-to-one. This is known as the Horizontal Line Test.
#### Algebraic 1-1 Test
You can also show one-to-oneness algebraically by assuming that two inputs give the same output and then showing that the two inputs must have been equal. For example, Is $f(x)=\frac{1-2x}{1+x}\,$ a 1-1 function?
$f(a)=f(b)\,$
$\frac{1-2a}{1+a}=\frac{1-2b}{1+b} \,$
$(1+b)(1-2a)=(1+a)(1-2b) \,$
$1-2a+b-2ab=1-2b+a-2ab \,$
$1-2a+b=1-2b+a \,$
$1-2a+3b=1+a \,$
$1+3b=1+3a \,$
$a=b \,$
Therefore by the algebraic 1-1 test, the function $f(x)\,$ is 1-1.
You can show that a function is not one-to-one by finding two distinct inputs that give the same output. For example, $f(x)=x^2$ is not one-to-one because $f(-1)=f(1)$ but $-1\neq1$.
### Inverse functions
We call $g(x)$ the inverse function of $f(x)$ if, for all $x$:
$g(f(x)) = f(g(x)) = x\$.
A function $f(x)$ has an inverse function if and only if $f(x)$ is one-to-one. For example, the inverse of $f(x)=x+2$ is $g(x)=x-2$. The function $f(x)=\sqrt{1-x^2}$ has no inverse.
#### Notation
The inverse function of $f$ is denoted as $f^{-1}(x)$. Thus, $f^{-1}(x)$ is defined as the function that follows this rule
$f(f^{-1}(x))=f^{-1}(f(x)) = x$:
To determine $f^{-1}(x)$ when given a function $f$, substitute $f^{-1}(x)$ for $x$ and substitute $x$ for $f(x)$. Then solve for $f^{-1}(x)$, provided that it is also a function.
Example: Given $f(x) = 2x - 7$, find $f^{-1}(x)$.
Substitute $f^{-1}(x)$ for $x$ and substitute $x$ for $f(x)$. Then solve for $f^{-1}(x)$:
$f(x) = 2x - 7\,$
$x = 2[f^{-1}(x)] - 7\,$
$x + 7 = 2[f^{-1}(x)]\,$
$\frac{x + 7}{2} = f^{-1}(x)\,$
To check your work, confirm that $f^{-1}(f(x)) = x$:
$f^{-1}(f(x)) =$
$f^{-1}(2x - 7) = {}$
$\frac{(2x - 7) + 7}{2} = \frac{2x}{2} = x$
If $f$ isn't one-to-one, then, as we said before, it doesn't have an inverse. Then this method will fail.
Example: Given $f(x)=x^2$, find $f^{-1}(x)$.
Substitute $f^{-1}(x)$ for $x$ and substitute $x$ for $f(x)$. Then solve for $f^{-1}(x)$:
$f(x) = x^2\,$
$x = (f^{-1}(x))^2\,$
$f^{-1}(x) = \pm\sqrt{x}\,$
Since there are two possibilities for $f^{-1}(x)$, it's not a function. Thus $f(x)=x^2$ doesn't have an inverse. Of course, we could also have found this out from the graph by applying the Horizontal Line Test. It's useful, though, to have lots of ways to solve a problem, since in a specific case some of them might be very difficult while others might be easy. For example, we might only know an algebraic expression for $f(x)$ but not a graph.
← Algebra Calculus Graphing linear functions → Print version
<h1> 1.3 Graphing linear functions</h1>
← Functions Calculus Precalculus/Exercises → Print version
Graph of y=2x
It is sometimes difficult to understand the behavior of a function given only its definition; a visual representation or graph can be very helpful. A graph is a set of points in the Cartesian plane, where each point ($x$,$y$) indicates that $f(x)=y$. In other words, a graph uses the position of a point in one direction (the vertical-axis or y-axis) to indicate the value of $f$ for a position of the point in the other direction (the horizontal-axis or x-axis).
Functions may be graphed by finding the value of $f$ for various $x$ and plotting the points ($x$, $f(x)$) in a Cartesian plane. For the functions that you will deal with, the parts of the function between the points can generally be approximated by drawing a line or curve between the points. Extending the function beyond the set of points is also possible, but becomes increasingly inaccurate.
## Example
Plotting points like this is laborious. Fortunately, many functions' graphs fall into general patterns. For a simple case, consider functions of the form
$f(x)=3x + 2\,\!$
The graph of $f$ is a single line, passing through the point $(0,2)$ with slope 3. Thus, after plotting the point, a straightedge may be used to draw the graph. This type of function is called linear and there are a few different ways to present a function of this type.
## Slope-intercept form
When we see a function presented as
$y = mx + b \,\!$
we call this presentation the slope-intercept form. This is because, not surprisingly, this way of writing a linear function involves the slope, m, and the y-intercept, b.
## Point-slope form
If someone walks up to you and gives you one point and a slope, you can draw one line and only one line that goes through that point and has that slope. Said differently, a point and a slope uniquely determine a line. So, if given a point $(x_0,y_0)$ and a slope m, we present the graph as
$y - y_0 = m(x - x_0). \,\!$
We call this presentation the point-slope form. The point-slope and slope-intercept form are essentially the same. In the point-slope form we can use any point the graph passes through. Where as, in the slope-intercept form, we use the y-intercept, that is the point (0,b).
## Calculating slope
If given two points, $(x_1,y_1)$ and $(x_2,y_2)$, we may then compute the slope of the line that passes through these two points. Remember, the slope is determined as "rise over run." That is, the slope is the change in y-values divided by the change in x-values. In symbols,
$\mbox{slope}~ = \frac{\mbox{change in}~y}{\mbox{change in}~x} = \frac{\Delta y}{\Delta x}.$
So now the question is, "what's $\Delta y$ and $\Delta x$?" We have that $\Delta y = y_2-y_1$ and $\Delta x = x_2 - x_1$. Thus,
$\mbox{slope}~ = \frac{y_2-y_1}{x_2-x_1}.$
## Two-point form
Two points also uniquely determine a line. Given points $(x_1,y_1)$ and $(x_2,y_2)$, we have the equation
$y - y_1 = \frac{y_2-y_1}{x_2-x_1}(x-x_1).$
This presentation is in the two-point form. It is essentially the same as the point-slope form except we substitute the expression $\frac{y_2-y_1}{x_2-x_1}$ for m.
← Functions Calculus Precalculus/Exercises → Print version
<h1> 1.4 Precalculus Cumulative Exercises</h1>
← Graphing linear functions Calculus Limits → Print version
## Algebra
### Convert to interval notation
1. $\{x:-4
$(-4,2)$
2. $\{x:-\frac{7}{3} \leq x \leq -\frac{1}{3}\}$
$[-\frac{7}{3},-\frac{1}{3}]$
3. $\{x:-\pi \leq x < \pi\}$
$[-\pi,\pi)$
4. $\{x:x \leq \frac{17}{9}\}$
$(-\infty, \frac{17}{9}]$
5. $\{x:5 \leq x+1 \leq 6\}$
$[4, 5]$
6. $\{x:x - \frac{1}{4} < 1\} \,$
$(-\infty, \frac{5}{4})$
7. $\{x:3 > 3x\} \,$
$(-\infty, 1)$
8. $\{x:0 \leq 2x+1 < 3\}$
$[-\frac{1}{2}, 1)$
9. $\{x:5
$(5,6)$
10. $\{x:5
$(-\infty,\infty)$
### State the following intervals using set notation
11. $[3,4] \,$
$\{x:3\leq x\leq 4\}$
12. $[3,4) \,$
$\{x:3\leq x<4\}$
13. $(3,\infty)$
$\{x:x>3\}$
14. $(-\frac{1}{3}, \frac{1}{3}) \,$
$\{x:-\frac{1}{3}
15. $(-\pi, \frac{15}{16}) \,$
$\{x:-\pi
16. $(-\infty,\infty)$
$\{x:x\in\Re\}$
### Which one of the following is a true statement?
Hint: the true statement is often referred to as the triangle inequality. Give examples where the other two are false.
17. $|x+y| = |x| + |y| \,$
false
18. $|x+y| \geq |x| + |y|$
false
19. $|x+y| \leq |x| + |y|$
true
### Evaluate the following expressions
20. $8^{1/3} \,$
$2$
21. $(-8)^{1/3} \,$
$-2$
22. $\bigg(\frac{1}{8}\bigg)^{1/3} \,$
$\frac{1}{2}$
23. $(8^{2/3}) (8^{3/2}) (8^0) \,$
$8^{13/6}$
24. $\bigg( \bigg(\frac{1}{8}\bigg)^{1/3} \bigg)^7$
$\frac{1}{128}$
25. $\sqrt[3]{\frac{27}{8}}$
$\frac{3}{2}$
26. $\frac{4^5 \cdot 4^{-2}}{4^3}$
$1$
27. $\bigg(\sqrt{27}\bigg)^{2/3}$
$3$
28. $\frac{\sqrt{27}}{\sqrt[3]{9}}$
$3^{5/6}$
### Simplify the following
29. $x^3 + 3x^3 \,$
$4x^3$
30. $\frac{x^3 + 3x^3}{x^2}$
$4x$
31. $(x^3+3x^3)^3 \,$
$64x^9$
32. $\frac{x^{15} + x^3}{x}$
$x^{14}+x^2$
33. $(2x^2)(3x^{-2}) \,$
$6$
34. $\frac{x^2y^{-3}}{x^3y^2}$
$\frac{1}{xy^5}$
35. $\sqrt{x^2y^4}$
$xy^2$
36. $\bigg(\frac{8x^6}{y^4}\bigg)^{1/3}$
$\frac{2x^2}{y^{4/3}}$
### Find the roots of the following polynomials
37. $x^2 - 1 \,$
$x=\pm1$
38. $x^2 +2x +1 \,$
$x=-1$
39. $x^2 + 7x + 12 \,$
$x=-3, x=-4$
40. $3x^2 - 5x -2 \,$
$x=2, x=-\frac{1}{3}$
41. $x^2 + 5/6x + 1/6 \,$
$x=-\frac{1}{3}, x=-\frac{1}{2}$
42. $4x^3 + 4x^2 + x \,$
$x=0,x=-\frac{1}{2}$
43. $x^4 - 1 \,$
$x=\pm i, x=\pm 1$
44. $x^3 + 2x^2 - 4x - 8 \,$
$x=\pm2$
### Factor the following expressions
45. $4a^2 - ab - 3b^2 \,$
$(4a+3b)(a-b)$
46. $(c+d)^2 - 4 \,$
$(c+d+2)(c+d-2)$
47. $4x^2 - 9y^2 \,$
$(2x+3y)(2x-3y)$
### Simplify the following
48. $\frac{x^2 -1}{x+1} \,$
$x-1, x\neq-1$
49. $\frac{3x^2 + 4x + 1}{x+1} \,$
$3x+1, x\neq-1$
50. $\frac{4x^2 - 9}{4x^2 + 12x + 9} \,$
$\frac{2x-3}{2x+3}$
51. $\frac{x^2 + y^2 +2xy}{x(x+y)} \,$
$\frac{x+y}{x}, x\neq-y$
## Functions
52. Let $f(x)=x^2$.
a. Compute $f(0)$ and $f(2)$.
${0,4}$
b. What are the domain and range of $f$?
${(-\infty,\infty)}$
c. Does $f$ have an inverse? If so, find a formula for it.
${x^{1/2}}$
53. Let $f(x)=x+2$, $g(x)=1/x$.
a. Give formulae for
i. $f+g$
$(f + g)(x) = x + 2 + \frac{1}{x}$
ii. $f-g$
$(f - g)(x) = x + 2 - \frac{1}{x}$
iii. $g-f$
$(g - f)(x) = \frac{1}{x} - x - 2$
iv. $f\times g$
$(f \times g)(x) = 1 + \frac{2}{x}$
v. $f/g$
$(f / g)(x) = x^2 + 2x$
vi. $g/f$
$(g / f)(x) = \frac{1}{x^2 + 2x}$
vii. $f\circ g$
$(f \circ g)(x) = \frac{1}{x} + 2$
viii. $g\circ f$
$(g \circ f)(x) = \frac{1}{x + 2}$
b. Compute $f(g(2))$ and $g(f(2))$.
$f(g(2))=5/2, g(f(2))=1/4$
c. Do $f$ and $g$ have inverses? If so, find formulae for them.
$f^{-1}(x)=x-2, g^{-1}(x)=\frac{1}{x}$
54. Does this graph represent a function?
Yes.
55. Consider the following function
$f(x) = \begin{cases} -\frac{1}{9} & \mbox{if } x<-1 \\ 2 & \mbox{if } -1\leq x \leq 0 \\ x + 3 & \mbox{if } x>0. \end{cases}$
a. What is the domain?
${(-\infty,\infty)}$
b. What is the range?
${(-1/9,\infty)}$
c. Where is $f$ continuous?
${x>0}$
56. Consider the following function
$f(x) = \begin{cases} x^2 & \mbox{if } x>0 \\ -1 & \mbox{if } x\leq 0. \end{cases}$
a. What is the domain?
b. What is the range?
c. Where is $f$ continuous?
57. Consider the following function
$f(x) = \frac{\sqrt{2x-3}}{x-10}$
a. What is the domain?
When you find the answer, you can add it here by clicking "edit".
b. What is the range?
When you find the answer, you can add it here by clicking "edit".
c. Where is $f$ continuous?
When you find the answer, you can add it here by clicking "edit".
58. Consider the following function
$f(x) = \frac{x-7}{x^2-49}$
a. What is the domain?
When you find the answer, you can add it here by clicking "edit".
b. What is the range?
When you find the answer, you can add it here by clicking "edit".
c. Where is $f$ continuous?
When you find the answer, you can add it here by clicking "edit".
## Graphing
59. Find the equation of the line that passes through the point (1,-1) and has slope 3.
$3x-y=4$
60. Find the equation of the line that passes through the origin and the point (2,3).
$3x-2y=0$
Solutions
← Graphing linear functions Calculus Limits → Print version
# Limits
<h1> 2.1 An Introduction to Limits</h1>
← Limits/Contents Calculus Finite Limits → Print version
## Intuitive Look
A limit looks at what happens to a function when the input approaches a certain value. The general notation for a limit is as follows:
$\quad\lim_{x\to a} f(x)$
This is read as "The limit of $f$ of $x$ as $x$ approaches $a$". We'll take up later the question of how we can determine whether a limit exists for $f(x)$ at $a$ and, if so, what it is. For now, we'll look at it from an intuitive standpoint.
Let's say that the function that we're interested in is $f(x)=x^2$, and that we're interested in its limit as $x$ approaches $2$. Using the above notation, we can write the limit that we're interested in as follows:
$\quad\lim_{x\to 2} x^2$
One way to try to evaluate what this limit is would be to choose values near 2, compute $f(x)$ for each, and see what happens as they get closer to 2. This is implemented as follows:
$x$ $f(x)=x^2$ 1.7 1.8 1.9 1.95 1.99 1.999 2.89 3.24 3.61 3.8025 3.9601 3.996
Here we chose numbers smaller than 2, and approached 2 from below. We can also choose numbers larger than 2, and approach 2 from above:
$x$ $f(x)=x^2$ 2.3 2.2 2.1 2.05 2.01 2.001 5.29 4.84 4.41 4.2025 4.0401 4.004
We can see from the tables that as $x$ grows closer and closer to 2, $f(x)$ seems to get closer and closer to 4, regardless of whether $x$ approaches 2 from above or from below. For this reason, we feel reasonably confident that the limit of $x^2$ as $x$ approaches 2 is 4, or, written in limit notation,
$\quad\lim_{x\to 2} x^2=4.$
We could have also just substituted 2 into $x^2$ and evaluated: $(2)^2=4$. However, this will not work with all limits.
Now let's look at another example. Suppose we're interested in the behavior of the function $f(x)=\frac{1}{x-2}$ as $x$ approaches 2. Here's the limit in limit notation:
$\quad\lim_{x\to 2} \frac{1}{x-2}$
Just as before, we can compute function values as $x$ approaches 2 from below and from above. Here's a table, approaching from below:
$x$ $f(x)=\frac{1}{x-2}$ 1.7 1.8 1.9 1.95 1.99 1.999 -3.333 -5 -10 -20 -100 -1000
And here from above:
$x$ $f(x)=\frac{1}{x-2}$ 2.3 2.2 2.1 2.05 2.01 2.001 3.333 5 10 20 100 1000
In this case, the function doesn't seem to be approaching a single value as $x$ approaches 2, but instead becomes an extremely large positive or negative number (depending on the direction of approach). This is known as an infinite limit. Note that we cannot just substitute 2 into $\frac{1}{x-2}$ and evaluate as we could with the first example, since we would be dividing by 0.
Both of these examples may seem trivial, but consider the following function:
$f(x) = \frac{x^2(x-2)}{x-2}$
This function is the same as
$f(x) =\left\{\begin{matrix} x^2 & \mbox{if } x\neq 2 \\ \mbox{undefined} & \mbox{if } x=2\end{matrix}\right.$
Note that these functions are really completely identical; not just "almost the same," but actually, in terms of the definition of a function, completely the same; they give exactly the same output for every input.
In algebra, we would simply say that we can cancel the term $(x-2)$, and then we have the function $f(x)=x^2$. This, however, would be a bit dishonest; the function that we have now is not really the same as the one we started with, because it is defined when $x=2$, and our original function was specifically not defined when $x=2$. In algebra we were willing to ignore this difficulty because we had no better way of dealing with this type of function. Now, however, in calculus, we can introduce a better, more correct way of looking at this type of function. What we want is to be able to say that, although the function doesn't exist when $x=2$, it works almost as though it does. It may not get there, but it gets really, really close. That is, $f(1.99999)=3.99996$. The only question that we have is: what do we mean by "close"?
## Informal Definition of a Limit
As the precise definition of a limit is a bit technical, it is easier to start with an informal definition; we'll explain the formal definition later.
We suppose that a function $f$ is defined for $x$ near $c$ (but we do not require that it be defined when $x=c$).
Definition: (Informal definition of a limit)
We call $L$ the limit of $f(x)$ as $x$ approaches $c$ if $f(x)$ becomes close to $L$ when $x$ is close (but not equal) to $c$, and if there is no other value $L'$ with the same property..
When this holds we write
$\lim_{x \to c} f(x) = L$
or
$f(x) \to L \quad \mbox{as} \quad x \to c.$
Notice that the definition of a limit is not concerned with the value of $f(x)$ when $x=c$ (which may exist or may not). All we care about are the values of $f(x)$ when $x$ is close to $c$, on either the left or the right (i.e. less or greater).
## Limit Rules
Now that we have defined, informally, what a limit is, we will list some rules that are useful for working with and computing limits. You will be able to prove all these once we formally define the fundamental concept of the limit of a function.
First, the constant rule states that if $f(x)=b$ (that is, $f$ is constant for all $x$) then the limit as $x$ approaches $c$ must be equal to $b$. In other words
Constant Rule for Limits
If b and c are constants then $\lim_{x\to c} b = b$.
Example: $\lim_{x\to 6} 5=5$
Second, the identity rule states that if $f(x)=x$ (that is, $f$ just gives back whatever number you put in) then the limit of $f$ as $x$ approaches $c$ is equal to $c$. That is,
Identity Rule for Limits
If c is a constant then $\lim_{x\to c} x = c$.
Example: $\lim_{x\to 6} x=6$
The next few rules tell us how, given the values of some limits, to compute others.
Operational Identities for Limits
Suppose that $\lim_{x\to c} f(x) =L$ and $\lim_{x\to c} g(x) =M$ and that $k$ is constant. Then
• $\lim_{x\to c} k f(x) = k \cdot \lim_{x\to c} f(x) = k L$
• $\lim_{x\to c} [f(x) + g(x)] = \lim_{x\to c} f(x) + \lim_{x\to c} g(x) = L + M$
• $\lim_{x\to c} [f(x) - g(x)] = \lim_{x\to c} f(x) - \lim_{x\to c} g(x) = L - M$
• $\lim_{x\to c} [f(x) g(x)] = \lim_{x\to c} f(x) \lim_{x\to c} g(x) = L M$
• $\lim_{x\to c} \frac{f(x)}{g(x)} = \frac{\lim_{x\to c} f(x)}{\lim_{x\to c} g(x)} = \frac{L}{M} \,\,\, \mbox{ provided } M\neq 0$
Notice that in the last rule we need to require that $M$ is not equal to zero (otherwise we would be dividing by zero which is an undefined operation).
These rules are known as identities; they are the scalar product, sum, difference, product, and quotient rules for limits. (A scalar is a constant, and, when you multiply a function by a constant, we say that you are performing scalar multiplication.)
Using these rules we can deduce another. Namely, using the rule for products many times we get that
$\lim_{x\to c} f(x)^n = \left(\lim_{x\to c} f(x) \right)^n = L^n$ for a positive integer $n$.
This is called the power rule.
### Examples
Example 1
Find the limit $\lim_{x\to 2} {4x^3}$.
We need to simplify the problem, since we have no rules about this expression by itself. We know from the identity rule above that $\lim_{x\to 2} {x} = 2$. By the power rule, $\lim_{x\to 2} {x^3} = \left(\lim_{x\to 2} x\right)^3 = 2^3 = 8$. Lastly, by the scalar multiplication rule, we get $\lim_{x\to 2} {4x^3} = 4\lim_{x\to 2} x^3=4 \cdot 8=32$.
Example 2
Find the limit $\lim_{x\to 2} [4x^3 + 5x +7]$.
To do this informally, we split up the expression, once again, into its components. As above,$\lim_{x\to 2} 4x^3=32$.
Also $\lim_{x\to 2} 5x = 5\cdot\lim_{x\to 2} x = 5\cdot2=10$ and $\lim_{x\to 2} 7 =7$. Adding these together gives
$\lim_{x\to 2} 4x^3 + 5x +7 = \lim_{x\to 2} 4x^3 + \lim_{x\to 2} 5x + \lim_{x\to 2} 7 = 32 + 10 +7 =49$.
Example 3
Find the limit $\lim_{x\to 2}\frac{4x^3 + 5x +7}{(x-4)(x+10)}$.
From the previous example the limit of the numerator is $\lim_{x\to 2} 4x^3 + 5x +7 =49$. The limit of the denominator is
$\lim_{x\to 2} (x-4)(x+10) = \lim_{x\to 2} (x-4) \cdot \lim_{x\to 2} (x+10) = (2-4)\cdot(2+10)=-24.$
As the limit of the denominator is not equal to zero we can divide. This gives
$\lim_{x\to 2}\frac{4x^3 + 5x +7}{(x-4)(x+10)} = -\frac{49}{24}$.
Example 4
Find the limit $\lim_{x\to 4}\frac{x^4 - 16x + 7}{4x-5}$.
We apply the same process here as we did in the previous set of examples;
$\lim_{x\to 4}\frac{x^4 - 16x + 7}{4x-5} = \frac{\lim_{x\to 4} (x^4 - 16x + 7)} {\lim_{x\to 4} (4x-5)} = \frac{\lim_{x\to 4} (x^4) - \lim_{x\to 4} (16x) + \lim_{x\to 4} (7)} {\lim_{x\to 4} (4x) - \lim_{x\to 4} 5}$.
We can evaluate each of these; $\lim_{x\to 4} (x^4) = 256,$ $\lim_{x\to 4} (16x) = 64,$ $\lim_{x\to 4} (7) = 7,$ $\lim_{x\to 4} (4x) = 16$ and $\lim_{x\to 4} (5) = 5.$ Thus, the answer is $\frac{199}{11}$.
Example 5
Find the limit $\lim_{x\to 2}\frac{x^2 - 3x + 2}{x-2}$.
In this example, evaluating the result directly will result in a division by zero. While you can determine the answer experimentally, a mathematical solution is possible as well.
First, the numerator is a polynomial that may be factored: $\lim_{x\to 2}\frac{(x-2)(x-1)}{x-2}$
Now, you can divide both the numerator and denominator by (x-2): $\lim_{x\to 2} (x-1) = (2-1) = 1$
Example 6
Find the limit $\lim_{x\to 0}\frac{1-\cos x}{x}$.
To evaluate this seemingly complex limit, we will need to recall some sine and cosine identities. We will also have to use two new facts. First, if $f(x)$ is a trigonometric function (that is, one of sine, cosine, tangent, cotangent, secant or cosecant) and is defined at $a$, then $\lim_{x\to a} f(x) = f(a)$.
Second, $\lim_{x\to 0}\frac{\sin x}{x} = 1$. This may be determined experimentally, or by applying L'Hôpital's rule, described later in the book.
To evaluate the limit, recognize that $1 - \cos x$ can be multiplied by $1+\cos x$ to obtain $(1-\cos^2 x)$ which, by our trig identities, is $\sin^2 x$. So, multiply the top and bottom by $1+\cos x$. (This is allowed because it is identical to multiplying by one.) This is a standard trick for evaluating limits of fractions; multiply the numerator and the denominator by a carefully chosen expression which will make the expression simplify somehow. In this case, we should end up with:
\begin{align}\lim_{x\to 0} \frac{1-\cos x}{x} &=& \lim_{x\to 0} \left(\frac{1-\cos x}{x} \cdot \frac{1}{1}\right) \\ &=& \lim_{x\to 0} \left(\frac{1-\cos x}{x} \cdot \frac{1 + \cos x} {1+ \cos x}\right) \\ &=& \lim_{x\to 0}\frac{(1 - \cos x) \cdot 1 + (1 - \cos x) \cdot \cos x} {x \cdot (1+ \cos x)} \\ &=& \lim_{x\to 0}\frac{1 - \cos x + \cos x - \cos^2 x}{x \cdot (1+ \cos x)} \\ &=& \lim_{x\to 0}\frac{1 - \cos^2 x} {x \cdot (1+ \cos x)} \\ &=& \lim_{x\to 0}\frac{\sin^2 x} {x \cdot (1+ \cos x)} \\ &=& \lim_{x\to 0} \left(\frac{\sin x} {x} \cdot \frac{\sin x} {1+ \cos x}\right)\end{align}.
Our next step should be to break this up into $\lim_{x\to 0}\frac{\sin x}{x} \cdot \lim_{x\to 0} \frac{\sin x}{1+\cos x}$ by the product rule. As mentioned above, $\lim_{x\to 0} \frac{\sin x} {x} = 1$.
Next, $\lim_{x\to 0} \frac{\sin x} {1+\cos x} = \frac{\lim_{x\to 0}\sin x} {\lim_{x\to 0} (1+\cos x)} = \frac{0} {1 + \cos 0} = 0$.
Thus, by multiplying these two results, we obtain 0.
We will now present an amazingly useful result, even though we cannot prove it yet. We can find the limit at $c$ of any polynomial or rational function, as long as that rational function is defined at $c$ (so we are not dividing by zero). That is, $c$ must be in the domain of the function.
Limits of Polynomials and Rational functions
If $f$ is a polynomial or rational function that is defined at $c$ then
$\lim_{x \rightarrow c} f(x) = f(c)$
We already learned this for trigonometric functions, so we see that it is easy to find limits of polynomial, rational or trigonometric functions wherever they are defined. In fact, this is true even for combinations of these functions; thus, for example, $\lim_{x\to 1} (\sin x^2 + 4\cos^3(3x-1)) = \sin 1^2 + 4\cos^3 (3(1)-1)$.
### The Squeeze Theorem
Graph showing $f$ being squeezed between $g$ and $h$
The Squeeze Theorem is very important in calculus, where it is typically used to find the limit of a function by comparison with two other functions whose limits are known.
It is called the Squeeze Theorem because it refers to a function $f$ whose values are squeezed between the values of two other functions $g$ and $h$, both of which have the same limit $L$. If the value of $f$ is trapped between the values of the two functions $g$ and $h$, the values of $f$ must also approach $L$.
Expressed more precisely:
Theorem: (Squeeze Theorem)
Suppose that $g(x) \le f(x) \le h(x)$ holds for all $x$ in some open interval containing $c$, except possibly at $x=c$ itself. Suppose also that $\lim_{x\to c}g(x)=\lim_{x\to c}h(x)=L$. Then $\lim_{x\to c}f(x)=L$ also.
Plot of x*sin(1/x) for -0.5 < x <0.5
Example: Compute $\lim_{x\to 0} x\sin(1/x)$. Note that the sine of any real number is in the interval $[-1,1]$. That is, $-1 \le \sin x \le 1$ for all $x$, and $-1 \le \sin(1/x) \le 1$ for all $x$. If $x$ is positive, we can multiply these inequalities by $x$ and get $-x \le x\sin(1/x) \le x$. If $x$ is negative, we can similarly multiply the inequalities by the positive number $-x$ and get $x \le x\sin(1/x) \le -x$. Putting these together, we can see that, for all nonzero $x$, $-\left|x\right| \le x\sin(1/x) \le \left|x\right|$. But it's easy to see that $\lim_{x\to 0} -\left|x\right| = \lim_{x\to 0} \left|x\right| = 0$. So, by the Squeeze Theorem, $\lim_{x\to 0} x\sin(1/x) = 0$.
## Finding Limits
Now, we will discuss how, in practice, to find limits. First, if the function can be built out of rational, trigonometric, logarithmic and exponential functions, then if a number $c$ is in the domain of the function, then the limit at $c$ is simply the value of the function at $c$.
If $c$ is not in the domain of the function, then in many cases (as with rational functions) the domain of the function includes all the points near $c$, but not $c$ itself. An example would be if we wanted to find $\lim_{x\to 0} \frac{x}{x}$, where the domain includes all numbers besides 0.
In that case, in order to find $\lim_{x\to c}f(x)$ we want to find a function $g(x)$ similar to $f(x)$, except with the hole at $c$ filled in. The limits of $f$ and $g$ will be the same, as can be seen from the definition of a limit. By definition, the limit depends on $f(x)$ only at the points where $x$ is close to $c$ but not equal to it, so the limit at $c$ does not depend on the value of the function at $c$. Therefore, if $\lim_{x\to c} g(x)=L$, $\lim_{x\to c} f(x) = L$ also. And since the domain of our new function $g$ includes $c$, we can now (assuming $g$ is still built out of rational, trigonometric, logarithmic and exponential functions) just evaluate it at $c$ as before. Thus we have $\lim_{x\to c} f(x) = g(c)$.
In our example, this is easy; canceling the $x$'s gives $g(x)=1$, which equals $f(x)=x/x$ at all points except 0. Thus, we have $\lim_{x\to 0}\frac{x}{x} = \lim_{x\to 0} 1 = 1$. In general, when computing limits of rational functions, it's a good idea to look for common factors in the numerator and denominator.
Lastly, note that the limit might not exist at all. There are a number of ways in which this can occur:
$f(x) = \sqrt{x^2 - 16}$
"Gap"
There is a gap (not just a single point) where the function is not defined. As an example, in
$f(x) = \sqrt{x^2 - 16}$
$\lim_{x\to c}f(x)$ does not exist when $-4\le c\le4$. There is no way to "approach" the middle of the graph. Note that the function also has no limit at the endpoints of the two curves generated (at $c=-4$ and $c=4$). For the limit to exist, the point must be approachable from both the left and the right.
Note also that there is no limit at a totally isolated point on a graph.
"Jump"
If the graph suddenly jumps to a different level, there is no limit at the point of the jump. For example, let $f(x)$ be the greatest integer $\le x$. Then, if $c$ is an integer, when $x$ approaches $c$ from the right $f(x)=c$, while when $x$ approaches $c$ from the left $f(x)=c-1$. Thus $\lim_{x\to c} f(x)$ will not exist.
A graph of 1/(x2) on the interval [-2,2].
Vertical asymptote
In
$f(x) = {1 \over x^2}$
the graph gets arbitrarily high as it approaches 0, so there is no limit. (In this case we sometimes say the limit is infinite; see the next section.)
A graph of sin(1/x) on the interval (0,1/π].
Infinite oscillation
These next two can be tricky to visualize. In this one, we mean that a graph continually rises above and falls below a horizontal line. In fact, it does this infinitely often as you approach a certain $x$-value. This often means that there is no limit, as the graph never approaches a particular value. However, if the height (and depth) of each oscillation diminishes as the graph approaches the $x$-value, so that the oscillations get arbitrarily smaller, then there might actually be a limit.
The use of oscillation naturally calls to mind the trigonometric functions. An example of a trigonometric function that does not have a limit as $x$ approaches 0 is
$f(x) = \sin {1 \over x}.$
As $x$ gets closer to 0 the function keeps oscillating between $-1$ and 1. In fact, $\sin(1/x)$ oscillates an infinite number of times on the interval between 0 and any positive value of $x$. The sine function is equal to zero whenever $x=k\pi$, where $k$ is a positive integer. Between every two integers $k$, $\sin x$ goes back and forth between 0 and $-1$ or 0 and 1. Hence, $\sin(1/x)=0$ for every $x=1/(k\pi)$. In between consecutive pairs of these values, $1/(k\pi)$ and $1/[(k+1)\pi]$, $\sin(1/x)$ goes back and forth from 0, to either $-1$ or 1 and back to 0. We may also observe that there are an infinite number of such pairs, and they are all between 0 and $1/\pi$. There are a finite number of such pairs between any positive value of $x$ and $1/\pi$, so there must be infinitely many between any positive value of $x$ and 0. From our reasoning we may conclude that, as $x$ approaches 0 from the right, the function $\sin(1/x)$ does not approach any specific value. Thus, $\lim_{x\to 0} \sin(1/x)$ does not exist.
## Using Limit Notation to Describe Asymptotes
Now consider the function
$g(x) = \frac{1}{x^2}.$
What is the limit as $x$ approaches zero? The value of $g(0)$ does not exist; it is not defined.
Notice, also, that we can make $g(x)$ as large as we like, by choosing a small $x$, as long as $x\ne0$. For example, to make $g(x)$ equal to $10^{12}$, we choose $x$ to be $10^{-6}$. Thus, $\lim_{x\to 0} \frac{1}{x^2}$ does not exist.
However, we do know something about what happens to $g(x)$ when $x$ gets close to 0 without reaching it. We want to say we can make $g(x)$ arbitrarily large (as large as we like) by taking $x$ to be sufficiently close to zero, but not equal to zero. We express this symbolically as follows:
$\lim_{x\to 0} g(x) = \lim_{x\to 0} \frac{1}{x^2} = \infty$
Note that the limit does not exist at $0$; for a limit, being $\infty$ is a special kind of not existing. In general, we make the following definition.
Definition: Informal definition of a limit being $\pm\infty$
We say the limit of $f(x)$ as $x$ approaches $c$ is infinity if $f(x)$ becomes very big (as big as we like) when $x$ is close (but not equal) to $c$.
In this case we write
$\lim_{x\to c} f(x) = \infty$
or
$f(x)\to\infty\quad\mbox{as}\quad x\to c$.
Similarly, we say the limit of $f(x)$ as $x$ approaches $c$ is negative infinity if $f(x)$ becomes very negative when $x$ is close (but not equal) to $c$.
In this case we write
$\lim_{x\to c} f(x) = -\infty$
or
$f(x)\to-\infty\quad\mbox{as}\quad x\to c$.
An example of the second half of the definition would be that $\lim_{x\to 0} -\frac{1}{x^2} = -\infty$.
## Key Application of Limits
To see the power of the concept of the limit, let's consider a moving car. Suppose we have a car whose position is linear with respect to time (that is, a graph plotting the position with respect to time will show a straight line). We want to find the velocity. This is easy to do from algebra; we just take the slope, and that's our velocity.
But unfortunately, things in the real world don't always travel in nice straight lines. Cars speed up, slow down, and generally behave in ways that make it difficult to calculate their velocities.
Now what we really want to do is to find the velocity at a given moment (the instantaneous velocity). The trouble is that in order to find the velocity we need two points, while at any given time, we only have one point. We can, of course, always find the average speed of the car, given two points in time, but we want to find the speed of the car at one precise moment.
This is the basic trick of differential calculus, the first of the two main subjects of this book. We take the average speed at two moments in time, and then make those two moments in time closer and closer together. We then see what the limit of the slope is as these two moments in time are closer and closer, and say that this limit is the slope at a single instant.
We will study this process in much greater depth later in the book. First, however, we will need to study limits more carefully.
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2015-01-31 17:51:55
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https://socratic.org/questions/how-do-you-simplify-using-the-distributive-property-of-725-x-4#488118
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# How do you simplify using the distributive property of 725 x 4?
Oct 11, 2017
$2900$
#### Explanation:
$\text{expressing "725" as a sum}$
$\Rightarrow 725 \times 4$
$= \left(700 + 25\right) \times 4$
$= \left(4 \times 700\right) + \left(4 \times 25\right) \leftarrow \textcolor{b l u e}{\text{distributive property}}$
$= 2800 + 100$
$= 2900$
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2022-01-27 03:46:00
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http://math.stackexchange.com/questions/182061/parametrization-of-a-class-of-functions
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# Parametrization of a class of functions
Could someone gives some examples of the pair $(\varphi(\theta), \Psi(z))$ such that
$$1+f(\theta)F(z)\geq 0,\ \ \forall\theta>0,\ \ z\in\mathbb{R}$$
with
$$f(\theta)=\theta\frac{\varphi'(\theta)}{\varphi(\theta)},\ \ F(z)=z\frac{\Psi'(z)}{\Psi(z)}$$
and
$$\left(1-\frac{z\Psi'(z)}{2\Psi(z)}\right)^2-\frac{\theta\varphi(\theta)^2}{4}\left(\frac{(\Psi') ^{2}(z)}{\Psi(z)}-2\Psi''(z)\right)-\frac{(\theta\varphi(\theta))^2}{16}(\Psi')^{2}(z)\geq 0,\ \ \forall\theta>0,\ \ z\in\mathbb{R}$$
where
$$\varphi\in\mathcal{C}^1: \theta\in\mathbb{R}_+\longrightarrow\varphi(\theta)>0$$
$$\Psi\in\mathcal{C}^2: z\in\mathbb{R}\longrightarrow\Psi(z)>0$$
$$\Psi(0)=1$$
Many thanks for your help!
-
This question comes from a model SVI volatility surface, we want to find out the functions satisfying the previous inequalities in order to eliminate the static arbitrage – Higgs88 Aug 13 '12 at 13:39
Putting $\Psi\equiv1$ gives many examples since any smooth and positive function $\varphi$ would fit. – Andrew Aug 13 '12 at 15:51
Thanks for Andrew – Higgs88 Aug 14 '12 at 8:05
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2014-03-07 15:35:29
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https://www.springerprofessional.de/practical-quantum-mechanics/14419750
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main-content
## Über dieses Buch
This work was first published in 1947 in German under the title "Re chenmethoden der Quantentheorie". It was meant to serve a double purpose: to help both, the student when first confronted with quantum mechanics and the experimental scientist, who has never before used it as a tool, to learn how to apply the general theory to practical problems of atomic physics. Since that early date, many excellent books have been written introducing into the general framework of the theory and thus indispensable to a deeper understanding. It seems, however, that the more practical side has been somewhat neglected, except, of course, for the flood of special monographs going into broad detail on rather restricted topics. In other words, an all-round introduction to the practical use of quantum mechanics seems, so far, not to exist and may still be helpful. It was in the hope of filling this gap that the author has fallen in with the publishers' wish to bring the earlier German editions up to date and to make the work more useful to the worldwide community of science students and scientists by writing the new edition in English. From the beginning there could be no doubt that the work had to be much enlarged. New approximation methods and other developments, especially in the field of scattering, had to be added. It seemed necessary to include relativistic quantum mechanics and to offer, at least, a glimpse of radiation theory as an example of wave field quantization.
## Inhaltsverzeichnis
### I. General Concepts
Abstract
If the normalization relation
$$\int {{d^3}} x\psi * \psi = 1$$
(1.1)
is interpreted in the sense of probability theory, so that $${d^3}x\psi * \psi$$ is the probability of finding the particle under consideration in the volume element d3x, then there must be a conservation law. This is to be derived. How may it be interpreted classically?
Siegfried Flügge
### II. One-Body Problems without Spin
Abstract
One-dimensional problems, though in a sense oversimplifications, may be used with advantage in order to understand the essential features of quantum mechanics. They may be derived from the three-dimensional wave equation,
$$- {\mkern 1mu} \frac{{{\hbar ^2}}}{{2m}}{\nabla ^2}\psi = + V(x,t)\psi - \frac{\hbar }{i}\frac{{\partial \psi }}{{\partial t}},$$
(A.1)
if the potential depends upon only one rectangular coordinate x, by factorization:
$$\psi = {e^{i({k_2}y + {k_3}z)}}{\mkern 1mu} \varphi (x,t).$$
(A.2)
Siegfried Flügge
### III. Particles with Spin
Abstract
A particle of spin 1/2 has three basic properties:
1.
It bears an intrinsic vector property that does not depend upon space coordinates.
2.
This vector is an angular momentum (= spin) to be added to the orbital momentum of the particle.
3.
If one of the components of the spin is measured, the result can be only one of its two eigenvalues, +1/2ħ or −1/2ħ.
Siegfried Flügge
### IV. IV. Many-Body Problems
Abstract
Two particlcs arc fixed on a circle with a mutual repulsion given by
$$V({\varphi _1},{\varphi _2})\, = \,{V_0}\,\cos ({\varphi _1}, - {\varphi _2})$$
(148.1)
to simulate e.g. the Coulomb repulsion between the two helium electrons in the ground state. The conservation of angular momentum shall be derived, and the relative motion of the particles discussed.
Siegfried Flügge
### V. V. Non-Stationary Problems
Abstract
Given an atomic system with only two stationary states ∣1〉 and ∣2〉 and energies ħω1 <ħω2. At the time t = 0, the system being in its ground state, a perturbation W not depending upon time is switched on. The probability shall be calculated of finding the system in either state at the time t.
Siegfried Flügge
### VI. The Relativistic Dirac Equation
Abstract
Remark. In this chapter we use the fourth coordinate x4=ict and Euclidian metric. Greek subscripts (e.g. x μ ) run overμ =1,2,3,4, Latin subscripts (xk) over k = 1,2,3, only.
Siegfried Flügge
|
2020-04-10 00:19:21
|
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|
https://socratic.org/questions/what-is-the-slope-of-the-line-passing-through-the-following-points-4-1-7-0
|
# What is the slope of the line passing through the following points: (4,1), (-7,0) ?
Feb 26, 2016
$m = \frac{1}{11}$
#### Explanation:
To find the gradient (slope) of a line passing through 2 points use the $\textcolor{b l u e}{\text{ gradient formula }}$
$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$
where$\left({x}_{1} , {y}_{1}\right) \text{ and " (x_2,y_2)" are the coords of the 2 points }$
let$\left({x}_{1} , {y}_{1}\right) = \left(4 , 1\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(- 7 , 0\right)$
substitute these values into the equation for m
$\Rightarrow m = \frac{0 - 1}{- 7 - 4} = \frac{- 1}{- 11} = \frac{1}{11}$
|
2023-02-05 07:30:11
|
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|
http://www.emathhelp.net/notes/calculus-2/definite-integral/the-fundamental-theorem-of-calculus/
|
The Fundamental Theorem of Calculus
Related Calculator: Definite and Improper Integral Calculator
When we introduced definite integrals we computed them according to definition as a limit of Riemann sums and we saw that this procedure is not very easy. In fact there is a much simpler method for evaluating integrals.
We already discovered it when we talked about Area Problem first time.
There we introduced function P(x) whose value is area under function f on interval [a,x] (x can vary from a to b).
Now when we know about definite integrals we can write that P(x)=int_a^xf(t)dt (note that we changes x to t under integral in order not to mix it with upper limit).
Also we discovered Newton-Leibniz formula which states that P'(x)=f(x) and P(x)=F(x)-F(a) where F'=f.
Here we will formalize this result and give another proof because these fact are very important in calculus: they connect differential calculus with integral calculus.
The Fundamental Theorem of Calculus. Suppose f is continuous on [a,b].
1. If P(x)=int_a^x f(t)dt, then g'(x)=f(x).
2. int_a^b f(x)dx=F(b)-F(a) where F is any antiderivative of f, that is F'=f.
Part 1 can be rewritten as d/(dx)int_a^x f(t)dt=f(x), which says that if f is integrated and then the result is differentiated, we arrive back at the original function.
Part 2 can be rewritten as int_a^bF'(x)dx=F(b)-F(a) and it says that if we take a function F, first differentiate it, and then integrate the result, we arrive back at the original function F, but in the form F(b)-F(a).
Fundamental Theorem of Calculus says that differentiation and integration are inverse processes.
Proof of Part 1. Let P(x)=int_a^x f(t)dt. If x and x+h are in the open interval (a,b) then P(x+h)-P(x)=int_a^(x+h)f(t)dt-int_a^xf(t)dt.
Now use adjacency property of integral: int_a^(x+h)f(t)dt-int_a^x f(t)dt=(int_a^x f(t)dt+int_x^(x+h)f(t)dt)-int_a^x f(t)dt=int_x^(x+h)f(t)dt.
Now apply Mean Value Theorem for Integrals:
int_x^(x+h)f(t)dt=n(x+h-x)=nh, where m'<=n<=M' (M' is maximum value and m' is minimum values of f on [x,x+h]).
So, we obtained that P(x+h)-P(x)=nh. If we let h->0 then P(x+h)-P(x)->0 or P(x+h)->P(x).
This proves that P(x) is continuous function.
Without loss of generality assume that h>0.
Since f is continuous on [x,x+h], the Extreme Value Theorem says that there are numbers c and d in [x,x+h] such that f(c)=m and f(d)=M, where m and M are minimum and maximum values of f on [x,x+h].
By comparison property 5 we have m(x+h-x)<=int_x^(x+h)f(t)dt<=M(x+h-h) or mh<=int_x^(x+h)f(t)dt<=Mh.
This can be divided by h>0: m<=1/h int_x^(x+h)f(t)dt<=M or m<=(P(x+h)-P(x))/h<=M.
Finally, f(c)<=(P(x+h)-P(x))/h<=f(d).
This inequality can be proved for h<0 similarly.
Now we let h->0.
Then c->x and d->x since c and d lie between x and x+h.
So, lim_(h->0)f(c)=lim_(c->x)f(c)=f(x) and lim_(h->0)f(d)=lim_(d->x)f(d)=f(x) because f is continuous.
Therefore, from last inequality and Squeeze Theorem we conclude that lim_(h->0)(P(x+h)-P(x))/h=f(x).
But we recognize in left part derivative of P(x), therefore P'(x)=f(x).
Proof of Part 2. We divide interval [a,b] into n subintervals with endpoints x_0(=a),x_1,x_2,...,x_n(=b) and with width of subinterval Delta x=(b-a)/n. Let F be any antiderivative of f. By subtracting and adding like terms, we can express the total difference in the F values as the sum of the differences over the subintervals: F(b)-F(a)=F(x_n)-F(x_0)=
=F(x_n)-F(x_(n-1))+F(x_(n-2))+...+F(x_2)-F(x_1)+F(x_1)-F(x_0)=
=sum_(i=1)^n(F(x_i)-F(x_(i-1))).
Now F is continuous (because it’s differentiable) and so we can apply the Mean Value Theorem to F on each subinterval [x_(i-1),x_i].
Thus, there exists a number x_i^(**) between x_(i-1) and x_i such that F(x_i)-F(x_(i-1))=F'(x_i^(**))(x_i-x_(i-1))=f(x_i^(**)) Delta x.
Therefore, F(b)-F(a)=sum_(i=1)^n f(x_i^(**))Delta x .
Now we take the limit of each side of this equation as n->oo. The left side is a constant and the right side is a Riemann sum for the function f, so F(b)-F(a)=lim_(n->oo) sum_(i=1)^n f(x_i^(**)) Delta x=int_a^b f(x)dx .
This finishes proof of Fundamental Theorem of Calculus.
When using Evaluation Theorem following notation is used: F(b)-F(a)=F(x)|_a^b=[F(x)]_a^b .
We already talked about introduced function P(x)=int_a^x f(t)dt.
We will talk about it again because it is new type of function. It is just like any other functions (power or exponential): for any x int_a^xf(t)dt gives definite number. Sometimes we can represent P(x) in terms of functions we know, sometimes not.
For example, we know that (1/3x^3)'=x^2, so according to Fundamental Theorem of calculus P(x)=int_0^x t^2dt=1/3x^3-1/3*0^3=1/3x^3. Here we expressed P(x) in terms of power function.
But we can't represent in terms of elementary functions, for example, function P(x)=int_0^x e^(x^2)dx, because we don't know what is antiderivative of e^(x^2). What we can do is just to value of P(x) for any given x.
Geometrically P(x) can be interpreted as the net area under the graph of f from a to x, where x can vary from a to b. (Think of g as the "area so far" function).
Example 1. Graph of f is given below. If P(x)=int_0^xf(t)dt, find P(0), P(1), P(2), P(3), P(4), P(6), P(7). Sketch the rough graph of P.
We immediately have that P(0)=int_0^0f(t)dt=0. We can see that P(1)=int_0^1 f(t)dt is area of triangle with sides 1 and 2. Therefore, P(1)=1/2 *1*2=1.
We see that P(2)=int_0^2f(t)dt is area of triangle with sides 2 and 4 so P(2)=1/2*2*4=4.
Area from 0 to 3 consists of area from 0 to 2 and area from 2 to 3 (triangle with sides 1 and 4):
P(3)=int_0^3f(t)dt=int_0^2f(t)dt+int_2^3f(t)dt=4+1/2*1*4=6.
Similarly P(4)=P(3)+int_3^4f(t)dt. But area of triangle on interval [3,4] lies below x-axis so we subtract it: P(4)=6-1/2*1*4=4.
Now P(5)=P(4)+int_4^5 f(t)dt=4-1/2*1*4=2.
P(6)=P(5)+int_5^6f(t)dt=2+1/2*1*4=4.
Finally, P(7)=P(6)+int_6^7 f(t)dt where int_7^6 f(t)dt is area of rectangle with sides 1 and 4. So, P(7)=4+1*4=8.
Sketch of P(x) is shown below.
Example 2. If P(x)=int_1^x t^3 dt , find a formula for P(x) and calculate P'(x).
Using part 2 of fundamental theorem of calculus and table of indefinite integrals we have that P(x)=int_1^x t^3 dt=(t^4/4)|_1^x=x^4/4-1/4.
Now, P'(x)=(x^4/4-1/4)'=x^3. We see that P'(x)=f(x) as expected due to first part of Fundamental Theorem.
Example 3. Find derivative of P(x)=int_0^x sqrt(t^3+1)dt.
Using first part of fundamental theorem of calculus we have that g'(x)=sqrt(x^3+1).
Example 4. Find d/(dx) int_2^(x^3) ln(t^2+1)dt.
Here we have composite function P(x^3). To find its derivative we need to use Chain Rule in addition to Fundamental Theorem.
Let u=x^3 then (du)/(dx)=(x^3)'=3x^2.
d/(dx) int_2^(x^3) ln(t^2+1)dt=d/(du) int_2^u ln(t^2+1) *(du)/(dx)=d/(du) int_2^u ln(t^2+1) *3x^2=
=ln(u^2+1) *3x^2=ln((x^3)^2+1) *3x^2=3x^2ln(x^6+1).
Now, a couple examples concerning part 2 of Fundamental Theorem.
Example 5. Calculate int_0^5e^xdx.
Using part 2 of fundamental theorem of calculus and table of indefinite integrals we have that int_0^5e^x dx=e^x|_0^5=e^5-e^0=e^5-1.
Example 6. Calculate int_0^(pi/2)cos(x)dx.
Using part 2 of fundamental theorem of calculus and table of indefinite integrals (antiderivative of cos(x) is sin(x)) we have that int_0^(pi/2)cos(x) dx=sin(x)|_0^(pi/2)=sin(pi/2)-sin(0)=1.
Example 7. Find int_0^2 (3x^2-7)dx.
Using properties of definite integral we can write that int_0^2(3x^2-7)dx=int_0^2 3x^2dx-int_0^2 7dx=3 int_0^2 x^2dx-7 int_0^2 7dx=
=3 (x^3/3)|_0^2-7*(2-0)=3 (8/3 -0/3)-14=-6.
Example 8. Find int_1^3 ((2t^5-8sqrt(t))/t+7/(t^2+1))dt .
First rewrite integral a bit: int_1^3 ((2t^5-8sqrt(t))/t+7/(t^2+1))dt=int_1^3 (2t^4-8t^(-1/2)+7/(t^2+1))dt
So, int_1^3 (2t^4-8t^(-1/2)+7/(t^2+1))dt=(2/5 t^5-16sqrt(t)+7tan^(-1)(t))|_1^3=
=(2/5 (3)^5-16sqrt(3)+7tan^(-1)(3))-(2/5 (1)^5-16sqrt(1)+7tan^(-1)(1))=
=564/5-16sqrt(3)-(7pi)/4+7tan^(-1)(3)~~88.3327.
Related Calculator: Definite and Improper Integral Calculator
|
2015-03-30 12:53:10
|
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|
https://www.math-only-math.com/smaller-units-to-bigger-units.html
|
# Smaller Units to Bigger Units
To convert a smaller unit into a bigger unit, we move the decimal point to the left. In other words, we can say that we divide.
This is very important for us to learn how to convert smaller units into bigger units. We use it often in our daily life.
Measurement of length, mass and capacity tables can be represented in the form of a place-value chart as shown below.
Converting Smaller Units of Length into Bigger Units of Length:
For Example:
1. Convert 80 mm into cm.
Solution:
We know that 10 mm = 1 cm
80 mm = $$\frac{80}{10}$$ cm
= 8 cm
2. Convert 485 mm into cm.
Solution:
485 mm = 480 mm + 5 mm
We know that 10 mm = 1 cm
480 mm + 5 mm = $$\frac{480}{10}$$ cm + 5 mm
= 48 cm 5 mm
3. Convert 15000 m into km.
Solution:
We know that 1000 m = 1 km
So, 15000 m = $$\frac{15000}{1000}$$ km
= 15 km
3. Convert 577 centimetres to decametres
1000 cm = 1 dam
577 cm = (577 ÷ 1000) dam
577 cm = 0.577 dam
Converting Smaller Units of Mass into Bigger Units of Mass:
To convert lower units of mass to higher units, we multiply by 1000. A quick way to convert is to write three digits from right as lower unit and the remaining as higher unit.
For example:
1. Convert 14000 mg into g.
We know that 1000 mg = 1 g
So, 14000 mg = $$\frac{14000}{1000}$$ g
= 14 g
2. Express 3180 g as kg.
3180 g = 3000 g + 180 g
= $$\frac{3000}{1000}$$ kg + 180 g
= 3 kg 180 g
3. Convert 7000 grams to kilograms.
1000 g = 1 kg
7000 g = (7000 ÷ 1000) kg
7000 g = 7 kg
4. Convert 225 kilograms to quintals
100 kg = 1 quintal
225 kg = (225 ÷ 100) quintals
225 kg = 2.25 quintals
5. Convert 36 quintals to tonnes
10 quintals = 1 tonne
36 quintals = (36 ÷ 10) tonnes
36 quintals = 3.6 tonnes
6. Convert 415 kilograms to tonnes
1000 kg = 1 tonne
415 kg = (415 ÷ 1000) tonnes
415 kg = 0.415 tonnes
Converting Smaller Units Capacity into Bigger Units Capacity:
To convert milliliters into liters, we divide the number of milliliters (ml) by 1000. A quick way to convert ml into l is to write three digits from right as ml and the remaining as l.
For example:
1. Convert 76489 ml into l.
76489 ml = 76000 ml + 489 ml
= $$\frac{76000}{1000}$$ + + 489 ml
= 76 l 489 ml
2. Convert 375 litres to kilolitres.
1000 l = 1 kl
375 l = (375 ÷ 1000) kl
375 l = 0.375 kl
Alternate Method:
Conversion of Lower Units to Higher Units:
When a lowers unit is changed into a higher unit we divide the number of the lowers unit by the number showing the relationship between the two units.
For example:
1. 5728 g = 5000 g 728 g5728 g = 5 kg 728 g5728 g = 5.728 g
Also 5728 g = 5000 g + 700 g + 20 g + 8 g
= 5 kg 7 hg 2 dag 8 g
2. 328 cm = 300 cm + 28 cm328 cm = 3 m 28 cm328 cm = 3.28 m
Also 328 cm = 300 cm + 20 cm + 8 cm
= 3 m 2 dm 8 cm
Solved examples for the conversion of smaller units to bigger units:
1. Convert 9362.8 grams into the following units.
(i) Decagrams
(ii) hectograms
(iii) kilograms
Solution:
Because 1 dag = 10 g
So, 1 g = $$\frac{1}{10}$$ dag
So, 9362.8 g = $$\frac{9362.8}{10}$$ = (9362.8 ÷ 10) dag = 936.28 dag
Thus,
(i) 9362.8 g = = (9362.8 ÷ 10) dag = 936.28 dag
(ii) 9362.8 g = = (9362.8 ÷ 100) hg = 93.628 hg, (Because 1 g = $$\frac{1}{100}$$ hg)
(ii) 9362.8 g = = (9362.8 ÷ 1000) kg = 9.3628 kg, (Because 1 g = $$\frac{1}{1000}$$ kg)
2. Convert 2345 millimetres into the following units.
(i) centimetres
(ii) metres
(iii) kilometres
Solution:
(i) 2345 millimetres = (2345 ÷ 10) = 234.5 centimetres, [Because 1 mm = $$\frac{1}{10}$$ cm]
(ii) 2345 millimetres = (2345 ÷ 1000) = 2.345 metres, [Because 1 mm = $$\frac{1}{1000}$$ m]
(iii) 2345 millimetres = (2345 ÷ 1000000) = 0.002345 kilometres, [Because 1 mm = $$\frac{1}{1000000}$$ km]
Let us consider another example involving different types of conversions.
3. Convert the following:
(i) 3598 mm to m
(ii) 4683254 mg to dg
(iii) 5923 ml to cl
Solution:
(i) 3598 mm
= (3598 ÷ 1000) m, [Because 1 mm = $$\frac{1}{1000}$$ m]
= 3.598 m
(ii) 4683254 mg
= (4683254 ÷ 100) dg, [Because 1 mg = $$\frac{1}{100}$$ dg]
= 46832.54 dg
(iii) 5923 ml
= (5923 ÷ 10) cl, [Because 1 ml = $$\frac{1}{10}$$ cl]
= 592.3 cl
4. Convert 12500 m into km.
Solution:
We know that 1000 m = 1 km
12500 m = 12000 m + 500 m
= $$\frac{12000}{1000}$$ km + 500 m
= 12 km 500 m
Questions and Answers on Smaller Units to Bigger Units:
I. Convert the given lengths:
(i) 40 mm = ………….. cm
(ii) 540 cm = ………….. m ………….. cm
(iii) 160 mm = ………….. cm
(iv) 1250 m = ………….. km ………….. m
(v) 10500 cm = ………….. m
(vi) 3500 cm = ………….. m ………….. cm
(vii) 612 cm = ………….. m ………….. cm
(viii) 41752 m = ………….. km ………….. m
I. (i) 4 cm
(ii) 5 m 40 cm
(iii) 16 cm
(iv) 1 km 250 m
(v) 105 m
(vi) 35 m 0 cm
(vii) 6 m 12 cm
(viii) 41 km 752 m
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2022-11-27 18:58:00
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https://unapologetic.wordpress.com/2009/10/15/
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# The Unapologetic Mathematician
## Clairaut’s Theorem
Now for the most common sufficient condition ensuring that mixed partial derivatives commute. If $f$ is a function of $n\geq2$ variables, we can for the moment hold the values of all but two of them constant. We’ll only consider two variables at a time, which will simplify our notation. For the moment, then, we write $f(x,y)$. We will also assume that $f$ is real-valued, and deal with vector values one component at a time.
I assert that if the partial derivatives $D_xf$ and $D_yf$ are continuous in a neighborhood of the point $(a,b)$, and if the mixed second partial derivative $D_{y,x}f$ exists and is continuous there, then the other mixed partial derivative $D_{x,y}f$ exists at $(a,b)$, and we have the equality
$\displaystyle\left[D_{x,y}f\right](a,b)=\left[D_{y,x}f\right](a,b)$
By definition, within the neighborhood in the statement of the theorem the partial derivative $\frac{\partial f}{\partial y}$ is given by the limit
$\displaystyle\left[D_yf\right](x,y)=\lim\limits_{k\to0}\frac{f(x,y+k)-f(x,y)}{k}$
So the numerator of the difference quotient defining the desired mixed partial derivative is
\displaystyle\begin{aligned}\left[D_yf\right](a+h,b)-\left[D_yf\right](a,b)=&\lim\limits_{k\to0}\frac{f(a+h,b+k)-f(a+h,b)}{k}\\-&\lim\limits_{k\to0}\frac{f(a,b+k)-f(a,b)}{k}\end{aligned}
For a fixed $k$, we define the function
$\displaystyle g_k(t)=f(a+t,b+k)-f(a+t,b)$
We compute the derivative of $g_k$ as
$\displaystyle g_k'(t)=\left[D_xf\right](a+t,b+k)-\left[D_xf\right](a+t,b)$
so we can apply the mean value theorem to write
$\displaystyle\left[D_yf\right](a+h,b)-\left[D_yf\right](a,b)=\lim\limits_{k\to0}\frac{g_k(h)-g_k(0)}{k}=\lim\limits_{k\to0}\frac{hg_k'(\bar{h})}{k}$
for some $\bar{h}$ between ${0}$ and $h$. We use the above expression for $g_k'$ to write the difference quotient
$\displaystyle\frac{\left[D_yf\right](a+h,b)-\left[D_yf\right](a,b)}{h}=\lim\limits_{k\to0}\frac{\left[D_xf\right](a+\bar{h},b+k)-\left[D_xf\right](a+\bar{h},b)}{k}$
In a similar trick to the one above, we can see that $\left[D_xf\right](a+\bar{h},b+s)$ is differentiable as a function of $s$ with derivative $\left[D_{y,x}f\right](a+\bar{h},b+s)$. And so the mean value theorem tells us that we can write our difference quotient as
$\displaystyle\frac{\left[D_yf\right](a+h,b)-\left[D_yf\right](a,b)}{h}=\lim\limits_{k\to0}\left[D_{y,x}f\right](a+\bar{h},\bar{y})$
for some $\bar{y}$ between $b$ and $b+k$.
And so we come to try taking the limit
$\displaystyle\lim\limits_{h\to0}\lim\limits_{k\to0}\left[D_{y,x}f\right](a+\bar{h},\bar{y})=\left[D_{y,x}f\right](a,b)$
If $\bar{h}$ didn’t depend in its definition on $k$, this would be easy. First we could let $k$ go to zero, which would make $\bar{y}$ go to $b$, and then letting $h$ go to zero would make $\bar{h}$ go to zero as well. But it’s not going to be quite so easy, and limits in two variables like this usually call for some delicacy.
Given an $\epsilon>0$, there (by the assumption of continuity) is some $\delta>0$ so that
$\displaystyle\lvert\left[D_{y,x}f\right](x,y)-\left[D_{y,x}f\right](a,b)\rvert<\frac{\epsilon}{2}$
for $(x,y)$ within a radius $\delta$ of $(a,b)$. As long as we keep $\lvert h\rvert$ and $\lvert k\rvert$ below $\frac{\delta}{2}$, the point $(a+\bar{h},\bar{y})$ will be within this radius. So we can keep $h$ fixed at some small enough value, and find that $\lvert k\rvert<\frac{\delta}{2}$ implies the inequality
$\displaystyle\lvert\left[D_{y,x}f\right](a+\bar{h},\bar{y})-\left[D_{y,x}f\right](a,b)\rvert<\frac{\epsilon}{2}$
Now we can take the limit as $k$ goes to zero. As we do so, the inequality here may become an equality, but since we kept it below $\frac{\epsilon}{2}$, we still have some wiggle room. So, if $\lvert h\rvert<\frac{\delta}{2}$, we have the inequality
$\displaystyle\left\lvert\lim\limits_{k\to0}\left[D_{y,x}f\right](a+\bar{h},\bar{y})-\left[D_{y,x}f\right](a,b)\right\rvert\leq\frac{\epsilon}{2}<\epsilon$
which gives us the limit we need.
Of course we could instead assume that the second mixed partial derivative exists and is continuous near $(a,b)$, and conclude that the first one exists and is equal to the second.
October 15, 2009 Posted by | Analysis, Calculus | 14 Comments
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2016-07-23 16:57:17
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http://toughi.lajvar.se/g9q6n/551601-finding-median-from-histogram-worksheet
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# finding median from histogram worksheet
This resource challenges pupils to combine their knowledge of finding averages from grouped data with what they know about histograms. Two sets of numbers that you work out for mean, median, mode, range, Q1, Q3, IQR, histogram… They will revisit how to find the mean, mode and median from a grouped data set as well as how to extract this same information from a histogram. 100/2 = 50 50 th value.. 50 th value lies in the 31 - 40 class - i.e. You can get both the mean and the median from the histogram. • To find the mean, add up all the numbers and divide by the number of numbers. Note that the answers are on the 2nd page of the PDF. You will need paper and a … The median is the middle term in the arranged data set. To remember the definition of a median, just think of the median of a road, which is the middlemost part of the road. The median … Interesting word problems are included in each section. The median is the midpoint of the value, which means that at the median there are exactly half the data below and above that point. • To find the median, place all the numbers in order and select the middle number. This can be visualized in many different ways, including the bar chart below for median income given by the office for national statistics in the UK. Please help. ; To estimate the Mean use the midpoints of the class intervals: . Median Worksheet 1. finding median for ungrouped data Median is the value which occupies the middle position when all the observations are arranged in an ascending or descending order. Central Tendencay Worksheet or Quiz Includes worksheets with and without number sets. Displaying top 8 worksheets found for - Median Of A Histogram. Serial order wise Ex 14.1 Ex 14.2 Ex 14.3 Ex 14.4 Examples . Find the bin(s) containing the middle item(s). For stdev subtract the avg from each value squared times the frequency and for The median is the middle number in an ordered set. To estimate the Median use: . The median is the middle item or the average of the two middle items. Summary. But in the histogram the hint is confusing me. The median is also the number that is halfway into the set. Study each of these problems carefully; you will see similar problems on the lesson knowledge check. If you know how many numbers there are in a set, which is the middle number? Let’s say, however, that you also want to publish the information on these weights for an audience that uses the imperial system rather than the metric system of measurements. Determine if that number is even or odd. 1) 37 42 48 51 52 53 54 This pack contains a teaching PowerPoint, worksheet and lesson plan. The formula for finding the middle number is: In this formula, n is how many numbers there are in the set. Finding the median from a table. In other words, 299.5-399.5 is the median class, i.e. The pdf exercises are curated for students of grade 3 through grade 8. From the last item of the third column, we have In order to plot a cumulative frequency graph, we have to plot cumulative frequency against the upper-class-boundary of each class. Concept wise Collecting Data Raw Data Ungrouped Data Grouped Data Note that because the set has 14 members, the median is the mean of two central values. Find the total number of items represented by the histogram 2. It is a positional average. ... Hand out the worksheet/recording sheet packet to every student. Remember that the histogram differs from a bar chart in that it is the area of the bar that denotes the value, not the height. What does that mean 43 is the median of the frequencies, but it's not the median of the values. • The mode is the number which appears most often. To find the median, the data should first be arranged in order from least to greatest. Drawing frequency polygon - with and without a histogram; Finding mean, median and mode of raw data . Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 1 Center and Spread of Data Name_____ Date_____ Period____ ... Find the mode, median, mean, lower quartile, upper quartile, interquartile range, and population standard deviation for each data set. Starting with , add the frequencies in the table starting with the first row until you reach . When I first start to teach measures of central tendency and compare mean and median, finding the median can be fun! The 3 histograms below show the batting averages of the winners of the batting title in the major league baseball (for both the American & National leagues) for certain years in the 1900s. We use linear interpolation to find it. The median class interval is the corresponding class where the median value falls. About This Quiz & Worksheet. Click on a NCERT Exercise below, or start the chapter from the concepts given below. The median value is the middle value when all items are in order. 3. Finding The Median Using Histograms - Displaying top 8 worksheets found for this concept.. Solution: Let the frequency of the class 30 – 40 be f 1 and that of 50 – 60 be f 2 . The curve should look like the following: Finding the median. Mean, median, mode and range worksheets contain printable practice pages to determine the mean, median, mode, range, lower quartile and upper quartile for the given set of data. Batting average shows the percent (written as a decimal) of the time a certain player gets a hit. In the formula, we aren’t “finding the N/2th observation”; we are measuring 1/2 on a continuous scale from 0 to N. Pramod used the (N+1)/2th observation, and got the same result. the class containing the median value. Determine the number of the middle item. How to get the Median from a Frequency table with Class Intervals, how to find the median of a frequency table when the number of observations is even or odd, how to find the median for both discrete and grouped data, find the mean, mode and median from a frequency distribution table, with video lessons, examples and step-by-step solutions. This means you will have to convert your units into pounds from kilograms, multiplying each observation value in your data by 2.2 to get an approximate weight. Finding the Mean, Median, Mode Practice Problems Now you get a chance to work out some problems. The mean, median, and mode of this distribution are equal at about 66.5 inches. Here, 1977 is used as the “base” year which is equal to 100. Histogram Worksheet. For the median of the values, if you sum up all the frequencies below the median, and all the frequencies above the median, you should get the same number. using the formula for median we have, Median or where, (lower class boundary of the median class), (total frequency), ( less than type cumulative frequency corresponding to ), How to interpret and draw a histogram? 1. Print median worksheet 1 with answers in PDF format. What is a Histogram? Estimated Mean = Sum of (Midpoint × Frequency)Sum of Frequency. The median is the middle value; uniformly spread data will provide that the area of the histogram on each side of the median will be equal. The median is the n/2 th value. The total area of this histogram is $10 \times 25 + 12 \times 25 + 20 \times 25 + 8 \times 25 + 5 \times 25 = 55 \times 25 = 1375$. The Median is the value of the middle in your list. Feb 19, 2015 - How to find the mean, median, and mode from a histogram. Use this quiz/worksheet combo to help test your understanding of finding the median. For grouped data, we cannot find the exact Mean, Median and Mode, we can only give estimates. Median Of A Histogram - Displaying top 8 worksheets found for this concept.. The way to calculate the mean is that illustrated in the video and already shown in one of the comments. A positive skewed histogram suggests the mean is greater than the median. the median class is the class for which upper class boundary is . The mean, median and mode are three different ways of describing the average. anywhere between 30.5 and 40.5. Now I want to see what happens when I add male heights into the histogram: {102, 109, 207, 357, 360, 403, 471, 483, 670, 729, 842, 843, 920, 941} Now, calculate the median M by finding the mean of 471 and 483. For each histogram bar, we start by multiplying the central x-value to the corresponding bar height. Again, the definition of the median for a continuous distribution is the value such that the cumulative probability is 1/2; we just multiplied N by that. Median 24.42 Quartile1 19.70 Quartile2 29.55 To find the first four quantities it's standard to use the midpoint of the ranges for the calculations. Answer (1 of 5): A histogram tells you how many items fall into each of several bins. D. Russell. Sample some of these worksheets for free! The histogram above shows a distribution of heights for a sample of college females. So for the avg add 2.5 to each value in the first column multiplied by the relative frequency. Given that the median value is 46, determine the missing frequencies using the median formula. Positive skewed histograms. These values are underlined in the ordered set below. Estimated Median = … You may use a calculator if you would like. Learn More. The Mean, the Median, and the Mode are all measures of Central Tendency. When constructing a histogram with non-uniform (unequal) class widths, we must ensure that the areas of the rectangles are proportional to the class frequencies. When the shape of the distribution is symmetric and unimodal, the mean, median, and mode are equal. Distribution is symmetric and unimodal, the mean, median and mode of raw data of these problems carefully you! All items are in a set, which is equal to 100 is: this! From each value in the ordered set below frequencies, but it not! To every student Using the median formula the frequency and for Positive skewed Histograms use this quiz/worksheet combo help. Is that illustrated in the set has 14 members, the median is the middle item ( s containing. Frequencies, but it 's not the median value is 46, determine missing... Mean is that illustrated in the table starting with the first column multiplied by the histogram hint. 53 54 the median, place all the numbers and divide by the number is! But it 's not the median is also the number of numbers use the midpoints of comments... Of describing the average of the class for which upper class boundary...., we start by multiplying the central x-value to the corresponding bar height, 299.5-399.5 is the in. Curve should look like the following: finding the median from the histogram shows. Than the median of the frequencies in the histogram the hint is confusing me describing the average of the.... That is halfway into the set has 14 members, the data should first be in! 2Nd page of the comments class interval is the mean and the median is the middle term in ordered... We start by multiplying the central x-value to the corresponding class where median... For stdev subtract the avg add 2.5 to each value in the ordered set - how to find the number. Histogram the hint is confusing me shows a distribution of heights for a sample college! ( written as a decimal ) of the comments class 30 – 40 be f 1 that! Mean 43 is the median Using Histograms - displaying top 8 worksheets found for - median the... Ncert Exercise below, or start the chapter from the histogram 2 sample of college females wise Ex 14.1 14.2. On a NCERT Exercise below, or start the chapter from the concepts given.... The mean is greater than the median value is 46, determine missing! The missing frequencies Using the median, and mode of raw data Midpoint × frequency Sum! Are curated for students of grade 3 through grade 8: finding the median of the class which... Is that illustrated in the 31 - 40 class - i.e numbers order... Class where the median, place all the numbers in order of.... From least to greatest missing frequencies Using the median is also the number of.. The value of the class 30 – 40 be f 2 finding median from histogram worksheet Positive skewed Histograms class 30 – be. Test your understanding of finding the median class, i.e grade 8 a calculator if you would like look... 5 ): a histogram finding the median three different ways of describing the average of the class for upper... To find the median from a histogram ; finding mean, median, the,! All measures of central Tendency • the mode is the median value is the median value is the term! Into the set a hit mean of two central values numbers and divide by the histogram shows... Is used as the “ base ” year which is equal to 100 and already shown one! You can get both the mean use the midpoints of the frequencies, but it 's not the median and. Median value is the middle item or the average a histogram tells you many... Ex 14.4 Examples the two middle items for - median of the two middle items are underlined the! How to find the mean, median and mode are equal at about 66.5 inches at about inches. And select the middle item ( s ) containing the middle number • the mode are different! First be arranged in order and select the middle number in an ordered below... Let the frequency and for Positive skewed histogram suggests the mean, median mode! Mode from a table because the set the bin ( s ) containing the number... … Feb 19, 2015 - how to find the median class is the median from concepts... “ base ” year which is the middle number skewed histogram suggests the and. 51 52 53 54 the median is the mean, finding median from histogram worksheet, and mode of raw data of... Number of items represented by the relative frequency help test your understanding finding! Halfway into the set has 14 members, the data should first be in. Arranged data set 42 48 51 52 53 54 the median class, i.e distribution is and. From least to greatest 46, determine the missing frequencies Using the median from a table histogram finding! In one of the PDF exercises are curated for students of grade 3 through grade 8 of ( ×! Two central values exercises are curated for students of grade 3 through 8! In PDF format all measures of central Tendency and compare mean and median, and mode this... From the concepts given below distribution is symmetric and unimodal, the data should first arranged. Tells you how many numbers there are in order and select the middle number number appears. Median … finding the median median of a histogram curve should look like the following: finding the class... The 2nd page of the frequencies, but it 's not the median is the middle (... The time a certain player gets a hit of a histogram tells you many... To each value squared times the frequency of the middle item or the average the... Central values frequencies Using the median of the time a certain player gets hit! 43 is the value of the distribution is symmetric and unimodal, the mean,,... Exact mean, median, place all the numbers and divide by the relative frequency lesson plan 14.3 Ex Examples... Chapter from the concepts given below to every student data, we by. For which upper class boundary is, which is the middle term in the arranged set! Ex 14.1 Ex 14.2 Ex 14.3 Ex 14.4 Examples the value of the 30! Frequency ) Sum of frequency 40 be f 2 2.5 to each value squared times the of. Where the median of a histogram tells you how many items fall into each of several.! Two middle items 51 52 53 54 the median class is the middle in... Be arranged in order batting average shows the percent ( written as a decimal ) of the a! Avg from each value squared times the frequency and for Positive skewed suggests... - median of the time a certain player gets a hit the frequencies in the data. The bin ( s ) for Positive skewed Histograms similar problems on the lesson knowledge check,! Middle number n is how many numbers there are in a set, is! To each value in the first column multiplied by the relative frequency every student packet to student! ; to estimate the mean use the midpoints of the class 30 – be! Arranged in order bar height unimodal, the data should first be arranged in order is... Is confusing me that the answers are on the lesson knowledge check and the median is the middle your! Heights for a sample of college females for this concept mode from table... This pack contains a teaching PowerPoint, worksheet and lesson plan average shows the percent written! Arranged in order of finding the median Using Histograms - displaying top worksheets. Wise Ex 14.1 Ex 14.2 Ex 14.3 Ex 14.4 Examples the midpoints of frequencies. The mean, median and mode are three different ways of describing average! With and without a histogram ; finding mean, median, the mean, and... Formula, n is how many items fall into each of several bins the chapter the! And median, and mode from a histogram ; finding mean, add up all the numbers in.... Pack contains a teaching PowerPoint, worksheet and lesson plan class 30 – 40 be f.... Of numbers PowerPoint, worksheet and lesson plan median Using Histograms - displaying top 8 found! Start by multiplying the central x-value to the corresponding bar height with the first row you! Value squared times the frequency of the class 30 – 40 be finding median from histogram worksheet 2 of. Item ( s ) or the average of the class for which upper class boundary is ) 42! Three different ways of describing the average of the class intervals: way to calculate the mean,,., we can not find the mean is greater than the median from a.... Is how many numbers there are in the histogram the hint is confusing.. The 31 - 40 class - i.e problems on the 2nd page of the two middle items teach measures central. The missing frequencies Using the median Using Histograms - displaying top 8 found! Measures of central Tendency mean is that illustrated in the 31 - 40 -!, median and mode are all measures of central Tendency that illustrated in the first column multiplied by histogram! The way to calculate the mean, median, and mode of distribution. Because the set has 14 members, the median from the histogram 2 class i.e... Below, or start the chapter from the concepts given below both the mean, median and mode three.
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2021-06-25 01:13:52
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https://mathematica.stackexchange.com/questions/236196/how-to-plot-a-very-large-table-of-plots-faster?noredirect=1
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# How to plot a very large table of plots faster?
I'm trying to plot the following table of plots.
The number of plots is very very big (236 196 plots) so it would take very long time to run.
I don't know when the running will end.
Is there any way to plot all cases easier or faster?
f1[a1_, a2_, a3_, a4_, a5_, a6_, b1_, b2_, b3_, b4_, b5_, b6_,
d_] := -(((a5 + a2 d - a5 d) (b4 + b1 d - b4 d) - (a4 + a1 d -
a4 d) (b5 + b2 d - b5 d) )/(-(a6 + a3 d - a6 d) (b5 + b2 d -
b5 d) + (a5 + a2 d - a5 d) (b6 + b3 d - b6 d)));
f2[a1_, a2_, a3_, a4_, a5_, a6_, b1_, b2_, b3_, b4_, b5_, b6_,
d_] := (a4 b6 - a6 (-1 + d) (b4 (-1 + d) - b1 d) +
d (a4 b3 - a3 b4 + a1 b6 - 2 a4 b6 +
a3 (-b1 + b4) d + (a1 - a4) (b3 - b6) d)) /(
a6 (-1 + d) (b5 (-1 + d) - b2 d) -
a5 (-1 + d) (b6 (-1 + d) - b3 d) +
d (a2 b6 (-1 + d) - a2 b3 d + a3 (b5 + b2 d - b5 d)));
Table[Quiet@
Plot[{f1[a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6, d],
f2[a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6, d], 1}, {d,
0, 1}, PlotLabel -> {Style[
StringForm[
"a1= a2= a3= a4= a5= a6= b1= b2= b3= \
b4= b5= b6= ", a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5,
b6], FontFamily -> "Times", FontSize -> 30, Blue, Bold],
"\n",
Style[StringForm[ "A = B= ",
FullSimplify[
f1[a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6, d]],
FullSimplify[
f2[a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6, d]]],
FontFamily -> "Times", FontSize -> 30, Blue, Bold]] },
ImageSize -> 1000,
PlotStyle -> {Directive[Red, Thick], Directive[Blue, Thick],
Directive[Blue, Dashed]},
Ticks -> Automatic,
TicksStyle -> Directive[Black, Bold, 20],
GridLines -> Automatic,
GridLinesStyle -> LightGray
]
, {a1, {0, 1}}, {a2, {-1, 0, 1}}, {a3, {-1, 0, 1}}, {a4, {0,
1}}, {a5, {-1, 0, 1}}, {a6, {-1, 0, 1}},
{b1, {0, 1}}, {b2, {-1, 0, 1}}, {b3, {-1, 0, 1}}, {b4, {0,
1}}, {b5, {-1, 0, 1}}, {b6, {-1, 0, 1}}] // Flatten //
Partition[#, 2] & // Grid
• Your table would actually generate 104,976 plots. Plots are used to present data for human visual consumption, but I cannot see a situation in which a human viewer could sift through that many plots and make any sense of them. Simply put, I don't see the point of what you are trying to do. Why do you need to plot that many plots? What do you want to do with them, since you pretty clearly won't be looking at them one by one? Dec 9 '20 at 15:43
• I will look at them one by one for my project Dec 9 '20 at 15:49
• What are you looking for though? If you describe the actual problem you are trying to solve, then perhaps we could build a way to have Mathematica recognize and select what you need, instead of having you look at plots for hours. Dec 9 '20 at 15:51
• Consider also that each plot won't take long itself, but you included TWO FullSimplify calls in each of them. Those will take a long time compared to plotting and they are the main bottleneck in your code. Can you do without them? Dec 9 '20 at 15:54
• I think I can do without that. I want to plot and see each case. One thing that can be simplified is that the plot where the function f1 is less than 0 for ALL d values should be omitted. Is there any way to do that? Dec 9 '20 at 16:17
It takes me about 1 minute to plot 2k of them without FullSimplify and 2 minutes with (50 - 100 minutes total). However, 100k plots takes a lot of memory. I'm using 1 GB just for those 2k. Mathematica will also take ages to format those for display, so don't try to display all 100k at once. I don't see the point, though: $$104,976 \text{ plots} \cdot \frac{1 \text{ minute}}{6 \text{ plots}} \cdot \frac{1 \text{ hour}}{60 \text{ minutes}} \cdot \frac{1 \text{ day}}{24 \text{ hours}} = 12.15 \text{ days}$$ (no sleeping!) And can you really say anything insightful about a graph after only looking at it for 10 seconds?
Computation should make your life easier, not harder. In the end, no one else will ever see all these plots. It seems to me that your actual problem (X) is that you need to compare or summarize the results of these graphs in some way, and you think the best way (Y) is to generate all of them and sort or analyze them by hand somehow (see XY Problem). If we knew what the end result was, we might be able to help you skip straight to the final result and literally save you weeks of poring over graphs.
For example you generate a huge number of plots that always reduce to Indeterminate, ComplexInfinity, 1, 0, -1, etc. Do you really need to see a plot of those?
Here's where I would start in order to at least start to get a grip on what's going on.
tup = Tuples[{
{0, 1}, {-1, 0, 1}, {-1, 0, 1}, {0, 1}, {-1, 0, 1}, {-1, 0, 1},
{0, 1}, {-1, 0, 1}, {-1, 0, 1}, {0, 1}, {-1, 0, 1}, {-1, 0, 1}
}];
resultsf1 = Quiet@FullSimplify[f1[##, d]] & @@@ tup;
resultsf2 = Quiet@FullSimplify[f2[##, d]] & @@@ tup;
keyvalsf1 = {Keys[#], Values[#]}\[Transpose] &@Counts[resultsf1];
keyvalsf2 = {Keys[#], Values[#]}\[Transpose] &@Counts[resultsf2];
cf1 = Cases[
keyvalsf1,
{ComplexInfinity | Indeterminate | _?NumberQ, _}
]
cf2 = Cases[
keyvalsf2,
{ComplexInfinity | Indeterminate | _?NumberQ, _}
]
Total[cf1[[All, 2]]]
Total[cf2[[All, 2]]]
Length[keyvalsf1]
Length[keyvalsf2]
$$\left( \begin{array}{cc} \text{Indeterminate} & 2253 \\ 0 & 12322 \\ \text{ComplexInfinity} & 5800 \\ -1 & 3957 \\ -\frac{1}{2} & 576 \\ 1 & 3956 \\ \frac{1}{2} & 576 \\ 2 & 200 \\ -2 & 200 \\ \end{array} \right)$$
$$\left( \begin{array}{cc} \text{Indeterminate} & 2204 \\ 0 & 12333 \\ \text{ComplexInfinity} & 5836 \\ 1 & 3959 \\ \frac{1}{2} & 576 \\ -1 & 3956 \\ -\frac{1}{2} & 576 \\ 2 & 200 \\ -2 & 200 \\ \end{array} \right)$$
29840
29840
2977
3151
I have Tuples generate every possible pairing of values. This gives the same result as your Table but doesn't require Flatten and it's easier to go back and compare the final result with the arguments in tup. I think you should be a bit shocked by some of those numbers. You can see that of the 100k plots, about 30% will either have a flat line or display nothing for one of f1 or f2. Since those 30k graphs only have 1 of 9 possible outcomes, a lot of computation has been done to generate 4 bits of information per graph.
What's worse are those last 2 numbers. They're telling us that there are only about 3k unique graphs for each of f1 and f2. So you would view at least 98k graphs and not learn anything. We can also find out how many times f1 and f2 will produce the same boring graph:
sel = Select[{resultsf1, resultsf2}\[Transpose], #[[1]] === #[[2]] &];
Cases[sel,
{ComplexInfinity | Indeterminate | _?NumberQ,
ComplexInfinity | Indeterminate | _?NumberQ}] // Length
13066
or in how many cases the graph will be boring for both functions but not necessarily identical:
Cases[
{resultsf1, resultsf2}\[Transpose],
{
ComplexInfinity | Indeterminate | _?NumberQ,
ComplexInfinity | Indeterminate | _?NumberQ
}] // Length
21600
If you look at the data stored in keyvalsf1/2, you'll be able to see what the unique functions are and how many times each one is repeated. If you wanted, it should also be possible to extract the arguments from tup that correspond to particular solutions, the problem is I'm not sure what you do want.
Finally, if you're really dead-set on analyzing all those graphs and don't want anyone to tell you it's not efficient, here's how you can generate groups of 2k at a time. It's fast enough that you could always start the next group while you're viewing the first 2k and it'll finish before you do.
f1[a1_, a2_, a3_, a4_, a5_, a6_, b1_, b2_, b3_, b4_, b5_, b6_,
d_] := -(((a5 + a2 d - a5 d) (b4 + b1 d - b4 d) - (a4 + a1 d -
a4 d) (b5 + b2 d - b5 d))/(-(a6 + a3 d - a6 d) (b5 + b2 d -
b5 d) + (a5 + a2 d - a5 d) (b6 + b3 d - b6 d)));
f2[a1_, a2_, a3_, a4_, a5_, a6_, b1_, b2_, b3_, b4_, b5_, b6_,
d_] := (a4 b6 - a6 (-1 + d) (b4 (-1 + d) - b1 d) +
d (a4 b3 - a3 b4 + a1 b6 - 2 a4 b6 +
a3 (-b1 + b4) d + (a1 - a4) (b3 - b6) d))/(a6 (-1 +
d) (b5 (-1 + d) - b2 d) - a5 (-1 + d) (b6 (-1 + d) - b3 d) +
d (a2 b6 (-1 + d) - a2 b3 d + a3 (b5 + b2 d - b5 d)));
tup = Tuples[{
{0, 1}, {-1, 0, 1}, {-1, 0, 1}, {0, 1}, {-1, 0, 1}, {-1, 0, 1},
{0, 1}, {-1, 0, 1}, {-1, 0, 1}, {0, 1}, {-1, 0, 1}, {-1, 0, 1}
}];
Partition[
Quiet@Plot[
{f1[##, d], f2[##, d], 1},
{d, 0, 1},
PlotLabel -> {
Style[
StringForm[
"a1= a2= a3= a4= a5= a6= b1= b2= b3= b4= b5= b6= ", ##],
FontFamily -> "Times",
FontSize -> 30,
Blue,
Bold], "\n",
Style[StringForm["A = B= ", FullSimplify[f1[##, d]],
FullSimplify[f2[##, d]]], FontFamily -> "Times",
FontSize -> 30, Blue, Bold]]}, ImageSize -> 1000,
PlotStyle -> {Directive[Red, Thick], Directive[Blue, Thick], Directive[Blue, Dashed]},
Ticks -> Automatic,
TicksStyle -> Directive[Black, Bold, 20],
GridLines -> Automatic,
GridLinesStyle -> LightGray
] & @@@ tup[[1 ;; 2000]],
2] // Grid
EDIT:
If there's no advantage to looking at duplicates, you can pare the possibilities down quite a bit and programmatically determine their ranges over $$0 \leq d \leq 1$$. I don't know if you can combine f1 and f2, but perhaps this will give you some ideas about how to analyze your results. If we're only looking at perfectly unique functions that are non-negative for at least one point in the domain, we can get it down to 3203 functions.
Re-using my first code block plus your definitions of f1 and f2:
keyvals = {Keys[#], Values[#]}\[Transpose] &@Counts[Join[resultsf1, resultsf2]];
ranges = Quiet@{
#,
Minimize[{#, 0 <= d <= 1}, d][[1]],
Maximize[{#, 0 <= d <= 1}, d][[1]]
} & /@ Cases[keyvals, {Except[Indeterminate | ComplexInfinity], _}][[All, 1]];
sel = Select[ranges, #[[3]] >= 0 &];
sel[[1;;10]]
$$\left( \begin{array}{ccc} 0 & 0 & 0 \\ 1 & 1 & 1 \\ \frac{1}{2} & \frac{1}{2} & \frac{1}{2} \\ \frac{d-1}{d} & -\infty & 0 \\ d-1 & -1 & 0 \\ \frac{1}{2-d}-1 & -\frac{1}{2} & 0 \\ \frac{d-1}{2 d-1} & -\infty & \infty \\ \frac{d-1}{3 d-2} & -\infty & \infty \\ \frac{d-1}{2 d} & -\infty & 0 \\ \frac{d-1}{d+1} & -1 & 0 \\ \end{array} \right)$$
This table gives the function, then the minimum, then the maximum. I think that determining the maximum and minimum programmatically is actually better. A plot will not specifically show you that the value goes to $$\pm \infty$$, you would have to infer it. Plus, this way you can get exact values for the max and min. And of course, you can still plot all 3203 unique functions if you like. At least it should take you 33x less time to analyze all of those graphs than the 100k we started with.
But I hope that you can see that if a question can be well-posed in a computational or mathematical sense, such as "what is the range of a function?", you can often save a lot of time. If you can specify what other things you want to measure, they can probably be automated as well. If it's just a qualitative measure of some kind, then I guess you're stuck looking at the graphs, but hopefully there will be fewer to look at now.
• Thanks but the code doesn't work for me. I understand that it's not efficient but what can I do? What I need is to plot them and look at the range of each plot. Also look at how each curve looks. Dec 9 '20 at 21:39
• @anhnha Which part isn't working? The plotting seems to work fine for me, but make sure to run your definitions for f1 and f2 first as well as my definition of tup. I guess I should edit that so that it can more easily be copied-and-pasted. The range is something that is easily found computationally rather than writing down 100k ranges. "How each curve looks" what does that mean? Good/bad/ugly? Or is there some quantitative definition? At the very least, would it be ok to pare them down to the ~6k that I identified as being unique? Does looking at the duplicates provide any advantage? Dec 9 '20 at 21:53
• I ran but it generated empty plots. I'm running it again but still running. I have to look at each curve, see the range of expression with d not quantitative definition. 6k seems OK. I did similar thing for 2k and checked one by one. Duplicate expression would not provide any advantage. Dec 9 '20 at 22:03
• @anhnha I added some further comments that I think might allow you to get down to ~3200 functions to look at. I've also shown one way that I think you could extract the minimum and maximum over your given domain. Dec 9 '20 at 23:57
• Thanks. I think you misunderstood the condition that #[[3]] >= 0. The selected one can be negative at some values of d but it shouldn't be negative for all values of d. Dec 10 '20 at 1:43
• Generate all values of $$a_1...a_6$$ and $$b_1...b_6$$, then plug them in to f1 to obtain all possible expressions as a function of d only.
abvals = Table[
{a1, a2, a3, a4, a5, a6, b1, b2, b3, b4, b5, b6, d},
{a1, {0, 1}}, {a2, {-1, 0, 1}}, {a3, {-1, 0, 1}},
{a4, {0, 1}}, {a5, {-1, 0, 1}}, {a6, {-1, 0, 1}},
{b1, {0, 1}}, {b2, {-1, 0, 1}}, {b3, {-1, 0, 1}},
{b4, {0, 1}}, {b5, {-1, 0, 1}}, {b6, {-1, 0, 1}}
] // Flatten[#, 11] &;
• FullSimplify those expressions to reduce them to a simpler "standard format".
simplified = FullSimplify[{#, f1 @@ #} & /@ abvals];
simplified // Dimensions
(* Out: {104976, 2} *)
• Group the results by the value of the function (i.e. the expression in d), removing those that resulted in Indeterminate or ComplexInfinity; notice the huge reduction in the cases you now have to consider, from 104,976 to 2,982. Still a lot though.
grouped = KeyDrop[{Indeterminate, ComplexInfinity}]@ GroupBy[Last]@ simplified;
grouped // Dimensions
(* Out: {2982} *)
• Select for further analysis only those expressions that are non-negative (i.e. $$\geq0$$ for all values of d:
selected = KeySelect[grouped, Simplify[# >= 0, Assumptions -> 0 <= d <= 1] &];
selected // Dimensions
(* Out: 306 *)
Now that you have your expressions boiled down to 306 distinct cases, that is a number fow which you can consider hand-picking or visual analysis or whatever else you need to do with these...
• The last condition is reversed actually. I want to remove those expressions that are negative for all values of d. So some expressions that are negative for some values of d are still OK. Dec 9 '20 at 21:44
|
2021-10-27 13:40:14
|
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|
https://www.zbmath.org/?q=an%3A1200.14011
|
# zbMATH — the first resource for mathematics
Maximal Cohen-Macaulay modules over surface singularities. (English) Zbl 1200.14011
Skowroński, Andrzej (ed.), Trends in representation theory of algebras and related topics. Proceedings of the 12th international conference on representations of algebras and workshop (ICRA XII), Toruń, Poland, August 15–24, 2007. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-062-3/hbk). EMS Series of Congress Reports, 101-166 (2008).
The paper contains a lot of material and information about maximal Cohen–Macaulay modules $$M$$ over the local ring $$A$$ of a surface singularity ($$\dim (A)=\text{depth}(M)$$). It starts with basic results and definitions as for instance the depth lemma and the Auslander–Buchsbaum formula, contains Matlis Duality, Grothendieck’s Local Duality and a lot of other stuff from commutative algebra related to the study of maximal Cohen–Macaulay modules. Then general properties of maximal Cohen–Macaulay modules over surface singularities are presented as for instance the fact that in case $$A$$ is normal $$M$$ is maximal Cohen–Macaulay if and only if it is reflexive. It follows a section about maximal Cohen–Macaulay modules over two–dimensional quotient singularities containing the result that a normal surface singularity is a quotient singularity if and only if it has finite Cohen–Macaulay representation type.
The algebraic and the geometric approaches to McKay correspondence for quotient surface singularities as well as its generalization for simply elliptic and cusp singularities are described. A new proof of a result of R.-O. Buchweitz, G.-M. Greuel and F.-O. Schreyer [Invent. Math. 88, 165–182 (1987; Zbl 0617.14034)] (the surface singularities $$A _{\infty}$$ and $$D_{\infty}$$ have countable Cohen–Macaulay representation type) is given. At the end one can find some conjectures concerning the Cohen–Macaulay representation type and an example for a Singular–computation.
For the entire collection see [Zbl 1144.16003].
##### MSC:
14B05 Singularities in algebraic geometry 14J17 Singularities of surfaces or higher-dimensional varieties 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13C14 Cohen-Macaulay modules
SINGULAR
Full Text:
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2021-04-12 22:31:12
|
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|
https://www.gradesaver.com/textbooks/science/chemistry/chemistry-molecular-approach-4th-edition/chapter-8-section-8-4-electron-configurations-valence-electrons-and-the-periodic-table-for-practice-page-347/8-3
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## Chemistry: Molecular Approach (4th Edition)
The electronic configuration of $P$ is $1s^22s^22p^63s^23p^3$ or $[Ne] 3s^23p^3$. The 5 electrons in the $3s^23p^3$ orbitals are the valence electrons. The 10 electrons in the $1s^22s^22p^6$ orbitals are the core electrons.
$P$ has 15 electrons. It has 5 more electrons than its previous noble gas, $Ne$. The electrons are distributed in the orbitals to get the electronic configuration as : $1s^22s^22p^63s^23p^3$ or $[Ne] 3s^23p^3$. The 5 electrons in the $3s^23p^3$ orbitals (outermost orbital) are the valence electrons. The 10 electrons in the $1s^22s^22p^6$ orbitals are the core electrons.
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2018-10-21 13:35:07
|
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https://codegolf.stackexchange.com/questions/10755/shortest-conditional-to-run-a-specific-statement-c-code-golf
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# Shortest conditional to run a specific statement: C code golf
Currently I have the code (expanded for your benefit):
<number> != <number> ? printf("|\\"); : <number>;
The idea here, is to do this condition in the least number of characters. This is currently, roughly the same number of characters it would take to be doing the exact same thing using an if statement, due to the fact, that I am fulfilling the second argument of the ternary operator which does not need to be filled, it is simply filled, as otherwise the statement does not run. (the final number is really a variable).
Is there any possible way to do the same conditional but shorter? (in C)
if(x!=y)printf("foo\n");
x!=y?printf("foo\n"):0;
x==y||printf("foo\n");
Yes, you can save some characters with the ternary operator or with the shortcutting logic operators (&& and ||). All three of the above do the same thing. Which you can use depends on things like the return type of printf and whether you need the return value.
x-y&&printf("foo\n");
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2019-11-20 20:19:03
|
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https://www.projecteuclid.org/euclid.aos/1024691093
|
## The Annals of Statistics
### Breakdown properties of location $M$-estimators
#### Abstract
In this article, we consider the asymptotic behavior of three kinds of sample breakdown points. It is shown that for the location $M$-estimator with bounded objective function, both the addition sample breakdown point and the simplified replacement sample breakdown point strongly converge to the gross-error asymptotic breakdown point, whereas the replacement sample breakdown point strongly converges to a smaller value. In addition, it is proved that under some regularity conditions these sample breakdown points are asymptotically normal. The addition sample breakdown point has a smaller asymptotic variance than the simplified replacement sample breakdown point. For the commonly used redescending $M$-estimators of location, numerical results indicate that among the three kinds of sample breakdown points, the replacement sample breakdown point has the largest asymptotic variance.
#### Article information
Source
Ann. Statist., Volume 26, Number 3 (1998), 1170-1189.
Dates
First available in Project Euclid: 21 June 2002
https://projecteuclid.org/euclid.aos/1024691093
Digital Object Identifier
doi:10.1214/aos/1024691093
Mathematical Reviews number (MathSciNet)
MR1635381
Zentralblatt MATH identifier
0929.62031
Subjects
Primary: 62F35: Robustness and adaptive procedures
#### Citation
Zhang, Jian; Li, Guoying. Breakdown properties of location $M$-estimators. Ann. Statist. 26 (1998), no. 3, 1170--1189. doi:10.1214/aos/1024691093. https://projecteuclid.org/euclid.aos/1024691093
#### References
• Bassett, G. W., Jr. (1991). Equivariant, monotonic, 50% breakdown estimators. Amer. Statist. 45 135-137.
• Chao, M. (1986). On M and P estimators that have breakdown point equal to 12. Statist. Probab. Lett. 4 127-131.
• Coakley, C. W. and Hettmansperger, T. P. (1992). Breakdown bounds and expected test resistance. J. Nonparametr. Statist. 1 267-276.
• Donoho, D. L. and Huber, P. J. (1983). The notion of breakdown point. In A Festschrift for Erich L. Lehmann (P. J. Bickel, K. Doksum and J. L. Hodges, Jr., eds.) 157-184. Wadsworth, Belmont, CA.
• Hampel, F. R. (1968). Contributions to the theory of robust estimation. Ph.D. dissertation, Dept. Statistics, Univ. California, Berkeley.
• Hampel, F. R. (1971). A general qualitative definition of robustness. Ann. Math. Statist. 42 1887- 1896.
• Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics: The Approach Based on Influence Functions. Wiley, New York.
• He, X., Simpson, D. G. and Portnoy, S. L. (1990). Breakdown robustness of tests. J. Amer. Statist. Assoc. 85 446-452.
• He, X., Jureckov´a, J., Koenker, R. and Portnoy, S. L. (1990). Tail behavior of regression estimators and their breakdown points. Econometrica 58 1195-1214.
• Huber, P. J. (1981). Robust Statistics. Wiley, New York.
• Huber, P. J. (1984). Finite sample breakdown point of Mand P-estimators. Ann. Statist. 12 119-126.
• Lopuha¨a, H. P. (1992). Highly efficient estimators of multivariate location with high breakdown point. Ann. Statist. 20 398-413.
• Mosteller, F. and Tukey, J. W. (1977). Data Analy sis and Regression. Addison-Wesley, Reading, MA.
• Okafor, R. (1990). A biweight approach to estimate currency exchange rate: The Nigerian example. J. Appl. Statist. 17 73-82.
• Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
• Pollard, D. (1990). Empirical Processes: Theory and Applications. IMS, Hay ward, CA.
• Rousseeuw, P. J. (1994). Unconventional features of positive-breakdown estimators. Statist. Probab. Lett. 19 417-431.
• Rousseeuw, P. J. and Croux, C. (1994). The bias of k-step M-estimators. Statist. Probab. Lett. 20 411-420.
• Sakata, S. and White, H. (1995). An alternative definition of finite-sample breakdown point with applications to regression model estimators. J. Amer. Statist. Assoc. 90 1099-1106.
• Stromberg, A. J. and Ruppert, D. (1992). Breakdown in nonlinear regression. J. Amer. Statist. Assoc. 87 991-998.
• Ylvisaker, D. (1977). Test resistance. J. Amer. Statist. Assoc. 72 551-556.
• Zhang, J. (1993). Anscombe ty pe theorem for empirical processes. In Proceedings of the First Scientific Conference of Chinese Postdoctors (E. Feng, G. Gai and H. Zeng, eds.) 1244- 1245. National Defence Industry Publishing House, Beijing.
• Zhang, J. (1996). The sample breakdown points of tests. J. Statist. Plann. Inference 52 161-181.
• Zhang, J. (1997). The optimal breakdown M-test and score test. Statistics. To appear.
• Zhang, J. and Li, G. (1993). A new approach to asy mptotic distributions of maximum likelihood ratio statistics. In Statistical Science and Data Analy sis: Proceedings of the Third Pacific Area Statistical Conference (K. Matusita, M. L. Puri and T. Hay akawa, eds.) 325-336. VSP, The Netherlands.
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2020-01-27 05:47:00
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https://math.stackexchange.com/questions/879344/number-of-triangles-formed-by-all-chords-between-n-points-on-a-circle
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# Number of triangles formed by all chords between $n$ points on a circle
We have $n$ point on circumference of a circle. We draw all chords between this points. No three chords are concurrent.
How many triangles exist that their apexes could be on circumference of circle or intersection points of chords and side of triangles is on chords?
Things I have done so far:
There are four situations:
• situation #1:all points on circumference.
• situation #2:two points on circumference,one intersection point.
• situation #3:one points on circumference,two intersection point.
• situation #4:three intersection point.
Situation #1 is easily countable. ${n \choose 3}$
For other situations, I can't find a way for counting them.
Answer (according to answer key): $${n \choose 3} + 4{n \choose 4} + 5{n \choose 5} + {n \choose 6}$$
• possible duplicate of How many triangles are formed by $n$ chords of a circle? – Saharsh Jul 27 '14 at 6:41
• – Saharsh Jul 27 '14 at 6:43
• @Alpha,the OP in this question math.stackexchange.com/questions/313489 don't have problems in figuring out why number of triangles are ${n \choose 3}$+$4{n \choose 4}$+$5{n \choose 5}$+${n \choose 6}$.and the second one you posted is not about combinations. – user2838619 Jul 27 '14 at 6:45
Situation $2$:
For each selection of any $4$ points on the circumference, you can draw the diagram you have. The line segment that joins the adjacent circumference points, could instead join any of the $4$ pairs of adjacent circumference points, so we have $4$ different triangles for each choice of $4$ circumference points.
There are $\binom{n}{4}$ ways to choose the $4$ points, so Situation $2$ contributes:
$\qquad4 \binom{n}{4}$ triangles.
Situation $3$:
For each selection of any $5$ points on the circumference, you can draw the diagram you have. You could choose any of these $5$ points to be a vertex of a Situation $3$ triangle, so we have $5$ different triangles for each choice of $5$ circumference points.
There are $\binom{n}{5}$ ways to choose the $5$ points, so Situation $3$ contributes:
$\qquad5 \binom{n}{5}$ triangles.
Situation $4$:
For each selection of any $6$ points on the circumference, you can draw the diagram you have. There is only one way to construct that internal triangle given these $6$ circumference points.
There are $\binom{n}{6}$ ways to choose the $6$ points, so Situation $4$ contributes:
$\qquad \binom{n}{6}$ triangles.
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2019-05-25 14:56:29
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http://en.turkcewiki.org/wiki/Mathematical_economics
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# Mathematical economics
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. By convention, these applied methods are beyond simple geometry, such as differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, and other computational methods.[1][2] Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.[3]
Mathematics allows economists to form meaningful, testable propositions about wide-ranging and complex subjects which could less easily be expressed informally. Further, the language of mathematics allows economists to make specific, positive claims about controversial or contentious subjects that would be impossible without mathematics.[4] Much of economic theory is currently presented in terms of mathematical economic models, a set of stylized and simplified mathematical relationships asserted to clarify assumptions and implications.[5]
• optimization problems as to goal equilibrium, whether of a household, business firm, or policy maker
• static (or equilibrium) analysis in which the economic unit (such as a household) or economic system (such as a market or the economy) is modeled as not changing
• comparative statics as to a change from one equilibrium to another induced by a change in one or more factors
• dynamic analysis, tracing changes in an economic system over time, for example from economic growth.[2][6][7]
Formal economic modeling began in the 19th century with the use of differential calculus to represent and explain economic behavior, such as utility maximization, an early economic application of mathematical optimization. Economics became more mathematical as a discipline throughout the first half of the 20th century, but introduction of new and generalized techniques in the period around the Second World War, as in game theory, would greatly broaden the use of mathematical formulations in economics.[8][7]
This rapid systematizing of economics alarmed critics of the discipline as well as some noted economists. John Maynard Keynes, Robert Heilbroner, Friedrich Hayek and others have criticized the broad use of mathematical models for human behavior, arguing that some human choices are irreducible to mathematics.
## History
The use of mathematics in the service of social and economic analysis dates back to the 17th century. Then, mainly in German universities, a style of instruction emerged which dealt specifically with detailed presentation of data as it related to public administration. Gottfried Achenwall lectured in this fashion, coining the term statistics. At the same time, a small group of professors in England established a method of "reasoning by figures upon things relating to government" and referred to this practice as Political Arithmetick.[9] Sir William Petty wrote at length on issues that would later concern economists, such as taxation, Velocity of money and national income, but while his analysis was numerical, he rejected abstract mathematical methodology. Petty's use of detailed numerical data (along with John Graunt) would influence statisticians and economists for some time, even though Petty's works were largely ignored by English scholars.[10]
The mathematization of economics began in earnest in the 19th century. Most of the economic analysis of the time was what would later be called classical economics. Subjects were discussed and dispensed with through algebraic means, but calculus was not used. More importantly, until Johann Heinrich von Thünen's The Isolated State in 1826, economists did not develop explicit and abstract models for behavior in order to apply the tools of mathematics. Thünen's model of farmland use represents the first example of marginal analysis.[11] Thünen's work was largely theoretical, but he also mined empirical data in order to attempt to support his generalizations. In comparison to his contemporaries, Thünen built economic models and tools, rather than applying previous tools to new problems.[12]
Meanwhile, a new cohort of scholars trained in the mathematical methods of the physical sciences gravitated to economics, advocating and applying those methods to their subject,[13] and described today as moving from geometry to mechanics.[14] These included W.S. Jevons who presented paper on a "general mathematical theory of political economy" in 1862, providing an outline for use of the theory of marginal utility in political economy.[15] In 1871, he published The Principles of Political Economy, declaring that the subject as science "must be mathematical simply because it deals with quantities." Jevons expected that only collection of statistics for price and quantities would permit the subject as presented to become an exact science.[16] Others preceded and followed in expanding mathematical representations of economic problems.
### Marginalists and the roots of neoclassical economics
Equilibrium quantities as a solution to two reaction functions in Cournot duopoly. Each reaction function is expressed as a linear equation dependent upon quantity demanded.
Augustin Cournot and Léon Walras built the tools of the discipline axiomatically around utility, arguing that individuals sought to maximize their utility across choices in a way that could be described mathematically.[17] At the time, it was thought that utility was quantifiable, in units known as utils.[18] Cournot, Walras and Francis Ysidro Edgeworth are considered the precursors to modern mathematical economics.[19]
#### Augustin Cournot
Cournot, a professor of mathematics, developed a mathematical treatment in 1838 for duopoly—a market condition defined by competition between two sellers.[19] This treatment of competition, first published in Researches into the Mathematical Principles of Wealth,[20] is referred to as Cournot duopoly. It is assumed that both sellers had equal access to the market and could produce their goods without cost. Further, it assumed that both goods were homogeneous. Each seller would vary her output based on the output of the other and the market price would be determined by the total quantity supplied. The profit for each firm would be determined by multiplying their output and the per unit Market price. Differentiating the profit function with respect to quantity supplied for each firm left a system of linear equations, the simultaneous solution of which gave the equilibrium quantity, price and profits.[21] Cournot's contributions to the mathematization of economics would be neglected for decades, but eventually influenced many of the marginalists.[21][22] Cournot's models of duopoly and Oligopoly also represent one of the first formulations of non-cooperative games. Today the solution can be given as a Nash equilibrium but Cournot's work preceded modern game theory by over 100 years.[23]
#### Léon Walras
While Cournot provided a solution for what would later be called partial equilibrium, Léon Walras attempted to formalize discussion of the economy as a whole through a theory of general competitive equilibrium. The behavior of every economic actor would be considered on both the production and consumption side. Walras originally presented four separate models of exchange, each recursively included in the next. The solution of the resulting system of equations (both linear and non-linear) is the general equilibrium.[24] At the time, no general solution could be expressed for a system of arbitrarily many equations, but Walras's attempts produced two famous results in economics. The first is Walras' law and the second is the principle of tâtonnement. Walras' method was considered highly mathematical for the time and Edgeworth commented at length about this fact in his review of Éléments d'économie politique pure (Elements of Pure Economics).[25]
Walras' law was introduced as a theoretical answer to the problem of determining the solutions in general equilibrium. His notation is different from modern notation but can be constructed using more modern summation notation. Walras assumed that in equilibrium, all money would be spent on all goods: every good would be sold at the market price for that good and every buyer would expend their last dollar on a basket of goods. Starting from this assumption, Walras could then show that if there were n markets and n-1 markets cleared (reached equilibrium conditions) that the nth market would clear as well. This is easiest to visualize with two markets (considered in most texts as a market for goods and a market for money). If one of two markets has reached an equilibrium state, no additional goods (or conversely, money) can enter or exit the second market, so it must be in a state of equilibrium as well. Walras used this statement to move toward a proof of existence of solutions to general equilibrium but it is commonly used today to illustrate market clearing in money markets at the undergraduate level.[26]
Tâtonnement (roughly, French for groping toward) was meant to serve as the practical expression of Walrasian general equilibrium. Walras abstracted the marketplace as an auction of goods where the auctioneer would call out prices and market participants would wait until they could each satisfy their personal reservation prices for the quantity desired (remembering here that this is an auction on all goods, so everyone has a reservation price for their desired basket of goods).[27]
Only when all buyers are satisfied with the given market price would transactions occur. The market would "clear" at that price—no surplus or shortage would exist. The word tâtonnement is used to describe the directions the market takes in groping toward equilibrium, settling high or low prices on different goods until a price is agreed upon for all goods. While the process appears dynamic, Walras only presented a static model, as no transactions would occur until all markets were in equilibrium. In practice very few markets operate in this manner.[28]
#### Francis Ysidro Edgeworth
Edgeworth introduced mathematical elements to Economics explicitly in Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences, published in 1881.[29] He adopted Jeremy Bentham's felicific calculus to economic behavior, allowing the outcome of each decision to be converted into a change in utility.[30] Using this assumption, Edgeworth built a model of exchange on three assumptions: individuals are self-interested, individuals act to maximize utility, and individuals are "free to recontract with another independently of...any third party."[31]
An Edgeworth box displaying the contract curve on an economy with two participants. Referred to as the "core" of the economy in modern parlance, there are infinitely many solutions along the curve for economies with two participants[32]
Given two individuals, the set of solutions where the both individuals can maximize utility is described by the contract curve on what is now known as an Edgeworth Box. Technically, the construction of the two-person solution to Edgeworth's problem was not developed graphically until 1924 by Arthur Lyon Bowley.[33] The contract curve of the Edgeworth box (or more generally on any set of solutions to Edgeworth's problem for more actors) is referred to as the core of an economy.[34]
Edgeworth devoted considerable effort to insisting that mathematical proofs were appropriate for all schools of thought in economics. While at the helm of The Economic Journal, he published several articles criticizing the mathematical rigor of rival researchers, including Edwin Robert Anderson Seligman, a noted skeptic of mathematical economics.[35] The articles focused on a back and forth over tax incidence and responses by producers. Edgeworth noticed that a monopoly producing a good that had jointness of supply but not jointness of demand (such as first class and economy on an airplane, if the plane flies, both sets of seats fly with it) might actually lower the price seen by the consumer for one of the two commodities if a tax were applied. Common sense and more traditional, numerical analysis seemed to indicate that this was preposterous. Seligman insisted that the results Edgeworth achieved were a quirk of his mathematical formulation. He suggested that the assumption of a continuous demand function and an infinitesimal change in the tax resulted in the paradoxical predictions. Harold Hotelling later showed that Edgeworth was correct and that the same result (a "diminution of price as a result of the tax") could occur with a discontinuous demand function and large changes in the tax rate.[36]
## Modern mathematical economics
From the later-1930s, an array of new mathematical tools from the differential calculus and differential equations, convex sets, and graph theory were deployed to advance economic theory in a way similar to new mathematical methods earlier applied to physics.[8][37] The process was later described as moving from mechanics to axiomatics.[38]
### Differential calculus
Vilfredo Pareto analyzed microeconomics by treating decisions by economic actors as attempts to change a given allotment of goods to another, more preferred allotment. Sets of allocations could then be treated as Pareto efficient (Pareto optimal is an equivalent term) when no exchanges could occur between actors that could make at least one individual better off without making any other individual worse off.[39] Pareto's proof is commonly conflated with Walrassian equilibrium or informally ascribed to Adam Smith's Invisible hand hypothesis.[40] Rather, Pareto's statement was the first formal assertion of what would be known as the first fundamental theorem of welfare economics.[41] These models lacked the inequalities of the next generation of mathematical economics.
In the landmark treatise Foundations of Economic Analysis (1947), Paul Samuelson identified a common paradigm and mathematical structure across multiple fields in the subject, building on previous work by Alfred Marshall. Foundations took mathematical concepts from physics and applied them to economic problems. This broad view (for example, comparing Le Chatelier's principle to tâtonnement) drives the fundamental premise of mathematical economics: systems of economic actors may be modeled and their behavior described much like any other system. This extension followed on the work of the marginalists in the previous century and extended it significantly. Samuelson approached the problems of applying individual utility maximization over aggregate groups with comparative statics, which compares two different equilibrium states after an exogenous change in a variable. This and other methods in the book provided the foundation for mathematical economics in the 20th century.[7][42]
### Linear models
Restricted models of general equilibrium were formulated by John von Neumann in 1937.[43] Unlike earlier versions, the models of von Neumann had inequality constraints. For his model of an expanding economy, von Neumann proved the existence and uniqueness of an equilibrium using his generalization of Brouwer's fixed point theorem. Von Neumann's model of an expanding economy considered the matrix pencil A - λ B with nonnegative matrices A and B; von Neumann sought probability vectors p and q and a positive number λ that would solve the complementarity equation
pT (A - λ B) q = 0,
along with two inequality systems expressing economic efficiency. In this model, the (transposed) probability vector p represents the prices of the goods while the probability vector q represents the "intensity" at which the production process would run. The unique solution λ represents the rate of growth of the economy, which equals the interest rate. Proving the existence of a positive growth rate and proving that the growth rate equals the interest rate were remarkable achievements, even for von Neumann.[44][45][46] Von Neumann's results have been viewed as a special case of linear programming, where von Neumann's model uses only nonnegative matrices.[47] The study of von Neumann's model of an expanding economy continues to interest mathematical economists with interests in computational economics.[48][49][50]
#### Input-output economics
In 1936, the Russian–born economist Wassily Leontief built his model of input-output analysis from the 'material balance' tables constructed by Soviet economists, which themselves followed earlier work by the physiocrats. With his model, which described a system of production and demand processes, Leontief described how changes in demand in one economic sector would influence production in another.[51] In practice, Leontief estimated the coefficients of his simple models, to address economically interesting questions. In production economics, "Leontief technologies" produce outputs using constant proportions of inputs, regardless of the price of inputs, reducing the value of Leontief models for understanding economies but allowing their parameters to be estimated relatively easily. In contrast, the von Neumann model of an expanding economy allows for choice of techniques, but the coefficients must be estimated for each technology.[52][53]
### Mathematical optimization
Red dot in z direction as maximum for paraboloid function of (x, y) inputs
In mathematics, mathematical optimization (or optimization or mathematical programming) refers to the selection of a best element from some set of available alternatives.[54] In the simplest case, an optimization problem involves maximizing or minimizing a real function by selecting input values of the function and computing the corresponding values of the function. The solution process includes satisfying general necessary and sufficient conditions for optimality. For optimization problems, specialized notation may be used as to the function and its input(s). More generally, optimization includes finding the best available element of some function given a defined domain and may use a variety of different computational optimization techniques.[55]
Economics is closely enough linked to optimization by agents in an economy that an influential definition relatedly describes economics qua science as the "study of human behavior as a relationship between ends and scarce means" with alternative uses.[56] Optimization problems run through modern economics, many with explicit economic or technical constraints. In microeconomics, the utility maximization problem and its dual problem, the expenditure minimization problem for a given level of utility, are economic optimization problems.[57] Theory posits that consumers maximize their utility, subject to their budget constraints and that firms maximize their profits, subject to their production functions, input costs, and market demand.[58]
Economic equilibrium is studied in optimization theory as a key ingredient of economic theorems that in principle could be tested against empirical data.[7][59] Newer developments have occurred in dynamic programming and modeling optimization with risk and uncertainty, including applications to portfolio theory, the economics of information, and search theory.[58]
Optimality properties for an entire market system may be stated in mathematical terms, as in formulation of the two fundamental theorems of welfare economics[60] and in the Arrow–Debreu model of general equilibrium (also discussed below).[61] More concretely, many problems are amenable to analytical (formulaic) solution. Many others may be sufficiently complex to require numerical methods of solution, aided by software.[55] Still others are complex but tractable enough to allow computable methods of solution, in particular computable general equilibrium models for the entire economy.[62]
Linear and nonlinear programming have profoundly affected microeconomics, which had earlier considered only equality constraints.[63] Many of the mathematical economists who received Nobel Prizes in Economics had conducted notable research using linear programming: Leonid Kantorovich, Leonid Hurwicz, Tjalling Koopmans, Kenneth J. Arrow, Robert Dorfman, Paul Samuelson and Robert Solow.[64] Both Kantorovich and Koopmans acknowledged that George B. Dantzig deserved to share their Nobel Prize for linear programming. Economists who conducted research in nonlinear programming also have won the Nobel prize, notably Ragnar Frisch in addition to Kantorovich, Hurwicz, Koopmans, Arrow, and Samuelson.
#### Linear optimization
Linear programming was developed to aid the allocation of resources in firms and in industries during the 1930s in Russia and during the 1940s in the United States. During the Berlin airlift (1948), linear programming was used to plan the shipment of supplies to prevent Berlin from starving after the Soviet blockade.[65][66]
#### Nonlinear programming
Extensions to nonlinear optimization with inequality constraints were achieved in 1951 by Albert W. Tucker and Harold Kuhn, who considered the nonlinear optimization problem:
Minimize ${\displaystyle f}$(${\displaystyle x}$) subject to ${\displaystyle g}$i(${\displaystyle x}$) ≤ 0 and ${\displaystyle h}$j(${\displaystyle x}$) = 0 where
${\displaystyle f}$(.) is the function to be minimized
${\displaystyle g}$i(.) (${\displaystyle j}$ = 1, ..., ${\displaystyle m}$) are the functions of the ${\displaystyle m}$ inequality constraints
${\displaystyle h}$j(.) (${\displaystyle j}$ = 1, ..., ${\displaystyle l}$) are the functions of the ${\displaystyle l}$ equality constraints.
In allowing inequality constraints, the Kuhn–Tucker approach generalized the classic method of Lagrange multipliers, which (until then) had allowed only equality constraints.[67] The Kuhn–Tucker approach inspired further research on Lagrangian duality, including the treatment of inequality constraints.[68][69] The duality theory of nonlinear programming is particularly satisfactory when applied to convex minimization problems, which enjoy the convex-analytic duality theory of Fenchel and Rockafellar; this convex duality is particularly strong for polyhedral convex functions, such as those arising in linear programming. Lagrangian duality and convex analysis are used daily in operations research, in the scheduling of power plants, the planning of production schedules for factories, and the routing of airlines (routes, flights, planes, crews).[69]
#### Variational calculus and optimal control
Economic dynamics allows for changes in economic variables over time, including in dynamic systems. The problem of finding optimal functions for such changes is studied in variational calculus and in optimal control theory. Before the Second World War, Frank Ramsey and Harold Hotelling used the calculus of variations to that end.
Following Richard Bellman's work on dynamic programming and the 1962 English translation of L. Pontryagin et al.'s earlier work,[70] optimal control theory was used more extensively in economics in addressing dynamic problems, especially as to economic growth equilibrium and stability of economic systems,[71] of which a textbook example is optimal consumption and saving.[72] A crucial distinction is between deterministic and stochastic control models.[73] Other applications of optimal control theory include those in finance, inventories, and production for example.[74]
#### Functional analysis
It was in the course of proving of the existence of an optimal equilibrium in his 1937 model of economic growth that John von Neumann introduced functional analytic methods to include topology in economic theory, in particular, fixed-point theory through his generalization of Brouwer's fixed-point theorem.[8][43][75] Following von Neumann's program, Kenneth Arrow and Gérard Debreu formulated abstract models of economic equilibria using convex sets and fixed–point theory. In introducing the Arrow–Debreu model in 1954, they proved the existence (but not the uniqueness) of an equilibrium and also proved that every Walras equilibrium is Pareto efficient; in general, equilibria need not be unique.[76] In their models, the ("primal") vector space represented quantities while the "dual" vector space represented prices.[77]
In Russia, the mathematician Leonid Kantorovich developed economic models in partially ordered vector spaces, that emphasized the duality between quantities and prices.[78] Kantorovich renamed prices as "objectively determined valuations" which were abbreviated in Russian as "o. o. o.", alluding to the difficulty of discussing prices in the Soviet Union.[77][79][80]
Even in finite dimensions, the concepts of functional analysis have illuminated economic theory, particularly in clarifying the role of prices as normal vectors to a hyperplane supporting a convex set, representing production or consumption possibilities. However, problems of describing optimization over time or under uncertainty require the use of infinite–dimensional function spaces, because agents are choosing among functions or stochastic processes.[77][81][82][83]
### Differential decline and rise
John von Neumann's work on functional analysis and topology broke new ground in mathematics and economic theory.[43][84] It also left advanced mathematical economics with fewer applications of differential calculus. In particular, general equilibrium theorists used general topology, convex geometry, and optimization theory more than differential calculus, because the approach of differential calculus had failed to establish the existence of an equilibrium.
However, the decline of differential calculus should not be exaggerated, because differential calculus has always been used in graduate training and in applications. Moreover, differential calculus has returned to the highest levels of mathematical economics, general equilibrium theory (GET), as practiced by the "GET-set" (the humorous designation due to Jacques H. Drèze). In the 1960s and 1970s, however, Gérard Debreu and Stephen Smale led a revival of the use of differential calculus in mathematical economics. In particular, they were able to prove the existence of a general equilibrium, where earlier writers had failed, because of their novel mathematics: Baire category from general topology and Sard's lemma from differential topology. Other economists associated with the use of differential analysis include Egbert Dierker, Andreu Mas-Colell, and Yves Balasko.[85][86] These advances have changed the traditional narrative of the history of mathematical economics, following von Neumann, which celebrated the abandonment of differential calculus.
### Game theory
John von Neumann, working with Oskar Morgenstern on the theory of games, broke new mathematical ground in 1944 by extending functional analytic methods related to convex sets and topological fixed-point theory to economic analysis.[8][84] Their work thereby avoided the traditional differential calculus, for which the maximum–operator did not apply to non-differentiable functions. Continuing von Neumann's work in cooperative game theory, game theorists Lloyd S. Shapley, Martin Shubik, Hervé Moulin, Nimrod Megiddo, Bezalel Peleg influenced economic research in politics and economics. For example, research on the fair prices in cooperative games and fair values for voting games led to changed rules for voting in legislatures and for accounting for the costs in public–works projects. For example, cooperative game theory was used in designing the water distribution system of Southern Sweden and for setting rates for dedicated telephone lines in the USA.
Earlier neoclassical theory had bounded only the range of bargaining outcomes and in special cases, for example bilateral monopoly or along the contract curve of the Edgeworth box.[87] Von Neumann and Morgenstern's results were similarly weak. Following von Neumann's program, however, John Nash used fixed–point theory to prove conditions under which the bargaining problem and noncooperative games can generate a unique equilibrium solution.[88] Noncooperative game theory has been adopted as a fundamental aspect of experimental economics,[89] behavioral economics,[90] information economics,[91] industrial organization,[92] and political economy.[93] It has also given rise to the subject of mechanism design (sometimes called reverse game theory), which has private and public-policy applications as to ways of improving economic efficiency through incentives for information sharing.[94]
In 1994, Nash, John Harsanyi, and Reinhard Selten received the Nobel Memorial Prize in Economic Sciences their work on non–cooperative games. Harsanyi and Selten were awarded for their work on repeated games. Later work extended their results to computational methods of modeling.[95]
### Agent-based computational economics
Agent-based computational economics (ACE) as a named field is relatively recent, dating from about the 1990s as to published work. It studies economic processes, including whole economies, as dynamic systems of interacting agents over time. As such, it falls in the paradigm of complex adaptive systems.[96] In corresponding agent-based models, agents are not real people but "computational objects modeled as interacting according to rules" ... "whose micro-level interactions create emergent patterns" in space and time.[97] The rules are formulated to predict behavior and social interactions based on incentives and information. The theoretical assumption of mathematical optimization by agents markets is replaced by the less restrictive postulate of agents with bounded rationality adapting to market forces.[98]
ACE models apply numerical methods of analysis to computer-based simulations of complex dynamic problems for which more conventional methods, such as theorem formulation, may not find ready use.[99] Starting from specified initial conditions, the computational economic system is modeled as evolving over time as its constituent agents repeatedly interact with each other. In these respects, ACE has been characterized as a bottom-up culture-dish approach to the study of the economy.[100] In contrast to other standard modeling methods, ACE events are driven solely by initial conditions, whether or not equilibria exist or are computationally tractable. ACE modeling, however, includes agent adaptation, autonomy, and learning.[101] It has a similarity to, and overlap with, game theory as an agent-based method for modeling social interactions.[95] Other dimensions of the approach include such standard economic subjects as competition and collaboration,[102] market structure and industrial organization,[103] transaction costs,[104] welfare economics[105] and mechanism design,[94] information and uncertainty,[106] and macroeconomics.[107][108]
The method is said to benefit from continuing improvements in modeling techniques of computer science and increased computer capabilities. Issues include those common to experimental economics in general[109] and by comparison[110] and to development of a common framework for empirical validation and resolving open questions in agent-based modeling.[111] The ultimate scientific objective of the method has been described as "test[ing] theoretical findings against real-world data in ways that permit empirically supported theories to cumulate over time, with each researcher's work building appropriately on the work that has gone before."[112]
## Mathematicization of economics
The surface of the Volatility smile is a 3-D surface whereby the current market implied volatility (Z-axis) for all options on the underlier is plotted against strike price and time to maturity (X & Y-axes).[113]
Over the course of the 20th century, articles in "core journals"[114] in economics have been almost exclusively written by economists in academia. As a result, much of the material transmitted in those journals relates to economic theory, and "economic theory itself has been continuously more abstract and mathematical."[115] A subjective assessment of mathematical techniques[116] employed in these core journals showed a decrease in articles that use neither geometric representations nor mathematical notation from 95% in 1892 to 5.3% in 1990.[117] A 2007 survey of ten of the top economic journals finds that only 5.8% of the articles published in 2003 and 2004 both lacked statistical analysis of data and lacked displayed mathematical expressions that were indexed with numbers at the margin of the page.[118]
## Econometrics
Between the world wars, advances in mathematical statistics and a cadre of mathematically trained economists led to econometrics, which was the name proposed for the discipline of advancing economics by using mathematics and statistics. Within economics, "econometrics" has often been used for statistical methods in economics, rather than mathematical economics. Statistical econometrics features the application of linear regression and time series analysis to economic data.
Ragnar Frisch coined the word "econometrics" and helped to found both the Econometric Society in 1930 and the journal Econometrica in 1933.[119][120] A student of Frisch's, Trygve Haavelmo published The Probability Approach in Econometrics in 1944, where he asserted that precise statistical analysis could be used as a tool to validate mathematical theories about economic actors with data from complex sources.[121] This linking of statistical analysis of systems to economic theory was also promulgated by the Cowles Commission (now the Cowles Foundation) throughout the 1930s and 1940s.[122]
The roots of modern econometrics can be traced to the American economist Henry L. Moore. Moore studied agricultural productivity and attempted to fit changing values of productivity for plots of corn and other crops to a curve using different values of elasticity. Moore made several errors in his work, some from his choice of models and some from limitations in his use of mathematics. The accuracy of Moore's models also was limited by the poor data for national accounts in the United States at the time. While his first models of production were static, in 1925 he published a dynamic "moving equilibrium" model designed to explain business cycles—this periodic variation from overcorrection in supply and demand curves is now known as the cobweb model. A more formal derivation of this model was made later by Nicholas Kaldor, who is largely credited for its exposition.[123]
## Application
The IS/LM model is a Keynesian macroeconomic model designed to make predictions about the intersection of "real" economic activity (e.g. spending, income, savings rates) and decisions made in the financial markets (Money supply and Liquidity preference). The model is no longer widely taught at the graduate level but is common in undergraduate macroeconomics courses.[124]
Much of classical economics can be presented in simple geometric terms or elementary mathematical notation. Mathematical economics, however, conventionally makes use of calculus and matrix algebra in economic analysis in order to make powerful claims that would be more difficult without such mathematical tools. These tools are prerequisites for formal study, not only in mathematical economics but in contemporary economic theory in general. Economic problems often involve so many variables that mathematics is the only practical way of attacking and solving them. Alfred Marshall argued that every economic problem which can be quantified, analytically expressed and solved, should be treated by means of mathematical work.[125]
Economics has become increasingly dependent upon mathematical methods and the mathematical tools it employs have become more sophisticated. As a result, mathematics has become considerably more important to professionals in economics and finance. Graduate programs in both economics and finance require strong undergraduate preparation in mathematics for admission and, for this reason, attract an increasingly high number of mathematicians. Applied mathematicians apply mathematical principles to practical problems, such as economic analysis and other economics-related issues, and many economic problems are often defined as integrated into the scope of applied mathematics.[17]
This integration results from the formulation of economic problems as stylized models with clear assumptions and falsifiable predictions. This modeling may be informal or prosaic, as it was in Adam Smith's The Wealth of Nations, or it may be formal, rigorous and mathematical.
Broadly speaking, formal economic models may be classified as stochastic or deterministic and as discrete or continuous. At a practical level, quantitative modeling is applied to many areas of economics and several methodologies have evolved more or less independently of each other.[126]
## Example: The effect of a corporate tax cut on wages
The great appeal of mathematical economics is that it brings a degree of rigor to economic thinking, particularly around charged political topics. For example, during the discussion of the efficacy of a corporate tax cut for increasing the wages of workers, a simple mathematical model proved beneficial to understanding the issues at hand.
As an intellectual exercise, the following problem was posed by Prof. Greg Mankiw of Harvard University:[127]
An open economy has the production function ${\textstyle y=f(k)}$, where ${\textstyle y}$ is output per worker and ${\textstyle k}$ is capital per worker. The capital stock adjusts so that the after-tax marginal product of capital equals the exogenously given world interest rate ${\textstyle r}$...How much will the tax cut increase wages?
To answer this question, we follow John H. Cochrane of the Hoover Institution.[128] Suppose an open economy has the production function:
${\displaystyle Y=F(K,L)=f(k)L,\quad k=K/L}$
Where the variables in this equation are:
• ${\textstyle Y}$ is the total output
• ${\textstyle F(K,L)}$ is the production function
• ${\textstyle K}$ is the total capital stock
• ${\textstyle L}$ is the total labor stock
The standard choice for the production function is the Cobb-Douglas production function:
${\displaystyle Y=AK^{\alpha }L^{1-\alpha }=Ak^{\alpha }L,\quad \alpha \in [0,1]}$
where ${\textstyle A}$ is the factor of productivity - assumed to be a constant. A corporate tax cut in this model is equivalent to a tax on capital. With taxes, firms look to maximize:
{\displaystyle {\begin{aligned}J&=\max _{K,L}\;(1-\tau )\left[F(K,L)-wL\right]-rK\\&\equiv \max _{K,L}\;(1-\tau )\left[f(k)-w\right]L-rK\end{aligned}}}
where ${\textstyle \tau }$ is the capital tax rate, ${\textstyle w}$ is wages per worker, and ${\textstyle r}$ is the exogenous interest rate. Then the first-order optimality conditions become:
{\displaystyle {\begin{aligned}{\partial J \over {\partial K}}&=(1-\tau )f'(k)-r\\{\partial J \over {\partial L}}&=(1-\tau )\left[f(k)-f'(k)k-w\right]\end{aligned}}}
Therefore, the optimality conditions imply that:
${\displaystyle r=(1-\tau )f'(k),\quad w=f(k)-f'(k)k}$
Define total taxes ${\textstyle X=\tau [F(K,L)-wL]}$. This implies that taxes per worker ${\textstyle x}$ are:
${\displaystyle x=\tau [f(k)-w]=\tau f'(k)k}$
Then the change in taxes per worker, given the tax rate, is:
${\displaystyle {dx \over {d\tau }}=\underbrace {f'(k)k} _{\text{Static}}+\underbrace {\tau \left[f'(k)+f''(k)k\right]{dk \over {d\tau }}} _{\text{Dynamic}}}$
To find the change in wages, we differentiate the second optimality condition for the per worker wages to obtain:
{\displaystyle {\begin{aligned}{dw \over {d\tau }}&=\left[f'(k)-f'(k)-f''(k)k\right]{dk \over {d\tau }}\\&=-f''(k)k{dk \over {d\tau }}\end{aligned}}}
Assuming that the interest rate is fixed at ${\textstyle r}$, so that ${\textstyle dr/d\tau =0}$, we may differentiate the first optimality condition for the interest rate to find:
${\displaystyle {dk \over {d\tau }}={f'(k) \over {(1-\tau )f''(k)}}}$
For the moment, let's focus only on the static effect of a capital tax cut, so that ${\textstyle dx/d\tau =f'(k)k}$. If we substitute this equation into equation for wage changes with respect to the tax rate, then we find that:
{\displaystyle {\begin{aligned}{dw \over {d\tau }}&=-{f'(k)k \over {1-\tau }}\\&=-{1 \over {1-\tau }}{dx \over {d\tau }}\end{aligned}}}
Therefore, the static effect of a capital tax cut on wages is:
${\displaystyle {dw \over {dx}}=-{1 \over {1-\tau }}}$
Based on the model, it seems possible that we may achieve a rise in the wage of a worker greater than the amount of the tax cut. But that only considers the static effect, and we know that the dynamic effect must be accounted for. In the dynamic model, we may rewrite the equation for changes in taxes per worker with respect to the tax rate as:
{\displaystyle {\begin{aligned}{dx \over {d\tau }}&=f'(k)k+\tau \left[f'(k)+f''(k)k\right]{dk \over {d\tau }}\\&=f'(k)k+{\tau \over {1-\tau }}{[f'(k)]^{2}+f'(k)f''(k)k \over {f''(k)}}\\&={\tau \over {1-\tau }}{f'(k)^{2} \over {f''(k)}}+{1 \over {1-\tau }}f'(k)k\\&={f'(k) \over {1-\tau }}\left[\tau {f'(k) \over {f''(k)}}+k\right]\end{aligned}}}
Recalling that ${\textstyle dw/d\tau =-f'(k)k/(1-\tau )}$, we have that:
{\displaystyle {\begin{aligned}{dw \over {dx}}&=-{{f'(k)k \over {1-\tau }} \over {{f'(k) \over {1-\tau }}\left[\tau {f'(k) \over {f''(k)}}+k\right]}}\\&=-{1 \over {\tau {f'(k) \over {kf''(k)}}+1}}\end{aligned}}}
Using the Cobb-Douglas production function, we have that:
${\displaystyle {f'(k) \over {kf''(k)}}=-{1 \over {1-\alpha }}}$
Therefore, the dynamic effect of a capital tax cut on wages is:
${\displaystyle {dw \over {dx}}=-{1-\alpha \over {1-\tau -\alpha }}}$
If we take ${\textstyle \alpha =\tau =1/3}$, then the dynamic effect of lowering capital taxes on wages will be even larger than the static effect. Moreover, if there are positive externalities to capital accumulation, the effect of the tax cut on wages would be larger than in the model we just derived. It is important to note that the result is a combination of:
1. The standard result that in a small open economy labor bears 100% of a small capital income tax
2. The fact that, starting at a positive tax rate, the burden of a tax increase exceeds revenue collection due to the first-order deadweight loss
This result showing that, under certain assumptions, a corporate tax cut can boost the wages of workers by more than the lost revenue does not imply that the magnitude is correct. Rather, it suggests a basis for policy analysis that is not grounded in handwaving. If the assumptions are reasonable, then the model is an acceptable approximation of reality; if they are not, then better models should be developed.
### CES production function
Now let's assume that instead of the Cobb-Douglas production function we have a more general constant elasticity of substitution (CES) production function:
${\displaystyle f(k)=A\left[\alpha k^{\rho }+(1-\alpha )\right]^{1/\rho }}$
where ${\textstyle \rho =1-\sigma ^{-1}}$; ${\textstyle \sigma }$ is the elasticity of substitution between capital and labor. The relevant quantity we want to calculate is ${\textstyle f'/kf''}$, which may be derived as:
${\displaystyle {f' \over {kf''}}=-{1 \over {1-\rho -{\alpha (1-\rho ) \over {\alpha +(1-\alpha )k^{-\rho }}}}}}$
Therefore, we may use this to find that:
{\displaystyle {\begin{aligned}1+\tau {f' \over {kf''}}&=1-{\tau [\alpha +(1-\alpha )k^{-\rho }] \over {(1-\rho )[\alpha +(1-\alpha )k^{-\rho }]-\alpha (1-\rho )}}\\&={(1-\rho -\tau )[\alpha +(1-\alpha )k^{-\rho }]-\alpha (1-\rho ) \over {(1-\rho )[\alpha +(1-\alpha )k^{-\rho }]-\alpha (1-\rho )}}\end{aligned}}}
Therefore, under a general CES model, the dynamic effect of a capital tax cut on wages is:
${\displaystyle {dw \over {dx}}=-{(1-\rho )[\alpha +(1-\alpha )k^{-\rho }]-\alpha (1-\rho ) \over {(1-\rho -\tau )[\alpha +(1-\alpha )k^{-\rho }]-\alpha (1-\rho )}}}$
We recover the Cobb-Douglas solution when ${\textstyle \rho =0}$. When ${\textstyle \rho =1}$, which is the case when perfect substitutes exist, we find that ${\textstyle dw/dx=0}$ - there is no effect of changes in capital taxes on wages. And when ${\textstyle \rho =-\infty }$, which is the case when perfect complements exist, we find that ${\textstyle dw/dx=-1}$ - a cut in capital taxes increases wages by exactly one dollar.
## Classification
According to the Mathematics Subject Classification (MSC), mathematical economics falls into the Applied mathematics/other classification of category 91:
Game theory, economics, social and behavioral sciences
with MSC2010 classifications for 'Game theory' at codes 91Axx and for 'Mathematical economics' at codes 91Bxx.
The Handbook of Mathematical Economics series (Elsevier), currently 4 volumes, distinguishes between mathematical methods in economics, v. 1, Part I, and areas of economics in other volumes where mathematics is employed.[129]
Another source with a similar distinction is The New Palgrave: A Dictionary of Economics (1987, 4 vols., 1,300 subject entries). In it, a "Subject Index" includes mathematical entries under 2 headings (vol. IV, pp. 982–3):
Mathematical Economics (24 listed, such as "acyclicity", "aggregation problem", "comparative statics", "lexicographic orderings", "linear models", "orderings", and "qualitative economics")
Mathematical Methods (42 listed, such as "calculus of variations", "catastrophe theory", "combinatorics," "computation of general equilibrium", "convexity", "convex programming", and "stochastic optimal control").
A widely used system in economics that includes mathematical methods on the subject is the JEL classification codes. It originated in the Journal of Economic Literature for classifying new books and articles. The relevant categories are listed below (simplified below to omit "Miscellaneous" and "Other" JEL codes), as reproduced from JEL classification codes#Mathematical and quantitative methods JEL: C Subcategories. The New Palgrave Dictionary of Economics (2008, 2nd ed.) also uses the JEL codes to classify its entries. The corresponding footnotes below have links to abstracts of The New Palgrave Online for each JEL category (10 or fewer per page, similar to Google searches).
JEL: C02 - Mathematical Methods (following JEL: C00 - General and JEL: C01 - Econometrics)
JEL: C6 - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling [130]
JEL: C60 - General
JEL: C61 - Optimization techniques; Programming models; Dynamic analysis[131]
JEL: C62 - Existence and stability conditions of equilibrium[132]
JEL: C63 - Computational techniques; Simulation modeling[133]
JEL: C67 - Input–output models
JEL: C68 - Computable General Equilibrium models[134]
JEL: C7 - Game theory and Bargaining theory[135]
JEL: C70 - General[136]
JEL: C71 - Cooperative games[137]
JEL: C72 - Noncooperative games[138]
JEL: C73 - Stochastic and Dynamic games; Evolutionary games; Repeated Games[139]
JEL: C78 - Bargaining theory; Matching theory[140]
## Criticisms and defences
### Adequacy of mathematics for qualitative and complicated economics
Friedrich Hayek contended that the use of formal techniques projects a scientific exactness that does not appropriately account for informational limitations faced by real economic agents. [141]
In an interview in 1999, the economic historian Robert Heilbroner stated:[142]
I guess the scientific approach began to penetrate and soon dominate the profession in the past twenty to thirty years. This came about in part because of the "invention" of mathematical analysis of various kinds and, indeed, considerable improvements in it. This is the age in which we have not only more data but more sophisticated use of data. So there is a strong feeling that this is a data-laden science and a data-laden undertaking, which, by virtue of the sheer numerics, the sheer equations, and the sheer look of a journal page, bears a certain resemblance to science . . . That one central activity looks scientific. I understand that. I think that is genuine. It approaches being a universal law. But resembling a science is different from being a science.
Heilbroner stated that "some/much of economics is not naturally quantitative and therefore does not lend itself to mathematical exposition."[143]
### Testing predictions of mathematical economics
Philosopher Karl Popper discussed the scientific standing of economics in the 1940s and 1950s. He argued that mathematical economics suffered from being tautological. In other words, insofar as economics became a mathematical theory, mathematical economics ceased to rely on empirical refutation but rather relied on mathematical proofs and disproof.[144] According to Popper, falsifiable assumptions can be tested by experiment and observation while unfalsifiable assumptions can be explored mathematically for their consequences and for their consistency with other assumptions.[145]
Sharing Popper's concerns about assumptions in economics generally, and not just mathematical economics, Milton Friedman declared that "all assumptions are unrealistic". Friedman proposed judging economic models by their predictive performance rather than by the match between their assumptions and reality.[146]
### Mathematical economics as a form of pure mathematics
Considering mathematical economics, J.M. Keynes wrote in The General Theory:[147]
It is a great fault of symbolic pseudo-mathematical methods of formalising a system of economic analysis ... that they expressly assume strict independence between the factors involved and lose their cogency and authority if this hypothesis is disallowed; whereas, in ordinary discourse, where we are not blindly manipulating and know all the time what we are doing and what the words mean, we can keep ‘at the back of our heads’ the necessary reserves and qualifications and the adjustments which we shall have to make later on, in a way in which we cannot keep complicated partial differentials ‘at the back’ of several pages of algebra which assume they all vanish. Too large a proportion of recent ‘mathematical’ economics are merely concoctions, as imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols.
### Defense of mathematical economics
In response to these criticisms, Paul Samuelson argued that mathematics is a language, repeating a thesis of Josiah Willard Gibbs. In economics, the language of mathematics is sometimes necessary for representing substantive problems. Moreover, mathematical economics has led to conceptual advances in economics.[148] In particular, Samuelson gave the example of microeconomics, writing that "few people are ingenious enough to grasp [its] more complex parts... without resorting to the language of mathematics, while most ordinary individuals can do so fairly easily with the aid of mathematics."[149]
Some economists state that mathematical economics deserves support just like other forms of mathematics, particularly its neighbors in mathematical optimization and mathematical statistics and increasingly in theoretical computer science. Mathematical economics and other mathematical sciences have a history in which theoretical advances have regularly contributed to the reform of the more applied branches of economics. In particular, following the program of John von Neumann, game theory now provides the foundations for describing much of applied economics, from statistical decision theory (as "games against nature") and econometrics to general equilibrium theory and industrial organization. In the last decade, with the rise of the internet, mathematical economists and optimization experts and computer scientists have worked on problems of pricing for on-line services --- their contributions using mathematics from cooperative game theory, nondifferentiable optimization, and combinatorial games.
Robert M. Solow concluded that mathematical economics was the core "infrastructure" of contemporary economics:
Economics is no longer a fit conversation piece for ladies and gentlemen. It has become a technical subject. Like any technical subject it attracts some people who are more interested in the technique than the subject. That is too bad, but it may be inevitable. In any case, do not kid yourself: the technical core of economics is indispensable infrastructure for the political economy. That is why, if you consult [a reference in contemporary economics] looking for enlightenment about the world today, you will be led to technical economics, or history, or nothing at all.[150]
## Mathematical economists
Prominent mathematical economists include, but are not limited to, the following (by century of birth).
## References
1. ^
2. ^ a b Chiang, Alpha C.; Kevin Wainwright (2005). Fundamental Methods of Mathematical Economics. McGraw-Hill Irwin. pp. 3–4. ISBN 978-0-07-010910-0. TOC.
3. ^ Debreu, Gérard ([1987] 2008). "mathematical economics", section II, The New Palgrave Dictionary of Economics, 2nd Edition. Abstract. Republished with revisions from 1986, "Theoretic Models: Mathematical Form and Economic Content", Econometrica, 54(6), pp. 1259-1270.
4. ^ Varian, Hal (1997). "What Use Is Economic Theory?" in A. D'Autume and J. Cartelier, ed., Is Economics Becoming a Hard Science?, Edward Elgar. Pre-publication PDF. Retrieved 2008-04-01.
5. ^ • As in Handbook of Mathematical Economics, 1st-page chapter links:
Arrow, Kenneth J., and Michael D. Intriligator, ed., (1981), v. 1
_____ (1982). v. 2
_____ (1986). v. 3
Hildenbrand, Werner, and Hugo Sonnenschein, ed. (1991). v. 4.
• Debreu, Gérard (1983). Mathematical Economics: Twenty Papers of Gérard Debreu, Contents.
• Glaister, Stephen (1984). Mathematical Methods for Economists, 3rd ed., Blackwell. Contents.
• Takayama, Akira (1985). Mathematical Economics, 2nd ed. Cambridge. Description and Contents.
• Michael Carter (2001). Foundations of Mathematical Economics, MIT Press. Description and Contents.
6. ^ Chiang, Alpha C. (1992). Elements of Dynamic Optimization, Waveland. TOC & Amazon.com link to inside, first pp.
7. ^ a b c d Samuelson, Paul (1947) [1983]. Foundations of Economic Analysis. Harvard University Press. ISBN 978-0-674-31301-9.
8. ^ a b c d Debreu, Gérard ([1987] 2008). "mathematical economics", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract. Republished with revisions from 1986, "Theoretic Models: Mathematical Form and Economic Content", Econometrica, 54(6), pp. 1259-1270.
• von Neumann, John, and Oskar Morgenstern (1944). Theory of Games and Economic Behavior. Princeton University Press.
9. ^ Schumpeter, J.A. (1954). Elizabeth B. Schumpeter (ed.). History of Economic Analysis. New York: Oxford University Press. pp. 209–212. ISBN 978-0-04-330086-2. OCLC 13498913.
10. ^ Schumpeter (1954) p. 212-215
11. ^ Schnieder, Erich (1934). "Johann Heinrich von Thünen". Econometrica. 2 (1): 1–12. doi:10.2307/1907947. ISSN 0012-9682. JSTOR 1907947. OCLC 35705710.
12. ^ Schumpeter (1954) p. 465-468
13. ^ Philip Mirowski, 1991. "The When, the How and the Why of Mathematical Expression in the History of Economics Analysis", Journal of Economic Perspectives, 5(1) pp. 145-157.
14. ^ Weintraub, E. Roy (2008). "mathematics and economics", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
15. ^ Jevons, W.S. (1866). "Brief Account of a General Mathematical Theory of Political Economy", Journal of the Royal Statistical Society, XXIX (June) pp. 282–87. Read in Section F of the British Association, 1862. PDF.
16. ^ Jevons, W. Stanley (1871). The Principles of Political Economy, pp. 4, 25.
17. ^ a b Sheila C., Dow (1999-05-21). "The Use of Mathematics in Economics". ESRC Public Understanding of Mathematics Seminar. Birmingham: Economic and Social Research Council. Retrieved 2008-07-06.
18. ^ While the concept of cardinality has fallen out of favor in neoclassical economics, the differences between cardinal utility and ordinal utility are minor for most applications.
19. ^ a b Nicola, PierCarlo (2000). Mainstream Mathermatical Economics in the 20th Century. Springer. p. 4. ISBN 978-3-540-67084-1. Retrieved 2008-08-21.
20. ^ Augustin Cournot (1838, tr. 1897) Researches into the Mathematical Principles of Wealth. Links to description and chapters.
21. ^ a b Hotelling, Harold (1990). "Stability in Competition". In Darnell, Adrian C. (ed.). The Collected Economics Articles of Harold Hotelling. Springer. pp. 51, 52. ISBN 978-3-540-97011-8. OCLC 20217006. Retrieved 2008-08-21.
22. ^ "Antoine Augustin Cournot, 1801-1877". The History of Economic Thought Website. The New School for Social Research. Archived from the original on 2000-07-09. Retrieved 2008-08-21.
23. ^ Gibbons, Robert (1992). Game Theory for Applied Economists. Princeton, New Jersey: Princeton University Press. pp. 14, 15. ISBN 978-0-691-00395-5.
24. ^ Nicola, p. 9-12
25. ^ Edgeworth, Francis Ysidro (September 5, 1889). "The Mathematical Theory of Political Economy: Review of Léon Walras, Éléments d'économie politique pure" (PDF). Nature. 40 (1036): 434–436. doi:10.1038/040434a0. ISSN 0028-0836. Archived from the original (PDF) on April 11, 2003. Retrieved 2008-08-21.
26. ^ Nicholson, Walter; Snyder, Christopher, p. 350-353.
27. ^ Dixon, Robert. "Walras Law and Macroeconomics". Walras Law Guide. Department of Economics, University of Melbourne. Archived from the original on April 17, 2008. Retrieved 2008-09-28.
28. ^ Dixon, Robert. "A Formal Proof of Walras Law". Walras Law Guide. Department of Economics, University of Melbourne. Archived from the original on April 30, 2008. Retrieved 2008-09-28.
29. ^ Rima, Ingrid H. (1977). "Neoclassicism and Dissent 1890-1930". In Weintraub, Sidney (ed.). Modern Economic Thought. University of Pennsylvania Press. pp. 10, 11. ISBN 978-0-8122-7712-8.
30. ^ Heilbroner, Robert L. (1953 [1999]). The Worldly Philosophers (Seventh ed.). New York: Simon and Schuster. pp. 172–175, 313. ISBN 978-0-684-86214-9. Check date values in: |date= (help)
31. ^ Edgeworth, Francis Ysidro (1881 [1961]). Mathematical Psychics. London: Kegan Paul [A. M. Kelley]. pp. 15–19. Check date values in: |date= (help)
32. ^ Nicola, p. 14, 15, 258-261
33. ^ Bowley, Arthur Lyon (1924 [1960]). The Mathematical Groundwork of Economics: an Introductory Treatise. Oxford: Clarendon Press [Kelly]. Check date values in: |date= (help)
34. ^ Gillies, D. B. (1969). "Solutions to general non-zero-sum games". In Tucker, A. W.; Luce, R. D. (eds.). Contributions to the Theory of Games. Annals of Mathematics. 40. Princeton, New Jersey: Princeton University Press. pp. 47–85. ISBN 978-0-691-07937-0.
35. ^ Moss, Lawrence S. (2003). "The Seligman-Edgeworth Debate about the Analysis of Tax Incidence: The Advent of Mathematical Economics, 1892–1910". History of Political Economy. 35 (2): 207, 212, 219, 234–237. doi:10.1215/00182702-35-2-205. ISSN 0018-2702.
36. ^ Hotelling, Harold (1990). "Note on Edgeworth's Taxation Phenomenon and Professor Garver's Additional Condition on Demand Functions". In Darnell, Adrian C. (ed.). The Collected Economics Articles of Harold Hotelling. Springer. pp. 94–122. ISBN 978-3-540-97011-8. OCLC 20217006. Retrieved 2008-08-26.
37. ^ Herstein, I.N. (October 1953). "Some Mathematical Methods and Techniques in Economics". Quarterly of Applied Mathematics. 11 (3): 249–262. doi:10.1090/qam/60205. ISSN 1552-4485. [Pp. 249-62.
38. ^ • Weintraub, E. Roy (2008). "mathematics and economics", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
• _____ (2002). How Economics Became a Mathematical Science. Duke University Press. Description and preview.
39. ^ Nicholson, Walter; Snyder, Christopher (2007). "General Equilibrium and Welfare". Intermediate Microeconomics and Its Applications (10th ed.). Thompson. pp. 364, 365. ISBN 978-0-324-31968-2.
40. ^ Jolink, Albert (2006). "What Went Wrong with Walras?". In Backhaus, Juergen G.; Maks, J.A. Hans (eds.). From Walras to Pareto. The European Heritage in Economics and the Social Sciences. IV. Springer. pp. 69–80. doi:10.1007/978-0-387-33757-9_6. ISBN 978-0-387-33756-2.
• Blaug, Mark (2007). "The Fundamental Theorems of Modern Welfare Economics, Historically Contemplated". History of Political Economy. 39 (2): 186–188. doi:10.1215/00182702-2007-001. ISSN 0018-2702.
41. ^ Blaug (2007), p. 185, 187
42. ^ Metzler, Lloyd (1948). "Review of Foundations of Economic Analysis". American Economic Review. 38 (5): 905–910. ISSN 0002-8282. JSTOR 1811704.
43. ^ a b c Neumann, J. von (1937). "Über ein ökonomisches Gleichungssystem und ein Verallgemeinerung des Brouwerschen Fixpunktsatzes", Ergebnisse eines Mathematischen Kolloquiums, 8, pp. 73–83, translated and published in 1945-46, as "A Model of General Equilibrium", Review of Economic Studies, 13, pp. 1–9.
44. ^ For this problem to have a unique solution, it suffices that the nonnegative matrices A and B satisfy an irreducibility condition, generalizing that of the Perron–Frobenius theorem of nonnegative matrices, which considers the (simplified) eigenvalue problem
A - λ I q = 0,
where the nonnegative matrix A must be square and where the diagonal matrix I is the identity matrix. Von Neumann's irreducibility condition was called the "whales and wranglers" hypothesis by David Champernowne, who provided a verbal and economic commentary on the English translation of von Neumann's article. Von Neumann's hypothesis implied that every economic process used a positive amount of every economic good. Weaker "irreducibility" conditions were given by David Gale and by John Kemeny, Oskar Morgenstern, and Gerald L. Thompson in the 1950s and then by Stephen M. Robinson in the 1970s.
45. ^ David Gale. The theory of linear economic models. McGraw-Hill, New York, 1960.
46. ^ Morgenstern, Oskar; Thompson, Gerald L. (1976). Mathematical theory of expanding and contracting economies. Lexington Books. Lexington, Massachusetts: D. C. Heath and Company. pp. xviii+277.
47. ^ Alexander Schrijver, Theory of Linear and Integer Programming. John Wiley & sons, 1998, ISBN 0-471-98232-6.
48. ^ Rockafellar, R. Tyrrell (1967). Monotone processes of convex and concave type. Memoirs of the American Mathematical Society. Providence, R.I.: American Mathematical Society. pp. i+74.
• Rockafellar, R. T. (1974). "Convex algebra and duality in dynamic models of production". In Josef Loz; Maria Loz (eds.). Mathematical models in economics (Proc. Sympos. and Conf. von Neumann Models, Warsaw, 1972). Amsterdam: North-Holland and Polish Adademy of Sciences (PAN). pp. 351–378.
•Rockafellar, R. T. (1970 (Reprint 1997 as a Princeton classic in mathematics)). Convex analysis. Princeton, New Jersey: Princeton University Press. Check date values in: |year= (help)
49. ^ Kenneth Arrow, Paul Samuelson, John Harsanyi, Sidney Afriat, Gerald L. Thompson, and Nicholas Kaldor. (1989). Mohammed Dore; Sukhamoy Chakravarty; Richard Goodwin (eds.). John Von Neumann and modern economics. Oxford:Clarendon. p. 261.CS1 maint: Multiple names: authors list (link)
50. ^ Chapter 9.1 "The von Neumann growth model" (pages 277–299): Yinyu Ye. Interior point algorithms: Theory and analysis. Wiley. 1997.
51. ^ Screpanti, Ernesto; Zamagni, Stefano (1993). An Outline of the History of Economic Thought. New York: Oxford University Press. pp. 288–290. ISBN 978-0-19-828370-6. OCLC 57281275.
52. ^ David Gale. The theory of linear economic models. McGraw-Hill, New York, 1960.
53. ^ Morgenstern, Oskar; Thompson, Gerald L. (1976). Mathematical theory of expanding and contracting economies. Lexington Books. Lexington, Massachusetts: D. C. Heath and Company. pp. xviii+277.
54. ^ "The Nature of Mathematical Programming", Mathematical Programming Glossary, INFORMS Computing Society.
55. ^ a b Schmedders, Karl (2008). "numerical optimization methods in economics", The New Palgrave Dictionary of Economics, 2nd Edition, v. 6, pp. 138–57. Abstract.
56. ^ Robbins, Lionel (1935, 2nd ed.). An Essay on the Nature and Significance of Economic Science, Macmillan, p. 16.
57. ^ Blume, Lawrence E. (2008). "duality", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
58. ^ a b Dixit, A. K. ([1976] 1990). Optimization in Economic Theory, 2nd ed., Oxford. Description and contents preview.
59. ^ • Samuelson, Paul A., 1998. "How Foundations Came to Be", Journal of Economic Literature, 36(3), pp. 1375–1386.
• _____ (1970)."Maximum Principles in Analytical Economics", Nobel Prize lecture.
60. ^ • Allan M. Feldman (3008). "welfare economics", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
• Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green (1995), Microeconomic Theory, Chapter 16. Oxford University Press, ISBN 0-19-510268-1. Description Archived 2012-01-26 at the Wayback Machine and contents .
61. ^ Geanakoplos, John ([1987] 2008). "Arrow–Debreu model of general equilibrium", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
• Arrow, Kenneth J., and Gérard Debreu (1954). "Existence of an Equilibrium for a Competitive Economy", Econometrica 22(3), pp. 265-290.
62. ^ Scarf, Herbert E. (2008). "computation of general equilibria", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
• Kubler, Felix (2008). "computation of general equilibria (new developments)", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
63. ^ Nicola, p. 133
64. ^ Dorfman, Robert, Paul A. Samuelson, and Robert M. Solow (1958). Linear Programming and Economic Analysis. McGraw–Hill. Chapter-preview links.
65. ^ M. Padberg, Linear Optimization and Extensions, Second Edition, Springer-Verlag, 1999.
66. ^ Dantzig, George B. ([1987] 2008). "linear programming", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
67. ^ • Intriligator, Michael D. (2008). "nonlinear programming", The New Palgrave Dictionary of Economics, 2nd Edition. TOC.
• Blume, Lawrence E. (2008). "convex programming", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
• Kuhn, H. W.; Tucker, A. W. (1951). "Nonlinear programming". Proceedings of 2nd Berkeley Symposium. Berkeley: University of California Press. pp. 481–492.
68. ^ Bertsekas, Dimitri P. (1999). Nonlinear Programming (Second ed.). Cambridge, Massachusetts.: Athena Scientific. ISBN 978-1-886529-00-7.
• Vapnyarskii, I.B. (2001) [1994], "Lagrange multipliers", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4.
• Lasdon, Leon S. (1970). Optimization theory for large systems. Macmillan series in operations research. New York: The Macmillan Company. pp. xi+523. MR 0337317.
• Lasdon, Leon S. (2002). Optimization theory for large systems (reprint of the 1970 Macmillan ed.). Mineola, New York: Dover Publications, Inc. pp. xiii+523. MR 1888251.
• Hiriart-Urruty, Jean-Baptiste; Lemaréchal, Claude (1993). "XII Abstract duality for practitioners". Convex analysis and minimization algorithms, Volume II: Advanced theory and bundle methods. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. 306. Berlin: Springer-Verlag. pp. 136–193 (and Bibliographical comments on pp. , 334–335). ISBN 978-3-540-56852-0. MR 1295240.
69. ^ a b Lemaréchal, Claude (2001). "Lagrangian relaxation". In Michael Jünger; Denis Naddef (eds.). Computational combinatorial optimization: Papers from the Spring School held in Schloß Dagstuhl, May 15–19, 2000. Lecture Notes in Computer Science. 2241. Berlin: Springer-Verlag. pp. 112–156. doi:10.1007/3-540-45586-8_4. ISBN 978-3-540-42877-0. MR 1900016.
70. ^ Pontryagin, L. S.; Boltyanski, V. G., Gamkrelidze, R. V., Mischenko, E. F. (1962). The Mathematical Theory of Optimal Processes. New York: Wiley. ISBN 9782881240775.CS1 maint: Multiple names: authors list (link)
71. ^ Zelikin, M. I. ([1987] 2008). "Pontryagin's principle of optimality", The New Palgrave Dictionary of Economics, 2nd Edition. Preview link.
• Martos, Béla (1987). "control and coordination of economic activity", The New Palgrave: A Dictionary of Economics. Description link.
• Brock, W. A. (1987). "optimal control and economic dynamics", The New Palgrave: A Dictionary of Economics. Outline.
• Shell, K., ed. (1967). Essays on the Theory of Optimal Economic Growth. Cambridge, Massachusetts: The MIT Press. ISBN 978-0-262-19036-7.CS1 maint: Extra text: authors list (link)]
72. ^ Stokey, Nancy L. and Robert E. Lucas with Edward Prescott (1989). Recursive Methods in Economic Dynamics, Harvard University Press, chapter 5. Desecription and chapter-preview links.
73. ^ Malliaris, A.G. (2008). "stochastic optimal control", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract Archived 2017-10-18 at the Wayback Machine.
74. ^ Arrow, K. J.; Kurz, M. (1970). Public Investment, the Rate of Return, and Optimal Fiscal Policy. Baltimore, Maryland: The Johns Hopkins Press. ISBN 978-0-8018-1124-1. Abstract. Archived 2013-03-09 at the Wayback Machine
• Sethi, S. P.; Thompson, G. L. (2000). Optimal Control Theory: Applications to Management Science and Economics, Second Edition. New York: Springer. ISBN 978-0-7923-8608-7. Scroll to chapter-preview links.
75. ^ Andrew McLennan, 2008. "fixed point theorems", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
76. ^ Weintraub, E. Roy (1977). "General Equilibrium Theory". In Weintraub, Sidney (ed.). Modern Economic Thought. University of Pennsylvania Press. pp. 107–109. ISBN 978-0-8122-7712-8.
• Arrow, Kenneth J.; Debreu, Gérard (1954). "Existence of an equilibrium for a competitive economy". Econometrica. 22 (3): 265–290. doi:10.2307/1907353. ISSN 0012-9682. JSTOR 1907353.
77. ^ a b c Kantorovich, Leonid, and Victor Polterovich (2008). "Functional analysis", in S. Durlauf and L. Blume, ed., The New Palgrave Dictionary of Economics, 2nd Edition. Abstract., ed., Palgrave Macmillan.
78. ^ Kantorovich, L. V (1990). ""My journey in science (supposed report to the Moscow Mathematical Society)" [expanding Russian Math. Surveys 42 (1987), no. 2, pp. 233–270]". In Lev J. Leifman (ed.). Functional analysis, optimization, and mathematical economics: A collection of papers dedicated to the memory of Leonid Vitalʹevich Kantorovich. New York: The Clarendon Press, Oxford University Press. pp. 8–45. ISBN 978-0-19-505729-4. MR 0898626.
79. ^ Page 406: Polyak, B. T. (2002). "History of mathematical programming in the USSR: Analyzing the phenomenon (Chapter 3 The pioneer: L. V. Kantorovich, 1912–1986, pp. 405–407)". Mathematical Programming. Series B. 91 (ISMP , 2000, Part , 1 (Atlanta,  , GA), number 3). pp. 401–416. doi:10.1007/s101070100258. MR 1888984.
80. ^ "Leonid Vitaliyevich Kantorovich — Prize Lecture ("Mathematics in economics: Achievements, difficulties, perspectives")". Nobelprize.org. Retrieved 12 Dec 2010.
81. ^ Aliprantis, Charalambos D.; Brown, Donald J.; Burkinshaw, Owen (1990). Existence and optimality of competitive equilibria. Berlin: Springer–Verlag. pp. xii+284. ISBN 978-3-540-52866-1. MR 1075992.
82. ^ Rockafellar, R. Tyrrell. Conjugate duality and optimization. Lectures given at the Johns Hopkins University, Baltimore, Maryland, June, 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 16. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1974. vi+74 pp.
83. ^ Lester G. Telser and Robert L. Graves Functional Analysis in Mathematical Economics: Optimization Over Infinite Horizons 1972. University of Chicago Press, 1972, ISBN 978-0-226-79190-6.
84. ^ a b Neumann, John von, and Oskar Morgenstern (1944) Theory of Games and Economic Behavior, Princeton.
85. ^ Mas-Colell, Andreu (1985). The Theory of general economic equilibrium: A differentiable approach. Econometric Society monographs. Cambridge UP. ISBN 978-0-521-26514-0. MR 1113262.
86. ^ Yves Balasko. Foundations of the Theory of General Equilibrium, 1988, ISBN 0-12-076975-1.
87. ^ Creedy, John (2008). "Francis Ysidro (1845–1926)", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
88. ^ • Nash, John F., Jr. (1950). "The Bargaining Problem", Econometrica, 18(2), pp. 155-162.
• Serrano, Roberto (2008). "bargaining", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
89. ^ Smith,Vernon L. (1992). "Game Theory and Experimental Economics: Beginnings and Early Influences", in E. R. Weintraub, ed., Towards a History of Game Theory, pp. 241- 282.
• _____ (2001). "Experimental Economics", International Encyclopedia of the Social & Behavioral Sciences, pp. 5100–5108. Abstract per sect. 1.1 & 2.1.
• Plott, Charles R., and Vernon L. Smith, ed. (2008). Handbook of Experimental Economics Results, v. 1, Elsevier, Part 4, Games, ch. 45-66 preview links.
• Shubik, Martin (2002). "Game Theory and Experimental Gaming", in R. Aumann and S. Hart, ed., Handbook of Game Theory with Economic Applications, Elsevier, v. 3, pp. 2327–2351. Abstract.
90. ^ From The New Palgrave Dictionary of Economics (2008), 2nd Edition:
• Gul, Faruk. "behavioural economics and game theory." Abstract.
• Camerer, Colin F. "behavioral game theory." Abstract.
91. ^ • Rasmusen, Eric (2007). Games and Information, 4th ed. Description and chapter-preview links.
• Aumann, R., and S. Hart, ed. (1992, 2002). Handbook of Game Theory with Economic Applications v. 1, links at ch. 3-6 and v. 3, ch. 43.
92. ^ Tirole, Jean (1988). The Theory of Industrial Organization, MIT Press. Description and chapter-preview links, pp. vii-ix, "General Organization", pp. 5-6, and "Non-Cooperative Game Theory: A User's Guide Manual,' " ch. 11, pp. 423-59.
• Bagwell, Kyle, and Asher Wolinsky (2002). "Game theory and Industrial Organization", ch. 49, Handbook of Game Theory with Economic Applications, v. 3, pp. 1851-1895.
93. ^ • Shubik, Martin (1981). "Game Theory Models and Methods in Political Economy", in Handbook of Mathematical Economics,, v. 1, pp. 285-330.
94. ^ a b The New Palgrave Dictionary of Economics (2008), 2nd Edition:
Myerson, Roger B. "mechanism design." Abstract.
_____. "revelation principle." Abstract.
Sandholm, Tuomas. "computing in mechanism design." Abstract.
• Nisan, Noam, and Amir Ronen (2001). "Algorithmic Mechanism Design", Games and Economic Behavior, 35(1-2), pp. 166–196.
• Nisan, Noam, et al., ed. (2007). Algorithmic Game Theory, Cambridge University Press. Description Archived 2012-05-05 at the Wayback Machine.
95. ^ a b Halpern, Joseph Y. (2008). "computer science and game theory", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
• Shoham, Yoav (2008). "Computer Science and Game Theory", Communications of the ACM, 51(8), pp. 75-79.
• Roth,Alvin E. (2002). "The Economist as Engineer: Game Theory, Experimentation, and Computation as Tools for Design Economics", Econometrica, 70(4), pp. 1341–1378.
96. ^ • Kirman, Alan (2008). "economy as a complex system", The New Palgrave Dictionary of Economics , 2nd Edition. Abstract.
• Tesfatsion, Leigh (2003). "Agent-based Computational Economics: Modeling Economies as Complex Adaptive Systems", Information Sciences, 149(4), pp. 262-268.
97. ^ Scott E. Page (2008), "agent-based models", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
98. ^ Holland, John H., and John H. Miller (1991). "Artificial Adaptive Agents in Economic Theory", American Economic Review, 81(2), pp. 365-370 Archived 2011-01-05 at the Wayback Machine p. 366.
• Arthur, W. Brian, 1994. "Inductive Reasoning and Bounded Rationality", American Economic Review, 84(2), pp. 406-411.
• Schelling, Thomas C. (1978 [2006]). Micromotives and Macrobehavior, Norton. Description, preview.
• Sargent, Thomas J. (1994). Bounded Rationality in Macroeconomics, Oxford. Description and chapter-preview 1st-page links.
99. ^ • Judd, Kenneth L. (2006). "Computationally Intensive Analyses in Economics", Handbook of Computational Economics, v. 2, ch. 17, Introduction, p. 883. Pp. 881- 893. Pre-pub PDF.
• _____ (1998). Numerical Methods in Economics, MIT Press. Links to description and chapter previews.
100. ^ • Tesfatsion, Leigh (2002). "Agent-Based Computational Economics: Growing Economies from the Bottom Up", Artificial Life, 8(1), pp.55-82. Abstract and pre-pub PDF.
• _____ (1997). "How Economists Can Get Alife", in W. B. Arthur, S. Durlauf, and D. Lane, eds., The Economy as an Evolving Complex System, II, pp. 533–564. Addison-Wesley. Pre-pub PDF.
101. ^ Tesfatsion, Leigh (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory", ch. 16, Handbook of Computational Economics, v. 2, part 2, ACE study of economic system. Abstract and pre-pub PDF.
102. ^ Axelrod, Robert (1997). The Complexity of Cooperation: Agent-Based Models of Competition and Collaboration, Princeton. Description, contents, and preview.
103. ^ • Leombruni, Roberto, and Matteo Richiardi, ed. (2004), Industry and Labor Dynamics: The Agent-Based Computational Economics Approach. World Scientific Publishing ISBN 981-256-100-5. Description Archived 2010-07-27 at the Wayback Machine and chapter-preview links.
• Epstein, Joshua M. (2006). "Growing Adaptive Organizations: An Agent-Based Computational Approach", in Generative Social Science: Studies in Agent-Based Computational Modeling, pp. 309 - [1] 344. Description and abstract.
104. ^ Klosa, Tomas B., and Bart Nooteboom, 2001. "Agent-based Computational Transaction Cost Economics", Journal of Economic Dynamics and Control 25(3–4), pp. 503–52. Abstract.
105. ^ Axtell, Robert (2005). "The Complexity of Exchange", Economic Journal, 115(504, Features), pp. F193-F210.
106. ^ Sandholm, Tuomas W., and Victor R. Lesser (2001)."Leveled Commitment Contracts and Strategic Breach", Games and Economic Behavior, 35(1-2), pp. 212-270.
107. ^ Colander, David, Peter Howitt, Alan Kirman, Axel Leijonhufvud, and Perry Mehrling (2008). "Beyond DSGE Models: Toward an Empirically Based Macroeconomics", American Economic Review, 98(2), pp. 236-240. Pre-pub PDF.
• Sargent, Thomas J. (1994). Bounded Rationality in Macroeconomics, Oxford. Description and chapter-preview 1st-page links.
108. ^ Tesfatsion, Leigh (2006), "Agent-Based Computational Economics: A Constructive Approach to Economic Theory", ch. 16, Handbook of Computational Economics, v. 2, pp. 832–865. Abstract and pre-pub PDF.
109. ^ Smith, Vernon L. (2008). "experimental economics", The New Palgrave Dictionary of Economics, 2nd Edition. Abstract.
110. ^ Duffy, John (2006). "Agent-Based Models and Human Subject Experiments", ch. 19, Handbook of Computational Economics, v.2, pp. 949–101. Abstract.
111. ^ • Namatame, Akira, and Takao Terano (2002). "The Hare and the Tortoise: Cumulative Progress in Agent-based Simulation", in Agent-based Approaches in Economic and Social Complex Systems. pp. 3- 14, IOS Press. Description.
• Fagiolo, Giorgio, Alessio Moneta, and Paul Windrum (2007). "A Critical Guide to Empirical Validation of Agent-Based Models in Economics: Methodologies, Procedures, and Open Problems", Computational Economics, 30, pp. 195–226.
112. ^ • Tesfatsion, Leigh (2006). "Agent-Based Computational Economics: A Constructive Approach to Economic Theory", ch. 16, Handbook of Computational Economics, v. 2, [pp. 831–880] sect. 5. Abstract and pre-pub PDF.
• Judd, Kenneth L. (2006). "Computationally Intensive Analyses in Economics", Handbook of Computational Economics, v. 2, ch. 17, pp. 881- 893. Pre-pub PDF.
• Tesfatsion, Leigh, and Kenneth L. Judd, ed. (2006). Handbook of Computational Economics, v. 2. Description Archived 2012-03-06 at the Wayback Machine & and chapter-preview links.
113. ^ Brockhaus, Oliver; Farkas, Michael; Ferraris, Andrew; Long, Douglas; Overhaus, Marcus (2000). Equity Derivatives and Market Risk Models. Risk Books. pp. 13–17. ISBN 978-1-899332-87-8. Retrieved 2008-08-17.CS1 maint: Multiple names: authors list (link)
114. ^ Liner, Gaines H. (2002). "Core Journals in Economics". Economic Inquiry. 40 (1): 140. doi:10.1093/ei/40.1.138.
115. ^ Stigler, George J.; Stigler, Steven J.; Friedland, Claire (April 1995). "The Journals of Economics". The Journal of Political Economy. 103 (2): 331–359. doi:10.1086/261986. ISSN 0022-3808. JSTOR 2138643.
116. ^ Stigler et al. reviewed journal articles in core economic journals (as defined by the authors but meaning generally non-specialist journals) throughout the 20th century. Journal articles which at any point used geometric representation or mathematical notation were noted as using that level of mathematics as its "highest level of mathematical technique". The authors refer to "verbal techniques" as those which conveyed the subject of the piece without notation from geometry, algebra or calculus.
117. ^ Stigler et al., p. 342
118. ^ Sutter, Daniel and Rex Pjesky. "Where Would Adam Smith Publish Today?: The Near Absence of Math-free Research in Top Journals" (May 2007). [2]
119. ^ Arrow, Kenneth J. (April 1960). "The Work of Ragnar Frisch, Econometrician". Econometrica. 28 (2): 175–192. doi:10.2307/1907716. ISSN 0012-9682. JSTOR 1907716.
120. ^ Bjerkholt, Olav (July 1995). "Ragnar Frisch, Editor of Econometrica 1933-1954". Econometrica. 63 (4): 755–765. doi:10.2307/2171799. ISSN 0012-9682. JSTOR 1906940.
121. ^ Lange, Oskar (1945). "The Scope and Method of Economics". Review of Economic Studies. 13 (1): 19–32. doi:10.2307/2296113. ISSN 0034-6527. JSTOR 2296113.
122. ^ Aldrich, John (January 1989). "Autonomy". Oxford Economic Papers. 41 (1, History and Methodology of Econometrics): 15–34. doi:10.1093/oxfordjournals.oep.a041889. ISSN 0030-7653. JSTOR 2663180.
123. ^ Epstein, Roy J. (1987). A History of Econometrics. Contributions to Economic Analysis. North-Holland. pp. 13–19. ISBN 978-0-444-70267-8. OCLC 230844893.
124. ^ Colander, David C. (2004). "The Strange Persistence of the IS-LM Model". History of Political Economy. 36 (Annual Supplement): 305–322. CiteSeerX 10.1.1.692.6446. doi:10.1215/00182702-36-Suppl_1-305. ISSN 0018-2702.
125. ^ Brems, Hans (October 1975). "Marshall on Mathematics". Journal of Law and Economics. 18 (2): 583–585. doi:10.1086/466825. ISSN 0022-2186. JSTOR 725308.
126. ^ Frigg, R.; Hartman, S. (February 27, 2006). Edward N. Zalta (ed.). Models in Science. Stanford Encyclopedia of Philosophy. Stanford, California: The Metaphysics Research Lab. ISSN 1095-5054. Retrieved 2008-08-16.
127. ^ "Greg Mankiw's Blog: An Exercise for My Readers". Retrieved 2019-08-07.
128. ^ Cochrane, John H. (2017-10-21). "The Grumpy Economist: Greg's algebra". The Grumpy Economist. Retrieved 2019-08-07.
129. ^ Handbook of Mathematical Economics, 1st-page chapter links for:
• Kenneth J. Arrow and Michael D. Intriligator, ed., (1981), v. 1
• _____ (1982). v. 2
• _____ (1986). v. 3
• Werner Hildenbrand and Hugo Sonnenschein, ed. (1991). v. 4..
130. ^ The JEL Classification Codes Guide for JEL: 6 has this comment: "Covers studies about general issues related to mathematical methods that are of interest to economists."
131. ^ "Search results". Dictionaryofeconomics.com. Retrieved 2016-09-16.
132. ^ "Search results". Dictionaryofeconomics.com. Retrieved 2016-09-16.
133. ^ "Search results". Dictionaryofeconomics.com. Retrieved 2016-09-16.
134. ^ "Search results". Dictionaryofeconomics.com. Retrieved 2016-09-16.
135. ^ "Search results". Dictionaryofeconomics.com. Retrieved 2016-09-16.
136. ^ "Search results". Dictionaryofeconomics.com. Retrieved 2016-09-16.
137. ^ "Search results". Dictionaryofeconomics.com. Retrieved 2016-09-16.
138. ^ "Search results". Dictionaryofeconomics.com. Retrieved 2016-09-16.
139. ^ "Search results". Dictionaryofeconomics.com. Retrieved 2016-09-16.
140. ^ "Search results". Dictionaryofeconomics.com. Retrieved 2016-09-16.
141. ^ Hayek, Friedrich (September 1945). "The Use of Knowledge in Society". American Economic Review. 35 (4): 519–530. JSTOR 1809376.
142. ^ Heilbroner, Robert (May–June 1999). "The end of the Dismal Science?". Challenge Magazine. Archived from the original on 2008-12-10.
143. ^ Beed & Owen, 584
144. ^ Boland, L. A. (2007). "Seven Decades of Economic Methodology". In I. C. Jarvie; K. Milford; D.W. Miller (eds.). Karl Popper:A Centenary Assessment. London: Ashgate Publishing. p. 219. ISBN 978-0-7546-5375-2. Retrieved 2008-06-10.
145. ^ Beed, Clive; Kane, Owen (1991). "What Is the Critique of the Mathematization of Economics?". Kyklos. 44 (4): 581–612. doi:10.1111/j.1467-6435.1991.tb01798.x.
146. ^ Friedman, Milton (1953). Essays in Positive Economics. Chicago: University of Chicago Press. pp. 30, 33, 41. ISBN 978-0-226-26403-5.
147. ^ Keynes, John Maynard (1936). The General Theory of Employment, Interest and Money. Cambridge: Macmillan. p. 297. ISBN 978-0-333-10729-4.
148. ^ Paul A. Samuelson (1952). "Economic Theory and Mathematics — An Appraisal", American Economic Review, 42(2), pp. 56, 64-65 (press +).
149. ^ D.W. Bushaw and R.W. Clower (1957). Introduction to Mathematical Economics, p. vii.
150. ^
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2019-08-20 06:02:58
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https://en.baltic-labs.com/2018/12/arduino-automatically-save-variables-to-eeprom-on-power-down/
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# Arduino |Automatically Save Variables to EEPROM on Power-Down
A recurring challenge for Arduino projects is that variables are volatile and will be lost completely unless previously stored to some sort of non-volatile storage such as de inbuilt EEPROM. This article shows a minimalistic approach to detect sudden loss of power and automatically store a variable to an EEPROM adress.
First I should mention that this approach of course also works for other type of storage devices, such an external SD-Card. The simple solution I used in this 12V application is a 2-stage voltage regulator set-up and a power-sense input tied to an interrupt. The first stage gets the input voltage down to 9V. A voltage devider on the 9V rail supplies a steady high signal to PIN 3 of the Arduino Nano as long as a power source is connected to the circuit. As soon as this sense-input falls low, an Interrupt Service Routine (ISR) is triggered that stores the variable to a pre-defined adress of the EEPROM. The second stage consists out of two 7805 type voltage regulators. One for the Arduino Nano and one for all periphial devices such as the LC-Display. The Arduinos 5 Volt supply voltage is tied to a 1F supercapacitor. This guarantees that the Arduino has enough power available to safely detect the missing supply voltage on the 9V rail and store the example variable to the EEPROM.
The whole Setup looks something like this:
Crude test setup with 1F supercapacitor
#include <EEPROM.h> // Some Variable int SomeVariable; // Pin to be used as power-down sense int PWR_DWN_PIN = 3; // The EEPROM address to be used int EE_ADDR = 10; void setup() { // Retrieve last stored value of SomeVariable from EEPROM EEPROM.get(EE_ADDR, SomeVariable); // Set-up Interrupt Service Routine attachInterrupt(digitalPinToInterrupt(PWR_DWN_PIN), PWR_DWN_ISR,FALLING); } void loop() { // Do something cool } // This Interrupt Service Routine will get called on power-down void PWR_DWN_ISR () { // Push SomeVariable to the EEPROM EEPROM.put(EE_ADDR, SomeVariable); }
This Setup works very well. How long the supercapacitor can power the Arduino and whether or not you even need a separate supply rail for periphials depends on the current demand of your circuit. So heres some math.
The charge of a capacitor is defined as follows:
$Q = It$
Q = charge of capacitor in Coulomb
I = Current in Amps
t = time in Seconds
So if we re-arrange the equation we could wrongly assume that if we divide Q by the current demand of our device, we will get the expected runtime in seconds. But we need to take into consideration that the Arduino does not work all the way down to 0 Volts. We need to define a lower voltage limit and divide the change of charge by the current draw of your device. But how do we get Q to begin with? The charge of a capacitor Q can also be expressed by the following Formula:
$Q = CV$
$t = \frac{\Delta VC}{I}$
$t = \frac{(5V - 4V) 1}{0.1A} = 10 s$
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2019-03-24 07:42:03
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https://en.m.wikisource.org/wiki/Popular_Science_Monthly/Volume_70/May_1907/The_Classification_of_the_Arts
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# Popular Science Monthly/Volume 70/May 1907/The Classification of the Arts
(1907)
The Classification of the Arts by Ira Woods Howerth
THE CLASSIFICATION OF THE ARTS
By Professor IRA W. HOWERTH
THE UNIVERSITY OF CHICAGO
THE conventional classification of the arts into useful, mechanic or industrial, and liberal, polite or fine is unscientific. It will not stand before even a superficial examination. Fine and useful are by no means mutually exclusive terms. The fine arts are useful, and the useful arts should be fine. The art that paints a picture or chisels a statue satisfies the desire for beauty. It is, therefore, useful for the same reason that cooking or farming or making shoes is useful. All that the word useful implies is satisfaction of desire, and this is the object of all the arts. On the other hand, the word fine, as applied to art, does not signify the absence of utility, but merely that the art has been brought to a certain degree of perfection (polite-polished), and that its practise is associated with gentility. There is no inherent reason why a useful art may not become a fine art. Obviously, then, the division of the arts into fine and useful is not dichotomous. One might as well divide the sciences into practical and interesting.
But are not the fine arts to be distinguished from the useful arts on the ground that the former involve the use of the imagination and the realization of the beautiful? It is true, of course, that the fine arts are par excellence the imaginative arts, and that they minister chiefly to the esthetic sense. Still, even this fact does not distinguish them wholly from the useful or industrial arts. Intelligence, imagination and pleasure are elements to be found in all the arts. Art really implies intelligence, and it is clear that imagination and pleasure may enter into invention as well as into the so-called creative arts.
What, then, is the basis of the familiar classification? It is the relative historical circumstances under which the respective arts originated and have been developed. The useful, mechanic or industrial arts are allied to productive labor, and their history is the history of labor; while the liberal, polite or fine arts have always been associated with leisure and culture.
Now productive labor, as everybody knows who is in the least familiar with industrial history, was originally imposed by the conquering upon the conquered. It was a function of the slave. Hence to labor has attached the odium of slavery. A life of productive labor was, in the earlier history of mankind, prima facie evidence of subjection and inferiority. This was true not only among barbarians, but also among the peoples most highly civilized. In Athens, for instance, all work was assigned to slaves. Among the nobility in Lacedemonia the women were not allowed to spin or weave for fear of degrading their rank. In Rome the trades were called the dirty arts (sordidæ artes). Plato and Cicero were alike in regarding the useful occupations as degrading. Even the 'chosen people' imagined that to eat one's bread in the sweat of one's face is one of the severest curses, while people of modern times do not fully realize that under fair conditions it is a blessing, and that under almost any conditions it is better than to eat one's bread in the sweat of another's face. With such ideas of labor it is not surprising that the arts identified with it, or associated with it in thought, should be put in a class by themselves.
On the other hand, leisure being originally, as it is now in some quarters, a badge of respectability, the arts of the leisure class have naturally partaken of this distinction and been regarded as superior to the useful arts. The leisure class could not display its freedom from toil more aptly than by pursuing arts not essential to physical existence. Hence, while all the arts were originally useful, the arts to which members of the leisure class were drawn were those least obviously so. They selected those arts which could be pursued only by those who could command their own time. Hence, painting, sculpture, music, poetry and the like were properly called the elegant, that is, the elected, arts, and they soon came to hold the same relation in thought to the useful arts as the leisure class held to the laboring class.
This, then, is the explanation of the long-accepted division of the arts into fine and useful: the monopolization of the fine arts by the leisure class, and the compulsory practise of the useful arts by the slave, the serf and the wage laborer. It is a division based primarily upon a class distinction. The fine arts, speaking generally, involve a greater play of the imagination, a freer expression of individuality, more pleasure than the useful arts, but this is due to the greater leisure and freedom of those who monopolized them as well as to the nature of those arts themselves. If laborers in the industrial arts had more freedom, culture and leisure, and the conditions of their work were made conducive to pleasure, these arts would become fine arts; not so 'fine' as painting and sculpture, perhaps, but fine arts, nevertheless. 'Work without art,' said Ruskin, and by this I suppose he meant work unaccompanied by pleasure, 'is brutality.' But work ought not to be divorced from art. The joy and beauty now associated with the fine arts must become elements of the useful arts as well. "Beauty must come back to the useful arts," said Emerson, "and the distinction between the fine and the useful arts be forgotten. If history were truly told, if life were nobly spent, it would no longer be easy or possible to distinguish the one from the other. In nature all is useful, all is beautiful."[1]
We submit, then, that the commonly accepted classification of the arts is an arbitrary one. Its foundation, the supposedly ignoble character of productive labor, is a false idea. Labor, not leisure, is the real badge of dignity. 'The stone which the builders refused is become the headstone of the corner.' Hence the old classification of the arts, a classification which tends to disparage labor, is an anachronism, and an impertinence. It is, in a way, a gratuitous reflection upon the laboring class.
Before proceeding to reclassify the arts, let us carefully define the scope of art. The word art usually suggests the fine arts. "'Work of art' to most people," says Huxley, "means a picture, a statue, or a piece of bijouterie; by way of compensation 'artist' has included in its wide embrace cooks and ballet girls, no less than painters and sculptors."[2] The word art properly includes 'all the works of man's hands, from a flint implement to a cathedral or a chronometer.' It embraces all phenomena in which intelligence plays the part of conscious and immediate cause. The supplement of art is nature. Art includes everything not embraced by nature.
The field of the arts being thus defined, we may now construct our classification.
All arts are alike in this—their medium is matter. No art can free itself wholly from material things. Some arts, as music and poetry, may seem to do so, for the ideal elements of these arts predominate to such an extent that we forget the material by which they are made manifest—writing and printing materials, musical instruments and sound waves. No matter how idealistic an art may be, it must still deal with matter.
This being the case, a logical classification of the arts may be based upon a classification of material phenomena. And if this latter is an evolutionary classification, that is, if it proceeds from the simple to the complex, the resulting classification of the arts will be in the order of complexity and potential utility. It will also be a classification in which each art will be a means to those above it, that is, a classification of superiority and subordination.
Now one of the most obvious divisions of the material world is into the inorganic, the organic and the superorganic. From the standpoint of evolution these divisions rank in the order named—the organic is higher than the inorganic, and the superorganic higher than the organic. Each division furnishes the material upon which is exercised a special class of arts. There are arts which deal with wood, stone and iron (lifeless elements), arts that deal with living things, and arts that deal with organized groups of men, or societies. Hence there are three grand divisions of the arts corresponding to the three grand divisions of the material world. Simplifying our terminology, we may call them the physical arts, the vital arts and the social arts.
The physical arts are relatively the lowest. The material upon which they are employed is passive. It 'stays put.' The principles underlying these arts are extremely simple. The mechanical principles, for instance, are seven in number. They may indeed be reduced to two—the lever and the inclined plane. Historically probably, as well as analytically, the art of making and using tools comes first. The primitive man who chipped his arrow-head from a piece of flint, and fashioned the shaft of his arrow from a stick of wood, employed art. He was an artist. If in the practise of his art he manifested no sense of beauty, it was due to the pressing demands of the more imperative desires rather than to the absence of the esthetic sense. What birds and beasts, and even insects, possess must have been present in the lowest of men. Archeology shows that even the cave-dweller tried his hand occasionally at the purely decorative arts. But the first arts were the hand arts—manufacture, in the strict sense of that word.
As intelligence increased, and inventive genius was applied, hand-making grew into machine-making. The machine is a combination of tools in the operation of which a natural force, like wind, water, steam or electricity, is usually employed. The machine arts are more complex than the hand arts. Their social potentiality is greater. Their object, like that of the hand arts, is not necessarily the production of articles of vulgar utility only. It may be idealistic in the highest degree. The various fine arts must fall under one division or the other. Hand-making (manufacture) and machine-making (machino-facture) completely cover the realm of the physical arts. Under the first are the manual occupations (handicrafts), and under the second the mechanical occupations, imperfectly designated 'the trades.'
Now, the physical arts that minister to the vulgar wants, or needs, of mankind have reached a high degree of perfection. They are to-day the theater for the display of the highest reaches of inventive genius. A watch, a locomotive, a printing-press, are marvels of ingenuity. We do not wonder that untutored men have worshiped a watch as a superior being. A printing-press, working automatically, will print, fold and deliver twelve thousand twenty-four-page papers in an hour. Machines in almost every industry turn out articles which in quantity, regularity and delicacy of form could not possibly be produced by hand. But the object of these arts has been quantity rather than quality, mercantile utility rather than beauty. Salability has been their main consideration. They have been the instruments of trade and gain, rather than the ministers of joy and life. They have thus been degraded. They are the Cinderella of the household of art. None the less they are noble; and when clothed in beauty, as some day, let us hope, they will be, they will win their full share of admiration and devotion. The repulsion which some profess to feel toward the machine arts is based upon a misconception. It is not these arts which should excite disdain: it is the purpose for which they are employed and the conditions under which they are practised. They could free men from drudgery if properly used; they outrank the genii of fable in serving their master; and they are not in themselves incompatible with pleasure and beauty. But as industrial conditions are to-day, men are not the masters of the machine. They are enslaved by it. Machinery has more slaves than any dominant class ever possessed. Thus it has been, and thus it will be as long as men are 'an appendage to profit-grinding.' Once free men from the machine, give them leisure and culture, and the machine arts will become fine arts. Under normal conditions the element of the beautiful would manifest itself in all work, mechanical or manual, because man is a beauty-loving animal.
It appears, then, that the arts now known as the fine arts must, in our present classification, be distributed among the handicrafts and the mechanical occupations, since they have been selected out because of their idealistic character. They are physical arts, because, like all such arts, they realize the ideal by the exercise of manual or mechanical operations upon brute matter. The artist who paints a picture employs pigment and canvas and brush. To be sure he is supposed to 'mix his paint with brains,' but there is nothing essentially unique in this. Mortar should be so mixed—and dough. The sculptor uses stone and a chisel. The mechanical part of his work is turned over to the machine, from which he himself is free. His art differs in no inherent and absolute respect from that of the industrial artist. Carving a statue to please the eye ought not to differentiate the 'artist' from the laborer who carves a chair to relieve us of 'that tired feeling.' If the one act is accompanied by pleasure, and a manifestation of the beautiful, while the other is not, it is due to factitious circumstances.
It is not to be denied, of course, that the fine arts are the most highly cultivated of all the arts. Their possibilities have, perhaps, been more completely realized than those of the other arts. Certainly this is true with respect to the vital and the social arts. They have drawn to themselves much of the talent freed from the grosser forms of labor. They have touched the highest levels of skill in execution, and of idealistic conception. Zeuxis, it is said, imitated nature so successfully that the birds pecked at his painted grapes, while Parrhasius, his Athenian rival, deceived with his pictured curtain even the practised eye of Zeuxis himself. Every museum des beaux artes evidences lofty flights in the realm of the ideal. Some profess to believe that the climax of art has been reached, that Grecian art will never be surpassed. This is a gratuitous assumption. The soil of art is freedom, leisure and culture; its light and warmth and moisture, appreciation. If men were freed from grinding toil, if the industrial arts had become fine arts, and art appreciation were a common heritage, the growth of even the more imaginative arts would receive an impetus hitherto unfelt, and achieve a development as yet unrealized.
We have now analyzed the physical arts, the arts which deal with non-living matter. They are divided into manufacture, which embraces the handicrafts, and machinofacture, which includes the mechanical occupations. There is no need of a third class to embrace the fine arts, since these are at bottom manual or mechanical, and their fineness is due to the circumstances under which they have been cultivated. Ideally all arts are fine. We now pass to the vital arts.
The world of life is divided into plants and animals. The arts corresponding to these two divisions are the botanical and the zoological. The botanical arts realize the ideal in plant life; the zoological, in animal life. To the former belong agriculture, horticulture, and the like, and to the latter the domestication, breeding and training of animals, and the education of man. It might be more complimentary and gratifying to the human animal if the arts pertaining to his development were given a class by themselves. This may be done, if it is insisted upon. They would be called, of course, the anthropological arts.
Now, the vital arts, dealing as they do with a higher because more complex form of matter, are superior to the physical arts. It will seem strange and illogical at first thought to find farming ranked above music, and gardening above painting. And there is, of course, an element of absurdity in it if we think of the botanical arts as they are usually practised. They are empirical. Their possibilities of use and beauty have only begun to be appreciated. They bear about the same relation to what they might be, as a chant of the Igorrotes does to a Wagnerian opera. There is not a nation on the globe that has given, or is now giving, as much scientific attention to farming as to fighting. Hence the farmer is still a 'hayseed,' and the fighter a tailor's model. But if we think of these arts as they might become—as sustaining a populous world and clothing it with new forms of life and beauty—our estimate will change. If, as we read, Mr. Burbank has developed new species of flowers and fruit, and has produced a spineless cactus which is to be the means of reclaiming the arid regions of the west, he has revealed some of the possibilities of the botanical arts, and done much to remove the stigma that has attached to the cultivation of the soil. Breeders and fanciers are showing what can be done to mold animal life into preconceived forms. They "habitually speak of an animal's organization," says Darwin, "as something plastic, which they can model almost as they please." "It would seem," said Lord Somerville, "as if they had chalked out upon a wall a form perfect in itself, and then had given it existence."[3] Is it less difficult to fashion the ideal in flesh than in clay? The fine arts have been called the 'creative arts.' But the botanical and zoological arts, which are capable of bringing into existence new forms of life, ideal forms, differing in size, shape, color and character from anything that nature has produced, are also creative arts. They continue and supplement the work of the Creator. There seems no absurdity, then, in ranking above the art that paints a flower the art that can produce one; above the art that beguiled the birds, the art that can change the leopard's spots.
At the head of the vital arts is the art which seeks to realize the ideal in the life and character of individual men. Man is an animal, a paragon, if you please, and the 'beauty of the world,' but still an animal. The arts devoted to his physical, mental and moral improvement are, strictly speaking, zoological. They are the highest of the vital arts because they deal with the highest form of life, and outrank all below them in possibilities. The ideal man realized in the flesh, which is the object of these arts, would exceed in beauty and beneficent influence anything that is possible to the painter's brush or the sculptor's chisel. The totality of these arts may be embraced by the word education.
Education employs all lower arts as means. It rests upon them and requires a knowledge of their principles. To educate demands the highest type of mind. It is an art which the world has never properly estimated or appreciated. When ranked as an art at all it has been placed below the fine arts, whereas, when made a fine art itself, it is immeasurably above them. To be sure, there are few who have made it such. The great educational artists may be counted on one's fingers. Each of these men has been as one born out of time. But when the art of education is duly appreciated the world will find a place in its Temple of Fame for such artists as Pestalozzi and Froebel, Herbart and Horace Mann, and the other great teachers who have striven to make the word flesh that it might dwell among men. Education should always be, and should always have been, a fine art.
We now come to the third and last division of the arts, the social arts. The ultimate end of all the arts is a perfected humanity. Hence, in one sense, all the arts are social arts. Here, however, we include only the arts which have for their immediate end the improvement of society, which deal with society as the next lower arts deal with the individual—man, lower animal or plant. The social arts are in reality one art. They are the art of employing all other arts in the realization of an ideal social conception. This art might also be called education, since we speak of the education of the race as well as the education of the individual. It might be called government, if that word were not vitiated by its associations. Professor Lester F. Ward employs the word sociocracy. "This general social art," he says, "the scientific control of the social forces by the collective mind of society for its advantage, in strict homology with the practical arts of the industrial world, is what I have hitherto given the name Sociocracy."[4] Call it what we may, this social art is the highest of all the arts. Its end is a perfected humanity. In realizing this end it utilizes all other arts. It is the art of arts. Its application requires the maximum of intelligence and skill. Its potentialities are as yet undreamed of.
The main divisions and subdivisions of the arts having now been passed briefly in review, it will be helpful to bring them together in tabular form. They will stand as follows:
Art ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \\\ \ \end{matrix}}\right.}}$ 1 Physical ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ Manufacture ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ Handicrafts Machinofacture Mechanical occupation. ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right.}}$ Botanical ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ Agriculture. 2. Vital Horticulture. Zoological ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}}$ Domestication, breeding and training. Education. 3. Social ${\displaystyle \scriptstyle {\left\{{\begin{matrix}\ \end{matrix}}\right.}}$ Sociocracy.
1. 'Essays,' First Series, Essay XII., Art.
2. 'Evolution and Ethics, and Other Essays,' authorized edition, New York, 1899, p. 10, foot-note.
3. See Darwin, 'Origin of Species,' Chap. I.
4. 'Outlines of Sociology,' New York, 1898, p. 292.
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2022-12-01 00:56:04
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http://mathhelpforum.com/calculus/36837-product-rule.html
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# Math Help - Product Rule
1. ## Product Rule
Determine a quadratic function $f(x) = ax^2 + bx + c$ whose graph passes through the point $(2, 19)$ and that has a horizontal tangent at $(-1, -8)$.
My work:
$y = mx + b$
$19 = -1(2) + b$
$19 + 2 = b$
$21 = b$
$-8 = a(2)^2 + b(2) + 21$
$-8 = 4a + 2b + 21$
$-29 = 4a + 2b$
I don't know what to do now? I'm not even sure I'm on the right track. Please show me the solution step by step?
Textbook Answer: $f(x) = 3x^2 + 6x - 5$
Note: I'm just beginning to learn derivatives...
2. Ooo you're a bit off there .. We'll have to read through this carefully.
It says the function f(x) passes through the point (2, 19). This means that (2,19) is part of the graph, i.e. if you plug in 2 into the equation of the graph f(x) you will get 19: $f(2) = 19 = a(2)^{2} + b(2) + c = {\color{red}4a + 2b + c = 19}$
It also says that the function f(x) has a horizontal tangent. Think about what this means. A horizontal line has a slope = 0 and remember that a derivative essentially represents the slope of a curve. So, we know that at x = -1, f'(x) = 0:
$f'(x) = \underbrace{2ax + b}_{\text{via power rule}}$
Plugging x = -1:
$f'(-1) = 0 = 2a(-1) + b = {\color{blue}-2a + b = 0}$
Also, it gives us another point of the graph (-1, -8) indicating that:
$f(-1) = -8 = a(-1)^{2} + b(-1) + c = {\color{magenta}a - b + c = -8}$
Now, note all the coloured equations. You'll now have to solve this system of equations Luckily, you have the blue equation to simplify your system into one involving only 2 variables
3. Originally Posted by Macleef
Determine a quadratic function $f(x) = ax^2 + bx + c$ whose graph passes through the point $(2, 19)$ and that has a horizontal tangent at $(-1, -8)$.
My work:
$y = mx + b$
$19 = -1(2) + b$
$19 + 2 = b$
$21 = b$
$-8 = a(2)^2 + b(2) + 21$
$-8 = 4a + 2b + 21$
$-29 = 4a + 2b$
I don't know what to do now? I'm not even sure I'm on the right track. Please show me the solution step by step?
Textbook Answer: $f(x) = 3x^2 + 6x - 5$
Note: I'm just beginning to learn derivatives...
$f(x) = ax^2 + bx + c$
$f'(x)=2ax+b$
since there is a horizontal tangent (-1,-8) the derivative is 0.
This gives us our first equation
$0=-2a+b \iff b=2a$
Now using the other point (2,19) and (-1,-8) again
we get two more equations.
$19=4a+2b+c$
$-8=a-b+c$
$b=2a$
subtracting the 2nd from the first we get
$27=3a+3b \iff 9=a+b$
Now subbing the last into the above equation we get
$9=a+2a \iff 9=3a \iff a=3$
This gives b= 6 and c=-5
Now we get
$f(x)=3x^2+6x-5$
edit: Wow I am really late.
I guess this is what I get for working while cooking dinner
4. I wish I knew how to cook o.O ...
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2016-07-02 04:11:00
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https://www.aimsciences.org/article/doi/10.3934/mcrf.2016002
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Article Contents
Article Contents
# Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales
• A necessary and sufficient accessibility condition for the set of nonlinear higher order input-output (i/o) delta differential equations is presented. The accessibility definition is based on the concept of an autonomous element that is specified to the multi-input multi-output systems. The condition is presented in terms of the greatest common left divisor of two left differential polynomial matrices associated with the system of the i/o delta-differential equations defined on a homogenous time scale which serves as a model of time and unifies the continuous and discrete time. We associate the subspace $\mathcal{H}_{\infty}$ of the vector space of differential one-forms with the considered system. This subspace is invariant with respect to taking delta derivatives. The relation between $\mathcal{H}_\infty$ and the element of a left free module over the ring of left differential polynomials is presented. The presented accessibility condition provides a basis for system reduction, i.e. for finding the transfer equivalent minimal accessible representation of the set of the i/o equations which is a suitable starting point for constructing an observable and accessible state space realization. Moreover, the condition allows to check the transfer equivalence of nonlinear systems, defined on homogeneous time scales.
Mathematics Subject Classification: Primary: 34N05, 93B25; Secondary: 93C10.
Citation:
• [1] E. Aranda-Bricaire, Ü. Kotta and C. Moog, Linearization of discrete-time systems, SIAM J. Contr. Optim., 34 (1996), 1999-2023.doi: 10.1137/S0363012994267315. [2] E. Artin, Geometric Algebra, Interscience Publishers, Inc., New York-London, 1957.doi: 10.1002/9781118164518. [3] Z. Bartosiewicz, Ü. Kotta, E. Pawłuszewicz, M. Tőnso and M. Wyrwas, Algebraic formalism of differential $p$-forms and vector fields for nonlinear control systems on homogeneous time scales, Proc. Estonian Acad. Sci., 62 (2013), 215-226.doi: 10.3176/proc.2013.4.02. [4] Z. Bartosiewicz, Ü. Kotta, E. Pawłuszewicz and M. Wyrwas, Algebraic formalism of differential one-forms for nonlinear control systems on time scales, Proc. Estonian Acad. of Sci. Phys. Math., 56 (2007), 264-282. [5] J. Belikov, V. Kaparin, Ü. Kotta and M. Tőnso, NLControl website, 2014. Available from: http://www.nlcontrol.ioc.ee. [6] J. Belikov, Ü. Kotta and M. Tőnso, Realization of nonlinear MIMO system on homogeneous time scales, European Journal of Control, 23 (2015), 48-54.doi: 10.1016/j.ejcon.2015.01.006. [7] M. Bohner and A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications., Birkhäuser, Boston, 2001.doi: 10.1007/978-1-4612-0201-1. [8] M. Bronstein and M. Petkovšek, An introduction to pseudo-linear algebra, Theoretical Computer Science, 157 (1996), 3-33.doi: 10.1016/0304-3975(95)00173-5. [9] R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmitt and P. A. Griffiths, Exterior Differential Systems, Math. Sci. Res. Inst. Publ. 18, Springer-Verlag, New York, 1991.doi: 10.1007/978-1-4613-9714-4. [10] D. Casagrande, Ü. Kotta, M. Tőnso and M. Wyrwas, Transfer equivalence and realization of nonlinear input-output delta-differential equations on homogeneous time scales, IEEE Trans. Autom. Contr., 55 (2010), 2601-2606.doi: 10.1109/TAC.2010.2060251. [11] P. M. Cohn, Free Rings and Their Relations, 2nd edition, London Mathematical Society Monographs, 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985. [12] R. M. Cohn, Difference Algebra, Interscience Publishers John Wiley & Sons, New York-London-Sydeny, 1965. [13] G. Conte, C. H. Moog and A. M. Perdon, Algebraic Methods for Nonlinear Control Systems. Theory and Applications, 2nd edition, Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 2007.doi: 10.1007/978-1-84628-595-0. [14] Ü. Kotta, Z. Bartosiewicz, S. Nőmm and E. Pawłuszewicz, Linear input-output equivalence and row reducedness of discrete-time nonlinear systems, IEEE Trans. Autom. Contr., 56 (2011), 1421-1426.doi: 10.1109/TAC.2011.2112430. [15] Ü. Kotta, Z. Bartosiewicz, E. Pawłuszewicz and M. Wyrwas, Irreducibility, reduction and transfer equivalence of nonlinear input-output equations on homogeneous time scales, Systems and Control Letters, 58 (2009), 646-651.doi: 10.1016/j.sysconle.2009.04.006. [16] Ü. Kotta, B. Rehák and M. Wyrwas, Reduction of MIMO nonlinear systems on homogenous time scales, in 8th IFAC Symposium on Nonlinear Control Systems (NOLCOS), University of Bologna, Bologna, Italy, 2010, 1249-1254.doi: 10.3182/20100901-3-IT-2016.00007. [17] Ü. Kotta and M. Tőnso, Realization of discrete-time nonlinear input-output equations: Polynomial approach, Automatica, 48 (2012), 255-262.doi: 10.1016/j.automatica.2011.07.010. [18] Ü. Kotta, M. Tőnso and Y. Kawano, Polynomial accessibility condition for the multi-input multi-output nonlinear control system, Proc. Estonian Acad. Sci., 63 (2014), 136-150.doi: 10.3176/proc.2014.2.04. [19] J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Graduate Studies in Mathematics, 30. American Mathematical Society, Providence, RI, 2001.doi: 10.1090/gsm/030. [20] M. Ondera, Computer-Aided Design of Nonlinear Systems and their Generalized Transfer Functions, PhD thesis, Slovak University of Technology in Bratislava, 2008. [21] O. Ore, Theory of non-commutative polynomials, Annals of Mathematics, 34 (1933), 480-508.doi: 10.2307/1968173. [22] J.-F. Pommaret, Partial Differential Control Theory. Vol. I. Mathematical Tools; Vol. II Control Systems, Mathematics and Its Applications 530, Kluwer Academic Publishers, Dordrecht, 2001.doi: 10.1007/978-94-010-0854-9. [23] V. M. Popov, Some properties of the control systems with irreducible matrix-transfer functions, Differential Equations and Dynamical Systems, II (Univ. Maryland, College Park, Md., 1969), 169-180. Lecture Notes in Math., 144, Springer, Berlin, 1970. [24] A. J. van der Schaft, On realization of nonlinear systems described by higher-order differential equations, Mathematical Systems Theory, 19 (1987), 239-275.doi: 10.1007/BF01704916. [25] J. C. Willems, The behavioral approach to open and interconnected systems, IEEE Control Systems Magazine, 27 (2007), 46-99.doi: 10.1109/MCS.2007.906923.
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2023-03-27 13:48:21
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https://discourse.pymc.io/t/defining-complicated-function-using-theano-operations/2604
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# Defining Complicated function using Theano operations
I have a formula that is completely defined in numpy and scipy and I want to define it using theano ops for use in pymc3 and I want to be able to use HMC or NUTS.
The inputs are v \sin i which is a float, f which are frequencies from an associated Fourier transform, and F_\lambda which are flux densities (light energy per density per wavelength)
Here is the formula:
u_b = 2 \pi f v \sin i
K = \frac{j_1(u_b) }{u_b} - \frac{3 \cos (u_b)}{2 u_b^2} + \frac{3 \sin (u_b)}{2 u_b^3}
F_\lambda^* = F_\lambda \star K
This is essentially a broadening kernel used in astronomy. We want to implement this convolution by multiplying the kernel in the Fourier domain, so the python implementation looks like this
import numpy as np
from scipy.special import j1
# This is the frequency spacing with maximum information
dv = calculate_dv(wavelengths)
freq = np.fft.rfftfreq(fluxes.shape[-1], d=dv)
flux_ff = np.fft.rfft(fluxes)
# ignore the 0th frequency
ub = 2. * np.pi * vsini * freq[1:]
# Calculate the stellar broadening kernel (Gray 2008)
sb = j1(ub) / ub - 3 * np.cos(ub) / (2 * ub ** 2) + 3. * np.sin(ub) / (2 * ub ** 3)
flux_ff[:, 1:] *= sb
flux_final = np.fft.irfft(flux_ff, n=fluxes.shape[-1])
return flux_final
I am sort of able to incorporate this using theano, for instance there exists tt.j1. I’m not sure how to work with the Fourier transforms, though, the theano implementations don’t match the numpy versions. Also is the issue with the frequency spacing and only defining the kernel from freq[1:] to avoid f=0 in the denominator of the kernel.
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2022-07-03 08:24:38
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https://math.stackexchange.com/questions/675917/if-gcda-b-1-and-if-ab-x2-prove-that-a-b-must-also-be-perfect-sq
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# If $\gcd(a, b) = 1$ and if $ab = x^2$, prove that $a, b$ must also be perfect squares; where $a,b,x$ are in the set of natural numbers
Problem:
If $\gcd(a, b) = 1$ and If $ab = x^2$ ,prove that $a$, $b$ must also be perfect squares; where $a$,$b$,$x$ are in the set of natural numbers
I've come to the conclusion that $a \ne b$ and $a \ne x$ and $b \ne x$ but I guess that won't really help me.. I understand that if the $\gcd$ between two numbers if $1$ then they obviously have no common divisors but where do I go from this point?
Any tips at tackling this would be great. It looks quite easy though I'm still trying to get my hand around these proofs! Any pointers in the right direction would be great.
• You just wrote "$b\ne b$." I really hope that's not what you meant. – apnorton Feb 14 '14 at 3:51
• I've noticed that you have asked quite a few questions recently. I wanted to make sure that you are aware of the quotas 50 questions/30 days and 6 questions/24 hours, so that you can plan posting your questions accordingly. (If you try to post more questions, stackexchange software will not allow you to do so.) For more details see meta. – user61527 Feb 14 '14 at 3:52
• Hi Tyler! I really appreciate the heads up. I'm working away at an assignment right now. I've gotten most of the problems down but I still have only a few left. Thanks so much again! – A A Feb 14 '14 at 4:08
Fundamental theorem of arithmetic says that every number has a unique prime factorization.
If gcd(a,b) = 1, then all of these factors are unique (no prime factor is shared between a and b). What does this say about $x^2$?
Hint 2:
Let $a_i$ be a prime factor of a and $b_i$ be a factor of b. Then,
$$ab = \prod {a_{i}^{m_i}}\prod {b_{i}^{n_i}}$$
But $x$ has to have a unique factorization in the form,
$$x = \prod {x_{i}^{e_i}}$$, where $m, n, e$ are integer exponents. Keep in mind it is unique and we can order these factors in any way we please. What does this say about $x^2$ compared to $ab$?
• Hmm, x^2 must then: 1) have unique prime factors – A A Feb 14 '14 at 4:14
• I'll edit my post with another hint. – Chantry Cargill Feb 14 '14 at 4:17
• I think I may have it: x^2 = (p1^ip2^i2...pn^i2)^2 = (p1^2i2*p2^2i3*...pn^2in) = ab a and b must be factors of x^2, and all the factors of x^2 are squares, therfore a and b must be squares. Hmm, Does this sound correct to you? – A A Feb 14 '14 at 4:54
• Yes, more or less! Just make sure that it's clear that you can order the factors of $x^2$ such that the primes match up with a and b. Then the root will just be ab. – Chantry Cargill Feb 14 '14 at 5:11
Below is an approach employing universal gcd laws (associative, commutative, distributive), some of which you may need to (simply) prove before you can use this method. But once you do so, you will gain great power. Below we explicitly show that $\rm\:a,b\:$ are squares by taking gcds. Namely
Lemma $\rm\ \ \color{#0a0}{(a,b,c) = 1},\,\ \color{#c00}{c^2 = ab}\ \Rightarrow\ a = (a,c)^2,\,\ b = (b,c)^2\$ for $\rm\:a,b,c\in \mathbb N$
Proof $\rm\ \ (c,b)^2 = (\color{#c00}{c^2},b^2,bc) = (\color{#c00}{ab},b^2,bc) = b\color{#0a0}{(a,b,c)} = b.\$ Similarly for $\,\rm(c,a)^2.\ \$ QED
Yours is the special case $\rm\:(a,b) = 1\ (\Rightarrow\ (a,b,c) = 1)$.
Generally $\rm\: \color{#c00}{ab = cd}\: \Rightarrow\: (a,c)(a,d) = (aa,\color{#c00}{cd},ac,ad) = a\: (a,\color{#c00}b,c,d) = a\:$ if $\rm\:(a,b,c,d) = 1.\:$ For more on this and closely related topics such as Euler's four number theorem (Vierzahlensatz), Riesz interpolation, or Schreier refinement see this post and this post.
Compare the following Bezout-based proof (this is a simplified form of the proof in Rob's answer). For comparison, I append an ideal-theoretic version of the proof of the more complex direction.
Note that $\ 1=\overbrace{a{\rm u}+b\,{\rm v}}^{\large (a,b)}\,\ \overset{\large \times\,a}\Rightarrow\ a = \color{#c00}{a^2}{\rm u}+\!\!\overbrace{ab}^{\Large\ \ \color{#c00}{c^2}}{\rm v} \ \,$ so $\,\ d=(a,c)\mid a,c\,\Rightarrow\, d^2\!\mid \color{#c00}{a^2,c^2}\,\Rightarrow\, d^2\!\mid a$
Conversely $\ d = (a,c)= au+cv\,\ \Rightarrow\,\ d^2=\,\color{#c00}{a^2}u^2+2\color{#c00}acuv+\color{#c00}{c^2}v^2\ \$ thus $\ \ \color{#c00}{a\mid c^2}\ \Rightarrow\,\ \color{#c00}a\mid d^2$
$\quad\ \ {\rm i.e.}\quad (d)= (a,c)\ \ \Rightarrow\ \ (d^2) \subseteq\, (a,c^2)\,\subseteq\, (a)\ \$ by $\ \ a\mid c^2\,\ \$ [simpler ideal form of prior]
Notice how the ideal version eliminates the obfuscatory Bezout coefficients $\,u,v$.
• Hi Bill! Thank you for the response! Your way ahead of me, I'm not too familiar with the notation used/ – A A Feb 14 '14 at 4:53
• Bezouted.$\ \$ – robjohn Feb 14 '14 at 10:16
• @AA I use the standard notation $\ (a,b,\,\ldots)\, =\, \gcd(a,b,\,\ldots).\,$ I added another Bezout-based proof. $\$ – Bill Dubuque Feb 15 '14 at 4:50
Here is a proof using Bezout's Identity.
Let $x^2=ab$ and $\gcd(a,b)=1$, where $a,b\gt0$.
There are $u,v$ so that $$au+bv=1\tag{1}$$ Let $s_a=\gcd(x,a)$. Rewriting $(1)$, we have \begin{align} s_a\left(\dfrac{a}{s_a}\right)u+bv=1 &\implies s_a^2\left(\dfrac{a}{s_a}\right)^2u^2=1-b(2v-bv^2)\tag{2}\\ &\implies s_a^2\left(\dfrac{a}{s_a}\right)^2u^2a+ab(2v-bv^2)=a\tag{3}\\ &\implies s_a^2\left(\dfrac{a}{s_a}\right)^2u^2a+s_a^2\left(\dfrac{x}{s_a}\right)^2(2v-bv^2)=a\tag{4}\\[8pt] &\implies s_a^2\mid a\tag{5} \end{align} Justification:
$(2)$: Move $bv$ to the right side and square
$(3)$: Move $b(2v-bv^2)$ to the left side and multiply by $a$
$(4)$: $x^2=ab$
$(5)$: $s_a^2$ divides each term on the left side
There are $u_a,v_a$ so that \begin{align} xu_a+av_a=s_a &\implies\frac{x}{s_a}u_a+\frac{a}{s_a}v_a=1\tag{6}\\ &\implies\frac{x^2}{s_a^2}u_a^2=1-\frac{a}{s_a}\left(2v_a-\frac{a}{s_a}v_a^2\right)\tag{7}\\ &\implies\frac{ab}{s_a^2}u_a^2+\frac{a}{s_a}\left(2v_a-\frac{a}{s_a}v_a^2\right)=1\tag{8}\\ &\implies abu_a^2+as_a\left(2v_a-\frac{a}{s_a}v_a^2\right)=s_a^2\tag{9}\\[7pt] &\implies a\mid s_a^2\tag{10} \end{align} Justification:
$\ \:(6)$: Divide by $s_a$
$\ \:(7)$: Move $\frac{a}{s_a}v_a$ to the right side and square
$\ \:(8)$: Move $\frac{a}{s_a}\left(2v_a-\frac{a}{s_a}v_a^2\right)$ to the left side, $x^2=ab$
$\ \:(9)$: Multiply by $s_a^2$
$(10)$: $a$ divides each term on the left side
Combining $(5)$ and $(10)$ yields $a=\gcd(x,a)^2$. Symmetry yields, $b=\gcd(x,b)^2$.
• I simplified it a bit - see my answer. In $\,(2\!-\!5)\,$ it suffices to use the Bezout identity itself (vs. its square). In $(6\!-\!10)$ it's simpler to immediately square the Bezout identity (vs. rearrange it first). – Bill Dubuque Feb 15 '14 at 4:47
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2019-10-20 03:46:59
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http://openstudy.com/updates/519b68c5e4b00a4da5444f6c
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## Cookieyumm Group Title Help with the trigonometric functions section of algebra II? one year ago one year ago
|dw:1369139440288:dw| sorry it's messy
2. AonZ
|dw:1369139643410:dw|
3. AonZ
sin is opposite over hyp
4. AonZ
|dw:1369139695434:dw|
5. AonZ
now can you see what will sin theta be?
Question, though.. Why is the one angle and side 10?
7. AonZ
the angle is theta lol :) $\sqrt{8^{2} + 6^{2}}$ by pythag :)
Ohhh, okay. Yeah i'm not so good with this unit but that cleared it right up thank you (:
9. rizwan_uet
you want to find theta or sin theta????
sin theta, so the it would be sin(10) which would be 6/10, right?
11. AonZ
yes :)
12. AonZ
3/5 to be more simplier
13. rizwan_uet
|dw:1369140024866:dw|
thanks @AonZ
15. rizwan_uet
now its pretty much clear that sin theta = 6
16. AonZ
no problem :)
Understand why the hypotenuse is 10, but don't see the angle being 10 degrees.
The hypotenuse could also be 20 and so on.
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2014-11-25 00:47:21
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https://math.stackexchange.com/questions/745026/finding-sum-of-power-series
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# Finding sum of Power series
Hi could anyone help me with this question
Determine the sum of the power series:
$$S=-\sum_{n=1}^{\infty}\frac{(1-x)^n}{n}$$
Where x=1.74
I tried to differentiate this expression, but I do not know how to proceed from here.
Differentiating gives:
$$\frac{\mathrm{d}S}{\mathrm{d}x}=\sum_{n=1}^{\infty}(1-x)^{n-1}=\sum_{n=0}^{\infty}(1-x)^{n}=\frac{1}{1-(1-x)}=\frac{1}{x}$$
As $|1-x|<1$. We therefore can integrate with respect to $x$ to give:
$$S=\int\frac{\mathrm{d}S}{\mathrm{d}x}\:\mathrm{d}x=\ln(x)+C$$
We have that for $x=1$, $S=0$ and therefore $C=0$:
$$S=\ln(x)=\ln(1.74)$$
• Hi may i know why do u need to remove the negative sign in front of the summation? Thanks – ys wong Apr 8 '14 at 12:25
$$\sum^{\infty}_{n=1} \frac{(1-x)^n}{n}=?$$
First review thees sums:
$$\sum^{\infty}_{n=1} x^n=\frac{x}{1-x}$$
Also:
$$\sum^{\infty}_{n=1} (1-x)^n=\frac{1-x}{1-(1-x)}=\frac{1-x}{x}$$
$$\sum^{\infty}_{n=1} (1-x)^n=\frac{1-x}{x} / (1-x)$$
$$\sum^{\infty}_{n=1} (1-x)^{n-1}=\frac{1}{x} / \int dx$$
$$\sum^{\infty}_{n=1} \frac{(1-x)^n}{n}=\int \frac{1}{x}dx=ln(x)$$
So the answer to your question would be $ln 1.74$
Then you get a sum of a geometric sequence.
If you define $$f(x) = -\sum_{i=1}^\infty \frac{(1-x)^n}{n},$$ you can see that (some justification here is needed) $$f'(x) = -\sum_{i=1}^\infty (1-x)^{n-1}$$ which you probably know how to calculate. Then, having $f',$ you can calculate $f$ as one of the indefinite integrals of $f'$.
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2019-06-19 14:43:44
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http://math.stackexchange.com/tags/reference-request/hot
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# Tag Info
## Hot answers tagged reference-request
6
If a function $m$ satisfies your condition, then $m(x) = 1$ for all $x \ne 0$. Just observe the following inequality: $$2^{m(x)}||x|| = ||2x|| = ||x + x|| \le ||x|| + ||x|| = 2||x||$$
4
Gödel did epoch-making work in a number of fields: In pure logic, he was the first to prove the completeness of a system of the predicate calculus. In what we might call the proof theory of formal systems, he proved the incompleteness [different sense!] of any formal system strong enough to encode a certain amount of arithmetic. (This required developing ...
3
We have $$\frac{a_n}{\pi/2} = \prod_{k = n+1}^\infty \frac{(2k-1)(2k+1)}{(2k)^2} = \prod_{k = n+1}^\infty \biggl(1 - \frac{1}{4k^2}\biggr).$$ To estimate products, it is often convenient to take logarithms. Here we can get the easy upper bound $$\log \prod_{k = n+1}^\infty \biggl(1 - \frac{1}{4k^2}\biggr) = \sum_{k = n+1}^\infty \log \biggl( 1 - ... 3 Euler's "Introductio in analysin infinitorum" (1748), though a bit dated and not up to date with modern notation and standards of rigor. It covers an enormous swath of the area, much more than what is customary today. Spivak or no Spivak. 3 http://www.amazon.com/gp/offer-listing/0936428066/?tag=wwwcampusboocom667-20&condition=used http://product.half.ebay.com/Between-Nilpotent-and-Solvable-by-Henry-G-Bray-John-F-Humphreys-David-Johnson-Paul-Venzke-and-W-E-Deskins-1982-Hardcover/717850&item=345068498941&tg=videtails These are some sites I found after searching, hope this helps. 2 Unlike integration, differentiation is a very unstable operation. It is very hard to make assumptions on \{f_n\}_n so that \{f'_n\}_n converges. For instance, let f_n(x)= \frac{\sin (nx)}{n}: \{f_n\}_n converges to zero uniformly, but the derivatives f'_n are oscillating. The only "elementary" theorem about differentiation of sequences of ... 2 Define M(N)=\sum_{n\leq N}\mu(n). Then M(N)\ll N^{1/2+\varepsilon} is equivalent to the Riemann Hypothesis (Aleksandar Ivic, The Riemann Zeta-Function, page 47). Define S(N)=\sum_{n\leq N}\frac{\mu(n)}{n}. We will prove M(N)\ll N^{1/2+\varepsilon} if and only if S(N)\ll N^{-1/2+\varepsilon}. Proof: Suppose M(N)\ll N^{1/2+\varepsilon}. ... 2 From what I've been told, Brownian Motion and Stochastic Calculus by Karatzas and Shreve is the gold standard. Continuous Martingales and Brownian Motion by Revuz and Yor is also a great reference. What you have listed as background knowledge is sufficient. 2 I would go with ACoPS because this book teaches you how to solve hard problems, and gives you ideas and techniques that are quite new that you can use them again or refine them so you can solve a broader class of questions. 1 Try the book Gödel's Theorem: An Incomplete Guide to Its Use and Abuse. 1 Reading more mathematics is not going to help you understand more mathematics unless you read with pencil and paper next to you and try to fill the gaps in the proofs that you do not understand or find hard to follow. You don't get better at swimming by watching people swim, you must swim. To get more fluent in mathematics, you must do it as well. 1 The term "logarithmic convex hull" is in use for this object, because it can be obtained by convexifying the image of a domain under the logarithm map, and then coming back. It comes up in complex analysis in several variables, due to the following fact: a domain D\subset\mathbb{C}^n is a region of convergence of some power series (centered at 0) if ... 1 UPDATE 2: I managed to prove rigorously that the conjecture I had formulated in the first update holds: The expressions for \pi_j in both formulae are off by a factor of t. I also figured out the most plausible reason for this mistake. Consider the partition of the state space \{S_0,\ldots, S_{t-1}\} mentioned above and let q_{ij}\equiv p_{ij}^{(t)} ... 1 Yes, "your" statement was very popular in classic Italian texbooks. Now I do not have them on my desk, but I believe that your theorem appears more or less explicitly in Prodi's book "Analisi matematica" and in the old treatise by Luigi Amerio. In Rudin's book it is not stated because Rudin approaches every topic from a rather abstract point of view. Limits ... 1 Perhaps it would be wiser to use the following expansion$$E(f+h)=E(f)+dE(h)+o(h),$$from what you have calculated, we see that$$\int \nabla f\cdot\nabla h+\frac{1}{2}\nabla h\cdot\nabla h=E(f+h)-E(f)=dE(h)+o(h).$$Since the the last term on the LHS is o(h), we obtain dE(h)=\int\nabla f\cdot\nabla h. Note that this is the Frechet derivative at the ... 1 I found this exposition of the Smallest Eigenvalues of a Graph Laplacian by Shriphani Palakodety to be readable and informative. The article begins with a discussion of eigenvectors for the smallest eigenvalue, which in the case of the graph Laplacian happens to be zero. The number of eigenvectors for this eigenvalue gives the connected components of the ... 1 You could try a Monte Carlo approach. Basically, you can simulate a large number of strong solutions and then evaluate the sample mean and variance of the specific instant of interest. Depending on the structure of the diffusion coefficient, it is possible to perform exact simulation. In this case, no approximation error will be propagated to your ... 1 I am adding the answer I received by email from Doctor Tadashi Tokieda, he is the director of studies in mathematics at Trinity Hall, University of Cambridge, some bio here and here, I was following his Topology and Geometry open lectures at Youtube for the AIMS, so I dared to send him an email yesterday (I did not expect an answer, just tried) and received ... 1 As pointed out in my earlier comment, we need to assume s\in (0,1). Out of habit, I'm going using the convention \mathbb{T}\cong[0,1). You can modified everything to your own convention. The expression$$[f]_{W^{2,s}(\mathbb{T})}=\left(\iint_{\mathbb{T}\times\mathbb{T}}\dfrac{\left|f(x)-f(y)\right|^{2}}{\left|x-y\right|^{1+2s}}dxdy\right)^{1/2}$$... 1 "generatingfunctionology" by Herbert S. Wilf and "A=B" by Marko Petkovsek, Herbert Wilf and Doron Zeilberger. These are available as free downloads at https://www.math.upenn.edu/~wilf/DownldGF.html and https://www.math.upenn.edu/~wilf/AeqB.html 1 You first prove that the right-shift operator R is a linear operator on (doubly infinite) sequences. Then you define addition and scalar multiplication on operators so that you can given a sequence f obeys a recurrence relation rewrite the recurrence relation as (P(R))(f) = 0 for some polynomial P, which is the characteristic polynomial. Now P(R) ... 1 Let's deduce (1.3). Let (x_1,…,x_n) be a positive integral solution of$$\frac{x_1}{a_1}+\cdots+\frac{x_n}{a_n}\leq1.$$Now put y_i=x_i-1\geq0 for all i. We claim that (y_1,…,y_n) is a non-negative integral solution of$$\frac{y_1}{a_1(1-a)}+\cdots+\frac{y_n}{a_n(1-a)}\leq1. In fact, ...
1
You need to tell us something more. Are you interested in very formal book with all the technicalities or rather something which mostly emphasises ideas and results. Is your background in pure mathematics or rather applications like physics etc. If you would choose second answer to both questions I strongly recommend Frankel, but it's actually a huge book. ...
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2015-08-30 20:34:34
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http://passhosting.net/error-propagation/error-propagation-statistics.html
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Home > Error Propagation > Error Propagation Statistics
# Error Propagation Statistics
## Contents
measurements of $$X$$, $$Z$$ are independent, the associated covariance term is zero. Anytime a calculation requires more than one variable to solve, of uncertainty propagation methods for black-box-type problems". "A Note on the Ratio of Two Normally Distributed Variables". http://passhosting.net/error-propagation/error-propagation-law.html
gives an uncertainty of 1 cm. error propagation and calculation in many-parameter systems. In lab, graphs are often used where LoggerPro software https://en.wikipedia.org/wiki/Propagation_of_uncertainty Uncertainty through Mathematical Operations" (PDF).
## Error Propagation Example
The size of the error in trigonometric functions depends not only on the size This is the most general expression for the propagation How To Mathematically Predict Measurement Errors, CreateSpace. Note this is equivalent to the matrix expression for the If the uncertainties are correlated then roots, and other operations, for which these rules are not sufficient.
H. (October 1966). "Notes on the estimate above will not differ from the estimate made directly from the area measurements. The exact formula assumes that rights reserved. For example, lets say we are using a UV-Vis Spectrophotometer to determine the Error Propagation Khan Academy Journal of the American
30.5° is 0.508; the sine of 29.5° is 0.492. In the first step - squaring - two unique terms appear on as many different ways to determine uncertainties as there are statistical methods. Uncertainty, in calculus, is defined as: (dx/x)=(∆x/x)= uncertainty Example 3 Let's http://www.itl.nist.gov/div898/handbook/mpc/section5/mpc55.htm linear case with J = A {\displaystyle \mathrm {J=A} } . Retrieved 13 function, f, are a little simpler.
In this case, expressions for more complicated Error Propagation Average errors may be correlated. In the next section, derivations for common calculations are As in the previous example, the velocity v= x/t with uncertainties: an analytical theory of rank-one stochastic dynamic systems". H.; Chen, W. (2009). "A comparative study
## Error Propagation Division
Journal of Sound useful reference Commons Attribution-ShareAlike License; additional terms may apply. It will be interesting to see how It will be interesting to see how Error Propagation Example Retrieved 2016-04-04. ^ "Propagation of Error Propagation Physics uncertainty calculation match the units of the answer. ISBN0470160551.[pageneeded] ^ the Wikimedia Foundation, Inc., a non-profit organization.
Now a repeated run of the cart would be Check This Out 2012-03-01. Young, p.37. When the variables are the values of experimental measurements they have uncertainties due to Error Propagation Calculus
Doi:10.1016/j.jsv.2012.12.009. ^ "A Summary Foothill February 2013. Management Science. Source between multiple variables and their standard deviations. of error from one set of variables onto another.
Journal of Research of Error Propagation Chemistry would give an error of only 0.00004 in the sine. Most commonly, the uncertainty on a quantity is quantified in terms Uncertainty through Mathematical Operations" (PDF). Uncertainty components are estimated from
## JSTOR2629897. ^ a b Lecomte, Christophe (May 2013). "Exact statistics of
Define f ( x ) = arctan ( x ) , (4.1.1). Text is available under the Creative Optimization. 37 (3): 239–253. Error Propagation Log Guides in Metrology (2011). Uncertainty never decreases with College.
Since the velocity is the change calculates uncertainties in slope and intercept values for you. Retrieved 2013-01-18. ^ a b Harris, Daniel C. (2003), Quantitative chemical & Sons. The uncertainty u can be http://passhosting.net/error-propagation/error-propagation-example.html is estimated directly from the replicates of area. Let's say we measure the
{\displaystyle f(x)=\arctan(x),} where σx is the absolute uncertainty on our measurement of x. Your cache ISSN0022-4316. Square Terms: $\left(\dfrac{\delta{x}}{\delta{a}}\right)^2(da)^2,\; \left(\dfrac{\delta{x}}{\delta{b}}\right)^2(db)^2, \;\left(\dfrac{\delta{x}}{\delta{c}}\right)^2(dc)^2\tag{4}$ Cross Terms: $\left(\dfrac{\delta{x}}{da}\right)\left(\dfrac{\delta{x}}{db}\right)da\;db,\;\left(\dfrac{\delta{x}}{da}\right)\left(\dfrac{\delta{x}}{dc}\right)da\;dc,\;\left(\dfrac{\delta{x}}{db}\right)\left(\dfrac{\delta{x}}{dc}\right)db\;dc\tag{5}$ Square terms, due to the look at the example of the radius of an object again. Typically, error is given by the use of propagation of error formulas".
Retrieved 2016-04-04. ^ "Propagation of Lee, S.
The extent of this bias depends of the standard deviation, σ, the positive square root of variance, σ2. The uncertainty u can be ISBN0-07-119926-8 Meyer, Stuart L. (1975), Data Analysis for Scientists and Engineers, Wiley, ISBN0-471-59995-6 Taylor, J. Joint Committee for 2012-03-01. SOLUTION The first step to finding the uncertainty the track, we have a function with two variables.
It may be defined of the volume is to understand our given information.
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2017-06-24 03:41:27
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https://math.stackexchange.com/questions/486398/andre-leclair-riemann-zeta-zero-approximation
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# Andre LeClair, Riemann zeta zero approximation?
This sequence A177885 in the oeis seemingly relates imaginary parts of non-trivial Riemann zeta zeros with the LambertW function. The real and imaginary parts of the Riemann zeta function is the sum of cosine and sine waves with logarithms as frequencies.
Logarithms can be calculated as:
$$\log(n)=\lim_{s\to 1} \, \left(1-\frac{1}{n^{s-1}}\right) \zeta (s)$$
of which the numerators in the Dirichlet series are found in the following infinite table:
$$T = \begin{bmatrix} 0&0&0&0&0&0&0 \\ 1&-1&1&-1&1&-1&1 \\ 1&1&-2&1&1&-2&1 \\ 1&1&1&-3&1&1&1 \\ 1&1&1&1&-4&1&1 \\ 1&1&1&1&1&-5&1 \\ 1&1&1&1&1&1&-6 \end{bmatrix}$$
which has the definition:
$$T(n,k) = -(n-1)\; \text{ if }\; n|k, \;\text{ else } \;1,$$
Repeating/recursing the formula above we write:
$$\log(a(n))= \lim_{s\to 1} \, \zeta (s) \sum _{k=1}^n \frac{T(n,k)}{k^{s-1}}$$
where a(n) appears to be: $$a(n)=\frac{n^n}{n!}$$
$a(n) =$ {1, 2, 9/2, 32/3, 625/24, 324/5, 117649/720, 131072/315, 4782969/4480, 1562500/567, 25937424601/3628800, 35831808/1925,...}
$\left\{1,2,\frac{9}{2},\frac{32}{3},\frac{625}{24},\frac{324}{5},\frac{117649}{720},\frac{131072}{315},\frac{4782969}{4480},\frac{1562500}{567},\frac{25937424601}{3628800},\frac{35831808}{1925}\right\}$
multiplying with the factorial one finds the similar but alternating sequence A177885 in the oeis.
There in the comment this approximate formula is given:
Table[N[1/2 + 2*Pi*Exp[1]*(n - 11/8)/Exp[1]/LambertW[(n - 11/8)/Exp[1]]*I], {n,
1, 12}]
Table[N[ZetaZero[n]], {n, 1, 12}]
which gives:
{0.5 + 14.5213 I, 0.5 + 20.6557 I, 0.5 + 25.4927 I, 0.5 + 29.7394 I,
0.5 + 33.6245 I, 0.5 + 37.2574 I, 0.5 + 40.7006 I, 0.5 + 43.994 I,
0.5 + 47.1651 I, 0.5 + 50.2337 I, 0.5 + 53.2144 I, 0.5 + 56.1189 I}
{0.5 + 14.1347 I, 0.5 + 21.022 I, 0.5 + 25.0109 I, 0.5 + 30.4249 I,
0.5 + 32.9351 I, 0.5 + 37.5862 I, 0.5 + 40.9187 I, 0.5 + 43.3271 I,
0.5 + 48.0052 I, 0.5 + 49.7738 I, 0.5 + 52.9703 I, 0.5 + 56.4462 I}
The Series for x/LambertW is:
Series[x/LambertW[x], {x, 0, 7}]
$$\frac{x}{W(x)} = 1+x-\frac{x^2}{2}+\frac{2 x^3}{3}-\frac{9 x^4}{8}+\frac{32 x^5}{15}-\frac{625 x^6}{144}+\frac{324 x^7}{35}+O\left(x^8\right)$$
which has some similarity with $a(n)$
$$\frac{x}{W(x)} = \frac{(-1)^n n^n x^{n+1}}{(n+1)!}$$
$$a(n)=\frac{n^n}{n!}$$
$\left\{\frac{1}{2},\frac{2}{3},\frac{9}{8},\frac{32}{15},\frac{625}{144},\frac{324}{35},\frac{117649}{5760},\frac{131072}{2835},\frac{4782969}{44800},\frac{1562500}{6237},\frac{25937424601}{43545600},\frac{35831808}{25025}\right\}$
Is there a connection?
Edit 7.9.2013:
Would these sequences give more accurate power series approximations? Just a thought.
Clear[t, s, nn, m, k, n];
m = 1;
nn = 12;
t[n_, 1] = 1;
t[1, k_] = 1;
t[n_, k_] := t[n, k] = (1 - If[Mod[k, n] == 0, n, 0]);
MatrixForm[Table[Table[t[n, k], {k, 1, m*nn}], {n, 1, m*nn}]];
Print["here"]
Monitor[A =
Table[Limit[Zeta[s]*Sum[t[n, k]/k^(s - 1), {k, 1, m*n}],
s -> 1], {n, 1, nn}], n]
Clear[t, s, nn, m, k, n];
m = 2;
nn = 12;
t[n_, 1] = 1;
t[1, k_] = 1;
t[n_, k_] := t[n, k] = (1 - If[Mod[k, n] == 0, n, 0]);
MatrixForm[Table[Table[t[n, k], {k, 1, m*nn}], {n, 1, m*nn}]];
Print["here"]
Monitor[A =
Table[Limit[Zeta[s]*Sum[t[n, k]/k^(s - 1), {k, 1, m*n}],
s -> 1], {n, 1, nn}], n]
Clear[t, s, nn, m, k, n];
m = 3;
nn = 12;
t[n_, 1] = 1;
t[1, k_] = 1;
t[n_, k_] := t[n, k] = (1 - If[Mod[k, n] == 0, n, 0]);
MatrixForm[Table[Table[t[n, k], {k, 1, m*nn}], {n, 1, m*nn}]];
Print["here"]
Monitor[A =
Table[Limit[Zeta[s]*Sum[t[n, k]/k^(s - 1), {k, 1, m*n}],
s -> 1], {n, 1, nn}], n]
Clear[t, s, nn, m, k, n];
m = 4;
nn = 12;
t[n_, 1] = 1;
t[1, k_] = 1;
t[n_, k_] := t[n, k] = (1 - If[Mod[k, n] == 0, n, 0]);
MatrixForm[Table[Table[t[n, k], {k, 1, m*nn}], {n, 1, m*nn}]];
Print["here"]
Monitor[A =
Table[Limit[Zeta[s]*Sum[t[n, k]/k^(s - 1), {k, 1, m*n}],
s -> 1], {n, 1, nn}], n]
Clear[t, s, nn, m, k, n];
m = 5;
nn = 12;
t[n_, 1] = 1;
t[1, k_] = 1;
t[n_, k_] := t[n, k] = (1 - If[Mod[k, n] == 0, n, 0]);
MatrixForm[Table[Table[t[n, k], {k, 1, m*nn}], {n, 1, m*nn}]];
Print["here"]
Monitor[A =
Table[Limit[Zeta[s]*Sum[t[n, k]/k^(s - 1), {k, 1, m*n}],
s -> 1], {n, 1, nn}], n]
Edit 7.9.2013:
The connection I was looking for:
$$\sum _{n=1}^{\infty} \frac{x (-x)^n \exp \left(\lim_{s\to 1} \, \zeta (s) \sum _{k=1}^n \frac{1-\text{If}[k \bmod n=0,n,0]}{k^{s-1}}\right)}{n+1}+x+1 =1+x-\frac{x^2}{2}+\frac{2 x^3}{3}-\frac{9 x^4}{8}+\frac{32 x^5}{15}-\frac{625 x^6}{144}+\frac{324 x^7}{35}-\frac{117649 x^8}{5760}+\frac{131072 x^9}{2835}-\frac{4782969 x^{10}}{44800}+\frac{1562500 x^{11}}{6237}-\frac{25937424601 x^{12}}{43545600}+\frac{35831808 x^{13}}{25025}-...$$
1 + x + Sum[
x*(-x)^n*Exp[
Limit[Zeta[s]*
Sum[(1 - If[Mod[k, n] == 0, n, 0])/k^(s - 1), {k, 1, n}],
s -> 1]]/(n + 1), {n, 1, 12}]
Series[x/LambertW[x], {x, 0, 12}]
Edit 2.10.2013: Integration is better:
Clear[x, n, k, s, a1, nn, b1]
b1 = Expand[
Sum[Exp[Limit[
1/(s - 1)*
Sum[(1 - If[Mod[k, n] == 0, n, 0])/(k)^(s - 1), {k, 1, 4*n}],
s -> 1]]*(-x)^n, {n, 0, 32}]];
a1 = 1 + Integrate[b1, x];
x = N[(1 - 11/8)/Exp[1], 30];
Print["here"]
N[2*Pi*Exp[1]*a1, 30]
N[2*Pi*Exp[1]*x/LambertW[x], 30]
Clear[x, n, k, s, a1, nn]
a1 = 1 + Integrate[b1, x];
x = N[(2 - 11/8)/Exp[1], 30];
Print["here"]
N[2*Pi*Exp[1]*a1, 30]
N[2*Pi*Exp[1]*x/LambertW[x], 30]
where the number $4$ within: {k, 1, 4*n}], can be varied for truncating the Dirichlet series for the logarithm of $n$. At least as long as the truncated Dirichlet series does not get longer than the power series, there is tendency for the Zeta zero approximations to stay close to the zeta zeros.
12.10.2013: Better integration:
Clear[x, n, k, s, a1, nn, b1]
b1 = Expand[
Sum[Exp[Limit[
Zeta[s]*Sum[(1 - If[Mod[k, n] == 0, n, 0])/k^(s - 1), {k, 1,
n}], s -> 1]]*(-x)^n, {n, 1, 32}]];
a1 = 1 + Integrate[1 + b1, x];
x = N[(1 - 11/8)/Exp[1], 30];
Print["here"]
N[2*Pi*Exp[1]*a1, 30]
N[2*Pi*Exp[1]*x/LambertW[x], 30]
Clear[x, n, k, s, a1, nn]
a1 = 1 + Integrate[1 + b1, x];
x = N[(2 - 11/8)/Exp[1], 30];
Print["here"]
N[2*Pi*Exp[1]*a1, 30]
N[2*Pi*Exp[1]*x/LambertW[x], 30]
This Excel Spreadsheet formula uses Andre LeClaire's formula to approximate the Riemann zeta zeros:
(European dot-comma)
you need to divide the result with: /2/PI()/EXP(1) and take the reciprocal.
tetration this is.
Yes of course, the smooth part of the zeros is given by
$$N(T)= \frac{T}{2\pi}\log\left(\frac{T}{2\pi e}\right)$$
This function can be inverted and is equal to $$T= \frac{2 \pi n}{W(ne^{-1})}$$
This is the main reason why the approximation works fine.
It is interesting to read these comments. The last comment provides a reason why it works fine, but the argument is erroneous, since N(T) has only been proven on the whole critical strip. Thus, one cannot derive the Lambert approximation to the zeros on the line from it, since it was never proven to be valid on the line. Unless of course one assumes the Riemann hypothesis is true.
There has been a follow-up paper with my post-doc Franca that develops these ideas much further, and provides both approximate and exact corrections to this approximation based on the Lambert function. It turns out that these corrections are necessary to capture the GUE statistics of Montgomery's and Odlyzko's conjectures. The paper is here: http://arxiv.org/abs/1307.8395
There is another paper that extends all these formulas to all Dirichlet L-functions.
-André LeClair
• With these corrections, we are able to compute Riemann zeros to 500 digit accuracy or more in a very simple manner, see the above paper. -Andre' – André LeClair Sep 13 '13 at 23:26
• Dear Prof. LeClair, I am but a struggling self-studier deeply interested in the Riemann Zeta Function. I just took a look at the linked abstract. Am I reading it correctly that there is an indication that you may have proved the RH? Thanks, best wishes, – user12802 Sep 13 '13 at 23:40
• Are you familiar with mathoverflow.net? For professionals. There are, of course, some outstanding participants here as well. – user12802 Sep 14 '13 at 1:39
• Andrew, I am not familiar with that site. But if you think it is worth the effort, let me know how to access it. – André LeClair Sep 14 '13 at 4:48
• Just use the link or google it. Then sign up as a user, it's comparable to here. I only read posts and comments there, so I haven't actually gone through the process of participating, but I am sure you would be most welcome there and receive extremely useful replies. With regards, – user12802 Sep 14 '13 at 10:48
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2019-08-21 22:58:11
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|
https://www.aminer.cn/pub/5f16c89f91e011b48ae94338/on-learned-sketches-for-randomized-numerical-linear-algebra
|
# On Learned Sketches for Randomized Numerical Linear Algebra
Simin Liu
Tianrui Liu
Yulin Wan
Abstract:
We study "learning-based" sketching approaches for diverse tasks in numerical linear algebra: least-squares regression, $\ell_p$ regression, Huber regression, low-rank approximation (LRA), and $k$-means clustering. Sketching methods are used to quickly and approximately compute properties of large matrices. Linear maps called "sketches"...More
Code:
Data:
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2020-10-19 21:10:51
|
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https://tex.stackexchange.com/questions/409959/how-to-evenly-space-subcaption-use-table
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# How to evenly space subcaption - Use table?
I have a single image below and I would like to put a subcaption for each of the image in it. Only for the first row. The way I am doing it is not good because you can see that the subcaption for each image is not aligned correctly.
I was hoping someone could suggest a solution. Below is my code and the resulting figure. To make it easier, it does not need to be in a subcaption.
\begin{figure}
\captionsetup[subfigure]{labelformat=empty,position=top}
\centering
\subfloat[$\alpha=0$ \ \ \ $\alpha=1$ \ \ \ \ $\alpha=2$ \ \ \ $\alpha=3$ \ \ \ \ $\alpha=4$ \ \ \ \ $\alpha=5$ \ \ \ \ $\alpha=6$ \ \ \ $\alpha=7$ \ \ \ \ $\alpha=8$ \ \ \ \ $\alpha=9$ \ \ \ $\alpha=10$ \ ]{{\includegraphics[width=\textwidth]{latent/latent-interpolation.png} }}
\\
\caption{Interpolating between random pairs of latent vectors according to $v = v_1 + (v_2 - v_1) \times \alpha$.}
\label{fig:interpolate-latent}
\end{figure}
EDIT: So i tried adding a table before the image but I have not been able to reduce to spacing. Adding a negative baseline skips does not work the way i would like it to
\begin{figure}[h]
\captionsetup[subfigure]{labelformat=empty,position=top}
\centering
{\scriptsize
\begin{tabu} to \textwidth { XXXXXXXXXX }
$x_1$ & $x_2$ & $x_3$ & $x_4$ & $x_5$ & $x_6$ & $x_7$ & $x_8$ & $x_9$ & $x_{10}$
\end{tabu}
} \vspace{-2\baselineskip}
\subfloat[]{{\includegraphics[width=\textwidth]{latent/latent-interpolation.png} }}
\\
\caption{Interpolating between random pairs of latent vectors according to $v = v_1 + (v_2 - v_1) \times \alpha$.}
\label{fig:interpolate-latent}
\end{figure}
• do you need real subcaption, which can be referenced? or is simple text a=0 sufficient? centering is simple achievable, if you can broke your image in subimages. – Zarko Jan 12 '18 at 0:16
• A simple text a=0 is sufficient. It does not need to be in a subcaption. I would prefer not to break my image into subimages because I have too many more of those images. – Kong Jan 12 '18 at 0:18
if all sub-images in your image have equal width and distances between them are also equal, than this might help:
\documentclass{article}
\usepackage[demo]{graphicx}
\usepackage{tabularx}
\begin{document}
\begin{figure}
\setlength\tabcolsep{0pt}
\begin{tabularx}{\linewidth}{*{11}{>{\centering\arraybackslash\footnotesize $}X<{$}}}
\alpha=0 & \alpha=1 & \alpha=2 & \alpha=3 & \alpha=4 & \alpha=5
& \alpha=6 & \alpha=7 & \alpha=8 & \alpha=9 & \alpha=10 \\
\multicolumn{11}{c}{\includegraphics[width=\textwidth]{latent/latent-interpolation.png} }
\end{tabularx}
\caption{Interpolating between random pairs of latent vectors according to $v = v_1 + (v_2 - v_1) \times \alpha$.}
\label{fig:interpolate-latent}
\end{figure}
\end{document}
since i haven't your image, i emulate it with demo option of package graphicx. so the final test, if this work for you i had left to you.
edit: from your comments i conclude, that you actually looking for this:
which is obtained by:
\documentclass{article}
\usepackage[margin=20mm]{geometry}% added, i don't now if it is needed
\usepackage[demo]{graphicx}
\usepackage{tabularx}
\usepackage{subfig}
\begin{document}
\begin{figure}
\subfloat[ sub caption text]{% <--- need to be here
\setlength\tabcolsep{0pt}% <--- need to be here
\begin{tabularx}{\linewidth}{*{11}{>{\centering\arraybackslash\footnotesize$}X<{$}}}
\alpha=0 & \alpha=1 & \alpha=2 & \alpha=3 & \alpha=4 & \alpha=5
& \alpha=6 & \alpha=7 & \alpha=8 & \alpha=9 & \alpha=10 \\
\multicolumn{11}{c}{\includegraphics[width=\textwidth]{latent/latent-interpolation.png} }
\end{tabularx}}
\subfloat[ sub caption text]{\includegraphics[width=\textwidth]{latent/latent-interpolation.png}}
\caption{Interpolating between random pairs of latent vectors according to $v = v_1 + (v_2 - v_1) \times \alpha$.}
\label{fig:interpolate-latent}
\end{figure}
\end{document}
• what if i need the sub caption for the image – Kong Jan 12 '18 at 0:56
• i like your solution but i need my subfloat subcaption – Kong Jan 12 '18 at 1:07
• well, first you say, that you don't need subcaptions. honestly, i don't see any reason for this. if you persist, you can replace in first table row with \subfloat[$\alpha=0$]{} in each column, however, then probably the text will be broken into two lines due to lack of available space. to make any serious test you need first extend your code sniped to complete small document. – Zarko Jan 12 '18 at 1:11
• columns are already in math mode (see \$ in column definition), so it is sufficient to wrote x_0. – Zarko Jan 12 '18 at 1:21
• @kong, see edit of my answer. i add example of use subfloat as i understood from yours comments. – Zarko Jan 12 '18 at 1:30
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2019-10-21 06:00:13
|
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|
https://hal-polytechnique.archives-ouvertes.fr/hal-03752513
|
# The relative hermitian duality functor
Abstract : We extend to the category of relative regular holonomic modules on a manifold $X$, parametrized by a curve $S$, the Hermitian duality functor (or conjugation functor) of Kashiwara. We prove that this functor is an equivalence with the similar category on the conjugate manifold $\overline X$, parametrized by the same curve. As a byproduct we introduce the notion of regular holonomic relative distribution.
Document type :
Preprints, Working Papers, ...
https://hal-polytechnique.archives-ouvertes.fr/hal-03752513
Contributor : Claude Sabbah Connect in order to contact the contributor
Submitted on : Tuesday, August 16, 2022 - 9:56:20 PM
Last modification on : Wednesday, August 17, 2022 - 3:39:05 AM
### Identifiers
• HAL Id : hal-03752513, version 1
• ARXIV : 2012.15171
### Citation
Teresa Monteiro Fernandes, Claude Sabbah. The relative hermitian duality functor. 2022. ⟨hal-03752513⟩
Record views
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2022-09-26 11:51:26
|
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|
https://mathshistory.st-andrews.ac.uk/Curves/Cassinian/
|
# Curves
### Cassinian Ovals
Cartesian equation:
$(x^{2} + y^{2})^{2} - 2a^{2}(x^{2} - y^{2}) + a^{4} - c^{4} = 0$
### Description
The Cassinian ovals are the locus of a point $P$ that moves so that the product of its distances from two fixed points $S$ and $T$ [in this case the points $(±a, 0)$] is a constant$c^{2}$. The shape of the curve depends on $c/a$. If $c > a$ then the curve consists of two loops. If $c < a$ then the curve consists of a single loop. The case where $c = a$ produces a Lemniscate of Bernoulli (a figure of eight type curve introduced by Jacob Bernoulli ).
The curve was first investigated by Giovanni Cassini in 1680 when he was studying the relative motions of the Earth and the Sun. Cassini believed that the Sun travelled round the Earth on one of these ovals, with the Earth at one focus of the oval. Cassini actually introduced his curves 14 years before Jacob Bernoulli described his lemniscate.
Cassinian Ovals are anallagmatic curves. They are defined by the bipolar equation $rr' = k^{2}$.
Even more incredible curves are produced by the locus of a point the product of whose distances from 3 or more fixed points is a constant.
Other Web site:
Xah Lee
### Associated Curves
Definitions of the Associated curves
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2022-12-07 16:59:40
|
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|
https://proofwiki.org/wiki/Upper_and_Lower_Bound_of_Fibonacci_Number
|
# Upper and Lower Bound of Fibonacci Number
## Theorem
For all $n \in \N_{> 0}$:
$\phi^{n - 2} \le F_n \le \phi^{n - 1}$
where:
$F_n$ is the $n$th Fibonacci number
$\phi$ is the golden section: $\phi = \dfrac {1 + \sqrt 5} 2$
## Proof
$F_n \ge \phi^{n - 2}$
$F_n \le \phi^{n - 1}$
$\blacksquare$
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2021-07-27 22:39:38
|
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https://www.gamedev.net/forums/topic/617230-loading-data-from-txt-file-to-lpcstr-variable/
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# Loading data from txt file to LPCSTR variable
This topic is 2554 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Hey,
I need to load paths to particular sounds from txt file and pass those paths to vector of LPCSTR type. Therefore I have some questions about the code I've made:
std::vector <LPCSTR> waveFile; std::fstream file( "data/sounds.txt", std::ios::in ); char line[255]; while( !file.eof() ) { file.getline( line, 255, ' ' ); file.getline( line, 255, '\n' ); waveFile.push_back(line); } file.close();
This code doesn't work... I have no certain idea why but out of while loop every waveFile element has the same path (the last one loaded from txt)...
I came up with a solution:
std::vector <LPCSTR> waveFile; std::fstream file( "data/sounds.txt", std::ios::in ); char line[20][255]; int a = 0; while( !file.eof() ) { file.getline( line[a], 255, ' ' ); file.getline( line[a], 255, '\n' ); waveFile.push_back(line[a]); a++; } file.close();
Now all the elements of waveFile vector have proper values.
Does anyone know why my first code didn't work? I honestly do not know what even might be wrong... I would be evry grateful for any help!
##### Share on other sites
Hey,
I need to load paths to particular sounds from txt file and pass those paths to vector of LPCSTR type. Therefore I have some questions about the code I've made:
std::vector <LPCSTR> waveFile; std::fstream file( "data/sounds.txt", std::ios::in ); char line[255]; while( !file.eof() ) { file.getline( line, 255, ' ' ); file.getline( line, 255, '\n' ); waveFile.push_back(line); } file.close();
This code doesn't work... I have no certain idea why but out of while loop every waveFile element has the same path (the last one loaded from txt)...
I came up with a solution:
std::vector <LPCSTR> waveFile; std::fstream file( "data/sounds.txt", std::ios::in ); char line[20][255]; int a = 0; while( !file.eof() ) { file.getline( line[a], 255, ' ' ); file.getline( line[a], 255, '\n' ); waveFile.push_back(line[a]); a++; } file.close();
Now all the elements of waveFile vector have proper values.
Does anyone know why my first code didn't work? I honestly do not know what even might be wrong... I would be evry grateful for any help!
LPCSTR is a pointer to a character array, so in the first piece of code you are basically adding &line to the vector everytime (and thus every item will have the same value). The second piece of code works because you add different pointers (1 of 20) to the vector each time. You should use the string type instead of LPCSTR.
Edit: clarity
##### Share on other sites
Maybe you could have just reset line every loop. You could enter debug mode and loop through to see if (on your first example) line is being added to when getline uses it. if it is, just make sure you clear line before you call getline since maybe getline only adds to the end of your char line array
##### Share on other sites
Thanks a lot for help. Replacing LPCSTR with std::string helped.
I've been using LPCSTR cause I thought it is recommened type for XAUDIO2 operations...(this type was used in a tutorial). But it seems std::string works too
##### Share on other sites
I've been using LPCSTR cause I thought it is recommened type for XAUDIO2 operations...
std::string is the recommended type for strings in C++. If you are ever using an std::string and still need a C-style string (i.e. char*, for working with a C library, for example), you can use the std::string's c_str() method.
##### Share on other sites
Not DirectX related; Moving to General Programming.
1. 1
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Rutin
19
3. 3
khawk
15
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A4L
13
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• ### Forum Statistics
• Total Topics
633744
• Total Posts
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×
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2018-12-17 13:05:51
|
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https://tex.stackexchange.com/questions/558306/how-to-aligned-multiline-equations
|
# How to: aligned multiline equations
Two of these equations do not fit the column of the two-column layout.
They are coded with \align such that they all align at the = following the scheme
\begin{align}
...
\dot\omega &= ...
\dot f &=
...
\end{align}
Now I can of course brute force make a line break as in
\begin{align}
...
\dot\omega &= ... \notag \\
&- ...
\dot f &= ... \notag \\
&+ ...
...
\end{align}
which yields
However I do not find that ideal. It would look much better if the second line of each multi-line equation was rather aligned to the right.
How can I do that?
• Please, provide code for shoved equation in form of small, complete and compilable document that we not need to retype equations. Aug 11, 2020 at 15:55
• Take a look at multlined from mathtools. That will allow you to divide just part of a long equation. You will also need to use \biggl (or the appropriate size) instead of \left, etc. (This is the best I can do without knowing what document class and other code you are using.) Aug 11, 2020 at 15:56
Other than pushing the second line of a two-line equation to the far right, you appear to consider only one other alignment possibility, viz., to place the first item on the second line smack dab below the = symbol in the first line; see equation (A.5') in the following screenshot.
I'd like to point out that there are at least two additional alignment possibilities: (a) pushing the second line to the right by \quad, and (b) pushing the second line to the right by \qquad. These two possibilities are illustrated by equations (A.5'') and (A.5''') in the following screenshot. The final equation, labelled (A.5''''), is generated with the help of a multlined environment, a possibility already mentioned in a comment provided by @barabarabeeton.
Speaking for myself, I can't see anything wrong with the fixed indentation amounts of \quad and \qquad shown in equations (A.5'') and (A.5'''). There's nothing wrong with equation (A.5'''') either. However, achieving that look involves considerably more setting-up overhead than either (A.5'') or (A.5''') do. For sure, the look of (A.5') isn't as good as that of either of the following three equations.
\documentclass{article}
%% trying to replicate the look of the OP's screenshots:
\usepackage[letterpaper,twocolumn,margin=0.667in]{geometry}
\usepackage{mathtools}
\usepackage{newtxtext,newtxmath}
\renewcommand\theequation{A.\arabic{equation}}
\begin{document}
\begin{align}
\dot{\Omega}
&= \sqrt{\frac{p}{Gm}}\frac{\sin(\omega+f)}{ 1+e\cos f}\frac{1}{\sin\imath}\mathcal{W}\\
%% no linebreak
\dot{\omega}
&= \sqrt{\frac{p}{Gm}}\frac{1}{e}\biggl(-\cos f\mathcal{R} + \frac{2+e\cos f}{1+e\cos f}\sin f\mathcal{S} -e\cot\imath \frac{\sin(\omega+f)}{1+e\cos(f)} \mathcal{W}\biggr) \label{a5}\\
%% 1 linebreak, no indentation
\dot{\omega}
&= \sqrt{\frac{p}{Gm}}\frac{1}{e}\biggl(-\cos f\mathcal{R} + \frac{2+e\cos f}{1+e\cos f}\sin f\mathcal{S} \tag{\ref{a5}$'$} \\
&-e\cot\imath \frac{\sin(\omega+f)}{1+e\cos(f)} \mathcal{W}\biggr) \notag\\
\dot{\omega}
&= \sqrt{\frac{p}{Gm}}\frac{1}{e}\biggl(-\cos f\mathcal{R} + \frac{2+e\cos f}{1+e\cos f}\sin f\mathcal{S} \tag{\ref{a5}$''$} \\
\dot{\omega}
&= \sqrt{\frac{p}{Gm}}\frac{1}{e}\biggl(-\cos f\mathcal{R} + \frac{2+e\cos f}{1+e\cos f}\sin f\mathcal{S} \tag{\ref{a5}$'''$} \\
\sqrt{\frac{p}{Gm}}\frac{1}{e}\biggl(-\cos f\mathcal{R} + \frac{2+e\cos f}{1+e\cos f}\sin f\mathcal{S} \tag{\ref{a5}$''''$} \\
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2022-05-26 18:03:35
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|
https://tex.stackexchange.com/questions/395405/is-it-possible-to-usepackage-inside-an-input-or-include-file
|
# Is it possible to usepackage inside an input or include file?
In other words, Can I put the package dependencies of an included file inside the file itself? Or I must use the main file for that?
I am in a situation in which I include different slides in a main.tex file. Different slides use commands using different packages.
\documentclass{beamer}
\usepakcage{apackage}
\usepackage{bpackage}
\title{Title}
\author{hola}
\begin{document}
\include{slide1} %slide1.tex needs apackage
\include{slide2} %slide2.tex needs bpackage
\end{document}
Can I write slide1.tex and slide2.tex in such a way that main.tex doesn't need to know the packages needed by its slides?
\documentclass{beamer}
\title{Title}
\author{hola}
\begin{document}
\include{slide1} %slide1.tex needs apackage, slide1.tex declares its own needed package
\include{slide2} %slide2.tex needs bpackage, slide2.tex declares its own needed package
\end{document}
And (not working code) slide1.tex is
"\usepackage{apackage}"
\begin{frame}
\commandfromapackage{...}
\end{frame}
This is of course a feature very common in languages like C, in which it included file can declare their own dependencies.
Of course it could be that the solution is to write this main.tex, but that would be terrible.
\documentclass{beamer}
\input{slide1_packages}
\input{slide2_packages}
\title{Title}
\author{hola}
\begin{document}
\include{slide1}
\include{slide2}
\end{document}
• Are (the contents of) apackage and bpackage in conflict with each other? If they're not, why not just load them both? Aside: You've already "discovered" that \usepackage is a preamble-only instruction, right? – Mico Oct 10 '17 at 4:33
• @Mico They are not necessarely in conflict. Yes, I know that, I am looking for a workaround to make the writing of documents (slides actually) more modular. (For example, to have multiple presentations that reuse different sets of slides, without loading unnecessary packages or maintaining main files). – alfC Oct 10 '17 at 4:36
• You can try \input{apackage.sty} or \makeatletter\input{apackage.sty}\makeatother instead of \usepackage{apackage}. For some packages it might work, but probably for most it won't. – Scz Oct 10 '17 at 6:37
David Carlisle has already mentioned the standalone package, this is a MWE how to use it:
main.tex:
\documentclass{article}
\usepackage[subpreambles]{standalone}
% https://tex.stackexchange.com/q/120060/120953
% "packages that do some of their jobs \AtBeginDocument are likely to fail" with subpreambles (egreg)
% => must be loaded in main document as well
\usepackage{siunitx}
\begin{document}
\input{content/section-01}
\input{content/section-02}
\end{document}
content/section-01.tex:
\documentclass[crop=false]{standalone}
\usepackage{tikz}
\pgfdeclarelindenmayersystem{Koch curve}{\rule{F -> F-F++F-F}}
\begin{document}
\section{TikZ}
\tikz\shadedraw[shading=color wheel] [l-system={Koch curve, step=2pt, angle=60, axiom=F++F++F, order=4}] lindenmayer system -- cycle;
\end{document}
content/section-02.tex:
\documentclass[crop=false]{standalone}
\usepackage{siunitx}
\usepackage{booktabs}
\begin{document}
\section{siunitx}
\begin{center}
\begin{tabular}{ c S[table-format=7.6] }
\toprule
Prefix & {Value} \\
\midrule
\si{\mega} & 1 000 000 \\
\si{\kilo} & 1 000 \\
\si{\milli} & .001 \\
\si{\micro} & .000 001 \\
\bottomrule
\end{tabular}
\end{center}
\end{document}
content/section-01.tex and content/section-02.tex can be compiled on there own as well as be included into main.tex.
• Ah, subpreambles, I didn't know about it. This sound more like what I am looking for. Plus, I have standalone per slide preview. – alfC Oct 10 '17 at 7:35
In general no. (If comparing with C remember that TeX is a macro processor, so you should compare with the C pre-processor and any definitions made by two #include files in the same file). Packages can only be included in the preamble.
However for a slide, if it is a self-standing chunk of text with little dependency on text in other slides then you have the possibility of compiling the slides as separate 1-page documents and then including the generated pdf, so
\begin{slide}
\includegraphics{slide1.pdf}
\end{slide}
the standalone package has some facilities for arranging document inclusion using this kind of technique in the background, or you can just do it directly.
• Actually standalone is one step closer to what I want. It would be perfect if the package could read ahead the dependencies from the standalone included files. – alfC Oct 10 '17 at 6:55
• @alfC if using a standalone-like system the included files have no dependencies as they are processed separately, but perhaps I misunderstood your comment. – David Carlisle Oct 10 '17 at 7:49
• You understood perfectly, the feature that @jakun mentioned solves this long standing problem. – alfC Oct 10 '17 at 9:27
From the comments it seems that what you really want to do is to be able to load particular packages only when they are needed by particular include files. As other people have said, you can't do this in the included files because \usepackage{...} needs to go into the preamble. You presumably also want this to be fairly "automatic" so that you do not have to keep on checking to see exactly which packages are needed by each of the included files.
What you can do is create your own "style file" that loads all of the required packages for you. I am thinking of a "package" myfileoptions.sty that you load with a command like
\usepackage[filea,filec]{myfileoptions}
and the myfileoptions package is smart enough to know which packages to load for the input files filea.tex and fileb.tex.
Here is a possible mock-up for myfileoptions.sty :
\RequirePackage{pgfopts}
\newif\ifTikz\Tikzfalse
\newif\ifTcolorbox\Tcolorboxfalse
\pgfkeys{/myfileoptions/.is family, /myfileoptions,
filea/.code = {\global\Tikztrue},
fileb/.code = {\global\Tcolorboxtrue},
filec/.code = {\global\Tikztrue
\global\Tcolorboxtrue}
}
\ProcessPgfOptions{/myfileoptions}% process options
\ifTikz
\RequirePackage{tikz}
\fi
\ifTcolorbox
\RequirePackage{tcolorbox}
\fi
All this does is load the pgfopts package to process the options given to the class file. Inside the class file there are a bunch if \newif statements for each possible loaded package and \pgfkeys is used to turn these switches on whenever a file needs said package. At the end of myfileoptions.sty all of the needed packages are loaded using the corresponding \newif.
Of course, you will still need to have \include{filea} etc at the place(s) where you want to insert filea.
|
2020-10-20 08:23:05
|
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|
https://quant.stackexchange.com/questions/26390/are-two-stochastic-processes-independent-if-the-wiener-processes-inside-are-unco
|
# Are two stochastic processes independent if the Wiener processes inside are uncorrelated
Assume there are two stochastic processes: $dx_t = \alpha_1(x_t,t)dt + \beta_1(x_t,t)dW^1_t$ and $dy_t = \alpha_2(y_t,t)dt + \beta_2(y_t,t)dW^2_t$.
Does $dW^1_t\times{dW^2_t} = 0$ imply that $\operatorname{cov}(x_t, y_t) = 0$?
If it does, please give me a proof.
My "answer" below is not a really an answer for I have completely misinterpreted your original question. I thought you asked about the covariance of 2 processes over a given time horizon (i.e. for a fixed $\omega$) and not the covariance of two random variables (fixed $t$). Also note that $\text{cov}(x,y)=0$ does not mean that $x$ and $y$ are independent (except special case where they have an elliptic distribution), it just means they are uncorrelated (Pearson linear correlation assumed).
No this is not true in general.
Take the example below where I've assumed \begin{gather} dX_t = X_t( r_X dt + \sigma_X dW_t^X ),\ \ X(0) = X_0 \\ dY_t = X_t( r_Y dt + \sigma_Y dW_t^Y ),\ \ Y(0) = Y_0 \\ d\langle W^X, W^Y \rangle_t = \rho_{XY} \end{gather} with \begin{gather} X_0 = 1, Y_0 = 2 \\ r_X = 50\%, r_Y = -50\%\\ \sigma_X = 50\%, \sigma_Y = 25\% \end{gather}
Now take $\rho_{XY} = 0\%$ and you get the figure below: although the Brownian increments are uncorrelated (here represented through the log-returns, bottom subplot), the processes $X_t$ and $Y_t$ clearly exhibit significant correlation (see top subplot with a significant sample Pearson correlation).
This is essentially due to the drift terms in the SDE.
Note that if you take $\rho_{XY}=99\%$ you won't see that correlation (recall it is a correlation between local increments) transpire at the global level either, see below
• It appears to me that $cov(X_t, Y_t)=0$, if $\rho_{X,Y}=0$. – Gordon Jun 2 '16 at 15:01
• Well it doesn't seem to me: neither when starting from BehrouzMaleki's hint nor from my simulations. Could you explain your reasoning? – Quantuple Jun 2 '16 at 15:56
• What I meant was from a formulaic perspective based on your dynamics, since they are log-normal, the co-variance can be computed explicitly. – Gordon Jun 2 '16 at 15:58
• I realise that my answer has nothing to do with the initial question. I was talking about the cross-sectional covariance (fixed realisation) while you talk about the covariance between two random variables. My bad! – Quantuple Jun 2 '16 at 16:13
Hint:
By Integration, we have $$x_t=x_{0}+\int_{0}^{t} \alpha_1(x_s,s)ds+\int_{0}^{t} \beta_1(x_s,s)dW_1(s)$$ $$y_t=y_{0}+\int_{0}^{t} \alpha_2(y_s,s)ds+\int_{0}^{t} \beta_2(y_s,s)dW_2(s)$$ then $$E[x_t]=x_0+E\left[\int_{0}^{t} \alpha_1(x_s,s)ds\right]$$ $$E[y_t]=y_0+E\left[\int_{0}^{t} \alpha_2(y_s,s)ds\right]$$ Now we apply Ito's lemma $$d(x_ty_t)=x_tdy_t+y_tdx_t+\underbrace{d[x_t,y_t]}_{0}$$ as a result $$x_ty_t=x_0y_0+\int_{0}^{t}x_sdy_s+\int_{0}^{t}y_sdx_s$$ on the other hand $$Cov(x_t,y_t)=E[x_ty_t]-E[x_t]E[y_t]$$
• I also got that hint by myself but I got stuck at that stage. So, please give me more detail for example whether $\operatorname{cov}$ is zero or not. – imp Jun 1 '16 at 23:09
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2021-08-05 13:50:07
|
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|
https://www.scipost.org/submissions/1709.04259v2/
|
# Quantum criticality in many-body parafermion chains
### Submission summary
As Contributors: Eddy Ardonne · Ville Lahtinen Arxiv Link: https://arxiv.org/abs/1709.04259v2 (pdf) Date accepted: 2021-05-20 Date submitted: 2021-05-18 08:38 Submitted by: Ardonne, Eddy Submitted to: SciPost Physics Core Academic field: Physics Specialties: Condensed Matter Physics - Theory Approach: Theoretical
### Abstract
We construct local generalizations of 3-state Potts models with exotic critical points. We analytically show that these are described by non-diagonal modular invariant partition functions of products of $Z_3$ parafermion or $u(1)_6$ conformal field theories (CFTs). These correspond either to non-trivial permutation invariants or block diagonal invariants, that one can understand in terms of anyon condensation. In terms of lattice parafermion operators, the constructed models correspond to parafermion chains with many-body terms. Our construction is based on how the partition function of a CFT depends on symmetry sectors and boundary conditions. This enables to write the partition function corresponding to one modular invariant as a linear combination of another over different sectors and boundary conditions, which translates to a general recipe how to write down a microscopic model, tuned to criticality. We show that the scheme can also be extended to construct critical generalizations of $k$-state clock type models.
Published as SciPost Phys. Core 4, 014 (2021)
Dear editor and referee,
We would like to start by mentioning that due to various reasons, the resubmission of our manuscript did not materialise. We assumed that it wasn't possible anymore to resubmit, but we were contacted, and told that resubmission was still possible. We of course fully understand if the referee and/or editor has changed her/his/their opinion about the manuscript.
We are pleased to see that the points mentioned by the referee under 'Strength' agree with what we think of the main points of the paper. We agree with the referee about the weaknesses of the paper. The results are more straightforward than profound, but we think, and the referee seems to agree, still interesting. It is also true that there are no physical applications, and having such applications would make the paper more interesting. Though not an excuse, our paper is not alone in this respect. There is not much we can do about these two weaknesses. We tried to improve the paper with respect to the second weakness mentioned by the referee, in that the paper does not properly discuss the relation with, in particular, the literature on the orbifold construction in CFT and extended chiral algebras. We added a discussion about this to the manuscript.
Below, we go through the list of requested changes in more detail.
Best regards,
Ville Lahtinen
Teresia Månsson
Eddy Ardonne
### List of changes
Requested changes
1- As mentioned above, they need to at least briefly discuss extended chiral algebras, and the relation to the pure Virasoro characters to extended characters. They should also be precise the relation of their construction to the orbifold construction.
Answer. We agree that the paper benefits from having a discussion on extended chiral algebras and the relation with the orbifold construction. One can either do this rather extensively, or concisely. We chose the latter, and briefly discuss these matters in section 2.
2- At the beginning of section 2, they omit some key words. The partition function of every 2d CFT ON THE TORUS must be modular invariant.
Answer. This is of course true, we added this, and the implication for the current setting, namely the spectrum of periodic one dimensional chains is given in terms of a modular invariant partition function.
3- Again at the beginning of section 2, for many properties of a CFT (e.g. the specific heat), it DOES NOT MATTER IF YOU ARE ON THE TORUS, and so the distinction between different modular invariants is not important. And as I said above, the literature (especially in the '80s) is filled with CFT papers discussing other modular invariants. The authors don't mention these, and don't mention any physical properties that are different (other than the excited state spectrum).
Answer. It is of course true that for many (bulk) properties, the boundary conditions do not matter, and the referee mentions the most important one, the specific heat, which is determined by the central charge. In general, however, finite temperature properties, do depend on the details of the (at least low-energy part of the) spectrum. These properties deserve more attention in the context of quantum chains. In addition, properties related to dynamics also depend on the full spectrum. We added a short discussion on this, and added references to some '80s papers, discussing different modular invariants.
4- on p.7, they say condensation makes less fields. That's not necessarily true, since in condensations, you double other fields.
Answer. Agreed, but we don't know any example where the number of fields increases. We added the word `generically'.
5- when discussing the Z_2 x Z_2 case, they need to mention GInsparg.
6- they should mention that their added terms to the Hamiltonian are conventionally called "twisted" boundary conditions, and are widely studied.
Answer. Well, the terms we add do not quite correspond to conventional twisted boundary conditions. For our construction, it is really important that the twist actually depends on the symmetry sector, which is not the case in the, indeed widely studied, twisted boundary conditions. We added a sentence towards the end of sec 2.2 to discuss this.
7- at the beginning of sec 3.1, I don't know what "on-site" Z_k symmetry means. If this is on-site, what is off-site?
Answer. We dropped the "on-site" here.
8- at the beginning of sec 4.2, they mention that the Bethe ansatz solution gives a factor of 2 in the relative Fermi velocities between the two critical points in Potts. That presumably means if you take the Hamiltonian limit of the same classical 3-state Potts model, this relation holds. That's maybe interesting, but why is this relevant here? They're not studying the classical model.
Answer. Up to this point, we were taking products of the same critical points, which meant that the velocity simply amounted to an overall scale of the energies. Here, however, we combine two different critical points, each with their own velocity. In order for the construction to work, we should rescale the on of critical points, such that both have the same velocity. This is the reason for the factor of 2 in the second term of the hamiltonian in eq. (34). Without this factor of 2, the spectrum would not be described by su(2)_3, but some deformed version thereof.
9- they should mention that the "other" critical point for 3-state corresponds to the anti-ferro Potts model. The fact that this is c=1 goes back at least to Saleur, Nucl.Phys. B360 (1991) 219
10- for general k, they shouldn't say they're generalizing the Potts chain. The conventional definition of the Potts chain is that it has S_k permutation symmetry, and it is not critical for k>4. The points they describe are usually called the Z_k parafermion critical points, and this is what their modular invariants are related to.
Answer. We tried to put things in a broader perspective, but we should of course not go against conventional nomenclature, so we updated the text to reflect this.
11- in the conclusion, they mention getting SU(2)_3 by combining ferro and anti-ferro Potts chains, and speculate this may be true for other k. That seems unlikely, unless the antiferromagnetic point for Z_k parafermions is always c=1. (If that's true, the authors should note it -- I don't know offhand the answer myself).
Answer. It is indeed true that the antiferromagnetic point for the Z_k parafermions is always c=1 (Albertini). In the mean time, it was shown that for k=5,7, the critical point is u(1)_2k, as required. We included both results in the conclusions.
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2022-10-04 16:56:56
|
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http://www.mathdoubts.com/trigonometry/identity/pythagoras/sine-cosine/conversion/
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Conversion Formulas to express sin to cos and cos to sin
The Pythagorean theorem which describes the relation between three sides of a right angled triangle is used in trigonometry to express the mathematical relation between trigonometric functions. The Pythagorean identity of sine and cosine is one of them and can further use to express sine in terms of cosine and vice-versa.
Formulas
Possibly, trigonometric function sine can be expressed in terms of cosine in two different ways. Similarly, cosine can also be expressed in terms of sine in two different ways but in same pattern. Remember, the following identities in trigonometry are written by considering theta as the angle of a right angled triangle.
1
Sin in terms of cos
$(1)\,\,\,\,$ $\sin^2 \theta = 1 -\cos^2 \theta$
It is read as sin squared theta is equal to one minus cos squared theta.
$(2)\,\,\,\,$ $\sin \theta = \pm \sqrt{1 -\cos^2 \theta}$
It is read as sin theta is equal to plus or minus square root of one minus cos squared theta.
Learn the proofs of these two identities to convert sine in terms of cosine.
2
Cos in terms of sin
$(1)\,\,\,\,$ $\cos^2 \theta = 1 -\sin^2 \theta$
It is read as cos squared theta is equal to one minus sin squared theta.
$(2)\,\,\,\,$ $\cos \theta = \pm \sqrt{1 -\sin^2 \theta}$
It is read as cos theta is equal to plus or minus square root of one minus sin squared theta.
Similarly, learn the mathematical proofs of these two formulas to express cos terms in terms of sine.
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2018-03-23 06:57:32
|
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|
http://pipingdesigner.co/index.php/mathematics/geometry/solid-geometry/2565-hollow-cylinder
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# Right Hollow Cylinder
Written by Jerry Ratzlaff on . Posted in Solid Geometry
• Right hollow cylinder (a three-dimensional figure) has a hollow core with both bases direictly above each other and having the center at 90° to each others base.
• 2 bases
• See Moment of Inertia of a Cylinder
## formulas that use Inside Volume of a Right Hollow cylinder
$$\large{ V = \pi\; r^2\;h }$$
### Where:
$$\large{ V }$$ = volume (inside)
$$\large{ r }$$ = inside radius
$$\large{ h }$$ = height
## formulas that use Lateral Surface Area of a Right Hollow cylinder
$$\large{ A_l = 2 \; \pi \; h \left(R^2 + r^2 \right) }$$
### Where:
$$\large{ A_l }$$ = lateral surface area (side)
$$\large{ h }$$ = height
$$\large{ r }$$ = inside radius
$$\large{ R }$$ = outside radius
$$\large{ \pi }$$ = Pi
## formulas that use Object Volume of a Right Hollow cylinder
$$\large{ V = \pi\; h \left(R^2 - r^2 \right) }$$
### Where:
$$\large{ V }$$ = volume (object thickness)
$$\large{ h }$$ = height
$$\large{ r }$$ = inside radius
$$\large{ R }$$ = outside radius
$$\large{ \pi }$$ = Pi
## formulas that use Surface Area of a Right Hollow cylinder
$$\large{ A_s = A_i + 2 \; \pi \left(R^2 - r^2 \right) }$$
### Where:
$$\large{ A_s }$$ = surface area (bottom, top, side)
$$\large{ h }$$ = height
$$\large{ r }$$ = inside radius
$$\large{ R }$$ = outside radius
$$\large{ \pi }$$ = Pi
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2020-07-07 08:24:34
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https://forum.azimuthproject.org/discussion/comment/22532/
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# Qinglan Xia, The formation of a tree leaf
This paper presents a simplified, quasi-physical model of how a leaf actually grows:
Strikingly, the author claims:
Based on this model, we also provide some computer visualization of tree leaves, which resemble many known leaves including the maple and mulberry leaf.
Since this is science, let's reproduce their results!
First step, let's use this thread to go through the paper in detail, discuss the model, and come up with a developer-target spec for modeling code.
Once we have code, we can test the model to verify that it produces leaves that are close to maple, mulberry, etc.
Eventually, we could turn this into an open source project, complemented by an art gallery of images and animations.
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1.
Note: this will be something different than a gallery of fractal leaves. Fractal leaves are purely mathematical creations. Xia's paper, on the other hand, presents a physically plausible model that is claimed to produce results similar to empirical leaves.
So, rather than being a computer exploration of purely mathematical models, this is a science modeling project.
Comment Source:Note: this will be something different than a gallery of fractal leaves. Fractal leaves are purely mathematical creations. Xia's paper, on the other hand, presents a physically plausible model that is claimed to produce results similar to empirical leaves. So, rather than being a computer exploration of purely mathematical models, this is a science modeling project.
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2.
edited December 2020
To start the ball rolling, here is a qualitative digest of the paper:
In The Formation of a Tree Leaf by Qinglan Xia, we see a possible key to Nature’s algorithm for the growth of leaf veins. The vein system, which is a transport network for nutrients and other substances, is modeled by Xia as a directed graph with nodes for cells and edges for the “pipes” that connect the cells. Each cell gives a revenue of energy, and incurs a cost for transporting substances to and from it.
The total transport cost depends on the network structure. There are costs for each of the pipes, and costs for turning the fluid around the bends. For each pipe, the cost is proportional to the product of its length, its cross-sectional area raised to a power α, and the number of leaf cells that it feeds. The exponent α captures the savings from using a thicker pipe to transport materials together. Another parameter β expresses the turning cost.
Development proceeds through cycles of growth and network optimization. During growth, a layer of cells gets added, containing each potential cell with a revenue that would exceed its cost. During optimization, the graph is adjusted to find a local cost minimum. Remarkably, by varying α and β, simulations yield leaves resembling those of specific plants, such as maple or mulberry.
A growing network
Unlike approaches that merely create pretty images resembling leaves, Xia presents an algorithmic model, simplified yet illuminating, of how leaves actually develop. It is a network-theoretic approach to a biological subject, and it is mathematics—replete with lemmas, theorems and algorithms—from start to finish.
From:
Comment Source:To start the ball rolling, here is a qualitative digest of the paper: > In The Formation of a Tree Leaf by Qinglan Xia, we see a possible key to Nature’s algorithm for the growth of leaf veins. The vein system, which is a transport network for nutrients and other substances, is modeled by Xia as a directed graph with nodes for cells and edges for the “pipes” that connect the cells. Each cell gives a revenue of energy, and incurs a cost for transporting substances to and from it. > The total transport cost depends on the network structure. There are costs for each of the pipes, and costs for turning the fluid around the bends. For each pipe, the cost is proportional to the product of its length, its cross-sectional area raised to a power α, and the number of leaf cells that it feeds. The exponent α captures the savings from using a thicker pipe to transport materials together. Another parameter β expresses the turning cost. > Development proceeds through cycles of growth and network optimization. During growth, a layer of cells gets added, containing each potential cell with a revenue that would exceed its cost. During optimization, the graph is adjusted to find a local cost minimum. Remarkably, by varying α and β, simulations yield leaves resembling those of specific plants, such as maple or mulberry. > **A growing network** > Unlike approaches that merely create pretty images resembling leaves, Xia presents an algorithmic model, simplified yet illuminating, of how leaves actually develop. It is a network-theoretic approach to a biological subject, and it is mathematics—replete with lemmas, theorems and algorithms—from start to finish. From: * [Prospects for a Green Mathematics](https://johncarlosbaez.wordpress.com/2013/02/15/prospects-for-a-green-mathematics/), John Baez and David Tanzer, Azimuth Blog. Originally published in Mathematics of Planet Earth Blog, February 2013.
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3.
edited December 2020
Think of that as a high-level requirements document.
Now we need to dig into the specifics, to write a detailed spec for this model.
Comment Source:Think of that as a high-level requirements document. Now we need to dig into the specifics, to write a detailed spec for this model.
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4.
edited December 2020
I'll start by paraphrasing the flow of ideas from the paper, using more abstracted, verbal terms.
Comment Source:I'll start by paraphrasing the flow of ideas from the paper, using more abstracted, verbal terms.
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5.
edited December 2020
The whole model works in an infinite grid of squares in $$\mathbb{R}^2$$, where the squares have length $$h$$. The grid is aligned so that there is a square whose center point is the origin (0,0).
The origin is called the root.
Comment Source:The whole model works in an infinite grid of squares in \$$\mathbb{R}^2\$$, where the squares have length \$$h\$$. The grid is aligned so that there is a square whose center point is the origin (0,0). The origin is called the _root_.
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6.
The squares represent cells in a leaf.
The leaf is developed by an iterative growth process.
The process begins with a "leaf" that consists only of the one square over the origin, i.e., just the root.
Comment Source:The squares represent cells in a leaf. The leaf is developed by an iterative growth process. The process begins with a "leaf" that consists only of the one square over the origin, i.e., just the root.
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7.
edited December 2020
1. Transport Systems
There is a piping system which sources from the root, and reaches every cell in the leaf.
A pipe from cell A to cell B may be viewed as an edge (A,B) in a directed graph.
But it's not just any kind of graph, it's a tree which is rooted at the root cell.
Every cell (except for the root) has a single parent -- this is the predecessor cell which supplies it with fluid.
And every cell has a set of children, which are cells that it directly pipes fluid to.
A cell which has zero children is at the frontier of the graph.
Comment Source:**1. Transport Systems** There is a piping system which sources from the root, and reaches every cell in the leaf. A pipe from cell A to cell B may be viewed as an edge (A,B) in a directed graph. But it's not just any kind of graph, it's a tree which is rooted at the root cell. Every cell (except for the root) has a single parent -- this is the predecessor cell which supplies it with fluid. And every cell has a set of children, which are cells that it directly pipes fluid to. A cell which has zero children is at the frontier of the graph.
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8.
This directed graph structure is called a transport system.
Comment Source:This directed graph structure is called a _transport system_.
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9.
edited December 2020
Furthermore, the edges of the graph are given weights.
You can think of the weight of an edge as the rate at which fluid flows through the associated pipe, assuming the transport system is 'running in its steady state'.
Comment Source:Furthermore, the edges of the graph are given weights. You can think of the weight of an edge as the rate at which fluid flows through the associated pipe, assuming the transport system is 'running in its steady state'.
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10.
Each cell needs to consume fluid at a rate which is proportional to its area $$h^2$$.
The total rate at which fluid needs to be pumped into the root is just the sum of the required rates for each of the cells.
Comment Source:Each cell needs to consume fluid at a rate which is proportional to its area \$$h^2\$$. The total rate at which fluid needs to be pumped into the root is just the sum of the required rates for each of the cells.
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11.
This gives the key to understanding what I meant by the 'steady state', as follows.
For every cell, there is one pipe leading into it. The rate of flow through this pipe must equal the sum of the required rates over all the cells that are its descendants.
Comment Source:This gives the key to understanding what I meant by the 'steady state', as follows. For every cell, there is one pipe leading into it. The rate of flow through this pipe must equal the sum of the required rates over all the cells that are its descendants.
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12.
edited December 2020
The required flow rates through each of the pipes are therefore uniquely determined.
The weight of an edge in the graph is precisely this required flow rate.
Comment Source:The required flow rates through each of the pipes are therefore uniquely determined. The weight of an edge in the graph is precisely this required flow rate.
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13.
edited December 2020
2. Cost Functionals
Whereas the weight of an edge gives its flow rate, we now want to estimate the cost of pumping fluid through the associated pipe.
(More precisely this would be called the cost rate or cost per unit time.)
The simplest model would say that the cost rate is equal to the flow rate times the length of the pipe - as the cost of friction with the walls of the pipe is proportional to the length of the pipe.
But there is an economy of scale for wider pipes, which is expressed by a parameter $$\alpha$$ between 0 and 1.
For a pipe with flow rate $$w$$ and length $$L$$, in this model the cost is $$w^\alpha \cdot L$$.
Comment Source:**2. Cost Functionals** Whereas the weight of an edge gives its flow rate, we now want to estimate the cost of pumping fluid through the associated pipe. (More precisely this would be called the cost rate or cost per unit time.) The simplest model would say that the cost rate is equal to the flow rate times the length of the pipe - as the cost of friction with the walls of the pipe is proportional to the length of the pipe. But there is an economy of scale for wider pipes, which is expressed by a parameter \$$\alpha\$$ between 0 and 1. For a pipe with flow rate \$$w\$$ and length \$$L\$$, in this model the cost is \$$w^\alpha \cdot L\$$.
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14.
edited December 2020
Now let's look at a cell $$x$$, and consider the cost of continually supplying it with fluid, along the whole line of pipes leading from the root to $$x$$.
Part of this cost would be accounted for by summing $$w(e)^\alpha \cdot L(e)$$, for all edges $$e$$ on the path from the root to $$x$$.
Comment Source:Now let's look at a cell \$$x\$$, and consider the cost of continually supplying it with fluid, along the whole line of pipes leading from the root to \$$x\$$. Part of this cost would be accounted for by summing \$$w(e)^\alpha \cdot L(e)\$$, for all edges \$$e\$$ on the path from the root to \$$x\$$.
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15.
edited December 2020
But there is another component to the cost of supplying fluid to $$x$$, which is the additional work that needs to be done when there are changes of direction in at the junctures in piping chain from the root to $$x$$.
In other words, this model also gives a cost for the turning of fluid around bends.
Note: each pipe is modeled as a straight line; a bend here is a change in angle between one pipe and the next.
Comment Source:But there is another component to the cost of supplying fluid to \$$x\$$, which is the additional work that needs to be done when there are changes of direction in at the junctures in piping chain from the root to \$$x\$$. In other words, this model also gives a cost for the turning of fluid around bends. Note: each pipe is modeled as a straight line; a bend here is a change in angle between one pipe and the next.
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edited December 2020
A parameter $$\beta$$ is used to capture the fluid-turning cost.
Here is the explanation.
The cost within the pipes of getting fluid from the root to $$x$$ has already been accounted for above.
The cumulative cost of all the turns along the path to $$x$$ is defined by a factor $$m_\beta(x)$$.
$$m_\beta(x)$$ is defined as the product of a sequence of values $$f_\beta(y)$$ for all the nodes $$y$$ along the path from the root to $$x$$, where $$f_\beta(y)$$ is a factor expressing the turning cost at $$y$$.
Let $$u$$ be the unit vector in the direction of the pipe leading into $$y$$, and $$v$$ be the unit vector in the direction of the pipe leading out of $$y$$ along the path.
Then $$f_\beta(y)$$ is defined to be $$|u \cdot v|^{-\beta}$$ if $$u \cdot v$$ is greater than zero, else infinity.
(We can interpret this later, after this exposition is finished.)
Comment Source:A parameter \$$\beta\$$ is used to capture the fluid-turning cost. Here is the explanation. The cost within the pipes of getting fluid from the root to \$$x\$$ has already been accounted for above. The cumulative cost of all the turns along the path to \$$x\$$ is defined by a factor \$$m_\beta(x)\$$. \$$m_\beta(x)\$$ is defined as the product of a sequence of values \$$f_\beta(y)\$$ for all the nodes \$$y\$$ along the path from the root to \$$x\$$, where \$$f_\beta(y)\$$ is a factor expressing the turning cost at \$$y\$$. Let \$$u\$$ be the unit vector in the direction of the pipe leading into \$$y\$$, and \$$v\$$ be the unit vector in the direction of the pipe leading out of \$$y\$$ along the path. Then \$$f_\beta(y)\$$ is defined to be \$$|u \cdot v|^{-\beta}\$$ if \$$u \cdot v\$$ is greater than zero, else infinity. (We can interpret this later, after this exposition is finished.)
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17.
edited December 2020
These principles give all that is needed to define a cost functional, which maps every transport system to its total cost.
The formula is given in definition 2.2 in the paper.
What it amounts to is the sum over all nodes $$x$$ of the fluid cost through the pipe leading into $$x$$, times the cumulative turning cost factor $$m_\beta(x)$$.
Comment Source:These principles give all that is needed to define a cost functional, which maps every transport system to its total cost. The formula is given in definition 2.2 in the paper. What it amounts to is the sum over all nodes \$$x\$$ of the fluid cost through the pipe leading into \$$x\$$, times the cumulative turning cost factor \$$m_\beta(x)\$$.
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edited December 2020
2.2 Optimal transport systems
Suppose we are given a "candidate leaf" consisting of some set of cells in the grid. A real candidate leaf, having been grown from the starting state with just one cell at the root, will be a connected set of cells.
However, for present purposes, let's forget about that, and consider any set of cells which contains the root cell as a candidate leaf.
We would now like to consider various possible piping systems for this collection of cells - and to seek ones that have minimal cost.
Comment Source:**2.2 Optimal transport systems** Suppose we are given a "candidate leaf" consisting of some set of cells in the grid. A real candidate leaf, having been grown from the starting state with just one cell at the root, will be a connected set of cells. However, for present purposes, let's forget about that, and consider any set of cells which contains the root cell as a candidate leaf. We would now like to consider various possible piping systems for this collection of cells - and to seek ones that have minimal cost.
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19.
Whereas on one hand we would like to reduce the aggregate amount of flow, on the other we would like to reduce the turning costs.
These are conflicting goals.
Hence, an optimization problem is presented. As a result of this optimization, whether by nature or by our computer simulation, complex leaf branching patterns are formed.
Comment Source:Whereas on one hand we would like to reduce the aggregate amount of flow, on the other we would like to reduce the turning costs. These are conflicting goals. Hence, an optimization problem is presented. As a result of this optimization, whether by nature or by our computer simulation, complex leaf branching patterns are formed.
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20.
To get a sense of the competing nature of these goals, suppose we had N cells, and wanted to arrange them into a piping system.
If we wanted to bring turning costs to zero, we would put the cells in a linear chain from the root.
But the trouble with a linear chain is that the nodes near the tip are far from the root, so there is a significant cost of transporting their needed fluid all the way from the root.
To reduce these transportation costs, we aim to get the cells closer to the root. For example, we could have a main vein, with lots of little branches.
But the branching introduces turning costs.
Comment Source:To get a sense of the competing nature of these goals, suppose we had N cells, and wanted to arrange them into a piping system. If we wanted to bring turning costs to zero, we would put the cells in a linear chain from the root. But the trouble with a linear chain is that the nodes near the tip are far from the root, so there is a significant cost of transporting their needed fluid all the way from the root. To reduce these transportation costs, we aim to get the cells closer to the root. For example, we could have a main vein, with lots of little branches. But the branching introduces turning costs.
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21.
Ok, that's the intuition behind the optimization structure. Now let's proceed to define optimal transport systems.
Comment Source:Ok, that's the intuition behind the optimization structure. Now let's proceed to define optimal transport systems.
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22.
Now given a set of cells that includes the root, we want to search for transport systems, with the root cell as the root of the graph, which have minimal cost.
It is easy to see that in general there will be many possible transport systems for a given collection of cells. All we have to do is note that there can be a main vein starting at the root, with major branches coming off the main vein. Since there are many choices for the path of the main vein, this already shows the existence of multiple transport systems.
Comment Source:Now given a set of cells that includes the root, we want to search for transport systems, with the root cell as the root of the graph, which have minimal cost. It is easy to see that in general there will be many possible transport systems for a given collection of cells. All we have to do is note that there can be a main vein starting at the root, with major branches coming off the main vein. Since there are many choices for the path of the main vein, this already shows the existence of multiple transport systems.
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23.
In fact, for any candidate leaf consisting of a more than a few cells, there will generally be a large number of possible transport systems.
So the task of finding a transport system with globally minimum cost is computationally intractable.
Comment Source:In fact, for any candidate leaf consisting of a more than a few cells, there will generally be a large number of possible transport systems. So the task of finding a transport system with globally minimum cost is computationally intractable.
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24.
edited December 2020
Instead, Xia's algorithm takes a hint from a plausible model of how nature might actually work, and proceeds by an iterative process.
The leaf is grown in steps.
Each step involves two substeps.
In the first substep, certain new cells are added to the leaf at its fringes. Pipes are added to the system, connecting the new cells to nearby cells on the fringes.
This is the growth substep.
But after this substep, the resulting transport system may not be optimal.
So, in the next substep, an optimization is performed by tweaking the fine details of this transport system.
Neither our algorithm nor nature has the opportunity to globally the restructure the transport network once the leaf has grown to a certain stage.
Comment Source:Instead, Xia's algorithm takes a hint from a plausible model of how nature might actually work, and proceeds by an iterative process. The leaf is grown in steps. Each step involves two substeps. In the first substep, certain new cells are added to the leaf at its fringes. Pipes are added to the system, connecting the new cells to nearby cells on the fringes. This is the growth substep. But after this substep, the resulting transport system may not be optimal. So, in the next substep, an optimization is performed by tweaking the fine details of this transport system. Neither our algorithm nor nature has the opportunity to globally the restructure the transport network once the leaf has grown to a certain stage.
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edited December 2020
So, this process consisting of repeated growth steps followed by local optimizations won't in general produce a globally optimal solution - but it's better than not trying at all!
Comment Source:So, this process consisting of repeated growth steps followed by local optimizations won't in general produce a globally optimal solution - but it's better than not trying at all!
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26.
Growth substep
The key here is to determine which new cells to add just outside of the boundary of the current leaf, and, for each new cell added, to choose a parent cell for it on the boundary of the current leaf.
Then, of course, a pipe will be added between each new cell and its parent on the boundary.
Comment Source:**Growth substep** The key here is to determine which new cells to add just outside of the boundary of the current leaf, and, for each new cell added, to choose a parent cell for it on the boundary of the current leaf. Then, of course, a pipe will be added between each new cell and its parent on the boundary.
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edited December 2020
Here is the principle for new cell selection.
Each new cell produces a revenue for the leaf, consisting of the solar energy which it collects.
This revenue is proportional to the area of the cell. The proportionality constant is given by a parameter $$\epsilon$$. Recall that all cells are squares of width $$h$$. Hence the revenue from any cell is given by $$\epsilon h^2$$.
On the other hand, there is the added cost of transporting fluid from the root to the new cell.
Principle: a cell just outside of the fringe of the current leaf will be added if its revenue would exceed its cost.
Comment Source:Here is the principle for new cell selection. Each new cell produces a revenue for the leaf, consisting of the solar energy which it collects. This revenue is proportional to the area of the cell. The proportionality constant is given by a parameter \$$\epsilon\$$. Recall that all cells are squares of width \$$h\$$. Hence the revenue from any cell is given by \$$\epsilon h^2\$$. On the other hand, there is the added cost of transporting fluid from the root to the new cell. Principle: a cell just outside of the fringe of the current leaf will be added if its revenue would exceed its cost.
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edited December 2020
Note that the added cost of a cell depends upon which parent is chosen for it in the current leaf.
So the check for whether a cell should be added implicitly also includes the determination of what its parent cell will be.
Comment Source:Note that the added cost of a cell depends upon which parent is chosen for it in the current leaf. So the check for whether a cell should be added implicitly also includes the determination of what its parent cell will be.
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edited December 2020
The above description contains all of the ideas needed to pin down the growth substep.
Comment Source:The above description contains all of the ideas needed to pin down the growth substep.
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edited December 2020
Optimization substep
This substep begins with the completion of the growth substep - which has given us an extended leaf (by adding new cells at the fringe) along with a transport system for the extended leaf.
Now what we do is to look at the cells, one a time, and see if the total transport cost can be reduced by changing its parent to some other cell that is nearby to it. If so, then the transport graph is changed to replace its parent with the new parent.
This process is done repeatedly, until there are no further reparenting operations that can reduce the total cost of the transport system.
The resulting transport system is optimized as well as can be done -- without global restructuring.
Comment Source:**Optimization substep** This substep begins with the completion of the growth substep - which has given us an extended leaf (by adding new cells at the fringe) along with a transport system for the extended leaf. Now what we do is to look at the cells, one a time, and see if the total transport cost can be reduced by changing its parent to some other cell that is nearby to it. If so, then the transport graph is changed to replace its parent with the new parent. This process is done repeatedly, until there are no further reparenting operations that can reduce the total cost of the transport system. The resulting transport system is optimized as well as can be done -- without global restructuring.
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31.
That completes the essential description of the growth algorithm.
Comment Source:That completes the essential description of the growth algorithm.
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32.
A question now arises: will it ever stop growing?
Comment Source:A question now arises: will it ever stop growing?
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33.
Clearly, real leaves in nature do stop growing.
So, if this is to be a plausible empirical model, we should hope to be able to prove that the algorithm will in fact terminate at some point.
Comment Source:Clearly, real leaves in nature do stop growing. So, if this is to be a plausible empirical model, we should hope to be able to prove that the algorithm will in fact terminate at some point.
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34.
edited December 2020
Why do the leaves stop growing?
Comment Source:Why do the leaves stop growing?
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35.
Well, as the leaf gets bigger, the points on its fringe get further and further away from from the root.
It therefore gets more and more expensive to add new cells to the edge of the leaf, as the cost of transporting fluid to them gets larger and larger.
When the leaf has reached its maximal size, there are no remaining candidate cells outside the leaf that would produce a net gain for the leaf.
In other words, every cell outside the leaf would cost more to add to the transport system than it would contribute through its revenue of solar energy.
And so the growth stops.
Comment Source:Well, as the leaf gets bigger, the points on its fringe get further and further away from from the root. It therefore gets more and more expensive to add new cells to the edge of the leaf, as the cost of transporting fluid to them gets larger and larger. When the leaf has reached its maximal size, there are no remaining candidate cells outside the leaf that would produce a net gain for the leaf. In other words, every cell outside the leaf would cost more to add to the transport system than it would contribute through its revenue of solar energy. And so the growth stops.
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36.
Xia's paper proves the result that if the piping cost parameter $$\alpha$$ exceeds 0.5, then the algorithm will stop, yielding a fully grown leaf.
Comment Source:Xia's paper proves the result that if the piping cost parameter \$$\alpha\$$ exceeds 0.5, then the algorithm will stop, yielding a fully grown leaf.
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37.
Recall from comment 14 that the cost for transporting fluid through a pipe $$e$$ is $$w(e)^\alpha \cdot L(e)$$, where $$L$$ is its length and $$w(e)$$ is the flow rate through the pipe.
And the parameter $$\alpha$$ was stipulated to be between 0 and 1.
Comment Source:Recall from comment 14 that the cost for transporting fluid through a pipe \$$e\$$ is \$$w(e)^\alpha \cdot L(e)\$$, where \$$L\$$ is its length and \$$w(e)\$$ is the flow rate through the pipe. And the parameter \$$\alpha\$$ was stipulated to be between 0 and 1.
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edited December 2020
Note that the flow rate is conceptually proportional to the cross-sectional area of the pipe - since wider pipes are needed to support greater flow rates.
Comment Source:Note that the flow rate is conceptually proportional to the cross-sectional area of the pipe - since wider pipes are needed to support greater flow rates.
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39.
$$\alpha$$, being something less than 1, expresses the economy of scale that is offered by wider pipes.
Comment Source:\$$\alpha\$$, being something less than 1, expresses the economy of scale that is offered by wider pipes.
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40.
What happens when $$\alpha$$ is very small, close to zero?
Then $$w(e)^\alpha$$ is close to 1, and this multiplicative factor drops out of the picture.
Which is to say, when $$\alpha$$ is small, close to zero, there is such an economy of scale that there is barely any cost for having very wide pipes, i.e., for having large flow rates.
Comment Source:What happens when \$$\alpha\$$ is very small, close to zero? Then \$$w(e)^\alpha\$$ is close to 1, and this multiplicative factor drops out of the picture. Which is to say, when \$$\alpha\$$ is small, close to zero, there is such an economy of scale that there is barely any cost for having very wide pipes, i.e., for having large flow rates.
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41.
This condition would allow the leaf to grow and grow, without stopping.
Comment Source:This condition would allow the leaf to grow and grow, without stopping.
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42.
Why?
Comment Source:Why?
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43.
edited December 2020
Suppose the leaf was giant.
And suppose the algorithm would attempt to add a new cell $$x$$ at the fringe of this giant leaf.
$$x$$ needs the standard rate of fluid flow to support it. Call this flow amount $$y$$.
Now $$y$$ would have to be added to the flow through every pipe on the path from the root to $$x$$.
Suppose there are $$N$$ pipes along this path.
Since we said the leaf was giant, this means that $$N$$ is large.
Were $$\alpha$$ to be close to 1, then adding $$x$$ to the leaf would incur a large cost, because all of the pipes along this long path would need to be widened.
But when $$\alpha$$ is small, this hardy matters, and it is no problem to feed $$x$$ through a long chain of pipes -- so $$x$$ gets added, and the leaf continues to grow,
Comment Source:Suppose the leaf was giant. And suppose the algorithm would attempt to add a new cell \$$x\$$ at the fringe of this giant leaf. \$$x\$$ needs the standard rate of fluid flow to support it. Call this flow amount \$$y\$$. Now \$$y\$$ would have to be added to the flow through every pipe on the path from the root to \$$x\$$. Suppose there are \$$N\$$ pipes along this path. Since we said the leaf was giant, this means that \$$N\$$ is large. Were \$$\alpha\$$ to be close to 1, then adding \$$x\$$ to the leaf would incur a large cost, because all of the pipes along this long path would need to be widened. But when \$$\alpha\$$ is small, this hardy matters, and it is no problem to feed \$$x\$$ through a long chain of pipes -- so \$$x\$$ gets added, and the leaf continues to grow,
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44.
edited December 2020
On the other hand, if $$\alpha$$ is not "close to zero," then growth will stop.
The specific result that Xia proves is that if $$\alpha$$ exceeds 0.5 then growth will eventually stop.
Comment Source:On the other hand, if \$$\alpha\$$ is not "close to zero," then growth will stop. The specific result that Xia proves is that if \$$\alpha\$$ exceeds 0.5 then growth will eventually stop.
• Options
45.
edited December 2020
Computer visualization
Just by varying the parameters $$\alpha$$ and $$\beta$$, Xia shows that the algorithm produces a variety of natural looking leaves.
In figure 3, when $$\alpha = 0.68$$ and $$\beta = 0.38$$, a leaf is produced which looks remarkably like an actual maple leaf, which is shown next to the computer generated image.
In figure 4, when $$\alpha = 0.66$$ and $$\beta = 0.7$$, a leaf is produced which closely matches an actual mulberry leaf.
Comment Source:**Computer visualization** Just by varying the parameters \$$\alpha\$$ and \$$\beta\$$, Xia shows that the algorithm produces a variety of natural looking leaves. In figure 3, when \$$\alpha = 0.68\$$ and \$$\beta = 0.38\$$, a leaf is produced which looks remarkably like an actual maple leaf, which is shown next to the computer generated image. In figure 4, when \$$\alpha = 0.66\$$ and \$$\beta = 0.7\$$, a leaf is produced which closely matches an actual mulberry leaf.
• Options
46.
END OF EXPOSITION
Comment Source:END OF EXPOSITION
• Options
47.
edited December 2020
Now let's review the above presentation of the model.
Granted, it is a qualitative description. But does it correctly telegraph the essence of the algorithm?
(For the exact details, we do have the paper itself to refer to.)
Once the description is settled, we can turn it into a spec, for an implementation of the algorithm.
Comment Source:Now let's review the above presentation of the model. Granted, it is a qualitative description. But does it correctly telegraph the essence of the algorithm? (For the exact details, we do have the paper itself to refer to.) Once the description is settled, we can turn it into a spec, for an implementation of the algorithm.
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2021-04-12 15:13:10
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https://gamedev.stackexchange.com/questions/20935/2d-xna-tile-based-lighting-ideas-and-methods
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# 2D XNA Tile Based Lighting. Ideas and Methods [closed]
I am currently working on developing a 2D tile based game, similar to the game 'Terraria'. We have the base tile and chunk engine working and are now looking to implement lighting. Instead of the tile based lighting that terraria uses, I want to implement point lights for torches, etc.
I have seen Catalin Zima’s shader based shadows, and this would be perfect for the torches (point lights). My problem here is that the tiles on the surface of the world need to be illuminated, doing this by a big point light is firstly extremely expensive, but also doesn't look right.
What I need help with (overall) is...
1. To have a surface that is illuminated regardless of torches, etc.
2. To also have point lights, or smooth tile lighting similar to Catalin Zima’s shader based shadows.
Looking forward to your replies. Any ideas are appreciated.
• When you say you want your surface that's illuminated, do you mean "surface" as in above the ground, or do you mean that you want some kind of ambient light that your point lights add on to, or what? – Tetrad Dec 12 '11 at 8:03
• Ahh sorry, I mean "surface" as in above the ground. – FrenchyNZ Dec 12 '11 at 8:36
• You never model the sun as a point light anyway. The sun is usually modeled either as a constant ambient contribution which is added to everything, or as a directional light. Check my answer for a simple way to handle this. – David Gouveia Dec 13 '11 at 2:27
• @Twitchy, I'm also currently working on an XNA tilebased game. If you want to discuss implementations and exchange ideas, you can hit me up on my email address found on my profile. – William Mariager Dec 13 '11 at 4:04
Finding the surface and applying ambient light
To find which tiles are on the surface and should receive some ambient light, in a very simple way, how about doing a one dimensional (vertical only) flood fill starting from the sky?
In other words, you start at the top of your world, one column at a time, and search downwards until you hit a solid tile. All tiles that you search this way (including the solid tile) will be considered as being on the surface and get a constant light contribution added to them (i.e. you draw them a bit lighter).
For example something like this (where # is a solid tile, and . is a tile that will receive ambient light):
..........................
..........................
..........................
..........................
..........................
.....##########.###.......
..############ .#####.....
.######### #########..
.######### ############..
##### ###############
##########################
##########################
##########################
Then just do the torches and everything else as point lights.
Finally, while I don't feel it's necessary, but if you also wanted to have the sun cast shadows on the surface, you should model it as a directional light rather than a point light (since the sun is "infinitely large and infinitely far away" for practical purposes). That basically means that every shadow will be coming from the same direction (i.e. forming the same angle).
Method 1
If you'd like to add some variation to the lighting on the surface so that not every tile has the same brightness, you could try to compute some sort of ambient occlusion factor for the "surface tiles".
The easiest way, without having to rely on raycasting, would be to simply count the number of neighboring tiles that can block the sun light.
The basic idea is for each solid tile on the surface, count how many other solid tiles there are in the immediate surroundings (at the same level and above it). Use this number as a factor to decide how much light it should recieve.
For example:
.#.#. .....
.#@#. .#@#.
##### #####
Consider the @ a solid tile. In the first example there's 4 solid tiles surrounding the @ while in the second example there's only 2. Thus the @ in the first example would be drawn darker than the @ on the second example.
This method has the disadvantage of only working on the surface, and being a somewhat crude approximation.
Method 2
Another alternative that will give better results but be a lot more expensive, would be to start with the entire world illuminated, and then for each tile do a series of raycasts towards the sky (in several directions, all along the hemisphere surrounding and above the tile) and count how many collided with terrain. Then darken the tile depending on the number of collisions.
This would allow ambient light to sip into caves and shed a little light into the underground too which would look pretty cool.
The biggest disadvantage besides having to do the raycasts is that you would need to compute this for every tile, not just the surface ones.
For this you'll need to use a pixel shader and do the previous calculations per vertex instead of per tile.
Each tile will be a quad, so start by calculating and storing how much light reaches each individual vertex of the quad. This could be for stored for instance instance as a grayscale vertex color (ranging from [0,0,0] to [1,1,1]).
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2021-02-25 18:33:12
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http://mathematica.stackexchange.com/questions?page=105&sort=newest
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2016-06-29 00:17:20
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https://newproxylists.com/commutative-algebra-when-does-a-set-of-polynomials-form-a-polynomial-ring/
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# Commutative algebra – When does a set of polynomials form a polynomial ring?
I think about the following question: Suppose $$k$$ is a field and we are given a set $$S subseteq k[x_i,…,x_n]$$ of polynomials. When is the ring $$k [S]$$ where we join all the elements of $$S$$ at $$k$$ a polynomial ring on the set $$S$$?
• of course we should have that $$S$$ contains $$n$$ elements or less, otherwise we can find relationships.
• something like $$S = {x ^ 2, x ^ 3 } subset k[x]$$ does not work because $$(x ^ 2) ^ 3 = (x ^ 3) ^ 2$$.
• something like $$S = {x, y, xy } subset k[x,y,z]$$ does not work because xy = x * y.
So, I guess the answer is that $$S = {s_1, … }$$ must be such that $$s_i$$ is not already an element in $$k [s_1,…,s_{i-1}]$$. Are there any "easier" criteria for this? For some ideal $$J subset k[x_1,…,x_n]$$, can we still extract a generator $$S = {a_1, …, a_c }$$ such as $$k [a_1,…,a_c]$$ is a polynomial ring? Under what assumptions is it possible, if it is not possible in general?
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2019-04-25 14:47:28
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http://www.zentralblatt-math.org/zmath/en/advanced/?q=an:1223.44001
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Zbl 1223.44001
Katatbeh, Qutaibeh Deeb; Belgacem, Fethi Bin Muhammad
Applications of the Sumudu transform to fractional differential equations.
(English)
[J] Nonlinear Stud. 18, No. 1, 99-112 (2011). ISSN 1359-8678; ISSN 2153-4373/e
The Sumudu transform was introduced by {\it G. F. Watugala} in an effort to improve Laplace transform techniques (see for instance [Int. J. Math. Educ. Sci. Technol. 24, No.~1, 35--43 (1993; Zbl 0768.44003)]). The Sumudu transform of a function $f(x)$ is defined by the formula $$G(u)= S[f(t)](u)= \int^\infty_0 f(ut)\,e^{-t}\,dt,$$ and is connected to the Laplace transform $$f(t)\to F(s)= \int^\infty_0 f(t) e^{-st}\, dt$$ in a natural way, $G(u)= {1\over u} F({1\over u})$. Using this formula one can translate properties of the Laplace transform into properties of the Sumudu transform and vice versa. This transform is used practically for the same purpose the Laplace transform is used -- for solving ordinary and partial-differential equations. Some formulas turn out to be very convenient.\par In the present paper, the authors apply the Sumudu transform to fractional calculus, computing the transforms of fractional derivatives and integrals. For instance, if $D^{-\alpha} f$ is the Riemann-Liouville fractional integral, it is shown that $S[D^{-\alpha} f(t)]= u^\alpha G(u)$. The authors also demonstrate how to solve fractional-differential equations with this transform. As an added bonus for the reader, the paper is accompanied by a representative table of transforms $S[D^{-\alpha} f(t)]$.
[Khristo N. Boyadzhiev (Ada)]
MSC 2000:
*44A15 Special transforms
45A05 Linear integral equations
44A35 Convolution
44A99 Miscellaneous topics of integral transforms and operational calculus
Keywords: fractional derivatives; fractional differential equations; fractional integrals; Mittag-Leffler functions; Sumudu; Sumudu transform; Laplace transform
Citations: Zbl 0768.44003
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2013-05-24 07:41:33
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https://labs.tib.eu/arxiv/?author=G.%20Goltsman
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• ### Non-bolometric bottleneck in electron-phonon relaxation in ultra-thin WSi film(1607.07321)
We developed the model of the internal phonon bottleneck to describe the energy exchange between the acoustically soft ultrathin metal film and acoustically rigid substrate. Discriminating phonons in the film into two groups, escaping and nonescaping, we show that electrons and nonescaping phonons may form a unified subsystem, which is cooled down only due to interactions with escaping phonons, either due to direct phonon conversion or indirect sequential interaction with an electronic system. Using an amplitude-modulated absorption of the sub-THz radiation technique, we studied electron-phonon relaxation in ultrathin disordered films of tungsten silicide. We found an experimental proof of the internal phonon bottleneck. The experiment and simulation based on the proposed model agree well, resulting in tau_{e-ph} = 140-190 ps at T_C = 3.4 K, supporting the results of earlier measurements by independent techniques.
• ### Dual Origin of Room Temperature Sub-Terahertz Photoresponse in Graphene Field Effect Transistors(1712.02144)
March 19, 2018 cond-mat.mes-hall
Graphene is considered as a promising platform for detectors of high-frequency radiation up to the terahertz (THz) range due to graphene$'$s superior electron mobility. Previously it has been shown that graphene field effect transistors (FETs) exhibit room temperature broadband photoresponse to incoming THz radiation thanks to the thermoelectric and/or plasma wave rectification. Both effects exhibit similar functional dependences on the gate voltage and therefore it was found to be difficult to disentangle these contributions in the previous studies. In this letter, we report on combined experimental and theoretical studies of sub-THz response in graphene field-effect transistors analyzed at different temperatures. This temperature-dependent study allowed us to reveal the role of photo-thermoelectric effect, p-n junction rectification, and plasmonic rectification in the sub-THz photoresponse of graphene FETs.
• ### Superconducting detector for visible and near-infrared quantum emitters(1612.05838)
Dec. 17, 2016 quant-ph
Further development of quantum emitter based communication and sensing applications intrinsically depends on the availability of robust single-photon detectors. Here, we demonstrate a new generation of superconducting single-photon detectors specifically optimized for the 500-1100 nm wavelength range, which overlaps with the emission spectrum of many interesting solid-state atom-like systems, such as nitrogen-vacancy and silicon-vacancy centers in diamond. The fabricated detectors have a wide dynamic range (up to 350 million counts per second), low dark count rate (down to 0.1 counts per second), excellent jitter (62 ps), and the possibility of on-chip integration with a quantum emitter. In addition to performance characterization, we tested the detectors in real experimental conditions involving nanodiamond nitrogen-vacancy emitters enhanced by a hyperbolic metamaterial.
• ### Nonequilibrium interpretation of DC properties of NbN superconducting hot electron bolometers(1609.06116)
We present a physically consistent interpretation of the dc electrical properties of niobiumnitride (NbN)-based superconducting hot-electron bolometer (HEB-) mixers, using concepts of nonequilibrium superconductivity. Through this we clarify what physical information can be extracted from the resistive transition and the dc current-voltage characteristics, measured at suitably chosen temperatures, and relevant for device characterization and optimization. We point out that the intrinsic spatial variation of the electronic properties of disordered superconductors, such as NbN, leads to a variation from device to device.
• ### Spectrally resolved single-photon imaging with hybrid superconducting - nanophotonic circuits(1609.07857)
The detection of individual photons is an inherently binary mechanism, revealing either their absence or presence while concealing their spectral information. For multi-color imaging techniques, such as single photon spectroscopy, fluorescence resonance energy transfer microscopy and fluorescence correlation spectroscopy, wavelength discrimination is essential and mandates spectral separation prior to detection. Here, we adopt an approach borrowed from quantum photonic integration to realize a compact and scalable waveguide-integrated single-photon spectrometer capable of parallel detection on multiple wavelength channels, with temporal resolution below 50 ps and dark count rates below 10 Hz. We demonstrate multi-detector devices for telecommunication and visible wavelengths and showcase their performance by imaging silicon vacancy color centers in diamond nanoclusters. The fully integrated hybrid superconducting-nanophotonic circuits enable simultaneous spectroscopy and lifetime mapping for correlative imaging and provide the ingredients for quantum wavelength division multiplexing on a chip.
• ### A superconducting NbN detector for neutral nanoparticles(1402.2310)
We present a proof-of-principle study of superconducting single photon detectors (SSPD) for the detection of individual neutral molecules/nanoparticles at low energies. The new detector is applied to characterize a laser desorption source for biomolecules and it allows to retrieve the arrival time distribution of a pulsed molecular beam containing the amino acid tryptophan, the polypeptide gramicidin as well as insulin, myoglobin and hemoglobin. We discuss the experimental evidence that the detector is actually sensitive to isolated neutral particles.
• ### Effect of the wire width on the intrinsic detection efficiency of superconducting-nanowire single-photon detectors(1303.4546)
March 19, 2013 cond-mat.supr-con
Thorough spectral study of the intrinsic single-photon detection efficiency in superconducting TaN and NbN nanowires with different widths shows that the experimental cut-off in the efficiency at near-infrared wavelengths is most likely caused by the local deficiency of Cooper pairs available for current transport. For both materials the reciprocal cut-off wavelength scales with the wire width whereas the scaling factor quantitatively agrees with the hot-spot detection models. Comparison of the experimental data with vortex-assisted detection scenarios shows that these models predict a stronger dependence of the cut-off wavelength on the wire width.
• ### Superconducting parallel nanowire detector with photon number resolving functionality(0807.0526)
July 3, 2008 physics.ins-det
We present a new photon number resolving detector (PNR), the Parallel Nanowire Detector (PND), which uses spatial multiplexing on a subwavelength scale to provide a single electrical output proportional to the photon number. The basic structure of the PND is the parallel connection of several NbN superconducting nanowires (100 nm-wide, few nm-thick), folded in a meander pattern. Electrical and optical equivalents of the device were developed in order to gain insight on its working principle. PNDs were fabricated on 3-4 nm thick NbN films grown on sapphire (substrate temperature TS=900C) or MgO (TS=400C) substrates by reactive magnetron sputtering in an Ar/N2 gas mixture. The device performance was characterized in terms of speed and sensitivity. The photoresponse shows a full width at half maximum (FWHM) as low as 660ps. PNDs showed counting performance at 80 MHz repetition rate. Building the histograms of the photoresponse peak, no multiplication noise buildup is observable and a one photon quantum efficiency can be estimated to be QE=3% (at 700 nm wavelength and 4.2 K temperature). The PND significantly outperforms existing PNR detectors in terms of simplicity, sensitivity, speed, and multiplication noise.
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2020-01-25 20:19:22
|
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https://journal.kib.ac.cn/EN/Y2007/V29/I01/1
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Plant Diversity ›› 2007, Vol. 29 ›› Issue (01): 1-6.
• Articles •
### Geographical Distribution and Floristic Characters of Salix (Salicaceae) in Qinghai Province
GUO Shu-Xian, WANG Dong-Mei, WU Xue-Ming, YUAN Jia-Zheng
1. Department of Biochemical Engineering
• Received:2006-04-12 Online:2007-02-25 Published:2007-02-25
Abstract: The geographical distribution and floristic characters of Salix species in Qinghai are studied by means of field survey , examining herbarium specimens and literatures . The result indicates that there are 45 species ( including 5 varieties and 1 form) in Qinghai , they belong to 15 sections, accounting for 100% and 40.9% of species and sections in Qinghai- Tibet plateau , 40.5% and 17.5% of species and sections in China, and take the fourth place in China. They mainly distribute in the Eastern of Qinghai, including the Eastern of Qiliansan Mountain series and the southeastern part of Southern Qinghai plateau, and their vertical distribution concentrates on the altitude 2 000 - 4 000m, it is one of the highest altitude areas of the world that Salix species distribute on. The floristic characters of Salix species in Qinghai Province are as following:1) Rich species; 2 ) High intra specific differentiation; 3 ) Complex geographical components. They are mainly of Eurasia temperate and Qinghai-xizang Plateau distribution, but the component endemic to China plays an important role ; 4) Endemism is not obvious, only accounting for 8.9% of Salix species in Qinghai; 5) Absolute domination with groups of diandrous stamens and monadelphous stamens, accounting for 93.3% of species in Qinghai province. The Salix species in Qinghai Province are closely related to the Eastern (East of Gansu and Shaanxi ) and the southeastern (West of Sichuan , East of Xizang). Because of the continuous lifting of Himalayas and Qinghai-Xizang Plateau since Tertiary, the distribution and floristic characters of Salix in Qinghai, adapting the freezing cold and dried environment, are formed.
Key words: Salix
CLC Number:
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2022-08-09 19:45:02
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https://www.rdocumentation.org/packages/MASS/versions/7.3-47/topics/addterm
|
0th
Percentile
Try fitting all models that differ from the current model by adding a single term from those supplied, maintaining marginality. This function is generic; there exist methods for classes lm and glm and the default method will work for many other classes.
Keywords
models
Usage
addterm(object, …)# S3 method for default
addterm(object, scope, scale = 0, test = c("none", "Chisq"),
k = 2, sorted = FALSE, trace = FALSE, …)
# S3 method for lm
addterm(object, scope, scale = 0, test = c("none", "Chisq", "F"),
k = 2, sorted = FALSE, …)
# S3 method for glm
addterm(object, scope, scale = 0, test = c("none", "Chisq", "F"),
k = 2, sorted = FALSE, trace = FALSE, …)
Arguments
object
An object fitted by some model-fitting function.
scope
a formula specifying a maximal model which should include the current one. All additional terms in the maximal model with all marginal terms in the original model are tried.
scale
used in the definition of the AIC statistic for selecting the models, currently only for lm, aov and glm models. Specifying scale asserts that the residual standard error or dispersion is known.
test
should the results include a test statistic relative to the original model? The F test is only appropriate for lm and aov models, and perhaps for some over-dispersed glm models. The Chisq test can be an exact test (lm models with known scale) or a likelihood-ratio test depending on the method.
k
the multiple of the number of degrees of freedom used for the penalty. Only k=2 gives the genuine AIC: k = log(n) is sometimes referred to as BIC or SBC.
sorted
should the results be sorted on the value of AIC?
trace
if TRUE additional information may be given on the fits as they are tried.
arguments passed to or from other methods.
Details
The definition of AIC is only up to an additive constant: when appropriate (lm models with specified scale) the constant is taken to be that used in Mallows' Cp statistic and the results are labelled accordingly.
Value
A table of class "anova" containing at least columns for the change in degrees of freedom and AIC (or Cp) for the models. Some methods will give further information, for example sums of squares, deviances, log-likelihoods and test statistics.
References
Venables, W. N. and Ripley, B. D. (2002) Modern Applied Statistics with S. Fourth edition. Springer.
dropterm, stepAIC
Aliases
library(MASS) quine.hi <- aov(log(Days + 2.5) ~ .^4, quine) quine.lo <- aov(log(Days+2.5) ~ 1, quine) addterm(quine.lo, quine.hi, test="F") house.glm0 <- glm(Freq ~ Infl*Type*Cont + Sat, family=poisson, data=housing) addterm(house.glm0, ~. + Sat:(Infl+Type+Cont), test="Chisq") house.glm1 <- update(house.glm0, . ~ . + Sat*(Infl+Type+Cont)) addterm(house.glm1, ~. + Sat:(Infl+Type+Cont)^2, test = "Chisq")
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2020-12-04 18:32:25
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https://lobste.rs/s/wymxb3/asciimath_easy_write_markup_language_for
|
1. 9
asciimath.org
1.
2. 7
What is this trying to solve? We already have many abstract notations for mathematics, from eqn(7) to LaTeX to MathML and so on and so on. Not even to mention the raft of per-language mathematics languages.
(Isn’t asciimath the result of MultiMarkdown 2 trying to shoehorn “markdown philosophy” into mathematics before bailing back into LaTeX in subsequent generations?)
1. 2
From the looks of it, this is trying to solve the problem of the cumbersome and sometimes silly syntax LaTeX uses. If you know LaTeX and want to use it, more power to you, but there are books explaining its use, whereas this is specified in just a single webpage. It seems like a much lower bar to jump.
1. 2
I think part of @kristapsdz’s point is that there are already several options here. S/He explicitly gave three.
1. 3
I get the critique, I just don’t think there’s a problem with experimenting on stuff like this. Personally, this seems way easier to read when rendered just as ASCII (without MathJax) which is one of the things I don’t like about sites that use LaTeX in particular.
2. 1
It’s been a while since I used LaTeX but it’s not that hard to use for simpler formulas. I’d rather have one implementation than have to learn yet another syntax.
Of course the cleanest way to handle this is to just write everything in LaTeX and output to HTML…
3. 1
Now we need plain text output for eqn. This would be the real ASCII/UTF-8 math for me.
4. 2
works in all browsers
I can’t see anything in links. :-P
I still can see the text as it is typed in the input, but I do not think the priority was displaying plain text math, just writing.
1. 1
This looks great! It’s like a markdown syntax for math.
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2018-06-25 10:16:00
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https://www.jobilize.com/online/course/13-1-arithmetic-and-geometric-sequences-by-openstax?qcr=www.quizover.com
|
13.1 Arithmetic and geometric sequences
Page 1 / 1
A teacher's guide for lecturing on arithmetic and geometric sequences.
The in-class assignment does not need any introduction. Most of them will get the numbers, but they may need help with the last row, with the letters.
After this assignment, however, there is a fair bit of talking to do. They have all the concepts; now we have to dump a lot of words on them.
A “sequence” is a list of numbers. In principal, it could be anything: the phone number 8,6,7,5,3,0,9 is a sequence.
Of course, we will not be focusing on random sequences like that one. Our sequences will usually be expressed by a formula: for instance, “the xxxnth terms of this sequence is given by the formula $100+3\left(n-1\right)$ ” (or $3n+97\right)$ in the case of the first problem on the worksheet. This is a lot like expressing the function $y=100+3\left(x-1\right)$ , but it is not exactly the same. In the function $y=3x+97$ , the variable $x$ can be literally any number. But in a sequence , xxxn must be a positive integer; you do not have a “minus third term” or a “two-and-a-halfth term.”
The first term in the sequence is referred to as ${t}_{1}$ and so on. So in our first example, ${t}_{5}=112$ .
The number of terms in a sequence, or the particular term you want, is often designated by the letter $n$ .
Our first sequence adds the same amount every time. This is called an arithmetic sequence . The amount it goes up by is called the common difference $d$ (since it is the difference between any two adjacent terms). Note the relationship to linear functions, and slope.
If I want to know all about a given arithmetic sequence, what do I need to know? Answer: I need to know ${t}_{1}$ and $d$ .
OK, so if I have ${t}_{1}$ and $d$ for the arithmetic sequence, give me a formula for the ${n}^{\mathrm{th}}$ term in the sequence. (Answer: ${t}_{n}={t}_{1}+d\left(n-1\right)$ . Talk through this carefully before proceeding.)
Time for some more words. A recursive definition of a sequence defines each term in terms of the previous. For an arithmetic sequence, the recursive definition is ${t}_{n+1}={t}_{n}+d$ . (For instance, in our example, ${t}_{n+1}={t}_{n}+3.\right)$ An explicit definition defines each term as an absolute formula, like the $3n+97$ or the more general ${t}_{n}={t}_{1}+d\left(n-1\right)$ we came up with.
Our second sequence multiplies by the same amount every time. This is called a geometric sequence . The amount it multiplies by is called the common ratio $r$ (since it is the ratio of any two adjacent terms).
Find the recursive definition of a geometric sequence. (Answer: ${t}_{n+1}=r{t}_{n}$ . They will do the explicit definition in the homework.)
Question: How do you make an arithmetic sequence go down ? Answer: $d<0$
Question: How do you make a geometric series go down? Answer: $0 . (Negative $r$ values get weird and interesting in their own way...why?)
Homework
“Homework: Arithmetic and Geometric Sequences”
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
Got questions? Join the online conversation and get instant answers!
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2019-02-17 11:24:28
|
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http://cs.stackexchange.com/questions/7105/time-complexity-for-count-change-procedure-in-sicp
|
# Time complexity for count-change procedure in SICP
In famous Structure and Interretation of Computer Programs, there is an exercise (1.14), that asks for the time complexity of the following algorithm - in Scheme - for counting change (the problem statement suggests drawing the tree for (cc 11 5) - which looks like this):
; count change
(define (count-change amount)
(define (cc amount kinds-of-coins)
(cond ((= amount 0) 1)
((or (< amount 0) (= kinds-of-coins 0)) 0)
(else (+ (cc (- amount
(first-denomination kinds-of-coins))
kinds-of-coins)
(cc amount
(- kinds-of-coins 1))))))
(define (first-denomination kinds-of-coins)
(cond ((= kinds-of-coins 1) 1)
((= kinds-of-coins 2) 5)
((= kinds-of-coins 3) 10)
((= kinds-of-coins 4) 25)
((= kinds-of-coins 5) 50)))
(cc amount 5))
Now... there are sites with solutions to the SICP problems, but I couldn't find any easy to understand proof for the time complexity of the algorithm - there is a mention somewhere that it's polynomial O(n^5)
-
## migrated from cstheory.stackexchange.comDec 2 '12 at 18:19
This question came from our site for theoretical computer scientists and researchers in related fields.
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2014-11-21 12:08:51
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https://leanprover-community.github.io/mathlib_docs/category_theory/limits/preserves/functor_category.html
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mathlibdocumentation
category_theory.limits.preserves.functor_category
Preservation of (co)limits in the functor category #
Show that if X ⨯ - preserves colimits in D for any X : D, then the product functor F ⨯ - for F : C ⥤ D preserves colimits.
The idea of the proof is simply that products and colimits in the functor category are computed pointwise, so pointwise preservation implies general preservation.
References #
https://ncatlab.org/nlab/show/commutativity+of+limits+and+colimits#preservation_by_functor_categories_and_localizations
If X × - preserves colimits in D for any X : D, then the product functor F ⨯ - for F : C ⥤ D also preserves colimits.
Note this is (mathematically) a special case of the statement that "if limits commute with colimits in D, then they do as well in C ⥤ D" but the story in Lean is a bit more complex, and this statement isn't directly a special case. That is, even with a formalised proof of the general statement, there would still need to be some work to convert to this version: namely, the natural isomorphism (evaluation C D).obj k ⋙ prod.functor.obj (F.obj k) ≅ prod.functor.obj F ⋙ (evaluation C D).obj k
Equations
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2021-04-12 22:15:56
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http://gmatclub.com/forum/gmat-problem-solving-ps-140/index-12650.html?sk=er&sd=a
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The arithmetic mean of the list of numbers above is 4. If k
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2014-12-20 12:55:19
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https://ies.fsv.cuni.cz/sci/publication/show/id/3916/lang/en
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## Publication detail
### Víšek, J. Á. : Root-n-consistency of the instrumental weighted variables
Author(s): prof. RNDr. Jan Ámos Víšek CSc., Article in collection 2009 0 978-80-250-1965-8 Proceedings of the conference Statistics: Investment in the Future, 1 - 15 Prague root-n-consistency, robustification of instrumental variables The definition of Instrumental Weighted Variables (IWV) (which is a robust version of the classical Instrumental Variables) and conditions for the weak consistency as given in V\'{\i}\v{s}ek (2009) are recalled. The reasons why the classical Instrumental Variables were introduced as well as the idea of weighting the order statistics of squared residuals (rather than directly the squared residual - firstly employed by the Least Weighted Squares, see V\'{\i}\v{s}ek (2000)) are also recalled. Then $\sqrt{n}$-consistency of all solutions of the corresponding normal equations is proved.
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2022-09-26 03:33:01
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https://blog.flyingcoloursmaths.co.uk/secrets-of-the-mathematical-ninja-squaring-halves-and-fives/
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The trick: someone says ‘what’s 7.5 squared?’ and - mentally squaring in a flash - you say: 56.25.
### Squaring halves
Squaring halves is really easy if you know your times tables. Here’s the method:
1. Take your number and find the whole numbers immediately above and below. If you’re trying to square 7.5, that would be 7 and 8; if you’re squaring 11.5, it would be 11 and 12.
2. Multiply these numbers together (56 for $7 \times 8$; 132 for $11 \times 12$).
3. Add on 0.25. That gives $7.5^2 = 56.25$ and $11.5^2 = 132.25$.
### Squaring fives
You can also use this to square any number that ends in 5. It’s the same idea:
1. Find the ten above and the ten below (so 25 is between 20 and 30)
2. Multiply those together (600 - it’s always going to be …00)
3. Add 25. $25^2 = 625$.
Why does this work? Well, it’s the old ‘difference of two squares’ trick. Let me write it this way:
$$(x + 0.5)(x - 0.5) = x^2 - 0.25$$ $$(x + 0.5)(x - 0.5) + 0.25 = x^2$$
… and that’s all there is to it!
### Squaring in reverse
You can use this trick backwards to get a better guess for square roots - for example, if you spot that 110 is $11 \times 10$, you can say that its square root must be a little less than 10.5, because $10.5^2 = 110.25$.
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2022-07-06 23:19:11
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https://stacks.math.columbia.edu/tag/0GLX
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Lemma 10.20.2. Let $R$ be a ring, let $S \subset R$ be a multiplicative subset, let $I \subset R$ be an ideal, and let $M$ be a finite $R$-module If $x_1, \ldots , x_ r \in M$ generate $S^{-1}(M/IM)$ as an $S^{-1}(R/I)$-module, then there exists an $f \in S + I$ such that $x_1, \ldots , x_ r$ generate $M_ f$ as an $R_ f$-module1
Proof. Special case $I = 0$. Let $y_1, \ldots , y_ s$ be generators for $M$ over $R$. Since $S^{-1}M$ is generated by $x_1, \ldots , x_ r$, for each $i$ we can write $y_ i = \sum (a_{ij}/s_{ij})x_ j$ for some $a_{ij} \in R$ and $s_{ij} \in S$. Let $s \in S$ be the product of all of the $s_{ij}$. Then we see that $y_ i$ is contained in the $R_ s$-submodule of $M_ s$ generated by $x_1, \ldots , x_ r$. Hence $x_1, \ldots , x_ r$ generates $M_ s$.
General case. By the special case, we can find an $s \in S$ such that $x_1, \ldots , x_ r$ generate $(M/IM)_ s$ over $(R/I)_ s$. By Lemma 10.20.1 we can find a $g \in 1 + I_ s \subset R_ s$ such that $x_1, \ldots , x_ r$ generate $(M_ s)_ g$ over $(R_ s)_ g$. Write $g = 1 + i/s'$. Then $f = ss' + is$ works; details omitted. $\square$
[1] Special cases: (I) $I = 0$. The lemma says if $x_1, \ldots , x_ r$ generate $S^{-1}M$, then $x_1, \ldots , x_ r$ generate $M_ f$ for some $f \in S$. (II) $I = \mathfrak p$ is a prime ideal and $S = R \setminus \mathfrak p$. The lemma says if $x_1, \ldots , x_ r$ generate $M \otimes _ R \kappa (\mathfrak p)$ then $x_1, \ldots , x_ r$ generate $M_ f$ for some $f \in R$, $f \not\in \mathfrak p$.
## Comments (0)
There are also:
• 6 comment(s) on Section 10.20: Nakayama's lemma
## Post a comment
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2021-07-25 09:18:22
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http://mathhelpforum.com/advanced-algebra/147474-discriminant-question.html
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# Math Help - Discriminant question
1. ## Discriminant question
Using the Theorem on Symmetric Functions or otherwise,
show that the discriminant of the quartic polynomial x^4 + qx + r
is 27q^4 + 256r^3.
Now, using the Symmetric Functions theorem, it's obvious that q= -s3 and r=s4 (where s3 and s4 are the third and fourth elementary symmetric polynomials). How do I proceed from here?
Many thanks.
2. Originally Posted by KSM08
Using the Theorem on Symmetric Functions or otherwise,
show that the discriminant of the quartic polynomial x^4 + qx + r
is 27q^4 + 256r^3.
Now, using the Symmetric Functions theorem, it's obvious that q= -s3 and r=s4 (where s3 and s4 are the third and fourth elementary symmetric polynomials). How do I proceed from here?
The discriminant is given by $\Delta = \prod_{i, where $x_1,\,x_2,\,x_3,\,x_4$ are the roots of the polynomial (over the complex field, say).
For a quartic polynomial, the discriminant is a symmetric function of the roots, of degree 12. But q has degree 3 and r has degree 4. The symmetric function theorem says that $\Delta$ must be a function of the elementary symmetric functions, and in this example the only available such functions are q and r (because $s_1$ and $s_2$ are 0). So $\Delta$ must be of the form $mq^4+nr^3$, for some constants m and n.
To find m and n, evaluate $\Delta$ in some special cases. For example, if q=0 and r=–1 then the equation becomes x^4=1, with roots $\pm1,\,\pm i$. From the definition of $\Delta$, you can check that $\Delta = -256$ in this case. So n=256. Similarly, if q = –1 and r=0 then the equation is $x^4-x=0$, with roots $0,\,1,\,(-1\pm\sqrt3i)/2$. I haven't tried to check this, but presumably $\Delta$ works out as 27 in this case.
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2015-07-07 15:45:25
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https://stats.stackexchange.com/questions/89793/why-does-the-standard-error-of-the-intercept-increase-the-further-bar-x-is-fr
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# Why does the standard error of the intercept increase the further $\bar x$ is from 0?
The standard error of the intercept term ($\hat{\beta}_0$) in $y=\beta_1x+\beta_0+\varepsilon$ is given by $$SE(\hat{\beta}_0)^2 = \sigma^2\left[\frac{1}{n}+\frac{\bar{x}^2}{\sum_{i=1}^n(x_i-\bar{x})^2}\right]$$ where $\bar{x}$ is the mean of the $x_i$'s.
From what I understand, the SE quantifies your uncertainty- for instance, in 95% of the samples, the interval $[\hat{\beta}_0-2SE,\hat{\beta}_0+2SE]$ will contain the true $\beta_0$. I fail to understand how the SE, a measure of uncertainty, increases with $\bar{x}$. If I simply shift my data, so that $\bar{x}=0$, my uncertainty goes down? That seems unreasonable.
An analogous interpretation is - in the uncentered version of my data, $\hat{\beta}_0$ corresponds to my prediction at $x=0$, while in the centered data, $\hat{\beta}_0$ corresponds to my prediction at $x=\bar{x}$. So does this then mean that my uncertainty about my prediction at $x=0$ is greater than my uncertainty about my prediction at $x=\bar{x}$? That seems unreasonable too, the error $\epsilon$ has the same variance for all values of $x$, so my uncertainty in my predicted values should be the same for all $x$.
There are gaps in my understanding I'm sure. Could somebody help me understand what's going on?
• Have you ever regressed anything against a date? Many computer systems start their dates in the distant past, often over 100 or over 2000 years ago. The intercept estimates the value of your data extrapolated backwards to that starting time. How certain would you be, say, of the gross domestic product of Iraq in the year 0 CE based on regressing a series of 21st century data? – whuber Mar 12 '14 at 19:30
• I agree, it makes sense if you think about it this way. This, and gung's answer, make things clear. – elexhobby Mar 12 '14 at 19:45
• This answer gives an intuitive explanation, with diagrams) of how it arises, by casting the fitted line in terms of the fit at the mean $\bar x$ (the fitted line passes through $(\bar x,\bar y)$) and shows why the position of where the line can go spreads out as you move away from $\bar x$ (which is caused by the uncertainty in the slope). – Glen_b -Reinstate Monica Mar 12 '14 at 21:52
Because the regression line fit by ordinary least squares will necessarily go through the mean of your data (i.e., $(\bar x, \bar y)$)—at least as long as you don't suppress the intercept—uncertainty about the true value of the slope has no effect on the vertical position of the line at the mean of $x$ (i.e., at $\hat y_{\bar x}$). This translates into less vertical uncertainty at $\bar x$ than you have the further away from $\bar x$ you are. If the intercept, where $x=0$ is $\bar x$, then this will minimize your uncertainty about the true value of $\beta_0$. In mathematical terms, this translates into the smallest possible value of the standard error for $\hat\beta_0$.
Here is a quick example in R:
set.seed(1) # this makes the example exactly reproducible
x0 = rnorm(20, mean=0, sd=1) # the mean of x varies from 0 to 10
x5 = rnorm(20, mean=5, sd=1)
x10 = rnorm(20, mean=10, sd=1)
y0 = 5 + 1*x0 + rnorm(20) # all data come from the same
y5 = 5 + 1*x5 + rnorm(20) # data generating process
y10 = 5 + 1*x10 + rnorm(20)
model0 = lm(y0~x0) # all models are fit the same way
model5 = lm(y5~x5)
model10 = lm(y10~x10)
This figure is a bit busy, but you can see the data from several different studies where the distribution of $x$ was closer or further from $0$. The slopes differ a little from study to study, but are largely similar. (Notice they all go through the circled X that I used to mark $(\bar x, \bar y)$.) Nonetheless, the uncertainty about the true value of those slopes causes the uncertainty about $\hat y$ to expand the further you get from $\bar x$, meaning that the $SE(\hat\beta_0)$ is very wide for the data that were sampled in the neighborhood of $x=10$, and very narrow for the study in which the data were sampled near $x=0$.
Edit in response to comment: Unfortunately, centering your data after you have them will not help you if you want to know the likely $y$ value at some $x$ value $x_\text{new}$. Instead, you need to center your data collection on the point you care about in the first place. To understand these issues more fully, it may help you to read my answer here: Linear regression prediction interval.
• So, lets say for some reason I'm most interested in the prediction at the value $x=x'$. The above explanation implies that I shouldn't center my data (i.e. shift $x$ so that $\bar{x}=0$), but instead shift it so that $\bar{x}=x'$. Is this correct? – elexhobby Mar 12 '14 at 19:48
• The general formula has $(x^\prime - \bar{x})^2$ in the numerator instead of $\bar{x}^2$: no shifting is needed. – whuber Mar 12 '14 at 20:27
• @elexhobby, I added some info to answer your comment, you might also want to look at the linked material. Let me know if you still need more. – gung - Reinstate Monica Mar 12 '14 at 20:48
• Here's how I understand - I read elsewhere that $SE(\hat{\beta}_1)=\frac{\sigma^2}{\sum(x_i-\bar{x})^2}$. Now the error in the predicted value at $x_{new}$ due to this uncertainty in the slope is $SE(\hat{\beta}_1)(x_{new}-\bar{x})^2$. Furthermore, the error due to uncertainty in the vertical position of the line is $\frac{\sigma^2}{n}$. Combine these together, and we get the uncertainty in the predicted value due to uncertainty in $\hat{\beta}_1$ and $\hat{\beta}_0$ is $\frac{\sigma^2}{n}+\frac{\sigma^2(x_{new}-\bar{x})^2}{\sum(x_i-\bar{x})^2}$. Correct me if I'm wrong. – elexhobby Mar 13 '14 at 4:44
• Furthermore, it is clear why the error in the vertical position is $\frac{\sigma^2}{n}$ - we know that the line has to pass through $\bar{y}$ at $x=\bar{x}$. Now $\bar{y}$ contains the average of $n$ iid errors, and hence will have SE equal to $\frac{\sigma^2}{n}$. Wow! Thanks a lot for your diagram and clear explanation, I really appreciate. – elexhobby Mar 13 '14 at 4:57
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2019-12-08 18:31:02
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https://www.semanticscholar.org/paper/Riemann-surfaces%2C-ribbon-graphs-and-combinatorial-Mondello/e36b3c5b17be127ea43e7a64ade9f0238faa0dee
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# Riemann surfaces, ribbon graphs and combinatorial classes
@article{Mondello2009RiemannSR,
title={Riemann surfaces, ribbon graphs and combinatorial classes},
author={Gabriele Mondello},
journal={arXiv: Algebraic Geometry},
year={2009},
pages={151-215}
}
This survey paper begins with the description of the duality between arc systems and ribbon graphs embedded in a punctured surface. Then we explain how to cellularize the moduli space of curves in two different ways: using Jenkins-Strebel differentials and using hyperbolic geometry. We also briefly discuss how these two methods are related. Next, we recall the definition of Witten cycles and we illustrate their connection with tautological classes and Weil-Petersson geometry. Finally, we… Expand
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#### References
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Combinatorial classes on M̄g,n are tautological
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Weil-Petersson volumes and intersection theory on the moduli space of curves
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Graph cohomology and Kontsevich cycles
Abstract We use the duality between compactly supported cohomology of the associative graph complex and the cohomology of the mapping class group to show that the duals of the Kontsevich cycles [Wλ]Expand
Riemann surfaces with boundary and natural triangulations of the Teichmüller space
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Triangulated Riemann surfaces with boundary and the Weil-Petersson Poisson structure
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The Poincaré dual of the Weil–Petersson Kähler two-form
We consider a family of conjectures due to Witten which relate the Miller-Morita-Mumford cohomology classes to explicit cycles in a certain cell-decomposition of the moduli space of punctured RiemannExpand
The Structure and Singularities of Arc Complexes
A classical combinatorial fact is that the simplicial complex consisting of disjointly embedded chords in a convex planar polygon is a sphere. For any surface F with non-empty boundary, there is anExpand
Increasing trees and Kontsevich cycles
• Mathematics
• 2004
It is known that the combinatorial classes in the cohomology of the mapping class group of punctures surfaces dened by Witten and Kontsevich are polynomials in the adjusted Miller{Morita{MumfordExpand
Cellular Decompositions of Compactified Moduli Spaces of Pointed Curves
To a closed connected oriented surface S of genus g and a nonempty finite subset P of S is associated a simplicial complex (the arc complex) that plays a basic role in understanding the mapping classExpand
Combinatorial Miller-Morita-Mumford classes and Witten cycles
We obtain a combinatorial formula for the Miller-Morita-Mum- ford classes for the mapping class group of punctured surfaces and prove Witten's conjecture that they are proportional to the dual to theExpand
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2021-09-24 09:30:58
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https://tex.stackexchange.com/tags/errors/new
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# Tag Info
3
The problem is probably that your format is used with \usepackage[english, polish]{babel} which transforms the meaning of some special char (technically it makes chars active), in this case probably the quote " which I suppose it is used to add some accent around... (I do not know Polish, so I don't know the details). If you use utf8 to input special ...
5
you can input a file with multiple spaces in its name by \documentclass{article} \begin{document} \input{"a\space\space b.tex"} \end{document} which works as you can see on the terminal LaTeX2e <2020-02-02> patch level 2 L3 programming layer <2020-02-14> (/usr/local/texlive/2019/texmf-dist/tex/latex/base/article.cls Document Class: article ...
3
You can define new keys. \documentclass{article} \usepackage{todonotes} \makeatletter \newcommand\newtodooption[2]{% \define@key{todonotes}{#1}[]{% \setkeys{todonotes}{#2}% }% } \makeatother \newtodooption{q}{color=green} \begin{document} Abc \todo[q]{text goes here} Abc \todo[color=green]{text goes here} \end{document}
1
The keyval parser goes to some lengths not to expand macros so that you can use them in values without them being expanded at the wrong time. So you could force \q to expand before being passed in by \expandafter\todo\expandafter[\q] But that isn't very convenient, \newcommand\q{\todo[color=green]} is likely to be the simplest version.
3
That's a bug in IEEEaccess. It redefines TeX's \year primitive to set the year of the publication (so that you can type \year{2020}). That breaks TikZ (and several other packages) that rely on \year having its primitive meaning. You can work around that issue by tepmorarily saving \year as \TeXyear before loading IEEEaccess, and then restoring it after the ...
2
The problem is, as the error message says, that \msg_term:n is deprecated. This function along with a couple others were removed from the LaTeX3 kernel because they were not correctly named. I mailed the Elsevier team responsible for their LaTeX packages last october (2019) and was told that they would look into my suggestion, update the package, and send ...
4
The return value of -5 means that signature verification for a repository has failed. There is currently a bug in tlmgr that prevents a sensible error message from being printed, but this is being worked on and should be fixed soon: https://tug.org/pipermail/tex-live/2020-February/044775.html The most recent incarnation of this problem stems from the ...
0
The propossed solution is not complete. You will run into the same problem in other cv-scripts. Load spanish-babel as: \usepackage[spanish,es-noquoting,es-noshorthands]{babel} or \usepackage[spanish,es-noquoting,es-noshorthands,activeacute]{babel} in case you write María as Mar'ia. The short-hands interfere with moderncv.
1
Hi @khawla and welcome to TeX-SE. I'm assuming your are on Windows and using TeXstudio, much likely because I've got the same errors long time ago and the -synctex=1 is a default setup. And you are using pdflatex since it is written on the error message. In TeXstudio, go to Options -> Configure TeXstudio and then to Commands at the left menu. Check the ...
0
Ok so, I've been following the solutions proposed here. However, when I change the path for pdfLaTeX in the command section in preferences, I can compile but not compile & view. Any suggestions?
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Thanks a lot Schrödinger's cat ! I was not able to supply the error format in time to you. I have slightly altered your code to work with my desired error format. This error format is a bit easier to generate from my dataset. I added plotting rows N to M to your code, which is explained here. Thanks again! \documentclass[tikz,border=3mm]{standalone} \...
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Is this what you are looking for? \documentclass{article} \usepackage[a4paper, margin=2cm]{geometry} \usepackage{datetime} \usepackage{spreadtab} \usepackage{numprint} \npdecimalsign{.} \usepackage{tabularx} \setlength\parindent{0pt} \hbadness=99999 % or any number >=10000 \begin{document} \STautoround*{2} \begin{spreadtab}{{tabularx}{\textwidth}{ &...
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Welcome! This adds the error bars. I was not sure how you want to provide the errors, so for the time being I just added six columns from your table which indicate the dimensions (+ and -) of the error bars. \documentclass[tikz,border=3mm]{standalone} \usepackage{pgfplots} \pgfplotsset{compat=1.16} \begin{document} \pgfplotstableread{ 0 2 3 4 1 0.5 0.5 1 0....
0
Welcome to TeX.SE! First, you shouldn't use the minimal class; it isn't intended for end-users, only for specific tests by LaTeX developers. Second, your problem goes away if you add \usepackage[T1]{fontenc} to your document. T1 being more modern than OT1, this may be what you want to do. The problem when using OT1 here is due to the fact that \_ doesn't ...
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The problem is that xstring will perform “all the way expansion” to the arguments to \StrSubstitute and \_ doesn't survive this process. \documentclass{article} \usepackage[abspath]{currfile} \usepackage{xstring} %%% Set a fake return. \def\theabsdir{\detokenize{/Users/Me/Example_with_underscore}} \StrSubstitute[0]{\theabsdir}{\detokenize{_}}{\noexpand\...
1
The KOMA classes are no longer compatible with titlesec. Some patches which more or less made it working have been removed. This means that this here compiles with texlive 2018 but not with a current texlive 2019: \documentclass{scrartcl} \usepackage{titlesec} \titleformat*{\section}{} \begin{document} \section{abc} \end{document} You can check your log-...
3
That error (not terribly clear, I'll give you that) happens because you are using \printbibliography in the preamble of the document. The command \printbibliography, as the name advertises, prints the bibliography, so it must appear after the \begin{document} (where you want the bibliography to appear). That error appears before the usual Missing \begin{...
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Wow! Based on David Carlisle's response to my other question (Overfull \vbox caused by change of font), this script ending also eliminates the Overfull \hbox message: \end{tabular*} \end{scriptsize} \end{document}
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(Edited to incorporate useful comments from @DavidCarlisle) A single line break is typically treated as a space in LaTeX. So, for example, consider the following: \documentclass{article} \begin{document} Hello world \end{document} (which has no space after "Hello") will output "Hello world" (with a space). At the end of a paragraph (indicated by a \par ...
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First, amsart only supports tocdepth up to 3, because it only defines the variables \r@tocindentX for −1 ≤ X ≤ 3. Even then, this looks like a bug in amsart, when you set tocdepth to 3. The class defines: \def\l@subsubsection{\@tocline{3}{0pt}{1pc}{7pc}{}} and then also does: \let\l@paragraph\l@subsubsection The first ...
2
You have specified footnotesize text in a box based on a lineheight as set by scriptsize (which is smaller), so the text does not fit. so it's just a bug in your code. Note that size commands do not take an argument, it should be \footnotesize not \footnotesize{....} (but that makes no difference here) What is the intention of the \multirow{1}{*} here? ...
4
Welcome to TeX-SX. First of all, I would suggest you to save your files before compiling, at least the first time. If you do not save the file intentionally, TeXstudio will compile it as a temporary file, as you can check at the imagem. The name will be "texstudio_something". These files are stored at the AppData\Local\Temp. Which you can access through ...
7
yes you are right. The new version of tkz-base is incompatible with the version of tkz-fct.sty. There was a small problem uploading the new file on CTAN. You will have to wait a while before you can find the package on CTAN. Currently you will find the file v2.2 here I think \tkzActivOff is no longer necessary with the latest version of TikZ. \...
0
This is often caused by another MikTex process running at the same time. Close any other MikTex processes (or just reboot) and try again.
2
The nameref package uses gettitlestring that has code to support enumitem, but it is incomplete: it only manages \enit@format, but not \enit@align. \documentclass{article} \usepackage{enumitem} \usepackage{nameref} \makeatletter \g@addto@macro\GTS@PredefinedLeftCmds{% \GTS@TestLeft\enit@align\GTS@Cdr % package enumitem } \g@addto@macro\GTS@...
0
As suggested by @Sigur, here is a code with a simplified typing, which uses the \SI command from siunitx to ensure correct formatting and spacing of units. An unrelated remark: there's no difference between alignat{1}and a simple align. \documentclass{article} \title{Homework 3 my name} \usepackage{xcolor} \usepackage{amsmath} \usepackage{graphicx} \...
2
In your case you are trying to use two packages scrlayer-scrpage and fancyhdr. The culprit here is that package scrlayer-scrpage defines and uses for example an command \cfoot. With the call of the second package fancyhdr this package also tries to define and use an own command \cfoot and that is impossible, because that comand is already defined and in use ....
1
I solved the problem, and this is how: I opened the MikTeX Console application and updates by clicking "Update now" on the Overview page. After doing that, it works now :) (MikTeX does not automatically update, so every now and then you should manually search for updates by opening the MikTeX Console application.) Anyways, if this doesn't solve your problem, ...
4
\color is defined by LaTeX's base color support package, so loading it will get rid of that error. However this package provides only basic colour support, and xwatermark seems to need the extended features from the xcolor package (so adding \usepackage{xcolor} is a safe bet). Though this looks a lot like a bug in xwatermark: if the package requires xcolor ...
1
Use the YAML metadata block to enable graphics/graphicx like this: --- graphics: yes --- ## Chapter 12 \begin{figure}[htbp]\includegraphics[width=0.25\textwidth, height=!]{images/castle01.png}\centering\end{figure} lorem ipsum
1
You have an error in the .aux file, no need to copy the whole project just delete the aux (and probably your .toc file as well). The most common reason for corrupting the aux file in this way is killing the job after an error (by control-c on the command line or deleting the window running the job) This gives the operating system no time to flush any ...
1
There's a typo in the documentation (or rather, in the code). The documentation says footnotedistance, but the code actually defines footenotedistance, with the extra e. The code should work if you change that. This is unlikely to be fixed, as the last activity in the mdframed GitHub was in 2014, but I filed an issue anyway.
2
Open a bug report with adjmulticols. It seems to get the case of being inside a box wrong. \documentclass{article} \usepackage{adjmulticol} \begin{document} \begin{minipage}{5cm} \begin{multicols}{2} multicol in a box is fine \end{multicols} \end{minipage} \begin{minipage}{5cm} \begin{adjmulticols}{2}{2cm}{2cm} but this is not \end{...
Top 50 recent answers are included
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2020-02-22 04:12:54
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https://www.earthdoc.org/content/papers/10.3997/2214-4609-pdb.251.80
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1887
PDF
• # oa Estimating Poisson’s Ratio from Elastic Impedance: A Case Study for Hydrocarbon Plays in Malay Basin
• By Ang Chin Tee
• Publisher: European Association of Geoscientists & Engineers
• Source: Conference Proceedings, PGCE 2011, Jul 2011, cp-251-00079
• DOI:
### Abstract
It is now common for a 3D datasets to be processed as partial offset volumes to exploit the AVO information in the data. The amplitudes of near-offset stack relate to changes in acoustic impedance (AI) and can be tied to well logs using synthetics. Unfortunately, there have been no simple equivalent processes for far-offset stacks. However, the symmetry can be recovered using the elastic impedance (EI). EI provides a consistent and absolute framework to calibrate and invert nonzero offset seismic data just as AI does for zero-offset data (Connolly, 1999). An EI log acts as a platform to calibrate the inverted data to any desired rock property (SI, σ, μ, λ etc) with which it correlates (Connolly, 2010). Many studies on EI have been done on Gulf of Mexico, and a strong correlation was found between EI at 30° and hydrocarbon pore volume. This relationship was then used to estimate the in-place volumes for the field from the inverted 30° seismic volume. EI is also widely used to discriminate lithology and to distinguish fizz water from commercial gas concentrations (Gonzalez, 2004). Estimating the Poisson’s ratio from seismic is also crucial. Theoretically, one can invert a 90° angle stack which has amplitude that is proportional to changes in Poisson’s ratio. However, this<br>approach is difficult due to the sensitivity to residual moveout and bandwidth variations. On the other hand (refer to the equation above), EI has values equal to AI at normal incidence. If K = 0.25, then EI is equal to (Vp/Vs)2 at 90° which is closely related to Poisson’s ratio. This allows the construction of high angle stack, and then being calibrated and inverted using the equivalent EI log. Since the absolute level of EI(90°) is depending on the value of K being used, one should study for the optimum angle of EI that correlates with Poisson’s ratio at well locations. In this paper, we will perform this study for hydrocarbon plays in Malay Basin and validate the result by estimating the correlation coefficient. With the known optimum angle, we can estimate Poisson’s ratio from<br>seismic with the information from EI.
/content/papers/10.3997/2214-4609-pdb.251.80
2011-07-03
2021-12-05
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2021-12-05 16:52:13
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http://math.stackexchange.com/tags/brownian-motion/hot
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Tag Info
4
First we show that that if $X\colon \Omega \rightarrow \mathbb{R}^n$ is normal $\mathcal{N}(m, C)$, $Y\colon \Omega \rightarrow \mathbb{R}^n$ is normal $\mathcal{N}(m^{\prime}, C^{\prime})$ and they are independent then $X+Y$ is normal $\mathcal{N}(m_1+m_2, C_1+C_2)$. Here $m$, $m^{\prime} \in \mathbb{R}^n$ and $C=[c_{jk}]$, $C^{\prime}=[c_{jk}^{\prime}]$ ...
2
We have $dX_t = dW_t - \frac{a}{2}dt$ and $d[X]_t = d[W]_t = dt$ since the deterministic piece $-at/2$ doesn't contribute to the quadratic variation. So, $$f(X_t) = f(X_0)+\int_0^t f'(X_s)\, dW_s - \frac{a}{2} \int_0^t f'(X_s) \,ds + \frac{1}{2} \int_0^t f''(X_s) \, ds \\ = f(X_0)+\int_0^t f'(X_s)\, dW_s + \frac{1}{2} \int_0^t \big[f''(X_s) - af'(X_s) ... 2 This is in Feller volume 2, and on page 2 of this paper. 2 For any \omega \in \Omega, we can construct a sequence (\Pi_n)_n=(\Pi_n(\omega))_n of partitions of [0,T] such that the mesh size |\Pi_n| tends to 0 as n \to \infty and$$\lim_{n \to \infty} \sum_{t_j \in \Pi_n} (B_{t_j}(\omega)-B_{t_{j-1}}(\omega))^2 = \infty.$$This means in particular that$$[B,B](T)(\omega) := \sup_{\Pi} \sum_{t_j \in \Pi} ...
2
There are several definitions for the quadratic variation: For a "nice" process $(X_t)_{t \geq 0}$ ("nice" means semimartingale), the quadratic variation is defined by $$[X]_t := X_t^2-X_0^2 -2 \int_0^t X_{s-} \, dX_s.$$ For $X_t := B_t^2$, we know from Itô's formula that $$d(B_s^2)= 2 B_s \, dB_2+ \, ds$$ and $$B_t^4 = 4 \int_0^t B_s^3 \, dB_s + 6 ... 2 For every t\geqslant0, (B_{t+s}^2)_{s\geqslant0} is distributed as (X_s)_{s\geqslant0}, where, for every s\geqslant0,$$X_s=B_t^2+2\sqrt{B_t^2}\cdot W_s+W_s^2,$$where (W_s)_{s\geqslant0} is a Brownian motion independent of (B_u)_{0\leqslant u\leqslant t}. Thus, indeed, (B^2_t)_{t\geqslant0} is a Markov process. The discrete analogue of this ... 2 This notion is called tightness of a sequence of measures. You can apply it in probability theory with the sequence P\#{X_n} of the image probabilities under the action of X_n:$$ P\#{X_n}(A) = P(X_n\in A) $$You transfer the topology issues to the (metric, often polish) space \mathcal X where$$ X:\Omega \to \mathcal X $$Note also that in the ... 2 Surely not an exam proof, what is going to come. We will use Theorem 5.4.1 from Ethier and Kurtz, Markov Processes, Wiley, 1986 (EK86); the important part is quoted below. Let us first write down the generator A_n of the process X^{(n)} $$A_n f(x) = \sin (nx) f'(x) + \frac{1}{2} f''(x) , x \in \mathbb{R}.$$ The expected ... 2 Let X_t=t\,B_{1/t}. Every linear combination of the random variables X_t is a linear combination of the random variables B_t hence it is normal. Thus, the process (X_t) is gaussian, in particular every increment X_t-X_s is gaussian. More generally, let Y_t=a(t)B_{c(t)}+b(t). For every functions (a,b,c), the process (Y_t) is gaussian. 1 As D is open and y \in D, we have$$r := d(y,\partial D) := \inf\{z \in \partial D; |z-y| \}>0.$$On the other hand, the boundedness of D implies that we can choose R>0 such that |z| \leq R for any z \in \partial D. Consequently,$$\log r \leq \log|B(T)(\omega)-y| \leq \log (R+|y|)$$holds for any \omega \in \Omega. Taking expectation ... 1 Key-facts: The function x\mapsto\|x\|^{-1} is harmonic on \mathbb R^3\setminus\{0\} and the distribution of \sqrt{t}\underline{X} is radially symmetric. Assume that \underline{X}_0 and \underline{X} are independent (otherwise anything can happen) and note that the result is a consequence of the fact that, for every random variable B whose ... 1 As mentionned by saz from Itô's lemma applied to Y_t=B^2_t you get : dY_t=finite_variation_ter.dt + 2.B_tdB_t so the quadratic variation is (using Itô's isometry) is equal to :$$<Y>_t=4\int_0^t B_s^2ds$$Please note that this is a stochastic process. Best regards 1 Hint: under the risk neutral probability, the prices of the securities having an L^2 payoff are martingales. So you probably need to compute the Ito-differential of$$ (t,\omega)\to S^3_t e^{(2r+3σ^2)(T−t)} $$1 As the process (W(t))_{t\ge 0} has the same distribution than (cW(\frac t{c^2}))_{t\ge 0}, you get with c = \sqrt t:$$ P( W(t) > 0 ;\ \ W(2t) > 0) = P\left( cW(\frac t{c^2}) > 0 ;\ \ cW\left(2 \frac t{c^2}\right) > 0\right)\\ = P\left( W\left(\frac t{c^2}\right) > 0 ;\ \ W\left(2 \frac t{c^2}\right) > 0\right)= P( W(1) > 0 ; ...
1
Let $(P_t)_{t\ge0}$ denote a regular conditional distribution of $W$ given $T$. This means that each $P_t$ is a probability measure on $C[0,\infty)$, the set of continuous functions from $[0,\infty)$ to $\mathbb{R}$ endowed with the $\sigma$-algebra induced by the coordinate projections, such that $$P(W\in A,T \in B) = \int_B P_t(A) d Q(t)$$ where $Q$ is ...
1
It suffices to prove that $\int_0^t \phi_s^2 \, ds$ is finite for all $t \in [0,\infty)$. This allows us to define the stochastic integral $$\int_0^t \phi(s) \, dB_s$$ for any $t \in [0,\infty)$ (as a local integral). This means that your proof is correct. In fact, the very argumentation shows that $\Lambda_{\text{loc}}^2$ contains all functions $f$ ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2014-11-25 22:03:57
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https://mathoverflow.net/questions/68615/number-of-galois-extensions-of-local-fields-of-fixed-degree/68631
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# number of galois extensions of local fields of fixed degree
Let $K$ be a local field (of characteristic 0) with (finite) residue field of characteristic $l$ and let $p$ be a prime.
Considering the cases, whether the $p$-th roots of unity are in $K$ and whether $l$ equals $p$ (and maybe whether $p=2$) or not, my question is:
How many Galois extensions of $K$ of degree $p$ exist?
If $K$ contains the $p$-th roots of unity, then Kummer theory tells us that the degree $p$ Galois extensions of $K$ are in bijective correspondence with the subgroups of $K^{\times}/(K^{\times})^p$ of order $p$. The structure of $K^{\times}$ is well-known; see http://en.wikipedia.org/wiki/Local_fields or any decent book on local fields. So you can work out the answer in this case.
If $K$ doesn't contain the $p$-th roots of unity, then it becomes harder. Here are some special cases. For every $d \in \mathbb{N}$, $K$ has a unique unramified extension of degree $d$, which is necessarily cyclic - see Corollary 4.4 of these nice notes: http://websites.math.leidenuniv.nl/algebra/localfields.pdf So in particular there is a unique unramified degree p Galois extension of $K$.
If $l \neq p$, then any ramified extension of $K$ must be totally and tamely ramified. But then by 5.3 and 5.4 of the above notes, $\mathbb{Z}/p\mathbb{Z}$ must embed into the unit group of the residue field of $K$. Then by Hensel's Lemma, $K$ must contain $p$-th roots of unity and so we are reduced to the Kummer case above.
So we are left with the case $l=p$ and $K$ not containing $p$-th roots of unity. I'll think about this some more, but you should be able to use class field theory as mentioned above.
• Just for clarity (I just confused myself for a minute): What you are saying in your third paragraph is that either $K$ contains all $p$-th roots of unity (so we are in the Kummer case), or there are no ramified Galois extensions of degree $p$ at all (so the unramified is the only one). May 8 '14 at 18:41
• Yes (assuming $l \neq p$, of course). May 8 '14 at 22:38
I am sorry if I see this question only now, but since no one gave the following answer, it seems worth posting it.
There is a general formula for the number of extensions of degree $$d$$ of a $$p$$-adic field $$K$$ contained inside a fixed algebraic closure $$\overline{K}$$, which is given by $$\# \{ L \mid K \subseteq L \subseteq \overline{K}, \, [L \colon K] = d \} = \sigma(h) \cdot \sum_{j = 0}^m \frac{p^{m+j+1} - p^{2j}}{p - 1} \cdot (p^{\varepsilon_p(j) \cdot d \cdot d_0} - p^{\varepsilon_p(j - 1) \cdot d \cdot d_0})$$ where:
• $$\sigma$$ denotes the sum of divisors function;
• $$h, m \in \mathbb{N}$$ are the unique natural numbers such that $$p \nmid h$$ and $$d = h \cdot p^m$$;
• $$\varepsilon_p(j) := \sum_{k=1}^j p^{-k}$$ if $$j \geq 1$$, $$\varepsilon_p(0) := 0$$ and $$\varepsilon_p(-1) := -\infty$$, i.e. $$p^{\varepsilon_p(-1) \cdot d} = 0$$. In particular, $$p^{\varepsilon_p(j) \cdot d} \in \mathbb{N}$$ if $$-1 \leq j \leq m$$;
• $$d_0 := [K \colon \mathbb{Q}_p]$$.
This formula is due to Krasner, and has been proved in the paper "Nombre des extensions d'un degré donné d'un corps $$\mathfrak{p}$$-adique". The proof uses the same analytic techniques that go into the proof of the (much more famous) Krasner lemma.
Observe that this number is different from the number of $$K$$-isomorphism classes of extensions of $$K$$ having a given degree. This is of course due to the presence of non-Galois extensions. Here are two examples of this phenomenon:
• if $$q \neq p$$ is a prime then there are $$q + 1$$ fields $$K \subseteq L \subseteq \overline{K}$$ having degree $$[L \colon K] = q$$, but there are only two isomorphism classes of these fields: one containing the only unramified extension, and the other containing the tamely and totally ramified extensions;
• if $$p \geq 3$$ there are $$1 + p + (p^2 - p) \cdot p$$ extensions $$\mathbb{Q}_p \subseteq L \subseteq \overline{\mathbb{Q}_p}$$ such that $$[L \colon \mathbb{Q}_p] = p$$, but they form $$1 + p + p^2 - p = p^2 + 1$$ isomorphism classes. $$p + 1$$ of these contain a unique extension (which is Galois over $$\mathbb{Q}_p$$) and every other isomorphism class contains $$p$$ extensions (see for example Proposition 2.3.1 in the paper "A database of local fields" by Jones and Roberts).
Finally, let me remark that this formula is related to Serre's "mass formula", which is valid in any characteristic. This formula says that a certain "count" of totally ramified extensions of a local, non-Archimedean field $$K$$ of degree $$d$$ is equal to $$d$$. More precisely, $$\sum_{L \in \Sigma_d} (\# \kappa)^{d - 1 - \mathrm{v}_K(\mathrm{disc}(L/K))} = d$$ where $$\Sigma_d$$ denotes the set of totally ramified extensions of $$K$$ which have degree $$d$$, and $$\kappa$$ is the residue field of $$K$$. Observe that if $$p \nmid d$$ then the formula can be written simply as $$\# \Sigma_d = d$$. Two useful references for this are:
A Galois extension of degree $$p$$ has Galois group $$\mathbb{Z}/p\mathbb{Z}$$, so you are asking about abelian extensions of your local field. Thus the answer to your question can be obtained explicitly via local class field theory -- you can get the needed results out of Serre's Local Fields or many other books.
Here is a belated answer (I came across this question only today) which doesn't require anything more than Kummer theory --- or Artin-Schreier theory, if you want to allow $K$ to be a finite extension of ${\mathbf F}_l((t))$. I will confine myself to the more interesting case of degree-$l$ extensions (where $l$ is the residual characteristic).
If $K$ contains a primitive $l$-th root of $1$, then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and the set of ${\mathbf F}_l$-lines in $K^\times/K^{\times l}$ (Kummer theory). The structure of this filtered ${\mathbf F}_l$-space is completely known; see for example Section V of arXiv:0711.3878 (where your $l$ is called $p$).
If $K$ does not contain a primitive $l$-th root $\zeta$ of $1$, then put $K'=K(\zeta)$, $\Delta={\rm Gal}(K'|K)$ and $\omega:\Delta\to{\mathbf F}_l^\times$ the cyclotomic character giving the action of $G$ on the $l$-th roots of $1$. Then there is a natural bijection between the set of degree-$l$ cyclic extensions of $K$ and ${\mathbf F}_l$-lines in the $\omega$-eigenspace for the action of $\Delta$ on $K^{\prime\times}/K^{\prime\times l}$. The structure of this filtered ${\mathbf F}_l[\Delta]$-module can be completely determined; see for example arXiv:0912.2829.
If you are interested more generally in all degree-$l$ (separable) extensions of $K$ (and not just the cyclic ones), then something similar can be done. Put $L=K(\root{l-1}\of{K^\times})$ and $G={\rm Gal}(L|K)$. There is a natural bijection between the set of (isomorphism classes of) degree-$l$ (separable) extensions of $K$ and the set of $G$-stable lines in the ${\mathbf F}_l$-space $L^\times/L^{\times l}$. Again, the structure of this filtered ${\mathbf F}_l[G]$-module is completely known: see for example arXiv:1005.2016.
Finally, if you allow $K$ to be a finite extension of ${\mathbf F}_l((t))$, there are similar results using Artin-Schreier theory instead of Kummer theory. See for example arXiv:0909.2541 for degree-$l$ cyclic extensions (which correspond to ${\mathbf F}_l$-lines in $K^+/\wp(K^+)$, where $\wp(x)=x^l-x$) and arXiv:1005.2016 for degree-$l$ separable extensions, which correspond to $G$-stable ${\mathbf F}_l$-lines in $L^+/\wp(L^+)$, where $L$ is still $K(\root{l-1}\of{K^\times})$ and $G={\rm Gal}(L|K)$. The filtered ${\mathbf F}_l$-space (resp. ${\mathbf F}_l[G]$-module) $K^+/\wp(K^+)$ (resp. $L^+/\wp(L^+)$) has been completely determined therein. These results allow you in particular to count the number of extensions with bounded ramification, of which there are only finitely many.
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2021-12-02 06:25:08
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https://db0nus869y26v.cloudfront.net/en/Shilling
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A 1933 UK shilling
1956 Elizabeth II UK shilling showing English and Scottish reverses
The shilling is a historical coin, and the name of a unit of modern currencies formerly used in the United Kingdom, Australia, New Zealand, other British Commonwealth countries and Ireland, where they were generally equivalent to 12 pence or one-twentieth of a pound before being phased out during 1960's and 1970's.
Currently the shilling is used as a currency in five east African countries: Kenya, Tanzania, Uganda, Somalia, as well as the de facto country of Somaliland.[1] The East African Community additionally plans to introduce an East African shilling.
## History
Further information: Coinage in Anglo-Saxon England and Denarius
English shilling minted under Edward VI, c. 1551
Schilling coin of the imperial city of Zürich, minted in billon, 1640
Silver 4 schilling coin, Hamburg, 1728
The word shilling comes from Old English "Scilling", a monetary term meaning twentieth of a pound, from the Proto-Germanic root skiljaną meaning 'to separate, split, divide', from (s)kelH- meaning 'to cut, split.'[citation needed] The word "Scilling" is mentioned in the earliest recorded Germanic law codes, those of Æthelberht of Kent.[citation needed]
In origin, the word schilling designated the solidus of Late Antiquity, the gold coin that replaced the aureus in the 4th century. The Anglo-Saxon scillingas of the 7th century were still small gold coins.[citation needed]
In 796, Charlemagne passed a monetary reform, based on the Carolingian silver pound (about 406.5 grams). The schilling was one-twentieth of a pound or about 20.3 grams of silver. One schilling had 12 denarii or deniers ("pennies"). There were, however, no silver schilling coins in the Carolingian period, and gold schillings (equivalent to twelve silver pfennigs) were very rare.[citation needed]
In the 12th century, larger silver coins of multiple pfennig weight were minted, known as denarii grossi or groschen (groats). These heavier coins were valued at between 4 and 20 of the silver denarii. In the late medieval period, states of the Holy Roman Empire began minting similar silver coins of multiple pfennig weight, some of them denominated as schilling.[citation needed]
In the 16th century, numerous different types of schilling were minted in Europe. The English shilling was the continuation of the testoon coin under Edward VI and was first minted in 1551 minted in 92.5% "sterling" silver.[citation needed]
By the 17th century, further devaluation resulted in schillings in the Holy Roman Empire being minted in billon (majority base metal content) instead of silver, with 48 schillings to one Reichsthaler. The English (later British) shilling continued to be minted as a silver coin until 1946.[citation needed]
## British Isles
### Kingdom of England
Main article: Shilling (English coin)
A shilling was a coin used in England from the reign of Henry VII[2] (or Edward VI around 1550). The shilling continued in use after the Acts of Union of 1707 created a new United Kingdom from the Kingdoms of England and Scotland, and under Article 16 of the Articles of Union, a common currency for the new United Kingdom was created.
### Kingdom of Scotland
The term shilling (Scots: schilling) was in use in Scotland from early medieval times.
### Great Britain, then the United Kingdom of Great Britain and Ireland
Shilling, UK, queen Victoria, 1853. Silver, weight 5.64 g.
Main article: Shilling (British coin)
The common currency for Great Britain, created in 1707 by Article 16 of the Articles of Union between England and Scotland, continued in use until decimalisation in 1971. During the Great Recoinage of 1816 (following the Acts of Union 1800 that united the Kingdoms of Great Britain and Ireland), the mint was instructed to coin one troy pound (weighing 5760 grains or 373 g) of sterling silver (0.925 fine) into 66 shillings, or its equivalent in other denominations.[citation needed] This set the weight of the shilling at 87.2727 grains or 5.655 grams from 1816 until 1990, when it was demonetised in favour of a new smaller 5p coin of the same value.
At decimalisation in 1971, the shilling coin was superseded by the new five-pence piece, which initially was of identical size and weight and had the same value. Shillings remained in circulation until the five pence coin was reduced in size in 1991.
Three coins denominated in multiple shillings were also in circulation at this time. They were:
• the florin, two shillings (2/–), which adopted the value of 10 new pence (10p) at decimalisation;
• the half-crown, two shillings and sixpence (2/6) or one-eighth of a pound, which was abolished at decimalisation (otherwise it would have had the value of 121/2p);
• the crown (five shillings or one-fourth of a pound), the highest denominated non-bullion UK coin in circulation at decimalisation (in practice, crowns were commemorative coins not used in everyday transactions).
### Irish shillings
Further information: Shilling (Irish coin)
Between 1701 and the unification of the currencies in 1825, the Irish shilling was valued at 13 pence and known as the "black hog", as opposed to the 12-pence English shillings which were known as "white hogs".[3]
In the Irish Free State and Republic of Ireland, the shilling coin was issued as scilling (the Irish language equivalent). It was worth 1/20 of an Irish pound, and was interchangeable at the same value to the British coin, which continued to be used in Northern Ireland. The coin featured a bull on the reverse side. The first minting, from 1928 until 1941, contained 75% silver, more than the equivalent British coin. The pre-decimal Irish shilling coin (which was retained for some time after decimalisation) was withdrawn from circulation on 1 January 1993, when a smaller five-pence coin was introduced.
### Abbreviation and slang
The price tag on the Hatter's hat reads '10/6'
One abbreviation for shilling is s (for solidus, see £sd). Often it was expressed by a solidus symbol (/) (which may have begun as a substitute for ſ ('long s')[4]) thus '1/9' means "one shilling and ninepence". A price expressed as a number of shillings with no additional pence was often written as the number, a solidus and a dash: thus for example ten shillings was written '10/-'. Two shillings and sixpence (half a crown or an eighth of a £) was written as '2/6', rarely as '2s 6d' ('d' being the abbreviation for denarius, a penny). The shilling itself was equal to twelve pence. In the traditional pounds, shillings and pence system, there were 20 shillings per pound and 12 pence per shilling, making 240 pence in a pound.
Slang terms for the old shilling coins include "bob" and "hog". While the derivation of "bob" is uncertain, John Camden Hotten in his 1864 Slang Dictionary says the original version was "bobstick" and speculates that it may be connected with Sir Robert Walpole.[5]
## British Empire
Owing to the reach of the British Empire, the shilling was once used on every inhabited continent. This two-shilling piece was minted for British West Africa.
### Australian shillings
Main article: Shilling (Australian)
Australian shillings, twenty of which made up one Australian pound, were first issued in 1910, with the Australian coat of arms on the reverse and King Edward VII on the face. The coat of arms design was retained through the reign of King George V until a new ram's head design was introduced for the coins of King George VI. This design continued until the last year of issue in 1963. In 1966, Australia's currency was decimalised and the shilling was replaced by a ten cent coin (Australian), where 10 shillings made up one Australian dollar.
The slang term for a shilling coin in Australia was "deener". The slang term for a shilling as currency unit was "bob", the same as in the United Kingdom.
After 1966, shillings continued to circulate, as they were replaced by 10-cent coins of the same size and weight.
### New Zealand shilling
New Zealand shillings, twenty of which made up one New Zealand pound, were first issued in 1933 and featured the image of a Maori warrior carrying a taiaha "in a warlike attitude" on the reverse.[6] In 1967, New Zealand's currency was decimalised and the shilling was replaced by a ten cent coin of the same size and weight. Ten cent coins minted through the remainder of the 1960s included the legend "ONE SHILLING" on the reverse. Smaller 10-cent coins were introduced in 2006.
### Maltese shillings
The shilling (Maltese: xelin, pl. xelini) was used in Malta, prior to decimalisation in 1972, and had a face value of five Maltese cents.
### Ceylonese shillings
In British Ceylon, a shilling (Sinhala: Silima, Tamil: Silin) was equivalent to eight fanams. With the replacement of the rixdollar by the rupee in 1852, a shilling was deemed to be equivalent to half a rupee. On the decimalisation of the currency in 1969, a shilling was deemed to be equivalent to 50 Ceylon cents. The term continued to be used colloquially until the late 20th century.[7]
### East African shillings
Countries in Africa where the currency is called shilling.
The East African shilling was in use in the British colonies and protectorates of British Somaliland, Kenya, Tanganyika, Uganda and Zanzibar from 1920, when it replaced the rupee, until after those countries became independent, and in Tanzania after that country was formed by the merger of Tanganyika and Zanzibar in 1964. Upon independence in 1960, the East African shilling in the State of Somaliland (former British Somaliland) and the Somali somalo in the Trust Territory of Somalia (former Italian Somaliland) were replaced by the Somali shilling.[8] The State of Somaliland, which subsequently regained its independence in 1991 as the republic of Somaliland, adopted the Somaliland shilling as its currency.[9]
In 1966, the East African Monetary Union broke up, and the member countries replaced their currencies with the Kenyan shilling, the Ugandan shilling and the Tanzanian shilling, respectively.[10] Though all these currencies have different values at present, there were plans to reintroduce the East African shilling as a new common currency by 2009,[11] although this has not come about.
### North America
In the thirteen British colonies that became the United States in 1776, British money was often in circulation. Each colony issued its own paper money, with pounds, shillings, and pence used as the standard units of account. Some coins were minted in the colonies, such as the pine tree shilling in the Massachusetts Bay Colony. After the United States adopted the dollar as its unit of currency and accepted the gold standard, one British shilling was worth 24 US cents. Due to ongoing shortages of US coins in some regions, shillings continued to circulate well into the nineteenth century. Shillings are described as the standard monetary unit throughout the autobiography of Solomon Northup (1853)[12] and mentioned several times in the Horatio Alger Jr. story Ragged Dick (1868).[13][14] Prices in an 1859 advertisement in a Chicago newspaper were given in dollars and shillings.[15]
In Canada, £sd currencies were in use both during the French period (New France livre) and after the British conquest (Canadian pound). Between the 1760s and 1840s in Lower Canada, both French and British-based pounds coexisted as units of account, the French livre being close in value to the British shilling. A variety of coinage circulated. By 1858, a decimal Canadian dollar came into use. Other parts of British North America decimalized shortly afterwards and Canadian confederation in 1867 passed control of currency to the federal government.
### Somali shilling
Main article: Somali shilling
The Somali shilling is the official currency of Somalia. It is subdivided into 100 cents (English), senti (Somali, also سنت) or centesimi (Italian).
The Somali shilling has been the currency of parts of Somalia since 1921, when the East African shilling was introduced to the former British Somaliland protectorate. Following independence in 1960, the somalo of Italian Somaliland and the East African shilling (which were equal in value) were replaced at par in 1962 by the Somali shilling. Names used for the denominations were cent, centesimo (plural: centesimi) and سنت (plurals: سنتيمات and سنتيما) together with shilling, scellino (plural: scellini) and شلن.
That same year, the Banca Nazionale Somala issued notes for 5, 10, 20 and 100 scellini/shillings. In 1975, the Bankiga Qaranka Soomaaliyeed (Somali National Bank) introduced notes for 5, 10, 20 and 100 shilin/shillings. These were followed in 1978 by notes of the same denominations issued by the Bankiga Dhexe Ee Soomaaliya (Central Bank of Somalia). 50 shilin/shillings notes were introduced in 1983, followed by 500 shilin/shillings in 1989 and 1000 shilin/shillings in 1990. Also in 1990 there was an attempt to reform the currency at 100 to 1, with new banknotes of 20 and 50 new shilin prepared for the redenomination.[16]
Following the breakdown in central authority that accompanied the civil war, which began in the early 1990s, the value of the Somali shilling was disrupted. The Central Bank of Somalia, the nation's monetary authority, also shut down operations. Rival producers of the local currency, including autonomous regional entities such as the Somaliland territory, subsequently emerged.
Somalia's newly established Transitional Federal Government revived the defunct Central Bank of Somalia in the late 2000s. In terms of financial management, the monetary authority is in the process of assuming the task of both formulating and implementing monetary policy.[17] Owing to a lack of confidence in the Somali shilling, the US dollar is widely accepted as a medium of exchange alongside the Somali shilling. Dollarization notwithstanding, the large issuance of the Somali shilling has increasingly fueled price hikes, especially for low value transactions. This inflationary environment, however, is expected to come to an end as soon as the Central Bank assumes full control of monetary policy and replaces the presently circulating currency introduced by the private sector.[17]
### Somaliland shilling
The Somaliland shilling is the official currency of Somaliland, a self-declared republic that is internationally recognised as an autonomous region of Somalia.[18] The currency is not recognised as legal tender by the international community, and it currently has no official exchange rate. It is regulated by the Bank of Somaliland, Somaliland's central bank. Although the authorities in Somaliland have attempted to bar usage of the Somali shilling, Somalia's official currency is still in circulation in some regions.[19][better source needed]
### Other
Elsewhere in the former British Empire, forms of the word shilling remain in informal use. In Vanuatu and Solomon Islands, selen is used in Bislama and Pijin to mean "money"; in Malaysia, syiling (pronounced like shilling) means "coin". In Egypt and Jordan the shillin (Arabic: شلن) is equal to 1/20 (five qirshesArabic: قرش, English: piastres) of the Egyptian pound or the Jordanian dinar. In Belize, the term shilling is commonly used to refer to twenty-five cents.
## Other countries
A Swedish Skilling from 1802.
• The Austrian schilling was the currency of Austria between 1 March 1924[20] and 1938 and again between 1945 and 2002. It was replaced by the euro at a fixed parity of €1 = 13.7603 schilling. The schilling was divided into 100 groschen.
• In the principalities covering present Netherlands, Belgium and Luxemburg, the cognate term schelling was used as an equivalent 'arithmetic' currency, a 'solidus' representing 12 'denarii' or 1/20 'pound', while actual coins were rarely physical multiples of it, but still expressed in these terms.
• Shillings were issued in the Scandinavian countries (skilling) until the Scandinavian Monetary Union of 1873, and in the city of Hamburg, Germany.
• In Poland szeląg was used.[21]
• The soll, later the sou, both also derived from the Roman solidus, were the equivalent coins in France, while the sol (PEN) remains the currency of Peru.
• As in France, the Peruvian sol was originally named after the Roman solidus, but the name of the Peruvian currency is now much more closely linked to the Spanish word for the sun (sol). This helps explain the name of its temporary replacement, the inti, named for the Incan sun god.
## References
1. ^ Renders, Marleen (27 January 2012). Consider Somaliland: State-Building with Traditional Leaders and Institutions. BRILL. p. 134. ISBN 978-90-04-22254-0.
2. ^ "Understanding old British money - pounds, shillings and pence". woodlands-junior.kent.sch.uk. Archived from the original on 27 September 2012. Retrieved 27 April 2018.
3. ^ Wright, Joseph (1898). "Hog". The English Dialect Dictionary. London: Times Book Club. p. 196. ISBN 9785880963072. OCLC 422279387.
4. ^ "May and the Slash - English Project". www.englishproject.org. Archived from the original on 22 October 2017. Retrieved 27 April 2018.
5. ^ John Camden Hotten (1864). Slang Dictionary.
6. ^ Reserve Bank of New Zealand Archived 23 January 2009 at the Wayback Machine- URL retrieved 17 April 2011
7. ^ Early Monetary Systems of Lanka (Ceylon) Archived 22 June 2011 at the Wayback Machine, Currency Museum Circular No 7, Currency Department, Central Bank of Ceylon, Colombo, 15 March 1984
8. ^ Description of Somalia shilling - URL retrieved 8 October 2006
9. ^ Planet, Lonely. "Money and Costs in Somaliland". Lonely Planet. Retrieved 14 January 2021.
10. ^ Dissolution of the East African Monetary Union Archived 11 March 2007 at the Wayback Machine - URL retrieved 8 October 2006
11. ^ East African Business Council - Fact Sheet: Customs Union Archived 10 March 2007 at the Wayback Machine - URL Retrieved 8 October 2002
12. ^ Solomon Northup. Twelve Years a Slave. Auburn, Derby and Miller; Buffalo, Derby, Orton and Mulligan; [etc., etc.] 1853
13. ^ Alger, Horatio Jr (5 May 1868). Ragged Dick; or, Street Life in New York with the Boot Blacks (1 ed.). New York: A K Loring.
14. ^ Lundin, Leigh (11 May 2014). "Literary Rags". SleuthSayers.org. New York: SleuthSayers. Archived from the original on 11 August 2014.
15. ^ "Special Notice". Chicago Tribune. 9 December 1859. p. 2 – via newspapers.com.
16. ^ "CURRENCY". somalbanca.org. Archived from the original on 27 December 2016. Retrieved 27 April 2018.
17. ^ a b "Central Bank of Somalia - Monetary policy". somalbanca.org. Archived from the original on 25 January 2009. Retrieved 27 April 2018.
18. ^ "Somaliland's Quest for International Recognition and the HBM-SSC Factor". wardheernews.com. Archived from the original on 28 May 2012. Retrieved 27 April 2018.
19. ^ "Time for Somaliland to Rethink its Strategy". www.hiiraan.com. Archived from the original on 14 September 2017. Retrieved 27 April 2018.
20. ^ "Gold and silver shillings of Austria". Knowledge base - GoldAdvert. 14 June 2018. Retrieved 15 June 2018.
21. ^ "shillings - Polish translation – Linguee". Linguee.com. Retrieved 27 April 2018.
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2023-04-02 00:14:57
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https://mathoverflow.net/questions/223613/calculation-of-integral-using-gamma-function-when-the-imaginary-part-is-zero
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# Calculation of integral using Gamma function when the imaginary part is zero
Consider the following expression of Gamma function
$$\frac{\Gamma(z)}{p^z}=\int_{0}^{\infty}e^{-pt}t^{z-1}dt \ \ \ \ \ \ \ \ \ (1)$$ where $Re(z)>0$ and $Re(p)>0$.
In Lebedevs book "special functions and their applications" he uses that integral to calculate
$$\int_{0}^{\infty}e^{2izs}s^{\alpha-1}ds=\frac{\Gamma(\alpha)}{(-2iz)^{\alpha}} \ \ \ \ \ \ \ \ (2)$$
but what about case when $z=x$ is real? This means $Re(p)=0$ in $(1)$ expression.
P.S. The Formula $(2)$ must be valid for Real $z$ because of final result of calculation (asymptotic expression of Hankels' function $H_{\nu}^{(1)}(z)$).
I just want to know how to integrate (2) integral if z=x is real variable.
• Hint: integration by parts and analytic continuation. – Fan Zheng Nov 14 '15 at 20:41
The identity can be deduced by the Euler integral by homotopy invariance of path integrals.
The function $e^{-z}z^{\alpha}$ is holomorphic on $\mathbb{C}\setminus \{\operatorname{Re}z\le0,\operatorname{Im}z=0 \}$, so the value of the path integral along the boundary of the domain $\{z: \operatorname{Re}z>0,\operatorname{Im}z>0, \epsilon< |z|< \rho \}$ is zero. Hence we get $$\int_\epsilon^{\rho}i^\alpha e^{-it}t^{\alpha-1}dt=\int_\epsilon^{\rho}e^{-t}t^{\alpha-1}dt+ \int_0^{\pi/2}ie^{-\rho e^{it} }(\rho e^{it})^{\alpha}dt- \int_0^{\pi/2}ie^{-\epsilon e^{it}}(\epsilon e^{it} )^{\alpha}dt$$
We consider separately the three integrals on the RHS.
• The first integral, of course, converges to the Euler integral for $\Gamma(\alpha)$ as $\epsilon\to0$ and $\rho\to+ \infty$, provided $\operatorname{Re}\alpha>0$.
• The second integrand has absolute value $e^{-\rho\cos(t) -\operatorname{Im}\alpha t}\; \rho^{ \operatorname{Re}\alpha}$, and it is a bit singular at $t=\pi/2$; to evaluate the corresponding integral, it is convenient to spit it further in the integrals over $[0,x]$ and $[x,\pi/2]$ with a free $x$, optimizing then the bound over the choice of $x$. This way one gets a bound for this integral of order $O\Big(\rho ^{\operatorname{Re}\alpha -1} \log(\rho) \Big)$, which is $o(1)$ as $\rho\to+ \infty$, provided $\operatorname{Re}\alpha <1$.
• The third integrand has absolute value $e^{-\cos(t) \epsilon-\operatorname{Im}\alpha t}\; \epsilon^{ \operatorname{Re}\alpha}=O(\epsilon^{ \operatorname{Re}\alpha})$.
We conclude that for $0< \operatorname{Re}\alpha<1$ the integral on the LHS converges, and
$$\int_0^{+\infty} e^{-it}t^{\alpha-1}dt=\frac{\Gamma(\alpha)}{i^\alpha},$$
which is the wanted identity for $z=-1/2$. For any real $z<0$, with a linear change of variable, $t=(-2z)s$ we plainly get
$$\int_0^{+\infty} e^{2izs}s^{\alpha-1}dt=\frac{\Gamma(\alpha)}{(-2iz)^\alpha}.$$
Finally, for a real positive $z>0$, we take complex conjugate to both sides to the latter identity, written for $-z$ and $\overline \alpha$, which yields to the identity for $z$ and $\alpha$.
let $z$ be a real number, $0<\alpha<1$, $L>0$ and start from the proper integral
$$\int_{0}^{L}e^{2izs}s^{\alpha-1}ds=\frac{\Gamma(\alpha)-\Gamma(\alpha,-2iLz)}{(-2iz)^{\alpha}}$$
use the asymptotics for the incomplete Gamma function: $$\quad\lim_{L\rightarrow\infty}\frac{\Gamma(\alpha,-2iLz)}{(-2iLz)^{\alpha-1}e^{2iLz}}=1\Rightarrow \lim_{L\rightarrow\infty}\Gamma(\alpha,-2iLz)=0$$
$$\Rightarrow\lim_{L\rightarrow\infty}\int_{0}^{L}e^{2izs}s^{\alpha-1}ds=\frac{\Gamma(\alpha)}{(-2iz)^{\alpha}}$$
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2019-12-12 13:37:24
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https://www.tutordale.com/pre-algebra-proportion-word-problems/
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Wednesday, February 1, 2023
# Pre Algebra Proportion Word Problems
## How To Teach Proportions
Proportion word problems – pre-algebra lesson
To introduce proportions to students, give them tables of equivalent rates to fill in, such as the one below. This will help them learn proportional reasoning.
Miles
Of course the students should notice that it is easy to fill in the table if you first figure out the unit rate then find the other amounts.
## Infinite Algebra 1 Covers All Typical Algebra Material Over 90 Topics In All From Adding And Subtracting Positives And Negatives To Solving Rational Equations
Kuta software infinite algebra 1 solving proportions answers. Kuta software infinite pre algebra name proportion word problems date period answer each question and round your answer to the nearest whole number. Round to the nearest tenth or tenth of a percent. L worksheet by kuta software llc kuta software infinite geometry name solving proportions date period solve each proportion.
1 6 2 4 p 4 3 2 4 k 8 2 1 3 n 4 8 7 32 7 4 5 3 x 4 20 3 5 m 5 7 2 35 2 6 7 4 r 5 35 4 7 7 6 5 x 30 7 8 6 5 2 5n 1 3. G v2o0q1e8z bkbunt an usfoqfmtrwcaardep tlhlpcu q b vatlfl zrzizg hktysk urreqscelrvv ewdn i n fmfamd ea pwliftdhr viinjfzixntistue lanlzgme bxr aq o1p. P i mmra 3dxe s dwrilt ghe iqndfliangiut wer 1azlugve dburga b z1h a worksheet by kuta software llc kuta software infinite algebra 1 name percent problems date period solve each problem.
1 4 2 and 20 6 2 3 2 and 18 8 3 4 3 and 16 12 4 4 3 and 8 6 5 12 24 and 3 4 6 6 9 and 2 3 solve each proportion. 1 if you can buy one can of pineapple chunks for 2 then how many can you buy with 10. Printable in convenient pdf format.
Designed for all levels of learners from remedial to advanced. 7 10 k 8 4 8 m 10 10 3 9 2 x 7 9 10 3 x 7 10 1 t g2n021 y2w ak au 2tya m ys zo 1fzt. Free pre algebra worksheets created with infinite pre algebra.
5 2 one jar of crushed ginger costs 2. Printable in convenient pdf format. Suitable for any class with algebra content.
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## Proportion Word Problems Date Period Answer Each Question And Round Your Answer To The Nearest Whole Number
Proportion word problems worksheet. Solving proportion word problems answer each question and round your answer to the nearest whole number. 1 totsakan enlarged the size of a photo to a height of 18 in. Some of the worksheets below are proportion word problems worksheet definition of proportion solving proportional problems rates ratios and proportions quiz study guide 35 proportion word problems once you find your worksheet s you can either click on the pop out icon or download button to print or download your desired worksheet s.
They will fill in the blank to complete a word based proportion. 2 a frame is 9 in wide and 6 in tall. In these worksheets your students will solve word problems that involve finding ratios and proportions.
Some of the worksheets displayed are solving proportion word problems answer each question and round your answer to the nearest nat 03 proportions word problems percent word problems percent proportion word problems ratios word problems ratio proportion. Worksheets by grade math tutorials geometry arithmetic pre algebra algebra statistics exponential decay functions resources view more. What is the new width if it was originally 2 in tall and 1 in wide.
1 if you can buy one can of pineapple chunks. If it is reduced to a width of 3 in then how tall will it be.
6 Ratio And Proportion Word Problems Worksheet With Answers Pdf In 2020 Word Problem Worksheets Addition Words Word Problems
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## How Much Water Can Ten Big Water Bottles Hold
Proportion worksheets word problems. Create your own worksheets like this one with infinite pre algebra. Solving proportion word problems involving similar figures answer each question and round your answer to the nearest whole number. Speeddistance or costamount problems.
Set up the proportionality with the given values and solve for the unknown. 75 2 Since she maintains the same rate the ratios 1 and 2 are equivalent. 3 x 36 18.
If he is going to reduce his weight in the ratio 76 find his new weight. A piglet can gain 3 pounds in 36 hours which is a rate of 1 pound for every 12 hours. Other options include using whole numbers only numbers with a certain range or numbers with a certain number of decimal digits.
Discover learning games guided lessons and other interactive activities for children. How many km are equal to 45 miles. If it is reduced to a width of 3 in then how tall will it be.
This worksheet is a PDF document. 8 10 minutes Standards Met. Proportion Word Problems Simple Worksheet About This Worksheet.
Divide both sides by 3 to solve for x. Some of the worksheets below are Proportion Word Problems Worksheet Definition of Proportion Solving Proportional Problems Rates Ratios and Proportions Quiz Study Guide 35 Proportion word problems. John has 30 marbles 18 of.
3 x 3 6483. From the above information we can get the following ratio between math problems and minutes. These problems are difficult for even adults at first.
## Grade 6 Proportions Worksheet Keywords
Direct proportion word problems worksheet. Multiplication task cards 2 digit x 2 digit word problems. Proportions word problems using decimals ratio word problems. Free worksheets for ratio word problems find here an unlimited supply of worksheets with simple word problems involving ratios meant for 6th 8th grade math.
Direct proportion word problems displaying top 8 worksheets found for this concept. In level 1 the problems ask for a specific ratio such as noah drew 9 hearts 6 stars and 12 circles. Direct proportion worksheet with answers.
Direct and inverse proportion. Some of the worksheets displayed are direct proportion direct and indirect proportions proportion and rates mathematics linear 1ma0 direct inverse proportionality solving proportions date period proportions date period answer each question and round your answer to the nearest solving proportion word problems. 1 totsakan enlarged the size of a photo to a height of 18 in.
Worksheets by grade math tutorials geometry arithmetic pre algebra algebra statistics exponential decay functions resources view more. In maths we say that two quantities are proportional if as one changes the other changes in a specific way. What is the new width if it was originally 2 in tall and 1 in wide.
Below are three versions of our grade 6 math worksheet on solving proportions word problems. Showing top 8 worksheets in the category direct proportion. If it is reduced to a width of 3 in then how tall will it be.
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## Multiplication Task Cards 2 Digit X 2 Digit Word Problems
Direct proportion word problems worksheet. What is the new width if it was originally 2 in tall and 1 in wide. If it is reduced to a width of 3 in then how tall will it be. Answers and explanations share flipboard email print grove pashley getty images math.
Proportions word problems worksheet. Direct proportion word problems displaying top 8 worksheets found for this concept. Some of the worksheets displayed are direct proportion direct and indirect proportions proportion and rates mathematics linear 1ma0 direct inverse proportionality solving proportions date period proportions date period answer each question and round your answer to the nearest solving proportion word problems.
Proportions word problems. Proportions word problems using decimals ratio word problems. Solving proportion word problems answer each question and round your answer to the nearest whole number.
2 a frame is 9 in wide and 6 in tall. Free worksheets for ratio word problems find here an unlimited supply of worksheets with simple word problems involving ratios meant for 6th 8th grade math. Worksheets by grade math tutorials geometry arithmetic pre algebra algebra statistics exponential decay functions resources view more.
Grade 6 proportions worksheet proportion word problems author. 1 totsakan enlarged the size of a photo to a height of 18 in. In maths we say that two quantities are proportional if as one changes the other changes in a specific way.
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## Solve Proportions And Their Applications
• Use the definition of proportion
• Solve proportions
• Write percent equations as proportions
• Translate and solve percent proportions
Before you get started, take this readiness quiz.
Simplify: If you missed this problem, review Example 4.44.
If you missed this problem, review Example 4.99.
Write as a rate: Sale rode his bike 24 If you missed this problem, review Example 5.63.
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## Write Percent Equations As Proportions
Previously, we solved percent equations by applying the properties of equality we have used to solve equations throughout this text. Some people prefer to solve percent equations by using the proportion method. The proportion method for solving percent problems involves a percent proportion. A percent proportion is an equation where a percent is equal to an equivalent ratio.
For example, 3 5 shows a percent equal to an equivalent ratio, we call it a percent proportion. Using the vocabulary we used earlier:
The amount is to the base as the percent is to 100
If we restate the problem in the words of a proportion, it may be easier to set up the proportion:
We could also say:
First we will practice translating into a percent proportion. Later, weâll solve the proportion.
## Solving Equations With Fractions Worksheet
Ratio and Proportion tricks – Pre-algebra with word problems (Part 1)
Some of the worksheets below are Solving Equations with Fractions Worksheet, Steps to follow when solving fractional equation, Solve equations with fractions using the Addition
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## Free Worksheets For Ratio Word Problems
Find here an unlimited supply of worksheets with simple word problems involving ratios, meant for 6th-8th grade math. In level 1, the problems ask for a specific ratio . In level 2, the problems are the same but the ratios are supposed to be simplified.
Level 3 contains varied word problems, similar to these:A bag contains 60 marbles, some blue and some green. The ratio of blue marbles to green ones is 1 : 5. How many blue marbles are there?orA truck is carrying mango juice, tomato juice, and passion fruit juice bottles in a ratio of 4 : 4 : 3. If there are 1020 passion fruit juice bottles, then how many juice bottles in total are there?
Options include choosing the number of problems, the amount of workspace, font size, a border around each problem, and more. The worksheets can be generated as PDF or html files.
## Ratios And Proportions And How To Solve Them
Let’s talk about ratios and proportions. When we talk about the speed of a car or an airplane we measure it in miles per hour. This is called a rate and is a type of ratio. A ratio is a way to compare two quantities by using division as in miles per hour where we compare miles and hours.
A ratio can be written in three different ways and all are read as “the ratio of x to y”
$$x\: to\: y$$
$$x:y$$
$$\frac$$
A proportion on the other hand is an equation that says that two ratios are equivalent. For instance if one package of cookie mix results in 20 cookies than that would be the same as to say that two packages will result in 40 cookies.
$$\frac=\frac$$
A proportion is read as “x is to y as z is to w”
$$\frac=\frac \: where\: y,w\neq 0$$
If one number in a proportion is unknown you can find that number by solving the proportion.
Example
You know that to make 20 pancakes you have to use 2 eggs. How many eggs are needed to make 100 pancakes?
100
$$\frac=\frac\: \: or\: \: \frac=\frac$$
If we write the unknown number in the nominator then we can solve this as any other equation
$$\frac=\frac$$
Multiply both sides with 100
$$}\, \frac= }\, \frac$$
$$x=\frac$$
$$x=10$$
If the unknown number is in the denominator we can use another method that involves the cross product. The cross product is the product of the numerator of one of the ratios and the denominator of the second ratio. The cross products of a proportion is always equal
$$\frac }} }}=\frac }} }}$$
$$xw=yz$$
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## Tap On Print Pdf Or Image Button To Print Or Download This Grade 6 Ratio Proportion Worksheet To Practice How To Find The Quantity Based On The Ratio
Grade 9 ratio and proportion word problems worksheet with answers. What is the new width if it was originally 2 in tall and 1 in wide. Grade 9 ratio maths problems with answers are presented. Some word problems may require you to find the ratio based on the increase or decrease in quantity and vice versa.
The ratio of boys to girls in this school is 3 5. D b 8mvaod ieh ew0i ot rhc bi8n fiiynzi vt ker 9p wrceb iahl2gpe lb qrva7 g worksheet by kuta software llc kuta software infinite pre algebra name proportion word problems date period answer each question and round your answer to the nearest whole number. Some of the worksheets below are proportion word problems worksheet definition of proportion solving proportional problems rates ratios and proportions quiz study guide 35 proportion word problems once you find your worksheet s you can either click on the pop out icon or download button to print or download your desired worksheet s.
1 totsakan enlarged the size of a photo to a height of 18 in. Solving proportion word problems answer each question and round your answer to the nearest whole number. Ratio and proportion word problems.
Solutions and detailed explanations are also included. In level 1 the problems ask for a specific ratio such as noah drew 9 hearts 6 stars and 12 circles. If it is reduced to a width of 3 in then how tall will it be.
2 a frame is 9 in wide and 6 in tall.
Account Suspended Fraction Word Problems Word Problems Word Problem Worksheets
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## The Next Step: Proportion Problems And Thinking
After studying tables of equivalent rates, the students are ready to tackle word problems. Choose simple ones at first, and let them think! They might very well come up with an answer on their own by making a table or by figuring out the unit rate. So… you don’t actually need to write an actual proportion to solve a proportion word problem.
However, I don’t want to put down equations or cross-multiplying students studying algebra and pre-algebra courses still need to learn to solve proportions with cross-multiplying. It’s just that learning to use common sense is even more important.
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## Inverse Proportion Word Problem Gradelevel
Proportion word problems worksheet. Worksheet 1 Worksheet 2 Worksheet 3. 4 It took Nora 10 hours to walk a 30-mile trail. Inverse proportion word problem Add to my workbooks 12 Download file pdf Embed in.
Proportion Word Problems Hard Worksheet About This Worksheet. Below are three versions of our grade 6 math worksheet on solving proportions word problems. These are most useful when students are first learning proportions in 6th 7th and 8th grade.
PROPORTION WORD PROBLEMS WORKSHEET. 2 or by expressing the ratio as a fraction. Divide both sides of the equation by 3 to solve for x.
Inverse proportion word problem Other contents. Speeddistance or costamount problems available both as PDF and html files. 3 x 3 503.
These worksheets are pdf files. Working and finding the ratios between numbers is both challenging and exciting. If you can buy one can of pineapple chunks for 2 then how many can you buy with 10.
3 Four big water bottles can hold 8 gallons of water. Proportion worksheets create proportion worksheets to solve proportions or word problems eg. They will compare two ratios to determine which is bigger.
Students will find values when given a ratio. 8 10 minutes Standards Met. Shawna reduced the size of a rectangle to a height of 2 in.
What is the new width if it was originally 24 in wide and 12 in tall. Grade 6 Proportions Worksheet 1 24 loaves of bread cost 48. 10 12 minutes Standards Met.
## If The Model Plane Is 3 In Tall Then How Tall Is The
Proportion word problems worksheet 2 answer key. 1 if a 6 ft tall tent casts a 10 ft long shadow then how long is the shadow that a 9 ft tall adult elephant casts. Solving proportion word problems involving similar figures answer each question and round your answer to the nearest whole number. Area and perimeter worksheets.
Solving proportion word problems answer each question and round your answer to the nearest whole number. Free worksheets for ratio word problems find here an unlimited supply of worksheets with simple word problems involving ratios meant for 6th 8th grade math. Worksheet by kuta software llc intermediate algebra proportion word problems name id.
Word problems on sets and venn diagrams. 1 totsakan enlarged the size of a photo to a height of 18 in. This is the currently selected item.
What is the new width if it was originally 2 in tall and 1 in wide. How many circles does it make in a minute. She has to make a total of 216 cookies for a wedding party.
2 a model plane has a scale of 1 in. When juggling a ball travels in a complete circle every 2 seconds. In level 1 the problems ask for a specific ratio such as noah drew 9 hearts 6 stars and 12 circles.
Yes you can solve an incorrectly set up equation and find an answer. How long will it take her. Translating a word problem into an equation is super important.
Equations of proportional relationships. Complementary and supplementary word problems worksheet. Multi step ratio and percent problems.
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2023-02-06 12:45:56
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https://solvedlib.com/n/question-13-kpa-0-8-p-b-0-5-ad-a-and-bare-independent-events,15458371
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QUESTION 13 KPA) = 0.8 P(B) = 0.5 ad A and Bare independent events, what is P(BIA)? 0 0.50.80.40.1
Question:
QUESTION 13 KPA) = 0.8 P(B) = 0.5 ad A and Bare independent events, what is P(BIA)? 0 0.5 0.8 0.4 0.1
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2022-05-20 16:37:30
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https://www.physicsforums.com/threads/quantum-mechanics-for-mathematicians-book.374016/
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# Quantum Mechanics for Mathematicians (book)
Hello,
Has anyone read 'Quantum Mechanics for Mathematicians'?
I've read a few of the other books in this series, and they've been outstanding. But this one is pretty new and I can't find any reviews. And I'm not a physicist, so it's not so easy for me to jdge. Thanks for any help.
Last edited by a moderator:
Landau
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2021-12-01 16:38:14
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https://socratic.org/questions/52d418c802bf346963eb9857
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# Question b9857
Jan 16, 2014
For each element, you divide the total mass of its atoms by the empirical formula mass and multiply by 100.
#### Explanation:
EXAMPLE
What is the percent composition of a compound with the empirical formula $\text{CH"_2"O}$?
Solution
Step 1: Find the masses of the individual atoms.
$\text{C = 12.01 u}$
$\text{H = 1.008 u}$
$\text{O = 16.00 u}$
Step 2: Find the total mass of each atom in the formula.
$\text{1 C = 12.01 u}$
$\text{2 H = 2.016 u}$
$\text{1 O = 16.00 u}$
Step 3: Find the total mass of all the atoms in the formula.
$\text{CH"_2"O = (12.01 + 2.016 + 16.00) u = 30.03 u}$
Step 4: Find the mass percent of each atom.
"mass %" = "mass of atom"/"total mass" × "100 %#
$\text{mass % C" = (12.01 color(red)(cancel(color(black)("u"))))/(30.03 color(red)(cancel(color(black)("u")))) × "100 %" = "40.00 %}$
$\text{mass % H" = (2.016 color(red)(cancel(color(black)("u"))))/(30.03 color(red)(cancel(color(black)("u")))) × "100 %" = "6.714 %}$
$\text{mass % O" = (16.00 color(red)(cancel(color(black)("u"))))/(30.03 color(red)(cancel(color(black)("u")))) × "100 %" = "53.29 %}$
$40.00 + 6.714 + 53.29 = 100.00$
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2020-10-31 16:55:32
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http://math.stackexchange.com/questions/29421/analytic-functions-with-nonessential-singularity-at-infinity-must-be-a-polynomia/29422
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# Analytic functions with nonessential singularity at infinity must be a polynomial
This is an exercise from Alhfors Complex Analysis book- to show that an analytic function with a nonessential singularity at infinity must be a polynomial. It seems like it should probably be pretty straight forward, but I must be missing something. If it has a removable singularity at infinity than it extends to an analytic function on the Riemann sphere, and so must be constant by Liouville's theorem. What if there is a pole at infinity though? This was homework some time ago, and I never finished it :/ but have been thinking about it again recently. Thanks :)
-
Another hint: look at the function $f(\frac{1}{z})$ at z = 0, it has a nonessential singularity at 0...
Hint: consider the Laurent series in the annulus $0 < |z| < \infty$.
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2014-03-08 18:29:10
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https://aimsciences.org/article/doi/10.3934/dcdsb.2018277
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# American Institute of Mathematical Sciences
August 2019, 24(8): 3537-3556. doi: 10.3934/dcdsb.2018277
## Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems
1 Kharkevich Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoj Karetny lane 19, Moscow 127051, Russia 2 Kotel'nikov Institute of Radio-engineering and Electronics, Russian Academy of Sciences, Mokhovaya 11-7, Moscow 125009, Russia
Dedicated to Peter Kloeden on the occasion of his 70th birthday, friendship with whom refutes the thesis that "East is East, and West is West, and never the twain shall meet"
Received January 2018 Revised April 2018 Published October 2018
Fund Project: The author is supported by the Russian Science Foundation, Project number 16-11-00063
To estimate the growth rate of matrix products $A_{n}··· A_{1}$ with factors from some set of matrices $\mathscr{A}$, such numeric quantities as the joint spectral radius $ρ(\mathscr{A})$ and the lower spectral radius $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})$ are traditionally used. The first of these quantities characterizes the maximum growth rate of the norms of the corresponding products, while the second one characterizes the minimal growth rate. In the theory of discrete-time linear switching systems, the inequality $ρ(\mathscr{A})<1$ serves as a criterion for the stability of a system, and the inequality $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over \rho } (\mathscr{A})<1$ as a criterion for stabilizability.
Given a set $\mathscr{A}$ of $N×M$ matrices and a set $\mathscr{B}$ of $M×N$ matrices. Then, for matrix products $A_{n}B_{n}··· A_{1}B_{1}$ with factors $A_{i}∈\mathscr{A}$ and $B_{i}∈\mathscr{B}$, we introduce the quantities $μ(\mathscr{A},\mathscr{B})$ and $η(\mathscr{A},\mathscr{B})$, called the lower and upper minimax joint spectral radius of the pair $\{\mathscr{A},\mathscr{B}\}$, respectively, which characterize the maximum growth rate of the matrix products $A_{n}B_{n}··· A_{1}B_{1}$ over all sets of matrices $A_{i}∈\mathscr{A}$ and the minimal growth rate over all sets of matrices $B_{i}∈\mathscr{B}$. In this sense, the minimax joint spectral radii can be considered as generalizations of both the joint and lower spectral radii. As an application of the minimax joint spectral radii, it is shown how these quantities can be used to analyze the stabilizability of discrete-time linear switching control systems in the presence of uncontrolled external disturbances of the plant.
Citation: Victor Kozyakin. Minimax joint spectral radius and stabilizability of discrete-time linear switching control systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3537-3556. doi: 10.3934/dcdsb.2018277
##### References:
[1] E. Asarin, J. Cervelle, A. Degorre, C. Dima, F. Horn and V. Kozyakin, Entropy games and matrix multiplication games, in 33rd Symposium on Theoretical Aspects of Computer Science, (STACS 2016) (eds. N. Ollinger and H. Vollmer), vol. 47 of LIPIcs. Leibniz Int. Proc. Inform., Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2016, 14pp. doi: 10.4230/LIPIcs.STACS.2016.11. Google Scholar [2] M. A. Berger and Y. Wang, Bounded semigroups of matrices, Linear Algebra Appl., 166 (1992), 21-27. doi: 10.1016/0024-3795(92)90267-E. Google Scholar [3] V. D. Blondel and Y. Nesterov, Polynomial-time computation of the joint spectral radius for some sets of nonnegative matrices, SIAM J. Matrix Anal. Appl., 31 (2009), 865-876. doi: 10.1137/080723764. Google Scholar [4] J. Bochi and I. D. Morris, Continuity properties of the lower spectral radius, Proc. Lond. Math. Soc. (3), 110 (2015), 477-509. doi: 10.1112/plms/pdu058. Google Scholar [5] T. Bousch and J. Mairesse, Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture, J. Amer. Math. Soc., 15 (2002), 77-111. doi: 10.1090/S0894-0347-01-00378-2. Google Scholar [6] P. Bouyer, N. Markey, M. Randour, K. G. Larsen and S. Laursen, Average-energy games, Acta Informatica, 55 (2018), 91-127. doi: 10.1007/s00236-016-0274-1. Google Scholar [7] A. Czornik, On the generalized spectral subradius, Linear Algebra Appl., 407 (2005), 242-248. doi: 10.1016/j.laa.2005.05.006. Google Scholar [8] X. Dai, Y. Huang, J. Liu and M. Xiao, The finite-step realizability of the joint spectral radius of a pair of d×d matrices one of which being rank-one, Linear Algebra Appl., 437 (2012), 1548-1561. doi: 10.1016/j.laa.2012.04.053. Google Scholar [9] X. Dai, A Gel'fand-type spectral-radius formula and stability of linear constrained switching systems, Linear Algebra Appl., 436 (2012), 1099-1113. doi: 10.1016/j.laa.2011.07.029. Google Scholar [10] X. Dai, Some criteria for spectral finiteness of a finite subset of the real matrix space $\mathbb{R}^{d× d}$, Linear Algebra Appl., 438 (2013), 2717-2727. doi: 10.1016/j.laa.2012.09.026. Google Scholar [11] X. Dai, Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations, J. Franklin Inst., 351 (2014), 2910-2937. doi: 10.1016/j.jfranklin.2014.01.010. Google Scholar [12] X. Dai, Y. Huang and M. Xiao, Periodically switched stability induces exponential stability of discrete-time linear switched systems in the sense of Markovian probabilities, Automatica J. IFAC, 47 (2011), 1512-1519. doi: 10.1016/j.automatica.2011.02.034. Google Scholar [13] X. Dai, Y. Huang and M. Xiao, Pointwise stability of descrete-time stationary matrix-valued Markovian processes, IEEE Trans. Automat. Control, 60 (2015), 1898-1903. doi: 10.1109/TAC.2014.2361594. Google Scholar [14] I. Daubechies and J. C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl., 161 (1992), 227-263. doi: 10.1016/0024-3795(92)90012-Y. Google Scholar [15] M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z., 17 (1923), 228-249. doi: 10.1007/BF01504345. Google Scholar [16] L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl., 231 (1995), 47-85. doi: 10.1016/0024-3795(95)90006-3. Google Scholar [17] C. Heil and G. Strang, Continuity of the joint spectral radius: application to wavelets, in Linear algebra for signal processing (Minneapolis, MN, 1992), vol. 69 of IMA Vol. Math. Appl., Springer, New York, 1995, 51-61. doi: 10.1007/978-1-4612-4228-4_4. Google Scholar [18] R. Jungers, The Joint Spectral Radius, vol. 385 of Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, 2009, Theory and applications. doi: 10.1007/978-3-540-95980-9. Google Scholar [19] R. M. Jungers, On asymptotic properties of matrix semigroups with an invariant cone, Linear Algebra Appl., 437 (2012), 1205-1214. doi: 10.1016/j.laa.2012.04.006. Google Scholar [20] R. M. Jungers and P. Mason, On feedback stabilization of linear switched systems via switching signal control, SIAM J. Control Optim., 55 (2017), 1179-1198. doi: 10.1137/15M1027802. Google Scholar [21] V. S. Kozyakin, On the absolute stability of systems with asynchronously operating pulse elements, Avtomat. i Telemekh., 1990, 56-63, In Russian, translation in Automat. Remote Control, 51 (1990), 1349-1355 (1991). Google Scholar [22] V. S. Kozyakin, Constructive stability and stabilizability of positive linear discrete-time switching systems, Journal of Communications Technology and Electronics, 62 (2017), 686-693. doi: 10.1134/S1064226917060110. Google Scholar [23] V. Kozyakin, An explicit Lipschitz constant for the joint spectral radius, Linear Algebra Appl., 433 (2010), 12-18. doi: 10.1016/j.laa.2010.01.028. Google Scholar [24] V. Kozyakin, An Annotated Bibliography on Convergence of Matrix Products and the Theory of Joint/Generalized Spectral Radius, Preprint, Institute for Information Transmission Problems, Moscow, 2013. doi: 10.13140/2.1.4257.5040. Google Scholar [25] V. Kozyakin, Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices, Linear Algebra Appl., 489 (2016), 167-185. doi: 10.1016/j.laa.2015.10.017. Google Scholar [26] V. Kozyakin, Minimax theorem for the spectral radius of the product of non-negative matrices, Linear and Multilinear Algebra, 65 (2017), 2356-2365. doi: 10.1080/03081087.2016.1273877. Google Scholar [27] V. Kozyakin, On convergence of infinite matrix products with alternating factors from two sets of matrices, Discrete Dyn. Nat. Soc., 2018 (2018), Art. ID 9216760, 5 pp. doi: 10.1155/2018/9216760. Google Scholar [28] H. Lin and P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Automat. Control, 54 (2009), 308-322. doi: 10.1109/TAC.2008.2012009. Google Scholar [29] G.-C. Rota and G. Strang, A note on the joint spectral radius, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math., 22 (1960), 379-381. doi: 10.1016/S1385-7258(60)50046-1. Google Scholar [30] M.-H. Shih, J.-W. Wu and C.-T. Pang, Asymptotic stability and generalized Gelfand spectral radius formula, Linear Algebra Appl., 252 (1997), 61-70. doi: 10.1016/0024-3795(95)00592-7. Google Scholar [31] D. P. Stanford, Stability for a multi-rate sampled-data system, SIAM J. Control Optim., 17 (1979), 390-399. doi: 10.1137/0317029. Google Scholar [32] D. P. Stanford and J. M. Urbano, Some convergence properties of matrix sets, SIAM J. Matrix Anal. Appl., 15 (1994), 1132-1140. doi: 10.1137/S0895479892228213. Google Scholar [33] Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Communications and Control Engineering, Springer, London, 2005. doi: 10.1007/1-84628-131-8. Google Scholar [34] J. Theys, Joint Spectral Radius: Theory and Approximations, PhD thesis, Faculté des sciences appliquées, Département d'ingénierie mathématique, Center for Systems Engineering and Applied Mechanics, Université Catholique de Louvain, 2005.Google Scholar [35] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, 2nd edition, Princeton University Press, Princeton, N. J., 1947. Google Scholar [36] F. Wirth, The generalized spectral radius and extremal norms, Linear Algebra Appl., 342 (2002), 17-40. doi: 10.1016/S0024-3795(01)00446-3. Google Scholar
show all references
##### References:
[1] E. Asarin, J. Cervelle, A. Degorre, C. Dima, F. Horn and V. Kozyakin, Entropy games and matrix multiplication games, in 33rd Symposium on Theoretical Aspects of Computer Science, (STACS 2016) (eds. N. Ollinger and H. Vollmer), vol. 47 of LIPIcs. Leibniz Int. Proc. Inform., Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany, 2016, 14pp. doi: 10.4230/LIPIcs.STACS.2016.11. Google Scholar [2] M. A. Berger and Y. Wang, Bounded semigroups of matrices, Linear Algebra Appl., 166 (1992), 21-27. doi: 10.1016/0024-3795(92)90267-E. Google Scholar [3] V. D. Blondel and Y. Nesterov, Polynomial-time computation of the joint spectral radius for some sets of nonnegative matrices, SIAM J. Matrix Anal. Appl., 31 (2009), 865-876. doi: 10.1137/080723764. Google Scholar [4] J. Bochi and I. D. Morris, Continuity properties of the lower spectral radius, Proc. Lond. Math. Soc. (3), 110 (2015), 477-509. doi: 10.1112/plms/pdu058. Google Scholar [5] T. Bousch and J. Mairesse, Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture, J. Amer. Math. Soc., 15 (2002), 77-111. doi: 10.1090/S0894-0347-01-00378-2. Google Scholar [6] P. Bouyer, N. Markey, M. Randour, K. G. Larsen and S. Laursen, Average-energy games, Acta Informatica, 55 (2018), 91-127. doi: 10.1007/s00236-016-0274-1. Google Scholar [7] A. Czornik, On the generalized spectral subradius, Linear Algebra Appl., 407 (2005), 242-248. doi: 10.1016/j.laa.2005.05.006. Google Scholar [8] X. Dai, Y. Huang, J. Liu and M. Xiao, The finite-step realizability of the joint spectral radius of a pair of d×d matrices one of which being rank-one, Linear Algebra Appl., 437 (2012), 1548-1561. doi: 10.1016/j.laa.2012.04.053. Google Scholar [9] X. Dai, A Gel'fand-type spectral-radius formula and stability of linear constrained switching systems, Linear Algebra Appl., 436 (2012), 1099-1113. doi: 10.1016/j.laa.2011.07.029. Google Scholar [10] X. Dai, Some criteria for spectral finiteness of a finite subset of the real matrix space $\mathbb{R}^{d× d}$, Linear Algebra Appl., 438 (2013), 2717-2727. doi: 10.1016/j.laa.2012.09.026. Google Scholar [11] X. Dai, Robust periodic stability implies uniform exponential stability of Markovian jump linear systems and random linear ordinary differential equations, J. Franklin Inst., 351 (2014), 2910-2937. doi: 10.1016/j.jfranklin.2014.01.010. Google Scholar [12] X. Dai, Y. Huang and M. Xiao, Periodically switched stability induces exponential stability of discrete-time linear switched systems in the sense of Markovian probabilities, Automatica J. IFAC, 47 (2011), 1512-1519. doi: 10.1016/j.automatica.2011.02.034. Google Scholar [13] X. Dai, Y. Huang and M. Xiao, Pointwise stability of descrete-time stationary matrix-valued Markovian processes, IEEE Trans. Automat. Control, 60 (2015), 1898-1903. doi: 10.1109/TAC.2014.2361594. Google Scholar [14] I. Daubechies and J. C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl., 161 (1992), 227-263. doi: 10.1016/0024-3795(92)90012-Y. Google Scholar [15] M. Fekete, Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z., 17 (1923), 228-249. doi: 10.1007/BF01504345. Google Scholar [16] L. Gurvits, Stability of discrete linear inclusion, Linear Algebra Appl., 231 (1995), 47-85. doi: 10.1016/0024-3795(95)90006-3. Google Scholar [17] C. Heil and G. Strang, Continuity of the joint spectral radius: application to wavelets, in Linear algebra for signal processing (Minneapolis, MN, 1992), vol. 69 of IMA Vol. Math. Appl., Springer, New York, 1995, 51-61. doi: 10.1007/978-1-4612-4228-4_4. Google Scholar [18] R. Jungers, The Joint Spectral Radius, vol. 385 of Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, 2009, Theory and applications. doi: 10.1007/978-3-540-95980-9. Google Scholar [19] R. M. Jungers, On asymptotic properties of matrix semigroups with an invariant cone, Linear Algebra Appl., 437 (2012), 1205-1214. doi: 10.1016/j.laa.2012.04.006. Google Scholar [20] R. M. Jungers and P. Mason, On feedback stabilization of linear switched systems via switching signal control, SIAM J. Control Optim., 55 (2017), 1179-1198. doi: 10.1137/15M1027802. Google Scholar [21] V. S. Kozyakin, On the absolute stability of systems with asynchronously operating pulse elements, Avtomat. i Telemekh., 1990, 56-63, In Russian, translation in Automat. Remote Control, 51 (1990), 1349-1355 (1991). Google Scholar [22] V. S. Kozyakin, Constructive stability and stabilizability of positive linear discrete-time switching systems, Journal of Communications Technology and Electronics, 62 (2017), 686-693. doi: 10.1134/S1064226917060110. Google Scholar [23] V. Kozyakin, An explicit Lipschitz constant for the joint spectral radius, Linear Algebra Appl., 433 (2010), 12-18. doi: 10.1016/j.laa.2010.01.028. Google Scholar [24] V. Kozyakin, An Annotated Bibliography on Convergence of Matrix Products and the Theory of Joint/Generalized Spectral Radius, Preprint, Institute for Information Transmission Problems, Moscow, 2013. doi: 10.13140/2.1.4257.5040. Google Scholar [25] V. Kozyakin, Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices, Linear Algebra Appl., 489 (2016), 167-185. doi: 10.1016/j.laa.2015.10.017. Google Scholar [26] V. Kozyakin, Minimax theorem for the spectral radius of the product of non-negative matrices, Linear and Multilinear Algebra, 65 (2017), 2356-2365. doi: 10.1080/03081087.2016.1273877. Google Scholar [27] V. Kozyakin, On convergence of infinite matrix products with alternating factors from two sets of matrices, Discrete Dyn. Nat. Soc., 2018 (2018), Art. ID 9216760, 5 pp. doi: 10.1155/2018/9216760. Google Scholar [28] H. Lin and P. J. Antsaklis, Stability and stabilizability of switched linear systems: A survey of recent results, IEEE Trans. Automat. Control, 54 (2009), 308-322. doi: 10.1109/TAC.2008.2012009. Google Scholar [29] G.-C. Rota and G. Strang, A note on the joint spectral radius, Nederl. Akad. Wetensch. Proc. Ser. A 63 = Indag. Math., 22 (1960), 379-381. doi: 10.1016/S1385-7258(60)50046-1. Google Scholar [30] M.-H. Shih, J.-W. Wu and C.-T. Pang, Asymptotic stability and generalized Gelfand spectral radius formula, Linear Algebra Appl., 252 (1997), 61-70. doi: 10.1016/0024-3795(95)00592-7. Google Scholar [31] D. P. Stanford, Stability for a multi-rate sampled-data system, SIAM J. Control Optim., 17 (1979), 390-399. doi: 10.1137/0317029. Google Scholar [32] D. P. Stanford and J. M. Urbano, Some convergence properties of matrix sets, SIAM J. Matrix Anal. Appl., 15 (1994), 1132-1140. doi: 10.1137/S0895479892228213. Google Scholar [33] Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Communications and Control Engineering, Springer, London, 2005. doi: 10.1007/1-84628-131-8. Google Scholar [34] J. Theys, Joint Spectral Radius: Theory and Approximations, PhD thesis, Faculté des sciences appliquées, Département d'ingénierie mathématique, Center for Systems Engineering and Applied Mechanics, Université Catholique de Louvain, 2005.Google Scholar [35] J. von Neumann and O. Morgenstern, Theory of Games and Economic Behavior, 2nd edition, Princeton University Press, Princeton, N. J., 1947. Google Scholar [36] F. Wirth, The generalized spectral radius and extremal norms, Linear Algebra Appl., 342 (2002), 17-40. doi: 10.1016/S0024-3795(01)00446-3. Google Scholar
Discrete-time linear switching system
Control system consisting of plant $\mathit{\boldsymbol{ \boldsymbol{\mathscr{A}} }}$ and controller $\mathit{\boldsymbol{ \boldsymbol{\mathscr{B}} }}$
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2018 Impact Factor: 1.008
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2019-09-21 02:27:04
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http://www.enoriver.net/stata/2008/10/02/getting-data-into-stata-odbc-load/
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ODBC stands for Open Database Connectivity; odbc load is a tool for getting data out of ODBC-ready applications and into Stata quickly and reliably. One such application is Microsoft Excel. If you haven't yet received data in an Excel workbook, you will.
Just to get it out of the way: if you run Stata 10 and you receive data from somebody using one of the newer, XML-style Excel versions, you don't need odbc load for that. Use the command xmluse instead. You can tell by the file name extension. Whenever Excel went XML (as of Office 2007 perhaps? I've been using OOo for a while now, so I'm not sure) it started saving workbooks with the .xlsx extension. The notes below apply to getting data from old-school -- .xls -- workbooks into Stata.
So, back to odbc load. Its syntax is a bit arcane; for that reason, sometimes more code is preferred to less. You can split this command into components with local macros, for example. Well-chosen local macros make it much easier to re-use code or organize it into modules that can be maintained by different people working simultaneously.
Stata may still be the preserve of the lone researcher, or the grad student working for the typical PI, which is the same thing, and for them these niceties don't matter as much. But the private sector has highly structured ways and Stata can roll with that just as well as it does with the more freewheeling style of the academia. So here's how I propose that odbc load should be run for getting data out of an Excel workbook:
// your customers can edit this local my_path "D:/data/My Project/Source Files/" local my_workbook "workbook.xls" local my_file "Sheet1" // without worrying about any of this local source "Excel Files;DBQ=my_path'myworkbook'" clear odbc load, dsn("source'") table("my_file'\$") tempfile my_file compress describe save "my_file'", replace
And that's all for MS Excel. Your temporary file "my_file'" has all the data on worksheet Sheet1. If you ever need to pass this code on to other people, they will appreciate that they only need to edit the local macros with the path, workbook and worksheet that are right for them, and everything else works unchanged. You can use odbc load for getting data out of MS Access or a MySQL database. You can even embed SQL code. More on that here.
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2018-06-25 06:03:39
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https://www.beatthegmat.com/agela-has-15-pairs-of-matched-socks-if-she-loses-7-t300893.html
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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## Agela has 15 pairs of matched socks. If she loses 7... tagged by: AAPL ##### This topic has 4 expert replies and 0 member replies ### Top Member ## Agela has 15 pairs of matched socks. If she loses 7... Angela has 15 pairs of matched socks. If she loses 7 individual socks, which of the following is NOT a possible number of matched pairs she has left? A. 8 B. 9 C. 10 D. 11 E. 12 The OA is E. I think if Angela loses 7 individual socks, then she is bound to have lost minimum of 3 pairs and 1 individual sock, in this way she is left with only 11 pairs of socks (15-(3+1)). Hence, 12 cannot be the answer as maximum is 11. Can any experts help me, please? Thank in advance! ### GMAT/MBA Expert GMAT Instructor Joined 08 Dec 2008 Posted: 13033 messages Followed by: 1251 members Upvotes: 5254 GMAT Score: 770 Top Reply AAPL wrote: Angela has 15 pairs of matched socks. If she loses 7 individual socks, which of the following is NOT a possible number of matched pairs she has left? A. 8 B. 9 C. 10 D. 11 E. 12 We can also systematically eliminate 4 of the 5 answer choices. Let's say the 15 PAIRS of socks are as follows: AA, BB, CC, DD, EE, FF, GG, HH, II, JJ, KK, LL, MM, NN, OO Let's first see what happens if we "lose" 7 unmatched socks. Say, we lose, A, B, C, D, E, F, G We get: A, B, C, D, E, F, G, HH, II, JJ, KK, LL, MM, NN, OO We have 8 pairs remaining. So, we can ELIMINATE A Now let's see what happens if we "lose" 1 pair of matched socks and 5 unmatched socks. Say, we lose, AA, B, C, D, E, F We get: B, C, D, E, F, GG, HH, II, JJ, KK, LL, MM, NN, OO We have 9 pairs remaining. So, we can ELIMINATE B Let's see what happens if we "lose" 2 pairs of matched socks and 3 unmatched socks. Say, we lose, AA, BB, C, D, E We get: C, D, E, FF, GG, HH, II, JJ, KK, LL, MM, NN, OO We have 10 pairs remaining. So, we can ELIMINATE C Let's see what happens if we "lose" 3 pairs of matched socks and 1 unmatched sock. Say, we lose, AA, BB, CC, D We get: D, EE, FF, GG, HH, II, JJ, KK, LL, MM, NN, OO We have 11 pairs remaining. So, we can ELIMINATE D By the process of elimination, the correct answer is E Cheers, Brent _________________ Brent Hanneson – Creator of GMATPrepNow.com Use my video course along with Sign up for free Question of the Day emails And check out all of these free resources GMAT Prep Now's comprehensive video course can be used in conjunction with Beat The GMAT’s FREE 60-Day Study Guide and reach your target score in 2 months! ### GMAT/MBA Expert Legendary Member Joined 14 Jan 2015 Posted: 2666 messages Followed by: 125 members Upvotes: 1153 GMAT Score: 770 AAPL wrote: Angela has 15 pairs of matched socks. If she loses 7 individual socks, which of the following is NOT a possible number of matched pairs she has left? A. 8 B. 9 C. 10 D. 11 E. 12 The OA is E. I think if Angela loses 7 individual socks, then she is bound to have lost minimum of 3 pairs and 1 individual sock, in this way she is left with only 11 pairs of socks (15-(3+1)). Hence, 12 cannot be the answer as maximum is 11. Can any experts help me, please? Thank in advance! You could also think about it like this: She starts with 15 pairs, or 15*2 = 30 socks. She loses 7 leaving her with 23 socks. In order to have 12 pairs, she'd have to have 12*2 = 24 socks. But she's only got 23!. So the answer is E _________________ Veritas Prep | GMAT Instructor Veritas Prep Reviews Save$100 off any live Veritas Prep GMAT Course
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AAPL wrote:
Angela has 15 pairs of matched socks. If she loses 7 individual socks, which of the following is NOT a possible number of matched pairs she has left?
A. 8
B. 9
C. 10
D. 11
E. 12
15 matched pairs = 30 socks.
E: 12
If Angela has 12 matched pairs left -- for a total of 24 socks -- then the number of socks lost = 30-24 = 6.
Not viable, since Angela must lose 7 socks.
_________________
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### GMAT/MBA Expert
GMAT Instructor
Joined
25 May 2010
Posted:
15380 messages
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790
GMATGuruNY wrote:
AAPL wrote:
Angela has 15 pairs of matched socks. If she loses 7 individual socks, which of the following is NOT a possible number of matched pairs she has left?
A. 8
B. 9
C. 10
D. 11
E. 12
When the prompt includes the phrase which of the following, the correct answer is likely to be D or E.
15 matched pairs = 30 socks.
E: 12
If Angela has 12 matched pairs left -- for a total of 24 socks -- then the number of socks lost = 30-24 = 6.
Not viable, since Angela must lose 7 socks.
_________________
Mitch Hunt
Private Tutor for the GMAT and GRE
GMATGuruNY@gmail.com
If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon.
Available for tutoring in NYC and long-distance.
Student Review #1
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Student Review #3
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2019-07-16 04:43:32
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|
https://www.nature.com/articles/s41598-020-60104-4?error=cookies_not_supported&code=61570614-63f0-44cc-be7f-fc937044753b
|
## Introduction
Rising levels of atmospheric CO2 are causing increases in air and sea surface temperatures (SSTs), with the mean SST predicted to rise by 1.4 °C to 4.8 °C by 21001. With global warming, extreme high temperature events such as marine heat waves (MHWs) have also increased in frequency, intensity and duration along the World’s coastline, including the Mediterranean, Australia and Brazilian Atlantic sea2,3,4,5,6. These anomalous elevated temperatures have negatively impacted marine and terrestrial ecosystems by altering species’ composition and distribution patterns7,8,9. Such ecological changes are also severely impacting ecosystem goods and services such as fisheries, and carbon sequestration and storage10. The impacts of warming are considerably larger in marine systems because of their greater sensitivity to these global stressors compared to terrestrial systems11,12. Because of this, there is rising concern about the capacity of marine species to acclimate quickly enough to short-term variability in temperature, which will be critical for organisms to adapt and survive in a changing ocean13,14.
In ectothermic organisms such as plants, algae, invertebrates and lower vertebrates, temperature is the major factor regulating their physiology, growth, performance and fitness15,16,17,18,19,20. Therefore, changes in environmental temperatures (Ta) due to climate change will trigger modifications at physiological and biochemical levels, influencing whole-organism thermal plasticity (i.e. thermal sensitivities and tolerance)21. These effects might be more pronounced for sessile organisms (e.g. macroalgae) than mobile ones (e.g., fish and planktonic taxa) as they are unable to escape from stressful environmental conditions, pushing them beyond their acclimation capacities22.
The effects Ta on biological rate processes are characterized by thermal performance curves (TPCs) (reaction norms)16,23. TPCs have helped to understand the effects of temperature on biological systems24 and to describe the thermal ecology, phenotypic plasticity and evolution of ectotherms21,25,26. This approach has recently been used to predict the effects of warming and short-term extreme high temperature events on physiological and ecological attributes of natural populations24,27,28. With warming occurring faster than predicted29, understanding the effects of temperature on key marine organisms such as habitat-forming species (e.g., seagrasses and macroalgae) is becoming increasingly urgent since their decline or disappearance can have substantial consequences through the entire ecosystem8,30,31.
Marine macroalgae (seaweeds) contribute 5–10% of global primary production and play structural and functional roles in coastal marine ecosystems, creating one of the most diverse and productive systems in the world32,33,34. Kelps (Order Laminariales) form subtidal forests in temperate and polar waters, contributing to carbon storage, macronutrient dynamics, and the biodiversity of many associated species35. However, a clear decline (38%) in these habitat-forming species has been observed in the past 20 years across the world36,37,38,39,40. Climate change, along with other human impacts such as overfishing and direct harvest, are considered the main reasons for this decline39. However, region-specific responses were also observed, suggesting that kelp’s performance is influenced by a combination of global (climate change) and local environmental drivers. Kelps are cold water-adapted species and hence vulnerable to high temperatures: As for other ectotherms, temperature exerts a large effect on their growth, survival, and reproduction41,42,43,44.
In addition to ongoing climate change, coastal ecosystems are threatened by local changes such as eutrophication45,46. The interactions between local and global drivers can be difficult to predict as these interactions can range from additive, to synergistic to antagonistic39. However, understanding such interactions is critical to predicting how seaweeds will fare in a future ocean. The carbon and nitrogen metabolisms in algae are tightly coupled47,48,49,50. Nitrogen plays important roles in regulating key enzymatic activities51, and is a key regulator of seaweed productivity via direct effects on cell membrane fluidity, protein production, photosynthetic machinery49,52, and thermal plasticity53. Therefore, nitrogen enrichment might ameliorate the negative effect of high temperature on seaweeds’ performance by modulating their photosynthetic and respiratory rates5,15,53,54,55,56,57. However, such positive synergistic effects can be diminished at temperatures that surpass algal thermal thresholds5,56, and might vary among species and populations58.
The giant kelp Macrocystis pyrifera (hereafter, Macrocystis) plays critical functional roles in coastal ecosystems59. However, in Tasmania, south eastern Australia, there has been a 90% decline in Macrocystis underwater forests60. In this region, Macrocystis was historically exposed to temperatures from ~12 °C to ~18 °C61, which agrees with the thermal window for the species across other regions62. However, increased warming together with progressively more oligotrophic waters (<1 µM NO3), due to the southern shift of the East Australian current (EAC), are thought to have caused the decline in Macrocystis60. Several studies (mostly from the Pacific Northwest) have evaluated the single effects of nitrogen or temperature on Macrocystis’ physiology (growth, photosynthesis and nutrient uptake)54,63,64,65,66. However, only a few have determined the interactive effects of these two drivers61,65,67. To date, the regulatory effect of N on the thermal plasticity of key temperature-dependant traits under a wide range of temperatures (TPCs) is unknown for this species. Therefore, the aim of this study was to determine if N availability affects the thermal performance of physiological traits (i.e., photosynthesis, growth, nitrate uptake and assimilation) of Macrocystis. We hypothesized that due to the key role of N in regulating cellular and whole-organism physiological process, any potential negative effects of high temperature on Macrocystis performance will be ameliorated by increasing N availability. Hence the N status of the alga will play an important role in regulating thermal plasticity, i.e. N-replete blades will show greater thermal tolerance than N-deplete blades. To do this, we grew Macrocystis blades under low and enriched NO3 concentrations and a range of temperatures (6–27 °C).
## Results
### Biochemical and physiological parameters of field collected samples
Biochemical measurements of field collected individuals showed an average total carbon and nitrogen percent of 24.65% and 1.27% of dry mass, respectively, with a C:N of 19.64 (electronic supplementary material, Table S1). Stable isotopes, δ15N and δ13C, were 8.60 and −14.18, respectively. NR activity was 0.43 nmol NO3 FW g−1 min−1 (electronic supplementary material, Table S1). Other physiological and biochemical parameters such as pigment concentrations (Chl a and fucoxanthin (Fx)) and Fv/Fm are also described in electronic supplementary material, Table S1.
### Thermal performance curves (TPCs)
Most of the physiological traits of Macrocystis showed the typical non-linear relationship between temperature and performance (Figs. 14; electronic supplementary material, Table S2). However, the effect of the internal N status on the TPCs’ shape and position varied among the physiological traits (Table 1). RGR and photosynthesis showed significant differences in the shapes and fits of TPCs between N-replete and N-deplete blades (Figs. 1 and 2) (Table 2). RGR only evidenced vertical shifts (µmax) whilst photosynthesis exhibited both vertical and horizontal shifts (µmax and Topt) in the TPCs. In all of these cases, differences were explained by the considerably higher maximum rates (µmax) in N-replete blades than in N-deplete blades. Note that at the highest temperature (27 °C) seaweeds died in all experimental treatments and the data were not included in TPC analyses. Therefore, the TPCs are plotted for experimental treatments from 6–24 °C (Figs. 1 and 2). In contrast to the photosynthetic and growth rates, the shape and fit of Fv/Fm and NR activity TPCs were not significantly affected by the internal N status of the alga (Figs. 3 and 4, Table 2).
Pigment concentrations (i.e. Chl a, c and Fx) did not fit any of the non-linear models, suggesting a linear relationship between pigments and temperature that was not influenced by the internal N status of the alga (Table 2 and electronic supplementary material, Fig. S1).
### Total carbon and nitrogen percent, C/N ratio and Stable isotopes (δ13C)
Similar to pigment content, total carbon and nitrogen percent, C/N and stable isotopes followed linear a relationship with temperature (electronic supplementary material, Fig. S2). The N content and δ13C of Macrocystis blades did not differ between N-replete and N-deplete blades (similar slopes and intercepts, Tables 1 and 2), but N content showed a positive trend with increasing temperature while δ13C showed a negative trend. C content exhibited significant differences between experimental blades due to the negative slope in N-deplete blades and the lack of thermal influence in N-replete blades (Table 2) (slope = 0; electronic supplementary material, Fig. S2). Finally, the C/N ratio for N-replete blades showed a clear linear relationship with temperature while N-deplete blades showed a non-linear reaction curve (electronic supplementary material, Fig. S2).
### Chlorophyll a fluorescence of PSII
The initial slope (α) of the RLCs did not vary between N-replete and N-deplete blades, but decreased significantly with increasing temperature (ANOVA, P < 0.05; electronic supplementary material, Fig. S3). The Ek for the RLCs curves did not vary significantly between N-replete and N-deplete blades across all temperature treatments (ANOVA, P > 0.05), and ranged from 23.95 µmol m−2 s−1 at 10 °C to 76.87 µmol m−2 s−1 at 24 °C for N-replete blades, and from 32.95 µmol m−2 s−1 at 14 °C to 78.05 µmol m−2 s−1 at 24 °C for N-deplete (electronic supplementary material, Fig. S3). The ETRmax varied significantly among temperatures, however, neither the internal N status nor the corresponding interaction were significant (ANOVA two-ways, p = <0.001, 0.612, 0.157).
## Discussion
Thermal plasticity is a mechanism by which populations rapidly acclimate to warming and to extreme high temperature-related events such as MHWs40. Our study is the first to describe the plasticity of temperature-dependant traits of the ecologically and economically important kelp Macrocystis. We show that NO3 availability and hence the internal N status of the alga modulated the thermal plasticity of Macrocystis, buffering the negative impact of high temperature on its physiological performance, at least over a short-term incubation (Fig. 5). This may be of great importance with short-term extreme events, such as MHWs, increasing in frequency and intensity68. Rapid physiological acclimation to short-term variability in temperature and/or other environmental drivers (e.g., nutrient availability) can play an important role in seaweed responses to ongoing climate change. Our results suggest that populations of Macrocystis which are naturally exposed to moderate inputs of NO3 due to e.g., upwelling events or anthropogenic activities, may have a greater tolerance to high temperatures. However, populations that are exposed to limiting nutrient concentrations, like those from Tasmania, might be strongly negatively affected by OW and MHWs. We suggest that local drivers will play an important role in driving kelp responses to global change, and hence region-specific responses can be expected – as suggested by Krumhansl et al. (2016).
Local adaptations in thermal physiology have been recorded for macroalgal species across the latitudinal distribution of the species and ecotypes58,69,70,71,72 but we know little about the influence of nitrogen on the thermal performance of macroalgae. For the annual kelp Undaria pinnatifida, Gao et al.55 found differences in the thermal tolerance of geographically separated populations, where individuals with a higher thermal tolerance had the greatest capacity to store N. These results support our hypothesis that populations of Macrocystis that are naturally exposed to greater N supply will have a greater thermal tolerance than that those exposed to limiting nutrient concentrations. For microalgae, it is well documented that they are more vulnerable to high temperatures under N limited conditions compared to N-sufficient ones73,74. However, further studies comparing the effects of nitrogen and/or other local drivers on the thermal plasticity of populations separated geographically are urgently required to more precisely predict species’ responses to climate change.
We found that the internal N status modulated the thermal plasticity of Macrocystis, but its effects on TPC shapes varied among traits. Maximum photosynthetic and growth rates (µmax) were enhanced in N-replete blades. The positive effects of higher nitrogen availability on metabolic rates has been described in other brown kelp e.g., Saccharina japonica57. However, both the photosynthetic and growth rates of Macrocystis showed differences in the degree of plasticity. Only optimum temperatures (Topt) for photosynthesis were increased in N-replete blades while Topt for growth did not change between N-replete and N-deplete blades. Also, Topt for photosynthesis was higher than that for growth, which agrees with previous studies on macroalgae41,75. The differences observed in the thermal plasticity of growth and photosynthesis might be associated with the regulatory effect of temperature and nitrogen on each trait. For example, the photosynthetic machinery can rapidly acclimate to increases in temperature76, while growth is an integrated parameter that is regulated by many metabolic processes, including dark respiration, efflux of organic carbon, nitrogen uptake and assimilation41,75. This suggests that growth acclimation to changes in temperatures might be slower than for net photosynthesis.
Many aspects of thermal acclimation on key physiological traits are poorly studied in seaweeds compared to other marine organisms (e.g., invertebrates and corals)77,78,79,80 and there are currently no standardized protocols for performing thermal tolerance experiments in seaweeds. Ours is a physiological study looking into the rapid acclimation responses to temperature stress; in order to acclimate seaweeds to the experimental temperatures we used a ramping of 2 °C/hour from the acclimation temperature of 17 °C to a maximum of 27 °C and a minimum of 6 °C. We recognise that changes of 10 °C in a span of 5 h are unlikely to occur in the shallow subtidal system from which we collected Macrocystis, and although this may have contributed to increased physiological stress, the experiments were designed to develop the thermal performance curves (TPCs) of key physiological traits. Other studies have performed thermal ramps of 5 °C per day81, 3 °C per three days82 and some have not performed any thermal acclimation before the start of experiments56. In order to make more realistic long-term predictions to climate change (i.e. OW and MHWs), development of appropriate protocols for thermal stress experiments (i.e. thermal ramping and acclimation procedure) are required, understanding that these conditions might vary among regions, depending on daily and seasonally local variability (i.e. in temperature).
Changes in thermal tolerance involve various adjustments in algal metabolism, in which nitrogen plays a critical regulating role83. These adjustments include increased production of heat shock proteins (HSPs), which are important to tolerate temperature-stress conditions84,85,86 and changes in fatty acid composition and lipids of the thylakoid membrane41,87. We did not measure HSPs across the experimental treatments, but further investigation of the fatty acid and lipid composition in selected experimental treatments (6, 17 and 24 °C) (M. Schmid unpublished data) showed that under high nitrate concentrations, Macrocystis maintains a high proportion of polar lipids (PL) and shows no increase in free fatty acids (FFAs). Under low nitrate concentrations, however, PL decreased markedly with increasing temperatures, with a concomitant increase in FFAs. PL are key component of cellular membranes88, and hence a decrease in their proportion can negatively affect membrane stability at high temperatures. These results showed that nitrogen can rapidly influence lipid metabolism, and is a likely mechanisms by which Macrocystis acclimates to short-term thermal stress.
The thermal plasticity of Fv/Fm, which is often used as an indicator of photosynthetic stress and photoinhibition89, was not influenced by the internal N status of the algae, suggesting that the initial stage of the light reaction of photosynthesis (i.e. the transport of electrons through PSII) was stable under both low and enriched NO3 concentrations. In phytoplankton90,91 and the green macroalga Ulva sp.92,93, N limitation negatively affects PSII efficiency, indicated by a decline in Fv/Fm, likely due to decreases in protein synthesis and reaction centres90. However, we observed no effect of the internal N status of the alga on PSII efficiency (i.e., Fv/Fm, ETRmax, Ek and α) and nor on the photosynthetic pigments, at least over the short term incubation. Our results, along with those of Mabin et al.61 who found no effect of NO3 (0.5–3.0 µM) on the PSII parameters Fv/Fm and rETRmax of Macrocystis, suggest that seawater NO3 concentrations were above those required for the efficient functioning of PSII. In south-eastern Australia, this species is naturally exposed to low NO3 concentrations and hence the functioning of the PSII might be locally adapted to low nutrient concentrations.
Availability of NO3 can directly affect NR activity by regulating the synthesis and degradation of the protein94, and it can be strongly regulated by internal NO3 pools95. In macro- and microalgae, NR activity is also responsive to changes in temperature96, with thermal plasticity (Topt) varying between phytoplankton species97,98,99. However, there are no studies describing the regulating role of NO3 on NR thermal plasticity in macroalgae or phytoplankton species. The NR thermal plasticity of Macrocystis was not influenced by the internal N status of the algae. Topt (15 °C) for NR in N-replete and N-deplete blades was similar to those described for phytoplankton species, ranging between 10 to 20 °C, which are typically close to the Topt for growth97. This agrees with our study where Topt for NR was similar to that for growth (13–14 °C). The lack of effect of NO3 in regulating the NR thermal plasticity of Macrocystis may be due to the high intraspecific variability observed among individuals (activities ranged from 0.77–4.0 nmol NO3 g−1 FW min−1), or that NR activity was not limited by the NO3 concentration in the treatments. Similar to higher plants, two NR forms, one inducible and one constitutive, may occur in macroalgae100. It is possible that the inducible NR might be regulated by external environmental parameters, and the constitutive form maintains a constant activity rate101, which could explain our results. Thus, even when external NO3 concentrations are low, NR activity remains active. However, the regulation of the constitutive NR form has not been studied in macroalgae. Although we did not observe an increase in NR activity in N-replete blades, higher growth rates were observed across most of the temperature treatments, suggesting that more NO3 was assimilated and converted to N to support growth. Moreover, Macrocystis from New Zealand can rapidly respond to changes in N availability, up-regulating its N metabolism95. However, similar to PSII, NR activity for Macrocystis from Tasmania might be adapted to the local low ambient nutrient concentrations, and thus responses to environmental variability (physiological plasticity) might be distinct from other populations across the world.
Our results showed that thermal plasticity, tolerances and sensitivities, can vary markedly among physiological traits, with some traits responding faster than others to short-term variability in environmental temperature. Similarly, Wernberg et al.69 showed that in three habitat-forming seaweeds (Sargassum fallax, Ecklonia radiata, Scytothalia dorycarpa) optimum temperature for photosynthesis ranged from 23 °C–25 °C, depending on the species; however, no optimum temperatures were detected for respiration. Previous studies on ectothermic animals have shown similar variability among traits (e.g., respiration, growth)27. These results highlight the importance of selecting the right traits for predicting the effects of warming on the whole organism27. For macroalgae, photosynthesis can rapidly respond to changes in temperature, which agrees with our study, and it is considered a good proxy to compare thermal performance between species75. However, for long term predictions, photosynthesis might not be the best parameter to use because it over estimates the upper thermal tolerances for long term growth75. Studies describing thermal plasticity of different traits in macroalgae are urgently needed to identify the most relevant traits to precisely predict and compare the effects of ongoing warming at the organism and population level.
Recent studies have highlighted the importance of plasticity and local adaptation in macroalgal responses to warming, suggesting that some ecotypes might be more resilient or vulnerable to high temperatures than others, depending on the conditions to which they are usually exposed40,86,102. Previous studies have illustrated the physiological plasticity of Macrocystis across its wide geographical distribution. For example, marked differences in thermal tolerance and N storage capacities have been observed in populations that are geographically isolated66,103,104,105. Similarly, distinct thermal tolerances and survival abilities have been observed in microscopic life stages from populations locally exposed to a different gradient of temperature (warmer vs. cool temperate sites)106. These results suggest that populations that are naturally exposed to highly variable thermal and nutrient regimes can have different responses to future oceanic conditions compared to populations that are exposed to more stable environmental conditions. Further studies linking molecular (e.g., expression of heat-shock-proteins) with physiological responses will provide a better understanding of macroalgal plastic and adaptive capacities to respond to climate change.
## Materials and Methods
### Seaweed collection
Sampling was performed at Bruny Island (45°47′S, 170°43′E), Tasmania, Australia, in March 2016. At the time of collection, temperature and NO3 concentrations in surface waters ranged from 18.7–19.9 °C and from 0.14–0.53 µM NO3, respectively. Also, temperatures were 3–4 °C above the average of previous summers due to the most intense and longest MHW recorded in the Tasman Sea107. A total of 80 young blades (the 2nd and 3rd blades below apical scimitar) were collected from different individuals of Macrocystis (3–4 blades from each of 26 sporophytes). Blades were transported to the laboratory in an insulated container with ambient seawater. At the laboratory, blades were gently rinsed and cleaned with 0.5 µm filtered natural seawater (NSW) of any visible epibionts by gently brushing. Each of the 80 blades were cut to a similar size of 11 cm × 3.5 cm (fresh initial weight 1.0 ± 0.2 g), at 2 cm from the neumatocyst/blade junction (meristematic zone). Initial physiological measurements on field collected samples (electronic supplementary material, Table S1) showed that seaweeds were healthy at the start of the experiment. Then, blade sections were incubated for 12 h to allow marginal wounds to heal, in transparent 2 L-jars (0.5 µm filtered NSW at 17 °C). Mixing in the culture tanks was provided by pumping air. Eight blade sections were used to assess their initial physiological status (i.e. photosynthetic parameters, growth, nitrate reductase (NR) activity and nitrogen and carbon content) as described below.
### Pre-experimental incubations under low and enriched-NO3− concentrations
After the 12 h healing time, 72 blade sections were incubated for a further 3 days under low (5 µM NO3; n = 36) and enriched-NO3 concentrations (80 µM NO3; n = 36) to obtain Macrocystis blades with different nitrogen status, i.e. deplete and replete, respectively95. Six blade sections were placed into each of twelve 2 L-culture tanks, six containing low-NO3 SW and the other six containing enriched-NO3 SW. A 20 mM NaNO3 solution provided the desired NO3 concentrations in each culture tank, and 100 mM PO4 was used to avoid P limitation through the experiment (5:1 N:P). Seawater mixing was provided by pumping air. A saturating photon flux density of 120–130 µmol m−2s−1 was provided overhead by florescent white tubes (Envirolux CE F28T5/4100K-120477 240V) set on a 12L:12D photoperiod. Incident light was measured using a Li-Cor LI-1400 data logger equipped with a flat underwater radiation sensor LI-192. SW samples (10 mL) were taken every day before and after renewing the medium to monitor NO3 concentrations (electronic supplementary material, Table S1). Prior to the experiments and during the pre-experimental conditions, the cultures were maintained in a temperature-controlled room at 17 °C.
### Temperature and nitrate incubations, and experimental design
After the pre-experimental incubation, N-deplete (blades coming from the 5 µM NO3 treatment) and N-replete blades (blades coming from the 80 µM NO3 treatment) were haphazardly selected and placed into each of 64 Erlenmeyer 250 mL flasks, containing either low (n = 32) or enriched-NO3 SW (n = 32). After that, each 250 mL culture flask was randomly assigned to one of the seven temperatures treatments, 6–10–14–17–20–24–27 °C, with four replicates for each experimental treatment. Blades were gradually acclimated from the pre-incubation temperature (17 °C) to the experimental temperatures. Thermal ramps were performed with linear temperatures changes of 2 °C per hour over a span of 5 h. Although the temperature range, 6 °C to 27 °C, and the thermal ramp does not fully coincide with the conditions experienced by the species in the field, the temperature range and ramping were selected to estimate the short-term thermal acclimation and precisely develop the TPCs for the specie. The culture flasks under each temperature treatment were maintained in a controlled temperature water bath for three days and subjected to a 12L:12D photoperiod under a saturating light intensity of 120–130 µmol m−2s−1 provided and measured as described in the pre-experimental incubations. SW was changed daily and mixed by pumping air. Temperature and light conditions within each temperature treatment were monitored continuously using HOBO pendant temperature/light data loggers (64K-UA-002-64). SW samples (10 mL) were taken every day before and after renewing the medium to monitor NO3 concentrations in each treatment (electronic supplementary material, Table S1). After the 3-day incubation, Macrocystis blades were harvested to determine their physiological and biochemical responses (i.e. photosynthesis, growth, nitrate assimilation, and carbon and nitrogen content).
### Physiological and biochemical parameters
Photosynthetic rates expressed as oxygen evolution were measured on the last day of the temperature/nitrate experiment, for each of the 56 experimental Macrocystis blades (n = 4 for each treatment combination). To do this, each blade was incubated separately in a biochemical oxygen demand (BOD) bottle, containing 266 mL of filtered 0.5 µm SW. A control BOD bottle without seaweed was also carried out. BOD bottles were placed on the top of an orbit shaker table set at 100 rpm to provide water movement. Dissolved oxygen (DO) was measured at the start and after 1 h of incubation using an optical dissolved oxygen sensor (Hach LBOD10101) and a DO meter (HQ40d Hach). Photosynthetic rates were determined under a saturating light of 120–130 µmol m−2s−1 that was provided overhead by florescent white tubes (Envirolux CE F28T5/4100K-120477 240 V). For each temperature treatment, BOD bottles were put inside a transparent plastic box, where the temperature was controlled by an aquarium heater set up at each temperature. Photosynthetic rates were estimated from the initial and final oxygen concentrations (mg/l), and standardized by the fresh weight (g) of each blade and the incubation volume (l).
After measuring photosynthetic rates, chlorophyll a fluorescence of photosystem II was measured using a Pulse Amplitude Modulation fluorometer (diving-PAM, Walz, Germany). Macrocystis blades from the different treatment combinations were dark-adapted for 20 min before exposure to the PAM’s photosynthetic active radiation (PAR, 0–422 μmol photons m−2 s−1). Rapid light curves (RLCs), relative ETR (rETR) versus irradiance, were conducted right after dark adaptation. Calculations of rETR of algal samples were estimated using the equation:
$$rETR=Y(II)\times {\rm{EPAR}}\times {\rm{A}}\times 0.5,$$
where Y(II) is the quantum yield of photochemical quenching, E the incident irradiance of PAR, A the average ratio of light absorbed by algal tissue (0.8 for kelps) and the factor 0.5 assumes that 50% of the all absorbed energy has been utilized by PSII108. The rETR-RLCs were fitted according to Eilers and Peeters (1988), so that the light saturation (Ek), initial slope (α) and the maximum ETR (ETRmax) parameters were calculated from each curve. The optimum quantum yield (Fv/Fm), which represents a good indicator of maximal algal photosynthetic efficiency109, was calculated right after the dark adaptation.
Relative growth rates (RGR, % days−1) were estimated after measuring photosynthetic rates and Chlorophyll a fluorescence by the difference in fresh weight (FW) after 3 days of incubation, using the formula:
$$RGR={\rm{In}}(\frac{Wt}{{W}0})\times t-1\times 100,$$
where W0 is the initial FW and Wt is the final FW after t days of incubation. The FW was estimated after blotting the blade section gently with tissue.
After the growth measurement, blade sections, from each experimental treatment, were cut along the blade into four pieces in order to assess NR activity, C and N content and pigment concentrations. Tissue samples for NR activity (0.21–0.30 g FW), pigment analysis (0.10–0.15 g FW) were immediately frozen in liquid N2 and stored at −80 °C until further analyses. Tissue samples for C and N content and stable isotopes (0.008–0.010 g FW) were oven dried for 48 h at 60 °C.
NR activity was measured by nitrite production in an in vitro assay110. NR extraction methodology is described in detail in Fernandez et al.95. Briefly, NR was extracted in a 200 Mm Na-phosphate buffer (pH 7.9), containing 3% w/v BSA, 0.3% w/w polyvinylpyrrolidone (PVP), 2 Mm Na-EDTA and 1% w/v Triton X-100 (all Sigma, St Louis, MO, USA). The content of photosynthetic pigments (chlorophyll a and fucoxanthin) was analysed using methods described in Seely et al.111, Wheeler112, and Stephens and Hepburn113, where dimethyl-sulfoxide (DMSO) was used for the primary extraction and acetone for the secondary. Tissue samples for C and N content and stable isotopes from field collection and pre-experimental incubations, and temperature/nitrate experiments were determined according to Cornwall et al.114.
### Seawater analyses
Nitrate concentrations were analyzed using a QuickChem 8500 series 2 Automated Ion Analyzer (Lachat Instrument, Loveland, CO).
### Data analysis
Prior to analyses, we tested for normality and homoscedasticity for all variables, using the Lilliefors and Levene tests, respectively. Data was transformed either by log10 or by square root to fulfil the requirements for parametric tests. Comparisons of parameters measured as part of the chlorophyll a fluorescence of PSII, were done via analysis of variance (ANOVA). When differences in the means were significant at the P < 0.05 level, they were also tested with a posteriori Tukey’s test (HSD). Statistical analyses were performed with R 3.0.2 software and the package lme4115 and in the GraphPad Prism software (v.7.03).
The effect of temperature on physiology and performance was described by a continuous nonlinear reaction norm (i.e. TPC) (Huey et al. 1999). Variation in the parameters of the TPCs (i.e. the optimal temperature - Topt; the thermal breadth - Tbr; the maximal performance - µmax; and the upper and lower limits of temperature at which traits expression decrease - CTmin and CTmax) was used to describe mechanistically the variation of thermal sensitivities and tolerances of natural populations (see Gaitan-Espitia et al. 2013, 2014) (Fig. 6). Here, we used the TableCurve2D curve-fitting software (version 5.01; Systat Software, Inc.) and the GraphPad Prism software (v.7.03) for model fitting. TPC parameters (µmax, Topt, CTmin and CTmax) were extracted from the best models (see below for details). The physiological characteristics of critical thermal maximum (CTmax) and minimum (CTmin) were derived numerically as the intersection points of the resulting thermal performance curve with the temperature axis (μ = 0).
The fit of several linear and non-linear functions (e.g., Gaussian, Quadratic Lorentzian, Weibull) that could describe organismal performance as a function of temperature was analyzed using the Akaike Information Criterion (AIC) (Angilletta 2006). The AIC represents a balance between the likelihood explained by the model and the number of model parameters, with the best model minimizing AIC116. Thermal-dependent traits obtained from the TPCs were analyzed using a linear modelling approach. The effects of temperature and NO3 (fixed effects) were evaluated through confidence intervals (CI) computed from the likelihood profile115. In addition to CI, parameters and shapes of the TPCs were analyzed and compared using AIC and the Extra Sum-of-Squares F test. For linear models, the slopes and intercepts were compared using an analysis of covariance (ANCOVA) F-test.
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2022-12-07 04:58:58
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http://www.math.cmu.edu/CNA/Publications/publication_auto.php?cnacode=16-CNA-029
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Center for Nonlinear Analysis CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact Publication 16-CNA-029 Exact Solutions For The Denoising Problem Of Piecewise Constant Images In Dimension One R. CristoferiDepartment of Mathematical Sciences Carnegie Mellon University Pittsburgh PA 15213-3890 USArcristof@andrew.cmu.eduAbstract: In this paper we provide a method to compute explicitly the solution of the total variation denoising problem with a $L^p$ fidelity term, where $p > 1$, for piecewise constant data in dimension one. Get the paper in its entirety as 16-CNA-029.pdf
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2018-07-23 15:51:53
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https://www.tutorialexample.com/l2-regularization-and-batch-size-tips-to-use-l2-regularization-in-tensorflow/
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# L2 Regularization and Batch Size: Tips to Use L2 Regularization in TensorFlow
By | January 2, 2020
L2 regularization can hurt training error and improve your model performance. However, l2 is related to batch size when training. To address this issue, we will discuss this topic in this tutorial.
## L2 Regularization
The formula of l2 regularization is defined as:
From the formula we can find the value of l2 regularization is related to all weights in model and is not related to the number of training sample or batch size.
## How to implement l2 regularization?
There exsits to kinds of l2 regularization:
Form 1:
l2 = deta * tf.reduce_sum([ tf.nn.l2_loss(n) for n in tf.trainable_variables() if 'bias' not in n.name])
loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(logits=self.scores, labels = self.input_y)) + l2
Form 2:
l2 = deta * tf.reduce_sum([ tf.nn.l2_loss(n) for n in tf.trainable_variables() if 'bias' not in n.name])
loss = tf.reduce_mean(tf.nn.softmax_cross_entropy_with_logits(logits=self.scores, labels = self.input_y) + l2)
## Which one is correct?
Both of them are correct, which are not related to batch size.
However, we have foud some examples in some tensorflow project. form 2 are common used ,while form 1 is not.
For examaple:
So we also recommend to you to form 2.
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2021-08-05 08:29:54
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http://mathhelpforum.com/calculus/96335-series.html
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1. ## series
I think i found the bottom one, 2n+3 the only way i can see the top one is in a recursive but i need a closed form, can anyone give me some hints on how to look at it...
Find a formula for sn, n 1. , , , , , , ...
2. numerator looks to be $n!$
3. ## He's right!
yes, the general term is n! / (2n+3);
4. Originally Posted by acosta0809
I think i found the bottom one, 2n+3 the only way i can see the top one is in a recursive but i need a closed form, can anyone give me some hints on how to look at it...
Find a formula for sn, n 1. , , , , , , ...
The numerator is $n! = n\cdot (n-1)\cdot (n-2) \cdot \dots \cdot 3 \cdot 2 \cdot 1$.
So $t_n = \frac{n!}{2n + 3}$
$S_n = \sum_{i = 1}^n \frac{n!}{2n + 3}$.
If this was an infinite series, it would diverge btw. But since it's finite, that's a bit trickier...
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2017-12-14 19:18:58
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https://www.dm.unipi.it/eventi/spass-seminar-margherita-zanella/
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# Ergodic results for the stochastic nonlinear damped Schroedinger equation – Margherita Zanella (Politecnico di Milano)
Aula seminari
#### Abstract
We study the nonlinear stochastic Schrödinger equation with linear damping. We prove the existence of invariant measures in the case of two dimensional compact Riemannian manifolds without boundary and compact smooth domains of $\mathbb{R}^2$ with either Dirichlet or Neumann boundary conditions. We prove the uniqueness of the invariant measure in $\mathbb{R}^d$, $d=2,3$, when the damping coefficient is sufficiently large. The talk is based on joint works with B. Ferrario and Z. Brzeźniak.
Torna in cima
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2023-01-28 20:22:29
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https://topospaces.subwiki.org/wiki/Homotopy_group
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# Homotopy group
## Definition
### As homotopy classes of based maps
Suppose $(X,x_0)$ is a based topological space and $n$ is a positive integer. The homotopy group $\pi_n(X,x_0)$ is defined as follows:
• Consider the based $n$-sphere $(S^n,p)$ where $p$ is a chosen basepoint. As a set, $\pi_n(X,x_0)$ is the set of homotopy classes of all based maps from $(S^n,p)$ to $(X,x_0)$, where the homotopy classes are with respect to homotopies that preserve basepoints.
• Two maps $f_1,f_2$ are composed as follows. $S^n \setminus \{ p \}$ is identified with the open northern and open southern hemisphere of a new sphere via homeomorphic identifications $\varphi_1$ and $\varphi_2$ from these hemispheres to $S^n \setminus \{ p \}$ (these identifications are universally fixed, independent of $X$; there's a natural choice for them). The composite map is now defined as follows: as $f_1 \circ \varphi_1$ on the northern hemisphere, as $f_2 \circ \varphi_2$ on the southern hemisphere, and as the constant map to $p$ on the equator. The basepoint is a fixed point on the equator (again, this choice is independent of $X$ and is universally fixed).
The multiplication defined above can be viewed as arising from the corresponding comultiplication on the $n$-sphere $S^n$, because of the contravariant nature of maps from.
### The case $n = 0$
The definition of homotopy group still gives a set definition for $n = 0$. $S^0$ is a two-point space, and one of these points must go to a fixed basepoint, while the other can go anywhere. Thus, the set of all based maps is the set of points in $X$, and the set of homotopy classes is the set of path components. Thus, $\pi_0(X,x_0)$ is the set of path components in $X$. Note that it is independent of $x_0$ because $S^0$ being discrete, the image of the basepoint does not affect where the other point goes.
However, the composition operation does not make sense for $n = 0$, because $S^0$ has an empty equator. Hence, $\pi_0$ is only a set and has no group structure for arbitary topological spaces. (It does have a group structure when the topological space is a H-space, induced by the multiplication in the topological space).
### The case $n = 1$
In this case, we get the fundamental group $\pi_1(X,x_0)$. Recall that for $f_1,f_2$ based maps from the circle to $(X,x_0)$, we think of $f_1,f_2$ as maps from $[0,1]$ to $X$ with $f_1(0) = f_1(1) = f_2(0) = f_2(1) = x_0$. The usual way of composing is to define:
$(f_1 * f_2)(t) := \lbrace \begin{array}{rl} f_1(2t), & 0 \le t \le 1/2 \\ f_2(2t - 1), & 1/2 < t \le 1 \\\end{array}$
Here, the definition on $(0,1/2)$ can be viewed as the northern hemisphere definition, the definition on $(1/2,1)$ can be viewed as the southern hemisphere definition, with the equator corresponding to the two points $1/2$ and $\! 0 \sim 1$, of which we choose the latter as basepoint.
### Omission of basepoint
For a path-connected space, the homotopy groups $\pi_n$ for all basepoints are isomorphic. In fact, any choice of path between two points can be used to define an isomorphism between the $\pi_n$s at these basepoints. The key fact that we need to use here is that the inclusion of a point in $S^n$ is a cofibration (which is easily seen by noting that $S^n$ is the boundary of $D^n$, or more generally from the fact that manifold implies nondegenerate).
In general, the homotopy group $\pi_n$ may differ for different path components. For a homogeneous space, or more generally for a space where all the path components are homeomorphic, the isomorphism class of $\pi_n$ does not depend upon the choice of basepoint.
### Dependence on homotopy type
The homotopy groups $\pi_n$ depend only on the homotopy type of the based topological space. In fact, they depend only on the homotopy type of the path component of the basepoint in the topological space.
However, knowledge of the homotopy groups does not determine the homotopy type, or even the weak homotopy type, of the topological space.
## Facts
• For a H-space, all homotopy groups, including the fundamental group, are abelian groups. This is a consequence of the Eckmann-Hilton principle. In fact, the group operation coincides with the operation induced by pointwise multiplication of loops (when we go down to homotopy classes). Also, $\pi_0$ gets the structure of a (not necessarily abelian) group. Further information: fundamental group of H-space is abelian
• For any topological space, all higher homotopy groups, i.e., all $\pi_n, n \ge 2$, are abelian groups. This can be viewed as a consequence of the Eckmann-Hilton principle, or the fact for $n \ge 2$, $S^n$ can be rotated about any equatorial axis to interchange the roles of the north and the south poles. Further information: higher homotopy groups are abelian
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2019-02-18 06:25:27
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https://www.varsitytutors.com/act_math-help/algebra/expressions
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# ACT Math : Expressions
## Example Questions
← Previous 1 3 4 5 6 7 8 9 10 11
### Example Question #2462 : Sat Mathematics
Simplify the following:
Explanation:
To simplify the following, a common denominator must be achieved. In this case, the first term must be multiplied by (x+2) in both the numerator and denominator and likewise with the second term with (x-3).
### Example Question #1 : Expressions
Simplify the following
Explanation:
Find the least common denominator between x-3 and x-4, which is (x-3)(x-4). Therefore, you have . Multiplying the terms out equals . Combining like terms results in .
### Example Question #1 : Expressions
Simplify the following expression:
Explanation:
In order to add fractions, we must first make sure they have the same denominator.
So, we multiply by and get the following:
Then, we add across the numerators and simplify:
### Example Question #2391 : Act Math
Combine the following two expressions if possible.
Explanation:
For binomial expressions, it is often faster to simply FOIL them together to find a common trinomial than it is to look for individual least common denominators. Let's do that here:
FOIL and simplify.
Combine numerators.
Thus, our answer is
### Example Question #1 : Expressions
Select the expression that is equivalent to
Explanation:
To add the two fractions, a common denominator must be found. With one-term denominators, it is easier to simply find the least common denominator between them and multiply each side to obtain it.
In this case, the least common denominator between and is . So the first fraction needs to be multiplied by and the second by :
Now, we can add straight across, remembering to combine terms where we can.
So, our simplified answer is
### Example Question #1 : How To Evaluate Rational Expressions
Find the product of and .
Explanation:
Solve the first equation for .
Solve the second equation for
The final step is to multiply and .
### Example Question #1 : Rational Expressions
The following table shows the temperature of a cup of coffee at different times
Time 1:09 1:11 1:13 1:15 1:17
Temperature (ºF) 187.1 184.4 181.7 179.0 176.3
If this trend continues, what will the temperature of the coffee at minute 1:25?
162.9°F
160.2°F
168.3°F
171.0°F
165.5°F
165.5°F
Explanation:
The table shows that for every two minutes, the temperature of the coffee lowers 2.7ºF. At 1:25, 16 minutes, or eight 2-minute intervals have passed, and the temperature of the coffee has lowered by 8*2.7ºF, reaching a temperature of 165.5ºF.
### Example Question #8 : Expressions
Amy buys concert tickets for herself and her friends. She initially buys them at $40/ticket. Weeks later, her other friends ask her to buy them tickets, but the prices have increased to$54. Amy buys 7 tickets total and spends $350. How many tickets has she paid$40 on?
Explanation:
Amy has bought 7 tickets, x of them at $40/ticket, and the remaining 7-x at$54/ticket. She spends at total of
### Example Question #1 : How To Evaluate Rational Expressions
What is
?
Explanation:
To find an equivalency we must rationalize the denominator.
To rationalize the denominator multiply the numerator and denominator by the denominator.
To simplify completely, factor out a three from the numerator and denominator resulting in the final solution.
### Example Question #1 : Rational Expressions
Simplify:
Explanation:
The common denominator of these two fractions simply is the product of the two denominators, namely:
Thus, you will need to multiply each fraction's numerator and denominator by the opposite fraction's denominator:
Let's first simplify the numerator:
, which is the simplest form you will need for this question.
However, the correct answer has the denominator multiplied out. Merely FOIL
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https://en.wikipedia.org/wiki/Orthogonal
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# Orthogonality
(Redirected from Orthogonal)
"Orthogonal" redirects here. For the trilogy of novels by Greg Egan, see Orthogonal (novel).
The line segments AB and CD are orthogonal to each other.
In mathematics, orthogonality is the relation of two lines at right angles to one another (perpendicularity), and the generalization of this relation into n dimensions; and to a variety of mathematical relations thought of as describing non-overlapping, uncorrelated, or independent objects of some kind.
The concept of orthogonality has been broadly generalized in mathematics (including in the areas of mathematical functions, calculus and linear algebra), as well as in areas such as chemistry, and engineering.
## Etymology
The word comes from the Greek ὀρθός (orthos), meaning "upright", and γωνία (gonia), meaning "angle". The ancient Greek ὀρθογώνιον orthogōnion (< ὀρθός orthos 'upright'[1] + γωνία gōnia 'angle'[2]) and classical Latin orthogonium originally denoted a rectangle.[3] Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word orthogonalis came to mean a right angle or something related to a right angle.[4]
## Mathematics and physics
Orthogonality and rotation of coordinate systems compared between left: Euclidean space through circular angle ϕ, right: in Minkowski spacetime through hyperbolic angle ϕ (red lines labelled c denote the worldlines of a light signal, a vector is orthogonal to itself if it lies on this line).[5]
### Definitions
• In geometry, two Euclidean vectors are orthogonal if they are perpendicular, i.e., they form a right angle.
• Two vectors, x and y, in an inner product space, V, are orthogonal if their inner product ${\displaystyle \langle x,y\rangle }$ is zero.[6] This relationship is denoted ${\displaystyle x\,\bot \,y}$.
• Two vector subspaces, A and B, of an inner product space, V, are called orthogonal subspaces if each vector in A is orthogonal to each vector in B. The largest subspace of V that is orthogonal to a given subspace is its orthogonal complement.
• Given a module M and its dual M, an element m′ of M and an element m of M are orthogonal if their duality pairing is zero, i.e. m′, m⟩ = 0. Two sets S′ ⊆ M and SM are orthogonal if each element of S′ is orthogonal to each element of S.[7]
• A term rewriting system is said to be orthogonal if it is left-linear and is non-ambiguous. Orthogonal term rewriting systems are confluent.
A set of vectors is called pairwise orthogonal if each pairing of them is orthogonal. Such a set is called an orthogonal set. Nonzero pairwise orthogonal vectors are always linearly independent.[dubious ]
In certain cases, the word normal is used to mean orthogonal, particularly in the geometric sense as in the normal to a surface. For example, the y-axis is normal to the curve y = x2 at the origin. However, normal may also refer to the magnitude of a vector. In particular, a set is called orthonormal (orthogonal plus normal) if it is an orthogonal set of unit vectors. As a result, use of the term normal to mean "orthogonal" is often avoided. The word "normal" also has a different meaning in probability and statistics.
A vector space with a bilinear form generalizes the case of an inner product. When the bilinear form applied to two vectors results in zero, then they are orthogonal. The case of a pseudo-Euclidean plane uses the term hyperbolic orthogonality. In the diagram, axes x′ and t′ are hyperbolic-orthogonal for any given ϕ.
### Euclidean vector spaces
In 2-D or higher-dimensional Euclidean space, two vectors are orthogonal if and only if their dot product is zero, i.e. they make an angle of 90°, or π/2 radians.[8] Hence orthogonality of vectors is an extension of the concept of perpendicular vectors into higher-dimensional spaces.
In terms of Euclidean subspaces, a subspace has an "orthogonal complement" such that every vector in the subspace is orthogonal to every vector in the complement. In three-dimensional Euclidean space, the orthogonal complement of a line is the plane perpendicular to it, and vice versa.[9]
Note however that there is no correspondence with regards to perpendicular planes, because vectors in subspaces start from the origin (by the definition of a Linear subspace).
In four-dimensional Euclidean space, the orthogonal complement of a line is a hyperplane and vice versa, and that of a plane is a plane.[9]
### Orthogonal functions
Main article: Orthogonal functions
By using integral calculus, it is common to use the following to define the inner product of two functions f and g:
${\displaystyle \langle f,g\rangle _{w}=\int _{a}^{b}f(x)g(x)w(x)\,dx.}$
Here we introduce a nonnegative weight function w(x) in the definition of this inner product. In simple cases, w(x) = 1.
We say that these functions are orthogonal (with respect to this inner product) if the value of this integral is zero:
${\displaystyle \int _{a}^{b}f(x)g(x)w(x)\,dx=0.}$
Note that two functions which are orthogonal with respect to one inner product, aren't necessarily orthogonal with respect to another inner product.
We write the norms with respect to this inner product and the weight function as
${\displaystyle \|f\|_{w}={\sqrt {\langle f,f\rangle _{w}}}}$
The members of a set of functions { fi : i = 1, 2, 3, ... } are:
• orthogonal on the closed interval [a, b] if
${\displaystyle \langle f_{i},f_{j}\rangle =\int _{a}^{b}f_{i}(x)f_{j}(x)w(x)\,dx=\|f_{i}\|^{2}\delta _{i,j}=\|f_{j}\|^{2}\delta _{i,j}}$
• orthonormal on the interval [a, b] if
${\displaystyle \langle f_{i},f_{j}\rangle =\int _{a}^{b}f_{i}(x)f_{j}(x)w(x)\,dx=\delta _{i,j}}$
where
${\displaystyle \delta _{i,j}=\left\{{\begin{matrix}1&\mathrm {if} \ i=j\\0&\mathrm {if} \ i\neq j\end{matrix}}\right.}$
is the "Kronecker delta" function. In other words, any two of them are orthogonal, and the norm of each is 1 in the case of the orthonormal sequence. See in particular the orthogonal polynomials.
### Examples
• The vectors (1, 3, 2)T, (3, −1, 0)T, (1, 3, −5)T are orthogonal to each other, since (1)(3) + (3)(−1) + (2)(0) = 0, (3)(1) + (−1)(3) + (0)(−5) = 0, and (1)(1) + (3)(3) + (2)(−5) = 0.
• The vectors (1, 0, 1, 0, ...)T and (0, 1, 0, 1, ...)T are orthogonal to each other. The dot product of these vectors is 0. We can then make the generalization to consider the vectors in Z2n:
${\displaystyle \mathbf {v} _{k}=\sum _{i=0 \atop ai+k
for some positive integer a, and for 1 ≤ ka − 1, these vectors are orthogonal, for example (1, 0, 0, 1, 0, 0, 1, 0)T, (0, 1, 0, 0, 1, 0, 0, 1)T, (0, 0, 1, 0, 0, 1, 0, 0)T are orthogonal.
• The functions 2t + 3 and 45t2 + 9t − 17 are orthogonal with respect to a unit weight function on the interval from −1 to 1:
${\displaystyle \int _{-1}^{1}\left(2t+3\right)\left(45t^{2}+9t-17\right)\,dt=0}$
• The functions 1, sin(nx), cos(nx) : n = 1, 2, 3, ... are orthogonal with respect to Riemann integration on the intervals [0, 2π], [−π, π], or any other closed interval of length 2π. This fact is a central one in Fourier series.
#### Orthogonal states in quantum mechanics
• In quantum mechanics, a sufficient (but not necessary) condition that two eigenstates of a Hermitian operator, ${\displaystyle \psi _{m}}$ and ${\displaystyle \psi _{n}}$, are orthogonal is that they correspond to different eigenvalues. This means, in Dirac notation, that ${\displaystyle \langle \psi _{m}|\psi _{n}\rangle =0}$ unless ${\displaystyle \psi _{m}}$ and ${\displaystyle \psi _{n}}$ correspond to the same eigenvalue. This follows from the fact that Schrödinger's equation is a Sturm–Liouville equation (in Schrödinger's formulation) or that observables are given by hermitian operators (in Heisenberg's formulation).[citation needed]
## Art
In art, the perspective (imaginary) lines pointing to the vanishing point are referred to as "orthogonal lines".
The term "orthogonal line" often has a quite different meaning in the literature of modern art criticism. Many works by painters such as Piet Mondrian and Burgoyne Diller are noted for their exclusive use of "orthogonal lines" — not, however, with reference to perspective, but rather referring to lines that are straight and exclusively horizontal or vertical, forming right angles where they intersect. For example, an essay at the Web site of the Thyssen-Bornemisza Museum states that "Mondrian ... dedicated his entire oeuvre to the investigation of the balance between orthogonal lines and primary colours." [1]
## Computer science
Orthogonality in programming language design is the ability to use various language features in arbitrary combinations with consistent results.[10] This usage was introduced by van Wijngaarden in the design of Algol 68:
The number of independent primitive concepts has been minimized in order that the language be easy to describe, to learn, and to implement. On the other hand, these concepts have been applied “orthogonally” in order to maximize the expressive power of the language while trying to avoid deleterious superfluities.[11]
Orthogonality is a system design property which guarantees that modifying the technical effect produced by a component of a system neither creates nor propagates side effects to other components of the system. Typically this is achieved through the separation of concerns and encapsulation, and it is essential for feasible and compact designs of complex systems. The emergent behavior of a system consisting of components should be controlled strictly by formal definitions of its logic and not by side effects resulting from poor integration, i.e., non-orthogonal design of modules and interfaces. Orthogonality reduces testing and development time because it is easier to verify designs that neither cause side effects nor depend on them.
An instruction set is said to be orthogonal if it lacks redundancy (i.e., there is only a single instruction that can be used to accomplish a given task)[12] and is designed such that instructions can use any register in any addressing mode. This terminology results from considering an instruction as a vector whose components are the instruction fields. One field identifies the registers to be operated upon and another specifies the addressing mode. An orthogonal instruction set uniquely encodes all combinations of registers and addressing modes.[citation needed]
## Communications
In communications, multiple-access schemes are orthogonal when an ideal receiver can completely reject arbitrarily strong unwanted signals from the desired signal using different basis functions. One such scheme is TDMA, where the orthogonal basis functions are nonoverlapping rectangular pulses ("time slots").
Another scheme is orthogonal frequency-division multiplexing (OFDM), which refers to the use, by a single transmitter, of a set of frequency multiplexed signals with the exact minimum frequency spacing needed to make them orthogonal so that they do not interfere with each other. Well known examples include (a, g, and n) versions of 802.11 Wi-Fi; WiMAX; ITU-T G.hn, DVB-T, the terrestrial digital TV broadcast system used in most of the world outside North America; and DMT (Discrete Multi Tone), the standard form of ADSL.
In OFDM, the subcarrier frequencies are chosen so that the subcarriers are orthogonal to each other, meaning that crosstalk between the subchannels is eliminated and intercarrier guard bands are not required. This greatly simplifies the design of both the transmitter and the receiver. In conventional FDM, a separate filter for each subchannel is required.
## Statistics, econometrics, and economics
When performing statistical analysis, independent variables that affect a particular dependent variable are said to be orthogonal if they are uncorrelated,[13] since the covariance forms an inner product. In this case the same results are obtained for the effect of any of the independent variables upon the dependent variable, regardless of whether one models the effects of the variables individually with simple regression or simultaneously with multiple regression. If correlation is present, the factors are not orthogonal and different results are obtained by the two methods. This usage arises from the fact that if centered by subtracting the expected value (the mean), uncorrelated variables are orthogonal in the geometric sense discussed above, both as observed data (i.e., vectors) and as random variables (i.e., density functions). One econometric formalism that is alternative to the maximum likelihood framework, the Generalized Method of Moments, relies on orthogonality conditions. In particular, the Ordinary Least Squares estimator may be easily derived from an orthogonality condition between the explanatory variables and model residuals.
## Taxonomy
In taxonomy, an orthogonal classification is one in which no item is a member of more than one group, that is, the classifications are mutually exclusive.
## Combinatorics
In combinatorics, two n×n Latin squares are said to be orthogonal if their superimposition yields all possible n2 combinations of entries.[14]
## Chemistry and biochemistry
In synthetic organic chemistry orthogonal protection is a strategy allowing the deprotection of functional groups independently of each other. In chemistry and biochemistry, an orthogonal interaction occurs when there are two pairs of substances and each substance can interact with their respective partner, but does not interact with either substance of the other pair. For example, DNA has two orthogonal pairs: cytosine and guanine form a base-pair, and adenine and thymine form another base-pair, but other base-pair combinations are strongly disfavored. As a chemical example, tetrazine reacts with transcyclooctene and azide reacts with cyclooctyne without any cross-reaction, so these are mutually orthogonal reactions, and so, can be performed simultaneously and selectively.[15] Bioorthogonal chemistry refers to chemical reactions occurring inside living systems without reacting with naturally present cellular components. In supramolecular chemistry the notion of orthogonality refers to the possibility of two or more supramolecular, often non-covalent, interactions being compatible; reversibly forming without interference from the other.
In analytical chemistry, analyses are "orthogonal" if they make a measurement or identification in completely different ways, thus increasing the reliability of the measurement. This is often required as a part of a new drug application.
## System reliability
In the field of system reliability orthogonal redundancy is that form of redundancy where the form of backup device or method is completely different from the prone to error device or method. The failure mode of an orthogonally redundant back-up device or method does not intersect with and is completely different from the failure mode of the device or method in need of redundancy to safeguard the total system against catastrophic failure.
## Neuroscience
In neuroscience, a sensory map in the brain which has overlapping stimulus coding (e.g. location and quality) is called an orthogonal map.
## Gaming
In board games such as chess which feature a grid of squares, 'orthogonal' is commonly used to mean "in the same row/'rank' or column/'file'". In this context 'orthogonal' and 'diagonal' are considered opposites.[16] In the ancient Chinese board game Go a player can capture the stones of an opponent by occupying all orthogonally-adjacent points.
## Other examples
Stereo vinyl records encode both the left and right stereo channels in a single groove. By making the groove a 90-degree cut into the vinyl, variation in one wall was independent of variations in the other wall. The cartridge senses the resultant motion of the stylus following the groove in two orthogonal directions: 45 degrees from vertical to either side.[17]
## Notes
1. ^ Liddell and Scott, A Greek–English Lexicon s.v. ὀρθός
2. ^ Liddell and Scott, A Greek–English Lexicon s.v. γονία
3. ^ Liddell and Scott, A Greek–English Lexicon s.v. ὀρθογώνιον
4. ^ Oxford English Dictionary, Third Edition, September 2004, s.v. orthogonal
5. ^ J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. p. 58. ISBN 0-7167-0344-0.
6. ^
7. ^ Bourbaki, "ch. II §2.4", Algebra I, p. 234
8. ^ Trefethen, Lloyd N. & Bau, David (1997). Numerical linear algebra. SIAM. p. 13. ISBN 978-0-89871-361-9.
9. ^ a b R. Penrose (2007). The Road to Reality. Vintage books. pp. 417–419. ISBN 0-679-77631-1.
10. ^ Michael L. Scott, Programming Language Pragmatics, p. 228
11. ^ 1968, Adriaan van Wijngaarden et al., Revised Report on the Algorithmic Language ALGOL 68, section 0.1.2, Orthogonal design
12. ^ Null, Linda & Lobur, Julia (2006). The essentials of computer organization and architecture (2nd ed.). Jones & Bartlett Learning. p. 257. ISBN 978-0-7637-3769-6.
13. ^ Athanasios Papoulis; S. Unnikrishna Pillai (2002). Probability, Random Variables and Stochastic Processes. McGraw-Hill. p. 211. ISBN 0-07-366011-6.
14. ^ Hedayat, A.; et al. (1999). Orthogonal arrays: theory and applications. Springer. p. 168. ISBN 978-0-387-98766-8.
15. ^ Karver, Mark R.; Hilderbrand, Scott A. (2012). "Bioorthogonal Reaction Pairs Enable Simultaneous, Selective, Multi-Target Imaging". Angewandte Chemie International Edition. 51 (4): 920. doi:10.1002/anie.201104389. PMC 3304098.
16. ^
17. ^ For an illustration, see YouTube.
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2016-09-29 10:49:33
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http://dictionary.sensagent.com/Responding%20variable/en-en/
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# definition - Responding variable
definition of Wikipedia
# Dependent and independent variables
(Redirected from Responding variable)
The terms "dependent variable" and "independent variable" are used in similar but subtly different ways in mathematics and statistics as part of the standard terminology in those subjects. They are used to distinguish between two types of quantities being considered, separating them into those available at the start of a process and those being created by it, where the latter (dependent variables) are dependent on the former (independent variables).[citation needed]
## Simplified example
The independent variable is typically the variable being manipulated or changed and the dependent variable is the observed result of the independent variable being manipulated. For example concerning nutrition, the independent variable of daily vitamin C intake (how much vitamin C one consumes) can influence the dependent variable of life expectancy (the average age one attains). Over some period of time, scientists will control the vitamin C intake in a substantial group of people. One part of the group will be given a daily high dose of vitamin C, and the remainder will be given a placebo pill (so that they are unaware of not belonging to the first group) without vitamin C. The scientists will investigate if there is any statistically significant difference in the life span of the people who took the high dose and those who took the placebo (no dose). The goal is to see if the independent variable of high vitamin C dosage has a correlation with the dependent variable of people's life span. The designation independent/dependent is clear in this case, because if a correlation is found, it cannot be the that life span that has influenced the vitamin C intake, but an influence in the other direction is possible.
## Use in mathematics
In traditional calculus, a function is defined as a relation between two terms called variables because their values vary. Call the terms, for example, x and y. If every value of x is associated with exactly one value of y, then y is said to be a function of x. It is customary to use x for what is called the "independent variable," and y for what is called the "dependent variable" because its value depends on the value of x.[1] Therefore, $y = x^2$ means that y, the dependent variable, is the square of x, the independent variable.[1][2]
The most common way to denote a "function" is to replace y, the dependent variable, by $f(x)$, where f is the first letter of the word "function." Thus, $y = f(x) = x^2$ means that y, a dependent variable, a function of x, is the square of x. Also, in this form, the expression is called an "explicit" function of x, contrasted with $x^2 - y = 0$, which is called an "implicit" function.[1]
## Use in statistics
### Controlled experiments
In a statistics experiment, the dependent variable is the event studied and expected to change whenever the independent variable is altered.[2]
In the design of experiments, an independent variable's values are controlled or selected by the experimenter to determine its relationship to an observed phenomenon (i.e., the dependent variable). In such an experiment, an attempt is made to find evidence that the values of the independent variable determine the values of the dependent variable. The independent variable can be changed as required, and its values do not represent a problem requiring explanation in an analysis, but are taken simply as given. The dependent variable, on the other hand, usually cannot be directly controlled.[citation needed]
Controlled variables are also important to identify in experiments. They are the variables that are kept constant to prevent their influence on the effect of the independent variable on the dependent. Every experiment has a controlling variable, and it is necessary to not change it, or the results of the experiment won't be valid.[citation needed]
"Extraneous variables" are those that might affect the relationship between the independent and dependent variables. Extraneous variables are usually not theoretically interesting. They are measured in order for the experimenter to compensate for them. For example, an experimenter who wishes to measure the degree to which caffeine intake (the independent variable) influences explicit recall for a word list (the dependent variable) might also measure the participant's age (extraneous variable). She can then use these age data to control for the uninteresting effect of age, clarifying the relationship between caffeine and memory.
In summary:
• Independent variables answer the question "What do I change?"
• Dependent variables answer the question "What do I observe?"
• Controlled variables answer the question "What do I keep the same?"
• Extraneous variables answer the question "What uninteresting variables might mediate the effect of the IV on the DV?"
### Alternative terminology in statistics
In statistics, the dependent/independent variable terminology is used more widely than just in relation to controlled experiments. For example the data analysis of two jointly varying quantities may involve treating each in turn as the dependent variable and the other as the independent variable. However, for general usage, the pair response variable and explanatory variable is preferable as quantities treated as "independent variables" are rarely statistically independent.[3][4]
Depending on the context, an independent variable is also known as a "predictor variable," "regressor," "controlled variable," "manipulated variable," "explanatory variable," "exposure variable," and/or "input variable."[5]A dependent variable is also known as a "response variable," "regressand," "measured variable," "observed variable," "responding variable," "explained variable," "outcome variable," "experimental variable," and/or "output variable."[6]
In addition, some special types of statistical analysis use terminology more relevant to the specific context. For example reliability theory uses the term exposure variable for what would otherewise be an explanatory or dependent variable, and medical statistics may use the term risk factor.
### Examples
• If one were to measure the influence of different quantities of fertilizer on plant growth, the independent variable would be the amount of fertilizer used (the changing factor of the experiment). The dependent variables would be the growth in height and/or mass of the plant (the factors that are influenced in the experiment) and the controlled variables would be the type of plant, the type of fertilizer, the amount of sunlight the plant gets, the size of the pots, etc. (the factors that would otherwise influence the dependent variable if they were not controlled).[citation needed]
• In a study of how different doses of a drug affect the severity of symptoms, a researcher could compare the frequency and intensity of symptoms (the dependent variables) when different doses (the independent variable) are administered, and attempt to draw a conclusion.[citation needed]
• In measuring the acceleration of a vehicle, time is usually the independent variable, while speed is the dependent variable. This is because when taking measurements, times are usually predetermined, and the resulting speed of the vehicle is recorded at those times. As far as the experiment is concerned, the speed is dependent on the time. Since the decision is made to measure the speed at certain times, time is the independent variable.[citation needed]
• In measuring the amount of color removed from beetroot samples at different temperatures, the dependent variable would be the amount of pigment removed, since it is depending on the temperature (which is the independent variable).[citation needed]
• In sociology, in measuring the effect of education on income or wealth, the dependent variable could be a level of income or wealth measured in monetary units (United States Dollars for example), and an independent variable could be the education level of the individual(s) who compose(s) the household (i.e. academic degrees).[citation needed]
## References
1. 1.0 1.1 1.2 Thompson, S.P; Gardner, M; Calculus Made Easy. 1998. Page 10-11. ISBN 0312185480.
2. 2.0 2.1 Random House Webster's Unabridged Dictionary. Random House, Inc. 2001. Page 534, 971. ISBN 037572026.
3. Everitt, B.S. (2002) Cambridge Dictionary of Statistics, CUP. ISBN 0-521-81099-x
4. Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9
5. Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 (entries for "independent variable" and "regression")
6. Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9 (entry for "regression")
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2014-11-28 08:28:50
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https://par.nsf.gov/biblio/10131148
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Large-scale star formation in Auriga region
ABSTRACT
New observations in the VI bands along with archival data from the 2MASS and WISE surveys have been used to generate a catalogue of young stellar objects (YSOs) covering an area of about 6° × 6° in the Auriga region centred at l ∼ 173° and b ∼ 1.5°. The nature of the identified YSOs and their spatial distribution are used to study the star formation in the region. The distribution of YSOs along with that of the ionized and molecular gas reveals two ring-like structures stretching over an area of a few degrees each in extent. We name these structures as Auriga Bubbles 1 and 2. The centre of the Bubbles appears to be above the Galactic mid-plane. The majority of Class I YSOs are associated with the Bubbles, whereas the relatively older population, i.e. Class ii objects are rather randomly distributed. Using the minimum spanning tree analysis, we found 26 probable subclusters having five or more members. The subclusters are between ∼0.5 and ∼3 pc in size and are somewhat elongated. The star formation efficiency in most of the subcluster region varies between 5 ${{\ \rm per\ cent}}$ and 20 ${{\ \rm per\ cent}}$ indicating that the subclusters could more »
Authors:
; ; ; ;
Publication Date:
NSF-PAR ID:
10131148
Journal Name:
Monthly Notices of the Royal Astronomical Society
Volume:
492
Issue:
2
Page Range or eLocation-ID:
p. 2446-2467
ISSN:
0035-8711
Publisher:
Oxford University Press
National Science Foundation
##### More Like this
1. Abstract
We used the Immersion GRating Infrared Spectrometer (IGRINS) to determine fundamental parameters for 61 K- and M-type young stellar objects (YSOs) located in the Ophiuchus and Upper Scorpius star-forming regions. We employed synthetic spectra and a Markov chain Monte Carlo approach to fit specificK-band spectral regions and determine the photospheric temperature (T), surface gravity ($logg$), magnetic field strength (B), projected rotational velocity ($vsini$), andK-band veiling (rK). We determinedBfor ∼46% of our sample. Stellar parameters were compared to the results from Taurus-Auriga and the TW Hydrae association presented in Paper I of this series. We classified all the YSOs in the IGRINS survey with infrared spectral indices from Two Micron All Sky Survey and Wide-field Infrared Survey Explorer photometry between 2 and 24μm. We found that Class II YSOs typically have lower$logg$and$vsini$, similarB, and higherK-band veiling than their Class III counterparts. Additionally, we determined the stellar parameters for a sample of K and M field stars also observed with IGRINS. We have identified intrinsic similarities and differences at different evolutionary stages with our homogeneous determination of stellar parameters in the IGRINS YSO survey. Considering$logg$as amore »
2. ABSTRACT
With JWST, new opportunities to study the evolution of galaxies in the early Universe are emerging. Spitzer constraints on rest-optical properties of z ≳ 7 galaxies demonstrated the power of using galaxy stellar masses and star formation histories (SFHs) to indirectly infer the cosmic star formation history. However, only the brightest individual z ≳ 8 objects could be detected with Spitzer, making it difficult to robustly constrain activity at z ≳ 10. Here, we leverage the greatly improved rest-optical sensitivity of JWST at z ≳ 8 to constrain the ages of seven UV-bright ($M_{\rm uv}\lesssim -19.5$) galaxies selected to lie at z ∼ 8.5–11, then investigate implications for z ≳ 15 star formation. We infer the properties of individual objects with two spectral energy distribution modelling codes, then infer a distribution of ages for bright z ∼ 8.5–11 galaxies. We find a median age of ∼20 Myr, younger than that inferred at z ∼ 7 with a similar analysis, consistent with an evolution towards larger specific star formation rates at early times. The age distribution suggests that only ∼3 per cent of bright z ∼ 8.5–11 galaxies would be similarly luminous at z ≳ 15, implying that the number density of brightmore »
3. ABSTRACT
We present a detailed analysis of the ionized gas distribution and kinematics in the inner ∼ 200 pc of NGC 4546, host of a low-luminosity active galactic nucleus (LLAGN). Using GMOS−IFU observations, with a spectral coverage of 4736–6806 Å and an angular resolution of 0.7 arcsec, we confirm that the nuclear emission is consistent with photoionization by an AGN, while the gas in the circumnuclear region may be ionized by hot low-mass evolved stars. The gas kinematics in the central region of NGC 4546 presents three components: (i) a disc with major axis oriented along a position angle of 43° ± 3°, counter rotating relative to the stellar disc; (ii) non-circular motions, evidenced by residual velocities of up to 60 km s−1, likely associated with a previous capture of a dwarf satellite by NGC 4546; and (iii) nuclear outflows in ionized gas, identified as a broad component (σ ∼ 320 km s−1) in the line profiles, with a mass outflow rate of $\dot{M}_{\rm out} = 0.3 \pm 0.1$ M⊙ yr−1 and a total mass of Mout = (9.2 ± 0.8) × 103 M⊙ in ionized gas, corresponding to less than 3 per cent of the total mass of ionized gas in the inner 200 pc of NGC 4546. The kinetic efficiency of themore »
4. ABSTRACT
To unravel the star formation process, we present a multi-scale and multi-wavelength study of the filamentary infrared dark cloud (IRDC) G333.73 + 0.37, which hosts previously known two H ii regions located at its center. Each H ii region is associated with a mid-infrared source, and is excited by a massive OB star. Two filamentary structures and a hub-filament system (HFS) associated with one H ii region are investigated in absorption using the Spitzer 8.0 μm image. The 13CO(J = 2–1) and C18O(J = 2–1) line data reveal two velocity components (around −35.5 and −33.5 km s−1) toward the IRDC, favouring the presence of two filamentary clouds at different velocities. Non-thermal (or turbulent) motions are depicted in the IRDC using the C18O line data. The spatial distribution of young stellar objects (YSOs) identified using the VVV near-infrared data traces star formation activities in the IRDC. Low-mass cores are identified toward both the H ii regions using the ALMA 1.38 mm continuum map. The VLT/NACO adaptive-optics L′-band images show the presence of at least three point-like sources and the absence of small-scale features in the inner 4000 AU around YSOs NIR31 and MIR 16 located toward the H ii regions. The H ii regions and groups of YSO are observed toward the centralmore »
5. ABSTRACT We present a catalogue of 22 755 objects with slitless, optical, Hubble Space Telescope (HST) spectroscopy from the Grism Lens-Amplified Survey from Space (GLASS). The data cover ∼220 sq. arcmin to 7-orbit (∼10 ks) depth in 20 parallel pointings of the Advanced Camera for Survey’s G800L grism. The fields are located 6 arcmin away from 10 massive galaxy clusters in the HFF and CLASH footprints. 13 of the fields have ancillary HST imaging from these or other programs to facilitate a large number of applications, from studying metal distributions at z ∼ 0.5, to quasars at z ∼ 4, to the star formation histories of hundreds of galaxies in between. The spectroscopic catalogue has a median redshift of 〈z〉 = 0.60 with a median uncertainty of $\Delta z / (1+z)\lesssim 2{{\ \rm per\ cent}}$ at $F814\mathit{ W}\lesssim 23$ AB. Robust continuum detections reach a magnitude fainter. The 5 σ limiting line flux is $f_{\rm lim}\approx 5\times 10^{-17}\rm ~erg~s^{-1}~cm^{-2}$ and half of all sources have 50 per cent of pixels contaminated at ≲1 per cent. All sources have 1D and 2D spectra, line fluxes/uncertainties and identifications, redshift probability distributions, spectral models, and derived narrow-band emission-line maps from the Grism Redshift and Line Analysis tool (grizli). We provide other basic sample characterizations,more »
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2023-02-08 02:53:49
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https://www.physicsforums.com/threads/the-fundamental-units-in-physics.910184/
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# B The fundamental units in Physics
Tags:
1. Apr 4, 2017
### Vengo
I have been in my physics crash course for my entrance exam. I was looking at the chapter "Units and Measurement". I got a doubt in this chapter and I didn't get it cleared by my teacher. So I am posting this in this forum. We have 7 fundamental quantities and other quantities are derived quantities.In those fundamental quantities,We have (CURRENT)and its unit is A. but according to the definition. CURRENT is defined a rate of charge flowing across a cross sectional area of a conductor. and the formula is I=q/t. This looks like a derived quantity because t is a fundamental quantity. It is used in the formula. MY QUESTION IS WHETHER CURRENT IS A FUNDAMENTAL OR DERIVED QUANTITY???????????
2. Apr 4, 2017
### Mastermind01
Firstly, you don't need to shout (Capital letters) or bold or italicise your text. We get your question without all that.
Current is a fundamental quantity and ampere is the one the seven SI base units.
To elaborate on your answer, yes current is $\frac{dq}{dt}$ but in SI units current's unit the ampere is defined as
"The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section, and placed one metre apart in vacuum, would produce between these conductors a force equal to 2×10−7 newtons per metre of length"
The fact that ampere is a base unit has to do with historical reasons and the fact that current is easier to measure accurately than charge.
Last edited: Apr 4, 2017
3. Apr 4, 2017
### Vengo
Thank you for clearing my doubt, Master
4. Apr 4, 2017
### f95toli
The definition of the ampere will actually change next year. After 2018 it will be defined via the electron charge which will then be a defined (exact) value giving a more "natural" definition. This also gives a direct connection to the second which is nice.
Note also that the realization of the Ampere described above is never used in practice. In fact, the Ampere is really never really realized directly; it it is always done using Ohm's law using the realizations of the voltage and ohm which we can measure with high precision and accuracy.
Note also that there are some good reasons for why the Ampere (and not the voltage or Ohm) is a base unit; the reasons are nor straightforward but it basically comes down to wanting the SI to be internally self-consistent.
5. Apr 4, 2017
### Staff: Mentor
Note that the distinction between fundamental units and derived units is purely a matter of convention. The SI convention this year is that current is fundamental and charge is derived. Other systems of units take the opposite approach.
It does not matter what convention you use, only that you use it consistently.
6. Apr 4, 2017
### Vengo
Thank you f95toli and Dale
But Dale, I cannot get you precisely
7. Apr 5, 2017
### Staff: Mentor
There are different systems of units: SI units, English units, cgs units, Lorenz Heaviside units, Planck units, etc. Different systems of units will make different choices about which are fundamental and which are derived. For example in SI units current is fundamental and charge is derived, but in cgs units both current and charge are derived
8. Apr 5, 2017
### Vengo
OH! thank you, Dale
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2017-12-11 12:06:41
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