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# Nerve Renew Review – Does It Work?
Nerve Renew is a natural supplement created by the Neuropathy Treatment Group to help combat the pain related to neuropathy. The Neuropathy Treatment Group specializes in finding natural ways to reduce these painful symptoms, working exclusively with this form of pain. Over the years, their trial and error, along with extensive research, has lead the group to create Nerve Renew.
When doctors diagnose neuropathy, they often suggest taking a vitamin B supplement, ones that contain highly concentrated levels of various vitamin B’s as well as alpha lipoic acid. Many users of these treatments have found relief from their neuropathy once they take them continuously. Unfortunately, these supplements are extremely expensive and often need to be mixed and matched to give even slight relief.
Nerve Renew was created to combat the pain that comes from neuropathy, but to do so in an affordable way. With no other supplement currently available that can be compared to Nerve Renew, it is truly a breakthrough for those who are tired of suffering from their neuropathy symptoms.
## What Is Nerve Renew?
Nerve Renew contains a mixture of B vitamins that are extremely powerful and have been clinically proven to support neuropathy. In addition to the B vitamins found in the supplement, Nerve Renew also contains a stabilized form of alpha lipoic acid. To further support the health of those who take the supplement, Nerve Renew contains antioxidants and other herbal extracts.
All these ingredients serve to make Nerve Renew the best natural supplement to combat the pains so often associated with neuropathy. And, to give this relief to as many people as possible, Nerve Renew has committed itself to keeping the price of the supplement as low as possible.
## What is Neuropathy?
Often associated with diabetes, neuropathy often begins as a slight tingling in certain body parts, usual extremities like toes or fingers. Eventually, this tingling sensation can transform into a numbing. While neuropathy may start in small areas of the body, it’s been known to spread, causing these feelings to take over entire limbs.
While tingling and numbing may not seem like critical problems, neuropathy often transforms in a burning sensation. It’s only at this point that medical doctors usually recognize these symptoms as being peripheral neuropathy. Unfortunately, even once diagnosed, many treatments and medications do not have the effect hoped for when it comes to this condition.
### Benefits of Nerve Renew
While understanding how Nerve Renew works is important, knowing its proven benefits is often more important to those who have been struggling with neuropathy pain. Over the years that Nerve Renew has been available to the public, there have been repeated praises for the supplement, with the same benefits being mentioned every single time. These benefits include:
• Reduced Numbness in the Hands, Feet, and Legs
• Reduced Tingling in the Hands, Feet, and Legs
• Less Pain
• Fewer Burning Sensations
• Support for the Nerves and Nerve Linings
• Strengthening of the Nerves and Nerve Linings
• Reduced Stress Levels
• Lower Anxiety Levels
• Increased Balance
• Increased Coordination
And, for those who still doubt that Nerve Renew will provide all these benefits for users, the supplement comes with a one year, money back guarantee.
### The Science Behind Nerve Renew
As mentioned above, using B vitamins to help treat neuropathy is a common practice. However, many of the supplements available today to treat neuropathy use thiamine, a form of vitamin B that the body struggles to absorb. Unfortunately, this means that when thiamine is taken, the body only absorbs a small amount of the supplement, leaving it with the pain that comes with neuropathy.
Unlike other supplements, Nerve Renew doesn’t use thiamine, but benfotiamine. This form of vitamin B1 is structured very uniquely, having an open ring that allows it to pass right into the cell membranes. This direct access to the cells allows the benfotiamine to be absorbed easily by the body, giving users faster and longer lasting relief.
In addition to containing benfotiamine, Nerve Renew also has vitamin B12 in its formula, or methylcobalamine. This form of vitamin B12 is also absorbed easier than the forms of B12 that are more frequently put in B vitamin supplements. In fact, a recent study of this form of B12 found that a high dose of the vitamin could produce the regeneration of nerves. This high dose of B12 is provided in Nerve Renew.
In addition to these amazing forms of vitamin B, Nerve Renew also contains 100% stabilized r-alpha lipoic acid, or R-ALA. This acid is a vital part of the formula, helping reduce neuropathy and general nerve pain. R-ALA works as an antioxidant that, due to its solubility in fat and water, is able to move throughout the body. And, because this acid can regenerate itself, R-ALA is able to keep working in the body long after the supplement has been taken.
### Ingredients in Nerve Renew
While a brief overview of the different vitamins in Nerve Renew has been provided above, Nerve Renew wants every customer to have a clear understanding of every single ingredient included in the supplement. Unlike other companies that want to keep their proprietary blends secret, Nerve Renew wants to be upfront about everything in its supplement. This will give customers a peace of mind they can’t get with other products.
The ingredients included in Nerve Renew are:
• Vitamin B2
• Vitamin B6: Supports nerve health
• Vitamin D
• Feverfew Extracts: Relieve pain and inflammation
• Oat Straw Extract: Helps sooth itchy skin
• Passion Flower: Aids in reducing stress and anxiety
• Skullcap Extract: Increases the blood supply to the brain. Also promotes a calming effect on the nervous system.
As mentioned above, Nerve Renew aims to keep its costs extremely affordable, so those struggling with neuropathy can find the relief they need without having to sacrifice financially. While a bottle of Nerve Renew has been valued at about $125, Nerve Renew offers its supplements for much less. In fact, there are several purchasing options offered by Nerve Renew, each one affordable and each one offer the relief that so many people need. The first option for purchasing Nerve Renew is by trying the Nerve Renew free trial. For the low cost of shipping and handling, customers can get a two week free trial of Nerve Renew. After the two week trial, a 30 day supply will be sent to the customer if they don’t cancel their supply sooner. This monthly supply will cost$43 a month, but can be canceled at any time.
For those who are interested in trying Nerve Renew for a month, but without the trial run, they can purchase a bottle with 120 capsules for $64 plus shipping and handling. This is considered a one month supply and there will be no automatic shipments included in this purchase. Finally, for those who are ready to fully commit to Nerve Renew, there is a three month supply option. While this would usually cost$189, it is currently being offered for \$129. Again, there are no automatic shipments with this purchase.
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## Asset Turnover
### Learning Outcomes
• Calculate the Asset Turnover ratio
The asset turnover ratio can be used as an indicator of how effectively a company uses its assets to generate revenue.
Use the following to calculate the asset turnover ratio: $\dfrac{\text{net sales or revenue}}{\text{average total assets}}$
Note: Long-term investments are not usually included in the calculation because they are not productivity assets used to generate sales to customers.
Jonick Company
Comparative Income Statement
For the Years Ended December 31, 2019 and 2018
Description 2019 2018
Sales $994,000$828,000
Cost of merchandise sold 414,000 393,000
Gross Profit Single Line$580,000 Single Line$435,000
Jonick Company
Comparative Balance Sheet
December 31, 2019 and 2018
2019 2018
Assets
Total current assets $911,000$800,000
Subcategory, Long-term investments:
Investment in equity securities $1,946,000$1,822,000
Subcategory, Property, plant and equipment:
Total property, plant and equipment $1,093,000$984,000
Total assets Single Line$3,950,000Double Line Single Line$3,606,000Double Line
In this case, we’ll reduce total assets by long-term investments. The adjusted long-term assets will be $2,004,000 for 2019 ($3,950,000-$1,946,000) and$1,784,000 ($3,606,000 –$1,822,000) for 2018, and the average of those two amounts is $1,894,000 (($2,004,000+$1,784,000)/2). Sales of$994,000 divided by average total assets of \$1,894,000 comes to 52.5%.
This ratio looks at the value of most of a company’s assets and how well they are leveraged to produce sales. The goal of owning the assets is to generate revenue that ultimately results in cash flow and profit.
The higher the asset turnover ratio, the more efficient a company is at generating revenue from its assets. Conversely, if a company has a low asset turnover ratio, it indicates it is not efficiently using its assets to generate sales.
Sometimes, investors and analysts are more interested in measuring how quickly a company turns its fixed assets or current assets into sales. In these cases, the analyst can use specific ratios, such as the fixed-asset turnover ratio or the working capital ratio to calculate the efficiency of these asset classes.
While the asset turnover ratio considers average total assets in the denominator, the fixed asset turnover ratio looks at only fixed assets. The fixed asset turnover ratio (FAT) is, in general, used by analysts to measure operating performance. This efficiency ratio compares net sales (income statement) to fixed assets (balance sheet) and measures a company’s ability to generate net sales from property, plant, and equipment (PP&E). The fixed asset balance is used net of accumulated depreciation. A higher fixed asset turnover ratio indicates that a company has more effectively utilized its investment in fixed assets to generate revenue.
The asset turnover ratio should be used to compare stocks that are similar and should be used in trend analysis to determine whether asset usage is improving or deteriorating.
The asset turnover ratio may be artificially deflated when a company makes large asset purchases in anticipation of higher growth. Likewise, selling off assets to prepare for declining growth will artificially inflate the ratio. Many other factors (such as seasonality) can also affect a company’s asset turnover ratio during interim periods (such as comparing quarterly results of a retailer).
Now, check your understanding of how to calculate the Asset Turnover ratio.
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# Use of fission products for electricity generation
Why can't we use fissions products for electricity production ?
As far has I know fissions products from current nuclear power plants create enough 'waste' heat to boil water; and temperature decreases too slowly for an human life. So why can't we design a reactor to use this energy.
• I assume that You mean spent fuel, right? Because actually the main role of fission products (or rather the energy emitted during fission) is to heat water and produce electricity. – Wojciech Mar 15 '14 at 21:55
• I assume you already looked up the thermal output of the reactor waste (typically expressed in units of W/Kg) and decided that it was economically viable, right? – DumpsterDoofus Mar 15 '14 at 21:56
• Yes. The wastes we are trying to put in the deep of the earth for long time. – Galigator Mar 15 '14 at 21:58
• @DumpsterDoofus economically viable is a point of view, because waste already exists and we spend a lot of money to manage them. – Galigator Mar 15 '14 at 22:00
• Actual spent fuel form only couple % of the fuel rod. Many countries reprocess spent fuel rods to retrieve uranium which didn't burn out and use it again as a fuel. I can guarantee that this is much more economic. – Wojciech Mar 15 '14 at 22:06
Here are some "order-of-magnitude" arguments:
After one year, typical spent nuclear fuel generates about 10 kW of decay heat per tonne, decreasing to about 1 kW/t after ten years
Now since this is heat, you can't convert it to electricity with 100% efficiency, the maximum possible efficiency is given by the Carnot efficiency $\eta$:
$$\eta \le 1 - \dfrac{T_\mathrm{cold}}{T_\mathrm{hot}}$$
where $T_\mathrm{hot}$ would be the temperature of the spent fuel rods (in Kelvin) and $T_\mathrm{cold}$ would be the temperature of a cold reservoir against which a generator would work. One would have to do another calculation what a reasonable temperature of the fuel rods would be (in practice they currently seem to be kept at 50 degrees C).
With 'primary' fuel typically 55 Gigawatt days per tonne can be produced, i.e. a 1 Gigawatt powerplant would use 365.25 / 55 = 6.6 tonnes per year.
Even assuming you would be able to convert this to electricity with 100% efficiency and assuming an average 5 kilowatts per tonne over 10 years, this would yield about 18'000 Kilowatt days or 0.018 Gigawatt days, about 0.03% of the primary energy production.
You'll also see from the Carnot efficiency above that higher temperatures imply a higher possible efficiency, i.e. if one can spend some energy to extract the still fissionable material to be used in a reactor again, that is likely to be more efficient in terms of electricity generation.
It's true on the other hand that radioisotope thermoelectric generators (radioactive sources combined with thermocouples) have been used on satellite missions.
• So the production per tonne could be about the same as a medium wind turbine. I have heard that radio-isotope thermoelectric generators are'nt popular because of security reason. End of life device need specials cares. – Galigator Oct 5 '14 at 7:55
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# Probability of multiple tries at once VS same amount of tries one at a time at different times?
For example, the chance of winning a game is 1%. Do I get the same outcome of playing 100 games in the following two ways.
1. Play one game a day for 100 days.
2. Play 100 games all at once, one after another until 100 games are played.
Note: The game is not lottery, you can only participate once per game.
From my experiences, it turns out I get better chances of winning the game when I play more games all at once compare to one game at a time spread through a long period of time. Is this correct? whether it's correct or not, any proofs to back it up?
• mathematically it should not make any difference, if the game tries are really independent of each other. when you play in a row, you play differently than over time... – gt6989b Jun 19 '17 at 21:22
• When you say play 100 games all at once, are you thinking of buying 100 tickets to the same lottery? Otherwise, what do you mean by playing all at once? – Ross Millikan Jun 19 '17 at 21:23
• If we were to assume that the results of the games played are independent from one another (which includes but is not limited to the result of the match and the time of day the game was played, etc...) then there shouldn't be any difference. An example of such a game would be rolling a $100$-sided die and trying to roll a 100. An example where it is not valid is as Ross points out playing the lottery. The result of the first game will influence the result of the second (if you guessed wrong you won't guess the same thing again the second time) but this violates the phrase "chance is 1%" – JMoravitz Jun 19 '17 at 21:28
• 100 games at once I meant playing it one after another for 100 games. – s-hunter Jun 19 '17 at 21:29
If you look at in in a purely probabilistic way, (i.e., the probability is $0.01$ every time) then the probability should be exactly the same either way. However, it may be true that the situation cannot be modeled so simply. Other variables often affect the gameplay. For example, if you play all $100$ games at the same time, you are more likely to get "in the zone" and play better and better each time, whereas if you wait, you let your experience dissolve in between games.
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# JEE Main & Advanced Mathematics Three Dimensional Geometry Angle of Intersection of Two Spheres
## Angle of Intersection of Two Spheres
Category : JEE Main & Advanced
If the angle of intersection of two spheres is a right angle, the spheres are said to be orthogonal.
Condition for orthogonality of two spheres :
Let the equation of the two spheres be
${{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2ux+2vy+2wz+d=0$ .....(i)
and ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}+2{u}'x+2{v}'y+2{w}'z+{d}'=0$ .....(ii)
If the sphere (i) and (ii) cut orthogonally, then $2u{u}'+2v{v}'+2w{w}'=d+{d}',$ which is the required condition.
• If the spheres ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}={{a}^{2}}$ and ${{x}^{2}}+{{y}^{2}}+{{z}^{2}}$ $+2ux+2vy+2wz+d=0$ cut orthogonally, then $d={{a}^{2}}$.
• Two spheres of radii ${{r}_{1}}$ and ${{r}_{2}}$ cut orthogonally, then the radius of the common circle is $\frac{{{r}_{1}}{{r}_{2}}}{\sqrt{r_{1}^{2}+r_{2}^{2}}}$.
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# Sign rank, VC dimension and spectral gaps
יום א', 11/01/2015 - 14:00
Speaker:
Seminar:
Place:
Abstract:
We study the maximum possible sign rank of $N \times N$ sign matrices with a given VC dimension $d$. For $d=1$, this maximum is $3$. For $d=2$, this maximum is $\tilde{\Theta}(N^{1/2})$. Similar (slightly less accurate) statements hold for $d>2$ as well. We discuss the tightness of our methods, and describe connections to combinatorics, communication complexity and learning theory.
We also provide explicit examples of matrices with low VC dimension and high sign rank. Let $A$ be the $N \times N$ point-hyperplane incidence matrix of a finite projective geometry with order $n \geq 3$ and dimension $d \geq 2$. The VC dimension of $A$ is $d$, and we prove that its sign rank is larger than $N^{\frac{1}{2}-\frac{1}{2d}}$. The large sign rank of $A$ demonstrates yet another difference between finite and real geometries.
To analyze the sign rank of $A$, we introduce a connection between sign rank and spectral gaps, which may be of independent interest. Consider the $N \times N$ adjacency matrix of a $\Delta$-regular graph with a second eigenvalue (in absolute value) $\lambda$ and $\Delta \leq N/2$. We show that the sign rank of the signed version of this matrix is at least $\Delta/\lambda$. A similar statement holds for all regular (not necessarily symmetric) sign matrices. We also describe limitations of this approach, in the spirit of the Alon-Boppana theorem.
Joint work with Noga Alon and Amir Yehudayoff.
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# Math Help - Normal distribution
1. ## Normal distribution
$\\\text{If }X\text{has a normal distribution with mean }1 \text{ and variance }4\text{, then }\\\\P[X^2-2X\leq 8]=?$
My question whether there is another way of approaching this problem other than realizing the probability in question can be rewritten as:
$\\P[X^2-2X\leq 8]= & P[X^2-2X+1\leq8+1]\\\\ & =P[(X-1)^2\leq 9]\\\\ & =P[-3\leq X-1\leq3]\\\\ & =P[-1.5\leq \frac{X-1}{2}\leq1.5]$
The reason that I ask is that I had a hard time seeing it.
2. ## Re: Normal distribution
Hey downthesun01.
The transformation you used is probably the best way to do it. Some alternatives include looking at (X-1)^2 [in terms of a chi-square distribution after centering) or using the transformation theorem to get the exact PDF of X^2 + 2X.
Again, the best thing in terms of simplicity is to do what you have done.
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# Unlike the Shiites, who constitute the other major branch of
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Unlike the Shiites, who constitute the other major branch of [#permalink] 06 Jul 2008, 18:17
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902.Unlike the Shiites, who constitute the other major branch of Islam, the Sunnites do not await the Mahdi as a messenger from God, nor do they endow him with divine qualities or immunity from failure in judgment.
(A) nor do they endow him
(B) but they do not endow him
(C) neither do they endow him
(D) and they neither endow him
(E) while endowing him neither
Friends, can you help me how to tell apart A nd C? Thanks!
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Re: 902.SC [#permalink] 06 Jul 2008, 18:25
Typically "nor" is used with the idiom "neither...nor", but it can be used alone, although rarely.
I think C is correct.
when the sentence uses "neither" is kind of like saying "also". This is not a very good explanation. I'm very tired right now.
Tough question. I hope I helped at least a little bit.
sondenso wrote:
902.Unlike the Shiites, who constitute the other major branch of Islam, the Sunnites do not await the Mahdi as a messenger from God, nor do they endow him with divine qualities or immunity from failure in judgment.
(A) nor do they endow him
(B) but they do not endow him
(C) neither do they endow him
(D) and they neither endow him
(E) while endowing him neither
Friends, can you help me how to tell apart A nd C? Thanks!
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Re: 902.SC [#permalink] 06 Jul 2008, 18:56
1
This post was
BOOKMARKED
I disagree. To me A is better than C. As JM explained nor can come up with out neither. I am not sure about the vice versa.
Does neither needs nor by all means? this has been bugging me for a while. Can some one explain?
Getting back to the Q, Look at the classic disclaimer of any website about their association with GMAC
The Graduate Management Admission Council does not endorse, nor is it affiliated in any way with the owner or any content of this Web site.
Now compare our Q
The Sunnites do not await the Mahdi as a messenger from God, nor do they endow him with divine qualities or immunity from failure in judgment.
I confidently feel that the answer is A. What is OA?
On a side note, I would like to get to the bottom of neither/nor, either/or and not only but also. How one part of them can appear with out other?
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Re: 902.SC [#permalink] 07 Jul 2008, 11:34
sondenso wrote:
902.Unlike the Shiites, who constitute the other major branch of Islam, the Sunnites do not await the Mahdi as a messenger from God, nor do they endow him with divine qualities or immunity from failure in judgment.
(A) nor do they endow him
(B) but they do not endow him
(C) neither do they endow him
(D) and they neither endow him
(E) while endowing him neither
Friends, can you help me how to tell apart A nd C? Thanks!
We need a coordinating conjunctions to join 2 independent sentence.
"Neither" is not a coordination conjunction
Nor, but, and - coordination conjunctions
We need to draw a contrast .Therefore " but" and " and" in B and D are not correct.
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Re: 902.SC [#permalink] 07 Jul 2008, 15:08
1
This post was
BOOKMARKED
It is A.
Here is a rule...
Neither must always be used before a nor...
Nor can be used without neither and as a co-ordinating conjunction.
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Re: 902.SC [#permalink] 07 Jul 2008, 17:44
Many many thanks!
Ashwin_Mohan wrote:
Neither must always be used before a nor...
Nor can be used without neither and as a co-ordinating conjunction
icandy wrote:
"Neither" is not a coordination conjunction
Nor, but, and - coordination conjunctions
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Re: Unlike the Shiites, who constitute the other major branch of [#permalink] 15 Apr 2015, 08:43
Hello from the GMAT Club VerbalBot!
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Unlike the Shiites, who constitute the other major branch of [#permalink] 15 Apr 2015, 13:44
presence of comma indicates that a FANBOY must be present.
would like to see the OA and OE.
Unlike the Shiites, who constitute the other major branch of [#permalink] 15 Apr 2015, 13:44
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# How do you factor y^4+6y^2+25?
Jul 23, 2016
$\left({y}^{4} + 6 {y}^{2} + 25\right)$
= $\left(y - 1 - 2 i\right) \left(y + 1 + 2 i\right) \left(y + 1 - 2 i\right) \left(y - 1 + 2 i\right)$
#### Explanation:
If we use ${y}^{2} = x$, the polynomial ${y}^{4} + 6 {y}^{2} + 25$ is equivalent to ${x}^{2} + 6 x + 25$, a quadratic polynomial. In this the discriminant $\left({b}^{2} - 4 a c\right)$ for $\left(a {x}^{2} + b x + c\right)$ is ${6}^{2} - 4 \times 1 \times 25 = 36 - 100 = - 64$ and hence ${x}^{2} + 6 x + 25$ does not have real zeros or factors.
Its zeros are given by quadratic formula $\frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$ are $\frac{- 6 \pm \sqrt{{6}^{2} - 4 \times 1 \times 25}}{2 \times 1} = \frac{- 6 \pm \sqrt{- 64}}{2}$ or
$\frac{- 6 \pm 8 i}{2} = - 3 \pm 4 i$
Hence ${x}^{2} + 6 x + 25 = \left(x + 3 - 4 i\right) \left(x + 3 + 4 i\right)$ and
${y}^{4} + 6 {y}^{2} + 25 = \left({y}^{2} + 3 - 4 i\right) \left({y}^{2} + 3 + 4 i\right)$
Now to factorize further, let us find zeros of $\left({y}^{2} + 3 - 4 i\right)$ and $\left({y}^{2} + 3 + 4 i\right)$
For $\left({y}^{2} + 3 - 4 i\right)$, quadratic formula gives its zeros as $\frac{\pm \sqrt{- 4 \times 1 \times \left(3 - 4 i\right)}}{2}$ or $\left(\pm \sqrt{- 3 + 4 i}\right)$ and assuming
$- 3 + 4 i = {\left(a + b i\right)}^{2}$, then
${a}^{2} - {b}^{2} + 2 a b i = - 3 + 4 i$ which gives us
${a}^{2} - {b}^{2} = - 3$ and $2 a b = 4$. As such ${a}^{2} + {b}^{2} = \sqrt{{\left({a}^{2} - {b}^{2}\right)}^{2} + 4 {a}^{2} {b}^{2}} = \sqrt{9 + 16} = 5$.
Hence ${a}^{2} = 1$ and ${b}^{2} = 4$, and possible solutions for $\left(a , b\right)$ are $\left(1 , 2\right)$ and $\left(- 1 , - 2\right)$ (as $a b$ is positive).
Hence zeros of $\left({y}^{2} + 3 - 4 i\right)$ are $1 + 2 i$ and $- 1 - 2 i$ and hence
$\left({y}^{2} + 3 - 4 i\right) = \left(y - 1 - 2 i\right) \left(y + 1 + 2 i\right)$.
Similarly $\left({y}^{2} + 3 + 4 i\right) = \left(y + 1 - 2 i\right) \left(y - 1 + 2 i\right)$ and hence
$\left({y}^{4} + 6 {y}^{2} + 25\right) = \left(y - 1 - 2 i\right) \left(y + 1 + 2 i\right) \left(y + 1 - 2 i\right) \left(y - 1 + 2 i\right)$
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## Math Monday
Next Monday, March 23, I'll be giving a Math Monday talk at Saint Mary's on Continued Fractions. The talk will be at 4 p.m. in Galileo 201. Students, etc. of all mathematical backgrounds are invited. Refreshments will be served.
Abstract: As we all learned long ago, any real number can be expressed using a decimal representation; for example, $\sqrt{2}=1.4142135623...$ Because $\sqrt{2}$ is irrational, we can be assured that its decimal expansion will not terminate, nor will it establish any sort of a pattern. However, upon expressing $\sqrt{2}$ as a continued fraction, a clear pattern becomes immediately apparent!
As it turns out, many of our favorite irrational numbers have lovely (or at least interesting) continued fraction expansions. In this Math Monday, we will discuss the mysterious history of continued fractions, learn a method for constructing them, and find out why they are useful.
## Broadcasting from the Bay Area!
For those of you who missed the news, I have accepted a tenure-track position at Saint Mary's College of California. My new contact information can be found by clicking the About tab.
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# I What's the significance of the phase in a coherent state?
1. May 17, 2017
### vancouver_water
In a coherent state defined by $$|\alpha\rangle = \exp{\left(-\frac{|\alpha|^2}{2}\right)}\exp{\left(\alpha \hat{a}^\dagger\right)} |0\rangle$$ there is a definite phase associated with the state by $$\alpha = |\alpha| \exp{\left(i\theta\right)}$$ where the number operator and phase operator are conjugates, $$-i\partial_{\theta} = \hat{n}.$$ The meaning of the number operator is obvious but what is the significance of the phase in this state? What would be a consequence of picking a new phase for this state?
2. May 17, 2017
### DrDu
Take the expectation value of x and p and you will see.
3. May 23, 2017
### vancouver_water
So i get $\langle x\rangle \propto \Re{(\alpha)}$ and $\langle p \rangle \propto \Im{(\alpha)}$. Which gives the relation $\langle p \rangle = m\omega\langle x \rangle \tan\theta$. So it gives the relation between expectations of x and p.
4. May 23, 2017
### DrDu
Yes. Are you familiar with the phase space from classical mechanics, or action-angle variables?
5. May 23, 2017
### vancouver_water
With phase space yes, but not with action angle variables. I'll read about them though, Thanks!
6. May 23, 2017
### DrDu
Also take in mind that alpha is time dependent as coherent states aren't energy eigenstates.
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## Exercises - Matrices
1. The following code creates in R the $5 \times 5$ matrix shown below.
$$\begin{bmatrix} 11 & 12 & 13 & 14 & 15\\ 21 & 22 & 23 & 24 & 25\\ 31 & 32 & 33 & 34 & 35\\ 41 & 42 & 43 & 44 & 45\\ 51 & 52 & 53 & 54 & 55 \end{bmatrix}$$
> m = rbind(c(11,12,13,14,15),
c(21,22,23,24,25),
c(31,32,33,34,35),
c(41,42,43,44,45),
c(51,52,53,54,55))
Using the above, write R code to generate the following sub-matrices of m:
1. the sub-matrix of elements in rows 3, 4, or 5 and in columns 1 or 2
2. the sub-matrix formed by deleting columns 2 and 4
3. the sub-matrix of rows whose first column element has an even ten's digit
4. the sub-matrix of elements in column 5 that correspond to an element in the same row and the first column that is less than 50. (Hint: the result should be a column matrix, not a vector.)
# (a)
> m[3:5,1:2]
[,1] [,2]
[1,] 31 32
[2,] 41 42
[3,] 51 52
# (b)
> m[,-c(2,4)]
[,1] [,2] [,3]
[1,] 11 13 15
[2,] 21 23 25
[3,] 31 33 35
[4,] 41 43 45
[5,] 51 53 55
# (c)
> m[(m[,1] %/% 10) %% 2 == 0,]
[,1] [,2] [,3] [,4] [,5]
[1,] 21 22 23 24 25
[2,] 41 42 43 44 45
# (d)
> m[m[,1]<50,5,drop=FALSE]
[,1]
[1,] 15
[2,] 25
[3,] 35
[4,] 45
2. Use R to determine the matrix product (in the traditional mathematical sense) of the following matrices:
$$A = \begin{bmatrix} 7 & 8 & 2 \\ 5 & 9 & 1 \\ 4 & 6 & 3 \end{bmatrix} \quad \textrm{ and } \quad B = \begin{bmatrix} 13 & -17\\ 0 & 19\\ -3 & -5 \end{bmatrix}$$
There are many ways to construct the matrices in question before multiplying them together. Below, we use the matrix() function to create matrix $A$ and the cbind() function to create matrix $B$. Note that byrow=FALSE in the matrix() function by default, so we don't have to specify it when we list the matrix contents by column. Also remember there is a rbind() function we could have used too.
> A = matrix(c(7,5,4,8,9,6,2,1,3),nrow=3)
> B = cbind(c(13,0,-3),c(-17,1,-5))
> A %*% B
[,1] [,2]
[1,] 85 -121
[2,] 62 -81
[3,] 43 -77
3. Matrices can be used to solve systems of linear equations. For example, one can solve for $x$, $y$, and $z$ in the system of equations \begin{align*} ax + by + cz &= p\\ dx + ey + fz &= q\\ gx + hy + iz &= r\\ \end{align*} by finding the product $$\begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i \end{bmatrix}^{-1} \cdot \begin{bmatrix}p\\q\\r\end{bmatrix}$$ Use a similar calculation to solve the following system:
\begin{align*} 2x-y+5z+w &= -3\\ 3x+2y+2z-6 &= -32\\ x+3y+3z-w &= -47\\ 5x-2y-3z+3w &= 49 \end{align*}
> M = rbind(c(2,-1,5,1),
c(3,2,2,-6),
c(1,3,3,-1),
c(5,-2,-3,3))
> C = rbind(-3,-32,-47,49)
> solve(M) %*% C
[,1]
[1,] 2
[2,] -12
[3,] -4
[4,] 1
Alternatively, one can simply execute the following and get the same answer:
> solve(M,C)
So $x=2, y=-12, z=-4, \textrm{ and } w=1$.
4. Suppose the following matrix is defined in R: $$\bf{A} = \begin{bmatrix}1 & 1 & 3\\5 & 2 & 6\\-2 & -1 & -3\end{bmatrix}$$
1. Use R to verify that $\bf{A}^3 = \bf{0}$ where $\bf{0}$ is a $3 \times 3$ matrix with every entry equal to $0$.
2. Use R to create a new matrix $\bf{B}$ equal to $\bf{A}$, except that its third column has been replaced by the sum of its second and third columns.
Check (a) with
A %*% A %*% A
For (b), the matrix B should display as the following:
> B
[,1] [,2] [,3]
[1,] 1 1 4
[2,] 5 2 8
[3,] -2 -1 -4
5. Create a $6 \times 10$ matrix of random integers chosen from $1,2,\ldots,10$ by executing the following two lines of code:
set.seed(75)
m = matrix( sample(10, size=60, replace=T), nrows=6)
Use R to find the number of entries in this matrix which are greater than 4.
6. sum(m>4)
7. Create the following patterned matrices. In each case your solution should make use of the special form of the matrix -- this means that the solution should easily generalize to creating a larger matrix with the same structure and should not involve typing in all the entries of the matrix.
(a) $\displaystyle{ \begin{bmatrix} 0 & 1 & 2 & 3 & 4\\ 1 & 2 & 3 & 4 & 0\\ 2 & 3 & 4 & 0 & 1\\ 3 & 4 & 0 & 1 & 2\\ 4 & 0 & 1 & 2 & 3 \end{bmatrix}}$ (b) $\displaystyle{ \begin{bmatrix} 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9\\ 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 0\\ \vdots & \vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\vdots &\\ 8 & 9 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7\\ 9 & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \end{bmatrix}}$
(c) $\displaystyle{ \begin{bmatrix} 0 & 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1\\ 1 & 0 & 8 & 7 & 6 & 5 & 4 & 3 & 2\\ 2 & 1 & 0 & 8 & 7 & 6 & 5 & 4 & 3\\ 3 & 2 & 1 & 0 & 8 & 7 & 6 & 5 & 4\\ 4 & 3 & 2 & 1 & 0 & 8 & 7 & 6 & 5\\ 5 & 4 & 3 & 2 & 1 & 0 & 8 & 7 & 6\\ 6 & 5 & 4 & 3 & 2 & 1 & 0 & 8 & 7\\ 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 & 8\\ 8 & 7 & 6 & 5 & 4 & 3 & 2 & 1 & 0 \end{bmatrix}}$
Hint: Use the outer() function and the %% operator.
8. Use R to calculate the following sums:
(a) $\displaystyle{\sum_{i=1}^{20} \sum_{j=1}^5 \frac{i^4}{(3+ij)}}$ (b) $\displaystyle{\sum_{i=1}^{10} \sum_{j=1}^i \frac{i^4}{(3+ij)}}$
(a) 89912.02; (b) 6944.743
9. Suppose one wishes to simulate how frequently a new dam will overflow, which depends on the number of rainy days seen in a row. We might start by making observations over some sequence of consecutive days of whether each day was rainy (R) or sunny (S). What we observed is shown below.
RRRRRRRRSSSSSSSRSSSSSRSSSSRRRRRRRRRRSRRRRSSSSSS
It appears that exactly half of the days observed were rainy. As such, one might try to simulate similar data by essentially "flipping a coin" for each day so that the chance that it rains on any particular day is 50%. This would be easy to do in R with the following:
s = sample(c("R","S"),size=28,replace=TRUE,prob=c(0.5,0.5))
paste0(s,collapse="")
Below is some data simulated in this way:
RRRSSSRRRRSSSRRRSRSRRRRSSSRSSRRRSRSSSSSRRSSRRRS
Notice how the simulated sequence doesn't look quite like the actual observed data. In the original data, there are long sequences of R's and S's, while the observed data has much shorter sequences. Granted, this is just one sequence, so additional observations and simulations are made. Strangely, the same pattern is seen in these as well. The observations always seem to have longer subsequences of R's and S's than the simulated data.
Upon reflection, you might realize that one possible reason for the discrepancy is that the probability that any particular day is sunny or rainy is not independent of whether or not the previous day was sunny or rainy. In particular, it appears that if a day is rainy, the probability the next day is rainy is higher than if it was sunny. Likewise, if a day is sunny, the probability the next day is sunny is higher than if it was rainy.
Based on all your observations, it appears that consecutive days have the same weather 90% of the time. With this, we can build the following model:
Such a model is called a Markov chain, named after Andrey Markov, a Russian mathematician who first published a paper on the topic in 1906.
In the model above, there are two "states" (R and S). However, a general Markov chain can have many more. One of the more interesting characteristics of a Markov chain is that the probability that you land in any one state after some number of iterations (i.e., days in the example discussed) can be found through a power of an associated transition matrix.
Let's see how this works...
Suppose it is rainy today (day 1), and you want to know what is the probability that it is rainy or sunny tomorrow (day 2). Of course, the answer is simple -- 90% chance of rain, 10% chance of sun.
Suppose however that you want to know the probability that it is rainy or sunny the day after that (day 3). We can find such a probability using our basic probability rules. For the calculations that follow, suppose $R_i$ is the event "day $i$ was rainy" and $S_i$ is the event "day $i$ was sunny". Then notice:
$$\begin{array}{rcl} P(R_3) &=& P((R_2 \, \& \, R_3) \, \textrm{or} \, (S_2 \, \& \, R_3))\\ &=& P(R_2 \, \& \, R_3) + P(S_2 \, \& \, R_3)\\ &=& P(R_2)P(R_3 | R_2) + P(S_2)P(R_3 | S_2)\\ &=& (0.90)(0.90) + (0.10)(0.10)\\ &=& 0.82 \end{array}$$
We could of course use the fact that R and S are complements to find $P(S_3)$, but let's find it in a manner similar to the above:
$$\begin{array}{rcl} P(S_3) &=& P((R_2 \, \& \, S_3) \, \textrm{or} \, (S_2 \, \& \, S_3))\\ &=& P(R_2 \, \& \, S_3) + P(S_2 \, \& \, S_3)\\ &=& P(R_2)P(S_3 | R_2) + P(S_2)P(S_3 | S_2)\\ &=& (0.90)(0.10) + (0.10)(0.90)\\ &=& 0.18 \end{array}$$
Similar calculations reveal the probabilities for day 4:
$$\begin{array}{rcll} P(R_4) &=& (0.90)(.82) + (0.10)(.18) \quad & [= 0.756]\\ P(S_4) &=& (0.10)(0.82) + (0.90)(0.18) \quad & [= 0.244] \end{array}$$
More generally, we have the following relationship:
$$\begin{array}{rcl} P(R_{i+1}) &=& 0.90 \cdot P(R_i) + 0.10 \cdot P(S_i)\\ P(S_{i+1}) &=& 0.10 \cdot P(R_i) + 0.90 \cdot P(S_i)\\ \end{array}$$
Which we could express using matrices with
$$\begin{bmatrix} P(R_{i+1})\\ P(S_{i+1}) \end{bmatrix} = \begin{bmatrix} 0.90 & 0.10\\ 0.10 & 0.90\\ \end{bmatrix} \cdot \begin{bmatrix} P(R_{i})\\ P(S_{i}) \end{bmatrix}$$
The middle matrix above (i.e., the one with the 0.90's and 0.10's in it) is called the transition matrix for this Markov chain. It can be applied over and over again to find the probabilities associated with both states after any number of days.
With all of this in mind, write a function p.of.rain() that if it rains on day 1, returns the probability that it also rains on both days 7 and 8.
Remember the events of raining on day 7 and raining on day 8 are not independent!
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# Python Program to Determine Whether a Given Number is Even or Odd Recursively
PythonServer Side ProgrammingProgramming
#### Beyond Basic Programming - Intermediate Python
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When it is required to check if a given number is an odd number or an even number using recursion, recursion can be used.
The recursion computes output of small bits of the bigger problem, and combines these bits to give the solution to the bigger problem.
## Example
Below is a demonstration for the same −
Live Demo
def check_odd_even(my_num):
if (my_num < 2):
return (my_num % 2 == 0)
return (check_odd_even(my_num - 2))
my_number = int(input("Enter the number that needs to be checked:"))
if(check_odd_even(my_number)==True):
print("The number is even")
else:
print("The number is odd!")
## Output
Enter the number that needs to be checked:48
The number is even
## Explanation
• A method named ‘check_odd_even’ is defined, that takes a number as parameter.
• If the number is less than 2, the remainder of the number when divided by 2 is computed, and checked wih 0.
• The function is called again, and this time, the parameter passed is the number decremented by 2.
• Outside the function, a number is taken as input by the user.
• The function is called, and checked to see if it is ‘True’, if yes, it is determined as an even number.
• Else it is considered an odd number.
• It is returned as output.
Updated on 12-Mar-2021 13:03:21
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시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율
2 초 256 MB 0 0 0 0.000%
## 문제
A $d$-dimensional collection is a set of vectors such that:
• Each coordinate of each vector in the set is a non-negative integer number.
• Let a vector $v = (v_1, \ldots, v_d)$ belong to the set. Then every vector $u = (u_1, \ldots, u_d)$ such that all $u_i$ are non-negative integers and $u$ does not exceed $v$ coordinate-wise (that is, for every index $i$ from $1$ to $d$ the inequality $0 \leq u_i \leq v_i$ must hold) must belong to the set as well.
For example, $\{(0, 0), (0, 1), (1, 0)\}$ and $\{(0, 0), (1, 0), (2, 0), (3, 0)\}$ are $2$-dimensional collections, while $\{(0, 1), (1, 0), (1, 1)\}$, $\{(0, 0), (-1, 0)\}$, and $\{(0, 0), (0, \frac{1}{2}), (0, 1)\}$ are not.
You have to store two identical $d$-dimensional collections, each of size $n$. In order to do that, you gathered $n$ identical $d$-dimensional containers each of which has a capacity given by the capacity vector $c = (c_1, \ldots, c_d)$. You decided to choose a container for each vector from each collection in such a way that every container would contain exactly one vector from the first collection and exactly one vector from the second collection. However, if two vectors $v = (v_1, \ldots, v_d)$ and $u = (u_1, \ldots, u_d)$ are placed in the same container, it must be verified that $v_i + u_i \leq c_i$ for every $i$; that is, the vector $v + u$ must not exceed the capacity vector $c$ coordinate-wise.
It is also guaranteed that each vector from each collection fits in any container by itself; that is, each vector $v$ of each collection does not exceed the capacity vector $c$ coordinate-wise.
Finding an arrangement to place all those vectors in containers turned out to be hard, so you decided to write a program that will solve this problem.
## 입력
The first line of input contains two space-separated positive integers $n$ and $d$ ($1 \leq n \cdot d \leq 10^5$) --- the size and the dimension of each collection.
The second line of input contains $d$ space-separated integers $c_1$, $\ldots$, $c_d$ ($0 \leq c_i \leq 10^9$).
Next $n$ lines of input contain description of the collections; $i$-th of these lines contains $d$ space-separated integers $v_{i 1}$, $\ldots$, $v_{i d}$ ($0 \leq v_{i j} \leq 10^9$) --- coordinates of $i$-th vector in the collection. It is guaranteed that the described set of vectors is indeed a $d$-dimensional collection according to the definition above, and $v_{i j} \leq c_j$.
## 출력
Output $n$ lines, with two space-separated integers in each --- the numbers of vectors from the first and the second collection respectively that must be put in the $i$-th container. The numeration of vectors in the input is 1-based. Every vector from every collection must be assigned to exactly one container.
If there is more than one possible arrangement, output any one of them. It is guaranteed that there is at least one arrangement that satisfies all the requirements.
## 예제 입력 1
4 2
1 1
0 0
0 1
1 0
1 1
## 예제 출력 1
1 4
2 3
3 2
4 1
## 예제 입력 2
4 2
2 1
0 0
1 0
2 0
0 1
## 예제 출력 2
1 4
2 2
3 1
4 3
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# PHP Array Operators
## Introduction
PHP defines following set of symbols to be used as operators on array data types
SymbolExampleNameResult
+$a +$bUnionUnion of $a and$b.
==$a ==$bEqualityTRUE if $a and$b have the same key/value pairs.
===$a ===$bIdentityTRUE if $a and$b have the same key/value pairs in the same order and of the same types.
!=$a !=$bInequalityTRUE if $a is not equal to$b.
<>$a <>$bInequalityTRUE if $a is not equal to$b.
!==$a !==$bNon-identityTRUE if $a is not identical to$b.
## Union of arrays
The Union operator appends the right-hand array appended to left-hand array. ; If a key exists in both arrays, the elements from the left-hand array will be used, and the matching elements from the right-hand array will be ignored.
Following example shows use of define() function to define constants
## Example
Live Demo
<?php
$arr1=array("phy"=>70, "che"=>80, "math"=>90);$arr2=array("Eng"=>70, "Bio"=>80,"CompSci"=>90);
$arr3=$arr1+$arr2; var_dump($arr3);
?>
## Output
Following result will be displayed
array(6) {
["phy"]=>
int(70)
["che"]=>
int(80)
["math"]=>
int(90)
["Eng"]=>
int(70)
["Bio"]=>
int(80)
["CompSci"]=>
int(90)
}
## comparison of arrays
Two arrays are said to be equal if they have same key-value pairs. Following example has an indexed array and other associative array with keys corresponding to index of elements in first. Hence both are equal
## Example
Live Demo
<?php
$arr1=array(0=>70, 2=>80, 1=>90);$arr2=array(70,90,80);
var_dump ($arr1==$arr2);
var_dump ($arr2!=$arr1);
?>
## Output
Following result will be displayed
bool(true)
bool(false)
## Identity operators
Arrays are identical if and only if both of them have same set of key-value pairs and in same order
## Example
Live Demo
<?php
$arr1=array(0=>70, 1=>80, 2=>90);$arr2=array(70,90,80);
var_dump ($arr1===$arr2);
$arr3=[70,80,90]; var_dump ($arr3===\$arr1);
?>
## Output
Following result will be displayed
bool(false)
bool(true)
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# AR development
Discussion in 'Instruments / Avionics / Electrical System' started by Hephaestus, Jun 3, 2019.
1. Jun 3, 2019
### Hephaestus
#### Well-Known Member
Joined:
Jun 25, 2014
Messages:
1,153
246
Location:
YMM
Anyone here a real programmer?
After playing with a DJI drone with the Epson AR glasses last week, I really am starting to wonder about the next step for GA...
I'm more of a copy and paste Arduino sketches to make a program work... So I'm asking what is way beyond my skillset this week.
Anyone have an idea what it would take to make use of something like the Epson glasses, and start adding in bits like a HUD?
Gonna start reading on arcore programming, see if I can start to wrap what's left of my mind around it lol.
2. Jun 3, 2019
### pwood66889
#### Well-Known Member
Joined:
Feb 10, 2007
Messages:
1,398
135
Location:
Sopchoppy, Florida, USA
Well, Heph, lessee here. Got my BS certified as computer science option of math degree. Programmed in CoBOL, FORTRAN, and several flavors of assembly languages. Some proprietary stuff (any one heard of PowerHouse by Cognos?) and BASIC of course. Last go round was Java on Linux/Unix. So you're looking at sending drone signals to Augmented Reality glasses? Like you can turn your head and the glasses adjust the image like you were in the cockpit? I think I'd need a bit better idea of what you were trying to accomplish before I start asking about inputs and outputs.
3. Jun 3, 2019
### FritzW
#### Well-Known Member
Joined:
Jan 31, 2011
Messages:
3,541
3,156
Location:
Las Cruces, NM
I've played with writing a few VR apps for Android using Unity VR. It's probably not that different than writing an AR app. (...except all the "augmented" part happens through the camera). Do Epson glasses have head tracking?
Just find an Epson SDK that has good youtube tutorials
4. Jun 3, 2019
### Hephaestus
#### Well-Known Member
Joined:
Jun 25, 2014
Messages:
1,153
246
Location:
YMM
Actually I want homebuilts guages, and my Mooney's g500 panel interpreted, then a basic aoa, speed slope info overlayed. Really for most it'd be an onboard black box to run it's own set of sensors, display them up on your headset so you're less prone to getting lost in the panel.
Adding adsb will turn into the killer app I'm sure, traffic highlighting within a mile - use the data highlight ballpark area to expect it. using the camera to pinpoint would be super cool but I expect that would be a processing nightmare.
These exist for drones already, not so much for GA.
The Epson do have a gyro, on the dji it's connected to the gymbal for the camera. Took about 10 min to get used to the overlays, even over my glasses. It was pretty much perfect.
5. Jun 3, 2019
### Hephaestus
#### Well-Known Member
Joined:
Jun 25, 2014
Messages:
1,153
246
Location:
YMM
*Shudder* yes... Actually - Yes I have... Back to the days of working at systemhouse crawling through crawlspaces in a suit and tie, because that's what IT schmucks got to do and how they were to dress. Lol.
At least the server rooms were clean. But...
6. Jun 3, 2019
### Hephaestus
#### Well-Known Member
Joined:
Jun 25, 2014
Messages:
1,153
246
Location:
YMM
That's what aero glass was headed. Some stuff I like... Some I hate.
I was thinking more like military HUD
[Stolen] from flyingmagazine.[/stolen]
Maybe you guys in busier and more zoned airspace would want the crazy zone walls etc.
I think the military has invested enough to make it clear what's needed, to me I'd be more interested in attitude, airspeed, direction & routing would be kinda cool. That 3D adsb output - that would be awesome as long as it doesn't overwhelm the screen like on the paragliding parts of aeroglass's video.
Do you really need boxes to fly through as your taking off? Probably not, the runway taxiway stuff could be cool though
7. Jun 3, 2019
### FritzW
#### Well-Known Member
Joined:
Jan 31, 2011
Messages:
3,541
3,156
Location:
Las Cruces, NM
The Auriga app has a very clean HUD. After a while you don't even see it anymore, it just sneaks the data into your brain You could (almost) just velcro a $35 flight controller to the airplane and run any of the play store apps. Some more info here 8. Jun 3, 2019 ### Hephaestus ### Hephaestus #### Well-Known Member Joined: Jun 25, 2014 Messages: 1,153 Likes Received: 246 Location: YMM That looks more correct. Now how do you take the drone out of the equation? I liked the Epson because they're see through, meant to go over glasses. So you don't really want the video feed just the data overlay... 9. Jun 3, 2019 ### FritzW ### FritzW #### Well-Known Member Joined: Jan 31, 2011 Messages: 3,541 Likes Received: 3,156 Location: Las Cruces, NM For prototyping/proof of concept it could be as simple as velcroing a drone flight controller to the airplane. You'd have 9 DOF and an OSD. If you didn't have it connected to a camera (or video receiver) the background would be null. ie. not black, just empty. A much better, and probably just as easy, way: Start poking around on sparkfun, there's a video of doing it (at least collecting the data) with a teensy and a 9 DoF sensor. This may not be the best hardware for the job, it was just the first stuff to show up on a google search. Teensy 9 DoF sensor 10. Jun 3, 2019 ### Hephaestus ### Hephaestus #### Well-Known Member Joined: Jun 25, 2014 Messages: 1,153 Likes Received: 246 Location: YMM The Epson glasses (300 I'm guessing 350s are 600$ more) have those sensors already built into the glasses + the base.
Getting the data isn't hard, it's that processing it into a overlay video that gets ugly
11. Jun 4, 2019
### Vigilant1
#### Well-Known MemberHBA Supporter
Joined:
Jan 24, 2011
Messages:
3,902
1,720
Location:
US
If a plane is in Class A airspace or otherwise in a place where all traffic is under positive control, then a HUD with all kinds of garbage on it is fine. Similarly, if a fighter is below FL180 but running his AI radar so his HUD will actually highlight my location so he can avoid me as I drone along, then that is great. Otherwise, I'd prefer that my little dot of a plane remain plainly visible to others and not hidden behind the fuel level depiction or other distractions. In VMC, all of us are supposed to be seeing and avoiding. ADS-B is not a requirement in most of the airspace below Class A. There's plenty of time for video games when at home. Thank you.
12. Jun 4, 2019
### pwood66889
#### Well-Known Member
Joined:
Feb 10, 2007
Messages:
1,398
135
Location:
Sopchoppy, Florida, USA
K. Won't have to relearn QTP
Now to get onboard with the Epson AR glasses - what do they take as inputs?
13. Jun 4, 2019
### Hephaestus
#### Well-Known Member
Joined:
Jun 25, 2014
Messages:
1,153
246
Location:
YMM
They're basically a full featured Android device on their own.
https://epson.ca/For-Home/Smart-Gla...-Glasses-(AR-Developer-Edition)-/p/V11H756020
My hunch would be use a separate microcontroller to poll can-bus or other sensors for data. I'd think you'd want that device building the video stream and sending to the glasses as a stream? Because you wouldn't want the glasses getting hung up or slowed down and lagging...
14. Jun 5, 2019
### pwood66889
#### Well-Known Member
Joined:
Feb 10, 2007
Messages:
1,398
135
Location:
Sopchoppy, Florida, USA
Epson says "Enhanced connectivity — dual-band wireless connectivity and Bluetooth Smart (BLE) enable integration with a wide range of accessories."
Also says "Operating Systems:Android 5.1" so I guess you just need to build on to the BLE and go from there.
15. Jun 5, 2019
Joined:
Sep 10, 2009
Messages:
288
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# Use ubtraction Formula to write the expression as a trigonometric function of one number.(cos)(13pi)/15cos(-pi/5)-(sin)(13pi)/15sin(-pi/5)
Use an Addition or Subtraction Formula to write the expression as a trigonometric function of one number, and then find its exact value.
$\left(\mathrm{cos}\right)\frac{13\pi }{15}\mathrm{cos}\left(-\frac{\pi }{5}\right)-\left(\mathrm{sin}\right)\frac{13\pi }{15}\mathrm{sin}\left(-\frac{\pi }{5}\right)$
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Our expression is
$\left(\mathrm{cos}\right)\frac{13\pi }{15}\mathrm{cos}\left(-\frac{\pi }{5}\right)-\left(\mathrm{sin}\right)\frac{13\pi }{15}\mathrm{sin}\left(-\frac{\pi }{5}\right)$
This kind of expression appears on the addition of angles formula for cosine
$\mathrm{cos}\left(x+y\right)=\mathrm{cos}x\mathrm{cos}y-\mathrm{sin}x\mathrm{sin}y$
with $x=\frac{13\pi }{15},y=-\frac{\pi }{5}$
So we have
$\left(\mathrm{cos}\right)\frac{13\pi }{15}\mathrm{cos}\left(-\frac{\pi }{5}\right)-\left(\mathrm{sin}\right)\frac{13\pi }{15}\mathrm{sin}\left(-\frac{\pi }{5}\right)=\mathrm{cos}\left(\frac{13\pi }{15}+\left(-\frac{\pi }{5}\right)\right)$
$=\mathrm{cos}\left(\frac{13\pi }{15}-\frac{3\pi }{15}\right)$
$=\left(\mathrm{cos}\right)\frac{10\pi }{15}$
$=\left(\mathrm{cos}\right)\frac{2\pi }{3}$
$=-\mathrm{cos}\left(\frac{\pi }{3}\right)\left[\mathrm{cos}\left(\pi -x\right)=-\mathrm{cos}x\right]$
$=-\frac{1}{2}$
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Obtaining the top matches from blast
5
2
Entering edit mode
8.3 years ago
S ▴ 100
Hi,
I have downloaded the current version of the stand-alone-blast (ncbi-blast-2.2.29+) and I am trying to use blast (blastn) to find similarity of of a group of nucleotide sequences that I have. However, I am interested on only the top 3 matches. I tried searching online and I saw some posts that suggests using -K, but I realized this does not work with the new version that I am using. I looked at the help document and I tried using (-max_target_seqs) and (-num_alignments) but none of them worked. The result contains all the matches found by blast.
Does anyone know how to limit the results to let say just top 3 matches?
Thanks!
blast • 18k views
0
Entering edit mode
Could you plz explain a bit more about the sorting technique that has been referred to in this thread?
8
Entering edit mode
8.3 years ago
hpmcwill ★ 1.2k
Depends what you are trying to do.
As Neilfws says, if you want to limit the number of hits reported you can use (from the NCBI BLAST+ help output):
-num_descriptions <Integer, >=0>
Number of database sequences to show one-line descriptions for
Not applicable for outfmt > 4
Default = 500'
* Incompatible with: max_target_seqs
-num_alignments <Integer, >=0>
Number of database sequences to show alignments for
Default = 250'
* Incompatible with: max_target_seqs
These correspond to the '-v' and '-b' options in legacy NCBI BLAST:
-v Number of database sequences to show one-line descriptions for (V) [Integer]
default = 500
-b Number of database sequence to show alignments for (B) [Integer]
default = 250
The '-K' option in legacy NCBI BLAST:
-K Number of best hits from a region to keep. Off by default.
If used a value of 100 is recommended. Very high values of -v or -b is also suggested [Integer]
Is slightly different and maps to the '-culling_limit' parameter in NCBI BLAST+:
-culling_limit <Integer, >=0>
If the query range of a hit is enveloped by that of at least this many
higher-scoring hits, delete the hit
* Incompatible with: best_hit_overhang, best_hit_score_edge
You may also want to limit the number of matches reported per hit (i.e. limit the number of HSPs):
-max_hsps <Integer, >=0>
Set maximum number of HSPs per subject sequence to save (0 means no limit)
Default = 0'
0
Entering edit mode
Thank you very much hpmcwill! I am sorry that my post was not clear enough, I was looking to limit the number of matches reported per hit so (-max_hsps) did the job.
4
Entering edit mode
5.2 years ago
shinken123 ▴ 110
Using the output of blast using the option -outfmt 6
awk '!seen[\$1]++' Blast_output_file.txt > Besthit_Blast_output_file.txt
0
Entering edit mode
Hello,
Could you explain your awk command please ? I am very interested by it !
2
Entering edit mode
8.3 years ago
Neilfws 49k
The relevant options are in the BLAST handbook:
num_descriptions integer 500 Show one-line descriptions for this number of database sequences.
num_alignments integer 250 Show alignments for this number of database sequences.
0
Entering edit mode
Thank you very much for the link!
1
Entering edit mode
8.3 years ago
edrezen ▴ 730
Hi,
What is the output format you use ? I think these options may not work with the default blast output format.
If you try the tabular output format (just add -outfmt 6 to your command), it may work better.
1
Entering edit mode
Changing the format to option 6 didn't help.
0
Entering edit mode
8.3 years ago
Whoknows ▴ 920
Hi
Please run your query with this parameter -outfmt 6 with this you can select those with highest similarity and also you can find out the number of mismatches, Then sort it .
But use this -best_hit_overhang` for finding best hit over the blast.
0
Entering edit mode
Hi,
Thanks for the response. I knew I could sort and pick the top hit but I just thought there should be a parameter while running blast that can limit the results (at least there was one for an older version).
Thanks!
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Pseudo-sphere
A surface of constant negative curvature formed by rotating a tractrix ( $x = u - \mathop{\rm tanh} u$, $y = \mathop{\rm sech} u$) around its asymptote ( $y= 0$; see Fig.).
Figure: p075840a
The line element in semi-geodesic coordinates has the form:
$$d s ^ {2} = d u ^ {2} + \cosh ^ {2} \frac{u}{a} \ d v ^ {2} ,\ a = \textrm{ const }$$
(the line $u = 0$ is a geodesic); while in isothermal coordinates it has the form:
$$d s ^ {2} = a ^ {2} \frac{d x ^ {2} + d y ^ {2} }{y ^ {2} } ,\ \ a = \textrm{ const } .$$
Every surface of constant negative curvature can be locally imbedded in the pseudo-sphere. The intrinsic geometry of a pseudo-sphere coincides locally with hyperbolic geometry (see Beltrami interpretation).
References
[1] M.Ya. Vygodskii, "Differential geometry" , Moscow-Leningrad (1949) (In Russian) [2] V.F. Kagan, "Foundations of the theory of surfaces in a tensor setting" , 2 , Moscow-Leningrad (1949) (In Russian)
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# Definition:Infinite Group
## Definition
A group which is not finite is an infinite group.
That is, an infinite group is a group of infinite order.
That is, a group $\struct {G, \circ}$ is an infinite group if and only if its underlying set $G$ is infinite.
That is, an infinite group is a group with an infinite number of elements.
### Countable
An infinite group whose underlying set $G$ is countable is a countably infinite group.
### Uncountable
An infinite group whose underlying set is uncountable is an uncountable group.
## Also see
• Results about the order of a group can be found here.
• Results about infinite groups can be found here.
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## Intermediate Algebra (6th Edition)
$-2\frac{5}{28}$
We have $\frac{-1-2}{2(-3)+10} - \frac{2(-5)}{-1(8)+1}$ Solving $$\frac{-3}{-6+10} - \frac{-10}{-8+1}$$ $$\frac{-3}{4} - \frac{-10}{-7}$$ $$\frac{-3}{4} - \frac{10}{7}$$ $$\frac{-21-40}{28}$$ $$\frac{-61}{28} = -2\frac{5}{28}$$
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# Seven numbers
Write seven 4-digit numbers that are divisible by 3 and at the same time by 4.
Result
a1 = 1008
a2 = 1020
a3 = 1032
a4 = 1044
a5 = 1056
a6 = 1068
a7 = 1080
#### Solution:
$3 \ ... \ prime number \ \\ 4=2^2 \ \\ LCM(3, 4)=2^2 \cdot 3=12 \ \\ \ \\ d=LCM(3,4)=12 \ \\ b=1000/d=1000/12 \doteq \dfrac{ 250 }{ 3 } \doteq 83.3333 \ \\ b_{1}=\lceil b \rceil=\lceil 83.3333 \rceil=84 \ \\ \ \\ a_{1}=b_{1} \cdot \ d=84 \cdot \ 12=1008$
$a_{2}=a_{1}+d=1008+12=1020$
$a_{3}=a_{2}+d=1020+12=1032$
$a_{4}=a_{3}+d=1032+12=1044$
$a_{5}=a_{4}+d=1044+12=1056$
$a_{6}=a_{5}+d=1056+12=1068$
$a_{7}=a_{6}+d=1068+12=1080$
Our examples were largely sent or created by pupils and students themselves. Therefore, we would be pleased if you could send us any errors you found, spelling mistakes, or rephasing the example. Thank you!
Leave us a comment of this math problem and its solution (i.e. if it is still somewhat unclear...):
Be the first to comment!
Tips to related online calculators
Do you want to calculate greatest common divisor two or more numbers?
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16. Crates 2
One crate will hold 50 oranges. If Bob needs to ship 932 oranges, how many crates will he need?
17. Round it
0.728 round to units, tenths, hundredths.
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# Recursive and Non-recursive SelectionSort
This is a naive attempt of writing a recursive and non-recursive versions for SelectionSort(). My goal is mainly to present an elegant, easy-to-understand, idiomatic code, and therefore, performance is a distant priority. Please comment away!
// Find the index at which the minimum value
// exists inside the Array
function findMinIndex(a){
return a.reduce((iMin, x, i, arr) => x < arr[iMin]? i : iMin, 0);
}
// Remove the minimum value from the array
// Return the value removed
function removeMin(a){
idx = findMinIndex(a);
minVal = a[idx]; // for [1, -5, 3], minVal = -5
a.splice(idx, 1); // [1, -5, 3] -> [1, 3]
return minVal;
}
// Selection sort, in recursive mode
// As the name suggests, we select and 'splice' it
// away from the array, and recursively
// concatenate it to get the final result
function selectRecursive(a) {
if (!a.length) return []; // terminating case
minVal = removeMin(a); // remove the smallest
console.log(v, a);
return [minVal].concat(selectRecursive(a));
}
var myl = [1, 2, 3, 99, 22, 55, 5];
selectRecursive(myl);
OUTPUT
1 [ 2, 3, 99, 22, 55, 5 ]
2 [ 3, 99, 22, 55, 5 ]
3 [ 99, 22, 55, 5 ]
5 [ 99, 22, 55 ]
22 [ 99, 55 ]
55 [ 99 ]
99 []
[ 1, 2, 3, 5, 22, 55, 99 ]
In addition, a non-recursive (less intuitive in my opinion) version of SelectionSort (using push, slice, splice and the spread operator) is presented below.
function selectionSort(a) {
var length = a.length;
for (var i = 0; i < length; i++) {
a.push( // select and append at the end
...a.splice(
findMinIndex( // get min in a sliced 'a'
a.slice(0, length - i)), 1)
);
}
return a;
}
I'll take this a function at a time before looking at your end goal.
function findMinIndex(a){
return a.reduce((iMin, x, i, arr) => x < arr[iMin]? i : iMin, 0);
}
The largest problem with this function is that it fails for empty arrays. findMinIndex([]) incorrectly returns 0, an index which does not exist. If the array is empty, I'd suggest returning a flag value of -1.
Since undefined is less than -Infinity, we then have to swap the comparison around to make it work with essentially the same logic.
function findMinIndex(a){
return a.reduce((iMin, x, i, arr) => x > arr[iMin]? iMin : i, -1);
}
I would argue that this is still more complex than it needs to be. There is no need to use the arr parameter as we can just use a from the function call. (Some may disagree with me here) While we are at it, it improves the clarity to rename a to arr.
function findMinIndex(arr){
return arr.reduce((iMin, x, i) => x > arr[iMin]? iMin : i, -1);
}
Next up:
function removeMin(a){
idx = findMinIndex(a);
minVal = a[idx]; // for [1, -5, 3], minVal = -5
a.splice(idx, 1); // [1, -5, 3] -> [1, 3]
return minVal;
}
How this works if pretty obvious, which is good. The one issue is that you create the global variable minVal. It looks like you missed a let. Some people would tell you to avoid functions which mutate their parameters, and while I generally do this myself, since this function is very clear about what it is doing it probably isn't an issue. However, I would probably rewrite this as a one liner, or just inline it as you did in your answer.
function removeMin(a) {
return a.splice(findMinIndex(a), 1)[0];
}
Now for your recursive function. It also creates a global variable minVal but besides that looks good to me! Really nothing to say here.
The non-recursive selection sort could certainly use some work.
function selectionSort(a) {
var length = a.length;
for (var i = 0; i < length; i++) {
a.push( // select and append at the end
...a.splice(
findMinIndex( // get min in a sliced 'a'
a.slice(0, length - i)), 1)
);
}
return a;
}
Personally, I would prefer just simple for loops in this case. No need for all those function calls.
function selectionSort(a) {
for (let i = 0; i < a.length; i++) {
let iLow = i;
for (let j = i + 1; j < a.length; j++) {
if (a[j] < a[iLow]) iLow = j
}
// Fancy destructuring to swap indexes
[a[i], a[iLow]] = [a[iLow], a[i]]
}
return a;
}
Lastly, I can't help but mention a.sort() though I know that's not the point of this exercise.
After I played around with splice a little bit more, I was able to refactor out the removeMin function completely. It seems like splice and recursive SelectionSort were made for each other! Of course, some may argue the code has become less readable, or has it really?
function selectRecursive(a) {
if (!a.length) return []; // terminal case
minVal = a.splice(findMinIndex(a), 1); // select and remove
console.log(minVal, a); // to witness the magic of recursion!
return minVal.concat(selectRecursive(a)); // concat recursively!
}
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Algebra
posted by on .
The value of y that minimizes the sum of the two distances from (3,5) to (1,y) and from (1,y) to (4,9) can be written as \frac{a}{b} where a and b are coprime positive integers. Find a + b
• Algebra - ,
d = √(4+(y-5)^2) + √(9+(y-9)^2)
dd/dy = (y-5)/√(4+(y-5)^2) + (y-9)/√(9+(y-9)^2)
= [(y-9)√(4+(y-5)^2) + (y-5)√(9+(y-9)^2)]/(√(4+(y-5)^2) * √(9+(y-9)^2))
dd/dy=0 when
(y-9)√(4+(y-5)^2) + (y-5)√(9+(y-9)^2) = 0
y = 33/5
a+b=38
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# Seminars & Events for 2011-2012
March 2, 2012
3:00pm - 4:00pm
##### Bocher Type Theorems for Degenerate Conformally Invariant Equations
###### Differential Geometry & Geometric Analysis Seminar
A classical theorem of Bocher states that a positive harmonic function in a punctured ball can be written as the sum of a multiple of the fundamental solution and a regular harmonic function. I will describe generalizations for degenerate $\sigma_k$ equations. Joint work with Yanyan Li.
Speaker: Luc Nguyen, Princeton University
Location:
Fine Hall 314
March 5, 2012
4:00pm - 5:00pm
##### Hardy Spaces With Variable Exponents and Generalized Campanato Spaces
###### Analysis Seminar
Hardy spaces play an important role not only in harmonic analysis but also in partial differential equations because singular integral operators are bounded on Hardy spaces. The Hardy space H1, which substitute for L1, and the Hardy spaces $H^p$ with $0 < p < 1$, are different in that the latter contains non-regular distributions.
Speaker: Yoshihiro Sawano, Kyoto University
Location:
Fine Hall 314
March 5, 2012
4:30pm - 5:30pm
##### Topological Landscape of Networks
###### PACM/Applied Mathematics Colloquium
We will discuss how one can endow a network with a landscape in a very simple and natural way. Critical point analysis is introduced for functions defined on networks. The concept of local minima/maxima and saddle points of different indices are defined, by extending the notion of gradient flows and minimum energy path to the network setting.
Speaker: Yuan Yao, Peking University
Location:
Fine Hall 214
March 6, 2012
4:30pm - 5:30pm
##### Tensor Products On Triangulated and Abelian Categories
###### Algebraic Geometry Seminar
Speaker: Yu-Han Liu, Princeton University
Location:
Fine Hall 322
March 7, 2012
4:30pm - 5:30pm
##### The Surface Subgroup Theorem and the Ehrenpreis Conjecture
###### Department Colloquium
We prove that there is a hyperbolic surface $S$ such that for any closed hy-perbolic 2 or 3-manifold M, and > 0, there is a nite cover ^ S of S, and a map f: ^ S ! M that is locally within of being an isometric immersion. When dimM = 3 this implies that 1(M) has a surface subgroup, and when dimM = 2 this is the Ehrenpreis conjecture.
Speaker: J. Kahn, Brown University
Location:
Fine Hall 314
March 8, 2012
2:00pm - 3:00pm
##### Effective bisector estimate for PSL(2,C) with applications to circle packings
###### Ergodic Theory & Statistical Mechanics
Let Gamma be a non-elementary discrete geometrically finite subgroup of PSL(2,C). Under the assumption that the critical exponent of Gamma is greater than 1 we prove an effective bisector counting theorem for Gamma. We then apply this Theorem to the Apollonian circle packing problem to get power savings and to compute the overall constant.
Location:
Fine Hall 601
March 8, 2012
2:15pm - 3:15pm
##### Pareto Optimal Solutions for Smoothed Analysts
###### Discrete Mathematics Seminar
Consider an optimization problem with n binary variables and $d+1$ linear objective functions. Each valid solution in ${0,1}^n$ gives rise to an objective vector in $R^{d+1}$, and one often wants to enumerate the Pareto optima among them.
Speaker: Ankur Moitra, IAS
Location:
Fine Hall 224
March 8, 2012
3:00pm - 4:00pm
##### Hermitian K-theory and Cobordism
###### Algebraic Topology Seminar
I will discuss Z/2-equivariant motivic spectra. As an example, I will talk about a Z/2-equivariant motivic spectrum representing Karoubi's Hermitian K-theory, and my joint solution with Kriz and Ormsby of Thomason's homotopy limit problem. As another example, I will talk about motivic Hermitian cobordism, and its topological realization, topological Hermitian cobordism.
Speaker: Po Hu, Wayne State University
Location:
Fine Hall 214
March 8, 2012
4:30pm - 5:30pm
##### Counting Connections and the Ehrenpreis Conjecture
###### Topology Seminar
Let $S$ be a closed hyperbolic surface. We review the theory of counting connections on $S$---between points, between horocycles, and between geodesic segments---and we explain how this relates to the proof of the Ehrenpreis conjecture.
Speaker: Jeremy Kahn, Brown University
Location:
Fine Hall 314
March 8, 2012
4:30pm - 5:30pm
##### Mod $p$ Points on Shimura Varieties
###### Princeton University/IAS Number Theory Seminar
A conjecture of Langlands-Rapoport predicts the structure of the mod $p$ points on a Shimura variety. The conjecture forms part of Langlands' program to understand the zeta function of a Shimura variety in terms of automorphic L-functions. I will report on progress towards the conjecture in the case of Shimura varieties attached to non-exceptional groups.
Speaker: Mark Kisin, Harvard University
Location:
Fine Hall 214
March 9, 2012
3:00pm - 4:00pm
##### The Spacetime Positive Mass Theorem in Dimensions Less Than 8
###### Differential Geometry & Geometric Analysis Seminar
After reviewing the proof of the Riemannian positive mass theorem in dimensions less than 8, I will briefly explain how to generalize the proof to slices of spacetime that are not time- symmetric. The basic idea is to replace minimal hypersurfaces by marginally outer-trapped hypersurfaces, and the main difficulty is to avoid using any minimization process.
Speaker: Dan Lee, CUNY
Location:
Fine Hall 314
March 12, 2012
4:00pm - 5:00pm
##### Finite Point Configurations, Incidence Theory and Multi-linear Operators
###### Analysis Seminar
A classical problem in geometric combinatorics is to determine how often a single distance may repeat among $n$ points in the plane. A related problem that has received much attention is how often a given triangle can repeat among $n$ points in the plane.
Speaker: Alex Iosevish, University of Rochester
Location:
Fine Hall 314
March 13, 2012
4:30pm - 4:30pm
##### A Frobenius Variant of Seshadri Constants
###### Algebraic Geometry Seminar
I will define a new variant of the Seshadri constant for ample line bundles in positive characteristic. We will then explore how lower bounds for this constant imply the global generation and/or very ampleness of the corresponding adjoint line bundle.
Speaker: Karl Schwede, Penn State University
Location:
Fine Hall 322
March 14, 2012
4:30pm - 5:30pm
##### A Case Study for Critical Non-linear Dispersive quations: The Energy Critical Wave Equation
###### Department Colloquium
We will discuss recent work on the energy critical wave equation. The issues studied are global existence, scattering, finite time blow-up, universal profiles at blow-up and soliton resolution. This is viewed not as an isolated series of results, but as a way of approaching many similar critical non-linear dispersive equations.
Speaker: Carlos Kenig , University of Chicago
Location:
Fine Hall 314
March 15, 2012
2:00pm - 3:00pm
##### Noncollision Singularities in Planar Two-center-two-body Problem
###### Ergodic Theory & Statistical Mechanics
In this work we study a model called planar 2-center-2-body problem. In the plane, we have two fixed centers Q_1=(-\chi,0), Q_2=(0,0) of masses 1, and two moving bodies Q_3 and Q_4 of masses \mu. They interact via Newtonian potential. Q_3 is captured by Q_2, and Q_4 travels back and forth between two centers.
Speaker: Jinxin Xue, Institute for Advanced Study
Location:
Fine Hall 601
March 15, 2012
2:15pm - 3:15pm
##### Tangles, Trees, and Flowers
###### Discrete Mathematics Seminar
Identifiable regions of high connectivity in a matroid or graph are captured by the notion of tangles''.
Speaker: Ben Clark, U. Wellington
Location:
Fine Hall 224
March 15, 2012
4:30pm - 5:30pm
##### Hypergeometric Motives
###### Princeton University/IAS Number Theory Seminar
The families of motives of the title arise from classical one-variable hypergeometric functions. This talk will focus on the calculation of their corresponding L-functions both in theory and in practice. These L-functions provide a fairly wide class which is numerically accessible. As an illustration we will consider the case of Artin L-functions.
Speaker: Fernando Villegas, University of Texas at Austin
Location:
IAS - West Building Lecture Hall
March 15, 2012
4:30pm - 5:30pm
##### A Khovanov Homotopy Type
###### Topology Seminar
We will start by describing Khovanov's categorification of the Jones polynomial from a cube of resolutions of a link diagram. We will then introduce the notion of a framed flow category, as defined by Cohen, Jones and Segal.
Speaker: Sucharit Sarkar, Columbia University
Location:
Fine Hall 314
March 16, 2012
3:00pm - 4:00pm
##### $L^p$ Bounds for Eigenfunctions on Locally Symmetric Spaces
###### Differential Geometry & Geometric Analysis Seminar
There is a classical theorem of Sogge which provides bounds for the $L^p$ norms of a Laplace eigenfunction on a compact Riemannian manifold, which are sharp on the sphere and for spectral clusters.
Speaker: Simon Marshall, Northwestern University
Location:
Fine Hall 314
March 26, 2012
4:00pm - 5:00pm
##### University of North Carolina at Chapel Hill
###### Analysis Seminar
It is well known that on $\reals^n$, the Schrödinger propagator is unitary on $L2$ based spaces, but that locally in space and on average in time there is a $1/2$ derivative smoothing effect. We consider a family of manifolds with trapped geodesics which are degenerately hyperbolic and prove a sharp local smoothing estimate with loss depending on the type of trapping.
Speaker: Hans Cristianson, University of North Carolina at Chapel Hill
Location:
Fine Hall 314
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# Free-Radical chlorination: Find all unique mono-chlorinated products
I'm having trouble with this problem. I suspect that there is a typo, as the methylcyclohexene described in the wording does not match the methylcyclopentene shown in the image(someone please confirm if my suspicion is correct or if I made a mistake in interpreting nomenclature, I am very new to Organic Chemistry). If it does appear to be a typo, I would like help in solving for the methylcyclopentene(the molecule described by the image).
I'm not quite sure how to tackle this problem, as this is actually the first time I've had to mono-chlorinate something with a double bond. I'm confused about how stereochemistry factors in, since there's no symmetry in the molecule, and the double bond onto one carbon suggests a sp2 triplanar configuration where there is only one H position that a radical chlorine can take. This problem isn't like anything I've seen so far, so I would appreciate any help from some veteran organic chemists.
• Well of course it's incorrect. Also you need to remember that it's two step - first alkenyl radical then halogenide – Mithoron Sep 4 '16 at 0:55
• @Mithoron halogenide is the German word for halide – DHMO Sep 4 '16 at 1:00
• Nice first question! – DHMO Sep 4 '16 at 1:33
• Hint for the stereochemistry part of the question: as soon as you have any cyclic $\mathrm{sp^3}$ carbon with two different substituents attached, stereochemistry comes into play. – Jan Sep 4 '16 at 21:53
• Can someone check these for me: 1. At the end of the methyl group, there is only one monochlorinated product because every hydrogen in the CH3 is identical to each other? 2. The cyclic carbon that is double bonded and is attached to the methyl group cannot be chlorinated 3. The cyclic carbon that is double bonded and is not attached to the methyl group can be chlorinated, but because of its sp2 configuration, there is only 1 chlorinated product – Garner Deng Sep 13 '16 at 23:03
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# Homework Help: Heat equation, Fourier cosine transform
1. Feb 9, 2012
### fluidistic
1. The problem statement, all variables and given/known data
Problem 8-17 from Mathew's and Walker's book:
Use a cosine transform with respect to y to find the steady-state temperature distribution in a semi-infinite solid $x>0$ when the temperature on the surface $x=0$ is unity for $-a<y<a$ and zero outside this strip.
2. Relevant equations
Heat equation: $\frac{\partial u}{ \partial t}+k \nabla u =0$.
Cosine Fourier transform: $f(x)=\frac {1} {\pi} \int _0 ^{\infty } g(y) \cos (xy )dy$.
3. The attempt at a solution
I've made a sketch of the situation, I don't think I have any problem figuring out the situation.
Now I'm stuck. Should I perform "brainlessly" a cosine transform to the heat equation as it is, or should I set $\frac{\partial u }{\partial t}=0$ since it's a steady state distribution of temperature? This would make $\nabla u =0 \Rightarrow \frac{\partial u }{\partial x }+\frac{\partial u }{\partial y }=0$ (Laplace equation). Should I apply now the cosine transform?
2. Feb 9, 2012
### vela
Staff Emeritus
That should be $\nabla^2 u$. You need to find the general solution first.
3. Feb 9, 2012
### fluidistic
Oops right, true.
Do you mean I should solve the heat equation with say for example separation of variables? If I get the general solution, then why would I need to perform a cosine tranform?
4. Feb 9, 2012
### vela
Staff Emeritus
Sorry, you're right. You want to set $\partial u/\partial t=0$ and then Fourier transform the equation.
5. Feb 9, 2012
### fluidistic
Ok I've checked another source for the definition of the cosine transform, I'll use it instead. I'm having a doubt however.
u depends on x,y and t.
The cosine transform with respect to y: $\mathbb{F_c}(u)=U_c (p,t)=\int _0 ^{\infty} u \cos (py)dy$. I have no problem with this. I notice that in my exercise the range of y is from negative to positive infinity rather than 0 to positive infinity; but it doesn't matter, I can solve it from 0 to infinity and then use the fact that the function u is symmetric with respect to the x axis, I believe.
$\mathbb{F_c} \left ( \frac{\partial u }{\partial x} \right )=\int _0 ^{\infty} \frac{\partial u }{\partial x } \cos (py)dy$. In order to solve this integral I know I can use integral by parts but I'm not 100% sure that it's worth $p U_s(p,t)-u(0,y,t)$ (where $U_s$ is the sine transform) because the derivative of u is with respect to x while the integration is with respect to y. This would also mean that $\mathbb{F_c} \left ( \frac{\partial ^2 u }{\partial x^2} \right )=-p^2U_c (p,t)-\frac{\partial u }{\partial x}(0,y,t)$.
Is this ok so far?
6. Feb 10, 2012
### vela
Staff Emeritus
It's actually only a function of x and y because you're looking for the steady-state solution.
That's right. You can use the cosine transform because the boundary condition is an even function of y.
You can't integrate by parts like that because the derivative is with respect to x, but the integration is with respect to y. What you can do is switch the order of integration and differentiation, so you'll end up with
$$\mathbb{F_c}\left[\frac{\partial^2 u(x, y)}{\partial x^2}\right] = \frac{\partial^2}{\partial x^2} \mathbb{F_c}[u(x,y)] = \frac{\partial^2}{\partial x^2} u(x,p)$$
7. Feb 10, 2012
### fluidistic
Ok thank you very much vela.
I have a little problem with the cosine transform of $\frac{\partial ^2 u }{\partial y^2}$. Is it $-p^2 U_c (p,x)-\frac{\partial u }{\partial y} \big | _{y=0}$? If so, I don't know how to evaluate the last term.
Edit: $\mathbb{F_c} \left ( \frac{\partial u (x,y)}{\partial y} \right )=[u(x,y)\cos (py)]^{y=\infty}_{y=0}+p\int _0^{\infty } u(x,y)\sin (py)dy=-u(x,0)+p U_s (p,x)$. Not sure how to evaluate $u(x,0)$ either here. I only know $u(0,0)$ which is worth 1, but no more than this, on the x-axis.
Last edited: Feb 10, 2012
8. Feb 11, 2012
### vela
Staff Emeritus
Yes, it is. From the symmetry of the physical problem, we know u(x,y) will be symmetric about the x-axis. What does this imply about the derivative at y=0?
This shouldn't matter because $\frac{\partial u (x,y)}{\partial y}$ isn't in the problem.
9. Feb 11, 2012
### fluidistic
Ok thank you vela!
This means that $\frac{\partial u }{\partial y} \big | _{y=0}=0$.
Thus the PDE is equivalent to $\frac{\partial ^2 U_c (p,x)}{\partial x^2}-p^2 U_c (p,x)=0$. Since $p>0$, $U_c(p,x)=Ae^{px}+Be^{-px}$. Now I think it's time to take the inverse cosine transform.
10. Feb 11, 2012
### fluidistic
So this gives me $\mathbb{F} _c ^{-1} [U_c(p,x)]=u(x,y)=\frac{2}{\pi} \int _0 ^{\infty} U_c (p,x) \cos (py)dp$. Is this ok?
$U_c(p,x)=Ae^{px}+Be^{-px}$. So that $u(x,y)=\frac{2}{\pi} \int _0 ^{\infty } (Ae^{px}+Be^{-px} ) \cos (py)dp$. This doesn't look a correct answer to me though, let alone how to simplify it and calculate A and B from the boundary conditions.
11. Feb 12, 2012
### vela
Staff Emeritus
Remember that the "constants" can still depend on p. That is,
$$U_c(x, p) = A(p)e^{px} + B(p)e^{-px}$$ You want a bounded solution as $x \to \infty$, so you can toss the first term.
Before you take the inverse transform, you want to incorporate the boundary condition for x=0 by doing essentially what was done on pages 242 and 243 in Mathews and Walker to determine B(p).
12. Feb 12, 2012
### fluidistic
I am a bit confused here. I want u(x,y) to be bounded when x tends to infinity. I guess you mean that this also imply that A(p) must be worth 0 in wich case it's something I have to digest.
I get $U_c(p,0)=B(p)=\int _0^{\infty} u(0,y) \cos (py)dp$.
I think something is wrong here.
13. Feb 12, 2012
### vela
Staff Emeritus
Why? That's correct. You were given what u(0,y) is equal to.
EDIT: Oops, missed that you were integrating with respect to p.
Last edited: Feb 13, 2012
14. Feb 13, 2012
### fluidistic
True but it's not single valued. It depends on y actually so this makes B depend on y too. Furthermore for $-a<y<a$, $B(p)=\int _0^{\infty } \cos (py ) dp$ which isn't definied.
I think that if the integration was with respect to y rather than p, I would have less problems.
If I integrate with respect to y rather than p, I get $B(p)=\frac{\sin (pa)}{p}$ so that $U_c(p,x)=\frac{e^{-px}\sin (pa)}{p}$. Now time to take the inverse transform.
15. Feb 13, 2012
### vela
Staff Emeritus
I didn't notice you were integrating with respect to p. Your latter result is correct. You're just setting x=0 in
$$U_c(x,p) = \int_0^\infty u(x,y)\cos py\,dy = B(p)e^{-px}.$$
16. Feb 13, 2012
### fluidistic
No problem vela, so far you've been of so much help for me...
I'm stuck at solving the integral when taking the inverse transform. $\mathbb{F_c}^{-1} [U_c (p,x)]=u(x,y)=\frac{2}{\pi} \int _0^{\infty} \frac{e^{-px}\sin (pa) \cos (py) dp}{p}$. This would be the answer to the problem but I'm hoping to simplify this result. Not sure how to tackle that integral.
17. Feb 13, 2012
### vela
Staff Emeritus
Use a trig identity on $2\sin (pa)\cos (py)$. You'll end up with two integrals of the form
$$I=\int_0^\infty \frac{\sin kp}{p} e^{-px}\,dp,$$ where k is a constant, which is the Laplace transform of (sin kp)/p.
18. Feb 13, 2012
### fluidistic
Right, now I get $u(x,y)=\frac{1}{\pi} \{ \int_0^{\infty} \frac{\sin [p(y+a)]e^{-px}}{p}dp + \int_0^{\infty} \frac{\sin [p(a-y)]e^{-px}}{p}dp \}$.
Now I have to use the residue theorem to calculate both integrals?
Edit: Hmm probably not... If I change p by z, the integral has no residue in z=0 which probably means it has no pole in z=0? Strange.
Last edited: Feb 13, 2012
19. Feb 13, 2012
### vela
Staff Emeritus
Hint:
$$\int_0^k \cos kp\,dk = \frac{\sin kp}{p}$$
Last edited: Feb 15, 2012
20. Feb 13, 2012
### fluidistic
Hmm, definitely not integration by parts. I don't really know right now.
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# What is the opposite of a cross term?
When we multiply out $(x + y)(x + y)$, we refer to the two $xy$ terms as "cross terms". Is there a corresponding term for the $x^2$ and $y^2$ terms?
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I'm not aware of standard terminology, but I'd call them "pure terms". – Matt Pressland Mar 28 '12 at 13:15
The happy terms?! – Ross Millikan Mar 28 '12 at 13:21
The "non-cross" terms? – David Mitra Mar 28 '12 at 13:26
Depending on the context, "diagonal terms" might work:
$$(x+y)(x+y)=\pmatrix{x&y}\pmatrix{1&1\\1&1}\pmatrix{x\\y}\;;$$
the cross-terms are the off-diagonal terms in this quadratic form and the other ones are the diagonal terms.
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+1, Came here to give this answer. I think I have seen diagonal used most often. – Eric Naslund Mar 28 '12 at 13:29
Direct or straight might be what you are looking for, as opposed to cross, crossed or mixed (since each resultant term has either one variable to a power or two different variables, a "mixture").
I was also taught that you can multiply $(a+b)(c+d)$ using the acronym FOIL for First, Inside, Outside, Last (which is mixing sequential and spatial metaphors).
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The squares or more general, the $n$th power.
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The aligned terms. ............
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# Limit of a sum using complex analysis.
I'm trying to find the limit of this sum: $$S_n =\frac{1}{n}\left(\frac{1}{2}+\sum_{k=1}^{n}\cos(kx)\right)$$ I tried to find a formula for the inner sum first and I ended up getting zero as an answer. The sum is supposed to converge to $\cot(x\over 2)$ and that appears in my last expression but it goes to zero. Here is what I got with a rather long and clumsy reasoning: $$S_n = \frac{1}{2n}\left(\cot\left(\frac{x}{2}\right)\sin(nx)+\cos(nx)\right)$$ Thanks in advance.
Just write $$\cos (kx)$$ as the real part of $$e^{ikx}$$.
• I already did that but it didn't work. Sep 30 '18 at 20:39
• Then the limit is 0. Why do you think the answer is a cotan ?
– user598294
Sep 30 '18 at 20:42
• I don't know either way, but it can't be zero because the question asks for an answer in terms of $x$ Sep 30 '18 at 20:43
• You must have forgotten something.
– user598294
Sep 30 '18 at 20:45
• TheSilverDoe's proof convinced me and it has two different answers depending on the value of $x$ which solves my troubles, so yes not $\cot$, my bad. Sep 30 '18 at 20:54
You have, for all $$x \neq 2k\pi$$, $$\sum_{k=1}^n \cos(kx) = \mathrm{Re}\left( \sum_{k=1}^n e^{ikx}\right) = \mathrm{Re}\left( e^{ix} \frac{1-e^{inx}}{1-e^{ix}} \right)$$
Moreover you have $$\left|e^{ix} \frac{1-e^{inx}}{1-e^{ix}}\right| \leq \frac{2}{|1-e^{ix}|}$$
So $$\left| \sum_{k=1}^n \cos(kx) \right| \leq \frac{2}{|1-e^{ix}|}$$
And therefore you get $$S_n \rightarrow 0$$.
If $$x = 2k\pi$$ for $$k \in \mathbb{Z}$$, you easily have $$S_n \rightarrow 1$$.
• Remember that $|e^{iz}|=1$ for all $z \in \mathbb{R}$. So by the triangle inequality, $|1-e^{inx}| \leq 2$. Sep 30 '18 at 20:48
• Thank you very much, this was very helpful. Sep 30 '18 at 20:51
Another view of the question.
Instead of taking the real part of $$e^{ikx}$$ as given by the answers from TheSilverDoe and AlexL, the complex exponential can be more symmetric:
\begin{align*} \cos kx &= \frac{e^{-ikx}+e^{ikx}}2\\ \frac12+\sum_{k=1}^n\cos kx &= \frac12\sum_{k=-n}^ne^{ikx}\\ &= \frac12 \cdot e^{-inx}\frac{e^{i(2n+1)x}-1}{e^{ix}-1}\\ &= \frac12 \cdot \frac{e^{i(n+1)x}-e^{-inx}}{e^{ix}-1}\\ &= \frac12 \cdot \frac{e^{i\left(n+\frac12\right)x}-e^{-i\left(n+\frac12\right)x}}{e^{i\frac12x}-e^{-i\frac12x}}\\ &= \frac12 \cdot \frac{\sin\left(n+\frac12\right)x}{\sin\frac12x}\\ S_n&= \frac{\sin\left(n+\frac12\right)x}{2n\sin\frac12x}\\ \end{align*}
When $$e^{ix} \ne 1$$, i.e. $$x\ne 2\pi m$$ for integer $$m$$, $$S_n \to 0$$.
For other $$x$$'s, i.e. when $$x=2\pi m$$,
\begin{align*} \cos kx = \cos 2\pi km &= 1\\ \sum_{k=1}^n\cos kx &= n\\ \frac12 + \sum_{k=1}^n\cos kx &= \frac12+ n\\ S_n &= \frac1n\left(\frac12 + n\right)\\ &= \frac1{2n} + 1\\ &\to 1 \end{align*}
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Prime Numbers
A prime number is a whole number that cannot be factored into two other whole numbers that are less than the number. In other words, a whole number is prime if the only other whole numbers that divide it are 1 and itself.
As an example, 3 is a prime number - there are no numbers $$a$$ and $$b$$ such that $$a \times b = 3$$, other than 1 and 3. On the other hand, 4 is not a prime number because $$2 \times 2 = 4$$. Whole numbers that are not prime are called composite numbers. On your own, try to think of 5 prime numbers and 5 composite numbers.
We have defined a isPrime() function that will test whether or not a whole number is prime. It will return true if the number is prime, and false otherwise. So for example, isPrime(3) will return true but isPrime(4) will return false.
print( isPrime(3) );
print( isPrime(4) );
This isPrime() function allows us to do some fun investigating with prime numbers! Instead of doing a bunch of computations, we can let the computer do that - while we make sense of the results! Let's check out an example.
Suppose we want to add together all of the prime numbers between 2 and 100. That would be a lot of work for us to do on paper, but the computer can do it in a fraction of a second! The code below shows this being done.
var sum = 0;
// Iterate from i=2 to i=100
for (var i = 2; i <= 100; i++){
// Check to see if i is prime...
if (isPrime(i)) {
// If so, add it to the sum.
sum = sum + i;
}
}
print(Sum: ${sum}); In Line (1) we create a variable sum and set its value to 0 - we will use this to keep track of the sum of the prime numbers. Then on Line (4) we start a for-loop and loop from i = 2 to i = 100, in increments of 1. For each value of i between 2 and 100, we will check to see if it is prime [Line (6)] and if so, we will add this value to our sum variable [Line (8)]. In Line (12) we print the sum. Here's the neat thing: on Line (4), change to 100 to a much bigger number, like 10000. This will tell the computer to compute the sum of all prime numbers between 2 and 10000 - and it can do this very quickly! Activities Activity: Counting primes Use the editor below to determine the number of prime numbers less than 1,000. You might want to use a loop similar to the one presented in the previous code snippet. print("Your code here."); Activity: Sorting based on primality An array a of random whole numbers between 2 and 100 has been defined behind the scenes. Sort the elements of a into two arrays based on whether or not they are prime. We've started you off with a little code. var primes = [], composites = []; for (var i = 0; i < a.length; i++) { // YOUR CODE HERE } print(Primes:${primes});
print(Composites: \${composites});
Challenge Activity: Sum of the first 100 primes
Use the editor below to determine the sum of the first 100 primes. Hint: this is different than computing the sum of all prime numbers less than or equal to 100.
print("Your code here.");
If done correctly, you should get a value of 24133. After doing that, find the sum of the first 1000 prime numbers.
Challenge Activity: Defining the isPrime function
We have been using an isPrime() function, which is defined behind the scenes. However, we actually wrote the code for that function - and you can too! In the editor below, create a function isPrime2() that determines whether or not a number is prime. The function should return true if the number is prime, and false otherwise.
function isPrime2(n) {
print( isPrime2(4) );
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40m QIL Cryo_Lab CTN SUS_Lab TCS_Lab OMC_Lab CRIME_Lab FEA ENG_Labs OptContFac Mariner WBEEShop
40m Log, Page 325 of 327 Not logged in
ID Date Author Type Category Subject
16230 Wed Jun 30 14:09:26 2021 Ian MacMillanUpdateCDSSUS simPlant model
I have looked at my code from the previous plot of the transfer function and realized that there is a slight error that must be fixed before we can analyze the difference between the theoretical transfer function and the measured transfer function.
The theoretical transfer function, which was generated from Photon has approximately 1000 data points while the measured one has about 120. There are no points between the two datasets that have the same frequency values, so they are not directly comparable. In order to compare them I must infer the data between the points. In the previous post [16195] I expanded the measured dataset. In other words: I filled in the space between points linearly so that I could compare the two data sets. Using this code:
#make values for the comparison tck_mag = splrep(tst_f, tst_mag) # get bspline representation given (x,y) values gen_mag = splev(sim_f, tck_mag) # generate intermediate values dif_mag=[] for x in range(len(gen_mag)): dif_mag.append(gen_mag[x]-sim_mag[x]) # measured minus predicted tck_ph = splrep(tst_f, tst_ph) # get bspline representation given (x,y) values gen_ph = splev(sim_f, tck_ph) # generate intermediate values dif_ph=[] for x in range(len(gen_ph)): dif_ph.append(gen_ph[x]-sim_ph[x])
At points like a sharp peak where the measured data set was sparse compared to the peak, the difference would see the difference between the intermediate “measured” values and the theoretical ones, which would make the difference much higher than it really was.
To fix this I changed the code to generate the intermediate values for the theoretical data set. Using the code here:
tck_mag = splrep(sim_f, sim_mag) # get bspline representation given (x,y) values gen_mag = splev(tst_f, tck_mag) # generate intermediate values dif_mag=[] for x in range(len(tst_mag)): dif_mag.append(tst_mag[x]-gen_mag[x])#measured minus predicted tck_ph = splrep(sim_f, sim_ph) # get bspline representation given (x,y) values gen_ph = splev(tst_f, tck_ph) # generate intermediate values dif_ph=[] for x in range(len(tst_ph)): dif_ph.append(tst_ph[x]-gen_ph[x])
Because this dataset has far more values (about 10 times more) the previous problem is not such an issue. In addition, there is never an inferred measured value used. That makes it more representative of the true accuracy of the real transfer function.
This is an update to a previous plot, so I am still using the same data just changing the way it is coded. This plot/data does not have a Q of 1000. That plot will be in a later post along with the error estimation that we talked about in this week’s meeting.
The new plot is shown below in attachment 1. Data and code are contained in attachment 2
Attachment 1: SingleSusPlantTF.pdf
Attachment 2: Plant_TF_Test.zip
16234 Thu Jul 1 11:37:50 2021 PacoUpdateGeneralrestarted c0rga
Physically rebooted c0rga workstation after failing to ssh into it (even as it was able to ping into it...) the RGA seems to be off though. The last log with data on it appears to date back to 2020 Nov 10, but reasonable spectra don't appear until before 11-05 logs. Gautam verified that the RGA was intentionally turned off then.
16235 Thu Jul 1 16:45:25 2021 YehonathanUpdateBHDSOS assembly
The bonding test passed - the weight still hangs from the dumbell. Unfortunately, I broke the bond trying to release the assembly from the bracket. I made another batch of 6 dumbell+magnet.
I used some of the leftover epoxy to bond an assembly from the previous batch to a bracket so I can test it.
16238 Tue Jul 6 10:47:07 2021 Paco, AnchalUpdateIOORestored MC
MC was unlocked and struggling to recover this morning due to misguided WFS offsets. In order to recover from this kind of issue, we
1. Cleared the bogus WFS offsets
2. Used the MC alignment sliders to change MC1 YAW from -0.9860 to -0.8750 until we saw the lowest order mode transmission on the video monitor.
3. With MC Trans sum at around ~ 500 counts, we lowered the C1:IOO-WFS_TRIGGER_THRESH_ON from 5000 to 500, and the C1:IOO-WFS_TRIGGER_MON from 3.0 to 0.0 seconds and let the WFS integrators work out some nonzero angular control offsets.
4. Then, the MC Trans sum increased to about 2000 counts but started oscillating slowly, so we restored the delayed loop trigger from 0.0 to 3.0 seconds and saw the MC Trans sum reach its nominal value of ~ 14000 counts over a few minutes.
The MC is now restored and the plan is to let it run for a few hours so the offsets converge; then run the WFS relief script.
16239 Tue Jul 6 16:35:04 2021 Anchal, Paco, GautamUpdateIOORestored MC
We found that megatron is unable to properly run scripts/MC/WFS/mcwfsoff and scripts/MC/WFS/mcwfson scripts. It fails cdsutils commands due to a library conflict. This meant that WFS loops were not turned off when IMC would get unlocked and they would keep integrating noise into offsets. The mcwfsoff script is also supposed to clear up WFS loop offsets, but that wasn't happening either. The mcwfson script was also not bringing back WFS loops on.
Gautam fixed these scripts temprorarily for running on megatron by using ezcawrite and ezcaswitch commands instead of cdsutils commands. Now these scripts are running normally. This could be the reason for wildly fluctuating WFS offsets that we have seen in teh past few months.
gautam: the problem here is that megatron is running Ubuntu18 - I'm not sure if there is any dedicated CDS group packaging for Ubuntu, and so we're using some shared install of the cdsutils (hosted on the shared chiara NFS drive), which is complaining about missing linked lib files. Depending on people's mood, it may be worth biting the bullet and make Megatron run Debian10, for which the CDS group maintains packages.
Quote: MC was unlocked and struggling to recover this morning due to misguided WFS offsets. In order to recover from this kind of issue, we Cleared the bogus WFS offsets Used the MC alignment sliders to change MC1 YAW from -0.9860 to -0.8750 until we saw the lowest order mode transmission on the video monitor. With MC Trans sum at around ~ 500 counts, we lowered the C1:IOO-WFS_TRIGGER_THRESH_ON from 5000 to 500, and the C1:IOO-WFS_TRIGGER_MON from 3.0 to 0.0 seconds and let the WFS integrators work out some nonzero angular control offsets. Then, the MC Trans sum increased to about 2000 counts but started oscillating slowly, so we restored the delayed loop trigger from 0.0 to 3.0 seconds and saw the MC Trans sum reach its nominal value of ~ 14000 counts over a few minutes. The MC is now restored and the plan is to let it run for a few hours so the offsets converge; then run the WFS relief script.
16243 Fri Jul 9 18:35:32 2021 YehonathanUpdateCDSOpto-isolator for c1auxey
Following Koji's channel list review, we made changes to the wiring spreadsheet.
Today, I made the changes real in the Acromag chassis. I went through the channel list one by one and made sure it is wired correctly. Additionally, since we now need all the channels the existing isolators have, I replaced the isolator with the defective channel with a new one.
The things to do next:
1. Create entries for the spare coil driver and satellite box channels in the EPICs DB.
2. Test the spare channels.
16244 Mon Jul 12 18:06:25 2021 YehonathanUpdateCDSOpto-isolator for c1auxey
I edited /cvs/cds/caltech/target/c1auxey1/ETMYaux.db (after creating a backup) and added the spare coil driver channels.
I tested those channels using caget while fixing wiring issues. The tests were all succesful. The digital output channel were tested using the Windows machine since they are locked by some EPICs mechanism I don't yet understand.
One worrying point is I found that the differential analog inputs to be unstable unless I connected a reference to some stable voltage source unlike previous tests showed. It was unstable (but less) even when I connected the ref to the ground connectors on the power supplies on the workbench. This is really puzzling.
When I say unstable I mean that most of the time the voltage reading shows the right value, but occasionly there is a transient sharp volage drop of the order of 0.5V. I will do a more quantitative analysis tomorrow.
16245 Wed Jul 14 16:19:44 2021 gautamUpdateGeneralBrrr
Since the repair work, the temperature is significantly cooler. Surprisingly, even at the vertex (to be more specific, inside the PSL enclosure, which for the time being is the only place where we have a logged temperature sensor, but this is not attributable to any change in the HEPA speed), the temperature is a good 3 deg C cooler than it was before the HVAC work (even though Koji's wind vane suggest the vents at the vertex were working). The setpoint for the entire lab was modified? What should the setpoint even be?
Quote: - I went to the south arm. There are two big vent ducts for the outlets and intakes. Both are not flowing the air. The current temp at 7pm was ~30degC. Max and min were 31degC and 18degC. - Then I went to the vertex and the east arm. The outlets and intakes are flowing.
Attachment 1: rmTemp.pdf
16246 Wed Jul 14 19:21:44 2021 KojiUpdateGeneralBrrr
Jordan reported on Jun 18, 2021:
"HVAC tech came today, and replaced the thermostat and a coolant tube in the AC unit. It is working now and he left the thermostat set to 68F, which was what the old one was set to."
16247 Wed Jul 14 20:42:04 2021 gautamUpdateLSCLocking
[paco, gautam]
we decided to give the PRFPMI lock a go early-ish. Summary of findings today eve:
1. Arms under ALS control display normal noise and loop UGFs.
2. PRMI took longer than usual to lock (when arms are held off resonance) - could be elevated sesimic, but warrants measuring PRMI loop TFs to rule out any funkiness. MICH loop also displayed some saturation on acquisition, but after the boosts and other filters were turned on, the lock seemed robust and the in-loop noise was at the usual levels.
3. We are gonna do the high bandwidth single arm locking experiments during daytime to rule out any issues with the CM board.
The ALS--> IR CARM handoff is the problematic step. In the past, getting over this hump has just required some systematic loop TF measurements / gain slider readjustments. We will do this in the next few days. I don't think the ALS noise is any higher than it used to be, and I could do the direct handoff as recently as March, so probably something minor has changed.
16248 Thu Jul 15 14:25:48 2021 PacoUpdateLSCCM board
[gautam, paco]
We tested the CM board by implementing the high bandwidth IR lock (single arm). In preparation for this test we temporarily connected the POY11_Q_MON output to the CM board IN1 input and checked the YARM POY transfer function by running the AA_YARM_TEMPLATE under users/Templates/LSC/LSC_loops/YARM_POY/. We made sure the YARM dither optimized TRY so as to maximize the optical gain stage. Then we proceeded as follows:
• From the LSC --> CM Servo screen, we controlled the REFL 1 Gain (dB) slider (nominal +25) and MC Servo IN2 Gain (dB) slider (nominal -32 dB) to transfer the low bandwidth (digital) control to the high bandwidth (analog) control of the YARM.
• During this game, we monitored the C1:LSC-POY11_I_ERR_DQ & C1:LSC-CM_SLOW_OUT_DQ error signal channels for saturation, oscillations, or stability.
• Once a set of gains was successful in maintaining a stable lock, we measured the OLTF using SR 785 to track the UGF as we mix the two paths.
• Once the gains have increased, a boost and super-boost stages may be enabled as well.
Ultimately, our ability to progressively increase the control bandwidth of the YARM is a proxy that the CM board is working properly. Attachment 1 shows the OLTF progression as we increased the loop's UGF. Note how as we approached the maximum measured UGF of ~ 22 kHz, our phase margin decreased signifying poor stability.
At the end of this measurement, at about ~ 15:45 I restored the CM board IN1 input and disconnected the POY11_Q_MON
gautam: the conclusion here is that the CM board seems to work as advertised, and it's not solely responsible for not being able to achieve the IR handoff.
Attachment 1: high_BW_TFs.pdf
16249 Fri Jul 16 16:26:50 2021 gautamUpdateComputersDocker installed on nodus
I wanted to try hosting some docker images on a "private" server, so I installed Docker on nodus following the instructions here. The install seems to have succeeded, and as far as I can tell, none of the functionality of nodus has been disturbed (I can ssh in, access shared drive, elog seems to work fine etc). But if you find a problem, maybe this action is responsible. Note that nodus is running Scientific Linux 7.3 (Nitrogen).
16250 Sat Jul 17 00:52:33 2021 KojiUpdateGeneralCanon camera / small silver tripod / macro zoom lens / LED ring light borrowed -> QIL
Canon camera / small silver tripod / macro zoom lens / LED ring light borrowed -> QIL
Attachment 1: P_20210716_213850.jpg
16251 Mon Jul 19 22:16:08 2021 pacoUpdateLSCPRFPMI locking
[gautam, paco]
Gautam managed to lock PRFPMI a little before ~ 22:00 local time. The ALS to RF handoff logic was found to be repeatable, which enabled us to lock a total of 4 times this evening. Under this nominal state, we can work on PRFPMI to narrow down less known issues and carry out systematic optimization. The second time we achieved lock, we ran sensing lines before entering the ASC stage (which we knew would destroy the lock), and offline analysis of the sensing matrix is pending (gpstime = 1310792709 + 5 min).
Things to note:
(a) there is an unexpected offset suggesting that the ALS and RF disagreed on what the lock setpoint should be, and it is still unclear where the offset is coming from.
(b) the first time the lock was reached, the ASC up stage destroyed it, suggesting these loops need some care (we were able to engage the ASC loops at low gains (0.2 instead of 1) but as soon as we enabled some integrators this consistently destroyed the lock
(c) gautam had (burt) restored to the settings from back in March when the PRFPMI was last locked, suggesting there was a small but somehow significant difference in the IFO that helped today relative to last week
Take home message--> The mere fact that we were able to lock PRFPMI rules out the considerably more serious problems with the signal chain electronics or processing. This should also be a good starting point for further debugging and optimization.
gautam: the circulating power, when the ASC was tweaked, hit 400 (normalized to single arm locked with a misaligned PRM) suggesting a recycling gain of 22.5, and an average arm loss of ~30ppm round trip (assuming 2% loss in the PRC).
16252 Wed Jul 21 14:50:23 2021 KojiUpdateSUSNew electronics
Jun 29, 2021 BIO I/F 6 units
Jul 19, 2021 PZT Drivers x2 / QPD Transimedance amp x2
Attachment 1: P_20210629_183950.jpeg
Attachment 2: P_20210719_135938.jpeg
16253 Wed Jul 21 18:08:35 2021 yehonathanUpdateLoss MeasurementLoss measurement
{Gautam, Yehonathan, Anchal, Paco}
We prepared for the loss measurement using DC reflection method. We did the following changes:
1. REFL55_Q was disconnected and replaced with MC_T cable coming from the PD on the MC2 table. The cable has a red tag on it. Consequently we lost the AS beam. We realigned the optics and regained arm locks. The spot on the AS QPD had to be corrected.
2. We tried using AS55 as the PD for the DC measurement but we got ratios of ~ 0.97 which implies losses of more than 100 ppm. We decided to go with the traditional PD520 used for these measurements in the past.
3. We placed the PD520 used for loss measurements in front of the AS55 PD and optimized its position.
4. AS110 cable was disconnected from the PD and connected to PD520 to be used as the loss measurement cable.
5. In 1Y2 rack, AS110 PD cable was disconnected, REFL55_I was disconnected and AS110 cable was connected to REFL55_I channel.
So for the test, the MC transmission was measured at REFL55_Q and the AS DC was measured at REFL55_I.
We used the scripts/lossmap_scripts/armLoss/measArmLoss.py script. Note that this script assumes that you begin with the arm locked.
We are leaving the IFO in the configuration described above overnight and we plan to measure the XARM loss early AM. After which we shall restore the affected electrical and optical paths.
We ran the /scripts/lossmap_scripts/armLoss/measureArmLoss.py script in pianosa with 25 repetitions and a 30 s "duty cycle" (wait time) for the Y arm. Preliminary results give an estimated individual arm loss of ~ 30 ppm (on both X/Y arms) but we will provide a better estimate with this measurement.
16254 Thu Jul 22 16:06:10 2021 PacoUpdateLoss MeasurementLoss measurement
[yehonathan, anchal, paco, gautam]
We concluded estimating the XARM and YARM losses. The hardware configuration from yesterday remains, but we repeated the measurements because we realized our REFL55_I_ERR and REFL55_Q_ERR signals representing the PD520 and MC_TRANS were scaled, offset, and rotated in a way that wasn't trivially undone by our postprocessing scripts... Another caveat that we encountered today was the need to add a "macroscopic" misalignment to the ITMs when doing the measurement to avoid any accidental resonances.
The final measurements were done with 16 repetitions, 30 second duration, and the logfiles are under scripts/lossmap_scripts/armLoss/logs/20210722_1423.txt and scripts/lossmap_scripts/armLoss/logs/20210722_1513.txt
Finally, the estimated YARM loss is 39$\pm$7 ppm, while the estimated XARM loss is 38$\pm$8 ppm. This is consistent with the inferred PRC gain from Monday and a PRM loss of ~ 2%.
Future measurements may want to look into slow drift of the locked vs misaligned traces (systematic errors?) and a better way of estimating the statistical uncertainty (e.g. by splitting the raw time traces into short segments)
16255 Sun Jul 25 18:21:10 2021 KojiUpdateGeneralCanon camera / small silver tripod / macro zoom lens / LED ring light returned / Electronics borrowed
Camera and accesories returned
One HAM-A coildriver and one sat amp borrowed -> QIL
https://nodus.ligo.caltech.edu:8081/QIL/2616
16256 Sun Jul 25 20:41:47 2021 ranaUpdateLoss MeasurementLoss measurement
What are the quantitative root causes for why the statistical uncertainty is so large? Its larger than 1/sqrt(N)
16257 Mon Jul 26 17:34:23 2021 PacoUpdateLoss MeasurementLoss measurement
[gautam, yehonathan, paco]
We went back to the loss data from last week and more carefully estimated the ARM loss uncertainties.
Before we simply stitched all N=16 repetitions into a single time-series and computed the loss: e.g. see Attachment 1 for such a YARM loss data. The mean and stdev for this long time series give the quoted loss from last time. We knew that the uncertainty was most certainly overestimated, as different realizations need not sample similar alignment conditions and are sensitive to different imperfections (e.g. beam angular motion, unnormalizable power fluctuations, etc...).
Today we analyzed the individual locked/misaligned cycles individually. From each cycle, it is possible to obtain a mean value of the loss as well as a std dev *across the duration of the trace*, but because we have a measurement ensemble, it is also possible to obtain an ensemble averaged mean and a statistical uncertainty estimate *across the independent cycle realizations*. While the mean values don't change much, in the latter estimate we find a much smaller statistical uncertainty. We obtain an XARM loss of 37.6 $\pm$ 2.6 ppm and a YARM loss of 38.9 $\pm$ 0.6 ppm. To make the distinction more clear, Attachment 2 and Attachment 3 the YARM and XARM loss measurement ensembles respectively with single realization (time-series) standard deviations as vertical error bars, and the 1 sigma statistical uncertainty estimate filled color band. Note that the XARM loss drifts across different realizations (which happen to be ordered in time), which we think arise from inconsistent ASS dither alignment convergence. This is yet to be tested.
For budgeting the excessive uncertainties from a single locked/misaligned cycle, we could look at beam pointing, angular drift, power, and systematic differences in the paths from both reflection signals. We should be able to estimate the power fluctuations by looking at the recorded arm transmissions, the recorded MC transmission, PD technical noise, etc... and we might be able to correlate recorded oplev signals with the reflection data to identify angular drift. We have not done this yet.
Attachment 1: LossMeasurement_RawData.pdf
Attachment 2: YARM_loss_stats.pdf
Attachment 3: XARM_loss_stats.pdf
16259 Tue Jul 27 17:14:18 2021 YehonathanUpdateBHDSOS assembly
Jordan has made 1/4" tap holes in the lower EQ stop holders (attachment). The 1/4" stops (schematics) fit nicely in them. Also, they are about the same length as the small EQ stops, so they can be used.
However, counting all the 1/4"-3/4" vented screws we have shows that we are missing 2 screws to cover all the 7 SOSs. We can either:
1. Order new vented screws.
2. Use 2 old (stained but clean) EQ stops.
3. Screw holes into existing 1/4"-3/4" screws and clean them.
4. Use small EQ stops for one SOS.
etc.
Also, I found a mistake in the schematics of the SOS tower. The 4-40 screws used to hold the lower EQ stop holders should be SS and not silver plated as noted. I'll have to find some (28) spares in the cleanroom or order new ones.
Attachment 1: 20210727_154506.png
16260 Tue Jul 27 20:12:53 2021 KojiUpdateBHDSOS assembly
1 or 2. The stained ones are just fine. If you find the vented 1/4-20 screws in the clean room, you can use them.
For the 28 screws, yeah find some spares in the clean room (faster), otherwise just order.
16261 Tue Jul 27 23:04:37 2021 AnchalUpdateLSC40 meter party
[ian, anchal, paco]
After our second attempt of locking PRFPMI tonight, we tried to resotre XARM and YARM locks to IR by clicking on IFO_CONFIGURE>Restore XARM (POX) and IFO_CONFIGURE>Restore YARM (POY) but the arms did not lock. The green lasers were locked to the arms at maximum power, so the relative alignments of each cavity was ok. We were also able to lock PRMI using IFO_CONFIGURE>Restore PRMI carrier.
This was very weird to us. We were pretty sure that the aligment is correct, so we decided to cehck the POX POY signal chain. There was essentially no signal coming at POX11 and there was a -100 offset on it. We could see some PDH signal on POY11 but not enough to catch the locks.
We tried running IFO_CONFIGURE>LSC OFFSETS to cancel out any dark current DC offsets. The changes made by the script are shown in attachment 1.
We went to check the tables and found no light visible on beam finder cards on POX11 or POY11. We found that ITMX was stuck on one of the coils. We unstuck it using the shaking method. The OPLEVs on ITMX after this could not be switched on as the OPLEV servo were railing to limits. But when we ran Restore XARM (POX) again, they started working fine. Something is done by this script that we are not aware of.
We're stopping here. We still can not lock any of the single arms.
Wed Jul 28 11:19:00 2021 Update:
[gautam, paco]
Gautam found that the restoring of POX/POY failed to restore the whitening filter gains in POX11 / POY11. These are meant to be restored to 30 dB and 18 dB for POX11 and POY11 respectively but were set to 0 dB in detriment of any POX/POY triggering/locking. The reason these are lowered is to avoid saturating the speakers during lock acquisition. Yesterday, burt-restore didn't work because we restored the c1lscepics.snap but said gains are actually in c1lscaux.snap. After manually restoring the POX11 and POY11 whitening filter gains, gautam ran the LSCOffsets script. The XARM and YARM were able to quickly lock after we restored these settings.
The root of our issue may be that we didn't run the CARM & DARM watch script (which can be accessed from the ALS/Watch Scripts in medm). Gautam added a line on the Transition_IR_ALS.py script to run the watch script instead.
Attachment 1: Screenshot_2021-07-27_22-19-58.png
16262 Wed Jul 28 12:00:35 2021 YehonathanUpdateBHDSOS assembly
After receiving two new tubes of EP-30 I resumed the gluing activities. I made a spreadsheet to track the assemblies that have been made, their position on the metal sheet in the cleanroom, their magnetic field, and the batch number.
I made another batch of 6 magnets yesterday (4th batch), the assembly from the 2nd batch is currently being tested for bonding strength.
One thing that we overlooked in calculating the amount of glue needed is that in addition to the minimum 8gr of EP-30 needed for every gluing session, there is also 4gr of EP-30 wasted on the mixing tube. So that means 12gr of EP-30 are used in every gluing session. We need 5 more batches so at least 60gr of EP-30 is needed. Luckily, we bought two tubes of 50gr each.
16263 Wed Jul 28 12:47:52 2021 YehonathanUpdateCDSOpto-isolator for c1auxey
To simulate a differential output I used two power supplies connected in series. The outer connectors were used as the outputs and the common connector was connected to the ground and used as a reference. I hooked these outputs to one of the differential analog channels and measured it over time using Striptool. The setup is shown in attachment 3.
I tested two cases: With reference disconnected (attachment 1), and connected (attachment 2). Clearly, the non-referred case is way too noisy.
Attachment 1: SUS-ETMY_SparePDMon0_NoRef.png
Attachment 2: SUS-ETMY_SparePDMon0_Ref_WithGND.png
Attachment 3: DifferentialOutputTest.png
16264 Wed Jul 28 17:10:24 2021 AnchalUpdateLSCSchnupp asymmetry
[Anchal, Paco]
I redid the measurement of Schnupp asymmetry today and found it to be 3.8 cm $\pm$ 0.9 cm.
### Method
• One of the arms is misalgined both at ITM and ETM.
• The other arm is locked and aligned using ASS.
• The SRCL oscillator's output is changed to the ETM of the chosen arm.
• The AS55_Q channel in demodulation of SRCL oscillator is configured (phase corrected) so that all signal comes in C1:CAL-SENSMAT_SRCL_AS55_Q_DEMOD_I_OUT.
• The rotation angle of AS55 RFPD is scanned and the C1:CAL-SENSMAT_SRCL_AS55_Q_DEMOD_I_OUT is averaged over 10s after waiting for 5s to let the transients pass.
• This data is used to find the zero crossing of AS55_Q signal when light is coming from one particular arm only.
• The same is repeated for the other arm.
• The difference in the zero crossing phase angles is twice the phase accumulated by a 55 MHz signal in travelling the length difference between the arm cavities i.e. the Schnupp Asymmetry.
I measured a phase difference of 5 $\pm$1 degrees between the two paths.
The uncertainty in this measurement is much more than gautam's 15956 measurement. I'm not sure yet why, but would look into it.
Quote: I used the Valera technique to measure the Schnupp asymmetry to be $\approx 3.5 \, \mathrm{cm}$, see Attachment #1. The data points are points, and the zero crossing is estimated using a linear fit. I repeated the measurement 3 times for each arm to see if I get consistent results - seems like I do. Subtle effects like possible differential detuning of each arm cavity (since the measurement is done one arm at a time) are not included in the error analysis, but I think it's not controversial to say that our Schnupp asymmetry has not changed by a huge amount from past measurements. Jamie set a pretty high bar with his plot which I've tried to live up to.
Attachment 1: Lsch.pdf
16265 Wed Jul 28 20:20:09 2021 YehonathanUpdateGeneralThe temperature sensors and function generator have arrived in the lab
I put the temperature sensors box on Anchal's table (attachment 1) and the function generator on the table in front of the c1auxey Acromag chassis (attachment 2).
Attachment 1: 20210728_201313.jpg
Attachment 2: 20210728_201607.jpg
16266 Thu Jul 29 14:51:39 2021 PacoUpdateOptical LeversRecenter OpLevs
[yehonathan, anchal, paco]
Yesterday around 9:30 pm, we centered the BS, ITMY, ETMY, ITMX and ETMX oplevs (in that order) in their respective QPDs by turning the last mirror before the QPDs. We did this after running the ASS dither for the XARM/YARM configurations to use as the alignment reference. We did this in preparation for PRFPMI lock acquisition which we had to stop due to an earthquake around midnight
16267 Mon Aug 2 16:18:23 2021 PacoUpdateASCAS WFS MICH commissioning
[anchal, paco]
We picked up AS WFS comissioning for daytime work as suggested by gautam. In the end we want to comission this for the PRFPMI, but also for PRMI, and MICH for completeness. MICH is the simplest so we are starting here.
We started by restoromg the MICH configuration and aligning the AS DC QPD (on the AS table) by zeroing the C1:ASC-AS_DC_YAW_OUT and C1:ASC-AS_DC_PIT_OUT. Since the AS WFS gets the AS beam in transmission through a beamsplitter, we had to correct such a beamsplitters's aligment to recenter the AS beam onto the AS110 PD (for this we looked at the signal on a scope).
We then checked the rotation (R) C1:ASC-AS_RF55_SEGX_PHASE_R and delay (D) angles C1:ASC-AS_RF55_SEGX_PHASE_D (where X = 1, 2, 3, 4 for segment) to rotate all the signal into the I quadrature. We found that this optimized the PIT content on C1:ASC-AS_RF55_I_PIT_OUT and YAW content on C1:ASC-AS_RF55_I_YAW_OUTMON which is what we want anyways.
Finally, we set up some simple integrators for these WFS on the C1ASC-DHARD_PIT and C1ASC-DHARD_YAW filter banks with a pole at 0 Hz, a zero at 0.8 Hz, and a gain of -60 dB (similar to MC WFS). Nevertheless, when we closed the loop by actuating on the BS ASC PIT and ASC YAW inputs, it seemed like the ASC model outputs are not connected to the BS SUS model ASC inputs, so we might need to edit accordingly and restart the model.
16268 Tue Aug 3 20:20:08 2021 AnchalUpdateOptical LeversRecentered ETMX, ITMX and ETMY oplevs at good state
Late elog. Original time 08/02/2021 21:00.
I locked both arms and ran ASS to reach to optimum alignment. ETMY PIT > 10urad, ITMX P > 10urad and ETMX P < -10urad. Everything else was ok absolute value less than 10urad. I recentered these three.
Than I locked PRMI, ran ASS on PRCL and MICH and checked BS and PRM alignment. They were also less than absolute value 10urad.
16269 Wed Aug 4 18:19:26 2021 pacoUpdateGeneralAdded infrasensing temperature unit to martian network
[ian, anchal, paco]
We hooked up the infrasensing unit to power and changed its default IP address from 192.168.11.160 (factory default) to 192.168.113.240 in the martian network. The sensor is online with user controls and the usual password for most workstations in that IP address.
16270 Thu Aug 5 14:59:31 2021 AnchalUpdateGeneralAdded temperature sensors at Yend and Vertex too
I've added the other two temperature sensor modules on Y end (on 1Y4, IP: 192.168.113.241) and in the vertex on (1X2, IP: 192.168.113.242). I've updated the martian host table accordingly. From inside martian network, one can go to the browser and go to the IP address to see the temperature sensor status . These sensors can be set to trigger alarm and send emails/sms etc if temperature goes out of a defined range.
I feel something is off though. The vertex sensor shows temperature of ~28 degrees C, Xend says 20 degrees C and Yend says 26 degrees C. I believe these sensors might need calibration.
• Modbus TCP solution:
• If we get it right, this will be easiest solution.
• We just need to add these sensors as streaming devices in some slow EPICS machine in there .cmd file and add the temperature sensing channels in a corresponding database file.
• Python workaround:
• Might be faster but dirty.
• We run a python script on megatron which requests temperature values every second or so from the IP addresses and write them on a soft EPICs channel.
• We still would need to create a soft EPICs channel fro this and add it to framebuilder data acquisition list.
• Even shorted workaround for near future could be to just write temperature every 30 min to a log file in some location.
[anchal, paco]
We made a script under scripts/PEM/temp_logger.py and ran it on megatron. The script uses the requests package to query the latest sensor data from the three sensors every 10 minutes as a json file and outputs accordingly. This is not a permanent solution.
16271 Fri Aug 6 13:13:28 2021 AnchalUpdateBHDc1teststand subnetwork now accessible remotely
c1teststand subnetwork is now accessible remotely. To log into this network, one needs to do following:
• Log into nodus or pianosa. (This will only work from these two computers)
• ssh -CY controls@192.168.113.245
• This will log you into c1teststand network.
• From here, you can log into fb1, chiara, c1bhd and c1sus2 which are all part of the teststand subnetwork.
Just to document the IT work I did, doing this connection was bit non-trivial than usual.
• The martian subnetwork is created by a NAT router which connects only nodus to outside GC network and all computers within the network have ip addresses 192.168.113.xxx with subnet mask of 255.255.255.0.
• The cloned test stand network was also running on the same IP address scheme, mostly because fb1 and chiara are clones in this network. So every computer in this network also had ip addresses 192.168.113.xxx.
• I setup a NAT router to connect to martian network forwarding ssh requests to c1teststand computer. My NAT router creates a separate subnet with IP addresses 10.0.1.xxx and suubnet mask 255.255.255.0 gated through 10.0.1.1.
• However, the issue is for c1teststand, there are now two networks accessible which have same IP addresses 192.168.113.xxx. So when you try to do ssh, it always search in its local c1teststand subnetwork instead of routing through the NAT router to the martian network.
• To work around this, I had to manually provide an ip router to c1teststand for connecting to two of the computers (nodus and pianosa) in martian network. This is done by:
ip route add 192.168.113.200 via 10.0.1.1 dev eno1
ip route add 192.168.113.216 via 10.0.1.1 dev eno1
• This gives c1teststand specific path for ssh requests to/from these computers in the martian network.
16272 Fri Aug 6 17:10:19 2021 PacoUpdateIMCMC rollercoaster
[anchal, yehonatan, paco]
For whatever reason (i.e. we don't really know) the MC unlocked into a weird state at ~ 10:40 AM today. We first tried to find a likely cause as we saw it couldn't recover itself after ~ 40 min... so we decided to try a few things. First we verified that no suspensions were acting weird by looking at the OSEMs on MC1, MC2, and MC3. After validating that the sensors were acting normally, we moved on to the WFS. The WFS loops were disabled the moment the IMC unlocked, as they should. We then proceeded to the last resort of tweaking the MC alignment a bit, first with MC2 and then MC1 and MC3 in that order to see if we could help the MC catch its lock. This didn't help much initially and we paused at about noon.
At about 5 pm, we resumed since the IMC had remained locked to some higher order mode (TEM-01 by the looks of it). While looking at C1:IOO-MC_TRANS_SUMFILT_OUT on ndscope, we kept on shifting the MC2 Yaw alignment slider (steps = +-0.01 counts) slowly to help the right mode "hop". Once the right mode caught on, the WFS loops triggered and the IMC was restored. The transmission during this last stage is shown in Attachment #1.
Attachment 1: MC2_trans_sum_2021-08-06_17-18-54.png
16273 Mon Aug 9 10:38:48 2021 AnchalUpdateBHDc1teststand subnetwork now accessible remotely
I had to add following two lines in the /etc/network/interface file to make the special ip routes persistent even after reboot:
post-up ip route add 192.168.113.200 via 10.0.1.1 dev eno1
post-up ip route add 192.168.113.216 via 10.0.1.1 dev eno1
16274 Tue Aug 10 17:24:26 2021 pacoUpdateGeneralFive day trend
Attachment 1 shows a five and a half day minute-trend of the three temperature sensors. Logging started last Thursday ~ 2 pm when all sensors were finally deployed. While it appears that there is a 7 degree gradient along the XARM it seems like the "vertex" (more like ITMX) sensor was just placed on top of a network switch (which feels lukewarm to the touch) so this needs to be fixed. A similar situation is observed in the ETMY sensor. I shall do this later today.
Done. The temperature reading should now be more independent from nearby instruments.
Wed Aug 11 09:34:10 2021 I updated the plot with the full trend before and after rearranging the sensors.
Attachment 1: six_day_minute_trend.png
16275 Wed Aug 11 11:35:36 2021 PacoUpdateLSCPRMI MICH orthogonality plan
[yehonathan, paco]
Yesterday we discussed a bit about working on the PRMI sensing matrix.
In particular we will start with the "issue" of non-orthogonality in the MICH actuated by BS + PRM. Yesterday afternoon we played a little with the oscillators and ran sensing lines in MICH and PRCL (gains of 50 and 5 respectively) in the times spanning [1312671582 -> 1312672300], [1312673242 -> 1312677350] for PRMI carrier and [1312673832 -> 1312674104] for PRMI sideband. Today we realize that we could have enabled the notchSensMat filter, which is a notch filter exactly at the oscillator's frequency, in FM10 and run a lower gain to get a similar SNR. We anyways want to investigate this in more depth, so here is our tentative plan of action which implies redoing these measurements:
Task: investigate orthogonality (or lack thereof) in the MICH when actuated by BS & PRM
1) Run sensing MICH and PRCL oscillators with PRMI Carrier locked (remember to turn NotchSensMat filter on).
2) Analyze data and establish the reference sensing matrix.
3) Write a script that performs steps 2 and 3 in a robust and safe way.
4) Scan the C1:LSC-LOCKIN_OUTMTRX, MICH to BS and PRM elements around their nominal values.
5) Scan the MICH and PRCL RFPD rotation angles around their nominal values.
We also talked about the possibility that the sensing matrix is strongly frequnecy dependant such that measuring it at 311Hz doesn't give us accurate estimation of it. Is it worthwhile to try and measure it at lower frequencies using an appropriate notch filter?
Wed Aug 11 15:28:32 2021 Updated plan after group meeting
- The problem may be in the actuators since the orthogonality seems fine when actuating on the ITMX/ITMY, so we should instead focus on measuring the actuator transfer functions using OpLevs for example (same high freq. excitation so no OSEM will work > 10 Hz).
16276 Wed Aug 11 12:06:40 2021 YehonathanUpdateCDSOpto-isolator for c1auxey
I redid the differential input experiment using the DS360 function generator we recently got. I generated a low frequency (0.1Hz) sine wave signal with an amplitude 0.5V and connected the + and - output to a differential input on the new c1auxcey Acromag chassis. I recorded a time series of the corresponding EPICS channel with and without the common on the DS360 connected to the Ref connector on the Acromag unit. The common connector on the DS360 is not normally grounded (there is a few tens of kohms between the ground and common connectors). The attachment shows that, indeed, the analog input readout is extremely noisy with the Ref being disconnected. The point where the Ref was connected to common is marked in the picture.
Conclusion: Ref connector on the analog input Acromag units must be connected to some stable voltage source for normal operation.
Attachment 1: SUS-ETMY_SparePDMon0_2.png
16277 Thu Aug 12 11:04:27 2021 PacoUpdateGeneralPSL shutter was closed this morning
Thu Aug 12 11:04:42 2021 Arrived to find the PSL shutter closed. Why? Who? When? How? No elog, no fun. I opened it, IMC is now locked, and the arms were restored and aligned.
16278 Thu Aug 12 14:59:25 2021 KojiUpdateGeneralPSL shutter was closed this morning
What I was afraid of was the vacuum interlock. And indeed there was a pressure surge this morning. Is this real? Why didn't we receive the alert?
Attachment 1: Screen_Shot_2021-08-12_at_14.58.59.png
16279 Thu Aug 12 20:52:04 2021 KojiUpdateGeneralPSL shutter was closed this morning
I did a bit more investigation on this.
- I checked P1~P4, PTP2/3, N2, TP2, TP3. But found only P1a and P2 were affected.
- Looking at the min/mean/max of P1a and P2 (Attachment 1), the signal had a large fluctuation. It is impossible to have P1a from 0.004 to 0 instantaneously.
- Looking at the raw data of P1a and P2 (Attachment 2), the value was not steadily large. Instead it looks like fluctuating noise.
So my conclusion is that because of an unknown reason, an unknown noise coupled only into P1a and P2 and tripped the PSL shutter. I still don't know the status of the mail alert.
Attachment 1: Screen_Shot_2021-08-12_at_20.51.19.png
Attachment 2: Screen_Shot_2021-08-12_at_20.51.34.png
16280 Mon Aug 16 23:30:34 2021 PacoUpdateCDSAS WFS commissioning; restarting models
[koji, ian, tega, paco]
With the remote/local assistance of Tega/Ian last friday I made changes on the c1sus model by connecting the C1:ASC model outputs (found within a block in c1ioo) to the BS and PRM suspension inputs (pitch and yaw). Then, Koji reviewed these changes today and made me notice that no changes are actually needed since the blocks were already in place, connected in the right ports, but the model probably just wasn't rebuilt...
So, today we ran "rtcds make", "rtcds install" on the c1ioo and c1sus models (in that order) but the whole system crashed. We spent a great deal of time restarting the machines and their processes but we struggled quite a lot with setting up the right dates to match the GPS times. What seemed to work in the end was to follow the format of the date in the fb1 machine and try to match the timing to the sub-second level. This is especially tricky when performed by a human action so the whole task is tedious. We anyways completed the reboot for almost all the models except the c1oaf (which tends to make things crashy) since we won't need it right away for the tasks ahead. One potential annoying issue we found was in manually rebooting c1iscey because one of its network ports is loose (the ethernet cable won't click in place) and it appears to use this link to boot (!!) so for a while this machine just wasn't coming back up.
Finally, as we restored the suspension controls and reopened the shutters, we noticed a great deal of misalignment to the point no reflected beam was coming back to the RFPD table. So we spent some time verifying the PRM alignment and TT1 and TT2 (tip tilts) and it turned out to be mostly the latter pair that were responsible for it. We used the green beams to help optimize the XARM and YARM transmissions and were able to relock the arms. We ran ASS on them, and then aligned the PRM OpLevs which also seemed off. This was done by giving a pitch offset to the input PRM oplev beam path and then correcting for it downstream (before the qpd). We also adjusted the BS OpLev in the end.
Summary; the ASC BS and PRM outputs are now built into the SUS models. Let the AS WFS loops be closed soon!
- Upon the RTS restarting,
• Date/Time adjustment
sudo date --set='xxxxxx'
• If the time on the CDS status medm screen for each IOP match with the FB local time, we ran
rtcds start c1x01
(or c1x02, etc)
• Every time we restart the IOPs, fb was restarted by
telnet fb1 8083 > shutdown
and restarted mx_stream from the CDS screen because these actions change the "DC" status.
- Today we once succeeded to restart the vertex machines. However, the RFM signal transmission did fail. So the end two machines were power cycled as well as c1rfm, but this made all the machines in RED again. Hell...
- We checked the PRM oplev. The spot was around the center but was clipped. This made us so confused. Our conclusion was that the oplev was like that before the RTS reboot.
16281 Tue Aug 17 04:30:35 2021 KojiUpdateSUSNew electronics
Aug 17, 2021 2x ISC Whitening
Delivered 2x Sat Amp board to Todd
Attachment 1: P_20210816_234136.jpg
Attachment 2: P_20210816_235106.jpg
Attachment 3: P_20210816_234220.jpg
16282 Wed Aug 18 20:30:12 2021 AnchalUpdateASSFixed runASS scripts
Late elog: Original time of work Tue Aug 17 20:30 2021
I locked the arms yesterday remotely and tried running runASS.py scripts (generally ran by clicking Run ASS buttons on IFO OVERVIEW screen of ASC screen). We have known for few weeks that this script stopped working for some reason. It would start the dithering and would optimize the alignment but then would fail to freeze the state and save the alignment.
I found the caget('C1:LSC-TRX_OUT') or caget('C1:LSC-TRY_OUT') were not working in any of the workstations. This is weird since caget was able to acquire these fast channel values earlier and we have seen this script to work for about a month without any issue.
Anyways, to fix this, I just changed the channel name to 'C1:LSC-TRY_OUT16' when the script checks in the end if the arm has indeed been aligned. It was only this step that was failing. Now the script is working fine and I tested them on both arms. On the Y arm, I misaligned the arm by adding bias in yaw by changing C1:SUS-ITMY_YAW_OFFSET from -8 to 22. The script was able to align the arm back.
16283 Thu Aug 19 03:23:00 2021 AnchalUpdateCDSTime synchornization not running
I tried to read a bit and understand the NTP synchronization implementation in FE computers. I'm quite sure that NTP synchronization should be 'yes' if timesyncd are running correctly in the output of timedatectl in these computers. As Koji reported in 15791, this is not the case. I logged into c1lsc, c1sus and c1ioo and saw that RTC has drifted from the software clocks too which does not happen if NTP synchronization was active. This would mean that almost certainly, if the computers are rebooted, the synchronization will be lost and the models will fail to come online.
My current findings are the following (this should be documented in wiki once we setup everything):
• nodus is running a NTP server using chronyd. One can check the configuration of this NTP serer in /etc/chornyd.conf
• fb1 is running an NTP server using ntpd that follows nodus and an IP address 131.215.239.14. This can be seen in /etc/ntp.conf.
• There are no comments to describe what this other server (131.215.239.14) is. Does the GC network have an NTP server too?
• c1lsc, c1sus and c1ioo all have systemd-timesyncd.service running with configuration file in /etc/systemd/timesyncd.conf.
• The configuration file set Servers=ntpserver but echo $ntpserver produces nothing (blank) on these computers and I've been unable to find anyplace where ntpserver is defined. • In chiara (our name server), the name server file /etc/hosts does not have any entry for ntpserver either. • I think the problem might be that these computers are unable to find the ntpserver as it is not defined anywhere. The solution to this issue could be as simple as just defining ntpserver in the name server list. But I'm not sure if my understanding of this issue is correct. Comments/suggestions are welcome for future steps. 16284 Thu Aug 19 14:14:49 2021 KojiUpdateCDSTime synchornization not running 131.215.239.14 looks like Caltech's NTP server (ntp-02.caltech.edu) https://webmagellan.com/explore/caltech.edu/28415b58-837f-4b46-a134-54f4b81bee53 I can't say it is correct or not as I did not make the survey at your level. I think you need a few tests of reconfiguring and restarting the NTP clients to see if time synchronization starts. Because the local time is not regulated right now anyway, this operation is safe I think. 16285 Fri Aug 20 00:28:55 2021 AnchalUpdateCDSTime synchornization not running I added ntpserver as a known host name for address 192.168.113.201 (fb1's address where ntp server is running) in the martian host list in the following files in Chiara: /var/lib/bind/martian.hosts /var/lib/bind/rev.113.168.192.in-addr.arpa Note: a host name called ntp was already defined at 192.168.113.11 but I don't know what computer this is. Then, I restarted the DNS on chiara by doing: sudo service bind9 restart Then I logged into c1lsc and c1ioo and ran following: controls@c1ioo:~ 0$ sudo systemctl restart systemd-timesyncd.service
controls@c1ioo:~ 0$sudo systemctl status systemd-timesyncd.service -l ● systemd-timesyncd.service - Network Time Synchronization Loaded: loaded (/lib/systemd/system/systemd-timesyncd.service; enabled) Active: active (running) since Fri 2021-08-20 07:24:03 UTC; 53s ago Docs: man:systemd-timesyncd.service(8) Main PID: 23965 (systemd-timesyn) Status: "Idle." CGroup: /system.slice/systemd-timesyncd.service └─23965 /lib/systemd/systemd-timesyncd Aug 20 07:24:03 c1ioo systemd[1]: Starting Network Time Synchronization... Aug 20 07:24:03 c1ioo systemd[1]: Started Network Time Synchronization. Aug 20 07:24:03 c1ioo systemd-timesyncd[23965]: Using NTP server 192.168.113.201:123 (ntpserver). Aug 20 07:24:35 c1ioo systemd-timesyncd[23965]: Using NTP server 192.168.113.201:123 (ntpserver). controls@c1ioo:~ 0$ timedatectl
Local time: Fri 2021-08-20 07:25:28 UTC
Universal time: Fri 2021-08-20 07:25:28 UTC
RTC time: Fri 2021-08-20 07:25:31
Time zone: Etc/UTC (UTC, +0000)
NTP enabled: yes
NTP synchronized: no
RTC in local TZ: no
DST active: n/a
The same output is shown in c1lsc too. The NTP synchronized flag in output of timedatectl command did not change to yes and the RTC is still 3 seconds ahead of the local clock.
Then I went to c1sus to see what was the status output before rstarting the timesyncd service. I got folloing output:
controls@c1sus:~ 0\$ sudo systemctl status systemd-timesyncd.service -l
● systemd-timesyncd.service - Network Time Synchronization
Active: active (running) since Tue 2021-08-17 04:38:03 UTC; 3 days ago
Docs: man:systemd-timesyncd.service(8)
Main PID: 243 (systemd-timesyn)
Status: "Idle."
CGroup: /system.slice/systemd-timesyncd.service
└─243 /lib/systemd/systemd-timesyncd
Aug 20 02:02:18 c1sus systemd-timesyncd[243]: Using NTP server 192.168.113.201:123 (ntpserver).
Aug 20 02:36:27 c1sus systemd-timesyncd[243]: Using NTP server 192.168.113.201:123 (ntpserver).
Aug 20 03:10:35 c1sus systemd-timesyncd[243]: Using NTP server 192.168.113.201:123 (ntpserver).
Aug 20 03:44:43 c1sus systemd-timesyncd[243]: Using NTP server 192.168.113.201:123 (ntpserver).
Aug 20 04:18:51 c1sus systemd-timesyncd[243]: Using NTP server 192.168.113.201:123 (ntpserver).
Aug 20 04:53:00 c1sus systemd-timesyncd[243]: Using NTP server 192.168.113.201:123 (ntpserver).
Aug 20 05:27:08 c1sus systemd-timesyncd[243]: Using NTP server 192.168.113.201:123 (ntpserver).
Aug 20 06:01:16 c1sus systemd-timesyncd[243]: Using NTP server 192.168.113.201:123 (ntpserver).
Aug 20 06:35:24 c1sus systemd-timesyncd[243]: Using NTP server 192.168.113.201:123 (ntpserver).
Aug 20 07:09:33 c1sus systemd-timesyncd[243]: Using NTP server 192.168.113.201:123 (ntpserver).
This actually shows that the service was able to find ntpserver correctly at 192.168.113.201 even before I changed the name server file in chiara. So I'm retracting the changes made to name server. They are probably not required.
The configuration files for timesynd.conf are read only even with sudo. I tried changing permissions but that did not work either. Maybe these files are not correctly configured. The man page of timesyncd says to use field 'NTP' to give the ntp servers. Our files are using field 'Servers'. But since we are not getting any error message, I don't think this is the issue here.
I'll look more into this problem.
16286 Fri Aug 20 06:24:18 2021 AnchalUpdateCDSTime synchornization not running
I read on some stack exchange that 'NTP synchornized' indicator turns 'yes' in the output of command timedatectl only when RTC clock has been adjusted at some point. I also read that timesyncd does not do the change if the time difference is too much, roughly more than 3 seconds.
So I logged into all FE machines and ran sudo hwclock -w to synchronize them all to the system clocks and then waited if the timesyncd does any correction on RTC. It did not. A few hours later, I found the RTC clocks drifitng again from the system clocks. So even if the timesynd service is running as it should, it si not performing time correction for whatever reason.
Maybe we should try to use some other service?
Quote: The NTP synchronized flag in output of timedatectl command did not change to yes and the RTC is still 3 seconds ahead of the local clock.
16288 Mon Aug 23 11:51:26 2021 KojiUpdateGeneralCampus Wide Power Glitch Reported: Monday, 8/23/21 at 9:30am
Campus Wide Power Glitch Reported: Monday, 8/23/21 at 9:30am (more like 9:34am according to nodus log)
nodus: rebooted. ELOG/apache/svn is running. (looks like Anchal worked on it)
chiara: survived the glitch thanks to UPS
fb1: not responding -> @1pm open to login / seemed rebooted only at 9:34am (network path recovered???)
megatron: not responding
optimus: no route to host
c1aux: ping ok, ssh not responding -> needed to use telnet (vme / vxworks)
c1auxex: ssh ok
c1auxey: ping ok, ssh not respoding -> needed to use telnet (vme / vxworks)
c1psl: ping NG, power cycled the switch on 1X2 -> ssh OK now
c1iscaux: ping NG -> rebooted the machine -> ssh recovered
c1iscaux2: does not exist any more
c1susaux: ping NG -> responds after 1X2 switch reboot
c1pem1: telnet ok (vme / vxworks)
c1iool0: does not exist any more
c1vac1: ethernet service restarted locally -> responding
ottavia: doesnot exist?
c1teststand: ping ok, ssh not respoding
3:20PM we started restarting the RTS
16289 Mon Aug 23 15:25:59 2021 Ian MacMillanUpdateCDSSUS simPlant model
I am adding a State-space block to the SimPlant cds model using the example Chris gave. I made a new folder in controls called SimPlantStateSpace. wI used the code below to make a state-space LTI model with a 1D pendulum then I converted it to a discrete system using c2d matlab function. Then I used these in the rtss.m file to create the state space code I need in the SimPlantStateSpace_1D_model.h file.
%sys_model.m
Q = 1000; phi = 1/Q; g = 9.806; m = 0.24; % mass of pendulum l = 0.248; %length of pendulum w_0 = sqrt(g/l);
f=16000 %this is the frequency of the channel that will be used
A = [0 1; -w_0^2*(1+1/Q*1i) -w_0/Q] B = [0; 1/m]; C = [1 0]; D = [0]; sys_dc = ss(A,B,C,D)
sys=c2d(sys_dc, 1/f)
This code outputs the discrete state space that is added to the header file attached.
Attachment 1: SimPlantStateSpace.zip
ELOG V3.1.3-
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{}
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## Linear Algebra: A Modern Introduction
$k=-2$ or $k=3$
We know that two vectors are orthogonal if their dot product is $0$. Hence: $u\cdot v=1\cdot k^2-1\cdot k+2\cdot(-3)=0\\k^2-k-6=0\\(k+2)(k-3)=0$ Thus: $k=-2$ or $k=3$
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{}
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# Equirectangular To Cubemap Photoshop
The OpenGL graphics system is a software interface to graphics hardware. png) and are essential for image-based lighting (IBL), professional rendering, architectural visualisation. Use Photoshop's "Offset" filter to shift your 360 photo around Changing projections for editing. tony wang skybox_cubemap. The update was released at Adobe MAX 2017 alongside new versions of Adobe's other Creative Cloud software, including Premiere Pro CC and Photoshop CC. The name of the command is the name of the option and the argument is the. Easily switch between editing formats and exporting to a variety of formats including: Fisheye, Cube-Map Facebook 3: 2, Cube-Map Pano 2VR 3: 2, Cube-Map GearVR 6: 1, Equirectangular 16: 9, Cube- Map 4: 3, Sphere Map, and Equirectangular 2: 1. The actions provide tools for converting images from several common panoramic formats such as angular fisheye, equirectangular, and cube map panoramas, and general utilities for VR and fulldome production. The Send Media to Photoshop LUA script will send all of the currently selected file loader or saver node files to Adobe Photoshop. 在各种编辑格式之间轻松切换，并导出为各种格式，包括：Fisheye、Cube-Map Facebook 3:2、Cube-Map Pano 2VR 3:2、Cube-Map GearVR 6:1、Equirectangular 16:9、Cube-Map 4:3、Sphere Map 和 Equirectangular 2:1。 VR 旋转球体. In short, a 360 camera captures two images or video files from dual lenses with a 180-degree field of view and either automatically stitches them together in-camera, or offers free companion software with which you can stitch the files together – often with one click. cloud/ajkWL. And after weeks of searching for panorama tools and Photoshop plugins I finally stumbled across a quite effortless solution. 53A and 53B may be used to reduce the width of some of the tiles of the tiling scheme 5300 of FIG. Hi, In Photoshop CC 20. but if you got ps-tutorials it might be interesting as well. Traditionally, most of us, panoramic photo app developers, preferred equirectangular projections. Some of the most popular capture solutions are listed below: 360 Panorama Capture for Unity : A free, easy-to-use 360° capture plugin for Unity. Panorama to cubemap - Grid to Equirectangular - Duration: 4:50. Couldn't find anything very interesting, so i made my way here to ask you, is there a way to do so ? Thanks very much ! Oliv. Find this utility tool & more on the Unity Asset Store. PAN, TILT, ROLL and VIEW CONTROL lets you adjust the virtual camera orientation. 在各种编辑格式之间轻松切换，并导出为各种格式，包括：Fisheye、Cube-Map FB 3:2、Cube-Map Pano 2VR 3:2、Cube-Map GearVR 6:1、Equirectangular 16:9、Cube-Map 4:3、Sphere Map 和 Equirectangular 2:1。 VR 旋转球体. Разрабатывайте титры для кинофильмов, заставки и переходы. There is no feature provided in PTGui for converting the 6 cube faces back to equirectangular. 0) You are free to: Share — copy and redistribute the material in any medium or format Adapt — remix, transform, and build upon the material for any purpose, even commercially. Maar we zullen het hebben over de meest efficiënte, Equirectangular virtual tour. How to add missing equirectangular photosphere metadata in Photoshop (CC 2015/2017/2018) Download the XMP template with required projection meta tag and unzip it. NVIDIA Texture Tools Exporter The NVIDIA Texture Tools Exporter allows users to create highly compressed texture files - that stay small both on disk and in memory - directly from image sources using NVIDIA’s CUDA-accelerated Texture Tools 3. png --ptofile=cube. Luego, hacemos el trabajo pesado (es decir, la reasignación con interpolación) usando OpenCV. You may feel free to use the code in any way, shape, or form and can modify it to your heart's content. png) and are essential for image-based lighting (IBL), professional rendering, architectural visualisation or for (mobile) games. For every pixel of output, it takes one sample from the input cube map. After Effects Latest Version 2018 is a new version of the special effects giant recently released by Adobe. Easypano provides users as many features as possible in Panoweaver. Windows x86, x64 Idiomas: inglés, alemán, griego, francés, español, italiano, portugués-portugués, portugués-brasileño, turco Tamaño del archivo: 5. A Skybox is basically a 360-degree background displayed around everything else in your Unity scene. A commercial Photoshop plugin for doing this is Flaming Pear's Flexify; a commercial panostitcher you can use is PTGui. Hi everyone, I've been playing with HDRI panoramas for a while ; the main goal is to use them for 3D rendering. Copy link to clipboard. Equirectangular to Cubic with point to point mapping? Ask Question Asked 4 years, 6 months ago. Could I paint directly on a sphere in Zbrush of Mudbox with the uv’s mapped in an equirectangular fashion, and then take this image and have it cut up by some magic tool into a cube map? No. The action we’ll need to use is called Revit Horizontal Strip Stereo to Cube Map Stereo. tony wang skybox_cubemap. • PanoMap - converts between latlong/equirectangular and various cubemap image formats, and also allows spherical rotation. com)Introduction. The quality isn't perfect, but unless you're making some sort of chrome-like material, the cubemap would most likely be blurred or distorted anyway, so it shouldn't matter much. Best would equirectangular, some moving clouds, bonus points if it includes a full sunrise to sunset. You can do this by using a scanner or just taking a good photograph of each. IMAGE-BASED LIGHTING 11. How to Create 360 Screenshots in Unreal Engine. Convert 2:1 equirectangular panorama to cube map (8). Panorama to cubemap - Grid to Equirectangular - Duration: 4:50. When you use a skybox, the most common method to create a 360-degree view is by using the cube map, and mapping it on a 3D cube. On my Mac I use the free app Hugin and some of the commandline tools included with that and the Panotools-Scripts commandline utilities. I warn you, this is a long procedure but I added some files at the. Fred's ImageMagick Scripts - cube2sphericalpano - transforms 6 cube face images into a spherical panorama image. north-korean-photoshop-fail/ Relighting is important! How to render an object inserted into an image? Traditional graphics way format (mirror ball, equirectangular, cube map, etc) - E. • PanoMap - converts between latlong/equirectangular and various cubemap image formats, and also allows spherical rotation. In CubeTheSphere open your Equirectangular. Equirectangular images are the most commonly used format for Game Development, You Tube, and Facebook. About Spacescape. I own the plugin and I can help convert your maps to cubemap format if the SE converter doesn't work, but it may take some time, especially when it comes to matching the. I actually build a physical room and place a SceneCaptureCube actor and then right click the CubeRenderTarget to create a static cubemap. First, make sure that your. Freelancer is the ultimate freelance jobs website with millions of freelance jobs and millions of professional freelancers ready to bid on your projects. Equirectangular images have a very large amount of data redundancy near the poles because they are stretched in the 'latitude' direction. Try this application : Camera360. Hey guys/girls, I've recently made a stereographic cube map in VRay, rendered/saved out as a jpeg and want to use my galaxy S6 and gear VR to view it, but every time I open it it is incredibly distorted - as if it's trying to load it as an equirectangular projection. Exif Fixer reads the metadata from JPEG panoramas and inserts the parameters needed for automatic detection and interactive playback in Facebook and Google. 7、VR转换器，在各种编辑格式之间轻松切换，并导出为各种格式包括：Fisheye、Cube-Map Facebook 3:2、Cube-Map Pano 2VR 3:2、Cube-Map GearVR 6:1、Equirectangular 16:9、Cube-Map 4:3、Sphere Map和Equirectangular 2:1。 8、VR旋转球体，轻松调整和旋转您的360素材，从而校准水平线、对齐视角等。. After Effects Latest Version 2018 is a new version of the special effects giant recently released by Adobe. The issue is, 3D renderers have specific "needs", and some of them want latitude/longitude (equirectangular), while others only want cubemaps or angular maps. wood abstract background wall pattern art paper texture nature wood texture marble sky black-and-white design night city texture paper vintage architecture concrete flowers paper street business beach creative metal texture technology blur grass space Life Of Pix. Показывать ленту постеров Скрыть постеры из разделов. Python fisheye to equirectangular. pto $nona -o cube_prefix cube. Recentelijk zijn de virtuele tours enorm populair geworden in verschillende bedrijven vanwege het hoge verlovingspotentieel met prospects. Equirectangular format. You can upload a spherical map (. , [16]), which can be compressed and decompressed. Click on the above cubemap image to open the full resolution version in a new tab and download it to your computer. SkyBox Creator supports more formats: Fisheye (FullDome), Cube-Map Facebook 3:2, Cube-Map Pano 2VR 3:2, Cube-Map GearVR 6:1, Equirectangular 16:9. Some options can be changed during the operation of the filter using a command. north-korean-photoshop-fail/ Relighting is important! How to render an object inserted into an image? Traditional graphics way format (mirror ball, equirectangular, cube map, etc) - E. Adobe After Effects CC 2019 Free Download Full Version, is a powerful software that can design your videos, logos, monograms etc plus best effects to edit a video. Разрабатывайте титры для кинофильмов, заставки и переходы. The Domemaster Photoshop Actions Pack is a collection of custom Adobe Photoshop actions – written by my friend Andrew Hazelden – designed to speed up the fulldome content creation workflow. The actions provide tools for converting images from several common panoramic formats such as angular fisheye, equirectangular, and cube map panoramas, and general utilities for fulldome production. It is also called the non-projection, or plate carre, since the horizontal coordinate is simply longitude, and the vertical coordinate is simply latitude, with no transformation or scaling applied In this tutorial you will see how to make a perfect unwrap of a. With the capability to I/O camera data from other packages (C4D, Unity, Maya, Nuke etc) Canvas 360 works with your current setup. Импортируйте свои работы из Photoshop, Illustrator и Audition. A 360° image built from stitching the set from step 1 together - usually a spherical, cyndrilical, or equirectangular. That brings up the effects panel, and you want to change the output of the effect from Equirectangular 2:1 to Cube-Map 4:3. A simple example render with the EXR as the environment map. The equirectangular projection (also called the equidistant cylindrical projection, geographic projection, or la carte parallélogrammatique projection, and which includes the special case of the plate carrée projection or geographic projection) is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. See ffmpeg -filters to view which filters have timeline support. pto$ nona -o cube_prefix cube. 在各种编辑格式之间轻松切换，并导出为各种格式，包括：Fisheye、Cube-Map FB 3:2、Cube-Map Pano 2VR 3:2、Cube-Map GearVR 6:1、Equirectangular 16:9、Cube-Map 4:3、Sphere Map 和 Equirectangular 2:1。 VR 旋转球体. Импортируйте свои работы из Photoshop, Illustrator и Audition. Hello, A client wants an existing cleared lot photographed to be used in renderings for an upcoming residential project we're designing for the site. This application for android. The issue is, 3D renderers have specific "needs", and some of them want latitude/longitude (equirectangular), while others only want cubemaps or angular maps. first of all check, your system is that 32-bit or 64-bit then start downloading Adobe After Effects CC 2018 Free Download Full Version. By looking at the scene, you can view the entire panorama. VR Rotate Sphere. tif$" <<<$1 if [ $? != 0 ]; then echo >&2 A. Fred's ImageMagick Scripts - cube2sphericalpano - transforms 6 cube face images into a spherical panorama image. 1 month ago ⋅ Sergej Majboroda. 10GB of space is free while 100GB will run you$10 a month. Esempi 2048x1001 per 6 immagini da 512x512, 4096x2002 per 6 immagini da 1024x1024. Go to the top menu bar and click on “Composition. I warn you, this is a long procedure but I added some files at the. tif) and PNG-files (. These options are marked 'T' on the output of ffmpeg -h filter=. Still though, the question remains. Easily switch between editing formats and export to a variety of formats including Fisheye, Cube-Map Facebook 3:2, Cube-Map Pano 2VR 3:2, Cube-Map GearVR 6:1, Equirectangular 16:9, Cube-Map 4:3, Sphere Map, and Equirectangular 2:1. Open the photo you wish to make into a panorama in Photoshop. A projection is performed when a cartographer maps a spherical globe of the earth onto a flat piece of paper, for example. Once those 6 images are complete, you will likely have to edit them in an image editor like Paint Shop, Photoshop, or Gimp. the code came from various sources. An alternative is to use a cubemap and treat it as if it was a point light. (Hugin, panorama tools, cubic2erect, erect2cube, )What is needed is:1. Extract Cubemap removes equirectangular distortion from 360-degree footage and extracts six separate camera views. This is a type of projection for mapping a portion of the surface of a sphere to a flat image. What is a cubemap? A cubemap is a type of media created using 3D Computer Graphic software. 10GB of space is free while 100GB will run you 10 a month. However, you can render to a cubemap (should be possible via script even without dedicated support) and then resample that into an equirectangular map (example program). Download the latest versions or free trials today. Equirectangular panorama to cube map (Python 3). 2: Render A Stereo Cube Map - YouTube. Panorama Miami Florida. You can even get free open source code to do it in a variety of environments. 在各种编辑格式之间轻松切换，并导出为各种格式，包括：Fisheye、Cube-Map Facebook 3:2、Cube-Map Pano 2VR 3:2、Cube-Map GearVR 6:1、Equirectangular 16:9、Cube-Map 4:3、Sphere Map 和 Equirectangular 2:1。 VR 旋转球体. I need to convert a panorama in equirectangular projection to 6 cubic faces and then to spherical projection and back, however I need to keep a track of how each point is mapped in each projection. Photoshop plug-in for panoramas, polyhedra, and maps. A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. GPU requirements for immersive video effects. 3 weeks ago ⋅ Andreas Mischok. There are plenty of contributing factors,. In video games, the parts of the environment that are very far away are usually part of a skybox, and in space that's usually the stars and nebulas. The Domemaster Photoshop Actions Pack is a collection of custom Adobe Photoshop actions - written by my friend Andrew Hazelden - designed to speed up the fulldome content creation workflow. This is the part where we actually "create" the cubemap from the images we have. Save the photo as a. Projected to reach 37. I have what is apparently called an equirectangular projection of the world. The problem is the inverse transformation, that takes you back to a distorted, equirectangular panorama. Whenever you still have a problem about this video (or other topics as long as they are still related to sketchup and vray) you can chat me on my page. Blender only accepts equirectangular maps • How to convert mirror ball to equirectangular? Mirror ball -> equirectangular. Equirectangular (0°) Equirectangular (0°) c Tobias Jung. Drawing Lessons Drawing Techniques Drawing Tips Drawing Reference Art Lessons Photoshop Illustrator renders out an equirectangular. That brings up the effects panel, and you want to change the output of the effect from Equirectangular 2:1 to Cube-Map 4:3. Imagery for 360° media isn't limited to captures from the real world. The Photoshop actions provide tools for converting images from several common panoramic formats such as angular fisheye, equirectangular, and cube map panoramas, and general utilities for fulldome production. Things get even better if you wear a headmount display, since the Adobe Immersive Environment in Adobe Premiere Pro now allows you to see your video as your audience will — while editing. 在各种编辑格式之间轻松切换，并导出为各种格式，包括：Fisheye、Cube-Map Facebook 3:2、Cube-Map Pano 2VR 3:2、Cube-Map GearVR 6:1、Equirectangular 16:9、Cube-Map 4:3、Sphere Map 和 Equirectangular 2:1。 VR 旋转球体 轻松调整和旋转您的 360 素材，从而校准水平线、对齐视角等。. I've been trying to use PTGui to "re-compose" an equirectangulat view, instead of having to go thru Pano2VR but haven't found how to !. The Spherical Track has one input, which requires a 360º equirectangular image. 0 International (CC BY 4. Thousands of new, high-quality pictures added every day. Freight Station. If a user is being abusive, please also submit an abuse report for our moderation team to review. exe from the Hugin distribution. Any help would be appreciated. The equirectangular projection (also called the equidistant cylindrical projection, geographic projection, or la carte parallélogrammatique projection, and which includes the special case of the plate carrée projection or geographic projection) is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. VR Rotate Sphere Easily adjust and rotate your 360 footage to level horizon lines, align viewpoints, and more. Dirt Bike Track 01. The HDRI measures 9000 x 4500 pixels (equirectangular) and is saved as an EXR-image to be used as an environment map. A filter that converts 360° panoramas into cubemaps, also allowing you to split the cylindrical panorama into six different images. Python fisheye to equirectangular. first of all check, your system is that 32-bit or 64-bit then start downloading Adobe After Effects CC 2018 Free Download Full Version. For my previous job (Oculus, google Cardbard), i export the render from the 3D software in equirectangular projection. Dirt Bike Track 01. Sharing project between multiple computers and easy access to shared files is another feature of Adobe After Effects CC. 2 months ago ⋅ Sergej Majboroda. Hi, I'm currently working on 360 images for a Samsung Gear VR Experience. The problem is making your source content. After you import your image into a texture, make sure that you disable compression for the image in the Texture Manager. When the image comes into Photoshop it comes in at default. Perspective projection; User-defined output projection (coef_out. A Cubemap is a collection of six square textures that represent the reflections on an environment. Forum rules For new users: this forum is moderated. 7、VR转换器，在各种编辑格式之间轻松切换，并导出为各种格式包括：Fisheye、Cube-Map Facebook 3:2、Cube-Map Pano 2VR 3:2、Cube-Map GearVR 6:1、Equirectangular 16:9、Cube-Map 4:3、Sphere Map和Equirectangular 2:1。 8、VR旋转球体，轻松调整和旋转您的360素材，从而校准水平线、对齐视角等。. com offers free software downloads for Windows, Mac, iOS and Android computers and mobile devices. 0 for Linux. and select or drop your Texture image into the Cubemap Starting with a 360 x 180 equirectangular panorama to make a skybox is the same method as above, but it. The process for assembling Cubemaps in Photoshop. A ReflexXTR típusú equirectangular panoráma képeknek hiányzik az alsó negyede. But equirectangular projections has a big disadvantage. These features are pretty straight forward and should be easy to add. wood abstract background wall pattern art paper texture nature wood texture marble sky black-and-white design night city texture paper vintage architecture concrete flowers paper street business beach creative metal texture technology blur grass space Life Of Pix. The quality of the preview is very low. Viewing options: South up Flat oceans Apply Equirectangular (0°) Creator: Marinus of Tyre (about AD 100). This produces superior-quality as it's based on the cube maps. The requirements in the documentation are as follows: Well. Once those 6 images are complete, you will likely have to edit them in an image editor like Paint Shop, Photoshop, or Gimp. I just noticed that Photoshop CC 14. 2: Render A Stereo Cube Map - YouTube. Go to your computer properties to check the system type. tif) and PNG-files (. Major video and audio updates to the Fall 2017 release of Creative Cloud — available now — give you the power to do more, collaborate easily, and deliver faster with a complete, integrated toolset. ) it could come in handy. Is it even possible to view a god damn cubemap image on the vive First Look: 'Batman' Gear VR Experience Featuring OTOY's OctaneRender for Maya Lesson 15. pto nona -o cube_prefix cube. 轻松调整和旋转您的360素材，从而校准水平线、对齐视角等。. This allows you to view and playback VR 360 video. Hi everyone, I've been playing with HDRI panoramas for a while ; the main goal is to use them for 3D rendering. bmp image and then select the individual cube face image size to be 2048px x 2048px under the Options toolbar. The only problem I have left is that the texture seems to be mirrored on the top and bottom of the inside of the sphere instead of wrapping the one texture all the way around:. png) and are essential for image-based lighting (IBL), professional rendering, architectural visualisation. a cube or sphere) around the camera, and setting up texture coordinates on the mesh such that the image gets wrapped around the mesh in the same mapping used to generate the original image. 13 Jun, 2013 # Re: 360 panorama in 3ds Max tutorial Nice demo, but the problem for me is Quicktime, and the time lost for building a nice scene in 360 using vray and fix all rendering issues. 360 Degree Space Nebula Panorama, Equirectangular Projection. (Hugin, panorama tools, cubic2erect, erect2cube, )What is needed is:1. These panoramic skies can be used as a background, source of lighting and reflection for your 3D scene. The quality isn't perfect, but unless you're making some sort of chrome-like material, the cubemap would most likely be blurred or distorted anyway, so it. — Supports fisheye, spherical (equirectangular 2:1), spherical (equirectangular 16:9), cube map, cube map Facebook, cube map Pano2VR, cube map GearVR, sphere Adobe After Effects CS3, CS4, CS5, CS6 — If you need to convert 360 video to fisheye so it can be watched in a dome, then After Effects can do the trick just using built-in effects. 1 month ago ⋅ Sergej Majboroda. Right click on the camera and go to the camera settings tab in your properties panel. - Another feature I have been trying to replicate in BGE is generation of cubemap environment maps in real-time. Cube mapping is the process of projecting images onto the six sides of a cube, giving the overall scene a feeling of depth (hence the 3D). 53A to obtain the reduced horizontal. VRWorks 360 Video SDK version 2. Upload Upload a panoramic image: The image should be formatted with the equirectangular projection. Is posible add the enhancement from transform Equirectangular to Cubemap? No. Still images were opened in Photoshop, offset losslessly to re-center to match, and saved with a text label applied (labels were sized to match the original photo resolution and warped into place using Flexify FB3x2 cube map -> equirectangular). 轻松调整和旋转您的 360 素材，从而校准水平线、对齐视角等。 VR 球体到平面. The equirectangular projection (also called the equidistant cylindrical projection, geographic projection, or la carte parallélogrammatique projection, and which includes the special case of the plate carrée projection or geographic projection) is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. VR Rotate Sphere Easily adjust and rotate your 360 footage to level horizon lines, align viewpoints, and more. Dirt Bike Track 01. Equirectangular images are the most commonly used format for Game Development, You Tube, and Facebook. SkyBox Creator supports more formats: Fisheye (FullDome), Cube-Map Facebook 3:2, Cube-Map Pano 2VR 3:2, Cube-Map GearVR 6:1, Equirectangular 16:9. First, make sure that your. Se que podría exportar cara x cara, y luego ensamblar, pero quiero saber si en PTGui u otro software se puede hacer de forma automatica. See the complete profile on LinkedIn and discover Ryan’s. A GPU -based, high-resolution, realtime compositing system that is a joy to experiment and explore ideas with. I am quite new to Blender rendering (my knowledge is limited to Cycles preview renders using simple materials) and I would like to replicate a handy rendering and baking trick I previously achieved in Max. psd), TIFF-files (. Whether you're working on a single gigapixel panorama or a virtual tour with thousands of scenes, Pano2VR can help you create an immersive experience for any modern browser. 360 VR Image : CubeMap vs Equirectangular Hi, I'm currently working on 360 images for a Samsung Gear VR Experience. tif image file must be selected. The six camera views are positioned in a cube formation. Well in this post I’m going to give you a step by step guide to how you can do just that. VR Rotate Sphere Easily adjust and rotate your 360 footage to level horizon lines, align viewpoints, and more. A cube map has to be created with some angular distortion in order to display a seamless panoramic image. In this section, you will get the Adobe After Effects CC 2018 for free for Windows and Mac. Dirt Bike Track 01. Although many cameras and tutorials refer to this as an "HDR" image, this type of image can be better described as a tonemapped image. edit: Or use this one I wrote and see why people don't use equirectangular in real time (nasty seam). A panorama viewer works by drawing a mesh of some shape (e. Can vary in number based upon the size of the original photo. Adobe Photoshop CC 2015; Domemaster Photoshop Actions Pack 2. photoshop(ps) cc 2018绿色破解版下载+破解补丁. 2 months ago ⋅ Sergej Majboroda. In this guide we will look at converting the Revit cube map render output to an equirectangular panorama. How do I do it in photoshop (ever image is 4096x4096) Correct answer by Semaphoric-PLXibM. Easily switch between editing formats and export to a variety of formats including:Fisheye, Cube-Map Facebook 3:2, Cube-Map Pano 2VR 3:2, Cube-Map GearVR 6:1, Equirectangular 16:9, Cube-Map 4:3, Sphere Map, and Equirectangular 2:1. pto $nona -o cube_prefix cube. Go to your computer properties to check the system type. first of all check, your system is that 32-bit or 64-bit then start downloading Adobe After Effects CC 2018 Free Download Full Version. The actions provide tools for converting images from several common panoramic formats such as angular fisheye, equirectangular, and cube map panoramas, and general utilities for VR and fulldome production. Of course if these are used as the cubemap source then the interactive player needs to be aware of the mapping. Re: 6 images (cube map) into equirectangular? Post by stuka » 2015-09-01T10:46:59+01:00 Yeah, you have some nice scripts there but none could be used in some way to turn 6 cubemap images into equirectangular sadly. VRWorks 360 Video SDK version 2. 2 Jelly Bean and above. For every pixel of output, it takes one sample from the input cube map. Clone via HTTPS Clone with Git or checkout with SVN using the repository’s web address. so read the article to download it for your PC. These are mostly useful as high dynamic range equirectangular panorama skies (the internet has plenty if you search for them), which replace Cubemaps in Godot 2. js, allowing very. The Spherical Track has one input, which requires a 360º equirectangular image. Используйте инструмент «Групповые проекты» для взаимодействия с редакторами независимо от их местонахождения. Note: If you purchased a new SkyBox Studio v1 license after Nov 8, the upgrade to v2. Easily switch between editing formats and export to a variety of formats including:Fisheye, Cube-Map Facebook 3:2, Cube-Map Pano 2VR 3:2, Cube-Map GearVR 6:1, Equirectangular 16:9, Cube-Map 4:3, Sphere Map, and Equirectangular 2:1. Our idea is to support the realization of CG artist’s projects. 4 Render the Scene as Illuminated by the IBL Environment 11. I have what is apparently called an equirectangular projection of the world. How a 360 degree camera works. 从 Photoshop、Illustrator 和 Audition 中轻松导入作品。 Adobe After Effects CC 2018 360/VR 编辑功能展示 无论您是要制作电影、视频游戏、培训视频还是模拟视频，360/VR 视频都能让您的观看者沉浸其中。. Hover over to select the correct projection (Equirectangular, Cubemap or Half Equirectangular) and layout (Mono, Side-by-Side or. See ffmpeg -filters to view which filters have timeline support. To convert an equirectangular image into a cubemap we need to render a (unit) cube and project the equirectangular map on all of the cube's faces from the inside and take 6 images of each of the cube's sides as a cubemap face. Thanks to the super high resolutions of up to 15,000 pixels wide, no separate backplates are needed. 360 VR Image : CubeMap vs Equirectangular Hi, I'm currently working on 360 images for a Samsung Gear VR Experience. I used a Photoshop plug in to make the "box" mode into the "globe" appearance. Some options can be changed during the operation of the filter using a command. ⭐ HOW TO REMOVE TRIPOD FROM 360 VIDEO NADIR WITHOUT PHOTOSHOP Don't have access to Photoshop? No problem! Today I will share with you how to remove the tripod from your 360 video nadir using a free open source photo editor called GIMP. D'abord: sauf si vous devez vraiment convertir les images vous-même (c. For example: increasing your "V" on a sphere might move you along a longitude line (north or south), while increasing your "U" might move you along a. Save the photo as a. 0 for Linux. This tutorial will explain how you can convert HDRIs (High Dynamic Range Images) to skyboxes for use in Roblox using Blender, however any other 3d software is capable of doing this. Alternative Methods. Adobe After Effects - ведущее приложение для создания анимации и творческих композиций, которое поможет воплотить самые смелые идеи. Hitfilm doesn't have anything inbuilt to deal with 360 degree video yet. Eventually, we might get support from Roblox with actual HDRIs, especially with the Future is Bright update, but for now this is the way to go. As well as spherical/equirectangular 360 panoramas, Exif fixer also supports cylinder panoramas (with custom horizons) and partial panoramas (less than 360 degrees). Panorama Miami Florida. Ez a terület, amikor gömbre kivetítjük, vagy transzformáljuk ”cubemap/SkyBox” képpé egy kis, fekete kör kerületű felületként jelentkezik. Before we get to installing and using Hugin, let me go over some alternatives - you may find something more amenable to your workflow. To be able to use bigger images on these devices, a different kind of image needs to be used, either a cube map or multires image instead of an equirectangular image. We have released the API and SDK for WebAPI with all RICOH THETA cameras, and provided a development source to remotely control RICOH THETA from a digital. A set of images (preferably very high resolution) shot in a way to cover a full 360° panorama from a single point of view. Sky Backgrounds (0) If you want get FREE HDRI Sky Maps please visit my new HDRI Skies site. See ffmpeg -filters to view which filters have timeline support. Perspective projection; User-defined output projection (coef_out. Adobe After Effects CC 2018 - популярная программа для редактирования видео и динамических изображений, создания композиций, различных эффектов и анимации. Equirectangular images have a very large amount of data redundancy near the poles because they are stretched in the 'latitude' direction. Blender only accepts equirectangular maps • How to convert mirror ball to equirectangular? Mirror ball -> equirectangular. The equirectangular projection (also called the equidistant cylindrical projection, geographic projection, or la carte parallélogrammatique projection, and which includes the special case of the plate carrée projection or geographic projection) is a simple map projection attributed to Marinus of Tyre, who Ptolemy claims invented the projection about AD 100. Now we can open and edit this texture in Photoshop. I am quite new to Blender rendering (my knowledge is limited to Cycles preview renders using simple materials) and I would like to replicate a handy rendering and baking trick I previously achieved in Max. Should be perfect. Equirectangular 360-degree footage is transformed into flat rectangular images that you can easily navigate through on your screen. Adobe After Effects - ведущее приложение для создания анимации и творческих композиций, которое поможет воплотить самые смелые идеи. Four across and one above and below. 2 will have two "stages" of sorts. First of all: I’m just a beginner, and only few days ago I wasn’t unable to create a stereo pano. a check box to indicate an image is an EQR and should be processed as such. Equirectangular and cube face projections are commonly used in 360 photo / VR panorama processing. Planet bump maps Thu Dec 22, 2016 9:30 pm There's a Photoshop plugin called Flexify which, among its huge number of options, can convert from a cylindrical equirectangular projection to a horizontal cross. Skybox to reverse process and put the patch back in 360 Skybox converter to re-position the Maya/3D titles & 3D clouds in post to give clients fast results without having to go back to 3D, and using. Viewing 12 posts - 1 through 12 (of 12 total) Author Posts 2018-03-02 at 3:35 pm #2610 Mikhail LuzyaninDeveloper For […]. 0 VRWorks 360 Video SDK 2. È una piccola utility che genera uno script da alimentare a hugin, in questo modo:$ erect2cubic --erect=input. ) it could come in handy. While these are complete solutions for Skies in Unity, many of you want to know what it takes to make your own. It is also called the non-projection, or plate carre, since the horizontal coordinate is simply longitude, and the vertical coordinate is simply latitude, with no transformation or scaling applied In this tutorial you will see how to make a perfect unwrap of a. What's the easiest way to create equirectangular photos? I'd like to create my own 360 photos for viewing in the Gear, especially ones from video games and such. Compositing Render Elements in Photoshop 8. We have released the API and SDK for WebAPI with all RICOH THETA cameras, and provided a development source to remotely control RICOH THETA from a digital. Используйте инструмент «Групповые проекты» для взаимодействия с редакторами независимо от их местонахождения. 3 weeks ago ⋅ Sergej Majboroda. 2 Jan , 2018 Others Dawei Chen image credit to valuebound. 在各种编辑格式之间轻松切换，并导出为各种格式，包括：Fisheye、Cube-Map Facebook 3:2、Cube-Map Pano 2VR 3:2、Cube-Map GearVR 6:1、Equirectangular 16:9、Cube-Map 4:3、Sphere Map 和 Equirectangular 2:1。 VR 旋转球体. Pano2VR is a powerful software that converts your panoramic or 360° photos and videos into interactive virtual experiences. Adobe After Effects CC 2018 - популярная программа для редактирования видео и динамических изображений, создания композиций, различных эффектов и анимации. Luego, hacemos el trabajo pesado (es decir, la reasignación con interpolación) usando OpenCV. I can output to cubemap, and even fix the naming automatically. Home › Forums › Tutorials › Environment and Scene Setting up Tutorial Tagged: Environment and Scene Setting This topic has 11 replies, 6 voices, and was last updated 1 year, 5 months ago by Yuri Kovelenov. And most 360 cameras, equirectangular is the go to format, since stitching is easy that way. Whenever you still have a problem about this video (or other topics as long as they are still related to sketchup and vray) you can chat me on my page. pto $nona -o cube_prefix cube. The spherical cubemap-compatible images are available as Photoshop-files (. This exporter combines four texture tools in a single plugin and standalone application, including flexible and. Free 360° HDR sky maps in 2K resolution for commercial use and paid full-res up to 20K. Cube Map, Equirectangular, Fish-Eye, and Dual Paraboloid are all as easy to access as setting a single parameter. 2 days ago ⋅ Sergej Majboroda. Home › Forums › Graphics / Blender › Enviorment Map Size This topic has 30 replies, 5 voices, and was last updated 10 months, 3 weeks ago by Yuri Kovelenov. Godot supports high dynamic range textures (as. …It will appeal to after effects artists…who are more experienced and prefer tools that are more…self contained. This item includes 3 equirectangular high quality skydome/skybox images artificially created in a resolution of 12866X6433. In Photoshop well want to edit the image to be usable in other applications. javascript - Convert 2:1 equirectangular panorama to cube map - Stack Overflow. (The GL stands for Graphics Library. 360 Degree Space Nebula Panorama, Equirectangular Projection. In CubeTheSphere open your Equirectangular. The problem is making your source content. tif$" <<<$1 if [$? != 0 ]; then echo >&2 A. and select or drop your Texture image into the Cubemap Starting with a 360 x 180 equirectangular panorama to make a skybox is the same method as above, but it. I just noticed that Photoshop CC 14. I tried stitching the textures together in Photoshop and applying them to the "Sky" in the Iray render parameters, using different set ups to correlate the different between the "box" mode and the "globe" mode. It also supports cube panoramas. 48 GB 免费高速下载。 AE 2020中文破解版是Adobe官方发布的一款专业化电影视觉效果和动态图形处理工具，AE 2020中文版功能强劲，支持透过图层控制音频与影片的合成，After Effects2020软件便捷高效，非常适合电视台、动画制作公司、个人后期制作. After Effects Latest Version 2018 is a new version of the special effects giant recently released by Adobe. This is the equi2cubic100. png) and are essential for image-based lighting (IBL), professional rendering, architectural visualisation or for (mobile) games. jpg (you may have success with other file types, I didn’t try). It is a technique that was used to create the underwater scene in the DreamWorksTV 360 Music Video Mama Said by Lukas Graham - Cover by Mackenzie Sol. A GPU -based, high-resolution, realtime compositing system that is a joy to experiment and explore ideas with. 360 Degree Space Nebula Panorama, Equirectangular Projection. Can be generated (not filmed) but should look fairly realistic. Right under that you should see Type: Then a box that says Fisheye, click on that then select "Equirectangular" 6. パノラマ画像をつかって Cube Map する。 2 については vvvv の標準のノードに Cube Map 用のノードが用意されているので比較的簡単に実現できます。 したがって、今回は 1 の部分を主に作りました。. Düzenleme formatları arasında kolayca geçiş yapın ve Fisheye, Cube-Map Facebook 3:2, Cube-Map Pano 2VR 3:2, Cube-Map GearVR 6:1, Equirectangular 16:9, Cube-Map 4:3, Sphere Map ve Equirectangular 2:1 gibi çeşitli formatlarda dışa aktarın. The quality of the preview is very low. After Effects Latest Version 2018 Free Full Version. The landing page for cubemaps: creation, export, import, and usage within Unreal Engine 4. Why hasn't VR taken off yet? For all the hype and buzz there has been little traction. The equirectangular format is widely used by a couple of Panorama Viewers as for example PTViewer and SPi-V. To use this cubemap texture in Unity as a skybox: 1. Equirectangular panorama to cube map (Python 3). Before its too late lets started. Home › Forums › Graphics / Blender › Enviorment Map Size This topic has 30 replies, 5 voices, and was last updated 10 months, 3 weeks ago by Yuri Kovelenov. Panorama Tools Plugins: Photoshop, GraphicConverter and Gimp plug-ins for image correction and remapping. tif image file must be selected. How to add missing equirectangular photosphere metadata in Photoshop (CC 2015/2017/2018) Download the XMP template with required projection meta tag and unzip it. In CubeTheSphere open your Equirectangular. Always sees every Individual Cubemap as a new single-point-perspective-grid. Photography, VR Photography, How To. fatBox Software is an independent game development studio specializing in the development of entertainment products for the IOS platform. This effect is known as subsurface scattering and is very slow to render. Of course if these are used as the cubemap source then the interactive player needs to be aware of the mapping. Where Simon was going with the 6 camera views was setting up a Cube Map, which other software can convert into an equirectangular projection. Our idea is to support the realization of CG artist’s projects. The Photoshop actions provide tools for converting images from several common panoramic formats such as angular fisheye, equirectangular, and cube map panoramas, and general utilities for fulldome production. It is also lightweight: 55KB when gzipped. ویژگیهای اصلی اضافه شده در این نسخه شامل. And most 360 cameras, equirectangular is the go to format, since stitching is easy that way. Here is the thing : i searched, a lot, here and across the web, to find a way to convert with PHP an equirectangular panorama (standard dimensions of 2 for 1), in a cubemap. FB is currently working on a work-in-progress title tentatively set for early winter release. Equirectangular to cubemap photoshop. Adobe also released several work-in-progress tools as full applications: Character Animator, the beta of which came with After Effects, becomes. Traditionally, most of us, panoramic photo app developers, preferred equirectangular projections. The Spherical Track has one input, which requires a 360º equirectangular image. js, allowing very. Also compatible to many other programs that can use Photoshop plugins. Otherwise you're suck with it I'm afraid. psd), TIFF-files (. The Domemaster Photoshop Actions Pack is a collection of custom Adobe Photoshop actions - written by Andrew Hazelden - designed to speed up the fulldome content creation workflow. but it works neatly. To use this cubemap texture in Unity as a skybox: 1. In the new series, the software also uses 4-D effects. Adobe After Effects - ведущее приложение для создания анимации и творческих композиций, которое поможет воплотить самые смелые идеи. After you import your image into a texture, make sure that you disable compression for the image in the Texture Manager. El código se puede hacer más compacto, si la legibilidad no es motivo de preocupación. Dirt Bike Track 01. TOP and BOTTOM go above and below FRONT. Bend spherical panoramas into dizzying new shapes. PTGui is panoramic image stitching software for Windows and macOS. Показывать ленту постеров Скрыть постеры из разделов. Let me do my best to help :) https://www. The idea is to take several photos with different exposures, and put them together so that all parts of the image are properly exposed (not too bright or too dark) - for example the sky is usually much brighter than the rest of an image, so the. The actions provide tools for converting images from several common panoramic formats such as angular fisheye, equirectangular, and cube map panoramas, and general utilities for VR and fulldome production. 1 has added support for 32-bit per channel images in the Polar Coordinates filter. The image should have an aspect ratio of 2:1 (the width must be exactly twice the height). The actions provide tools for converting images from several common panoramic formats such as angular fisheye, equirectangular, and cube map panoramas, and general utilities for fulldome production. panoramic image projections An image projection occurs whenever a flat image is mapped onto a curved surface, or vice versa, and is particularly common in panoramic photography. Typically, if we want to use an equirectangular image in an interactive scene, we'll convert it into a cubemap first, and we'll sacrifice some image fidelity in the process. The basic elements used to render a Skybox environment map, an upper and lower dome representing 'sky' and 'ground', and a reference point, usually located grid-centre, which forms the point of view from which rendering takes place ("Camera" and "Lamp" objects visible in the Scene are not explicitly used for Skybox rendering). To find out how to render VR ready cube maps with Revit, look at my previous guide, here. The quality isn't perfect, but unless you're making some sort of chrome-like material, the cubemap would most likely be blurred or distorted anyway, so it. Make a plane in Blender that is the size of the diffuse texture you're making. jpg (you may have success with other file types, I didn’t try). The issue is, 3D renderers have specific "needs", and some of them want latitude/longitude (equirectangular), while others only want cubemaps or angular maps. See the complete profile on LinkedIn and discover Ryan’s. PTStitcher can be used stand alone or with some other (non-GUI) Tool: together with some tool that separate color channels (ImageMagick [*] ) to correct Chromatic aberration assembling 6 cube faces to an equirectangular panorama can be done by GUI but is more conveniently done by script, batch file or PanoCube Plus [*]. 07 I've been experiencing problems with 3D panorama workflow. You've got that technique pretty well under control (better than me ). Like so: If you chose to render a cubemap instead of a spherical panorama, it should look something like this: Finally, here's a view of the render. - [Narrator] We focused in these lessons on VR Comp Viewer…to setup and manage the multiple compositions…required for 360 video and after effects. Hi, In Photoshop CC 20. The HDRI measures 9000 x 4500 pixels (equirectangular) and is saved as an EXR-image to be used as an environment map. Jul 09, 2018. Re: 6 images (cube map) into equirectangular? Post by stuka » 2015-09-01T10:46:59+01:00 Yeah, you have some nice scripts there but none could be used in some way to turn 6 cubemap images into equirectangular sadly. From first cut to final credits, Adobe video tools give you everything you need to bring your stories to life on film, TV, and the web. Also compatible to many other programs that can use Photoshop plugins. pto $nona -o cube_prefix cube. 从 Photoshop、Illustrator 和 Audition 中轻松导入作品。 Adobe After Effects CC 2018 360/VR 编辑功能展示 无论您是要制作电影、视频游戏、培训视频还是模拟视频，360/VR 视频都能让您的观看者沉浸其中。. Equirectangular (0°) Equirectangular (0°) c Tobias Jung. After that,. Unity (unity3d. 360 VR Image : CubeMap vs Equirectangular Hi, I'm currently working on 360 images for a Samsung Gear VR Experience. 3 weeks ago ⋅ Sergej Majboroda. Perspective projection; User-defined output projection (coef_out. 3 weeks ago ⋅ Andreas Mischok. 2 Model the Geometry and Reflectance of the Scene 11. 07 I've been experiencing problems with 3D panorama workflow. A Skybox A special type of Material used to represent skies. so read the article to download it for your PC. 360 VR Image : CubeMap vs Equirectangular. Photoshop: Cubemap to Equirectangular; Highlighted. Equirectangular images are the most commonly used format for Game Development, You Tube, and Facebook. Remember, all we need to do is convert from a rectilinear image (or a cubemap) into an equirectangular projection. The OpenGL graphics system is a software interface to graphics hardware. We did this with a After Effects Roundtrip and the Plugin Skybox by Mettle. CubeMap Drawing to the Web Converting photos/scans of CubeMap squares to Equirectangular Creating a "VR" webpage on TerpConnect Scan/Photograph the 6 CubeMap sheets The first step is to digitize the 6 CubeMap sheets you've drawn. Create the node and connect it to the clip. Can be generated (not filmed) but should look fairly realistic. A 360° image built from stitching the set from step 1 together - usually a spherical, cyndrilical, or equirectangular. Our idea is to support the realization of CG artist’s projects. When we saw the cubemap image from within the Skybox conversion, we figured why not export out that image and use it as a guide to build handmade cubemaps in Photoshop and then swap out the cubemap within the. For mobile performance reasons I'm using the three. Uo 31 9N sB 82 8p eZ gw 5N Ro q5 XX ib TQ qb 6Z T8 cH UW ir Nm uq hL de 14 iF Al ah aa VW nz cA 6c Yi QG oJ Hg O9 ra 9u Pd Rt vF dK nz mw Jl 8s XO 5f 53 aO Xx Z8 MB. 轻松调整和旋转您的360素材，从而校准水平线、对齐视角等。. When the in and out points of the composition are adjusted,. The creators of Octane renderer put there own Samsung Gear VR application in the store and those skybox images are incredible sharp. 1 Basic IBL 11. I did it by simply rendering textures from 6 cameras and then combining them in a shader. Well, that seem a cool way to present a 360° pano, but this option doesn’t exist in Hugin , so I had to dig to find how they all have done it : some are using flexify ( a proprietary photoshop plugin ) , other are « geek »( or scientists) enough to make it the mathematical way with matlab, but I use GIMP , and I’m not a student and have. The Domemaster Photoshop Actions Pack is a collection of custom Adobe Photoshop actions - written by my friend Andrew Hazelden - designed to speed up the fulldome content creation workflow. For practical reasons they are in fact asking for cylindrical projections, so you simply resample your original 3. so read the article to download it for your PC. VRWorks 360 Video features are aimed at content creators and VR application developers and bring a new level of performance, immersiveness and responsiveness to capture, stitch and stream 360 mono or stereo in real-time or offline. Es una pequeña utilidad que genera una secuencia de commands para alimentar a hugin, de esta manera:$ erect2cubic --erect=input. Scrolling down through the “Effects & Presets” I will open the “Mettle” folder and drag the Mettle SkyBox Converter and drop it onto the 360 video clip. The cubemap is 6 times 2048×2048 so it might take a while for it to load. UPDATE 2: found the definitive tool for equirectangular to cubemap conversion, and it's called erect2cubic. Some of the most popular capture solutions are listed below: 360 Panorama Capture for Unity : A free, easy-to-use 360° capture plugin for Unity. Use Unity to build high-quality 3D and 2D games, deploy them across mobile, desktop, VR/AR, consoles or the Web, and connect with loyal and enthusiastic players and customers. Adobe After Effects CC 2018 - популярная программа для редактирования видео и динамических изображений, создания композиций, различных эффектов и анимации. Photoshop plug-in for panoramas, polyhedra, and maps. How to add missing equirectangular photosphere metadata in Photoshop (CC 2015/2017/2018) Download the XMP template with required projection meta tag and unzip it. Movie files of above types (except cube map) Output file format (png, bmp, jpg, tif, jp2, ppm, mp4, avi) Dome master (equidistant projection) Equirectangular projection. This is a simple MATLAB script that takes an equirectangular version of a scene that is provided as an image with a 2:1 width and height ratio and creates six cube faces that represents the scene. Whether you're working on a single gigapixel panorama or a virtual tour with thousands of scenes, Pano2VR can help you create an immersive experience for any modern browser. 2为max2020版本英文的，中文爱好者不要下。1。下载并安装v-ray。无需安装许可证服务器。. DCCツール 3ds Max 2017, V-Ray, Unity 2018, Babylon. GPU requirements for immersive video effects. Urban Street 04. From another post I read about using the Photoshop's "offset" filter to align the compass points of different nodes' source equirectangular images. Use MathJax to format equations. Category: Skies. 53A to obtain the reduced horizontal. The Google VR aplication you want to use says that need a cylindrical projection, but asks for a proportion of 2:1 on your image. pto $nona -o cube_prefix cube. The spherical cubemap-compatible images are available as Photoshop-files (. Ez a terület, amikor gömbre kivetítjük, vagy transzformáljuk ”cubemap/SkyBox” képpé egy kis, fekete kör kerületű felületként jelentkezik. We’ll take this 360-degree photo of the Amazon. com) is a popular tool for creating 3D (and 2D) content for Virtual Reality, video games, and simulations. Still images were opened in Photoshop, offset losslessly to re-center to match, and saved with a text label applied (labels were sized to match the original photo resolution and warped into place using Flexify FB3x2 cube map -> equirectangular). You can do this by using a scanner or just taking a good photograph of each. Equirectangular image presents a sphere, it's distorted so that meridians and parallels on the sphere are shown as straight vertical and horizontal lines. Source of hemicube render - Source of idea for explanation. PlaySys - Interactive Digital Media PlaySys is a software engineering company based in the center of Milan since 2007. 0 for Windows VRWorks 360 Video SDK 2. There's a Photoshop plugin called Flexify which, among its huge number of options, can convert from a cylindrical equirectangular projection to a horizontal cross. PTFilter photoshop plugins A fully 16bit colordepth compatible version of the Panorama Tools Photoshop plugins. The VR Converter lets you switch on-the-fly between different immersive formats so you can. It's really cool, but for most games you'll probably stick with non-HDR because you just don't need the extra data. How a 360 degree camera works. In this preview mode I cannot use photoshop tool to retouche the image, because I cannot see details of image. cloud/ajkWL. wood abstract background wall pattern art paper texture nature wood texture marble sky black-and-white design night city texture paper vintage architecture concrete flowers paper street business beach creative metal texture technology blur grass space Life Of Pix. 在各种编辑格式之间轻松切换，并导出为各种格式，包括：Fisheye、Cube-Map FB 3:2、Cube-Map Pano 2VR 3:2、Cube-Map GearVR 6:1、Equirectangular 16:9、Cube-Map 4:3、Sphere Map 和 Equirectangular 2:1。 VR 旋转球体. Godot supports high dynamic range textures (as. Easily switch between editing formats and export to a variety of formats including:Fisheye, Cube-Map Facebook 3:2, Cube-Map Pano 2VR 3:2, Cube-Map GearVR 6:1, Equirectangular 16:9, Cube-Map 4:3, Sphere Map, and Equirectangular 2:1. To find out how to render VR ready cube maps with Revit, look at my previous guide, here. In video games, the parts of the environment that are very far away are usually part of a skybox, and in space that's usually the stars and nebulas. I'm recording the images on the iPhone with the Google Photosphere app, or similar apps that create 2:1 equirectangular panoramas. How to Design VR Skyboxes. This produces superior-quality as it's based on the cube maps. Another common type of panorama is a cube map. Heightmaps and color images for curved surfaces should be equirectangular projections of a portion of a sphere. so read the article to download it for your PC. The quality isn't perfect, but unless you're making some sort of chrome-like material, the cubemap would most likely be blurred or distorted anyway, so it. Using the Edge tool in this application keeps more detail when removing. These options are marked 'T' on the output of ffmpeg -h filter=. Start Fusion and open a new comp. Use the Panorama To Cubemap from ft-lab on your next project. The action we’ll need to use is called Revit Horizontal Strip Stereo to Cube Map Stereo. The actions provide tools for converting images from several common panoramic formats such as angular fisheye, equirectangular, and cube map panoramas, and general utilities for fulldome production. 360Photos takes an equirectangular image and makes it a sphere you can look around inside of. Although many cameras and tutorials refer to this as an "HDR" image, this type of image can be better described as a tonemapped image. Photo Sphere Viewer is pure JS and based on Three. Perspective projection; User-defined output projection (coef_out. And most 360 cameras, equirectangular is the go to format, since stitching is easy that way. How a 360 degree camera works. Equirectangular 360-degree footage is transformed into flat rectangular images that you can easily navigate through on your screen. And this is exactly what the core does: the ColorMix (3) node is used to apply the color suitable for this particular metal to the cubemap. A GPU -based, high-resolution, realtime compositing system that is a joy to experiment and explore ideas with. first of all check, your system is that 32-bit or 64-bit then start downloading Adobe After Effects CC 2018 Free Download Full Version. Click on the above cubemap image to open the full resolution version in a new tab and download it to your computer. Cube mapping is the process of projecting images onto the six sides of a cube, giving the overall scene a feeling of depth (hence the 3D). A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Greenwich Park 02. Two extremely useful Adobe Photoshop additions for editing 360 photos are Flexify 2 for Panoramas (Photoshop plugin) and Domemaster Photoshop Actions Pack. This code is protected under the MIT License. A 2:1 projection normally is a spherical one that has a diferent deformation, but it is harder to achive in a video for example. Cube faces to equirectangular in PTGui ?. I've made it possible to create 3 cubemap layouts : cross inline separate The ressolution option is meant for a single piece so if You set …. The only difference I've found so far, is that I render an equirectangular, maybe an cubemap render would reduce some distortions. 在各种编辑格式之间轻松切换，并导出为各种格式，包括：Fisheye、Cube-Map Facebook 3:2、Cube-Map Pano 2VR 3:2、Cube-Map GearVR 6:1、Equirectangular 16:9、Cube-Map 4:3、Sphere Map 和 Equirectangular 2:1。 08、VR 旋转球体. The panorama-tools gimp plug-in has a separate Mercurial module. The tiling scheme 5300 is a uniform equirectangular tiling scheme, and the tiling scheme is an equirectangular tiling scheme with reduced horizontal sampling at the top and bottom. 此次与大家分享的after effects cc 2018 mac 破解版，内附ae cc 2018 激活工具与ae mac破解方法供您破解，让您可以第一时间体验adobe after effects cc 2018新功能。. Drawing Lessons Drawing Techniques Drawing Tips Drawing Reference Art Lessons Photoshop Illustrator renders out an equirectangular. Cubemaps in Photoshop CC 2015. Viewing 15 posts - 1 through 15 (of 31 total) 1 2 3 → Author Posts 2019-04-04 at 1:02 pm #13607 elkLicensee Is there a …. ویژگیهای اصلی اضافه شده در این نسخه شامل. For instance if you are uploading this to Artstation you will need to convert it into a. Godot supports high dynamic range textures (as. Easypano provides users as many features as possible in Panoweaver. 在各種編輯格式之間輕鬆切換，並匯出為各種格式，包括：Fisheye、Cube-Map Facebook 3:2、Cube-Map Pano 2VR 3:2、Cube-Map GearVR 6:1、Equirectangular 16:9、Cube-Map 4:3、Sphere Map 和 Equirectangular 2:1。 8、VR旋轉球體 輕鬆調整和旋轉您的360素材，從而校準水平線、對齊視角等。. Clone via HTTPS Clone with Git or checkout with SVN using the repository’s web address. tif) and PNG-files (. I'm thinking I'll photograph the site from a fixed central point, taking shots while rotating the 360…. 360Photos takes an equirectangular image and makes it a sphere you can look around inside of. Cube Map to Equirectangular (LatLong Map) If you did this 6 times along each axis it would be called a cube map, which means you can easily use Photoshop's vanishing point tool on walls and floors. You can even get free open source code to do it in a variety of environments. first of all check, your system is that 32-bit or 64-bit then start downloading Adobe After Effects CC 2018 Free Download Full Version. È una piccola utility che genera uno script da alimentare a hugin, in questo modo:$ erect2cubic --erect=input. Now, I use Promenadd plugin for 3DS Max and it changed my life. Create the node and connect it to the clip. The spherical cubemap-compatible images are available as Photoshop-files (. 5 Apr 2020 - HDRI Dome: loc00213 - HDR dome for visualizations and automotive renderings. The cubemap is 6 times 2048×2048 so it might take a while for it to load. On my Mac I use the free app Hugin and some of the commandline tools included with that and the Panotools-Scripts commandline utilities. 1 Like Bone-Studio (Juan Gea). StreetView requires that photos are at least 5300×2650 and in a 2:1 ratio. CGI software can generate 360° images and videos for everything from architectural walkthroughs to movie previews. Cube maps can be generated by many different software packages, while multires images are generated using Pannellum's generate. 2 Model the Geometry and Reflectance of the Scene 11. 360Photos takes an equirectangular image and makes it a sphere you can look around inside of. Please fix this as well!. These features are pretty straight forward and should be easy to add. If your settings are set to Mercury Software Only, the effects do not render and display a warning banner - This effect requires GPU acceleration. tif) and PNG-files (. Students will use their skills to create a panoramic 360º seascape. Cubemaps are generated based on camera rotation (cubemap view). i guess i found some code to do that. The image should have an aspect ratio of 2:1 (the width must be exactly twice the height). And most 360 cameras, equirectangular is the go to format, since stitching is easy that way. So far I have not succeeded doing the same. it's a single rectangular image that wraps 360 degrees horizontally and 180 degrees vertically. Easily switch between editing formats and exporting to a variety of formats including: Fisheye, Cube-Map Facebook 3: 2, Cube-Map Pano 2VR 3: 2, Cube-Map GearVR 6: 1, Equirectangular 16: 9, Cube- Map 4: 3, Sphere Map, and Equirectangular 2: 1. The action we’ll need to use is called Revit Horizontal Strip Stereo to Cube Map Stereo. We did this with a After Effects Roundtrip and the Plugin Skybox by Mettle. 2 will have two "stages" of sorts. VR Döner Küre.
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# Have I just proven $0=1$?
A long time ago I noticed that $$2+2 = 2 \times 2 = 2^2$$, which is pretty cool because it’s the 3 basic arithmetical operations. It then recently occurred to me to try to prove that $$2$$ is the only real number for which this is true. Here is what I came up with:
$$r+r = r \times r = r^r$$ or rewritten as $$r+r = r^2 = r^r$$.
Just examining the first part of the double equality:
$$r+r=r^2,$$ I divide by $$r$$ and get:
$$1+1 = r \implies r=2.$$
Looking at the second part of the double equality:
$$r^2=r^r$$
I divide by $$r^2$$ and get:
$$1=r^{r-2}$$
Next, I take the logarithm of both sides:
$$\ln(1) = \ln\left(r^{r-2}\right) \implies 0 = (r-2)\ln(r).$$
The only numbers that make this true are $$r=2$$ and $$r=1$$, since substituting in any other real number would mean that two non-zero numbers multiplied together would make $$0$$, which is clearly false. Furthermore $$r=1$$ does not satisfy the first part of the double equality so it has to be $$2$$. QED.
I was pretty proud of myself for solving this (and yes I'm sure to most of you this is no big deal but I'm not a math person). However a few hours later a serious problem occurred to me. Going back to this step:
$$0=(r-2)\ln(r).$$
What if I divide both sides by $$(r-2)\ln(r)$$, then I get:
$$\dfrac{0}{(r-2)\ln(r)} = \dfrac{(r-2)\ln(r)}{(r-2)\ln(r)} \implies 0 = 1.$$
I can't explain this away as division by zero since it's in the numerator.
Can someone tell me what I'm doing wrong?
Thank you
• Why not just define $x=0$ and divide both sides by $x$ to get $1=0$? – Michael Oct 15 '18 at 2:02
If $$0=(r-2)\ln(r)$$, then you can't divide by $$(r-2)\ln(r)$$, since it is equal to $$0$$ so you would be dividing by $$0$$.
More generally, any time you divide by an expression, that step is only valid under the assumption that the expression is not equal to $$0$$. If the expression involves a variable, this may be true for some values of the variable.
• I knew the explanation had to be something really basic that I was missing. Thank you – Martino Ciaramidaro Oct 14 '18 at 23:28
You have not proven that $$0=1$$
you have divided $$0$$ by $$0$$ and the result was $$1$$
As you know $$0/0$$ is not a real number because you can assign whatever value that you like to it.
For example you may argue that $$0/0=5$$ and cross multiply to get the correct result $$5\times 0=0$$
The same goes for any other number.
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8th HEAD Meeting, 8-11 September, 2004
Session 26 AGN/Galactic Nuclei
Poster, Friday, September 10, 2004, 9:00am-10:00pm
## [26.23] Continuum-Driven High-Velocity X-ray Winds
J.E. Everett, D.R. Ballantyne (CITA)
Reports of blueshifted X-ray absorption lines with v\rm outflow \gtrsim 0.1c have been reported by Pounds et al. (PG 1211+143, PG 0844+349), Reeves et al. (PDS 456), and Chartas et al. (APM 08279+5255, PG 1115+080). For the case of PG 1211+143, King & Pounds (2003) have proposed that the large column (NH \approx 5 \times 1023~{\rm cm-2}) of highly ionized gas is accelerated by continuum-driving, and launched from r \approx 150~R\rm Schwarzschild, where the escape velocity is of order 0.1c. We test simple models to determine whether such columns can be accelerated to the observed velocities by using Cloudy photoionization simulations and numerical integrations of a continuum-driven radial outflow. We also report on the observational signatures of such an wind. This work is supported by the Natural Sciences and Engineering Research Council of Canada.
Bulletin of the American Astronomical Society, 36 #3
© 2004. The American Astronomical Soceity.
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# How far could be an object from the Sun and still be under the influence of its gravitational field?
I'm trying to see how far can our star reaches with its gravity. I'm asking if anyone could give info as to what's our star's limit or the furthest object found in our solar system.
## 4 Answers
The Sun's gravity extends infinitely, but eventually solar objects would be unstable due to the influence of other stars. The minor planet "Sedna" has an orbit which takes it nearly 1000 AU (0.016 light years) from the sun at its furthest point (but now it is a lot closer)
It is also thought that billions of comets must orbit in the outer part of the solar system, out to 50000AU, or 0.8 light years, (or possibly further) forming the Oort Cloud. However, at such distances, they could not be directly observed. This marks the greatest distance at which orbiting solar system bodies can be found.
• And for example, is there any limit of mass and distance? For example, jupiter can't orbit at 1000AU or it will fly away, or Saturn cant orbit at 2000AU or it fly away, is there any mass-distance orbit relation? – Alberto Martínez May 29 '17 at 8:07
• @AlbertoMartínez No, the mass of a planet has no influence on its orbit. The only thing you need to create a stable orbit is the right velocity. If you want you could place jupiter 1km aboves sun surfaces if you make sure jupiter has the right velocity – RononDex May 29 '17 at 8:27
• @RononDex thanks for the info, appreciate it! :) – Alberto Martínez May 29 '17 at 10:43
• The word "infinitely" doesn't really have meaning here. Even gravitational waves propagate, so despite the simple 1/R^2 model, one can claim that the Sun's field reaches no farther than the speed of light times the age of the Sun. (yes, I know that's inaccurate since the sun didn't pop into existence) – Carl Witthoft May 30 '17 at 13:24
• Corrected, it was meant to be 0.015 ly and 50000 AU – James K May 30 '17 at 16:37
There's no straight forward answer. In the solar-system, which is well ordered, objects that are in stable orbits, and not too elliptical, have well defined spheres of influence. Planet 9, if/when it's discovered, will probably have the largest sphere of influence for known solar-system objects. Currently, Neptune has the largest.
If the stars near the sun were static relative to each other, the Sun's sphere of influence could be calculated and it would probably extend between 2 and 3 light years. But because the stars are not static, the sphere of influence is constantly changing and stars (probably) exchange outer, loosely orbiting debris fairly frequently.
The Oort cloud by this article is thought to extend to almost 2 light years, so that's one possible answer to your question. If you want to know the most distant aphelion of an object currently orbiting the sun, James Ks answer is good, but I think the outer most aphelion is a bit further than the 0.8 light years that he suggests. At least 2 light-years, possibly even 3. The problem is, an orbit that distant, such an object has a good chance of being deflected before it reaches it's perihelion, a journey that takes over 10 million years. Orbits that distant are likely not very stable. A lot depends on how close other stars get to our sun. A star that passes too close would likely throw everything in the vicinity that it passes through out of wack.
See chart and Wikipedia.
The tiny Scholz's star is thought to have passed within 1 light year of our sun about 70,000 years ago. Stars passing that close are quite rare, but, from the link above
A star is expected to pass through the Oort Cloud every 100,000 years or so. An approach as close or closer than 52,000 AU is expected to occur about every 9 million years.
This does make defining an outermost orbit somewhat difficult, as the most distant orbits take millions of years to reach their closest point from their most distant point, and they run a pretty good chance of being perturbed within a single orbit. Stars likely play Frisbee with their outer-most orbiting objects all the time. Picking an outermost stable orbit is impossible.
A curious sidebar on Scholz, is that, it may have sent a bunch of outer comets and oort cloud objects heading towards the inner solar system. We wont find out how many for another 2 million years or so. That's how long it will take any objects that were sent towards the inner solar-system to reach it.
• thanks for your long and understandable explanation, really appreciate it, so there is no real formula to caluclate it then? – Alberto Martínez May 29 '17 at 10:47
• @AlbertoMartínez In theory, an object could be determined to be orbiting the sun up to perhaps about 2 light years based on tangental velocity, but there's no guarantee it would complete a single orbit before being orbitally disturbed. That far out the gravity from nearby stars is pretty close to our sun's gravity. It's very much a grey area, so to my knowledge, no, there's no formula and no clear distance to say this is where orbits stop being stable. – userLTK May 29 '17 at 13:20
Since the effect on space-time curvature (gravity) of the Sun propagates through space at the speed of light, a observer beyond the Suns Cosmological horizon, or it's age in light years away, will never be able to feel it.
• +1 This is always important to point out! Light is so darn slow. – uhoh Dec 2 '20 at 22:45
Take a simple case where we know the mass of two solar systems (M_1 and M_2) and the distance between their centers of gravity (x). We want to find the location between them where the two forces of gravity from each system cancel out. Where an object placed on one or the other side of that point would eventually fall into one or the other the star system.
To find the point of equilibrium between 2 systems, first we need the formula for the force of gravity:
F = GmM/R^2 (this is Newtonian gravity, so it is ultimately wrong but a fair approximation anyhow).
I am using 'M_1' for the mass of the first system, 'R_1' for the distance from the center of the first system to the placed object, and 'm' for the mass of the placed object.
F_1 is the force from the first system acting on the placed object:
F_1 = GmM_1/R_1^2
We do the same thing for the second system:
F_2 = GmM_2/R_2^2
And then we set the forces equal eachother to find the point where they will cancel out:
F_1 = F_2
At this exact point an object is pulled equally by the forces from both systems and will stay motionless. We can see that several variables cancel out, the mass of the object (m) and the gravitational constant (G), and we are left with:
M_1/R_1^2 = M_2/R_2^2
Since we know the distance between the systems (x) we can make a substitution using the formula:
R_1 + R_2 = x
What we are left with (after some algebra) is the infamous quadratic equation solution, where:
a = 1 - M_1/M_2
b = - 2*x
c = x^2
Lastly, we need the quadratic equation:
R_2 = [-b +/- sqrt(b^2 - 4ac)]/[2a]
Skipping the algebra, simply plug your 3 knowns into a, b, and c and apply these to the quadratic solution.
You can then find R_1 using the formula:
R_1 = x - R_2
The concept is similar to rain basins, these are space basins. In the case of a basin in space we merely create a point and draw a line perpendicular to an imaginary line connecting the 2 systems. In a more complex set of multiple systems we find the equilibrium points between all neighboring systems and extend the perpendicular basin lines to where they first touch one another.
Sample problem:
M_1 = 1 solar masses
M_2 = 2 solar masses
x = 100 au
Solution:
R_2 = 58.5786437626905 au
• This doesn't seem to answer this question at all. Is there a chance that you have posted this under the wrong question by accident? – uhoh Nov 28 '20 at 5:37
• A more direct answer to the question is extremely complex. If we assume the universe is correctly 78 billion light years across and in fact 13.5 billion years old then there is expansion to contend with. What is meant by influence? If we mean even the slightest influence then gravity travels at the speed of light so what influence the sun has deoends on when we say the sun in fact came into existence. – Philip Moseman Dec 2 '20 at 21:53
• Nobody understands dark energy. We don't know what future expansion will look like. What the sun from the past is influencing now or will influence in the future is out of reachof the sun of right now; it will likely not reach those same objects. There is no concrete answer. Mind you, the top answer in this stack is relating the "influence" of the sun, meaning the furthest distance an object might be in orbit around it. My answer creates the rough/newtonian estimate for that distance, allowing someone to get their own idea based on locations of neighboring stars. – Philip Moseman Dec 2 '20 at 22:08
• The 2nd top voted answer also answers the question by considering the furthest an object would orbit our sun, contesting the first answer, while giving various estimates. I don't see anyone using math or physics equations to answer the question about astronomical distance, which seems a fairly vague way to answer the question. – Philip Moseman Dec 2 '20 at 22:27
• Consider 2 stars of the same mass. An object will orbit whichever star it is closer to. Now consider the likely possibility that the 2 suns have different masses. The orbital dividing line is no longer merely the halfway point between the 2 stars and it is not a simple proportion of the 2 masses. An object in orbit about one star will remain in orbit about that star until the influence of its neighbor captures that objecr. Beyond that distance the star will continue to have some influence, changing the shape of the orbit by pulling the orbiter toward itself. Where that ends has no equation. – Philip Moseman Dec 2 '20 at 22:45
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# 21.3 Kirchhoff’s rules (Page 2/8)
Page 2 / 8
Kirchhoff’s second rule requires $\text{emf}-\text{Ir}-{\text{IR}}_{1}-{\text{IR}}_{2}=0$ . Rearranged, this is $\text{emf}=\text{Ir}+{\text{IR}}_{1}+{\text{IR}}_{2}$ , which means the emf equals the sum of the $\text{IR}$ (voltage) drops in the loop.
## Applying kirchhoff’s rules
By applying Kirchhoff’s rules, we generate equations that allow us to find the unknowns in circuits. The unknowns may be currents, emfs, or resistances. Each time a rule is applied, an equation is produced. If there are as many independent equations as unknowns, then the problem can be solved. There are two decisions you must make when applying Kirchhoff’s rules. These decisions determine the signs of various quantities in the equations you obtain from applying the rules.
1. When applying Kirchhoff’s first rule, the junction rule, you must label the current in each branch and decide in what direction it is going. For example, in [link] , [link] , and [link] , currents are labeled ${I}_{1}$ , ${I}_{2}$ , ${I}_{3}$ , and $I$ , and arrows indicate their directions. There is no risk here, for if you choose the wrong direction, the current will be of the correct magnitude but negative.
2. When applying Kirchhoff’s second rule, the loop rule, you must identify a closed loop and decide in which direction to go around it, clockwise or counterclockwise. For example, in [link] the loop was traversed in the same direction as the current (clockwise). Again, there is no risk; going around the circuit in the opposite direction reverses the sign of every term in the equation, which is like multiplying both sides of the equation by $–1.$
[link] and the following points will help you get the plus or minus signs right when applying the loop rule. Note that the resistors and emfs are traversed by going from a to b. In many circuits, it will be necessary to construct more than one loop. In traversing each loop, one needs to be consistent for the sign of the change in potential. (See [link] .)
• When a resistor is traversed in the same direction as the current, the change in potential is $-\text{IR}$ . (See [link] .)
• When a resistor is traversed in the direction opposite to the current, the change in potential is $+\text{IR}$ . (See [link] .)
• When an emf is traversed from $–$ to + (the same direction it moves positive charge), the change in potential is +emf. (See [link] .)
• When an emf is traversed from + to $–$ (opposite to the direction it moves positive charge), the change in potential is $-$ emf. (See [link] .)
Calculate the work done by an 85.0-kg man who pushes a crate 4.00 m up along a ramp that makes an angle of 20.0º20.0º with the horizontal. (See [link] .) He exerts a force of 500 N on the crate parallel to the ramp and moves at a constant speed. Be certain to include the work he does on the crate an
What is thermal heat all about
why uniform circular motion is called a periodic motion?.
when a train start from A & it returns at same station A . what is its acceleration?
what is distance of A to B of the stations and what is the time taken to reach B from A
BELLO
the information provided is not enough
aliyu
Hmmmm maybe the question is logical
yusuf
where are the parameters for calculation
HENRY
there is enough information to calculate an AVERAGE acceleration
Kwok
mistake, there is enough information to calculate an average velocity
Kwok
~\
Abel
what is the unit of momentum
Abel
wha are the types of radioactivity ?
what are the types of radioactivity
Worku
what is static friction
It is the opposite of kinetic friction
Mark
static fiction is friction between two surfaces in contact an none of sliding over on another, while Kinetic friction is friction between sliding surfaces in contact.
MINDERIUM
I don't get it,if it's static then there will be no friction.
author
It means that static friction is that friction that most be overcome before a body can move
kingsley
static friction is a force that keeps an object from moving, and it's the opposite of kinetic friction.
author
It is a force a body must overcome in order for the body to move.
Eboh
If a particle accelerator explodes what happens
Eboh
why we see the edge effect in case of the field lines of capacitor?
Arnab
what is wave
what is force
Muhammed
force is something which is responsible for the object to change its position
MINDERIUM
more technically it is the product of mass of an object and Acceleration produced in it
MINDERIUM
wave is disturbance in any medium
iqra
energy is distributed in any medium through particles of medium.
iqra
If a particle accelerator explodes what happens
we have to first figure out .... wats a particle accelerator first
Teh
What is surface tension
The resistive force of surface.
iqra
Who can tutor me on simple harmonic motion
on both a string and peldulum?
Anya
spring*
Anya
Yea
yusuf
Do you have a chit-chat contact
yusuf
I dont have social media but i do have an email?
Anya
Which is
yusuf
Where are you chatting from
yusuf
I don't understand the basics of this group
Jimmy
teach him SHM init
Anya
Simple harmonic motion
yusuf
how.an.equipotential.line is two dimension and equipotential surface is three dimension ?
definition of mass of conversion
Force equals mass time acceleration. Weight is a force and it can replace force in the equation. The acceleration would be gravity, which is an acceleration. To change from weight to mass divide by gravity (9.8 m/s^2).
Marisa
how many subject is in physics
the write question should be " How many Topics are in O- Level Physics, or other branches of physics.
effiom
how many topic are in physics
Praise
Praise what level are you
yusuf
If u are doing a levels in your first year you do AS topics therefore you do 5 big topic i.e particles radiation, waves and optics, mechanics,materials, electricity. After that you do A level topics like Specific Harmonic motion circular motion astrophysics depends really
Anya
Yeah basics of physics prin8
yusuf
Heat nd Co for a level
yusuf
yh I need someone to explain something im tryna solve . I'll send the question if u down for it
a ripple tank experiment a vibrating plane is used to generate wrinkles in the water .if the distance between two successive point is 3.5cm and the wave travel a distance of 31.5cm find the frequency of the vibration
Tamdy
hallow
Boniface
Boniface
the range of objects and phenomena studied in physics is
Boniface
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# How to get the derivation for mean-field approximation?
In page 487/1266 of [probabilistic graphic model: principal and technique] book, the authors give the following Lagrange equation:
$$L_i[Q]=\sum_{\phi \in \Phi} \mathbf{E}_{\mathbf{U}_\phi \sim Q}[\ln \phi] + \mathbf{H}_Q (X_i) + \lambda (\sum_{x_i} Q(x_i) - 1)$$
And in page 42/86 of [http://web.eng.tau.ac.il/deep_learn/wp-content/uploads/2017/01/Semantic-Segmentation.pdf], the authors derived the following Lagrange equation for solving a maximum problem for KL divergence using mean-field approximation:
$$\mathbf{D}(Q\|P) = \sum_{\mathbf{x}} Q(\mathbf{x}) \log \frac{Q(\mathbf{x})}{P(\mathbf{x})}=\mathbf{E}_{U\sim Q}[E(\mathbf{U})]+\sum \mathbf{E}_{U_i \sim Q_i }[\log Q_i(U_i)]+\log Z + \lambda (\sum_{x_i} Q(x_i) - 1)$$ Then take the derivation with respect to $Q(x_i)$.
How to get the derivation with respect to $Q(x_i)$. I'm not sure if my understanding is right. Suppose we have $N$ pixels and the label is from $L$ classes. The random variable is $X=(X_1,X_2,\cdots,X_N)$, where $X_i \in {\{0,1\}}^{L}$, i.e., for each pixel, the possible label is from $0,1,2,\cdots,L-1$. The energy is defined:
$$P(\mathbf X) = \frac{1}{Z(\mathbf X)} \exp \{{-E(\mathbf X)}\}$$ $$E(\mathbf X) = \sum_i \phi_u (x_i) + \sum_i \sum_{j\neq i} \phi_p (x_i,x_j)$$.
What does it mean for $Q(\mathbf X)$? How to get the derivation for $Q(x_i)$?
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# S=I in a closed economy
In a closed economy, the national income identities imply that $$S=I$$
However, the intuition is not clear to me. The saving is the part of national disposable income that is not consumed. $$I$$ is the sum of residential and nonresidential investment. When firms invest, they generally borrow to finance that investment. However, why is it the case that this would necessarily equal aggregate savings? Individuals might save, but firms (or households) need not borrow. Does $$i$$ include idle funds as well?
## 1 Answer
This is a macroeconomic tautology, and like many others, the words often do not have their commonly expected meaning. In particular saving and investment are not directly about money but about stuff measured in money terms
Suppose you had an economy based purely on grains of wheat. The real activities are:
• Plant wheat grains, grow them and harvest them, getting more grains as a result
• Consume wheat as food
• Store unconsumed wheat for future planting or consumption
In this case, macroeconomic saving is the amount that production exceeds consumption while macroeconomic investment is either increasing stored stocks or the planting of grains for future production. (If you were wondering, grains which rot after having been purchase by consumers still count as consumption, while grains which rot while in producers' storage count as negative investment.) So in this case, both saving and investment measure the same thing, seen from different perspectives.
Clearly you could measure this particular economy purely in grains of wheat, but if you want to do so in money terms then you would need to know the price of wheat, where say a million grains are worth $$\5$$. This would then allow you to incorporate the production and consumption of other stuff into your calculations, such as barley where say a million grains are worth $$\3$$ and yields are different, or beer produced from barley and bread from wheat, or anything else. But saving representing unconsumed production, and investment representing the other non-consumption uses, would remain equal, almost by definition
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My Math Forum 1/4 circle and rectangle - Differentiation - Calculus
Pre-Calculus Pre-Calculus Math Forum
July 23rd, 2015, 11:59 PM #1
Newbie
Joined: Jun 2015
From: South Africa
Posts: 27
Thanks: 0
1/4 circle and rectangle - Differentiation - Calculus
The cross-section of an object has the shape of a quarter-circle of radius r adjoining a rectangle of width x and height r. As shown in the diagram (attached).
a) The perimeter and area of the cross-section are P and A respectively. Express each of P and A in terms of r and x, and hence show that
A=1/2Pr - r^2.
This I worked out, no trouble:
P=2πr/4 + 2r + 2x = 2πr/2 + 2r + 2x
A= πr^2/4 + xr
x= A/r - πr/4 Put this into P
P= 2r + 2A/r
Pr= 2r^2 + 2A
Pr/2= r^2 + A
A=pr/2 - r^2
b) Talking the perimeter P of the cross-section as fixed, find x in terms of r for the case when the area A of the cross-section is a maximum, and show that, for this value of x, A is a maximum and not a minimum.
I have no idea where to start..
Thanks!
Attached Images
p.png (2.7 KB, 15 views)
July 24th, 2015, 01:01 AM #2 Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 Find $\dfrac{dA}{dr}$, set it equal to 0 and solve for $r$. This will give you a critical point. To show that it is a maximum, either investigate the sign of $\dfrac{d^2A}{dr^2}$ or note that $A$ is a concave down parabola. The rest is fairly simple.
July 24th, 2015, 02:16 AM #3 Newbie Joined: Jun 2015 From: South Africa Posts: 27 Thanks: 0 So, dA/dr = pr/2 - r^2 What about P?
July 24th, 2015, 03:08 AM #4 Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 You have $A = \dfrac{Pr}{2} - r^2$. If $P$ is fixed, we can treat it like a constant. So $\dfrac{dA}{dr} = \dfrac{P}{2} - 2r$. Now, set $\dfrac{dA}{dr} = 0$ and solve for $r$. $\dfrac{P}{2} - 2r = 0$ $2r = \dfrac{P}{2}$ $r = \dfrac{P}{4}$ So we know there is a critical point of $A$ at $r = \dfrac{P}{4}$. Since the graph of $A$ is a concave down parabola, the critical point must be a maximum. To find the maximum value of $A$, we plug in $r = \dfrac{P}{4}$ into our equation for $A$. $A_{\text{max}} = \dfrac{P^2}{8} - \dfrac{P^2}{16} = \dfrac{P^2}{16}$ But you have shown another formula for $A$: $A = \dfrac{\pi r^2}{4} + xr$ At the point where $A$ is maximum (that is, $A = A_{\text{max}}$) we have. $\dfrac{\pi r^2}{4} + xr = \dfrac{P^2}{16}$ Now find $x$ in terms of $r$.
July 24th, 2015, 05:13 AM #5 Newbie Joined: Jun 2015 From: South Africa Posts: 27 Thanks: 0 Okay great! I understand everything. The part to find x in terms of r though, the answer in the book is x=1/4(r(4-π)) I almost get there, but something i'snt right. The P in that last equation, am i meant to use P=4r? I get to x=(1-πr)/4 using that. which simplifies to 1/4(4-πr) obviously. But thats not right! Last edited by Jmun; July 24th, 2015 at 05:18 AM.
July 24th, 2015, 08:29 PM #6 Math Team Joined: Nov 2014 From: Australia Posts: 689 Thanks: 244 Using $P = 4r$ is correct. Check your algebra again. I seem to get the right answer.
July 25th, 2015, 03:26 AM #7 Newbie Joined: Jun 2015 From: South Africa Posts: 27 Thanks: 0 okay great! thanks so much!
Tags 1 or 4, calculus, circle, differentiation, rectangle
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# A plate of cookies on the right has the shape of a circle.The radius(r)of this plate is 8 cm as shown onn the picture.calculate the perimeter of the plate
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# Sub-Doppler laser cooling of fermionic 40 K atoms
@article{Modugno1999SubDopplerLC,
title={Sub-Doppler laser cooling of fermionic 40 K atoms},
author={Giovanni Modugno and Craig Benko and Peter Hannaford and G. Roati and M. Inguscio},
journal={Physical Review A},
year={1999},
volume={60}
}
• Published 6 August 1999
• Physics, Materials Science
• Physical Review A
We report laser cooling of fermionic ${}^{40}\mathrm{K}$ atoms, with temperatures down to $(15\ifmmode\pm\else\textpm\fi{}5) \ensuremath{\mu}\mathrm{K},$ for an enriched sample trapped in a magneto-optical trap and additionaly cooled in optical molasses. This temperature is a factor of 10 below the Doppler-cooling limit and corresponds to an rms velocity within a factor of 2 of the lowest realizable rms velocity $(\ensuremath{\sim}{3.5v}_{\mathrm{rec}})$ in three-dimensional optical molasses…
26 Citations
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# Journal of Time Series Analysis
## Maximum entropy models for general lag patterns
### Journal Article
The maximum entropy problem for autocovariances given over a class of subsets of is solved. A more general problem when prediction coefficients and prediction error variances are given instead of covariances is considered and solved, as well. Two notions about maximum entropy in time series context are introduced and some misconceptions in the literature are discussed.
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# Re: Footer in Koma letter - \lfoot - \cfoot - \rfoot
On Tue, 24 Jun 2014 21:29:59 +0200
Jean-Marie Pacquet <j...@pacquet.net> wrote:
> Le 24/06/2014 16:20, Jean-Marie Pacquet a écrit :
> > [...] It won't be very difficult to add a center column.
> >
> If you have a 2.9x version of KomaScript (like me) the following
> modified example will do what you want. If your version is > 3.x you
> should replace the first line "\firstfoot{%" with
> "\setkomavar{firstfoot}{%".
> All this has to be inserted in your .lco parameter file.
>
>
> % Define a new letter foot
>
> \firstfoot{%
> \parbox[t]{\textwidth}{\footnotesize
> \rule{\linewidth}{2pt}
> \begin{tabular}[t]{l@{}}%
> \multicolumn{1}{@{}l@{}}{Partners:}\\
> Jim Smith\\
> Russ Mayer
> \end{tabular}%
> \hfill
> \begin{tabular}[t]{l@{}}%
> \multicolumn{1}{@{}l@{}}{Banks:}\\
> Citigroup\\
> Deutsche Bank
> \end{tabular}%
> \hfill
> \begin{tabular}[t]{l@{}}%
> \multicolumn{1}{@{}l@{}}{Manager:}\\
> Jane Fonda\\[1ex]
> \multicolumn{1}{@{}l@{}}{Court Of Jurisdiction:}\\
> Great Plains
> \end{tabular}%
> \ifkomavarempty{frombank}{}{%
> \hfill
> \begin{tabular}[t]{l@{}}%
> \multicolumn{1}{@{}l@{}}{\usekomavar*{frombank}:}\\
> \usekomavar{frombank}
> \end{tabular}%
> }%
> }%
> }
>
> Cheers
Thank you Jean-Marie,
The first example you pointed me to on page 178 works, and I was working
on tweaking it, and then I saw you had sent this email, which works
perfectly.
I feel like such an idiot. My only excuse is that I had looked on the
net and through other documentation that I didn't fine that reference
in the scrguien.pdf.
It's not unusual for me to go the hard way first.
I did look through the scrguien.pdf that you have directed me to in the
past and I've got in my LyX information file, but maybe I was already
overloaded with footnote information? I really have no excuse.
Anyway, as ever, and as previously thank you for your help and taking
the time to create the result. It's much appreciated and works
a treat.
Thank you once again.
Stay well,
Charlie
--
Registered Linux User:- 329524
***********************************************
The smallest seed of faith is better than the largest fruit of
happiness. .............................Henry David Thoreau
***********************************************
Debian GNU/Linux - just the best way to create magic
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# Documentation
### This is machine translation
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## Numerical Estimation of Pi Using Message Passing
This example shows the basics of working with spmd statements, and how they provide an interactive means of performing parallel computations. We do this by performing relatively simple computations to approximate pi.
Related Documentation:
Related Examples:
The code shown in this example can be found in this function:
function paralleldemo_quadpi_mpi
### Introduction
We intend to use the fact that
to approximate pi by approximating the integral on the left.
We intend to have the parallel pool perform the calculations in parallel, and to use the spmd keyword to mark the parallel blocks of code. We first look at the size of the parallel pool that is currently open.
p = gcp; p.NumWorkers
ans = 12
### Parallelize the Computations
We approximate pi by the numerical integral of 4/(1 + x^2) from 0 to 1.
type pctdemo_aux_quadpi.m
function y = pctdemo_aux_quadpi(x) %PCTDEMO_AUX_QUADPI Return data to approximate pi. % Helper function used to approximate pi. This is the derivative % of 4*atan(x). % Copyright 2008 The MathWorks, Inc. y = 4./(1 + x.^2);
We divide the work between the workers (labs) by having each worker calculate the integral of the function over a subinterval of [0, 1] as shown in the picture.
We define the variables a and b on all the workers, but let their values depend on labindex so that the intervals [a, b] correspond to the subintervals shown in the figure. We then verify that the intervals are correct. Note that the code in the body of the spmd statement is executed in parallel on all the workers in the parallel pool.
spmd a = (labindex - 1)/numlabs; b = labindex/numlabs; fprintf('Subinterval: [%-4g, %-4g]\n', a, b); end
Lab 1: Subinterval: [0 , 0.0833333] Lab 2: Subinterval: [0.0833333, 0.166667] Lab 3: Subinterval: [0.166667, 0.25] Lab 4: Subinterval: [0.25, 0.333333] Lab 5: Subinterval: [0.333333, 0.416667] Lab 6: Subinterval: [0.416667, 0.5 ] Lab 7: Subinterval: [0.5 , 0.583333] Lab 8: Subinterval: [0.583333, 0.666667] Lab 9: Subinterval: [0.666667, 0.75] Lab 10: Subinterval: [0.75, 0.833333] Lab 11: Subinterval: [0.833333, 0.916667] Lab 12: Subinterval: [0.916667, 1 ]
We let all the workers now use a MATLAB quadrature method to approximate each integral. They all operate on the same function, but on the different subintervals of [0,1] shown in the figure above.
spmd myIntegral = integral(@pctdemo_aux_quadpi, a, b); fprintf('Subinterval: [%-4g, %-4g] Integral: %4g\n', ... a, b, myIntegral); end
Lab 1: Subinterval: [0 , 0.0833333] Integral: 0.332565 Lab 2: Subinterval: [0.0833333, 0.166667] Integral: 0.32803 Lab 3: Subinterval: [0.166667, 0.25] Integral: 0.31932 Lab 4: Subinterval: [0.25, 0.333333] Integral: 0.307088 Lab 5: Subinterval: [0.333333, 0.416667] Integral: 0.292162 Lab 6: Subinterval: [0.416667, 0.5 ] Integral: 0.275426 Lab 7: Subinterval: [0.5 , 0.583333] Integral: 0.257707 Lab 8: Subinterval: [0.583333, 0.666667] Integral: 0.239713 Lab 9: Subinterval: [0.666667, 0.75] Integral: 0.221994 Lab 10: Subinterval: [0.75, 0.833333] Integral: 0.204949 Lab 11: Subinterval: [0.833333, 0.916667] Integral: 0.188836 Lab 12: Subinterval: [0.916667, 1 ] Integral: 0.173804
The workers have all calculated their portions of the integral of the function, and we add the results together to form the entire integral over [0, 1]. We use the gplus function to add myIntegral across all the workers and return the sum on all the workers.
spmd piApprox = gplus(myIntegral); end
### Inspect Results in the Client
Since the variable piApprox was assigned to inside an spmd statement, it is accessible on the client as a Composite. Composite objects resemble cell arrays with one element for each worker. Indexing into a Composite brings back the corresponding value from the worker to the client.
approx1 = piApprox{1}; % 1st element holds value on worker 1. fprintf('pi : %.18f\n', pi); fprintf('Approximation: %.18f\n', approx1); fprintf('Error : %g\n', abs(pi - approx1))
pi : 3.141592653589793100 Approximation: 3.141592653589792700 Error : 4.44089e-16
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# How can I attach a file to a question? [duplicate]
I want to attach a code file to my question, but I am not able to find a link to do it. Is attaching files possible at all?
## migrated from stackoverflow.comJul 27 '11 at 12:17
This question came from our site for professional and enthusiast programmers.
• And yes, I'm aware that he's asking about attaching a file, but I still think the FAQ makes it clear that it should be inlined. – Time Traveling Bobby Jul 27 '11 at 13:17
• People won't bother to sip through big files anyway, and those who are willing will ask you to send it to them personally via email. – Shadow Jul 27 '11 at 13:57
• @Shadow Wizard: Don't give him the wrong ideas. Nobody on SE will do that. – Time Traveling Bobby Jul 27 '11 at 14:50
• Not sure this is a duplicate of that question. Related and informative, yes, but not a dupe, IMO. – Jon Seigel Jul 28 '11 at 4:14
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# Violation of the classical Pigeonhole Principle in Neutron Optics
September 8, 2017 2:16 pm
The pigeonhole principle, first stated by Dirichlet in 1834 and therefore also reffered to as Dirichlet’s box principle or Dirichlet‘s drawer principle (“Schubfachprinzip”), states the following: “If you put three pigeons in two pigeonholes, at least two of the pigeons end up in the same hole.” In our neutron interferometric experiment 1 the role of the two holes or boxes is replaced by the two spin eigenstates and . Using our neutron optical approach for weak measurements of the (see here for details) allows for an experimental violation of the classical pigeonhole principle – demonstrating the quantum pigeonhole effect 2. Our pre-selected (initial) product state of the three neutron spins is given by , with and the post-selected (final) product state of spins is , with (see here for details of the theory). A product operator for any two spins of the three spins, has a spectral decomposition as , where , which accounts for projections onto the same state (and therefore for correlation of the two spins and ) and for projections onto different states (anti-correlation). A simple calculation reveals and . Applying the AharonovBergmannLebowitz (ABL) formula 3, which gives the probability of obtaining a particular strong measurement outcome , between a preparation and a postselection , where the outcome corresponds to a projection operator . The ABL formula can be expressed in terms of weak values as with , which gives and and finally yields . This pairwise constraint is the quantum pigeonhole effect, with the spin eigenstates correspond to two boxes in which pigeons may be placed – the pigeonhole principle states that if pigeons are placed in two boxes, then at least one box must contain multiple pigeons. However, the constraint for all pairs implies that, regardless of how many pigeons are placed in the two boxes, no two pigeons are ever in the same box!
The experiment was carried out at the neutron interferometer instrument S18 at the high-flux reactor of the Institute Laue-Langevin (ILL) in Grenoble, France, a schematical illustration is depicted above. In our setup we measure the weak value of the neutron spin in the -direction applying an interferometer. The neutron’s path is used as a pointer to measure both the real and imaginary parts of , which has already been successfully used to completely determine weak values of massive systems 4. In our setup two magnetically birefringent prisms (Polarizer) split the unpolarized beam in two sub-beams, one with the neutron spin aligned parallel to the positive z-direction and one aligned antiparallel. Even though the angular separation is just four seconds of arc (exaggerated in illustration), only the beam with spin up component fulfills Bragg’s condition at the interferometer’s first plate. The other spin-down sub-beam passes through the first plate of the interferometer unaffected and does not further contribute to the experiment since it is absorbed by a small Cadmium slab. next the coil DC-Coil 1 rotates the initial spin form to due to Larmor precession within the coils’s magnetic field pionting in -direction in inducing a spin-rotation, therby completing the spin preselection, preparating the state . At the first interferometer plate, the beam is coherently split by amplitude division. In each path a spin rotator coil (in a Helmholtz configuration) produces a weak magnetic field in the – direction, causing (weak) entanglement between the spin and path degrees of freedom of each neutron. Between the second and final interferometer plate, a sapphire phase shifter is inserted, which in combination with a Cadmium beam block mounted on a rotational stage provides full control over the neutron’s path for the pointer readout (phase shifter changees the path state in the equatorial plane of the Bloch sphere, while the beam block permits access to the path eigenstates at the poles). At the third interferometer plate, the two paths are recombined and DC coil 2, in combination with a array allows for post-selection of the spin state . Finally the neutrons are detected by .
The measured values of are used to construct the pairwise anti-correlations . In general the weak value of a product of operators is not equal to the product of weak values of the operators. However, if initial and final states are given by product states, as it the the case ion our experiment holds. In addition, according to the ABL rule, we have . Our final experimental results are plotted above. All pairs violate the classical pigeonhole principle, which is clearly illustrated by our experiment, demonstrating the quantum pigeonhole effect.
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# Contributing Datasets
### Introduction
This page explains how to set up your coding environment to create, develop, and test your dataset before you contribute to the Dataset Market.
### Prerequisites
Working knowledge of C#. You also need to install .NET 6.0 Runtime and initialize the LEAN CLI.
### Create Data Source Repository
1. Open the Lean.DataSource.SDK repository and click .
2. Start with the SDK repository instead of existing data source implementations because we periodically update the SDK repository.
3. On the Create a new repository from Lean.DataSource.SDK page, set the repository name to Lean.DataSource.<vendorNameDatasetName> (for example, Lean.DataSource.XYZAirlineTicketSales).
4. Click .
5. Clone the Lean.DataSource.<vendorNameDatasetName> repository.
6. $git clone https://github.com/username/Lean.DataSource.<vendorNameDatasetName>.git 7. If you're on a Linux terminal, in your Lean.DataSource.<vendorNameDatasetName> directory, change the access permissions of the bash script. 8. $ chmod +x ./renameDataset
9. In your Lean.DataSource.<vendorNameDatasetName> directory, run the renameDataset.sh bash script.
10. \$ renameDataset.sh
The bash script replaces some placeholder text in the Lean.DataSource.<vendorNameDatasetName> directory and renames some files according to your dataset's <vendorNameDatasetName>.
You can also see our Videos. You can also get in touch with us via Discord.
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Definitions
# Ideal gas law
The ideal gas law is the equation of state of a hypothetical ideal gas, first stated by Benoît Paul Émile Clapeyron in 1834.
The state of an amount of gas is determined by its pressure, volume, and temperature according to the equation:
$PV = nRT$
where
$P$ is the absolute pressure of the gas,
$V$ is the volume of the gas,
$n$ is the number of moles of gas,
$R$ is the universal gas constant,
$T$ is the absolute temperature.
The value of the ideal gas constant, R, is found to be as follows.
R 8.314472 J·mol−1·K−1
8.314472 m3·Pa·K−1·mol−1
8.314472 kPa·L·mol-1·K-1
0.08205746 L·atm·K−1·mol−1
62.36367 mmHg·K−1·mol−1
10.73159 ft3·psi·°R−1·lb-mol−1
53.34 ft·lbf·°R−1·lbm−1 (for air)
The ideal gas law mathematically follows from a statistical mechanical treatment of primitive identical particles (point particles without internal structure) which do not interact, but exchange momentum (and hence kinetic energy) in elastic collisions.
Since it neglects both molecular size and intermolecular attractions, the ideal gas law is most accurate for monoatomic gases at high temperatures and low pressures. The neglect of molecular size becomes less important for larger volumes, i.e., for lower pressures. The relative importance of intermolecular attractions diminishes with increasing thermal kinetic energy i.e., with increasing temperatures. More sophisticated equations of state, such as the van der Waals equation, allow deviations from ideality caused by molecular size and intermolecular forces to be taken into account.
## Alternative forms
As the amount of substance could be given in mass instead of moles, sometimes an alternative form of the ideal gas law is useful. The number of moles ($n,$) is equal to the mass ($, m$) divided by the molar mass ($, M$):
$n = \left\{frac\left\{m\right\}\left\{M\right\}\right\}$
By replacing $, n$, we get:
$PV = frac\left\{m\right\}\left\{M\right\}RT$
from where
$P = rho frac\left\{R\right\}\left\{M\right\}T$.
This form of the ideal gas law is particularly useful because it links pressure, density $rho = m/V$, and temperature in a unique formula independent from the quantity of the considered gas. In statistical mechanics the following molecular equation is derived from first principles:
$PV = NkT .$
Here $,k$ is Boltzmann's constant, and $,N$ is the actual number of molecules, in contrast to the other formulation, which uses $,n$, the number of moles. This relation implies that $N,k = nR$, and the consistency of this result with experiment is a good check on the principles of statistical mechanics.
From here we can notice that for an average particle mass of $mu$ times the atomic mass constant $m_mathrm\left\{u\right\}$ (i.e., the mass is $mu$ u)
$N = frac\left\{m\right\}\left\{mu m_mathrm\left\{u\right\}\right\}$
and since $rho = m/V$, we find that the ideal gas law can be re-written as:
$P = frac\left\{1\right\}\left\{V\right\}frac\left\{m\right\}\left\{mu m_mathrm\left\{u\right\}\right\} kT = frac\left\{k\right\}\left\{mu m_mathrm\left\{u\right\}\right\} rho T .$
## Calculations
Process Constant Known ratio P2 V2 T2
Isobaric process
Pressure
V2/V1
P2 = P1 V2 = V1 (V2/V1) T2 = T1 (V2/V1)
"
"
T2/T1
P2 = P1 V2 = V1 (T2/T1) T2 = T1 (T2/T1)
Isochoric process
Volume
P2/P1
P2 = P1 (P2/P1) V2 = V1 T2 = T1 (P2/P1)
"
"
T2/T1
P2 = P1 (T2/T1) V2 = V1 T2 = T1 (T2/T1)
Isothermal process
Temperature
P2/P1
P2 = P1 (P2/P1) V2 = V1 / (P2/P1) T2 = T1
"
"
V2/V1
P2 = P1 / (V2/V1) V2 = V1 (V2/V1) T2 = T1
Isentropic process
Entropy
P2/P1
P2 = P1 (P2/P1) V2 = V1 (P2/P1) -1/$gamma$ T2 = T1 (P2/P1)($gamma$-1)/$gamma$
"
"
V2/V1
P2 = P1 (V2/V1) -$gamma$ V2 = V1 (V2/V1) T2 = T1 (V2/V1) 1-$gamma$
"
"
T2/T1
P2 = P1 (T2/T1) $gamma$/($gamma$-1) V2 = V1 (T2/T1) 1/(1-$gamma$) T2 = T1 (T2/T1)
a. In an isentropic process, system entropy (Q) is constant. Under these conditions, P1 V1$gamma$ = P2 V2$gamma$, where $gamma$ is defined as the heat capacity ratio, which is constant for an ideal gas.
## Derivations
### Empirical
The ideal gas law can be derived from combining two empirical gas laws: the combined gas law and Avogadro's law. The combined gas law states that
$frac \left\{pV\right\}\left\{T\right\}= C$
where C is a constant which is directly proportional to the amount of gas, n (Avogadro's law). The proportionality factor is the universal gas constant, R, i.e. $C=nR$.
Hence the ideal gas law
$pV = nRT ,$
### Theoretical
The ideal gas law can also be derived from first principles using the kinetic theory of gases, in which several simplifying assumptions are made, chief among which are that the molecules, or atoms, of the gas are point masses, possessing mass but no significant volume, and undergo only elastic collisions with each other and the sides of the container in which both linear momentum and kinetic energy are conserved.
### Derivation from the statistical mechanics
Let q = (qx, qy, qz) and p = (px, py, pz) denote the position vector and momentum vector of a particle of an ideal gas,respectively, and let F denote the net force on that particle, then
begin{align} langle mathbf{q} cdot mathbf{F} rangle &= Bigllangle q_{x} frac{dp_{x}}{dt} Bigrrangle + Bigllangle q_{y} frac{dp_{y}}{dt} Bigrrangle + Bigllangle q_{z} frac{dp_{z}}{dt} Bigrrangle &=-Bigllangle q_{x} frac{partial H}{partial q_x} Bigrrangle - Bigllangle q_{y} frac{partial H}{partial q_y} Bigrrangle - Bigllangle q_{z} frac{partial H}{partial q_z} Bigrrangle = -3k_{B} T, end{align} where the first equality is Newton's second law, and the second line uses Hamilton's equations and the equipartition theorem. Summing over a system of N particles yields
3Nk_{B} T = - biggllangle sum_{k=1}^{N} mathbf{q}_{k} cdot mathbf{F}_{k} biggrrangle.
By Newton's third law and the ideal gas assumption, the net force on the system is the force applied by the walls of their container, and this force is given by the pressure P of the gas. Hence
-biggllanglesum_{k=1}^{N} mathbf{q}_{k} cdot mathbf{F}_{k}biggrrangle = P oint_{mathrm{surface}} mathbf{q} cdot mathbf{dS},
where dS is the infinitesimal area element along the walls of the container. Since the divergence of the position vector q is
boldsymbolnabla cdot mathbf{q} = frac{partial q_{x}}{partial q_{x}} + frac{partial q_{y}}{partial q_{y}} + frac{partial q_{z}}{partial q_{z}} = 3,
the divergence theorem implies that
P oint_{mathrm{surface}} mathbf{q} cdot mathbf{dS} = P int_{mathrm{volume}} left(boldsymbolnabla cdot mathbf{q} right) dV = 3PV,
where dV is an infinitesimal volume within the container and V is the total volume of the container.
Putting these equalities together yields
3Nk_{B} T = -biggllangle sum_{k=1}^{N} mathbf{q}_{k} cdot mathbf{F}_{k} biggrrangle = 3PV,
which immediately implies the ideal gas law for N particles:
PV = Nk_{B} T = nRT,,
where n=N/NA is the number of moles of gas and R=NAkB is the gas constant.
The readers are referred to the comprehensive article Configuration integral (statistical mechanics) where an alternative statistical mechanics derivation of the ideal-gas law, using the relationship between the Helmholtz free energy and the partition function, but without using the equipartition theorem, is provided.
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# What justifies the differential in an indefinite integral?
I saw this question and found the accepted answer to be very satisfying. However I thought about indefinite integrals and the differential in them (eg. $dx$). And a central part of the answer's argument is that $\dfrac{dy}{dx}$ is no longer a ratio, but a limit of a ratio. Seeing as there is no limit form of the antiderivative, I thought "what justifies the differential in the integral then". If $y = F(x)$ and $f(x) = F'(x)$ then $$\int f(x)dx = \int \dfrac{dy}{dx} \cdot dx$$ Obviously if we treat $dx$ like a fraction then they cancel out, and we end up summing $dy$, which is what we actually do when we take an antiderivative. This is a very compelling argument, but only in the context of hyperreals, where treating infinitesimals like fractions is justified. However, seeing as calculus was developed and made rigorous by people like Euler and the Bernoulli's much before hyperreals were made rigorous (by people like Abraham Robinson), what was the original justification for the differential? (I'm looking for an answer that doesn't delve into hyperreals or non-standard analysis)
Note:
Statements like "variable of integration" and "with respect to $x$" are irrelevant. Let me put that into context. When we do u-substitution, a central part is treating differentials like fractions. That is, Let $u = y(x)$. Then $\dfrac{du}{dx} = y'$ Then we take $du = y'dx$ or $dx = \dfrac{du}{y'}$. You see where I'm getting at.
Further Note:
I know that there have been similar questions asked before, but I feel like their context wasn't exactly identical to mine, and furthermore, none of the answers were as solid or convincing as Arturo Magidin's.
• The "reason" for the symbolism is mainly historical (as per Arturo's answer); the success of Leibniz's notation was exactly due to his "algebraic" nature, that allowed for "easy manipulations". Of course, those manipulations must be justified, and Leibniz - as well as the otehrs "founding fathers", like Euler, Bernoulli, l'Hopital - was well aware of this. – Mauro ALLEGRANZA Mar 19 '16 at 15:54
• @MauroALLEGRANZA That is in part the motivation of my question. As you say his notation was successful due to its algebraic nature, that must mean that later mathematicians agreed to some extent with his algebraic notation. I want to understand how said mathematicians justified the use of algebraic manipulations in integrals. Arturo's answer has given me a thorough understanding for derivatives, namely because of his discussion on the limit. – Airdish Mar 19 '16 at 15:56
• I think the reason is because it happens to work just like a fraction. In reality, we cannot directly say that $\int \frac{dy}{dx}dx=\int dy$ if we didnt have the fundamental theorem of calculus. – lEm Mar 19 '16 at 16:16
About Euler's definition of integral, see:
Definitio I. Calculus integralis est methodus ex data differentialia relatione inveniendi relationem ipsarum quantitaum: et operatio, qua hoc prestatur, integratio vocari solet.
Thus, the starting point is the (already known; see: Leonhard Euler, Institutiones calculi differentialis (1755), page 100) Leibnizian notion of differential, i.e. "differentia infinite parvas".
Coroll.I. Cum igitur calculus differentialis ex data relatione quatitate variabilium, relatione differentialium investigare doceat: calculus integralis methodum inversam suppeditat.
Again, the "conputational" aspect of the Leibnizian method is stressed: integration is the "inverse operation" withe respect to differentatition.
Coroll.III. Proposita relatione quacunque inter binas quantitates variabiles $x$ et $y$, in calculo differentiali methodus traditur rationem differentialium $dy : dx$ investigandi: sin autem vicissim ex hac differentialium ratione ipsa quantitaum $x$ et $y$ relatio sit definienda, hoc opus calculo integrali tribuitur.
Thus, having set up in the "calculo differentiali methodus" the way to investigate the behaviour of the rationem differentialium $dy : dx$ generated by two "related" variables quantities $x$ and $y$, the calculo integralis is aimed at "retrieving" from the rationem differentialium the relation between the two original quantities $x$ and $y$.
Definitio 2 [ page 4 ] Cum functionis cuiuscunque ipsius $x$ differentiale huiusmodi habet formam $Xdx$, proposita tali formam differentiali $Xdx$, in qua $X$ sit functio quaecunque ipsius $x$, illa functio, cuius differentiale est $=Xdx$, huius vocatur integrale, et prefixo signo signo $\int$ indicari solet, ita ut $\int Xdx$ eam denotet quantitatem variabilem, cuius differentiale est $=Xdx$.
In other words, if $F(x)$ is a function whatever of $x$ whose differential is $f(x)dx$, the said $F(x)$ is said the integral of the above differential and we denote it with $\int f(x)dx$.
There is a basic difference here: for us (from Weierstrass and on) $\dfrac{dy}{dx}$ is not a fraction, but one symbol.
For Euler, following Leibniz, $dy \div dx$ is a ratio between differentials.
• "whose differential is $f(x) d x$" -- this point arises because $$y = F(x) \Longleftrightarrow F'(x) = \dfrac{d y}{d x} = f(x) \rightarrow d y = f(x) d x$$ I know that the explanation behind the differential is intuitive, but what is the formality behind it. In other words, what justifies our taking $dx$ to the other side, since we can't treat it as an algebraic quantity? – Airdish Mar 20 '16 at 14:47
• Let me put my last statement into context. $$\dfrac{dy}{dx} = \lim_{h \rightarrow 0} \dfrac{F(x+h) - F(x)}{h}$$ If we take $dx$ to the other side as illustrated in my comment above, then $$\lim_{h \rightarrow 0} (F(x+h) - F(x)) = f(x) \lim_{h \rightarrow 0} h$$ This clearly cannot be done, because everything evaluates to $0$. – Airdish Mar 20 '16 at 14:54
• @TheOddbodNumber - Euler does not "take $dx$ to the other side"... He consider a function $F$ of $x$ whose differential is expressible as $Xdx$. He does not state -in modern terms - the conditions for a function to be "integrable", but his assumption holds of course for many known functions, like polynomials. – Mauro ALLEGRANZA Mar 20 '16 at 16:00
• My last comments were outside of the context of Euler. Sorry if I misunderstand (I'm a very novice student of math) but in modern times, $$F'(x) = f(x)$$ implies $$d(F(x)) = f(x)dx$$ right? – Airdish Mar 20 '16 at 17:47
• @TheOddbodNumber - in your previous to last comment, you cannot do it; you cannot take $h$ outside od $\lim$, bcause it is not an algebraic expression. – Mauro ALLEGRANZA Mar 20 '16 at 18:18
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# Weighted clustering of lat lon coordinates
I have millions of lat long points that have been grouped into squares. Some squares have thousands of points, others have a couple of points. The idea is that we have one set of lat long for the square with a weighting related to the square based on the number of items it has instead of having millions of rows of data to the cluster.
I was originally using the leadercluster algorithim which allowed me to specify the distance each of my clusters should cover. This is ideal for my use case but now I would like to use it to cluster squares factoring in the weight of the squares. This is basically a learning experience for me and any help would be greatly appreciated
I have seen this which could be useful but I'm not really able to make headway with it
Below is some sample data in R about airports and arrivals
df <- read.csv('https://raw.githubusercontent.com/plotly/datasets/master/2011_february_us_airport_traffic.csv')
ggplot(df, aes(x = long, y = lat)) + geom_point()
sample <- df %>% select(long, lat, arrivals = cnt)
## 1 Answer
Leader clustering is so simple, the weight does not make a difference.
It assigns points to a cluster if the distance is less than a threshold. It does not matter whether the point has 1 weight or is a "square" of 100.
• Hi @Anony-Mousse, I like the algorithm very much because it scales well. However our IT architects have reasonably told me they cannot give me a million data points every 1/4 or so for a learning experience. What they can give me is a grid of 10*10 square meters of the area I'm interested in tell me how many points are contained within each of those squares. This will reduce the data by almost a factor of 20. I want to cluster these squares instead where the weighting will be attached to the number of items contained within a square........ Sep 19, 2017 at 7:50
• ......The square itself will have a centroid of a single lat-lon coordinate. In the example above i have approximated the problem by using an airport dataset where each airport has a number of arrivals. The more arrivals in that airport, the higher the pulling power of that airport/square Sep 19, 2017 at 7:52
• Well, you can't use Leader well on a grid map. Because Leader only uses distance (not weight), and you only have weight information. What you should probably do instead is find local maxima. Sep 19, 2017 at 18:49
• Scalability is also a pretty bad reason to like an algorithm. Random partitions scale even better. Sep 19, 2017 at 18:49
• Hi @Anony-Mousse, thank you again for the feedback. Agreed on the scalability aspect. It worked well for us but also gave good results. Could you recommend an algorithim that would be useful for my use case using weights for lat-lon. Sep 27, 2017 at 17:27
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# fastlogranktest v0.1.0
0
0th
Percentile
## A Fast Way to Calculate the p-Value of One or Multiple Log-Rank-Tests
It is for all people who have to compute many Log-Rank-Tests and the 'survival' package is not fast enough for their purpose. Therefore this small package provides two functions, one for a single Log-Rank-Test and another which calculates many Log-Rank-Tests at a time using threading. Both runs very fast because it uses C++ code with 'Rcpp'.
## Functions in fastlogranktest
Name Description logrank_test Calculate the Log-Rank-Test very fast multi_logrank_test Calculate multiple Log-Rank-Tests very fast No Results!
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# Fielding percentage
In baseball statistics, fielding percentage, also known as fielding average, is a measure that reflects the percentage of times a defensive player properly handles a batted or thrown ball. It is calculated by the sum of putouts and assists, divided by the number of total chances (putouts + assists + errors).
While a high fielding percentage is regarded as a sign of defensive skill, it is also possible for a player of lesser defensive skill to have a high fielding percentage, as it does not reflect or take into account a player's defensive range; a player who cannot get to a ball surrenders a hit instead of having an opportunity to make an out or an error. Conversely, a highly skilled fielder might have a comparatively low fielding percentage by virtue of reaching, and potentially missing, a greater number of balls.
In order to qualify for the league lead in fielding percentage, an infielder or outfielder must appear at the specific position in at least two-thirds of his team's games (games in the outfield are not separated by position). A catcher must appear in at least half his team's games. A pitcher must pitch at least one inning for each of his team's scheduled games (however, a pitcher with fewer innings may qualify if they have more total chances and a higher average). In order to qualify for major league career records for fielding average, a player must appear in 1,000 games at the position; pitchers must have at least 1,500 innings.
The MLB record for team fielding percentage is currently held by the 2013 Baltimore Orioles, with a .99104 fielding percentage.
## Footnotes
1. ^ Rule 10.21(d). "Official Rules". Major League Baseball (MLB.com). Retrieved 2010-06-02.
2. ^ Center, Bill (March 31, 2010). "In defense of the Padres". The San Diego Union-Tribune. Archived from the original on April 3, 2010.
3. ^ Fitzpatrick, Frank (September 30, 2011). "Phillies can rely on their defense ... or maybe not". The Philadelphia Inquirer. Retrieved October 6, 2011. But there's a lot more to defense, obviously, than just not making errors. You have to get to the ball to not make an error in the first place.
4. ^ Rule 10.22(c)(2). "Official Rules". Major League Baseball (MLB.com). Retrieved 2010-06-02.
5. ^ Rule 10.22(c)(1). "Official Rules". Major League Baseball (MLB.com). Retrieved 2010-06-02.
6. ^ Rule 10.22(c)(3). "Official Rules". Major League Baseball (MLB.com). Retrieved 2010-06-02.
This page was last updated at 2022-05-27 22:30 UTC. . View original page.
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# QInterface¶
Defined in qinterface.hpp.
This provides a basic interface with a wide-ranging set of functionality
class Qrack::QInterface
A “Qrack::QInterface” is an abstract interface exposing qubit permutation state vector with methods to operate on it as by gates and register-like instructions.
See README.md for an overview of the algorithms Qrack employs.
Subclassed by Qrack::QEngine, Qrack::QPager, Qrack::QStabilizerHybrid, Qrack::QUnit
## Creating a QInterface¶
There are five primary implementations of a QInterface:
enum Qrack::QInterfaceEngine
Enumerated list of supported engines.
Use QINTERFACE_OPTIMAL for the best supported engine.
Values:
QINTERFACE_CPU = 0
Create a QEngineCPU leveraging only local CPU and memory resources.
QINTERFACE_OPENCL
Create a QEngineOCL, leveraging OpenCL hardware to increase the speed of certain calculations.
QINTERFACE_HYBRID
Create a QHybrid, switching between QEngineCPU and QEngineOCL as efficient.
QINTERFACE_STABILIZER_HYBRID
Create a QStabilizerHybrid, switching between a QStabilizer and a QHybrid as efficient.
QINTERFACE_QPAGER
Create a QPager, which breaks up the work of a QEngine into equally sized “pages.”.
QINTERFACE_QUNIT
Create a QUnit, which utilizes other QInterface classes to minimize the amount of work that’s needed for any given operation based on the entanglement of the bits involved.
This, combined with QINTERFACE_OPTIMAL, is the recommended object to use as a library consumer.
QINTERFACE_QUNIT_MULTI
Create a QUnitMulti, which distributes the explicitly separated “shards” of a QUnit across available OpenCL devices.
QINTERFACE_FIRST = QINTERFACE_CPU
QINTERFACE_OPTIMAL_SCHROEDINGER = QINTERFACE_CPU
QINTERFACE_OPTIMAL_SINGLE_PAGE = QINTERFACE_CPU
QINTERFACE_OPTIMAL_G0_CHILD = QINTERFACE_STABILIZER_HYBRID
QINTERFACE_OPTIMAL_G1_CHILD = QINTERFACE_CPU
QINTERFACE_OPTIMAL_G2_CHILD = QINTERFACE_CPU
QINTERFACE_OPTIMAL = QINTERFACE_QUNIT
QINTERFACE_OPTIMAL_MULTI = QINTERFACE_QUNIT_MULTI
QINTERFACE_MAX
These enums can be passed to an allocator to create a QInterface of that specified implementation type:
template <typename… Ts>
QInterfacePtr Qrack::CreateQuantumInterface(QInterfaceEngine engine, QInterfaceEngine subengine1, QInterfaceEngine subengine2, Ts... args)
Factory method to create specific engine implementations.
## Constructors¶
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::QInterface” with arguments (bitLenInt, qrack_rand_gen_ptr, bool, bool, bool, real1) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- Qrack::QInterface::QInterface()
- Qrack::QInterface::QInterface(bitLenInt, qrack_rand_gen_ptr, bool, bool, bool, real1_f)
## Members¶
StateVectorPtr Qrack::QEngineCPU::stateVec
## Configuration Methods¶
bitLenInt Qrack::QInterface::GetQubitCount()
Get the count of bits in this register.
bitCapInt Qrack::QInterface::GetMaxQPower()
Get the maximum number of basis states, namely $$n^2$$ for $$n$$ qubits.
## State Manipulation Methods¶
virtual void Qrack::QInterface::SetPermutation(bitCapInt perm, complex phaseFac = CMPLX_DEFAULT_ARG) = 0
Set to a specific permutation.
virtual void Qrack::QInterface::SetQuantumState(const complex *inputState) = 0
Set an arbitrary pure quantum state representation.
Warning
PSEUDO-QUANTUM
virtual bitLenInt Qrack::QInterface::Compose(QInterfacePtr toCopy)
Combine another QInterface with this one, after the last bit index of this one.
“Compose” combines the quantum description of state of two independent QInterface objects into one object, containing the full permutation basis of the full object. The “inputState” bits are added after the last qubit index of the QInterface to which we “Compose.” Informally, “Compose” is equivalent to “just setting another group of qubits down next to the first” without interacting them. Schroedinger’s equation can form a description of state for two independent subsystems at once or “separable quantum subsystems” without interacting them. Once the description of state of the independent systems is combined, we can interact them, and we can describe their entanglements to each other, in which case they are no longer independent. A full entangled description of quantum state is not possible for two independent quantum subsystems until we “Compose” them.
“Compose” multiplies the probabilities of the indepedent permutation states of the two subsystems to find the probabilites of the entire set of combined permutations, by simple combinatorial reasoning. If the probablity of the “left-hand” subsystem being in |00> is 1/4, and the probablity of the “right-hand” subsystem being in |101> is 1/8, than the probability of the combined |00101> permutation state is 1/32, and so on for all permutations of the new combined state.
If the programmer doesn’t want to “cheat” quantum mechanically, then the original copy of the state which is duplicated into the larger QInterface should be “thrown away” to satisfy “no clone theorem.” This is not semantically enforced in Qrack, because optimization of an emulator might be acheived by “cloning” “under-the-hood” while only exposing a quantum mechanically consistent API or instruction set.
Returns the quantum bit offset that the QInterface was appended at, such that bit 5 in toCopy is equal to offset+5 in this object.
virtual bitLenInt Qrack::QInterface::Compose(QInterfacePtr toCopy, bitLenInt start) = 0
virtual void Qrack::QInterface::Decompose(bitLenInt start, QInterfacePtr dest) = 0
Minimally decompose a set of contiguous bits from the separably composed unit, into “destination”.
Minimally decompose a set of contigious bits from the separably composed unit. The length of this separable unit is reduced by the length of bits decomposed, and the bits removed are output in the destination QInterface pointer. The destination object must be initialized to the correct number of bits, in 0 permutation state. For quantum mechanical accuracy, the bit set removed and the bit set left behind should be quantum mechanically “separable.”
Like how “Compose” is like “just setting another group of qubits down next to the first,” then “Decompose” is like “just moving a few qubits away from the rest.” Schroedinger’s equation does not require bits to be explicitly interacted in order to describe their permutation basis, and the descriptions of state of separable subsystems, those which are not entangled with other subsystems, are just as easily removed from the description of state. (This is equivalent to a “Schmidt decomposition.”)
If we have for example 5 qubits, and we wish to separate into “left” and “right” subsystems of 3 and 2 qubits, we sum probabilities of one permutation of the “left” three over ALL permutations of the “right” two, for all permutations, and vice versa, like so:
$$P(|1000>|xy>) = P(|1000 00>) + P(|1000 10>) + P(|1000 01>) + P(|1000 11>).$$
If the subsystems are not “separable,” i.e. if they are entangled, this operation is not well-motivated, and its output is not necessarily defined. (The summing of probabilities over permutations of subsytems will be performed as described above, but this is not quantum mechanically meaningful.) To ensure that the subsystem is “separable,” i.e. that it has no entanglements to other subsystems in the QInterface, it can be measured with M(), or else all qubits other than the subsystem can be measured.
virtual void Qrack::QInterface::Dispose(bitLenInt start, bitLenInt length) = 0
Minimally decompose a set of contiguous bits from the separably composed unit, and discard the separable bits from index “start” for “length.”.
Minimally decompose a set of contigious bits from the separably composed unit. The length of this separable unit is reduced by the length of bits decomposed, and the bits removed are output in the destination QInterface pointer. The destination object must be initialized to the correct number of bits, in 0 permutation state. For quantum mechanical accuracy, the bit set removed and the bit set left behind should be quantum mechanically “separable.”
Like how “Compose” is like “just setting another group of qubits down next to the first,” then “Decompose” is like “just moving a few qubits away from the rest.” Schroedinger’s equation does not require bits to be explicitly interacted in order to describe their permutation basis, and the descriptions of state of separable subsystems, those which are not entangled with other subsystems, are just as easily removed from the description of state. (This is equivalent to a “Schmidt decomposition.”)
If we have for example 5 qubits, and we wish to separate into “left” and “right” subsystems of 3 and 2 qubits, we sum probabilities of one permutation of the “left” three over ALL permutations of the “right” two, for all permutations, and vice versa, like so:
$$P(|1000>|xy>) = P(|1000 00>) + P(|1000 10>) + P(|1000 01>) + P(|1000 11>).$$
If the subsystems are not “separable,” i.e. if they are entangled, this operation is not well-motivated, and its output is not necessarily defined. (The summing of probabilities over permutations of subsytems will be performed as described above, but this is not quantum mechanically meaningful.) To ensure that the subsystem is “separable,” i.e. that it has no entanglements to other subsystems in the QInterface, it can be measured with M(), or else all qubits other than the subsystem can be measured.
virtual void Qrack::QInterface::Dispose(bitLenInt start, bitLenInt length, bitCapInt disposedPerm) = 0
Dispose a a contiguous set of qubits that are already in a permutation eigenstate.
virtual real1_f Qrack::QInterface::Prob(bitLenInt qubitIndex) = 0
Direct measure of bit probability to be in |1> state.
Warning
PSEUDO-QUANTUM
virtual real1_f Qrack::QInterface::ProbAll(bitCapInt fullRegister) = 0
Direct measure of full permutation probability.
Warning
PSEUDO-QUANTUM
real1_f Qrack::QInterface::ProbReg(const bitLenInt &start, const bitLenInt &length, const bitCapInt &permutation)
Direct measure of register permutation probability.
Returns probability of permutation of the register.
Warning
PSEUDO-QUANTUM
real1_f Qrack::QInterface::ProbMask(const bitCapInt &mask, const bitCapInt &permutation)
Direct measure of masked permutation probability.
Returns probability of permutation of the mask.
“mask” masks the bits to check the probability of. “permutation” sets the 0 or 1 value for each bit in the mask. Bits which are set in the mask can be set to 0 or 1 in the permutation, while reset bits in the mask should be 0 in the permutation.
Warning
PSEUDO-QUANTUM
virtual void Qrack::QInterface::GetProbs(real1 *outputProbs) = 0
Get the pure quantum state representation.
Warning
PSEUDO-QUANTUM
virtual void Qrack::QInterface::Swap(bitLenInt qubitIndex1, bitLenInt qubitIndex2) = 0
Swap values of two bits in register.
virtual void Qrack::QInterface::Swap(bitLenInt start1, bitLenInt start2, bitLenInt length)
Bitwise swap.
virtual void Qrack::QInterface::ISwap(bitLenInt qubitIndex1, bitLenInt qubitIndex2) = 0
Swap values of two bits in register, and apply phase factor of i if bits are different.
virtual void Qrack::QInterface::ISwap(bitLenInt start1, bitLenInt start2, bitLenInt length)
Bitwise swap.
virtual void Qrack::QInterface::SqrtSwap(bitLenInt qubitIndex1, bitLenInt qubitIndex2) = 0
Square root of Swap gate.
virtual void Qrack::QInterface::SqrtSwap(bitLenInt start1, bitLenInt start2, bitLenInt length)
Bitwise square root of swap.
virtual void Qrack::QInterface::CSwap(const bitLenInt *controls, const bitLenInt &controlLen, const bitLenInt &qubit1, const bitLenInt &qubit2) = 0
Apply a swap with arbitrary control bits.
virtual void Qrack::QInterface::AntiCSwap(const bitLenInt *controls, const bitLenInt &controlLen, const bitLenInt &qubit1, const bitLenInt &qubit2) = 0
Apply a swap with arbitrary (anti) control bits.
virtual void Qrack::QInterface::CSqrtSwap(const bitLenInt *controls, const bitLenInt &controlLen, const bitLenInt &qubit1, const bitLenInt &qubit2) = 0
Apply a square root of swap with arbitrary control bits.
virtual void Qrack::QInterface::AntiCSqrtSwap(const bitLenInt *controls, const bitLenInt &controlLen, const bitLenInt &qubit1, const bitLenInt &qubit2) = 0
Apply a square root of swap with arbitrary (anti) control bits.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::FSim” with arguments (real1, real1, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::FSim(real1_f, real1_f, bitLenInt, bitLenInt) = 0
- void Qrack::QInterface::FSim(real1_f, real1_f, bitLenInt, bitLenInt, bitLenInt)
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::FSim” with arguments (real1, real1, bitLenInt, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::FSim(real1_f, real1_f, bitLenInt, bitLenInt) = 0
- void Qrack::QInterface::FSim(real1_f, real1_f, bitLenInt, bitLenInt, bitLenInt)
virtual void Qrack::QInterface::Reverse(bitLenInt first, bitLenInt last)
Reverse all of the bits in a sequence.
virtual bool Qrack::QInterface::TrySeparate(bitLenInt start, bitLenInt length = 1, real1_f error_tol = REAL1_EPSILON)
Qrack::QUnit types maintain explicit separation of representations of qubits, which reduces memory usage and increases gate speed.
This method is used to manually attempt internal separation of a QUnit subsytem. We attempt a Decompose() operation, on a state which might not be separable. If the state is not separable, we abort and return false. Otherwise, we complete the operation, add the separated subsystem back in place into the QUnit “shards,” and return true.
This should never change the logical/physical state of the QInterface, only possibly its internal representation, for simulation optimization purposes. This is not a truly quantum computational operation, but it also does not lead to nonphysical effects.
Warning
PSEUDO-QUANTUM
std::map<bitCapInt, int> Qrack::QInterface::MultiShotMeasureMask(const bitCapInt *qPowers, const bitLenInt qPowerCount, const unsigned int shots)
Statistical measure of masked permutation probability.
“qPowers” contains powers of 2^n, each representing QInterface bit “n.” The order of these values defines a mask for the result bitCapInt, of 2^0 ~ qPowers[0] to 2^(qPowerCount - 1) ~ qPowers[qPowerCount - 1], in contiguous ascending order. “shots” specifies the number of samples to take as if totally re-preparing the pre-measurement state. This method returns a dictionary with keys, which are the (masked-order) measurement results, and values, which are the number of “shots” that produced that particular measurement result. This method does not “collapse” the state of this QInterface. (The idea is to efficiently simulate a potentially statistically random sample of multiple re-preparations of the state right before measurement, and to collect random measurement resutls, without forcing the user to re-prepare or “clone” the state.)
Warning
PSEUDO-QUANTUM
## Quantum Gates¶
Note
Most gates offer both a single-bit version taking just the index to the qubit, as well as a register-spanning variant for convienence and performance that performs the gate across a sequence of bits.
### Single Gates¶
virtual void Qrack::QInterface::ApplySingleBit(const complex *mtrx, bitLenInt qubitIndex) = 0
Apply an arbitrary single bit unitary transformation.
virtual void Qrack::QInterface::ApplyControlledSingleBit(const bitLenInt *controls, const bitLenInt &controlLen, const bitLenInt &target, const complex *mtrx) = 0
Apply an arbitrary single bit unitary transformation, with arbitrary control bits.
void Qrack::QInterface::AND(bitLenInt inputBit1, bitLenInt inputBit2, bitLenInt outputBit)
Quantum analog of classical “AND” gate.
(Assumes the outputBit is in the 0 state)
void Qrack::QInterface::CLAND(bitLenInt inputQBit, bool inputClassicalBit, bitLenInt outputBit)
Quantum analog of classical “AND” gate.
Takes one qubit input and one classical bit input. (Assumes the outputBit is in the 0 state)
void Qrack::QInterface::OR(bitLenInt inputBit1, bitLenInt inputBit2, bitLenInt outputBit)
Quantum analog of classical “OR” gate.
(Assumes the outputBit is in the 0 state)
void Qrack::QInterface::CLOR(bitLenInt inputQBit, bool inputClassicalBit, bitLenInt outputBit)
Quantum analog of classical “OR” gate.
Takes one qubit input and one classical bit input. (Assumes the outputBit is in the 0 state)
void Qrack::QInterface::XOR(bitLenInt inputBit1, bitLenInt inputBit2, bitLenInt outputBit)
Quantum analog of classical “XOR” gate.
(Assumes the outputBit is in the 0 state)
void Qrack::QInterface::CLXOR(bitLenInt inputQBit, bool inputClassicalBit, bitLenInt outputBit)
Quantum analog of classical “XOR” gate.
Takes one qubit input and one classical bit input. (Assumes the outputBit is in the 0 state)
virtual bool Qrack::QInterface::M(bitLenInt qubitIndex)
Measurement gate.
Measures the qubit at “qubitIndex” and returns either “true” or “false.” (This “gate” breaks unitarity.)
All physical evolution of a quantum state should be “unitary,” except measurement. Measurement of a qubit “collapses” the quantum state into either only permutation states consistent with a |0> state for the bit, or else only permutation states consistent with a |1> state for the bit. Measurement also effectively multiplies the overall quantum state vector of the system by a random phase factor, equiprobable over all possible phase angles.
Effectively, when a bit measurement is emulated, Qrack calculates the norm of all permutation state components, to find their respective probabilities. The probabilities of all states in which the measured bit is “0” can be summed to give the probability of the bit being “0,” and separately the probabilities of all states in which the measured bit is “1” can be summed to give the probability of the bit being “1.” To simulate measurement, a random float between 0 and 1 is compared to the sum of the probability of all permutation states in which the bit is equal to “1”. Depending on whether the random float is higher or lower than the probability, the qubit is determined to be either |0> or |1>, (up to phase). If the bit is determined to be |1>, then all permutation eigenstates in which the bit would be equal to |0> have their probability set to zero, and vice versa if the bit is determined to be |0>. Then, all remaining permutation states with nonzero probability are linearly rescaled so that the total probability of all permutation states is again “normalized” to exactly 100% or 1, (within double precision rounding error). Physically, the act of measurement should introduce an overall random phase factor on the state vector, which is emulated by generating another constantly distributed random float to select a phase angle between 0 and 2 * Pi.
Measurement breaks unitary evolution of state. All quantum gates except measurement should generally act as a unitary matrix on a permutation state vector. (Note that Boolean comparison convenience methods in Qrack such as “AND,” “OR,” and “XOR” employ the measurement operation in the act of first clearing output bits before filling them with the result of comparison, and these convenience methods therefore break unitary evolution of state, but in a physically realistic way. Comparable unitary operations would be performed with a combination of X and CCNOT gates, also called “Toffoli” gates, but the output bits would have to be assumed to be in a known fixed state, like all |0>, ahead of time to produce unitary logical comparison operations.)
virtual bool Qrack::QInterface::ForceM(bitLenInt qubit, bool result, bool doForce = true, bool doApply = true) = 0
Act as if is a measurement was applied, except force the (usually random) result.
Warning
PSEUDO-QUANTUM
virtual void Qrack::QInterface::H(bitLenInt qubitIndex)
Applies a Hadamard gate on qubit at “qubitIndex.”
virtual void Qrack::QInterface::X(bitLenInt qubitIndex)
X gate.
Applies the Pauli “X” operator to the qubit at “qubitIndex.” The Pauli “X” operator is equivalent to a logical “NOT.”
virtual void Qrack::QInterface::Y(bitLenInt qubitIndex)
Y gate.
Applies the Pauli “Y” operator to the qubit at “qubitIndex.” The Pauli “Y” operator is similar to a logical “NOT” with permutation phase effects.
virtual void Qrack::QInterface::Z(bitLenInt qubitIndex)
Z gate.
Applies the Pauli “Z” operator to the qubit at “qubitIndex.” The Pauli “Z” operator reverses the phase of |1> and leaves |0> unchanged.
void Qrack::QInterface::S(bitLenInt qubitIndex)
S gate.
Apply 1/4 phase rotation.
Applies a 1/4 phase rotation to the qubit at “qubitIndex.”
void Qrack::QInterface::IS(bitLenInt qubitIndex)
Inverse S gate.
Apply inverse 1/4 phase rotation.
Applies an inverse 1/4 phase rotation to the qubit at “qubitIndex.”
void Qrack::QInterface::T(bitLenInt qubitIndex)
T gate.
Apply 1/8 phase rotation.
Applies a 1/8 phase rotation to the qubit at “qubitIndex.”
void Qrack::QInterface::IT(bitLenInt qubitIndex)
Inverse T gate.
Apply inverse 1/8 phase rotation.
Applies an inverse 1/8 phase rotation to the qubit at “qubitIndex.”
virtual void Qrack::QInterface::SqrtX(bitLenInt qubitIndex)
Square root of X gate.
Applies the square root of the Pauli “X” operator to the qubit at “qubitIndex.” The Pauli “X” operator is equivalent to a logical “NOT.”
virtual void Qrack::QInterface::ISqrtX(bitLenInt qubitIndex)
Inverse square root of X gate.
Applies the (by convention) inverse square root of the Pauli “X” operator to the qubit at “qubitIndex.” The Pauli “X” operator is equivalent to a logical “NOT.”
virtual void Qrack::QInterface::SqrtY(bitLenInt qubitIndex)
Square root of Y gate.
Applies the square root of the Pauli “Y” operator to the qubit at “qubitIndex.” The Pauli “Y” operator is similar to a logical “NOT” with permutation phase effects.
virtual void Qrack::QInterface::ISqrtY(bitLenInt qubitIndex)
Square root of Y gate.
Applies the (by convention) inverse square root of the Pauli “Y” operator to the qubit at “qubitIndex.” The Pauli “Y” operator is similar to a logical “NOT” with permutation phase effects.
virtual void Qrack::QInterface::SqrtH(bitLenInt qubitIndex)
Applies the square root of the Hadamard gate on qubit at “qubitIndex.”
virtual void Qrack::QInterface::SqrtXConjT(bitLenInt qubitIndex)
Phased square root of X gate.
Applies T.SqrtX.IT to the qubit at “qubitIndex.”
virtual void Qrack::QInterface::ISqrtXConjT(bitLenInt qubitIndex)
Inverse phased square root of X gate.
Applies IT.ISqrtX.T to the qubit at “qubitIndex.”
void Qrack::QInterface::CNOT(bitLenInt control, bitLenInt target)
Controlled NOT gate.
Controlled not.
If the control is set to 1, the target bit is NOT-ed or X-ed.
void Qrack::QInterface::AntiCNOT(bitLenInt control, bitLenInt target)
Anti controlled NOT gate.
“Anti-controlled not” - Apply “not” if control bit is zero, do not apply if control bit is one.
If the control is set to 0, the target bit is NOT-ed or X-ed.
void Qrack::QInterface::CCNOT(bitLenInt control1, bitLenInt control2, bitLenInt target)
Doubly-controlled NOT gate.
Doubly-controlled not.
If both controls are set to 1, the target bit is NOT-ed or X-ed.
void Qrack::QInterface::AntiCCNOT(bitLenInt control1, bitLenInt control2, bitLenInt target)
Anti doubly-controlled NOT gate.
“Anti-doubly-controlled not” - Apply “not” if control bits are both zero, do not apply if either control bit is one.
If both controls are set to 0, the target bit is NOT-ed or X-ed.
void Qrack::QInterface::CY(bitLenInt control, bitLenInt target)
Controlled Y gate.
Apply controlled Pauli Y matrix to bit.
If the “control” bit is set to 1, then the Pauli “Y” operator is applied to “target.”
void Qrack::QInterface::CZ(bitLenInt control, bitLenInt target)
Controlled Z gate.
Apply controlled Pauli Z matrix to bit.
If the “control” bit is set to 1, then the Pauli “Z” operator is applied to “target.”
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::RT” with arguments (real1, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::RT(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::RT(real1_f, bitLenInt)
void Qrack::QInterface::RTDyad(int numerator, int denomPower, bitLenInt qubitIndex)
Dyadic fraction “phase shift gate” - Rotates as e^(i*(M_PI * numerator) / 2^denomPower) around |1> state.
Rotates as $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ around |1> state.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::CRT” with arguments (real1, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::CRT(real1_f, bitLenInt, bitLenInt, bitLenInt)
- void Qrack::QInterface::CRT(real1_f, bitLenInt, bitLenInt)
void Qrack::QInterface::CRTDyad(int numerator, int denomPower, bitLenInt control, bitLenInt target)
Controlled dyadic fraction “phase shift gate”.
Controlled dyadic “phase shift gate” - if control bit is true, rotates target bit as e^(i*(M_PI * numerator) / 2^denomPower) around |1> state.
If control bit is set to 1, rotates target bit as $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ around |1> state.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::RX” with arguments (real1, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::RX(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::RX(real1_f, bitLenInt)
void Qrack::QInterface::RXDyad(int numerator, int denomPower, bitLenInt qubitIndex)
Dyadic fraction X axis rotation gate.
Dyadic fraction x axis rotation gate - Rotates around Pauli x axis.
Rotates $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ on Pauli x axis.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::CRX” with arguments (real1, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::CRX(real1_f, bitLenInt, bitLenInt, bitLenInt)
- void Qrack::QInterface::CRX(real1_f, bitLenInt, bitLenInt)
void Qrack::QInterface::CRXDyad(int numerator, int denomPower, bitLenInt control, bitLenInt target)
Controlled dyadic fraction X axis rotation gate.
Controlled dyadic fraction x axis rotation gate - Rotates around Pauli x axis.
If “control” is 1, rotates as $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ around Pauli x axis.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::RY” with arguments (real1, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::RY(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::RY(real1_f, bitLenInt)
void Qrack::QInterface::RYDyad(int numerator, int denomPower, bitLenInt qubitIndex)
Dyadic fraction Y axis rotation gate.
Dyadic fraction y axis rotation gate - Rotates around Pauli y axis.
Rotates as $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ around Pauli Y axis.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::CRY” with arguments (real1, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::CRY(real1_f, bitLenInt, bitLenInt, bitLenInt)
- void Qrack::QInterface::CRY(real1_f, bitLenInt, bitLenInt)
void Qrack::QInterface::CRYDyad(int numerator, int denomPower, bitLenInt control, bitLenInt target)
Controlled dyadic fraction y axis rotation gate.
Controlled dyadic fraction y axis rotation gate - Rotates around Pauli y axis.
If “control” is set to 1, rotates as $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ around Pauli Y axis.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::RZ” with arguments (real1, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::RZ(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::RZ(real1_f, bitLenInt)
void Qrack::QInterface::RZDyad(int numerator, int denomPower, bitLenInt qubitIndex)
Dyadic fraction Z axis rotation gate.
Dyadic fraction y axis rotation gate - Rotates around Pauli y axis.
Rotates as $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ around Pauli Z axis.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::CRZ” with arguments (real1, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::CRZ(real1_f, bitLenInt, bitLenInt, bitLenInt)
- void Qrack::QInterface::CRZ(real1_f, bitLenInt, bitLenInt)
void Qrack::QInterface::CRZDyad(int numerator, int denomPower, bitLenInt control, bitLenInt target)
Controlled dyadic fraction Z axis rotation gate.
Controlled dyadic fraction z axis rotation gate - Rotates around Pauli z axis.
If “control” is set to 1, rotates as $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ around Pauli Z axis.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::Exp” with arguments (real1, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::Exp(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::Exp(bitLenInt *, bitLenInt, bitLenInt, complex *, bool)
- void Qrack::QInterface::Exp(real1_f, bitLenInt)
void Qrack::QInterface::ExpDyad(int numerator, int denomPower, bitLenInt qubitIndex)
Dyadic fraction (identity) exponentiation gate - Applies exponentiation of the identity operator.
Applies $$\exp\left(-i * \pi * numerator * I / 2^{denomPower}\right)$$, exponentiation of the identity operator
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::ExpX” with arguments (real1, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::ExpX(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::ExpX(real1_f, bitLenInt)
void Qrack::QInterface::ExpXDyad(int numerator, int denomPower, bitLenInt qubitIndex)
Dyadic fraction Pauli X exponentiation gate.
Dyadic fraction Pauli X exponentiation gate - Applies exponentiation of the Pauli X operator.
Applies $$\exp\left(-i * \pi * numerator * \sigma_x / 2^{denomPower}\right)$$, exponentiation of the Pauli X operator
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::ExpY” with arguments (real1, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::ExpY(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::ExpY(real1_f, bitLenInt)
void Qrack::QInterface::ExpYDyad(int numerator, int denomPower, bitLenInt qubitIndex)
Dyadic fraction Pauli Y exponentiation gate.
Dyadic fraction Pauli Y exponentiation gate - Applies exponentiation of the Pauli Y operator.
Applies $$\exp\left(-i * \pi * numerator * \sigma_y / 2^{denomPower}\right)$$, exponentiation of the Pauli Y operator
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::ExpZ” with arguments (real1, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::ExpZ(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::ExpZ(real1_f, bitLenInt)
void Qrack::QInterface::ExpZDyad(int numerator, int denomPower, bitLenInt qubitIndex)
Dyadic fraction Pauli Z exponentiation gate.
Dyadic fraction Pauli Z exponentiation gate - Applies exponentiation of the Pauli Z operator.
Applies $$\exp\left(-i * \pi * numerator * \sigma_z / 2^{denomPower}\right)$$, exponentiation of the Pauli Z operator
void Qrack::QInterface::Exp(bitLenInt *controls, bitLenInt controlLen, bitLenInt qubit, complex *matrix2x2, bool antiCtrled = false)
Imaginary exponentiation of arbitrary 2x2 gate.
Imaginary exponentiate of arbitrary single bit gate.
Applies $$e^{-i*Op}$$, where “Op” is a 2x2 matrix, (with controls on the application of the gate).
virtual void Qrack::QInterface::UniformlyControlledSingleBit(const bitLenInt *controls, const bitLenInt &controlLen, bitLenInt qubitIndex, const complex *mtrxs)
Apply a “uniformly controlled” arbitrary single bit unitary transformation.
A different unitary 2x2 complex matrix is associated with each permutation of the control bits. The first control bit index in the “controls” array is the least significant bit of the permutation, proceeding to the most significant bit. “mtrxs” is a flat (1-dimensional) array where each subsequent set of 4 components is an arbitrary 2x2 single bit gate associated with the next permutation of the control bits, starting from 0. All combinations of control bits apply one of the 4 component (flat 2x2) matrices. For k control bits, there are therefore 4 * 2^k complex components in “mtrxs,” representing 2^k complex matrices of 2x2 components. (The component ordering in each matrix is the same as all other gates with an arbitrary 2x2 applied to a single bit, such as Qrack::ApplySingleBit.)
void Qrack::QInterface::UniformlyControlledRY(const bitLenInt *controls, const bitLenInt &controlLen, bitLenInt qubitIndex, const real1 *angles)
Apply a “uniformly controlled” rotation of a bit around the Pauli Y axis.
Uniformly controlled y axis rotation gate - Rotates as e^(-i*/2) around Pauli y axis for each permutation “k” of the control bits.
A different rotation angle is associated with each permutation of the control bits. The first control bit index in the “controls” array is the least significant bit of the permutation, proceeding to the most significant bit. “angles” is an array where each subsequent component is rotation angle associated with the next permutation of the control bits, starting from 0. All combinations of control bits apply one of rotation angles. For k control bits, there are therefore 2^k real components in “angles.”
void Qrack::QInterface::UniformlyControlledRZ(const bitLenInt *controls, const bitLenInt &controlLen, bitLenInt qubitIndex, const real1 *angles)
Apply a “uniformly controlled” rotation of a bit around the Pauli Z axis.
Uniformly controlled z axis rotation gate - Rotates as e^(-i*/2) around Pauli z axis for each permutation “k” of the control bits.
A different rotation angle is associated with each permutation of the control bits. The first control bit index in the “controls” array is the least significant bit of the permutation, proceeding to the most significant bit. “angles” is an array where each subsequent component is rotation angle associated with the next permutation of the control bits, starting from 0. All combinations of control bits apply one of rotation angles. For k control bits, there are therefore 2^k real components in “angles.”
### Register-wide Gates¶
virtual void Qrack::QInterface::AND(bitLenInt inputStart1, bitLenInt inputStart2, bitLenInt outputStart, bitLenInt length)
Bitwise “AND”.
“AND” registers at “inputStart1” and “inputStart2,” of “length” bits, placing the result in “outputStart”.
virtual void Qrack::QInterface::CLAND(bitLenInt qInputStart, bitCapInt classicalInput, bitLenInt outputStart, bitLenInt length)
Classical bitwise “AND”.
“AND” registers at “inputStart1” and the classic bits of “classicalInput,” of “length” bits, placing the result in “outputStart”.
virtual void Qrack::QInterface::OR(bitLenInt inputStart1, bitLenInt inputStart2, bitLenInt outputStart, bitLenInt length)
Bitwise “OR”.
virtual void Qrack::QInterface::CLOR(bitLenInt qInputStart, bitCapInt classicalInput, bitLenInt outputStart, bitLenInt length)
Classical bitwise “OR”.
virtual void Qrack::QInterface::XOR(bitLenInt inputStart1, bitLenInt inputStart2, bitLenInt outputStart, bitLenInt length)
Bitwise “XOR”.
virtual void Qrack::QInterface::CLXOR(bitLenInt qInputStart, bitCapInt classicalInput, bitLenInt outputStart, bitLenInt length)
Classical bitwise “XOR”.
virtual bitCapInt Qrack::QInterface::MReg(bitLenInt start, bitLenInt length)
Measure permutation state of a register.
virtual void Qrack::QInterface::H(bitLenInt start, bitLenInt length)
virtual void Qrack::QInterface::X(bitLenInt start, bitLenInt length)
Bitwise Pauli X (or logical “NOT”) operator.
virtual void Qrack::QInterface::Y(bitLenInt start, bitLenInt length)
Bitwise Pauli Y operator.
virtual void Qrack::QInterface::Z(bitLenInt start, bitLenInt length)
Bitwise Pauli Z operator.
virtual void Qrack::QInterface::S(bitLenInt start, bitLenInt length)
Bitwise S operator (1/4 phase rotation)
virtual void Qrack::QInterface::IS(bitLenInt start, bitLenInt length)
Bitwise inverse S operator (1/4 phase rotation)
virtual void Qrack::QInterface::T(bitLenInt start, bitLenInt length)
Bitwise T operator (1/8 phase rotation)
virtual void Qrack::QInterface::IT(bitLenInt start, bitLenInt length)
Bitwise inverse T operator (1/8 phase rotation)
virtual void Qrack::QInterface::SqrtX(bitLenInt start, bitLenInt length)
Bitwise square root of Pauli X operator.
virtual void Qrack::QInterface::ISqrtX(bitLenInt start, bitLenInt length)
Bitwise inverse square root of Pauli X operator.
virtual void Qrack::QInterface::SqrtY(bitLenInt start, bitLenInt length)
Bitwise square root of Pauli Y operator.
virtual void Qrack::QInterface::ISqrtY(bitLenInt start, bitLenInt length)
Bitwise inverse square root of Pauli Y operator.
virtual void Qrack::QInterface::SqrtH(bitLenInt start, bitLenInt length)
virtual void Qrack::QInterface::SqrtXConjT(bitLenInt start, bitLenInt length)
Bitwise phased square root of Pauli X operator.
virtual void Qrack::QInterface::ISqrtXConjT(bitLenInt start, bitLenInt length)
Bitwise inverse phased square root of Pauli X operator.
virtual void Qrack::QInterface::CNOT(bitLenInt inputBits, bitLenInt targetBits, bitLenInt length)
Bitwise controlled-not.
virtual void Qrack::QInterface::AntiCNOT(bitLenInt inputBits, bitLenInt targetBits, bitLenInt length)
Bitwise “anti-“controlled-not.
virtual void Qrack::QInterface::CCNOT(bitLenInt control1, bitLenInt control2, bitLenInt target, bitLenInt length)
Bitwise doubly controlled-not.
virtual void Qrack::QInterface::AntiCCNOT(bitLenInt control1, bitLenInt control2, bitLenInt target, bitLenInt length)
Bitwise doubly “anti-“controlled-not.
virtual void Qrack::QInterface::CY(bitLenInt control, bitLenInt target, bitLenInt length)
Bitwise controlled Y gate.
If the “control” bit is set to 1, then the Pauli “Y” operator is applied to “target.”
virtual void Qrack::QInterface::CZ(bitLenInt control, bitLenInt target, bitLenInt length)
Bitwise controlled Z gate.
If the “control” bit is set to 1, then the Pauli “Z” operator is applied to “target.”
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::RT” with arguments (real1, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::RT(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::RT(real1_f, bitLenInt)
virtual void Qrack::QInterface::RTDyad(int numerator, int denomPower, bitLenInt start, bitLenInt length)
Bitwise dyadic fraction phase shift gate.
Rotates as $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ around |1> state.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::RX” with arguments (real1, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::RX(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::RX(real1_f, bitLenInt)
virtual void Qrack::QInterface::RXDyad(int numerator, int denomPower, bitLenInt start, bitLenInt length)
Bitwise dyadic fraction X axis rotation gate.
Rotates $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ on Pauli x axis.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::CRX” with arguments (real1, bitLenInt, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::CRX(real1_f, bitLenInt, bitLenInt, bitLenInt)
- void Qrack::QInterface::CRX(real1_f, bitLenInt, bitLenInt)
virtual void Qrack::QInterface::CRXDyad(int numerator, int denomPower, bitLenInt control, bitLenInt target, bitLenInt length)
Bitwise controlled dyadic fraction X axis rotation gate.
If “control” is 1, rotates as $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ around Pauli x axis.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::RY” with arguments (real1, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::RY(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::RY(real1_f, bitLenInt)
virtual void Qrack::QInterface::RYDyad(int numerator, int denomPower, bitLenInt start, bitLenInt length)
Bitwise dyadic fraction Y axis rotation gate.
Rotates as $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ around Pauli Y axis.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::CRY” with arguments (real1, bitLenInt, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::CRY(real1_f, bitLenInt, bitLenInt, bitLenInt)
- void Qrack::QInterface::CRY(real1_f, bitLenInt, bitLenInt)
virtual void Qrack::QInterface::CRYDyad(int numerator, int denomPower, bitLenInt control, bitLenInt target, bitLenInt length)
Bitwise controlled dyadic fraction y axis rotation gate.
If “control” is set to 1, rotates as $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ around Pauli Y axis.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::RZ” with arguments (real1, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::RZ(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::RZ(real1_f, bitLenInt)
virtual void Qrack::QInterface::RZDyad(int numerator, int denomPower, bitLenInt start, bitLenInt length)
Bitwise dyadic fraction Z axis rotation gate.
Rotates as $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ around Pauli Z axis.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::CRZ” with arguments (real1, bitLenInt, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::CRZ(real1_f, bitLenInt, bitLenInt, bitLenInt)
- void Qrack::QInterface::CRZ(real1_f, bitLenInt, bitLenInt)
virtual void Qrack::QInterface::CRZDyad(int numerator, int denomPower, bitLenInt control, bitLenInt target, bitLenInt length)
Bitwise controlled dyadic fraction Z axis rotation gate.
If “control” is set to 1, rotates as $$\exp\left(i*{\pi * numerator} / 2^{denomPower}\right)$$ around Pauli Z axis.
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::Exp” with arguments (real1, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::Exp(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::Exp(bitLenInt *, bitLenInt, bitLenInt, complex *, bool)
- void Qrack::QInterface::Exp(real1_f, bitLenInt)
virtual void Qrack::QInterface::ExpDyad(int numerator, int denomPower, bitLenInt start, bitLenInt length)
Bitwise Dyadic fraction (identity) exponentiation gate.
Applies $$\exp\left(-i * \pi * numerator * I / 2^{denomPower}\right)$$, exponentiation of the identity operator
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::ExpX” with arguments (real1, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::ExpX(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::ExpX(real1_f, bitLenInt)
virtual void Qrack::QInterface::ExpXDyad(int numerator, int denomPower, bitLenInt start, bitLenInt length)
Bitwise Dyadic fraction Pauli X exponentiation gate.
Applies $$\exp\left(-i * \pi * numerator * \sigma_x / 2^{denomPower}\right)$$, exponentiation of the Pauli X operator
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::ExpY” with arguments (real1, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::ExpY(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::ExpY(real1_f, bitLenInt)
virtual void Qrack::QInterface::ExpYDyad(int numerator, int denomPower, bitLenInt start, bitLenInt length)
Bitwise Dyadic fraction Pauli Y exponentiation gate.
Applies $$\exp\left(-i * \pi * numerator * \sigma_y / 2^{denomPower}\right)$$, exponentiation of the Pauli Y operator
Warning
doxygenfunction: Unable to resolve multiple matches for function “Qrack::QInterface::ExpZ” with arguments (real1, bitLenInt, bitLenInt) in doxygen xml output for project “qrack” from directory: /tmp/qrack/doc/xml. Potential matches:
- virtual void Qrack::QInterface::ExpZ(real1_f, bitLenInt, bitLenInt)
- void Qrack::QInterface::ExpZ(real1_f, bitLenInt)
virtual void Qrack::QInterface::ExpZDyad(int numerator, int denomPower, bitLenInt start, bitLenInt length)
Bitwise Dyadic fraction Pauli Z exponentiation gate.
Applies $$\exp\left(-i * \pi * numerator * \sigma_z / 2^{denomPower}\right)$$, exponentiation of the Pauli Z operator
## Algorithmic Implementations¶
void Qrack::QInterface::QFT(bitLenInt start, bitLenInt length, bool trySeparate = false)
Quantum Fourier Transform - Apply the quantum Fourier transform to the register.
Quantum Fourier Transform - Optimized for going from |0>/|1> to |+>/|-> basis.
“trySeparate” is an optional hit-or-miss optimization, specifically for QUnit types. Our suggestion is, turn it on for speed and memory effciency if you expect the result of the QFT to be in a permutation basis eigenstate. Otherwise, turning it on will probably take longer.
void Qrack::QInterface::IQFT(bitLenInt start, bitLenInt length, bool trySeparate = false)
Inverse Quantum Fourier Transform - Apply the inverse quantum Fourier transform to the register.
Inverse Quantum Fourier Transform - Quantum Fourier transform optimized for going from |+>/|-> to |0>/|1> basis.
“trySeparate” is an optional hit-or-miss optimization, specifically for QUnit types. Our suggestion is, turn it on for speed and memory effciency if you expect the result of the QFT to be in a permutation basis eigenstate. Otherwise, turning it on will probably take longer.
virtual bitCapInt Qrack::QInterface::IndexedLDA(bitLenInt indexStart, bitLenInt indexLength, bitLenInt valueStart, bitLenInt valueLength, unsigned char *values, bool resetValue = true) = 0
Set 8 bit register bits by a superposed index-offset-based read from classical memory.
“inputStart” is the start index of 8 qubits that act as an index into the 256 byte “values” array. The “outputStart” bits are first cleared, then the separable |input, 00000000> permutation state is mapped to |input, values[input]>, with “values[input]” placed in the “outputStart” register. FOR BEST EFFICIENCY, the “values” array should be allocated aligned to a 64-byte boundary. (See the unit tests suite code for an example of how to align the allocation.)
While a QInterface represents an interacting set of qubit-based registers, or a virtual quantum chip, the registers need to interact in some way with (classical or quantum) RAM. IndexedLDA is a RAM access method similar to the X addressing mode of the MOS 6502 chip, if the X register can be in a state of coherent superposition when it loads from RAM.
The physical motivation for this addressing mode can be explained as follows: say that we have a superconducting quantum interface device (SQUID) based chip. SQUIDs have already been demonstrated passing coherently superposed electrical currents. In a sufficiently quantum-mechanically isolated qubit chip with a classical cache, with both classical RAM and registers likely cryogenically isolated from the environment, SQUIDs could (hopefully) pass coherently superposed electrical currents into the classical RAM cache to load values into a qubit register. The state loaded would be a superposition of the values of all RAM to which coherently superposed electrical currents were passed.
In qubit system similar to the MOS 6502, say we have qubit-based “accumulator” and “X index” registers, and say that we start with a superposed X index register. In (classical) X addressing mode, the X index register value acts an offset into RAM from a specified starting address. The X addressing mode of a LoaD Accumulator (LDA) instruction, by the physical mechanism described above, should load the accumulator in quantum parallel with the values of every different address of RAM pointed to in superposition by the X index register. The superposed values in the accumulator are entangled with those in the X index register, by way of whatever values the classical RAM pointed to by X held at the time of the load. (If the RAM at index “36” held an unsigned char value of “27,” then the value “36” in the X index register becomes entangled with the value “27” in the accumulator, and so on in quantum parallel for all superposed values of the X index register, at once.) If the X index register or accumulator are then measured, the two registers will both always collapse into a random but valid key-value pair of X index offset and value at that classical RAM address.
Note that a “superposed store operation in classical RAM” is not possible by analagous reasoning. Classical RAM would become entangled with both the accumulator and the X register. When the state of the registers was collapsed, we would find that only one “store” operation to a single memory address had actually been carried out, consistent with the address offset in the collapsed X register and the byte value in the collapsed accumulator. It would not be possible by this model to write in quantum parallel to more than one address of classical memory at a time.
virtual bitCapInt Qrack::QInterface::IndexedADC(bitLenInt indexStart, bitLenInt indexLength, bitLenInt valueStart, bitLenInt valueLength, bitLenInt carryIndex, unsigned char *values) = 0
Add to entangled 8 bit register state with a superposed index-offset-based read from classical memory.
“inputStart” is the start index of 8 qubits that act as an index into the 256 byte “values” array. The “outputStart” bits would usually already be entangled with the “inputStart” bits via a IndexedLDA() operation. With the “inputStart” bits being a “key” and the “outputStart” bits being a value, the permutation state |key, value> is mapped to |key, value + values[key]>. This is similar to classical parallel addition of two arrays. However, when either of the registers are measured, both registers will collapse into one random VALID key-value pair, with any addition or subtraction done to the “value.” See IndexedLDA() for context.
FOR BEST EFFICIENCY, the “values” array should be allocated aligned to a 64-byte boundary. (See the unit tests suite code for an example of how to align the allocation.)
While a QInterface represents an interacting set of qubit-based registers, or a virtual quantum chip, the registers need to interact in some way with (classical or quantum) RAM. IndexedLDA is a RAM access method similar to the X addressing mode of the MOS 6502 chip, if the X register can be in a state of coherent superposition when it loads from RAM. “IndexedADC” and “IndexedSBC” perform add and subtract (with carry) operations on a state usually initially prepared with IndexedLDA().
virtual bitCapInt Qrack::QInterface::IndexedSBC(bitLenInt indexStart, bitLenInt indexLength, bitLenInt valueStart, bitLenInt valueLength, bitLenInt carryIndex, unsigned char *values) = 0
Subtract from an entangled 8 bit register state with a superposed index-offset-based read from classical memory.
“inputStart” is the start index of 8 qubits that act as an index into the 256 byte “values” array. The “outputStart” bits would usually already be entangled with the “inputStart” bits via a IndexedLDA() operation. With the “inputStart” bits being a “key” and the “outputStart” bits being a value, the permutation state |key, value> is mapped to |key, value - values[key]>. This is similar to classical parallel addition of two arrays. However, when either of the registers are measured, both registers will collapse into one random VALID key-value pair, with any addition or subtraction done to the “value.” See QInterface::IndexedLDA for context.
FOR BEST EFFICIENCY, the “values” array should be allocated aligned to a 64-byte boundary. (See the unit tests suite code for an example of how to align the allocation.)
While a QInterface represents an interacting set of qubit-based registers, or a virtual quantum chip, the registers need to interact in some way with (classical or quantum) RAM. IndexedLDA is a RAM access method similar to the X addressing mode of the MOS 6502 chip, if the X register can be in a state of coherent superposition when it loads from RAM. “IndexedADC” and “IndexedSBC” perform add and subtract (with carry) operations on a state usually initially prepared with IndexedLDA().
virtual void Qrack::QInterface::Hash(bitLenInt start, bitLenInt length, unsigned char *values) = 0
Transform a length of qubit register via lookup through a hash table.
The hash table must be a one-to-one function, otherwise the behavior of this method is undefined. The value array definition convention is the same as IndexedLDA(). Essentially, this is an IndexedLDA() operation that replaces the index register with the value register, but the lookup table must therefore be one-to-one, for this operation to be unitary, as required.
void Qrack::QInterface::TimeEvolve(Hamiltonian h, real1_f timeDiff)
To define a Hamiltonian, give a vector of controlled single bit gates (“HamiltonianOp” instances) that are applied by left-multiplication in low-to-high vector index order on the state vector.
As a general point of linear algebra, where A and B are linear operators,
$$$e^{i (A + B) t} = e^{i A t} e^{i B t}$$$
might NOT hold, if the operators A and B do not commute. As a rule of thumb, A will commute with B at least in the case that A and B act on entirely different sets of qubits. However, for defining the intended Hamiltonian, the programmer can be guaranteed that the exponential factors will be applied right-to-left, by left multiplication, in the order
$$$e^{-i H_{N - 1} t} e^{-i H_{N - 2} t} \ldots e^{-i H_0 t} \left|\psi \rangle\right. .$$$
(For example, if A and B are single bit gates acting on the same bit, form their composition into one gate by the intended right-to-left fusion and apply them as a single HamiltonianOp.)
Warning
Hamiltonian components might not commute.
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{}
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Quantum Affine Algebras, Extended Affine Lie Algebras, and Their Applications
Quantum Affine Algebras, Extended Affine Lie Algebras, and Applications, March 2-7, 2008, Banff International Research Station, Banff, Canada
Author: Yun Gao
Publisher: American Mathematical Soc.
ISBN: 0821845071
Category: Mathematics
Page: 302
View: 4636
This volume contains the proceedings of the conference on Quantum Affine Algebras, Extended Affine Lie Algebras, and Applications, which was held at the Banff International Research Station, Banff, Canada, from March 2-7, 2008. Many of the papers include new results on different aspects of quantum affine algebras, extended affine Lie algebras, and their applications in other areas of mathematics and physics. Any reader interested in learning about the recent developments in quantum affine algebras and extended affine Lie algebras will benefit from this book.
Posted in Mathematics
Recent Developments in Quantum Affine Algebras and Related Topics
Representations of Affine and Quantum Affine Algebras and Their Applications, North Carolina State University, May 21-24, 1998
Author: Naihuan Jing,Kailash C. Misra
Publisher: American Mathematical Soc.
ISBN: 0821811991
Category: Mathematics
Page: 469
View: 5324
This volume reflects the proceedings of the International Conference on Representations of Affine and Quantum Affine Algebras and Their Applications held at North Carolina State University (Raleigh). In recent years, the theory of affine and quantum affine Lie algebras has become an important area of mathematical research with numerous applications in other areas of mathematics and physics. Three areas of recent progress are the focus of this volume: affine and quantum affine algebras and their generalizations, vertex operator algebras and their representations, and applications in combinatorics and statistical mechanics. Talks given by leading international experts at the conference offered both overviews on the subjects and current research results. The book nicely presents the interplay of these topics recently occupying 'center stage' in the theory of infinite dimensional Lie theory.
Posted in Mathematics
Recent Developments in Quantum Affine Algebras and Related Topics
Representations of Affine and Quantum Affine Algebras and Their Applications, North Carolina State University, May 21-24, 1998
Author: Naihuan Jing,Kailash C. Misra
Publisher: American Mathematical Soc.
ISBN: 0821811991
Category: Mathematics
Page: 469
View: 2431
This volume reflects the proceedings of the International Conference on Representations of Affine and Quantum Affine Algebras and Their Applications held at North Carolina State University (Raleigh). In recent years, the theory of affine and quantum affine Lie algebras has become an important area of mathematical research with numerous applications in other areas of mathematics and physics. Three areas of recent progress are the focus of this volume: affine and quantum affine algebras and their generalizations, vertex operator algebras and their representations, and applications in combinatorics and statistical mechanics. Talks given by leading international experts at the conference offered both overviews on the subjects and current research results. The book nicely presents the interplay of these topics recently occupying 'center stage' in the theory of infinite dimensional Lie theory.
Posted in Mathematics
Representations of Lie Algebras, Quantum Groups and Related Topics
Author: Naihuan Jing,Kailash C. Misra
Publisher: American Mathematical Soc.
ISBN: 1470436965
Category: Algebra
Page: 233
View: 2667
This volume contains the proceedings of the AMS Special Session on Representations of Lie Algebras, Quantum Groups and Related Topics, held from November 12–13, 2016, at North Carolina State University, Raleigh, North Carolina. The articles cover various aspects of representations of Kac–Moody Lie algebras and their applications, structure of Leibniz algebras and Krichever–Novikov algebras, representations of quantum groups, and related topics.
Posted in Algebra
Symmetries, Integrable Systems and Representations
Author: Kenji Iohara,Sophie Morier-Genoud,Bertrand Rémy
Publisher: Springer Science & Business Media
ISBN: 1447148630
Category: Mathematics
Page: 638
View: 4224
This volume is the result of two international workshops; Infinite Analysis 11 – Frontier of Integrability – held at University of Tokyo, Japan in July 25th to 29th, 2011, and Symmetries, Integrable Systems and Representations held at Université Claude Bernard Lyon 1, France in December 13th to 16th, 2011. Included are research articles based on the talks presented at the workshops, latest results obtained thereafter, and some review articles. The subjects discussed range across diverse areas such as algebraic geometry, combinatorics, differential equations, integrable systems, representation theory, solvable lattice models and special functions. Through these topics, the reader will find some recent developments in the field of mathematical physics and their interactions with several other domains.
Posted in Mathematics
Recent Developments in Algebraic and Combinatorial Aspects of Representation Theory
Author: Vyjayanthi Chari,Jacob Greenstein,Kailash C. Misra,K. N. Raghavan,Sankaran Viswanath
Publisher: American Mathematical Soc.
ISBN: 0821890379
Category: Mathematics
Page: 210
View: 5034
This volume contains the proceedings of the International Congress of Mathematicians Satellite Conference on Algebraic and Combinatorial Approaches to Representation Theory, held August 12-16, 2010, at the National Institute of Advanced Studies, Bangalore, India, and the follow-up conference held May 18-20, 2012, at the University of California, Riverside, CA. It contains original research and survey articles on various topics in the theory of representations of Lie algebras, quantum groups and algebraic groups, including crystal bases, categorification, toroidal algebras and their generalizations, vertex algebras, Hecke algebras, Kazhdan-Lusztig bases, $q$-Schur algebras, and Weyl algebras.
Posted in Mathematics
Recent Developments in Infinite-dimensional Lie Algebras and Conformal Field Theory
Proceedings of an International Conference on Infinite-dimensional Lie Theory and Conformal Field Theory, May 23-27, 2000, University of Virginia, Charlottesville, Virginia
Author: Stephen Berman
Publisher: American Mathematical Soc.
ISBN: 9780821856338
Category: Mathematics
Page: 334
View: 8168
Because of its many applications to mathematics and mathematical physics, the representation theory of infinite-dimensional Lie and quantized enveloping algebras comprises an important area of current research. This volume includes articles from the proceedings of an international conference, ''Infinite-Dimensional Lie Theory and Conformal Field Theory'', held at the University of Virginia. Many of the contributors to the volume are prominent researchers in the field. Thisconference provided an opportunity for mathematicians and physicists to interact in an active research area of mutual interest. The talks focused on recent developments in the representation theory of affine, quantum affine, and extended affine Lie algebras and Lie superalgebras. They also highlightedapplications to conformal field theory, integrable and disordered systems. Some of the articles are expository and accessible to a broad readership of mathematicians and physicists interested in this area; others are research articles that are appropriate for more advanced readers.
Posted in Mathematics
Affine Lie Algebras and Quantum Groups
An Introduction, with Applications in Conformal Field Theory
Author: Jürgen Fuchs
Publisher: Cambridge University Press
ISBN: 9780521484121
Category: Mathematics
Page: 433
View: 4748
This is an introduction to the theory of affine Lie algebras and to the theory of quantum groups. It is unique in discussing these two subjects in a unified manner, which is made possible by discussing their respective applications in conformal field theory. The description of affine algebras covers the classification problem, the connection with loop algebras, and representation theory including modular properties. The necessary background from the theory of semisimple Lie algebras is also provided. The discussion of quantum groups concentrates on deformed enveloping algebras and their representation theory, but other aspects such as R-matrices and matrix quantum groups are also dealt with.
Posted in Mathematics
The Monster and Lie Algebras
Proceedings of a Special Research Quarter at the Ohio State University, May 1996
Publisher: Walter de Gruyter
ISBN: 3110801892
Category: Mathematics
Page: 262
View: 1602
This series is devoted to the publication of monographs, lecture resp. seminar notes, and other materials arising from programs of the OSU Mathemaical Research Institute. This includes proceedings of conferences or workshops held at the Institute, and other mathematical writings.
Posted in Mathematics
Geometric Representation Theory and Extended Affine Lie Algebras
Author: Erhard Neher,Alistair Savage,Weiqiang Wang
Publisher: American Mathematical Soc.
ISBN: 0821871617
Category: Mathematics
Page: 213
View: 3595
This text presents lectures given at the Fields Institute Summer School on Geometric Representation Theory and Extended Affine Lie Algebras held at the University of Ottawa in 2009. It provides a systematic account by experts of some of the developments in Lie algebras and representation theory in the last two decades.
Posted in Mathematics
Kac-Moody Lie Algebras and Related Topics
Ramanujan International Symposium on Kac-Moody Lie Algebras and Applications, January 28-31, 2002, Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai, India
Author: Neelacanta Sthanumoorthy,Kailash C. Misra
Publisher: American Mathematical Soc.
ISBN: 0821833375
Category: Mathematics
Page: 370
View: 5127
This volume is the proceedings of the Ramanujan International Symposium on Kac-Moody Lie algebras and their applications. The symposium provided researchers in mathematics and physics with the opportunity to discuss new developments in this rapidly-growing area of research. The book contains several excellent articles with new and significant results. It is suitable for graduate students and researchers working in Kac-Moody Lie algebras, their applications, and related areas of research.
Posted in Mathematics
Infinite Dimensional Groups and Algebras in Quantum Physics
Author: Johnny T. Ottesen
Publisher: Springer Science & Business Media
ISBN: 3540589147
Category: Science
Page: 218
View: 2194
The idea of writing this book appeared when I was working on some problems related to representations of physically relevant infinite - mensional groups of operators on physically relevant Hilbert spaces. The considerations were local, reducing the subject to dealing with representations of infinite-dimensional Lie algebras associated with the associated groups. There is a large number of specialized articles and books on parts of this subject, but to our suprise only a few represent the point of view given in this book. Moreover, none of the written material was self-contained. At present, the subject has not reached its final form and active research is still being undertaken. I present this subject of growing importance in a unified manner and by a fairly simple approach. I present a route by which students can absorb and understand the subject, only assuming that the reader is familliar with functional analysis, especially bounded and unbounded operators on Hilbert spaces. Moreover, I assume a little basic knowledge of algebras , Lie algebras, Lie groups, and manifolds- at least the definitions. The contents are presented in detail in the introduction in Chap. The manuscript of this book has been succesfully used by some advanced graduate students at Aarhus University, Denmark, in their "A-exame'. I thank them for comments.
Posted in Science
Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics
Author: Josi A. de Azcárraga,Josi M. Izquierdo
Publisher: Cambridge University Press
ISBN: 9780521597005
Category: Mathematics
Page: 455
View: 1975
Now in paperback, this book provides a self-contained introduction to the cohomology theory of Lie groups and algebras and to some of its applications in physics. No previous knowledge of the mathematical theory is assumed beyond some notions of Cartan calculus and differential geometry (which are nevertheless reviewed in the book in detail). The examples, of current interest, are intended to clarify certain mathematical aspects and to show their usefulness in physical problems. The topics treated include the differential geometry of Lie groups, fibre bundles and connections, characteristic classes, index theorems, monopoles, instantons, extensions of Lie groups and algebras, some applications in supersymmetry, Chevalley-Eilenberg approach to Lie algebra cohomology, symplectic cohomology, jet-bundle approach to variational principles in mechanics, Wess-Zumino-Witten terms, infinite Lie algebras, the cohomological descent in mechanics and in gauge theories and anomalies. This book will be of interest to graduate students and researchers in theoretical physics and applied mathematics.
Posted in Mathematics
Affine Lie Algebras, Weight Multiplicities, and Branching Rules
Author: Sam Kass,R. V. Moody,J. Patera,R. Slansky
Publisher: Univ of California Press
ISBN: 9780520067684
Category: Science
Page: 893
View: 6367
00 This practical treatise is an introduction to the mathematics and physics of affine Kac-Moody algebras. It is the result of an unusual interdisciplinary effort by two physicists and two mathematicians to make this field understandable to a broad readership and to illuminate the connections among seemingly disparate domains of mathematics and physics that are tantalizingly suggested by the ubiquity of Lie theory. The book will be useful to Lie algebraists, high energy physicists, statistical mechanics, and number theorists. Volume One contains a description of Kac-Moody Lie algebras, and especially the affine algebras and their representations; the results of extensive computations follow in Volume Two, which is spiral bound for easy reference. This practical treatise is an introduction to the mathematics and physics of affine Kac-Moody algebras. It is the result of an unusual interdisciplinary effort by two physicists and two mathematicians to make this field understandable to a broad readership and to illuminate the connections among seemingly disparate domains of mathematics and physics that are tantalizingly suggested by the ubiquity of Lie theory. The book will be useful to Lie algebraists, high energy physicists, statistical mechanics, and number theorists. Volume One contains a description of Kac-Moody Lie algebras, and especially the affine algebras and their representations; the results of extensive computations follow in Volume Two, which is spiral bound for easy reference.
Posted in Science
Kac-Moody Groups, their Flag Varieties and Representation Theory
Author: Shrawan Kumar
Publisher: Springer Science & Business Media
ISBN: 9780817642273
Category: Mathematics
Page: 606
View: 6170
This is the first monograph to exclusively treat Kac-Moody (K-M) groups, a standard tool in mathematics and mathematical physics. K-M Lie algebras were introduced in the mid-sixties independently by V. Kac and R. Moody, generalizing finite-dimensional semisimple Lie algebras. K-M theory has since undergone tremendous developments in various directions and has profound connections with a number of diverse areas, including number theory, combinatorics, topology, singularities, quantum groups, completely integrable systems, and mathematical physics. This comprehensive, well-written text moves from K-M Lie algebras to the broader K-M Lie group setting, and focuses on the study of K-M groups and their flag varieties. In developing K-M theory from scratch, the author systematically leads readers to the forefront of the subject, treating the algebro-geometric, topological, and representation-theoretic aspects of the theory. Most of the material presented here is not available anywhere in the book literature. {\it Kac--Moody Groups, their Flag Varieties and Representation Theory} is suitable for an advanced graduate course in representation theory, and contains a number of examples, exercises, challenging open problems, comprehensive bibliography, and index. Research mathematicians at the crossroads of representation theory, geometry, and topology will learn a great deal from this text; although the book is devoted to the general K-M case, those primarily interested in the finite-dimensional case will also benefit. No prior knowledge of K-M Lie algebras or of (finite-dimensional) algebraic groups is required, but some basic knowledge would certainly be helpful. For the reader's convenience some of the basic results needed from other areas, including ind-varieties, pro-algebraic groups and pro-Lie algebras, Tits systems, local cohomology, equivariant cohomology, and homological algebra are included.
Posted in Mathematics
Introduction to Vertex Operator Algebras and Their Representations
Author: James Lepowsky,Haisheng Li
Publisher: Springer Science & Business Media
ISBN: 0817681868
Category: Mathematics
Page: 318
View: 2986
* Introduces the fundamental theory of vertex operator algebras and its basic techniques and examples. * Begins with a detailed presentation of the theoretical foundations and proceeds to a range of applications. * Includes a number of new, original results and brings fresh perspective to important works of many other researchers in algebra, lie theory, representation theory, string theory, quantum field theory, and other areas of math and physics.
Posted in Mathematics
Lie Algebras, Part 2
Finite and Infinite Dimensional Lie Algebras and Applications in Physics
Author: E.A. de Kerf,G.G.A. Bäuerle,A.P.E. ten Kroode
Publisher: Elsevier
ISBN: 9780080535463
Category: Science
Page: 553
View: 6225
This is the long awaited follow-up to Lie Algebras, Part I which covered a major part of the theory of Kac-Moody algebras, stressing primarily their mathematical structure. Part II deals mainly with the representations and applications of Lie Algebras and contains many cross references to Part I. The theoretical part largely deals with the representation theory of Lie algebras with a triangular decomposition, of which Kac-Moody algebras and the Virasoro algebra are prime examples. After setting up the general framework of highest weight representations, the book continues to treat topics as the Casimir operator and the Weyl-Kac character formula, which are specific for Kac-Moody algebras. The applications have a wide range. First, the book contains an exposition on the role of finite-dimensional semisimple Lie algebras and their representations in the standard and grand unified models of elementary particle physics. A second application is in the realm of soliton equations and their infinite-dimensional symmetry groups and algebras. The book concludes with a chapter on conformal field theory and the importance of the Virasoro and Kac-Moody algebras therein.
Posted in Science
Publications Update
Author: World Bank
Publisher: N.A
ISBN: N.A
Category:
Page: N.A
View: 3422
Posted in
Representation Theory and Noncommutative Harmonic Analysis I
Fundamental Concepts. Representations of Virasoro and Affine Algebras
Author: Alexandre Kirillov
Publisher: Springer Science & Business Media
ISBN: 9783540186984
Category: Mathematics
Page: 236
View: 8191
This two-part survey provides a short review of the classical part of representation theory, carefully exposing the structure of the theory without overwhelming readers with details, and deals with representations of Virasoro and Kac-Moody algebra. It presents a wealth of recent results on representations of infinite-dimensional groups.
Posted in Mathematics
Tokyo Journal of Mathematics
Author: N.A
Publisher: N.A
ISBN: N.A
Category: Mathematics
Page: N.A
View: 2970
Posted in Mathematics
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# logdet
Tags:
Updated:
logdet is used to define objective functions in determinant maximization problems.
### Syntax
p = logdet(x)
If you have a logdet operator on a variable $$X$$, the constraint $$X \succeq 0$$ should not be added to the model. It is automatically assumed by the solver.
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{}
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Article | Open | Published:
# Determination of total and unbound docetaxel in plasma by ultrafiltration and UPLC-MS/MS: application to pharmacokinetic studies
Scientific Reportsvolume 7, Article number: 14609 (2017) | Download Citation
## Abstract
A sensitive and specific liquid chromatographic/tandem mass spectrometric (LC-MS/MS) method was developed and validated for quantifying total and unbound docetaxel drug concentrations in plasma. Calibration curves for unbound and total docetaxel were linear over the respective ranges of 0.108~10.8 and 0.54~216 ng/mL. The intra- and interday assay accuracy and precision did not exceed 15%. The methods were validated to show the standard range linearity, sensitivity, selectivity, accuracy, precision, and stability of docetaxel in the matrices tested. In addition, this method is fast and simple with a short run time of 4.5 min and a small plasma sample volume (500 µL). The validated method was successfully applied to a pharmacokinetic study of a docetaxel micelle formulation in rat plasma after intravenous administration at a dose of 10 mg/kg. Docetaxel micelles slowly released their drug payload, and protein-bound, unbound, and micellar drug pools existed simultaneously. These various forms in plasma pools were also measured in the study. We confirmed that most of the docetaxel in plasma was micelle-associated (96.52% at 24 h and 83.14% at 72 h) after micellar docetaxel administration, as a result of sequestration of the drug in long-circulating micelles.
## Introduction
Docetaxel is an antineoplastic agent that shows significant activity in ovarian, non-small-cell lung, breast, and head and neck cancers1. Several factors influence its efficacy and toxicity, and assessing an optimal dose is difficult due to inter-patient variability in pharmacokinetics (PKs)2. One of the factors inducing this variation is protein binding. Docetaxel is highly bound to proteins, with unbound fractions generally < 10% in plasma. Interestingly, it was recently reported that exposure to unbound docetaxel rather than the total drug level was more closely related than the total drug to drug-induced hematological toxicity3. The unbound, active fraction is more closely related to the pharmacological and/or toxic effect, and measuring the unbound fraction may therefore be more useful than the total plasma concentration. Numerous analytical methods with high-performance liquid chromatography (LC; HPLC) and LC-tandem mass spectrometry (LC-MS/MS) were developed to determine docetaxel in plasma samples4,5,6,7,8,9,10. These methods quantified the total amount of docetaxel in circulation. However, unbound concentrations of docetaxel may be better correlated with treatment outcomes (i.e., efficacy and toxicity) than is the total drug11. All methods use either equilibrium dialysis or ultrafiltration to separate the unbound drug from the protein-bound or encapsulated fractions. Further, adopting ultrafiltration for separating unbound drugs in PK studies has gained attention in recent years12,13 due to its rapidity, simplicity, and higher analytical throughput. Analytical methods using ultrafiltration followed by HPLC-MS/MS were reported for determining unbound docetaxel in biological samples2,14. Nevertheless, those methods have several shortcomings, such as a long single run time (10 min), a large volume of biological sample required (2 mL), poor sensitivity (lower limit of quantification (LOQ; LLOQ) of 10 or 0.4 ng/mL)14,15,16, and a restricted formulation of docetaxel14. Traditionally, docetaxel has been used in high doses every third week to treat cancers. Lately, there has been a trend towards giving lower weekly doses to improve the therapeutic index8. It is important to develop suitable HPLC methods to determine docetaxel concentrations in patients receiving low-dose therapy.
In addition, docetaxel is greatly limited in clinical use due to its high toxicity and poor water solubility. Ethanol and Tween 80 are often added as solubilizers, which can induce some unpredictable toxicities such as diarrhea, cumulative fluid retention, hypersensitivity, neurotoxicity, and neutropenia17,18. They also affect the binding of docetaxel in the plasma in a concentration-dependent manner, and influence the disposition of intravenously (iv) administered solubilized drugs. Therefore, there is a need to develop better drug delivery systems to overcome these shortcomings and allow determination of unbound drug concentrations to assess the PKs of docetaxel. Several drug carriers, including liposomes, microspheres, micelles, nanocapsules, etc., have been used to deliver docetaxel. An HPLC method was developed and applied to the PKs of docetaxel liposomes and a docetaxel injection after iv administration to rabbits. Calibration curves were linear over the range of 0.025~2.525 µg/mL for total docetaxel in plasma. The LOQ was 10.0 ng/mL. These results showed that liposome carriers led to a significant difference in tissue distributions and PK profiles compared to a traditional docetaxel injection19. However, several factors contributed to the complexity of the pharmacologic effects of the drug delivered by carriers after iv administration, including that the circulating drug was present in three different forms (e.g., protein-unbound, protein-bound, and carrier-associated). Because these pools differed in their PKs, efficacy, and safety, quantification of the total drug concentration alone was insufficient to characterize the biopharmaceutic properties of drugs loaded or encapsulated in carriers. In our previous studies, docetaxel-loaded mixed polymeric micelles showed improved efficacies in cancer therapy and reduced side effects20. It is necessary to establish a method for determining unbound docetaxel concentrations in plasma after administration by micelles, since we are unaware of any published literature describing this.
LC-MS/MS generally has better selectivity and sensitivity than HPLC. It is widely used because of its ability to accurately quantitate analytes with minimal sample clean-up and rapid separation. But LC-MS/MS methodologies occasionally encounter some problems caused by matrix effects21,22. Current US Food and Drug Administration (FDA) guidance documents now require that these effects be evaluated as a part of quantitative LC-MS/MS method development, validation, and routine use23,24,25. However, no analytical method describing the quantitation of unbound docetaxel in plasma has discussed matrix effects to any degree. The FDA documents further emphasize the urgent need for a more-thorough and -complete evaluation and solution to the problem of matrix effects. In the present study, the authors present the development and validation of a versatile and sensitive UPLC-MS/MS method to determine unbound and total docetaxel in plasma with the advantages of a short run time and small plasma sample volume. We also evaluated matrix effects (MEs) with the UPLC-MS/MS method. The method was used to carry out PK studies for docetaxel micelles in rats after iv administration. Furthermore, these data were used to estimate the docetaxel concentrations of plasma in various pools (e.g., unbound, protein-bound, and micellar-associated) after administration of docetaxel micelles.
## Results
### Chromatographic and MS conditions
To evaluate the sensitivity of both docetaxel and the IS, different solvent types and additives in various concentrations and ratios were tested. Better peak shapes and optimal sensitivity were obtained with the addition of 0.1% formic acid and acetonitrile as the organic solvents. Using the gradient elution mobile phase, all analytes were rapidly eluted within 6.0 min (Fig. 1S). Docetaxel and paclitaxel (the IS) were respectively eluted at 1.58 and 1.63 min.
Docetaxel and the IS could be ionized under either positive (ESI+) or negative (ESI) ESI conditions26. In this study, ESI+ was used for detection due to its superior sensitivity over ESI. Depending on different experimental conditions, docetaxel and the IS may tend to generate different adducts when ionized through an ESI source. Although there are reports that exploited protonated adduct MRM transitions with good sensitivity27, in our studies, spectra revealed that docetaxel formed Na+ adducts more easily than H+ adducts even in the presence of formic acid. Under ESI+ conditions respective precursor ions for docetaxel and the IS were [docetaxel + Na]+ at m/z 830.0 and [paclitaxel + Na]+ at m/z 876.2 as shown in Fig. 1. In the spectra, m/z 304.0 and 308.0 were the most sensitive product ions for docetaxel and the IS, respectively, and these were chosen as quantifier ions in subsequent detections. Product ions of m/z 549.1 and 591.0 produced the next highest intensities for docetaxel and the IS, respectively, and these were chosen as qualifier ions.
### Method validation
Chromatographic conditions had adequate specificity for docetaxel and the IS, while no endogenous interference peaks were observed at retention times of docetaxel (1.58 min) or the IS (1.63 min). Docetaxel and the IS were clearly separated from endogenous peaks originating from the blank matrix. Representative MRM chromatograms of blank plasma in mass transition showed that there was no background interference in the docetaxel or IS peak windows (data not shown).
Results of the attained intra- and inter-day precision and accuracy for docetaxel in ultrafiltered plasma and total plasma are summarized in Table 1. The intra- and inter-day precision levels of the analysis for docetaxel in ultrafiltered plasma were < 11.92%, with an accuracy of < ± 12.72% for both (n = 6). And the intra- and inter-day precision levels of the analysis for docetaxel in total plasma were < 10.66%, with an accuracy of < ± 12.70% for both (n = 6). These results showed that the analytical method is precise, accurate, and reproducible for detecting both docetaxel and the IS. The method exhibited excellent linearity over the ranges of 0.108~10.8 and 0.54~216 ng/ml in ultrafiltered and total plasma, respectively, with reproducible correlation coefficients of r2 > 0.99. The LLOQ concentration under the optimized condition was 0.108 ng/ml for docetaxel with a signal-to-noise (S/N) ratio of 19.6. The response was reproducible, and a stable baseline was maintained during the analysis with an S/N ratio of ≥ 10-fold the baseline. A chromatogram of docetaxel at the LLOQ concentration is presented in Fig. 2S.
It is important that the carry-over effect and dilution integrity applied to measure drug concentrations in biological matrices be validated for bioanalytical methods. Results showed that carry-over was avoidable, and it did not affect the accuracy or precision of the bioanalytical methods. In addition, results of dilution integrity showed that the accuracy and precision were within set criteria, i.e., within ± 15% (Table 1). Dilution of samples did not affect the accuracy or precision.
### Assessment of ME/extraction efficiency
The purpose of sample preparation was to remove interferences and extract the analytes from the plasma samples. An economical, clean, and simple SPE method was developed to achieve requirements of clinical PKs in this study. As stated in the 2001 FDA guidelines, “In the case of LC-MS/MS-based procedures, appropriate steps should be taken to ensure the lack of matrix effects throughout the application of the method”23. Matuszewski et al.25 developed a method to quantitatively evaluate MEs in the analytical process. The analytical quality of the entire experimental process (i.e., process efficiency, PE) is related to the presence of interference during sample preparation and assay (e.g., ion transmission, ionization, etc.). Interference might also affect sample preparation, and lead to a decrease in the extraction yield (EY) compared to standards. The extraction recovery (ER) in the presence of the matrix should therefore be determined. Thus, a complete investigation of MEs in the studies required the evaluation of four ratios (ME, PE, ER, and EY), and these results are listed in Table 2. The ME (i.e., potential ion enhancement or suppression) of docetaxel, calculated as the area ratio with and without matrix ions present, was between 103.5 and 87.7 for ultrafiltered plasma, and between 111.5 and 99.4 for total plasma. These results showed that impurities, degradation products, or co-eluting matrix components did not affect the ionization of docetaxel. Elimination of MEs is critical to generate reliable bioanalytical and PK data. In the range of concentrations studied, the PEs for docetaxel in ultrafiltered and total plasma were > 90.0% and > 87.6%, respectively. As shown in Table 2, the ER and EY of docetaxel were about 89.4%~104.8% for ultrafiltered plasma and 88.1%~109.0% for total plasma, and were within acceptable ranges. Therefore, the extraction efficiency of docetaxel in ultrafiltered and total plasma was constant, precise, and reproducible.
### Correlation between unbound and total docetaxel concentrations in vitro
Rat ultrafiltered and unfiltered plasma samples were spiked with docetaxel over a range of concentrations, concentrations of unbound (CUF) and total (CTOT) docetaxel were measured, and these results are listed in Table 3. Equation 1 was obtained using a nonlinear, least-squares parameter estimation method to fit these data:
$${{\rm{C}}}_{{\rm{UF}}}=35.03\,{\rm{x}}\,{{\rm{C}}}_{{\rm{TOT}}}/776.87+{{\rm{C}}}_{{\rm{TOT}}{\rm{.}}}$$
(1)
### PK studies
The validated UPLC-MS/MS method was applied to PK studies of the iv administration of Tynen and docetaxel micelles to SD rats. Profiles of the mean plasma docetaxel concentration versus time are illustrated in Fig. 3S. Figure 3S shows that the drug concentration in plasma rapidly decreased during the first 1.0 h, which is consistent with Zhao’s studies.ref? Docetaxel plasma concentrations obtained at 0.083 h after iv administration were 25,365.00 ± 8342.03 ng/mL for micelles and 7519.00 ± 1955.05 ng/mL for Tynen. The docetaxel concentration of micelles in plasma showed an insignificant difference with Tynen at each time point except for the first 0.083 h. And the drug concentration in plasma was detectable at up to 72 h using the analytical method established in this study. The main PK parameters are presented in Table 4. The respective half-lives of Tynen and micelles were 26.47 ± 22.54 and 24.56 ± 8.19 h. The AUC of micelles was about 2.0 times greater than that of Tynen, and the clearance decreased after iv administration of micelles. The results indicated that micelles could postpone elimination and lead to a longer blood circulating effect of the drug in rats.
### Concentration profiles of docetaxel micelles in various plasma pools
Profiles of docetaxel concentrations in various plasma pools after iv administration of 10 mg/kg docetaxel micelles in SD rats versus time are shown in Fig. 2. Concentrations of mean unbound drug fell from 3.91 ± 3.29 ng/mL at 0.083 h after the end of the infusion, to < 0.2 ng/mL by 24 h, and they remained between 0.1 and 0.2 ng/mL for the rest of the 72-h study. Concentrations of unbound docetaxel did not exceed 10 ng/mL at any time point in any subject administrated micellar docetaxel. Concentrations of non-micellar (protein-bound plus unbound) docetaxel were estimated from the measured unbound concentrations and percent binding data from the in vitro binding study; however, the non-micellar pool of docetaxel present in the plasma after micellar docetaxel administration was almost entirely protein-bound. Concentrations of non-micellar docetaxel after administration of 10 mg/kg micellar docetaxel remained at < 100 ng/mL in all subjects, with concentrations falling from 80.39 ng/mL immediately after dosing to < 5 ng/mL after 72 h (Fig. 2). Micellar docetaxel was then estimated by the difference between total and non-micellar docetaxel. Figure 3 shows that the percentage of docetaxel in plasma that was micelle-associated (%) slowly fell over the course of the study, from 99.68% at the end of the infusion to 83.14% at the end of the 72-h study. Overall, these results evidence that micellar docetaxel is a stable, long-circulating micellar delivery system that sequesters its docetaxel payload in the plasma, slowly releasing it over time as well as targeting the drug to cancer cells via the uptake of intact micelles.
## Conclusions
A specific and sensitive analytical method with ultrafiltration and UPLC-MS/MS for determining both unbound and total concentrations of docetaxel in plasma was developed. The method can monitor plasma concentrations of as low as 0.108 ng/mL, and it is fast and simple with a short run time of 4.5 min and a small plasma sample volume (500 µL). In the study, this method was successfully validated and applied to determine both total and unbound docetaxel plasma concentrations from rats receiving a docetaxel micelle formulation at a dose of 10 mg/kg. Furthermore, concentrations of non-micellar and micellar docetaxel were also determined, and results confirmed that docetaxel micelles were highly associated (>80%) in plasma as a result of sequestration of the drug in long-circulating micelles.
## Methods
### Calibration and method validation
Two separate stock solutions of docetaxel from independent weighings were prepared in methanol at a concentration of 1080 µg/mL. One set of stock solutions was used to prepare calibration standards in methanol, while the other set was used to prepare calibration curves of total and unbound docetaxel in plasma. The IS stock solution was 1180 µg/mL paclitaxel in methanol. A calibration curve of docetaxel in methanol was prepared at concentrations of 21.6, 54, 810, 1080, and 1800 ng/mL by further dilution of the stock solution with methanol. Calibration curves for the total and unbound forms of docetaxel in plasma were prepared using the same procedures. First, appropriate amounts of the docetaxel stock solution and IS were spiked into drug-free rat plasma (for total docetaxel) or control ultrafiltered plasma (for the unbound form of docetaxel), then 1% formic acid was added and mixed well. Second, these mixtures were deproteinized using SPE as follows. With the help of a vacuum, the mixture was then passed through preconditioned (with 1 mL of methanol and 2 mL water) Waters Oasis HLB solid-phase cartridges (Milford, MA, USA). The cartridges were then washed with 1 mL of 1% formic acid and 1 mL of 50% methanol (containing 0.1% formic acid). Then, docetaxel and the IS were eluted with methanol (containing 0.1% formic acid). After an SPE, the organic layer was evaporated, and the residue was redissolved in 120 µL of 90% MeOH (containing 0.1% formic acid); it was then ready for the UPLC-MS/MS analysis. Finally, calibration curves of the total and unbound forms of docetaxel in plasma were prepared at concentrations of 0.54, 1.08, 5.4, 54, 108, and 216 ng/mL and 0.108, 0.27, 0.54, 1.08, 5.4, and 10.8 ng/mL, respectively. All stock and working solutions were stored at −20 °C and brought to ambient temperature before the analysis.
Responses of plasma samples were determined using the ratio of the peak area of each analyte to that of the IS. The ratio of the peak area (y) was plotted against analyte concentrations (x), and calibration curves were constructed in the form of y = A + Bx. Plasma calibration curves were prepared and assayed in triplicate on three different days to demonstrate the linearity of this method. Recovery was assessed by comparing the means of sample replicates to the aforementioned expected concentrations. The coefficient of variation (CV) and relative error (RE) of the mean were used to validate the intra- and inter-day precision and accuracy by determining standard samples of docetaxel in plasma. The LLOQ, defined as the lowest concentration measurable on the calibration curve that could be measured with acceptable accuracy and precision, was determined in six replicates on three consecutive days.
During validation, the carry-over effect was assessed by injecting blank samples after a high concentration sample or calibration standard at the upper limit of quantification (ULOQ). Dilution integrity was demonstrated by spiking the matrix (ultrafiltered plasma and total plasma, respectively) with docetaxel concentration above the ULOQ and diluting this sample with blank matrix (at least five determinations per dilution factor).
### ME characterization and extraction efficiency
To evaluate the absolute ME, blank and ultrafiltered plasma samples were extracted and then spiked with docetaxel at three quality control (QC) concentrations. Corresponding peak areas of docetaxel in post-extraction spiked plasma (B) were then compared to those of neat standards in the mobile phase (A) at equivalent concentrations. The ME was calculated by the ratio (B/A × 100). The process efficiency (PE) of docetaxel was estimated by comparing the corresponding peak areas of docetaxel in pre-extraction spiked plasma (C) with those obtained from the neat standard prepared in the mobile-phase solvent (A) at equivalent concentrations. PE was determined by the ratio (C/A × 100). The plasma extraction recovery (ER) of docetaxel was obtained by comparing the peak area of samples spiked before extraction (C, pre-extraction spiked plasma) to those of plasma samples spiked after extraction (B, post-extraction spiked plasma): ER = C/B × 100 according to Matuszewski et al.24,25. The extraction yield (EY) was determined with the corresponding peak areas of docetaxel in the neat extraction standard (D) to those of the neat standard in the mobile phase (A) at equivalent concentrations. The ratio (D/A × 100) was defined as the EY.
### Establishment of the correlation between unbound and total docetaxel concentrations in vitro
Unfiltered and ultrafiltered plasma samples were assayed by LC-MS/MS to obtain the total (CTOT) and unbound (CUF) concentrations of docetaxel and establish a correlation of total and unbound docetaxel concentrations in plasma. Sample processing procedures are depicted in Fig. 4S. Unfiltered and ultrafiltered plasma samples were both analyzed using the same analytical conditions described above.
For plasma samples spiked with docetaxel, the unbound docetaxel concentration (CUF, ultrafilterable) in ultrafiltered plasma plotted versus the total docetaxel concentration in plasma (CTOT, total) was fit to the CUF = (A × CTOT)/(B + CTOT) equation using a nonlinear, least-squares parameter estimation method (Sigmaplot Software, Salt Lake City, UT, USA). The calculated parameters, A and B, were used to obtain docetaxel concentrations in various plasma pools in docetaxel micelle-treated rats.
### Applications to PK studies
SD rats (male, 7 weeks old) were purchased from BioLASCO Taiwan (Taipei, Taiwan) and used for PK studies. They were housed under a 12-h light/dark cycle with free access to food and water. All animal experimental methods were conducted in accordance with guidelines and regulations established by Taipei Medical University (Taipei, Taiwan). The experimental protocols in the animal study (approval no. LAC-2013-0137) were approved by the Laboratory Animal Center of Taipei Medical University.
Formulations were administered through iv routes with each drug administered at 10 mg/kg for Tynen and docetaxel micelles. Whole-blood samples were collected at predetermined time points. Plasma samples were obtained by centrifuging blood samples at 3000 rpm and 4 °C for 10 min; the plasma was then transferred to a new tube and stored at −80 °C until the analysis. These plasma samples were divided into two groups. One was for measuring total docetaxel concentrations, and the other was for unbound docetaxel concentrations in plasma. Total docetaxel plasma concentrations in rats administrated Tynen or docetaxel micelles were analyzed as follows. Plasma (100 µL) was mixed with 500 ng/mL paclitaxel (10 µL) as the IS and 1% formic acid and vigorously mixed. The sample was further deproteinized using SPE as described above. After the SPE, the organic layer was evaporated, and the residue was redissolved in 120 µL of 90% MeOH (containing 0.1% formic acid) and was then ready for injection into the UPLC-MS/MS instrument. Procedures for analyzing unbound docetaxel plasma concentrations in rat administered Tynen or docetaxel micelles were as follows. Plasma (500 µL) was placed inside a collection cup with a 30-kDa cutoff and centrifuged at 4 °C by rotating at 5270 g for 45 min in a model Allegra X-12R centrifuge (Beckman Coulter). Then, 150 µL of the supernatant was transferred to a microcentrifuge tube and mixed well with 59 ng/mL paclitaxel (15 µL) as the IS and 1% formic acid (150 µL). The next steps included deproteination and redissolvation based on the analytical procedures of total docetaxel plasma concentrations described above.
The related PK parameters, including the half-life of the drug (T1/2), maximum concentration of drug in blood plasma (Cmax), area under the receiver operating curve at 0~72 h (AUC0~72), AUC at 0 to infinity (AUC0~inf), volume distribution (V), clearance (Cl), and mean residence time (MRT), were analyzed using noncompartmental methods provided by WinNonlin software (vers. 6.3.0.395, Pharsight®, Princeton, NJ, USA).
### Estimation of docetaxel in various plasma pools
When conventional docetaxel is applied to rats, the majority of docetaxel is bound to plasma proteins, and the remaining unbound fraction is available to the target site. Therefore, the total concentration (CTOT) of docetaxel is the sum of that bound to proteins (CPT) and the unbound fraction (ultrafilterable, CUF) (shown in Equation 2). However, when micelles are used to deliver docetaxel to SD rats, non-micellar (the sum of protein-bound and unbound, Cnon-micellar) and micellar (Cmicellar) drug pools may exist simultaneously within the body. Since these pools differ in their PK, safety, and efficacy profiles, measuring every pool’s concentration is important. Equation 3 shows the relationship of docetaxel concentrations after administration of docetaxel micelles in plasma pools.
$${{\rm{C}}}_{{\rm{TOT}}}={{\rm{C}}}_{{\rm{PT}}}+{{\rm{C}}}_{{\rm{UF}}}({\rm{for}}\,{\rm{conventional}}\,{\rm{drug}})$$
(2)
$${{\rm{C}}}_{{\rm{TOT}}}={{\rm{C}}}_{{\rm{non}}-{\rm{micellar}}}+{{\rm{C}}}_{{\rm{micellar}}}={{\rm{C}}}_{{\rm{PT}}}+{{\rm{C}}}_{{\rm{UF}}}+{{\rm{C}}}_{{\rm{micellar}}}({\rm{for}}\,{\rm{micellar}}\,{\rm{carriers}})$$
(3)
Total (CTOT) and unbound (ultrafilterable, CUF) docetaxel concentrations were measured from the analyzed total and ultrafiltered rat plasma, respectively. The concentration of non-micellar docetaxel (Cnon-micellar) was defined as the sum of protein-bound (CPT) and unbound (CUF) drug in each micellar docetaxel-treated SD rat’s plasma. The above calculated parameters, A and B, were used to estimate concentrations of non-micellar docetaxel in plasma (Cnon-micellar) after administration of docetaxel micelles. The micellar docetaxel concentration was then calculated as the difference between the total and non-micellar docetaxel concentrations. The micelle-associated percentage (%) was defined as the percentage of docetaxel concentration in micelles after iv administration of docetaxel micelles and calculated using the following formula:
$${\rm{Micelle}} \mbox{-} {\rm{associated}}\,( \% )=100\times ({{\rm{C}}}_{{\rm{TOT}}}-\,{{\rm{C}}}_{{\rm{non}}-{\rm{micellar}}})/{{\rm{C}}}_{{\rm{TOT}}}.$$
Change History: A correction to this article has been published and is linked from the HTML version of this paper. The error has been fixed in the paper.
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
## Change history
• ### 18 January 2018
A correction to this article has been published and is linked from the HTML version of this paper. The error has been fixed in the paper.
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Zhao, X. et al. A UFLC-MS/MS method coupled with one-step protein precipitation for determination of docetaxel in rat plasma: comparative pharmacokinetic study of modified nanostructured lipid carrier. J. Pharm. Biomed. Anal. 83, 202–208 (2013).
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Du, P. et al. Development and validation of a rapid and sensitive UPLC-MS/MS method for determination of total docetaxel from a lipid microsphere formulation in human plasma. J. Chromatogr. B 926, 101–107 (2013).
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Hou, W. Y., Watters, J. W. & McLeod, H. L. Simple and rapid docetaxel assay in plasma by protein precipitation and high-performance liquid chromatography-tandem mass spectrometry. J. Chromatogra. B 804, 263–267 (2004).
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Yamaguchi, H. et al. A rapid and sensitive LC/ESI-MS/MS method for quantitative analysis of docetaxel in human plasma and its application to a pharmacokinetic study. J. Chromatogra. B 893-894, 157–161 (2012).
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Corona, G. et al. High-throughput plasma docetaxel quantification by liquid chromatography-tandem mass spectrometry. Clinica Chimica Acta 412, 358–364 (2011).
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Parise, R. A. et al. Sensitive liquid chromatography–mass spectrometry assay for quantitation of docetaxel and paclitaxel in human plasma. J. Chromatogra. B 783, 231–236 (2003).
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Andersen, A. et al. High sensitivity assays for docetaxel and paclitaxel in plasma using solid-phase extraction and high-performance liquid chromatography with UV detection. BMC Clin. Pharmacol. 6, 2, https://doi.org/10.1186/1472-6904-6-2 (2006).
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Kalra, N., Nagpal, M. & Nyola, N. K. Sensitive high-Performance liquid chromatographic method for the determination of taxol category drug in plasma. Int. J. Pharm. Sci. Res. 2, 65–68 (2013).
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Acharya, M. R. et al. Determination of fraction unbound docetaxel using microequilibrium dialysis. Anal. Biochem. 331, 192–194 (2004).
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Wang, C. & Williams, N. S. A mass balance approach for calculation of recovery and binding enables the use of ultrafiltration as a rapid method for measurement of plasma protein binding for even highly lipophilic compounds. J. Pharm. Biomed. Anal. 75, 112–117 (2013).
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Arellano, C. et al. Determination of unbound fraction of imatinib and N-desmethyl imatinib, validation of an UPLC–MS/MS assay and ultrafiltration method, J. Chromatogra. B 907, 94–100 (2012).
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Mortier, K. A. et al. Development and validation of a liquid chromatography−tandem mass spectrometry assay for the quantification of docetaxel and paclitaxel in human plasma and oral fluid. Anal. Chem. 77, 4677–4683 (2005).
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Du, P. et al. Development and validation of an ultrafiltration-UPLC-MS/MS method for rapid quantification of unbound docetaxel in human plasma. J. Chromatogr. B 967, 28–35 (2014).
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Zhang, J. & Musson, D. G. Investigation of high-throughput ultrafiltration for the determination of an unbound compound in human plasma using liquid chromatography and tandem mass spectrometry with electrospray ionization. J. Chromatogra. B 843, 47–56 (2006).
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Zhao, L. et al. PK and tissue distribution of docetaxel in rabbits after i.v. administration of liposomal and injectable formulations. J. Pharm. Biomed. Anal. 49, 989–996 (2009).
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Sheu, M. T. et al. Co-delivery of doxorubicin-containing thermosensitive hydrogels incorporated with docetaxel-loaded mixed micelles to enhance local cancer therapy. Colloids Surf. B 143, 260–270 (2016).
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Chambers, E. et al. Systematic and comprehensive strategy for reducing matrix effects in LC/MS/MS analyses. J. Chromatogr. B 852, 22–34 (2007).
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Marchi, I. et al. Characterization and classification of matrix effects in biological samples analyses. J. Chromatogra. A 1217, 4071–4078 (2010).
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Zhang, W. et al. Quantitation of paclitaxel and its two major metabolites using a liquid chromatography-electrospray ionization tandem mass spectrometry. J. Chromatogr. B 879, 2018–2022 (2011).
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Matuszewski, B. K., Constanzer, M. I. & Chavez-Eng, C. M. Strategies for the assessment of matrix effect in quantitative bioanalytical methods based on HPLC-MS/MS. Anal. Chem. 75, 3019–3030 (2003).
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Huang, Q. et al. Simultaneous determination of docetaxel and ketoconazole in rat plasma by liquid chromatography/electrospray ionization tandem mass spectrometry. Rapid Commun. Mass Spectrom. 21, 1009–1018 (2007).
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Rafiei, P., Michel, D. & Haddadi, A. Application of a rapid ESI-MS/MS method for quantitative analysis of docetaxel in polymeric matrices of PLGA and PLGA-PEG nanoparticles through direct Injection to mass spectrometer. American Journal of Analytical Chemistry 6, 164–175 (2015).
## Acknowledgements
Financial support by the Ministry of Science and Technology, Taiwan, ROC (MOST 103-2320-B-038-007) is gratefully acknowledged.
## Author information
### Affiliations
1. #### School of Pharmacy, College of Pharmacy, Taipei Medical University, 250 Wu-Hsing Street, Taipei, 11031, Taiwan
• Ming-Thau Sheu
• , Chen-Yuan Wu
• , Chia-Yu Su
• & Hsiu-O Ho
### Contributions
M.S. and H.H. conceived the experiments; C.W. and C.S. conducted the experiments and analyzed the data; M.S. and H.H. wrote the manuscript. All authors reviewed the manuscript.
### Competing Interests
The authors declare that they have no competing interests.
### Corresponding author
Correspondence to Hsiu-O Ho.
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# In a particular gumball machine, there are 4 identical blue gumballs,
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In a particular gumball machine, there are 4 identical blue gumballs, [#permalink]
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In a particular gumball machine, there are 4 identical blue gumballs, 3 identical red gumballs, 2 identical green gumballs, and 1 yellow gumball. In how many different ways can the gumballs be dispensed, 1 at a time, if the 3 red gumballs are dispensed last?
A. 105
B. 210
C. 315
D. 420
E. 630
[Reveal] Spoiler: OA
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Since the red gumballs are being dispensed last they will not impact the number of permutations of the other gumballs, so we can effectively ignore them. (There is only one way for the three red gumballs to be dispensed last (RRR), so we will multiply the number of permutations of the other colours by 1).
The remaining gumballs are: 4 blue, 2 green, 1 yellow. 7 gumballs in total, so they can be arranged in 7! ways. But since the gumballs of each colour are identical, we must divide by the arrangements of each colour.
Total arrangements of the 7 gumballs = $$\frac{7!}{4!*2!*1!} = \frac{7*6*5}{2} = 105$$
So total arrangements of the 10 gumballs, with all 3 reds chosen last is 105 *1 = 105
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# zbMATH — the first resource for mathematics
## Tarasov, Vitaly Olegovich
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Author ID: tarasov.vitaly-o Published as: Tarasov, V.; Tarasov, V. O.; Tarasov, Vitali; Tarasov, Vitaly; Tarasov, Vitaly O. Homepage: http://math.iupui.edu/people/vitaly-tarasov External Links: MGP
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Rs Aggarwal 2019 2020 Solutions for Class 8 Math Chapter 9 Percentage are provided here with simple step-by-step explanations. These solutions for Percentage are extremely popular among Class 8 students for Math Percentage Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the Rs Aggarwal 2019 2020 Book of Class 8 Math Chapter 9 are provided here for you for free. You will also love the ad-free experience on Meritnation’s Rs Aggarwal 2019 2020 Solutions. All Rs Aggarwal 2019 2020 Solutions for class Class 8 Math are prepared by experts and are 100% accurate.
#### Question 1:
Express each of the following as a fraction:
(i) 48%
(ii) 220%
(iii) 2.5%
#### Question 2:
Express each of the following as a decimal:
(i) 6%
(ii) 72%
(iii) 125%
#### Question 3:
Express each of the following as a percentage:
(i) $\frac{9}{25}$
(ii) $\frac{3}{125}$
(iii) $\frac{12}{5}$
#### Question 4:
Convert the ratio 4 : 5 to percentage.
#### Question 5:
Express 125% as a ratio.
#### Question 6:
Which is largest in $6\frac{2}{3}%,\frac{3}{20}$ and 0.14?
#### Question 7:
(i) What per cent of 150 is 96?
(ii) What per cent of 5 kg is 200 g?
(iii) What per cent of 2 litres is 250 mL?
#### Question 8:
Find $4\frac{1}{2}%$ of Rs 3600.
#### Question 9:
If 16% of a number is 72, find the number.
#### Question 10:
A man saves 18% of his monthly income. If he saves Rs 1890 per month, what is his monthly income?
#### Question 11:
A football team wins 7 games, which is 35% of the total games played. How many games were played in all?
$=\left(x×\frac{35}{100}\right)\phantom{\rule{0ex}{0ex}}=\frac{35x}{100}$
#### Question 12:
Amit was given an increment of 20% on his salary. If his new salary is Rs 15300, what was his salary before the increment?
#### Question 13:
Sonal attended her school on 204 days in a full year. If her attendance is 85%, find the number of days on which the school was opened.
$=\left(\mathrm{x}×\frac{85}{100}\right)\phantom{\rule{0ex}{0ex}}=\frac{85\mathrm{x}}{100}$
#### Question 14:
A's income is 20% less than that of B. By what per cent is B's income more than A's?
$=Rs125$
#### Question 15:
The price of petrol goes up by 10%. By how much per cent must a motorist reduce the consumption of petrol so that the expenditure on it remains unchanged?
$=\frac{1}{11}\mathrm{unit}$
$=9\frac{1}{11}%$
#### Question 16:
The population of a town increases by 8% annually. If the present population is 54000, what was it a year ago?
$=\left(x×\frac{108}{100}\right)\phantom{\rule{0ex}{0ex}}=\frac{27x}{25}$
#### Question 17:
The value of a machine depreciates every year by 20%. If the present value of the machine be Rs 160000, what was its value last year?
$=\mathrm{Rs}\left(\mathrm{x}×\frac{80}{100}\right)\phantom{\rule{0ex}{0ex}}=\mathrm{Rs}\frac{4\mathrm{x}}{5}$
#### Question 18:
An alloy contains 40% copper, 32% nickel and rest zinc. Find the mass of zinc in one kg of the alloy.
$=28%$
#### Question 19:
Balanced diet should contain 12% of proteins, 25% of fats and 63% of carbohydrates. If a child needs 2600 calories in his food daily, find in calories the amount of each of these in his daily food intake.
#### Question 20:
Gunpowder contains 75% nitre and 10% sulphur. Find the amount of gunpowder which carries 9 kg nitre. What amount of gunpowder would contain 2.5 kg sulphur?
#### Question 21:
Divide Rs 7000 among A, B and C such that A gets 50% of what B gets and B gets 50% of what C gets.
#### Question 22:
Find the percentage of pure gold in 22-carat gold, if 24-carat gold is 100% pure.
$=91\frac{2}{3}%$
#### Question 23:
The salary of an officer is increased by 25%. By what per cent should the new salary be decreased to restore the original salary?
Let the original salary be Rs 100
Then, after increment of 25% the salary becomes
=
To restore the original salary, let the new salary be decreased by x%.
Thus, we get
Therefore, the new salary must be reduced by 20% to restore the original salary.
#### Question 1:
$\frac{3}{5}=?$
(a) 30%
(b) 40%
(c) 45%
(d) 60%
(d) 60%
#### Question 2:
0.8% when expressed as a decimal, is
(a) 0.08
(b) 0.008
(c) 8
(d) 0.8
(b) 0.008
#### Question 3:
6 : 5 when expressed as a percentage, is
(a) $83\frac{1}{3}%$
(b) 90%
(c) 120%
(d) 6.5%
(c) 120%
#### Question 4:
5% of a number is 9. The number is
(a) 45
(b) 90
(c) 135
(d) 180
(d) 180
#### Question 5:
What per cent of 92 is 120?
(a) 75%
(b) $33\frac{1}{3}%$
(c) $133\frac{1}{3}%$
(d) none of these
(c) $133\frac{1}{3}%$
#### Question 6:
What per cent of 10 kg is 250 g?
(a) 25%
(b) 5%
(c) 10%
(d) 2.5%
(d) 2.5%
40% of ? = 240
(a) 60
(b) 600
(c) 6000
(d) 960
(b) 600
?% of 400 = 60
(a) 6
(b) 12
(c) 15
(d) 20
(c) 15
#### Question 9:
(180% of ?) ÷ 2 = 504
(a) 400
(b) 480
(c) 600
(d) 560
(d) 560
#### Question 10:
20% of Rs 800 = ?
(a) Rs 160
(b) Rs 16
(c) Rs 1600
(d) none of these
(a) Rs 160
#### Question 11:
In an examination, Nitin gets 98 marks. This amounts to 56% of the maximum marks. What are the maximum marks?
(a) 75
(b) 150
(c) 175
(d) 225
(c) 175
$=\frac{56x}{100}$
#### Question 12:
A number is first increased by 10% and then reduced by 10%. The number
(a) does not change
(b) decrease by 1 %
(c) increases by 1 %
(d) none of these
(b) decrease by 1 %
#### Question 13:
A period of 4 hours 30 min is what per cent of a day?
(a) $18\frac{3}{4}%$
(b) 20%
(c) $16\frac{2}{3}%$
(d) 19%
(a) $18\frac{3}{4}%$
#### Question 14:
In an examination, 65% of the total examinees passed. If the number of failures is 420, the total number of examinees is
(a) 500
(b) 1000
(c) 1200
(d) 1625
(c) 1200
$=\left(x×\frac{35}{100}\right)\phantom{\rule{0ex}{0ex}}=\frac{35x}{100}$
#### Question 15:
A number exceeds 20% of itself by 40. The number is
(a) 50
(b) 60
(c) 80
(d) 320
(a) 50
#### Question 16:
A number decreased by $27\frac{1}{2}%$ gives 87. The number is
(a) 58
(b) 110
(c) 120
(d) 135
(c) 120
#### Question 17:
0.05 is what per cent of 20?
(a) 25%
(b) 2.5%
(c) 0.25%
(d) 0.025%
(c) 0.25%
#### Question 18:
One-third of 1206 is what per cent of 134?
(a) 3%
(b) 30%
(c) 20%
(d) 300%
(d) 300%
#### Question 19:
x% of y is y% of?
(a) x
(b) 100x
(c) $\frac{x}{100}$
(d) $\frac{y}{100}$
(a) x
What per cent of
(a) 2.5%
(b) 10%
(c) 20%
(d) 25%
(a) x
#### Question 1:
Express:
(i) 24% as a fraction;
(ii) 105% as a decimal;
(iii) 4 : 5 as a percentage;
(iv) 56% as a ratio.
#### Question 2:
If 34% of a number is 85, find the number.
#### Question 3:
The value of a machine depreciates every year by 10%. If the present value of the machine is Rs 54000, what was its value last year?
#### Question 4:
An alloy contains 30% copper, 42% nickel and rest zinc. Find the mass of zinc in 1 kg of alloy.
$=28%$
#### Question 5:
In a class, 60% of the total number of students are boys and there are 14 girls. How many students are there in the class?
$=\left(x×\frac{40}{100}\right)\phantom{\rule{0ex}{0ex}}=\frac{40x}{100}\phantom{\rule{0ex}{0ex}}$
#### Question 6:
Which is largest in $8\frac{1}{3}%,\frac{4}{25}$ and 0.15?
#### Question 7:
Mark (✓) against the correct answer:
What per cent of
(a) 2.5%
(b) 5%
(c) 7.5%
(d) 10%
(d) 10%
#### Question 8:
Mark (✓) against the correct answer:
A number decreased by 30% gives 84. The number is
(a) 90
(b) 110
(c) 120
(d) 135
(c) 120
#### Question 9:
Mark (✓) against the correct answer:
(?)% of 320 is 48?
(a) 25%
(b) 15%
(c) 14%
(d) 9%
(b) 15%
#### Question 10:
Mark (✓) against the correct answer:
What per cent of 45 is 54?
(a) $83\frac{1}{3}%$
(b) 104%
(c) 108%
(d) 120%
(d) 120%
#### Question 11:
Mark (✓) against the correct answer:
A number exceeds 25% of itself by 60. The number is
(a) 75
(b) 45
(c) 80
(d) 65
(c) 80
#### Question 12:
Mark (✓) against the correct answer:
5% of which number is 12?
(a) 120
(b) 180
(c) 240
(d) 320
(c) 240
#### Question 13:
Fill in the blanks.
(i) $7\frac{1}{2}%$ of Rs 1200 = .........
(ii) 240 mL is ......... % of 3 L.
(iii) If x% of 35 is 42, then x = .........
(iv) $\frac{12}{5}$ = .........%.
(v) 120 = (.........)% of 80.
#### Question 14:
Write 'T' for true and 'F' for false for each of the following:
(i) 6% of 8 is 48.
(ii) 6 : 5 = 30%.
(iii) $\frac{3}{5}=60%.$
(iv) 6 hours = 25% of a day.
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# Trig Identity proof
1. Apr 6, 2013
### trollcast
1. The problem statement, all variables and given/known data
Prove the identity:
$$\csc(2\theta)-\cot(2\theta)\equiv\tan(\theta)$$
2. Relevant equations
3. The attempt at a solution
Starting with the LHS:
$$\csc(2\theta)-\cot(2\theta)$$
$$\frac{1}{\sin(2\theta)}-\frac{\cos(2\theta)}{\sin(2\theta)}$$
$$\frac{1-\cos(2\theta)}{sin(2\theta)}$$
And thats as far as I can see to rearrange it.
2. Apr 6, 2013
### rock.freak667
Now just substitute your identities for cos2θ and sin2θ and it should work out easily.
3. Apr 6, 2013
### trollcast
Woops I forgot about the double angle formulae
so the rest of it is:
$$\frac{1-(1-2\sin^2(\theta))}{2sin(\theta)\cos(\theta)}$$
$$\frac{2\sin^2(\theta)}{2sin(\theta)\cos(\theta)}$$
$$\frac{sin(\theta)}{\cos(\theta)}$$
$$=\tan(\theta)$$
∴ LHS = RHS
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Just lost a round $32 to the Atlantis Casino Reno. I had wanted to come to a corner of this casino and mope about landscaping. That’s right, landscaping. Today, I finally admitted to my wife that I have never before understood why so many people plant so much vegetation near their houses, and by near I mean touching. Why do people do so when it has been well known for decades that trees pull up moisture and rots wood used to build the house? The phenomenon is quite prevalent in the Bay Area. People do landscaping to their houses all the time like people do drugs to their own bodies. And then it dawned on me today that they do it out of hedonic needs. The trees and flowers, so close you can touch and smell, they make people happy. So what if the house will fall down in 20 to 30 years? What comparison can you make with the happiness of the house’s residents for those number of years? A modern human being would always choose happiness of people over the happiness of dead wood. An ancient human being may also argue that since god created everything for humans, that this destructive servitude is most holy. There is no why. It just happens, by will of God and Humans. There is no why. But I choose not to mope about soiled by my realization of how normal people think. Another idea to write about is the wonderful buildings of casinos. The Atlantis has two gigantic torches outside that blows 5-stories-tall flames every 30 minutes. The other casinos are also decadently extravagant. The artwork, the fanciful lighting, the domes, arches, fountains,… a single casino could have many architectural and distinctive features that come from centuries of history and art. Combined with modern day lighting, they look fantastic. The casinos here look like the originals of buildings and paintings that belong to a future museum. How do they get white walls so white? How do they make yellow walls work? But this fantasizing is colored with the lens of twenty-teens. We go to museums to see these historical buildings because they are beautiful, difficult to create, and original (when they were created) in a few centuries time, what will actually attract humans to museums? It is not going to be casinos for sure. I mean, they won’t be able to get the smoke out of these buildings for another millennium. Occasionally, some of us has the experience of being reminded that classic, museum-worthy, privately collectible and perpetually appreciating, transcendental art works from the past, they might have been nouvo and scandalous at their time of production, and they might have taken hundreds of years to gain their due desert of acceptance and appreciation. What we know for sure is that some sufficient number of human beings, at some remarkable times in the past, found such works appealing and decided to preserve them and propagate appreciation for them. Here and now I remark upon Reno casinos… I appreciate these whole buildings that art art. Museum put on display historical intellectual arguments: who said what and influenced whom… visualized or represented in multi-media ways. We can watch great historical debates about politics and sciences. We can watch humanity decide what it wants to believe and wants to do. We can put on display Creationist Science versus Evolutionary Sciences, Climate change spectrum, Bayesian versus Frequentists, Empirical risk minimization versus… We can put on display the battle among titans of industries–IBM versus Apple, Tesla versus the rest, Intel versus Asia, AC versus DC, Amd versus NVIDIA, Ford versus Ferrari, … possibly many other struggle for dominance or survival. Perhaps these will be the legacies we will be remembered by. Our modern marvels will be the process that got us to that time and place where we are museum-worthy relics. # Right Quit-right Right During my thanksgiving-induced food coma, I had run into a problem in my mind. Recall that we had begun to think about right and wrong, just and unjust, should and shouldn’t, could and couldn’t, could do to and could do onto, etc. in an abstract mind conception of the Action Space. Roughly speaking, with the rigor required at the outset of Nichomachean Ethics, Action Space is a set of all possible actions anybody or any body could do to another at a specified time. For a significant portion of this space, a subspace, we can describe actions in English sentence: “Eve gives Adam a forbidden fruit(at any time).” But we do not unnecessarily restrict us to these at this stage. Sets have interesting but practical operators that we use to model other aspects of our world, including: membership and subset relations, union and disjunction operators, etc. A hope is that using sets of actions we can both cover a lot of ground in representing our real world, and we leverage our innate understanding of these concepts to interpret the matters of Actions. This framing gives us an immediate idea to compare the size of action spaces. Suppose there is an Action Space that represent the actions permitted by the U.S. constitution L. Now, we can also have the action space specified by the United State Code(USC). We can very safely demand that when interpreted their action spaces $USC \subset C$. And we say that the Constitution of the United States of America grants strictly more freedom than the USC. The set of actions permitted by the prior is a strict superset of those permitted by the latter. More actions means more rights, liberties and freedoms. Fewer actions means more constrained and fewer choices. Strictly more free is a partial ordering of all action spaces. In a less strict sense of freedom, we can also compare cardinalities of two action spaces. But clearly this ordering is not very useful: have all the rights to sneeze in various poses is not nearly as important as the right to take a sip of water. Of course that can too be ameliorated with utilitarian’s individual utility function or the social welfare function, and other such attempts so as to produce a useful ordering of preference over action spaces. Having considered many perspectives on permissibility and selection of actions, and considering conservative believes about our physical universe and all that we could possibly be concerned with, we have come to designate an API with which thinking and controlling systems may interact with our faculties that deal with rules of law and right and wrong. We suspend our fear of making a homunculus argument as we do not say we have found or made such modules of this artificial intelligence, but merely that we want to separate these concerns to reduce the complexity of reasoning. The separation is not physical, all the thinking could be produced on the same gray matter or CPU. The interface can also be defined implicitly, for visualization, consider looking at a hyperplane through which these two separate functions connect. For the purpose of characterizing whether an action is permitted by a determining agent or subcomponent $E$, we ask that a permission function$latest P_E$to be accessible. $P_E(a, t, H, n=N) -> [permissible|impermissible]$ The parameters are typed: • $a \in A$ is an action of the action space • $t \in T$ is a set of timestamps in question. •$latex T” is a predesignated time indexing which is a set of objects known as timestamps. It is is totally ordered. For convenience we also include the open and closed contiguous sets of timestamp called intervals or ranges using ‘[]()’ symbols. We use the symbol < to mean before, > to mean after and = to mean at the same time.
• time $t$ can be a single timestamp or a set of timestamps. The function is polymorphic.
• The type of time parameters should be be inferred from context if ambiguous: happening at “a time $t$” means a single timestamp, “happening at time/times $t$” means occurring for all time in set $t$.
• Often $T$ is specified to be the real numbers or integers. In this case a reference must be set for the time 0, as well as scale explaining what duration of 1 means in the physical world.
• $H$ is the whole history of the world up to $t$.
• History has, among other information, the timestamp of now $H_n$ which is the maximal time about which we have information through $H$. Calling it now is more positive than the end of history.
• Regarding performed actions, $H$ is a log of actions that have been taken each with timestamp of when they were taken. We use a convenience expression $did(a,h,t=T,n=N)$ to check if an action was reportedly taken in $h$ at time $t$. Not specifying $t$ asks if the action was ever taken. $did(a,h=h)$ is an injective function. An action is taken or not taken, it cannot be unknown.
• $n$ is the nature of the world. It may contain matter such as the laws of physics, existence of god, etc. Since we care most about the nature that we are in, by default this parameter is specified as the nature of our world. We should be able to query for information such as number: $\pi$, e, c, $N_A$ etc.
$P_E$ therefore yields the result that we use to decide whether an action is permissible or not under some system of determination for propriety and preference. The answer, as given by $E$, is E’s answer at time $t$. An agent, upon receiving the permissible result from $E$ will understand that the action they asked about is permitted at the time in question given history of the world leading up to time $t$, and our nature.
….The E is member of world and accessible as part of nature. We could also imagine historical E’s that are result of history: made computer, wrote programs, program decides…..
In considering the permissibility of actions we should also for functional purposes suppose the existence of the doit function. $doit(a, t, h, n=N)=(h',n')$. All that doit does is that it instantaneously adds the action to history and nature at time $latext t$. and reports the results of that insertion. When specified, a natural action is one which does not change nature: $doit(a, t, h, n=N)=(h',n)$ and a supernatural action is one that changes nature: $doit(a, t, h, n=N)=(h,n')$.
Two actions $a_1, a_2$ are homopotent if $doit(a_1, t, h, n=N) =doit(a_2, t, h, n=N)$. This equivalence relation creates equivalent classes of actions. Such classes exist even the Natural language descriptions can have many descriptions of the exactly same action. We will prefix homopotency with historic and natural for equivalence that match only history and only nature respectively.
For convenience of notation, we can query nature and history for, among other things, the deed of past actions: $did(a, h, t=\emptyset, n=N)$
We are conscientious of many other potential problems of our present endeavors. Mathematicians has given us many concerning thoughts about sets of things. One example of a problem with these innings is that most of our computational machinery have known limitation that terminability of a function is unknowable—eg the Halting Problem.
In practical implementation the function may produce a response that is either permissible nor impermissible. If we have to wait for ever, then this function is not useful. If we do not know whether it terminates or not, then we do not know if we can use the result or not. Of course, Software Engineers have long worked around this issue by creating time-box around functions. Each function evaluation is surrounded by machinery that will wait patiently for a result, but if some preset time box is exceeded, the efforts to evaluate said function is suspended and the invoking agent is informed that the function did not function as expected. Since time pass as surely as we can time it, this time boxing wrapper approach guarantees us that we can implement a function with this signature:
$P_E(a, t, H, n=N) -> [permissible|impermissible|indiscernible]$
Users of our API are warned and required to handle the case when such a component fails to function. Such demand is not unreasonable as there are many such safety implements in most modern artificial computational systems. The result of indiscernible expresses no opinion regarding the action. The user of this API may choose conventions on how to react to the result. A information security implementation may choose to be conservative by reacting to indiscernible as if the answer is impermissible to ensure security. Where as human legal system may choose to be liberal and interpret indiscernible reaction as permissible granting maximal freedom when in doubt.
Another oft-used software engineering safety technique is that of rate limiting. The provider of $P_e$ API may choose to rate limit how much any single agent may query the API. Rate limiting helps to mitigate finial of service(DOS) attacks on the permissions system. In reality, this rate limit is enforced by our limited implementations. In theory, a rate limit on API invocation allows to analyze the ability of a real agent to follow the directions of a permission function under realistic constraints. Rate limits can be expressed as a limit on requests can be made within any contiguous interval of a certain set period of time (ie queries per second (qps)), or it may be a rough restriction in the form of interval between requests, among many other choices.
For a third problematic example, we shall eponymously name it the quit-right problem. It is a shady imitator of the Russel Paradox. The problem is self explanatory: Are quit-right actions members of our action space? Can one consider the right to give up a right? If so, can we quit all quit-rights rights? Can we quit a quit right the action itself?
Legal theory has a convenient solution to this problem. In legal arrangements, one can make something called a default rule and another that is called mandatory rule. A default rule applies if there is no forceful contract or declaration to its contrary. Mandatory rules, on the other hand, are those rules that cannot be overwritten irrespective of contracting or forceful declarations. Certainly quit right is an action we can imagine to be part of a legal action space, but a legal action space will not contain quit right actions for actions that are mandatorily protected. Some commonplace examples are the potency of Nondisclosure Agreements (NDA) in the rules of law. In this case a natural person or other legal entities may contract away their right of speech and other expressions—they quit their right of speech and freedom of expression. However, no matter if you sign with a in $2^10000000$ bits of cryptographic signature carved into stone, you can not sign away your life to be taken by another individual. It will always be called into question whether that other individual is responsible for advertently or inadvertently cause your loss of life irrespective of your renunciation thereof. The force of such system is infinite, the person may not change his right to change his right to life, he may not give himself permission to give himself permission to contract or declare away his life, and is on and so forth.
Now, those are a subject itself quitting its own right. Again using a easy target of human life, the action space still contain actions such as the state killing you. Under some circumstance states maintain the right to kill you in its action space for purpose of capital punishment. The American government actually also has the right to modify its own right regarding capital punishment within the confines of its constitution.
But what does the quit-right action look like in the action space? Let’s for simplicity of expression designate a macro $q(e, a, s, t=\infty)$ to mean the action:
quit the right to take action $a$ in the permissibility determining system $e$ on all such times on or after timestamp $s$ and before time $t$.
The meaning of macro $q(e, a, s, t)$
If $P_e(q(e, a, s, t), s_<, u)$ is permissible, and if $q(e, a, s, t)$ was successively taken in history $u\in h$ then $P_e(a, s_>, h,...)$ returns impermissible.
• Time is a totally ordered set of timestamps. These corresponds to wall clock time in our world. The set has membership as well as open and closed interval as.
• Action Space actions has success and failure return codes.
• doit succeeds only when action is permitted.
• doit returns a history. Suppose we can query that history for whether an action was taken in time range. The behavior of doit is then definable on the function’s input and output.
• other agents can be invoking doit as well, it does not affect present agent…\$
• permit, forbid only when the stated modifications to subactionspace is permitted
The resignation to these rights are targeted for a specific permission function $P_e$ to allow us to perform activities permitted by one system and disallowed by another, e.g. law and conscience, rationality and greed, etc. Since we have not introduced macro and action variables or even functions within the action space, we skirt issues like writing a macro that when expanded produces $q(q(q(q($ ad infinitum. But even when that is enabled, it will not be a problem because for uninterpretable actions we have a convenient indiscernible result to resort to when we receive obnoxious or pathological questions and actions that we can certainly deem unreasonable, irrelevant, or useless.
Now then, we may say that if an agent has taken an action $q(e, a, s, t)$ then we expect $P_e(a, u, H_u)$ to return impermissible $\forall u.s\leq u < t$.
So far we have not distinguished actors(subjects) and objects of action. But it does not hinder our efforts. An action space built constructively using verbs and nouns into a transitive action space. We can also explore by building increasingly more complex action space, for example by increasing valency of verbs used to construct a action space. In such spaces, the action passed to the quit-right macro may contain a subject not covariant with the object. In such a situations, the action $a q(e, b does...)$ in which actor $a$ quits an action for $b$. The DMV($a$), for example, has the right to take an action that forbids a person($b$) from driving according to traffic law($e$) according to $e$: A motorist has the API to ask the question $P_e(q^a(e, b drives...)...)$ and receive an affirmative answer of permissible.
More to come…
# That spider nursery rhyme
Eentsy weensy spider climbed the dragon’s ’nout,
Down came the snots and washed the spider out,
Out came the fire that dries up the snots, and the
Entsy weensy spider went up the snout again.
# Academic freedom with Goog
Just saw NPR news about Gebru being fired/resigning for reason related to publication dispute with Google the company. Some many number of people signed letter to ask for transparency and reconsideration.
Honestly, HR is not the most customer friendly or innovative department of silicone valley companies—on average. There are certainly awesome HR individuals and HR leaders that I’ve encountered, but there surely are some seriously uncaring individuals, senior leaders and policies that acted inhumanely and unreasonably. This quotation from Gebru, as published by a very sympathetic review in the Washington Post, is something that perhaps most minorities can write with significant degree of sincerity:
Gebru recounted her most recent experience in the email as an example of why she had given up on advocating for diversity inside Google. “[S]top writing your documents because it doesn’t make a difference,” she wrote. “[Y]our life gets worse when you start advocating for underrepresented people, you start making the other leaders upset when they don’t want to give you good ratings during calibration
This advocacy she speaks of represent a swath of disparate and dissenting opinions regarding various modes in which minorities are treated. When “reasoning” with the employee through normal management chain fails, the mighty HR steps in and uses company business related performance reviews (known as calibrations at Google) to enforce the company’s stance. The only exception here is that one of Gebru’s job is to improve minority inclusion at Google—by losing her own inclusion she has created a self-fulfilling failure to perform her duties to the company. Also, her declaration that writing documents is useless is self defeating as well. As a scientist, a big part of her work will be to document and publish her learnings and believes. Quit writing documents is quitting the job. Sigh… the troubles we have in the computer industry.
It is certainly not surprising that Gebru had to separate from Google. Recall recent episode of her very civilized and reasonable discussion with LeCunn on Twitter. Sadly, I empathize with both of them. Being a minority who likes to think about the reasonable, I definitely feel her frustration from lack of acknowledgement and consideration. But I also feel the scientific curiosity that I imagine LeCunn has for the science.
The problem here is what we do not know. 3 hours after the NPR articles published, I do not see the paper whose quality is in dispute on pre-publication sites. It is highly unlikely that Google will respond publicly to explain why it does not want Gebru at al. to talk about why Google’s core technology is inherently racist and environmentally damaging.
The next day, Jeff Dean published a google doc presenting his view of this incident. One obvious takeaway from Dean’s postmortem write up is that, somewhere in some of her communications, Gebru challenged the mystery, anonymity and opacity of “the official Google internal review process” that assesses scientific qualifications of prospective publications. Her challenge may have been that the review process has excessive and unaudited (white/discriminative and profit-only focused) power. Google apparently has kept that process a secret even to its subjects and in light of this very public revolt.
Interestingly, many academicians and institutions come out in support of Gebru. Some many dozens of hours later MIT tech review published writing based on draft of the paper. MIT seems to also take the position that there is unclear and inconsistent behavior on the part of Google.
This is not a company. This is a center of a civilization. While it is very brave for a few researchers to stand up to it to demand something, the outcome and benefit of this divisive exercise is not clear at all. The signatories of the complaint letter certainly cannot all resign from their jobs under leads who wishes to uphold the values and methods that worked so great at Google.
My firm belief is that we need to build more common ground by working on creating the common ground. We need for people of all kinds to come closer and closer to discuss and improve shared core principles. And I definitely mean that the shared principles are truly shared: when presented with similar situation, different people owning the same principles react and decide essentially for the same reasons.
Despite our biological similarities, despite our common humanity, common ground does not come for free. Despite our shared computer protocols for global exchange of information. All that needs to be said must still be said to build understanding. Demands for apologies and submission to a point of view, while righteous to do for the righteous, doesn’t really build common ground, it doesn’t improve shared understandings.
But I will always end my commentary on this subject by saying that my people, my ancestors were not enslaved for centuries. My direct ancestors had not had the pleasure of being subjects or objects of European men. My parents were not firehosed or beaten to shot or burned or segregated for the color of their skin. I will humbly acknowledge that I have lesser cultural and genetic sensitivity and immunity to racism and imperialism. This leaves me with a gaping chasm of doubt about my views regarding the forgoing news. Maybe I really don’t know how bad things really are and what radical means of resistance or revolution are required for a true change for the better.
😬😱
A week later, some big wigs have weighed in. One unavoidable observation is that University of Washington prof is taking Jeff Dean’s side. Perhaps Dean’s matriculation there had this kind of politics. behind it as well?!
Here is one question is if you were behind the veil of ignorance, and you don’t know if you are minority or not, you don’t know if you’re rich or not and you don’t know if you have knowledge or not, would you:
• Want to work with someone like Gebru or Dean.
• Would you want to manage Gebru or Dean?
• Would you want to be managed by Gebru or Dean?
• Would you cite Gebru or Dean regarding the safety and ethics of Google inc taking on the same position as their paper?
• Would you follow either one’s leadership in terms of ethics or social justice.
I cannot imagine myself wanting to be any where near either of these two characters. I would probably have to cite both opposing opinions since BOTH of them have knowledge about the matter far beyond my cognition and experience. But honestly, I do have to factor into consideration that Dean is really protecting a possibly very evil industry that he was instrumental in creating. I also must factor into the citation that Gebru is highly leveraged in the Identity Politics market. One cannot conscientiously discuss the very polarizing political topic without acknowledging these objectively existing aspects of these characters. In either case, although I fear for my own personal safety and sanity to come into proximity of these people, I do have to say that they are highly valuable assets of our society. Their existence enriches us all, and quite possibly quantified in similar orders of magnitude. IMMHO, this is great!
… some time later… 🎇 The formation of a minority union is a nice touch. Hope to see more…
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The Rapid Cycling Synchrotron (RCS), whose beam energy ranges from 400 MeV to 3 GeV and which is located in the Japan Proton Accelerator Research Complex, is a kicker-impedance-dominated machine, which violates the impedance budget from a classical viewpoint. Contrary to conventional understanding, we have succeeded in accelerating a 1 MW equivalent beam. The machine has some interesting features: e.g., the beam tends to be unstable for the smaller transverse beam size and the beam is stabilized by increasing the peak current. Space charge effects play an important role in the beam instability at the RCS. In this study, a new theory has been developed to calculate the beam growth rate with the head-tail and coupled-bunch modes $(m,μ)$ while taking space charge effects into account. The theory sufficiently explains the distinctive features of the beam instabilities at the RCS.
## 1. Introduction
The 3 GeV Rapid Cycling Synchrotron (RCS) at the Japan Proton Accelerator Research Complex [1] aims to achieve a megawatt-class beam. Two bunched beams ($4.15×1013$ particles per bunch) are accelerated from 400 MeV to 3 GeV with a repetition rate of 25 Hz. To avoid the effects of eddy currents on metal chambers [2,3], ceramic chambers are adopted instead [4,5]. Accordingly, the resistive wall impedance [510] is negligible in the RCS [1113].
However, there has been some concern that the kicker impedance limits the beam intensity of the RCS [14] by exciting beam instabilities [6]. Precise offline and online measurements of the impedance show that the kicker at the RCS has a huge impedance [15]. The offline measurement is conducted using the standard wire method [7], while the online one is conducted by observing the beam induced-voltage at the end of the power cable [15]. The results of these two independent measurements agree with each other. Finally, we demonstrate that the RCS is a kicker-impedance-dominated machine; we show this by suppressing the beam growth rate in accordance with the reduction of the kicker impedance [12,13].
In general, when we design accelerators, a lower impedance source along the rings is preferable for achieving a high intensity beam. Concretely, ceramic chambers are adopted, the chambers are connected as smoothly as possible over the rings [16], and significant efforts are made to lower the kicker impedances [12,17].
The conventional Sacherer formula [18,19] estimates the beam growth rate by using the impedances as an input parameter. However, such estimation differs significantly from the measured results at the RCS. We suspected the main reason for this is that the formula neglects space charge effects. Because the RCS covers the intermediate energy region (from 400 MeV to 3 GeV), space charge should have an effect on the beam instability.
Other theories exist to assess beam instability that includes space charge effects [2022]. However, those theories assume simple forms of impedance, e.g., resistive wall impedance, resonator type impedance with a single resonance frequency, and constant wakes. Moreover, the theories do not include coupled-bunch-type instabilities.1
In this paper, we develop a new theory that includes coupled-bunch and head-tail instabilities with space charge effects based on the Vlasov equation [6,23]. Using this theory, we try to understand the parameter dependence (the transverse emittance dependence, the beam peak current dependence, the tune dependence, etc.) of the beam instability observed at the RCS.
In Sect. 2, we start with the Hamiltonian and construct the Vlasov equation [6]. In Sect. 2.1, we derive a dispersion relation with the head-tail and coupled-bunch modes $(m,μ)$ that includes space charge effects. In Sect. 2.2, we reproduce the previous Sacherer formula by neglecting the space charge effects.
In Sect. 3, typical parameters at the RCS are shown, and we show that the observed beam instability cannot be explained at all using the classical theory, i.e., Sacherer's theory [18,19], where space charge effects are neglected.
In Sect. 4, the beam instability observed at the RCS is analyzed using our new theory. In Sect. 4.1, the space charge effects on the beam instability are investigated by comparing the measurements with the theoretical results. In Sect. 4.2, tune manipulations are discussed from both the theoretical and the experimental viewpoints. The paper is summarized in Sect. 5.
In Appendix A, the scalar potential describing the space charge effect is calculated by solving the Poisson equation with the boundary condition of being surrounded by a perfectly conductive cylindrical chamber. In Appendix B, we explain canonical transformations to derive the Hamiltonian describing nonlinear betatron oscillation by using action-angle variables from the original Hamiltonian given by Eq. (1) in the next section.
## 2. Linearized Vlasov approach
The linearized Vlasov approach is a standard theoretical method to analyze beam instabilities [6]. In Sect. 2.1, the linearized Vlasov equation converts to a dispersion relation as a powerful tool to discuss the space charge effect on the beam instability. In Sect. 2.2, the classical Sacherer formula is reproduced, based on the linearized Vlasov equation.
### 2.1. Dispersion relation with the head-tail and coupled-bunch modes that include space charge effects
Here, we present the dispersion relation, by which we calculated the beam growth rate; this method takes into account the coupled mode $μ$, head-tail mode $m$, and space charge effects.
The original Hamiltonian is given by [24,25]
(1)
$Ho=−ps(1+xρ)ΔEpsβsc+ps2γs2(ΔEpsβsc)2+px2+py22ps+ps2Kx(s)(1−ΔEEs)x2 +ps2Ky(s)(1−ΔEEs)y2−psxEsFx+eΦc(x,y,s−cβst)βsγs2c −eVrfωRFδp(s)cos(ωRFt−hsR+φs)+…,$
where $φs$ is the synchronous phase; $ps$ is the constant longitudinal momentum of the synchronous particle; $Es=cps/βs$ is the particle energy on the designed orbit; $βs$ and $γs$ are the Lorentz-$β$ and the Lorentz-$γ$ of the designed particle, respectively; $ΔE$ is given by $ΔE=E−Es$; $Fx$ is the transverse wake force; $δp(s)$ is the periodic $δ$-function; $c$ is the velocity of light; $Kx$ and $Ky$ are the periodic focusing forces in the horizontal and the vertical directions, respectively; $Φc$ is the space charge potential felt by the bunch center [26]; $h$ is harmonic number; $Vrf$ is the amplitude of the radio frequency (RF) voltage; $1/ρ$ is the local curvature around the machine; $R$ is the average radius of the machine; and $ωRF$ is the angular frequency of the RF voltage, which is expressed as
(2)
$ωRF=cβshR.$
The orbit length $s$ is used as an independent variable. The canonical variables are $(x,px)$, $(y,py)$, and $(t,−E)$ for the horizontal, vertical, and longitudinal directions, respectively. The scalar potential $Φc$ is obtained by solving the Poisson equation with the boundary condition that the beam is surrounded by a cylindrical, perfectly conductive chamber with radius $a$. The solution is expressed in Appendix A.
Successive canonical transformations convert Eq. (1) to the Hamiltonian with action-angle variables, to describe the nonlinear betatron motions. The derivation is explained in detail in Appendix B. From now on, we consider only the horizontal and the longitudinal motions of the beam, for simplicity. Finally, the Hamiltonian is given by
(3)
$H≃QxJx+νs0JL+Ux+Y′,$
with the horizontal $(Jx,ψx)$ and the longitudinal action-angle variables $(JL,ϕL)$, and its independent variable $θ=s/R$, where $Qx$ and $νs0$ are the horizontal and the synchrotron tunes, and
(4)
$Ux=−2βsRc((βx(s)Jx2ps)1/2cos(ψx+ϕx(s)−QxRs)−D(s)βs(ω0νs0JL2Es|η|)1/2cosϕL)Fx,$
(5)
$ϕx(s)=∫sdsβx(s),$
(6)
$Y′=Ycoh,0′(JL)+Y′coh,2(JL)βx(s)Jxps+Y′coh,4(JL)3βx2(s)Jx22,$
(7)
$Ycoh,0′(JL)=eRZ0eNb(γ˜−Ei[−a22σx2]+log[a22σx2])4π2βsγs2σz(π2)1/2exp(−c2JL|η|2Esh2νs0σz2ω0) ×I0(c2JL|η|2Esh2νs0σz2ω0),$
(8)
$Y′coh,2(JL)=−eRZ0eNb8π2βsγs2σx2(π2)1/2exp(−c2JL|η|2Esh2νs0σz2ω0)[−σx2(γ˜−Ei[−a22σx2]+log[a22σx2])2γs2σz5 ×((σz2−c2JL|η|Esh2νs0ω0)I0(c2JL|η|2Esh2νs0σz2ω0)+c2JL|η|I1(c2JL|η|2Esh2νs0σz2ω0)Esh2νs0ω0) +(1−exp(−a22σx2))I0(c2JL|η|2Esh2νs0σz2ω0)σz]+eRZ0eNbexp(−c2JL|η|2Esh2νs0σz2ω0)42ππβsγs2σx4σz ×(σx2+2σx4(exp(−a22σx2)−1)a2)I0(c2JL|η|2Esh2νs0σz2ω0) −eRZ0eNb8π2βsγs2σx2σz(π2)1/2exp(−c2JL|η|2σz2h2ω0Esνs0)I0(c2JL|η|2σz2h2ω0Esνs0),$
(9)
$Y′coh,4(JL) =−eRZ0eNbexp(−c2JL|η|2Esh2νs0σz2ω0)16π2σx2βsγs2ps2(π2)1/2{−σx2(γ˜−Ei[−a22σx2]+log[a22σx2])2 ×[(3σz4−6c2JLσz2|η|Esh2νs0ω0+2c4JL2|η|2Es2h4νs02ω02)I0(c2JL|η|2Esh2νs0σz2ω0)−2c2JL|η|(−2σz2+c2JL|η|Esh2νs0ω0)I1(c2JL|η|2Esh2νs0σz2ω0)Esh2νs0ω08γs4σz9 −(σz2−c2JL|η|Esh2νs0ω0)I0(c2JL|η|2Esh2νs0σz2ω0)+c2JL|η|I1(c2JL|η|2Esh2νs0σz2ω0)Esh2νs0ω0σx2γs2σz5+I0(c2JL|η|2Esh2νs0σz2ω0)σx4σz] +[1−exp(−a22σx2)−γ˜+Ei[−a22σx2]−log[a22σx2]] ×[(σz2−c2JL|η|Esh2νs0ω0)I0(c2JL|η|2Esh2νs0σz2ω0)+c2JL|η|Esh2νs0ω0I1(c2JL|η|2Esh2νs0σz2ω0)2γs2σz5−I0(c2JL|η|2Esh2νs0σz2ω0)σx2σz] +(6−(a2σx2+6)exp(−a22σx2)−4γ˜+4Ei[−a22σx2]−4log[a22σx2])I0(c2JL|η|2Esh2νs0σz2ω0)8σx2σz} +eRZ0eNbexp(−c2JL|η|2Esh2νs0σz2ω0)8π2σx4βsγs2ps2(π2)1/2 ×{(8σx2−exp(−a22σx2)(a4σx2+4a2+8σx2))I0(c2JL|η|2Esh2νs0σz2ω0)4a2σz +σx2[2σx2−(a2+2σx2)exp(−a22σx2)][(14σx2+1a2)I0(c2JL|η|2Esh2νs0σz2ω0)σx2σz −(σz2−c2JL|η|Esh2νs0ω0)I0(c2JL|η|2Esh2νs0σz2ω0)+c2JL|η|I1(c2JL|η|2Esh2νs0σz2ω0)Esh2νs0ω04a2γs2σz5]+σx2[1−exp(−a22σx2)] ×[−I0(c2JL|η|2Esh2νs0σz2ω0)σx2σz+(σz2−c2JL|η|Esh2νs0ω0)I0(c2JL|η|2Esh2νs0σz2ω0)+c2JL|η|I1(c2JL|η|2Esh2νs0σz2ω0)Esh2νs0ω04γs2σz5]} +eRZ0eNbexp(−c2JL|η|2Esh2νs0σz2ω0)32π2σx4βsγs2σzps2(π2)1/2(1−8σx4−exp(−a22σx2)(4a2σx2+8σx4)a4) ×I0(c2JL|η|2Esh2νs0σz2ω0)+eRZ0eNbβsγs2π2σx2ps2(π2)1/2exp(−c2JL|η|2h2ω0Esνs0σz2) ×[(−1128γs2σz3+164σx2σz+c2JL|η|128γs2σz5h2ω0Esνs0)I0(c2JL|η|2h2ω0Esνs0σz2) −c2JL|η|I1(c2JL|η|2h2ω0Esνs0σz2)128γs2σz5h2ω0Esνs0];$
$η$ is slippage factor; $Z0=120π Ω$ is the impedance of free space; $ω0$ is the angular revolution frequency; $βx(s)$ is the Twiss parameter; $D(s)$ is the dispersion function; $In(x)$ is the modified Bessel function; $Ei[z]$ is the exponential integral function [27]; $γ˜$ is Euler-$γ$; and $σx$ and $σz$ are the root mean square (rms) horizontal and longitudinal beam sizes, respectively. The potential functions $Ux$ and $Y′$ originate from the horizontal wake and the space charge forces, respectively.
Here, let us consider the situation that $M$ buckets are filled with $M$ bunches in a ring. We denote by $Ψn(θ,Jx,ψx,JL,ϕL)$ the phase space distribution function of the $n$th bunch among $M$ bunches.
The Vlasov equation is expressed as
(10)
$∂Ψn∂θ+Jx′∂Ψn∂Jx+ψx′∂Ψn∂ψx∂ψy+JL′∂Ψn∂JL+ϕL′∂Ψn∂ϕL=0,$
where the prime denotes differentiation with respect to the variable $θ$.
The distribution function $Ψn$ is decoupled into an unperturbed part $F0(Jx)G0(JL)$ and a perturbed part $f1(Jx,ψx)g1(JL,ϕL)$ as
(11)
$Ψn=F0(Jx)G0(JL)+f1(Jx,ψx)g1(JL,ϕL)exp(jνθ−jν2πnM−jQxξxϕcphη−j2πμnM),$
where $Qxξx$ is the chromaticity in the horizontal direction and $Ψn$ is normalized according to
(12)
$∫0∞dJx∫−ππdψx∫0∞dJL∫−ππdψLΨn(Jx,ψx,JL,ϕL)=1.$
In advance, let us formulate the dipole current $Dp$ of the beam (and its Fourier transform $D˜(p)$) and the horizontal wake force $Fx$ (i.e., the potential $Ux$) not only in the time domain but also in the frequency domain. The horizontal wake force $Fx$ is obtained by the summation of the wake force induced by the previous passage of beams. It is expressed as
(13)
$Fx=e2Nb2πR∫0∞dJ′x∫−ππdψ′x∫0∞dJ′L∫−ππdϕ′Lx′f1(J′x,ψ′x)g1(J′L,ϕ′L) ×∑n′=0M−1∑k=−∞∞exp(jνθ+jν(−2πn′M−2πk)−jQxξxϕcp′hη)exp(−jμ2πn′M) ×WT(ϕcp′h−ϕcph+2πk+2π(n′−n)M),$
where $WT(s)$ is the horizontal wake function, which has the causality condition $WT(s)=0$ for $s≤0$, and $ϕcp$ denotes the longitudinal position of beam, which is related to $JL$ and $ϕL$ as
(14)
$ϕcp=(2JLh2|η|ω0cpsβsνs0)1/2sinϕL,$
(see Eq. (B31)). The effect of the wake excited by all previous revolutions of beams is included as a summation over $k$ in Eq. (13). The dipole current $Dp$, its Fourier transform $D˜(p)$, and the horizontal impedance $ZT(ω)$ are defined as
(15)
$Dp(JL,ϕL)=∫0∞dJx∫−ππdψxx(Jx,ψx,JL,ϕL)f1(Jx,ψx)g1(JL,ϕL)βx,$
(16)
$D˜(p)=12π∫0∞dJL∫−ππdϕLDp(JL,ϕL)exp(jpϕcph),$
and
(17)
$WT(ω0t)=−∫−∞∞dω2πjZT(ω)exp(jωω0ω0t),$
respectively, where
(18)
$x(Jx,ψx,JL,ϕL)=(2βx(s)Jxps)1/2cos(ψx+ϕx(s)−Qxθ) −D(s)psc(2νs0ω0EsJL|η|)1/2cosϕL,$
and $j$ is the imaginary unit (the definitions of the causality condition of the wake function and of the impedance in Ref. [6] are different from those in this paper (see Eqs. (13) and (17))).
Substituting Eqs. (15) and (17) into Eq. (13) and using Poisson's sum rule [6],
(19)
$∑k=−∞∞exp(jkx)=2π∑p=−∞∞δ(x−2πp),$
where $δ(x)$ is the $δ$-function, the wake force $Fx$ is rewritten as
(20)
$Fx=−je2NbβxT0Rexp(jνθ−jν2πnM)∑n′=0M−1∑p=−∞∞D˜(ν+p−Qxξxη) ×exp(−j2πμn′M+jp2π(n′−n)M)ZT(ω0(Qx+p))exp(−j(ν+p)ϕcph),$
in frequency domain, where $T0$ is the revolution time of the designed particle. Now, the potential $Ux$ in Eq. (4) can be expressed as
(21)
$Ux=je2Nbβs2βxπRexp(jνθ−jν2πnM) ×((βx(s)Jx2ps)1/2cos(ψx+ϕx(s)−QxRs)−Dβs(ω0νs0JL2Es|η|)1/2cosϕL) ×∑n′=0M−1∑p=−∞∞D˜(ν+p−Qxξxη)exp(−j2πμn′M+jp2π(n′−n)M) ×ZT(ω0(ν+p))exp(−j(ν+p)ϕcph).$
Second, we introduce the Fourier transforms of the perturbed parts $f1$ and $g1$ as
(22)
$f1(Jx,ψx)=∑qf˜1,q(Jx)exp(−jqψx),$
(23)
$g1(JL,ϕL)=∑mg˜1,m(JL)exp(−jmϕL).$
Substituting Eqs. (22)–(23) into Eq. (15), the dipole current $Dp$ is rewritten as
(24)
$Dp(JL,ϕL) =∑q′,m′∫0∞dJx∫−ππdψxx(Jx,ψx,JL,ϕL)f˜1,q′(Jx)g˜1,m′(JL)βxexp(−jq′ψx−jm′ϕL).$
Here, we introduce $Dpq′,l′,m′(JL)$ as
(25)
$Dp(JL,ϕL)=∑q′,m′Dpq′,m′(JL)exp(−jm′ϕL),$
so that
(26)
$Dpq′,m(JL)=∑m′∫−ππdϕL∫0∞dJx∫−ππdψx ×f˜1,q′(Jx)g˜1,m′(JL)x(Jx,ψx,JL,ϕL)2πβxexp(−jq′ψx−j(m′−m)ϕL).$
Then, Eq. (16) is expanded by $Dpq,m$ as
(27)
$D˜(p)=12π∫0∞dJL∫−ππdϕLDp(JL,ϕL)exp(jpϕcph)=∑q,m∫0∞dJ′LDpq,m(J′L)Jm*[pωRFh(2J′L|η|cpsβsνs0ω0)1/2] ,$
by using the relation
(28)
$∫−ππexp(−jqϕcpωRF+jmϕL)dϕL=∫−ππexp(−jq(2JL|η|cpsβsνs0ω0)1/2sinϕL+jmϕL)dϕL=2πJm[q(2JL|η|cpsβsνs0ω0)1/2],$
where $∗$ denotes the complex conjugate and $Jm[x]$ is the Bessel function [27].
The equations of motion are given by
(29)
$dJxdθ=−∂H∂ψx,$
(30)
$dψxdθ=∂H∂Jx,$
(31)
$dJLdθ=−∂H∂ϕL,$
(32)
$dϕLdθ=∂H∂JL,$
where the Hamiltonian is given by Eq. (3). By substituting Eqs. (11) and (29)–(32) into Eq. (10), and by retaining only the perturbed parts, the linearized Vlasov equation is obtained as
(33)
$(jν−jQxξxhηνL(Jx,JL)ωRF(2|η|RJLβs3cEsνs0)1/2cosϕL)f1(Jx,ψx)g1(JL,ϕL) +je2Nbβs2βx(s)2psπRJxsin(ψx+ϕx(s)−Qxθ) ×∑n′=0M−1∑p=−∞∞D˜(ν+p−Qxξxη)exp(−j2πμn′M+jp2π(n′−n)M)ZT(ω0(ν+p)) ×exp(−j(ν+p)ϕcph)∂F0(Jx)∂JxG0(JL)exp(jQxξxϕcphη+j2πμnM) +νx(Jx,JL)∂f1(Jx,ψx)∂ψxg1(JL,ϕL)+νL(Jx,JL)f1(Jx,ψx)∂g1(JL,ϕL)∂ϕL=0,$
where we assume
(34)
$dG0(JL)dJL≃0,$
and
(35)
$νL(Jx,JL)=νs0+dY′dJL,$
(36)
$νx(Jx,JL)=Qx+dY′dJx.$
Here, let us substitute Eqs. (22)–(23) into Eq. (33) before it is multiplied by $exp(jq′ψx+jm′ϕL)$. By integrating the result over $ψx$ and $ϕL$, we obtain approximately
(37)
$f˜1,q′(Jx)g˜1,m′(JL) ≃je2Nbβs2βx(s)Jx(exp(j(ϕx(s)−Qxθ))δq′,−1−exp(−j(ϕx(s)−Qxθ))δq′,1)exp(j2πμnM)22psπR[ν−m′νL(Jx,JL)−q′νx(Jx,JL)] ×∑n′=0M−1∑p=−∞∞D˜(ν+p−Qxξxη)exp(−j2πμn′M+jp2π(n′−n)M)ZT(ω0(ν+p)) ×Jm′[((ν+p)ωRFh−QxξxωRFhη)(2JL|η|cpsβsνs0ω0)1/2]∂F0(Jx)∂JxG0(JL),$
under the condition
(38)
$1≫(−2ω0JLν0sβs2Esη)1/2|Qxξx|QxνL(Jx,JL),$
where we use the relation Eq. (28).
Multiplying the factor
(39)
$x(Jx,ψx,JL,ϕL)2πβxexp(−jq′ψx−j(m′−m)ϕL)$
by Eq. (37), before integrating the result over $ϕL$, $Jx$, $ψx$, $Jy$, $ψy$, and summing it over $m′$, we derive
(40)
$Dpq′,m(JL) =∑m′∫−ππdϕL∫0∞dJx∫−ππdψxx(Jx,ψx,JL,ϕL)2πβxexp(−jq′ψx−j(m′−m)ϕL) ×je2Nbβs2βx(s)Jx(exp(j(ϕx(s)−Qxθ))δq′,−1−exp( −j(ϕx(s)−Qxθ))δq′,1)exp( j2πμnM )22psπR[ν−m′νL(Jx,JL)−q′νx(Jx,JL)] ×∑n′=0M−1∑p=−∞∞D˜(ν+p−Qxξxη)exp(−j2πμn′M+jp2π(n′−n)M)ZT(ω0(ν+p)) ×Jm′[((ν+p)ωRFh−QxξxωRFhη)(2JL|η|cpsβsνs0ω0)1/2]∂F0(Jx)∂JxG0(JL).$
Then, by substituting Eqs. (18) and (27) into Eq. (40), Eq. (40) is rewritten as
(41)
$Dpq′,m′(JL) =∑q,m∫0∞dJxje2Nbβs2βx(s)Jx(δq′,−1−δq′,1)exp(j2πμnM)2psR[ν−m′νL(Jx,JL)−q′νx(Jx,JL)] ×∑n′=0M−1∑p=−∞∞∫0∞dJ′LDpq,m(J′L)exp(−j2πμn′M+jp2π(n′−n)M)ZT(ω0(ν+p)) ×∂F0(Jx)∂JxG0(JL)Jm*[(ν+p−Qxξxη)ωRFh(2J′L|η|cpsβsνs0ω0)1/2] ×Jm′[(ν+p−Qxξxη)ωRFh(2JL|η|cpsβsνs0ω0)1/2] .$
Here, let us introduce the function $Dm$ as
(42)
$Dm(JL)=Dp1,m(JL)+Dp−1,m(JL).$
When we retain only the diagonal terms, Eq. (41) is simplified by using the function $Dm$ as
(43)
$Dm(JL) =−je2NbMβs2βx(s)G0(JL)2psR ×∫0∞dJx[1[ν−mνL(Jx,JL)−νx(Jx,JL)]−1[ν−mνL(Jx,JL)+νx(Jx,JL)]] ×Jx∂F0(Jx)∂Jx∑p=−∞∞ZT(ω0(ν+μ+pM))Jm[(ν+μ+pM−Qxξxη)(2JLω0|η|cpsβsνs0)1/2] ×∫0∞dJ′LDm(J′L)Jm*[(ν+μ+pM−Qxξxη)(2J′Lω0|η|cpsβsνs0)1/2] .$
If we choose the functions $Dm(JL)$, $G0(JL)$, and $F0(JL)$ as
(44)
$Dm(JL)=Bmδ(JL−JL0),$
(45)
$G0(JL)=12πδ(JL−JL0),$
(46)
$F0(Jx)=12πJx0exp(−JxJx0),$
where
(47)
$Jx0=βsEsϵx,rmsc,$
and $ϵx,rms$ is the root mean square (rms) emittance of the beam, and expand Eq. (43) around small $Jx$ before integrating Eq. (43) over $Jx$, we finally obtain the dispersion relation as
(48)
$1≃ −je2NbMπβs2〈βx(s)〉[1+(ν−mνL0−νX0)(mdνLdJx+dνxdJx)Jx0exp(−(ν−mνL0−νX0)(mdνLdJx+dνxdJx)Jx0)Γ[0,−(ν−mνL0−νX0)(mdνLdJx+dνxdJx)Jx0]]8π3Jx0psR(mdνLdJx+dνxdJx) ×∑p=−∞∞|Jm[(ν+μ+pM−Qxξxη)(2ω0JL0|η|cpsβsνs0)1/2]|2ZT(ω0(ℜ[ν]+μ+pM)) ,$
where
(49)
$νL0=νs0+dYcoh,0′(JL)dJL|JL=JL0,$
(50)
$νX0=Qx+〈βx(s)〉psYcoh,2′(JL0),$
(51)
$mdνLdJx+dνxdJx≃mdYcoh,2′(JL)dJL|JL=JL0〈βx(s)〉ps+3〈βx2(s)〉Ycoh,4′(JL0),$
$νL0$ and $νX0$ are the coherent synchrotron and the betatron tunes, respectively, $Γ[0,z]$ is the incomplete $Γ$-function [27], and $〈⋯〉$ denotes the average value around the ring. Here we assume the rms beam sizes $σx$ and $σz$ are given as
(52)
$σx=(〈βx(s)〉ϵx,rms+〈D2(s)〉(Δpp)2)1/2=(〈βx(s)〉cJx0βsEs+〈D2(s)〉2JL0νs0ω0Esβs2|η|)1/2,$
(53)
$σz=cω0(2JL0|η|ω0Esνs0)1/2,$
considering Eqs. (14), (47), (B17), (B18), and (B32). For reference, Fig. 1 illustrates typical behavior of the functions $dY′coh,0(JL0)/dJL0$, $Y′coh,2(JL0)$, $dY′coh,2(JL0)/dJL0$, and $Y′coh,4(JL0)$ in Eq. (48), which are calculated by using the beam parameters at the ramping time 15 ms in the RCS (refer to Sect. 3).
Fig. 1.
Typical behavior of the functions $dY′coh,0(JL0)/dJL0$ (left top), $Y′coh,2(JL0)$ (right top), $dY′coh,2(JL0)/dJL0$ (left bottom), and $Y′coh,4(JL0)$ (right bottom) calculated by using the beam parameters at the ramping time 15 ms in the RCS.
Fig. 1.
Typical behavior of the functions $dY′coh,0(JL0)/dJL0$ (left top), $Y′coh,2(JL0)$ (right top), $dY′coh,2(JL0)/dJL0$ (left bottom), and $Y′coh,4(JL0)$ (right bottom) calculated by using the beam parameters at the ramping time 15 ms in the RCS.
The beam growth rate and the coherent tune affected by the wake force, which are given by the real parts of $jω0ν$ and of $ν$, respectively, are solved by Eq. (48) as a function of nominal tune $Qx$. The differences among $Qx$, $νX0$, and $ℜ[ν]$ are tiny ($≲0.01$) in a practical situation.
### 2.2. The Sacherer formula
Here, we reproduce the classical Sacherer formula [18,19,23], where the space charge effect on the beam oscillations is neglected. In this case, the $Jx$ and $JL$ dependence of the tunes $νL$ and $νx$ vanishes. If we confine ourselves to the case, the $Jx$-integration in Eq. (43) can be performed for the distribution Eq. (46). Consequently, Eq. (43) is simplified as
(54)
$Dm(JL)=je2NbMβx(s)G0(JL)4πpsR(1ν−mνs0−Qx−1ν−mνs0+Qx) ×∑p=−∞∞ZT(ω0(ν+μ+pM))Jm[(ν+μ+pM−Qxξxη)(2JLω0|η|cpsνs0)1/2] ×∫0∞dJ′LDm(J′L)Jm*[(ν+μ+pM−Qxξxη)(2J′Lω0|η|cpsνs0)1/2],$
for an ultra-relativistic beam ($βs=1$). Here, following the conventional manner, let us replace the action-variable $JL$ with the amplitude-variable $rs$:
(55)
$rs=1ω0(2JLω0|η|cpsνs0)1/2,$
and assume the unperturbed distribution function $G0(JL(rs))$ as
(56)
$G0(JL(rs))=|η|πω0cpsτ0s2νs0Θ(τ0s−rs),$
where $Θ(x)$ is the step function and $τs0$ denotes the half-bunch length. Accordingly, Eq. (54) is rewritten as
(57)
$Dm(JL(rs))=jce2NbM4π2EsQxτ0s2Θ(τ0s−rs)(1ν−mνs0−Qx−1ν−mνs0+Qx) ×∑p=−∞∞ZT(ω0(ν+μ+pM))Jm[ω0(ν+μ+pM−Qxξxη)rs] ×∫0∞dr′sr′sDm(J′L(r′s))Jm*[ω0(ν+μ+pM−Qxξxη)r′s] ,$
where $βx(s)$ is replaced by $R/Qx$.
Let us expand the function $Dm(JL(rs))$ using a complete set of orthogonal functions $fk(m)(rs)$ as
(58)
$Dm(JL(rs))=W˜(rs)∑k=0∞ak(m)fk(m)(rs)≡∑k=0∞ak(m)gm,k(rs),$
where $k$ is the radial mode and $W˜(rs)$ is the weight function. The functions $fk(m)(r)$ and $gm,k(r)$ satisfy the orthogonality relationship
(59)
$∫0∞W˜(rs)fk(m)(rs)fl(m)(rs)rsdrs=δkl,$
(60)
$∫0∞gm,k(rs)gm,l(rs)W˜(rs)rsdrs=δkl,$
respectively. Here, the weight function $W˜(rs)$ is defined as
(61)
$W˜(rs)=CηNbπτ0s2ω0νs0Θ(τ0s−rs),$
where $C$ is a normalization constant. Accordingly, the functions $fk(m)(rs)$ or $gm,l(rs)$ can be revealed as
(62)
$fk(m)(rs)=(2W˜)1/2Jm(μmkrsτ0s)τ0sJm+1(μmk), for rs<τ0s,$
(63)
$gm,l(rs)=(2W˜(rs))1/2Jm(μmlrsτ0s)τ0sJm+1(μml)=(2CηRNbπcνs0τ0s2)1/2Jm(μmlrsτ0s)τ0sJm+1(μml)(1−Θ(rs−τ0s)),$
where $μmk$ is the $k$th zero of $Jm(x)$.
Let us introduce the particle distribution function $ρm,l(τ)$ with head-tail mode $m$ and radial mode $l$ in real space as
(64)
$ρm,l(τ)=−∫−∞∞gm,l(rs)exp(−jmϕL)ωRFdWEs≡∫−∞∞gm,l(rs)exp(−jmϕL)dδ,$
and its Fourier transform
(65)
$ρ˜m,l(k)=∫−∞∞dτ2πexp(jkτ)ρm,l(τ),$
where $τ=ϕcp/ωRF$ and $W$ is the momentum conjugate to $ϕcp$ (see Eq. (B17)).
Substituting Eq. (64) into Eq. (65), Eq. (65) is written as
(66)
$ρ˜m,l(k)=∫0∞gm,l(r)ω0νs0ηjmJm(kr)r dr,$
(see Eqs. (55), (B31) and (B32)) by using the relation
(67)
$12π∫02πdφexp(ilφ−ixcosφ)=j−lJl[x],$
while its inverse transform $ρm,l(τ)$ is given by
(68)
$ρm,l(τ)=∫gm,l(r)ω0νs0ηjmJm(kr)exp(−jkτ)r dr dk,$
which is expressed as
(69)
$ρm,l(τ)=(2CRNbνs0πcη)1/2ω0μmljm∫0∞dk[((−1)m+1)coskτ+((−1)m−1)jsinkτ]Jm(kτ0s)(μml2−k2τ0s2),$
for the function given by Eq. (63). Equation (66) satisfies the relationship $ρ˜m,l*(k)=(−1)mρ˜m,l(k)$ for real $gm,l(r)$ (see Eq. (91)).
By substituting Eq. (58) into Eq. (57), in combination with Eqs. (60) and (66), Eq. (57) is solved as
(70)
$νm,l=Qx+mνs0+je2cηCm,l8π2EsQxCω02νs0 ×∑p=−∞∞ZT(ω0(ν+μ+Mp))h′m,l(ω0(Qx+mνs0+μ+Mp−Qxξη))∑p=−∞∞h′m,l(ω0(Qx+mνs0+μ+Mp−Qxξη)) ,$
in the lowest-order approximation, where we define the constant $Cm,l$ and the function $h′m,l(ω)$ as
(71)
$Cm,l=∫−∞∞dτ|ρm,l(τ)|2=2π∫−∞∞dω|ρ˜m,l(ω)|2,$
(72)
$h′m,l(ω)≡|ρ˜m,l(ω)|2,$
respectively. In Eq. (70), the $ω$-integration is approximated by the summation of $p$.
The constants $C$ and $Cml$ are determined as follows. If we impose the condition $ρm,l(±τ0s)=0$ on the distribution function, the function $ρm,l(τ)$ should be written as
(73)
$ρm,l(τ)=∑p=1,3,5,…Dpcosπp2τ0sτ+∑p=2,4,6,…Epsinπp2τ0sτ,$
where $Dp$ and $Ep$ are expansion coefficients. By equating Eq. (69) to Eq. (73), they are expressed as
(74)
$Dp=−pπω0μmlτ0s4(2CRNbνs0πcη)1/2(−1)m/2 (−1)(p+1)/2ℜ[∫−∞∞dkJm(kτ0s)exp(jkτ0s)(k2−μml2τ0s2)(k2−(πp2τ0s)2)],for m=0,2,4,…,$
(75)
$Ep=−pπω0μmlτ0s4(2CRNbνs0πcη)1/2(−1)(m−1)/2 (−1)p/2ℑ[∫−∞∞dkJm(kτ0s)exp(jkτ0s)(k2−μml2τ0s2)(k2−(πp2τ0s)2)],for m=1,3,5,…,$
where we use
(76)
$∫−τ0sτ0scoskτcosπpτ2τ0sdτ=(−1)(p+1)/2pπτ0s(k2−(πp2τ0s)2)coskτ0s, for odd p,$
(77)
$∫−τ0sτ0ssinkτsinπpτ2τ0sdτ=(−1)p/2pπτ0s(k2−(πp2τ0s)2)sinkτ0s, for even p,$
(78)
$Jm(−x)=(−1)mJm(x).$
By picking up the residues, the $k$-integration in Eqs. (15) and (75) is performed. As a result, we obtain
(79)
$Dp=ω0τ0s(πCRNbνs0cη)1/2(−1)m/222μmlJm(πp2)(μml2−(πp2)2), for even m,$
(80)
$Ep=ω0τ0s(πCRNbνs0cη)1/2(−1)(m−1)/222μmlJm(πp2)(μml2−(πp2)2), for odd m.$
Finally, $ρm,l(τ)$ is summarized as
(81)
$ρm,l(τ)=∑pAlpmbp(τ),$
(82)
$bp(τ)={cosπpτ2τ0s,for p=1,3,5,…,sinπpτ2τ0s,for p=2,4,6,…,$
(83)
$Alpm=ω0τ0s(πCRNbνs0cη)1/2Pm,p,l{(−1)m/2,for m=0,2,4,…,(−1)(m−1)/2,for m=1,3,5,…,$
(84)
$Pm,p,l=22μmlJm(πp2)(μml2−π2p24),$
where $p$ runs $1,3,5,…$ for $m=0,2,4,…,$ and $p$ runs $2,4,6,…$ for $m=1,3,5,….$
Here, let us focus on the lowest-order term for the radial mode $l=1$. The factor
(85)
$8μm12Jm2(π(m+1)2)(μm,12−π2(m+1)24)2,$
which appears in Eq. (70), dominates for the component $m+1=p$. Then, the function $ρm,l=1(τ)$, its Fourier transform $ρ˜m,l=1(ω)$, and the factor $Pm,p=m+1,l=1$ are approximated as
(86)
$ρm,l=1(τ)={(−1)m2cosπ(m+1)τ2τ0s,for m=0,2,4,…,(−1)m−12sinπ(m+1)τ2τ0s,for m=1,3,5,…,$
(87)
$ρ˜m,l=1(ω)={(−1)m/2∫−τ0sτ0sdτ2πexp(jωτ)cosπ(m+1)τ2τ0s=2τ0s(1+m)cosωτ0sπ2[(1+m)2−4ω2τ0s2π2], for m=0,2,4,…,(−1)(m−1)/2∫−τ0sτ0sdτ2πexp(jωτ)sinπ(m+1)τ2τ0s=j(1+m)2τ0ssinωτ0sπ2[(1+m)2−4ω2τ0s2π2], for m=1,3,5,…,$
and
(88)
$Pm,p=m+1,l=1=22μm,1Jm(π(m+1)2)(μm,12−π2(m+1)24)≃16(3+2m)(5+4m)π2m+1∼1m+1,$
respectively. Substituting Eq. (86) into Eq. (71), we finally obtain
(89)
$Cm,l=τ0s.$
The constant $C$ is determined by the condition $Al=1,p=m+1m=1$. As a result, it is calculated as
(90)
$C=ηc(m+1)τ0s2ω02νs0πRNb.$
Accordingly, the function $gm,l(rs)$ is described as
(91)
$gm,l(rs)=2(m+1)|η|Jm(μmlrsτ0s)πω0νs0τ0sJm+1(μml)(1−Θ(rs−τ0s)),$
owing to Eqs. (63) and (90).
By summarizing all these results (by substituting Eqs. (89) and (90) into Eq. (70), and by calculating Eq. (72) with Eq. (87)), we finally derive the conventional Sacherer formula:
(92)
$τm−1=−cIc4πQx(m+1)Es/e∑p=−∞∞ℜ[ZT(ω′p)]F′m(ω′p−ωξ),$
where $τm−1$ is the growth rate,
(93)
$F′m(ω)=h′m(ω)B′f∑p=−∞∞h′m(ω′p−ωξ),$
(94)
$h′m(ω)=(2τ0s)22π4(m+1)2[1+(−1)mcos(ω2τ0s)][(ω2τ0sπ)2−(m+1)2]2,$
(95)
$ω′p=ω0(Qx+mνs0+μ+Mp),$
(96)
$ωξ=ω0Qxξη,$
(97)
$Ic=eMNbT0,$
(98)
$B′f=M2τ0sc2πR.$
In this paper's calculations, the factor $Bf′$ is approximated by the typical bunching factor $Bf$ defined by the average current divided by the peak current (see Eq. (101)).
## 3. RCS parameters and the beam growth rate estimated by the Sacherer formula
At the RCS, the bunched beams are formed by accumulating the injection beam from the LINAC with a painting scheme [28,29]. They are accelerated from 400 MeV to 3 GeV over 20 ms. Figure 2 shows the typical patterns of the acceleration voltage $Vrf$ (red), and of the synchronous phase $φs$ (blue) in that period. Table 1 shows typical machine and beam parameters for the RCS, which were used in this paper's calculations. The average chamber radius $a$ around the ring is determined to be 145 mm, in order that the coherent betatron tune shift reproduces the measured date for a 400 MeV beam.
Fig. 2.
Typical pattern of the acceleration voltage $Vrf$ (red), and the synchronous phase $φs$ (blue) during the ramping time.
Fig. 2.
Typical pattern of the acceleration voltage $Vrf$ (red), and the synchronous phase $φs$ (blue) during the ramping time.
Table 1.
Typical parameter list
$T$ (kinetic energy, GeV) 0.4 3 $f0$ (revolution frequency, MHz) 0.61 0.84 $η$ (slippage factor) -0.478 -0.047 $Ic$ (average current, A) 8.1 11.2 $νs0$ (synchrotron tune) 0.0053 0.0005 $〈βx(s)〉$ (m) 11.6 $〈βx2(s)〉$ (m2) 172.3 $〈D2(s)〉$ (m)2 3.46 $ϵx,rms$ (mmrad) $100/6βsγs$ $JL0$ (eV $⋅$ s) $0.1645$
$T$ (kinetic energy, GeV) 0.4 3 $f0$ (revolution frequency, MHz) 0.61 0.84 $η$ (slippage factor) -0.478 -0.047 $Ic$ (average current, A) 8.1 11.2 $νs0$ (synchrotron tune) 0.0053 0.0005 $〈βx(s)〉$ (m) 11.6 $〈βx2(s)〉$ (m2) 172.3 $〈D2(s)〉$ (m)2 3.46 $ϵx,rms$ (mmrad) $100/6βsγs$ $JL0$ (eV $⋅$ s) $0.1645$
(Circumference $C=348.333 m$, harmonic number $h=2$, repetition rate = 25 Hz, particles per bunch $Nb=4.15×1013$, and the average chamber radius $a=145 mm$).
Here, $βx(s)$ and $D(s)$ are the $β$-function and the dispersion function, respectively; $ϵx,rms$ and $JL0$ are the root mean square (rms) horizontal and the longitudinal emittances, respectively; $〈⋯〉$ denotes the average value around the ring; and $βs$ and $γs$ are the Lorentz-$β$ and Lorentz-$γ$ on the designed particle.
Eight kickers are installed in the RCS. The real and the imaginary parts of the horizontal impedance $ZT(ω)$ for one kicker are shown in the left and the middle panels of Fig. 3, respectively. The red and blue lines show the impedances at $βs=0.7$ and $βs=1$, respectively. The impedance is roughly proportional to the Lorentz-$β$ [15]. The corresponding wake function $WT(ω0t)$ calculated by Eq. (17) is denoted by the same color in the right-hand figure. The reflection wave excited at the end of the power cable of the kicker creates the spike structure of the kicker impedance.
Fig. 3.
Dependence of the horizontal kicker impedance $ZT(ω)$ (left/middle) and of the wake function $WT(ω0t)$ (right) on the Lorentz-$β$. The red and blue lines show the results at $βs=0.7$ and $βs=1$, respectively. The wave propagation speeds in the kicker magnet and in the power cable are about $0.02×c$ and $0.57×c$, respectively. The magnet length and the cable length are 705 mm and 130 m, respectively.
Fig. 3.
Dependence of the horizontal kicker impedance $ZT(ω)$ (left/middle) and of the wake function $WT(ω0t)$ (right) on the Lorentz-$β$. The red and blue lines show the results at $βs=0.7$ and $βs=1$, respectively. The wave propagation speeds in the kicker magnet and in the power cable are about $0.02×c$ and $0.57×c$, respectively. The magnet length and the cable length are 705 mm and 130 m, respectively.
As shown in the left and middle panels, the impedance is very large indeed. We have demonstrated that the RCS is a kicker-impedance-dominated machine by stabilizing unstable beams by temporarily reducing the impedance [12,13]. For simplicity, we assume in this paper that the only source of impedance in the RCS is kicker impedance.
Mostly (except the discussion about chromaticity dependence of beam growth rates shown in Figs. 16 and 17), let us consider a case in which the chromaticity $ξQx$ is activated by a DC-power supply at the injection energy. In this case the chromaticity approaches the natural chromaticity $(ξQx=−10.3)$ [30] as the beam energy is increased, as shown in Fig. 4.
Fig. 4.
Calculated chromaticity $ξQx$ change during the ramping time.
Fig. 4.
Calculated chromaticity $ξQx$ change during the ramping time.
We have observed beam instabilities at the J-PARC RCS, where the chromaticity was fully corrected only at the injection energy. The blue line of Fig. 5 shows an example of the results of the horizontal beam position for a 750 kW equivalent beam ($3.10×1013$ particles per bunch). For reference, the green line shows the results where only one bucket among the two is filled with one bunched beam. Since no instability occurs on the green line, we have judged that the instabilities on the blue line are the coupled-bunch instabilities.
Fig. 5.
Measured horizontal beam positions for the case of $3.10×1013$ particles per bunch and $Qx=6.45$. The blue line shows the results where two buckets are perfectly filled with two bunches, and the green line shows the results where only one bucket among the two is filled with one bunch. The momentum spread of the injection beam from LINAC is 0.18%.
Fig. 5.
Measured horizontal beam positions for the case of $3.10×1013$ particles per bunch and $Qx=6.45$. The blue line shows the results where two buckets are perfectly filled with two bunches, and the green line shows the results where only one bucket among the two is filled with one bunch. The momentum spread of the injection beam from LINAC is 0.18%.
Figure 6 shows the measured results for a 1 MW equivalent beam ($4.15×1013$ particles per bunch), where the chromaticity was fully corrected only at the injection energy. Both results for 750 kW equivalent and 1 MW equivalent beams have demonstrated that the beam is stable at low energies, while they tend to be unstable at high energies.
Fig. 6.
Measured horizontal beam positions for the case of $4.15×1013$ particles per bunch and $Qx=6.45$. The momentum spread of the injection beam is 0.18%.
Fig. 6.
Measured horizontal beam positions for the case of $4.15×1013$ particles per bunch and $Qx=6.45$. The momentum spread of the injection beam is 0.18%.
Here, let us investigate whether the conventional Sacherer formula Eq. (92) can explain the measured beam behavior. From now on, we assume that the maximum number of the head-tail mode $m$ is 5, and that the coupled mode $μ$ runs from 0 to 1. Figure 7 shows the theoretical results for the case. The results predict that the beam is unstable at low energies, while it is stable at high energies. These results suggest that a partial chromaticity correction at low energies should enhance the beam instability at low energies. However, these theoretical results (Fig. 7) differ significantly from the measured data (Figs. 5 and 6).
Fig. 7.
The maximum beam growth rate among with different modes $(m,μ)$ estimated by using the Sacherer formula, Eq. (92), for $Qx=6.45$.
Fig. 7.
The maximum beam growth rate among with different modes $(m,μ)$ estimated by using the Sacherer formula, Eq. (92), for $Qx=6.45$.
The measurement results indicate that space charge stabilizes the beam instability at low energies. Note that Eq. (92) is derived by neglecting this effect. In the next section, let us theoretically examine the space charge effect on the beam instability at the RCS.
## 4. Investigation of the beam instability at the RCS
### 4.1. Space charge effects on the beam growth rate
The Landau damping caused by the space charge effect appears in Eq. (51). Because this equation depends only on the longitudinal emittance $JL0$ in our model, only the longitudinal size of the beam is likely to affect the effect, significantly. However, the true space charge effect is revealed in Eq. (48) after integration with respect to $Jx$ according to Eq. (43). In particular, the damping effect is neglected for a beam with infinitesimal transverse beam size, and Eq. (48) is sufficiently well approximated by the analytical formula
(99)
$ν≃mνL0+νX0+je2NbMπcβs〈βx(s)〉8π3EsR ×∑p=−∞∞Fm(JL0,mνL0+νX0+μ+pM−Qxξxη)ZT(ω0(mνL0+νX0+μ+pM)),$
where
(100)
$Fm(JL0,x)=|Jm[x(2ω0JL0|η|cpsβsνs0)1/2]|2.$
Figure 8 shows the maximum beam growth rate among different modes $(m,μ)$ estimated according to Eq. (99). As in the results obtained using the conventional formula (shown in Fig. 7), these results show that the beam is unstable at low energies. However, this result successfully explains the beam instability of the measured results at high energies, which the conventional formula does not explain. To understand the beam stabilization at low energies, the Landau damping effects owing to space charge must be taken into account.
Fig. 8.
The maximum beam growth rate among different modes $(m,μ)$ estimated by Eq. (99) for $Qx=6.45$.
Fig. 8.
The maximum beam growth rate among different modes $(m,μ)$ estimated by Eq. (99) for $Qx=6.45$.
Here, let us investigate the effect more closely. First, we present the theoretical results of taking the space charge effect into account for the maximum beam growth rate by solving Eq. (48). The results are shown in Fig. 9. Comparing the results shown in Fig. 8 with the present results, we find that the beam is stabilized at low energies and that the theoretical results explain well the characteristic of the measurement ones (shown in Figs. 5 and 6). We can see a sharp rise at $t=13 ms$ only in the measured data of the 1 MW-equivalent beam (Fig. 6). The space charge damping effect seems to be drastically reduced for a beam with larger oscillation amplitudes. If this is a kind of nonlinear phenomenon, our theory, based on the linearized Vlasov equation, has a limit to explain it.
Fig. 9.
The maximum beam growth rate among different modes $(m,μ)$ estimated by solving Eq. (48) for $Qx=6.45$.
Fig. 9.
The maximum beam growth rate among different modes $(m,μ)$ estimated by solving Eq. (48) for $Qx=6.45$.
Figure 10 shows the theoretical results of the transverse beam emittance dependence of the beam growth rate. The red, black, and purple lines are the beam growth rates excited by $(m=0,μ=1)$, $(m=2,μ=1)$, and $(m=4,μ=1)$ modes, respectively (the growth rate excited by the other modes is negligibly low.). As already explained, the Landau damping effect becomes ineffective for all modes, as the transverse emittance decreases. Figure 11 illustrates the measured beam positions for different transverse emittances. The red, blue, black, and yellow lines show the results for the cases that the injection painting areas are $0π$ (center injection), $100π$, $150π$, and $200π$ mmrad, respectively [28,29]. The emittance dependence is clearly observable in the results. As the painting area is larger at the injection period, the beam tends to be more stabilized at high energies.
Fig. 10.
Theoretical results of the transverse beam emittance $ϵx,rms$ dependence of the beam growth rate at 15 ms for $Qx=6.45$. The red, black, and purple lines are the beam growth rates excited by $(m=0,μ=1)$, $(m=2,μ=1)$, and $(m=4,μ=1)$ modes, respectively.
Fig. 10.
Theoretical results of the transverse beam emittance $ϵx,rms$ dependence of the beam growth rate at 15 ms for $Qx=6.45$. The red, black, and purple lines are the beam growth rates excited by $(m=0,μ=1)$, $(m=2,μ=1)$, and $(m=4,μ=1)$ modes, respectively.
Fig. 11.
Measurement results ($Nb=4.15×1013$) of the horizontal beam positions for different transverse painting areas, where the chromaticity was fully corrected only at the injection energy. The red, blue, black, and yellow lines show the results for $0π$ (center injection), $100π$, $150π$, and $200π$ mmrad injection painting schemes, respectively, where the tune $Qx$ changes during the ramping time following the black line in the right panel of Fig. 14. The momentum spread of the injection beam from LINAC is 0.18%.
Fig. 11.
Measurement results ($Nb=4.15×1013$) of the horizontal beam positions for different transverse painting areas, where the chromaticity was fully corrected only at the injection energy. The red, blue, black, and yellow lines show the results for $0π$ (center injection), $100π$, $150π$, and $200π$ mmrad injection painting schemes, respectively, where the tune $Qx$ changes during the ramping time following the black line in the right panel of Fig. 14. The momentum spread of the injection beam from LINAC is 0.18%.
Thus, we find that the Landau damping effect owing to the space charge (depending on the longitudinal beam size) is enhanced by enlarging the transverse beam size. From a phenomenological point of view, the space charge damping effect is easily activated for the lower-energy beam, as a result of the larger transverse beam emittance.
Now, let us closely investigate the bunching factor $Bf$ (longitudinal beam size) dependence of the beam growth rate for different head-tail and coupled-bunch modes $(m,μ)$. Figure 12 shows the theoretical results of the beam growth rate at 15 ms, where the bunching factor $Bf$ is evaluated by using
(101)
$Bf=43(π−ϕe−φs)(2JL,0h2|η|ω0Esβs2νs0)1/2,$
where $ϕe$ is the solution of
(102)
$cosϕe+ϕesinφs+cosφs−(π−φs)sinφs=0,$
which satisfies the condition $−π<ϕe<0$ [31]. The left and the middle panels of Fig. 12 illustrate the beam growth rates with space charge effects for the chamber radii $a=145 mm$ and $a=160 mm$, respectively. The right panel illustrates the beam growth rate without space charge effects calculated by using Eq. (99). The red, blue, black, green, purple, and brown lines are the beam growth rates excited by $(m=0,μ=1)$, $(m=1,μ=1)$, $(m=2,μ=1)$, $(m=3,μ=1)$, $(m=4,μ=1)$, and $(m=5,μ=1)$ modes, respectively (the other modes do not excite the beam instabilities).
Fig. 12.
Theoretical results of the beam growth rate (at 15 ms for $Qx=6.45$) with (left/middle) and without (right) space charge effects, dependence on the bunching factor. The red, blue, black, green, purple, and brown lines are the beam growth rates excited by $(m=0,μ=1)$, $(m=1,μ=1)$, $(m=2,μ=1)$, $(m=3,μ=1)$, $(m=4,μ=1)$, and $(m=5,μ=1)$ modes, respectively. The left and the middle panels show the results for the chamber radii $a=145 mm$ and $a=160 mm$, respectively.
Fig. 12.
Theoretical results of the beam growth rate (at 15 ms for $Qx=6.45$) with (left/middle) and without (right) space charge effects, dependence on the bunching factor. The red, blue, black, green, purple, and brown lines are the beam growth rates excited by $(m=0,μ=1)$, $(m=1,μ=1)$, $(m=2,μ=1)$, $(m=3,μ=1)$, $(m=4,μ=1)$, and $(m=5,μ=1)$ modes, respectively. The left and the middle panels show the results for the chamber radii $a=145 mm$ and $a=160 mm$, respectively.
The conventional Sacherer formula (92) indicates that the beam growth rate without space charge is roughly inversely proportional to the bunching factor $Bf$. The left and middle panels demonstrate that the overall behavior of the beam growth rate including space charge effect is also roughly inversely proportional to the bunching factor $Bf$. However, the beam is ultimately stabilized in the extremely compressed beam (with the extremely small bunching factor). In this case, the Landau damping due to the space charge force absolutely stabilizes the beam instability.
The beam growth rates for the different modes ($m,μ$) in all panels of Fig. 12 reveal the respective comb-like structures along the bunching factor. The behavior originates from the head-tail motion of the beam, as shown in the form factor $Fm(JL,x)$ in Eq. (99). Thus, when we fix a mode, the beam growth rate for the mode follows the characteristic comb-like behavior, even in the results without space charge effect (right). However, because the growth rate patterns are overlapped for the different modes ($m,μ$) in the results without space charge effect, it is hard to specify the optimized point along the bunching factor from the viewpoint of beam instability. Thus, we reach the conventional conclusion that the larger bunching factor (smaller peak current) is preferable for beam stabilization, when the space charge effect is neglected.
Contrary to the such conventional understanding, beam stabilized regions emerge along the bunching factor in the results with the space charge effects for $a=145 mm$ (e.g., around the area $A$). Comparing both the results for $a=145 mm$ (left) and for $a=160 mm$ (middle), we find that the bandwidth of the stabilized region caused by the space charge effect significantly depends on the chamber radius $a$. Though the difference between the chamber radii is only 15 mm, the beam stabilization area $A$ in the results for $a=145 mm$ (left) disappears in the results for $a=160 mm$ (middle). In conclusion, the smaller chamber radius is preferable in view of the beam stabilization to make maximum use of the space charge damping effect.
The existence of such a beam stabilization region, stemming from the space charge effects, along the bunching factor can be demonstrated at a low-energy proton ring like the RCS. At the RCS, the bunching factor can be changed by changing the momentum spread of the injection beam from the LINAC. We can prepare two types of injection beams: $dp/p=0.08%$ and $dp/p=0.18%$. The injection beam with the smaller momentum spread creates an accumulated beam with a smaller bunching factor. The measurement results for the beam positions and their corresponding bunching factors are illustrated in Fig. 13 with the same colors, where the number of particles per bunch is $3.10×1013$. It is observable that the beam can be more stabilized with the smaller bunching factor, contrary to conventional understanding. Theoretically, this stabilization is caused by the dip around the area $A$ in Fig. 12.
Fig. 13.
Beam growth rate (for $3.10×1013$ particles per bunch) dependence on the bunching factor, where the tune is fixed to 6.45. The chromaticity was fully corrected only at the injection energy. The left panel shows the measured beam positions for two different bunching factors. The right panel shows the measured bunching factor. The two lines with the same color in both figures denote identical situations.
Fig. 13.
Beam growth rate (for $3.10×1013$ particles per bunch) dependence on the bunching factor, where the tune is fixed to 6.45. The chromaticity was fully corrected only at the injection energy. The left panel shows the measured beam positions for two different bunching factors. The right panel shows the measured bunching factor. The two lines with the same color in both figures denote identical situations.
### 4.2. The effects of tune manipulation on beam growth rate
Here, let us illustrate the tune dependence of the beam growth rate. The measurement results are shown in the left panel of Fig. 14. The tracking pattern of the tune during the acceleration period is shown in the right panel of Fig. 14 using the same color. The results represented by the red line correspond to the highest beam growth rate case. The second highest case is represented by the yellow line. The most stable case is indicated by the black line, which is sandwiched by these two unstable cases (the red and the yellow lines). Figure 15 shows the theoretical results of the beam growth rate at 15 ms, which are obtained by solving Eq. (48). The red, black, and purple lines are the beam growth rates excited by the $m=0$, $m=2$, and $m=4$ modes, respectively. The solid and dashed lines show the $μ=1$ and $μ=0$ modes, respectively (the other head-tail modes do not excite the beam instabilities). The theoretical calculation explains the characteristic of the tune dependence of the beam growth rate sufficiently well, as revealed by the measured results (Fig. 14). The tune dependence of the beam growth rate originates from the spike structure of the kicker impedance (see Fig. 3).
Fig. 14.
The left panel shows the measured ($4.15×1013$ particles per bunch) beam positions for five different tune tracking patterns, where the chromaticity was fully corrected only at the injection energy. The right panel shows the measured tune tracking patterns under the condition that the space charge effect is negligible. The lines of matching color in each panel denote identical situations. The momentum spread of the injection beam is 0.08%.
Fig. 14.
The left panel shows the measured ($4.15×1013$ particles per bunch) beam positions for five different tune tracking patterns, where the chromaticity was fully corrected only at the injection energy. The right panel shows the measured tune tracking patterns under the condition that the space charge effect is negligible. The lines of matching color in each panel denote identical situations. The momentum spread of the injection beam is 0.08%.
Fig. 15.
Dependence of the theoretical results of the beam growth rate at 15 ms on the tune $Qx$. The red, black, and purple lines are the beam growth rates excited by the $m=0$, $m=2$, and $m=4$ modes, respectively. The solid and dashed lines show the $μ=1$ and $μ=0$ modes, respectively.
Fig. 15.
Dependence of the theoretical results of the beam growth rate at 15 ms on the tune $Qx$. The red, black, and purple lines are the beam growth rates excited by the $m=0$, $m=2$, and $m=4$ modes, respectively. The solid and dashed lines show the $μ=1$ and $μ=0$ modes, respectively.
Finally, we illustrate the chromaticity dependence of the beam growth rate. Figure 16 shows the theoretical results of the beam growth rate at 15 ms for $Qx=6.45$. The red, blue, black, green, purple, and brown lines are the beam growth rates excited by the $(m=0,μ=1)$, $(m=1,μ=1)$, $(m=2,μ=1)$, $(m=3,μ=1)$, $(m=4,μ=1)$, and $(m=5,μ=1)$ modes, respectively (the other modes do not excite the beam instabilities). We expect that the beam growth rate will be drastically suppressed, as the chromaticity correction is weakened.
Fig. 16.
Theoretical results of the beam growth rate at 15 ms dependence on the chromaticity $ξQx$ for $Qx=6.45$. The red, blue, black, green, purple, and brown lines are the beam growth rates excited by the $(m=0,μ=1)$, $(m=1,μ=1)$, $(m=2,μ=1)$, $(m=3,μ=1)$, $(m=4,μ=1)$, and $(m=5,μ=1)$ modes, respectively.
Fig. 16.
Theoretical results of the beam growth rate at 15 ms dependence on the chromaticity $ξQx$ for $Qx=6.45$. The red, blue, black, green, purple, and brown lines are the beam growth rates excited by the $(m=0,μ=1)$, $(m=1,μ=1)$, $(m=2,μ=1)$, $(m=3,μ=1)$, $(m=4,μ=1)$, and $(m=5,μ=1)$ modes, respectively.
The measured results are shown in Fig. 17. To clearly observe the chromaticity dependence of the beam growth rate, let us study the highest growth rate case (the tracking pattern of the tune is designated by the red line in the right panel of Fig. 14). The red, blue, and black lines in Fig. 17 show, respectively, the results for which the chromaticity was fully corrected only at the injection energy by the DC-power supply, half corrected compared to the full correction, and quarter corrected in the same manner. Concretely, the chromaticity values at 15 ms are $−7.46$ for the red line, $−8.92$ for the blue line, and $−9.64$ for the black line. As expected, the beam is drastically suppressed by an increase in chromaticity in the negative direction.
Fig. 17.
Measured beam positions ($Nb=4.15×1013$, $dp/p=0.08%$) with different chromaticity, where the tune changes, following the red line in the right panel of Fig. 14. The red, blue, and black lines show the results for which, at the injection energy only, the chromaticity was fully, half, and quarter corrected, respectively.
Fig. 17.
Measured beam positions ($Nb=4.15×1013$, $dp/p=0.08%$) with different chromaticity, where the tune changes, following the red line in the right panel of Fig. 14. The red, blue, and black lines show the results for which, at the injection energy only, the chromaticity was fully, half, and quarter corrected, respectively.
## 5. Summary
The RCS in J-PARC, where kicker impedance dominates, is a special machine from an impedance viewpoint, which means that the RCS violates the impedance budget from a classical viewpoint [6,18,19]. Nevertheless, we have successfully accelerated a 1 MW equivalent beam ($4.15×1013$ particles per bunch). The RCS is an accelerator covering the intermediate beam energy region (from 400 MeV to 3 GeV). Thus, it is pertinent to study the space charge effects on the beam instability.
The machine has some interesting characteristics: e.g., the beam can be stabilized by reducing the bunching factor (increasing the peak current) and the beam tends to be unstable when reducing the transverse beam size. The classical theory, i.e., Sacherer's theory, fails to explain these characteristics by neglecting the space charge effects.
Recently, there has been a significant development in the field of computer technologies. Numerical computer simulations are powerful tools to quantitatively estimate the beam behavior associated with space charge effects [3234]. It may seem that a numerical simulation study is sufficient to accelerate beams from a practical viewpoint.
However, such simulations take excessive CPU time and memory for one set of fixed parameters. If we theoretically understand what conditions (parameters sets) excite beam instabilities in combination with space charge effects in advance, numerical studies are more efficiently performed by selecting the appropriate parameters sets, based on the theoretical comprehension. Consequently, we can focus on the quantitative discussion about the issues concerning beam commissioning (beam loss, beam halo, etc.). Moreover, the theoretical study is vital to understand the nature of the phenomena concerning beams in accelerators.
In this paper, we try to understand the beam instabilities associated with the space charge effects by developing a new theory. And, we have clarified the parameters (such as the transverse emittance, the bunching factor, etc.) dependence on the beam growth rate.
The space charge damping effect is significant at low energies, not only due to the smaller Lorentz-$γ$ but also due to the larger transverse beam size. The large transverse emittance is essential to activate the Landau damping owing to the space charge effect.
It is of interest that the beam growth rate is suppressed by increasing the peak current (shortening the bunch length, or reducing the bunching factor) at the RCS. Theoretically, the beam growth rate for different modes $(m,μ)$ follows different characteristic comb-like structures along the bunching factor. The dependence of the beam growth rate on the bunching factor originates from the head-tail motion of the beam. Thus, even in the case without the space charge effect, the beam growth rate for one fixed mode can be suppressed by increasing the peak current (shortening the bunch length, or reducing the bunching factor).
However, the beam growth rates excited by different modes $(m,μ)$ are sufficiently overlapped along the bunching factor in the case. Finally, the theory reproduces the conventional conclusion that the maximum beam growth rate among different modes $(m,μ)$ is reduced by increasing the bunch length (reducing the peak current or increasing the bunching factor) when the space charge effect is neglected.
On the contrary, if we take the space charge effect into consideration, the overlap of the beam growth rates for different modes ($m,μ$) is separated over the axis of the bunching factor, and some beam stabilization regions emerge on the axis. The optimization of bunching factor enables the beam to be stabilized, regardless of the amount of the bunching factor, in a lower-energy proton synchrotron like the RCS.
The space charge damping effect is quite sensitive to the chamber radius. Consequently, a smaller radius chamber is preferable from a beam instability point of view. As the beam energy becomes higher, the space charge damping effect becomes less effective, and the beam stabilization region diminishes along the bunching factor.
In a low-energy proton machine, such as the RCS, the violation of the impedance budget from a classical viewpoint is not vital to achieve high intensity beams. They can be realized by optimizing the machine's (beam) parameters, i.e., the bunching factor, transverse emittance, tune, chromaticity, etc.
## Acknowledgement
The authors would like to thank Kazuhito Ohmi, Jie Wei, Katsunobu Oide, Yoshishige Yamazaki, Tadashi Koseki, Kazuo Hasegawa, and Michikazu Kinsho for fruitful discussions. They also would like to thank all members of the J-PARC Accelerator Technical Advisory Committee, which was led by Steve Holmes until 2009, and has been led by Thomas Roser since 2010. The authors would also like to thank all the members of the J-PARC project at JAEA/KEK.
#### Appendix A. A solution of the Poisson equation with cylindrical chamber
In this section, we show how to solve the Poisson equation for an axisymmetric beam that is surrounded by a perfectly conductive cylindrical chamber with radius $a$. The Poisson equation in the rest frame of the beam $(ct¯,x,y,z¯)$ is described by
(A1)
$∂2Φ¯∂x2+∂2Φ¯∂y2+∂2Φ¯∂z¯2=−cZ0ρ¯p(x,y,z¯),$
with
(A2)
$ρ¯p(x,y,z¯)=eNbρ^(z¯)exp(−(ρcosφ−r0cosθ0)2+(ρsinφ−r0sinθ0)22σx2)2πσx2,$
(A3)
$ρ^(z¯)=exp(−z¯22σ¯z2)2πσ¯z,$
(A4)
$σ¯z=γsσz,$
where $γs$ is the Lorentz-$γ$ of the reference particle, $c$ is light velocity, $Z0$ is the impedance of free space, $σx$ is the rms transverse beam size, and $Nb$ is the number of particle per bunch. Polar coordinates are introduced as
(A5)
$x=ρcosφ,$
(A6)
$y=ρsinφ,$
and the center of the bunch on the horizontal plane is given by $(r0cosθ0,r0sinθ0)$. From now on, the condition $σx≪a$ is assumed.
When a perfectly conductive chamber with radius $a$ exists, the Green function $G(r→,r→′)$ that satisfies the boundary condition $G=0$ at $ρ=a$, is given by [35]
(A7)
$G(r→,r→′)=∑m=0∞ϵm2π2cosm(φ−φ′) ×{∫0∞dλ[Km(λρ′)−Km(λa)Im(λa)Im(λρ′)]Im(λρ)cosλ(z¯−z¯′),for ρ′>ρ,∫0∞dλ[Km(λρ)−Km(λa)Im(λa)Im(λρ)]Im(λρ′)cosλ(z¯−z¯′),for ρ′>ρ,$
where $Im(z)$ and $Km(z)$ are the modified Bessel functions, $r→=(ρ,φ,z¯),r→′=(ρ′,φ′,z¯′)$, $ϵm=2−δm0$ and $δmn$ is the Kronecker-$δ$. By using the Green function, the solution $Φ¯$ is approximated as
(A8)
$Φ¯(ρ,φ,z¯)≃∫0∞dλ∫−∞∞dz¯′∫ρaρ′dρ′∫02πdφ′∑m=0∞cZ0ϵm2π2cosm(φ−φ′) ×[Km(λρ′)−Km(λa)Im(λa)Im(λρ′)]Im(λρ)cosλ(z¯−z¯′) ×eNbexp(−z¯′22σ¯z2)2πσ¯zexp(−(ρ′cosφ′−r0cosθ0)2+(ρ′sinφ′−r0sinθ0)22σx2)2πσx2 +∫0∞dλ∫−∞∞dz¯′∫0ρρ′dρ′∫02πdφ′∑m=0∞cZ0ϵm2π2cosm(φ−φ′) ×[Km(λρ)−Km(λa)Im(λa)Im(λρ)]Im(λρ′)cosλ(z¯−z¯′) ×eNbexp(−z¯′22σ¯z2)2πσ¯zexp(−(ρ′cosφ′−r0cosθ0)2+(ρ′sinφ′−r0sinθ0)22σx2)2πσx2.$
By using the formulae
(A9)
$∫02πdφ′cosm(φ−φ′)exp(ρ′r0cos(φ′−θ0)σx2)=2πIm(ρ′r0σx2)cos(m(φ−θ0)),$
(A10)
$12πσ¯z∫−∞∞dz¯′exp(−z¯′22σ¯z2)cosλ(z¯−z¯′)=exp(−λ2σ¯z22)cosλz¯,$
the integrations in the azimuthal and longitudinal directions in Eq. (A8) are performed, so that we get
(A11)
$Φ¯(ρ,φ,z¯)=∫ρaρ′dρ′∑m=0∞cZ0ϵm2π2eNbexp(−ρ′2+r022σx2)σx2Im(ρ′r0σx2)cos(m(φ−θ0)) ×∫0∞dλ[Km(λρ′)−Km(λa)Im(λa)Im(λρ′)]Im(λρ)cosλz¯exp(−λ2σ¯z22) +∫0ρρ′dρ′∑m=0∞cZ0ϵm2π2eNbexp(−ρ′2+r022σx2)σx2Im(ρ′r0σx2)cos(m(φ−θ0)) ×∫0∞dλ[Km(λρ)−Km(λa)Im(λa)Im(λρ)]Im(λρ′)cosλz¯exp(−λ2σ¯z22).$
The potential $Φ¯c$ felt at the bunch center is calculated by plugging in $ρ=r0$ and $φ=θ0$ [26]. Figure 18 illustrates typical behavior of the potential $Φ¯c(z¯=0)$ calculated by using the beam parameters at the ramping time 15 ms in the RCS.
Fig. 18.
Typical behavior of the potential $Φ¯c(z¯=0)$ calculated by using the beam parameters at the ramping time 15 ms in the RCS.
Fig. 18.
Typical behavior of the potential $Φ¯c(z¯=0)$ calculated by using the beam parameters at the ramping time 15 ms in the RCS.
Here, let us expand the result for small $ρ$ around zero. As a result, it is expressed as
(A12)
$Φ¯c(x,y,z¯)≃Φ¯coh,0(z¯)+Φ¯coh,2(z¯)(x2+y2)+Φ¯coh,4(z¯)(x2+y2)2,$
where
(A13)
$Φ¯coh,0(z¯)=cZ0eNb2π2σx2∫0adρ′ρ′exp(−ρ′22σx2)∫0∞dλexp(−λ2σ¯z22) ×[K0(λρ′)−K0(λa)I0(λa)I0(λρ′)]cosλz¯,$
(A14)
$Φ¯coh,2(z¯)=cZ0eNb2π2σx2∫0adρ′ρ′exp(−ρ′22σx2)∫0∞dλ(λ24+ρ′24σx4−12σx2)exp(−λ2σ¯z22) ×[K0(λρ′)−K0(λa)I0(λa)I0(λρ′)]cosλz¯ +cZ0eNb4σx4π2∫0adρ′ρ′2exp(−ρ′22σx2)∫0∞dλλexp(−λ2σ¯z22) ×[K1(λρ′)−K1(λa)I1(λa)I1(λρ′)]cosλz¯−cZ0eNb8π2σx2σ¯z(π2)1/2exp(−z¯22σ¯z2),$
(A15)
$Φ¯coh,4(z¯)=cZ0eNb16π2σx2∫0adρ′ρ′exp(−ρ′22σx2)∫0∞dλ(λ48+ρ′2λ22σx4+ρ′48σx8−λ2σx2−ρ′2σx6+1σx4) ×exp(−λ2σ¯z22)[K0(λρ′)−K0(λa)I0(λa)I0(λρ′)]cosλz¯ +cZ0eNb8π2σx4∫0adρ′exp(−ρ′22σx2)∫0∞dλ(ρ′4λ4σx4+ρ′2λ34−ρ′2λσx2)exp(−λ2σ¯z22) ×[K1(λρ′)−K1(λa)I1(λa)I1(λρ′)]cosλz¯ +cZ0eNb64π2σx6∫0adρ′exp(−ρ′22σx2)ρ′3∫0∞dλλ2exp(−λ2σ¯z22) ×[K2(λρ′)−K2(λa)I2(λa)I2(λρ′)]cosλz¯ +cZ0eNbπ2σx2(−(σ¯z2−z¯2)128σ¯z5+164σx2σ¯z)(π2)1/2exp(−z¯22σ¯z2).$
The terms $Φ¯coh,2(z¯)$ and $Φ¯coh,4(z¯)$ contribute to the coherent space charge tune shift, and to the nonlinear motion of the beam, respectively.
The scalar potential $Φ$ and the vector potential $Az$ in the lab-frame $(ct,x,y,z)$ are given by
(A16)
$Φ(x,y,z−βsct)=γsΦ¯(x,y,γs(z−βsct)),$
(A17)
$Az(x,y,z−βsct)=βscγsΦ¯(x,y,γs(z−βsct)),$
respectively, where $βs$ is the Lorentz-$β$ of the reference particle.
#### Appendix B. Derivation of the Hamiltonian with action-angle variables including horizontal wake and space charge effects
In this section, we will obtain the Hamiltonian Eq. (B50) with action-angle variables, by successively canonically transforming Hamiltonians.
The original Hamiltonian in an electromagnetic field is approximately given by [24,25]
(B1)
$Ho=−ps(1+xρ)ΔEpsβsc+ps2γs2(ΔEpsβsc)2+px2+py22ps+ps2Kx(s)(1−ΔEEs)x2 +ps2Ky(s)(1−ΔEEs)y2−psxEsFx+eΦc(x,y,s−cβst)βsγs2c −eVrfωRFδp(s)cos(ωRFt−hsR+φs)+…,$
where $φs$ is the synchronous phase; $ps$ is the constant momentum on the synchronous particle; $Es=cps/βs$ is the particle energy on the designed orbit; $βs$ and $γs$ are the Lorentz-$β$ and $γ$, respectively; $ΔE$ is given by $ΔE=E−Es$; $Fx$ is the horizontal wake force; $δp(s)$ is the periodic $δ$-function; $c$ is the velocity of light; $Kx$ and $Ky$ are the periodic focusing forces in the horizontal and vertical directions, respectively; $Φc$ is the space charge potential; $h$ is harmonic number; $Vrf$ is the amplitude of the radio frequency (RF) voltage; $1/ρ$ is the local curvature around the machine; $R$ is the average radius of the machine; and $ωRF$ is the angular frequency of the RF voltage, which is expressed as
(B2)
$ωRF=cβshR.$
The orbit length $s$ is used as an independent variable. The canonical variables are $(x,px),(y,py)$, and $(t,−E)$ for the horizontal, vertical, and the longitudinal directions, respectively. It is noticeable that the contribution from the vector potentials is included in the Hamiltonian, where the contributions from both the scalar and vector potentials are confined to the scalar potential only with Eqs. (A16) and (A17).
Using the generating function $F1$,
(B3)
$F1(x,p¯x,y,p¯y,t,−ΔE¯)=(x−ΔE¯psβscD)p¯x+yp¯y−t(ΔE¯+Es) +ΔE¯βscdDdsx−ps2(ΔE¯psβsc)2DdDds,$
we make a canonical transformation from the variables $(x,px),(y,py)$, and $(t,−E)$ to $(x¯,p¯x),(y¯,p¯y)$, and $(t¯,−ΔE¯)$, respectively, according to
(B4)
$px=∂F1∂x=p¯x+ΔE¯βscdDds,$
(B5)
$x¯=∂F1∂p¯x=x−ΔE¯psβscD,$
(B6)
$py=p¯y,$
(B7)
$y¯=y,$
(B8)
$−E=∂F1∂t=−ΔE¯−Es,$
(B9)
$t¯=−∂F1∂ΔE¯=Dp¯xpsβsc+t−1βscdDdsx¯,$
where the dispersion function $D(s)$ in the horizontal direction satisfies the relation
(B10)
$d2Dds2+KxD=1ρ.$
The new Hamiltonian $H1$ is obtained as
(B11)
$H1≃p¯x2+p¯y22ps+ps2(1−ΔE¯Es)(Kxx¯2+Kyy¯2)−psEs(x¯+ΔE¯psβscD)Fx +eΦc[x¯+ΔE¯psβscD,y¯,s−cβs(t¯−Dp¯xpsβsc+1βscdDdsx¯)]βsγs2c −ΔE¯βsc+ps2γs2(ΔE¯psβsc)2−ps2(ΔE¯psβsc)2Dρ−KxDpsx¯ΔE¯2c2−12KxD2βsΔE¯3ps2c3 −eVrfωRFδp(s)cos(ωRFt¯−hsR+φs) .$
In the above derivation, the assumption is made that $D(s=0)=dD(s=0)/ds=0$ at the RF cavity.
Next, using the generating function $F2$,
(B12)
$F2(x¯,px,y¯,py,W,t¯)=x¯p˜x+y¯p˜y+W(ωRFt¯−hsR) ,$
we make a canonical transform from $(x¯,p¯x),(y¯,p¯y),(t¯,−ΔE¯)$ to $(x˜,p˜x),(y˜,p˜y),(ϕcp,W)$, respectively, using
(B13)
$p¯x=∂F2∂x¯=p˜x,$
(B14)
$x˜=∂F2∂p˜x=x¯,$
(B15)
$p¯y=p˜y,$
(B16)
$y˜=y¯,$
(B17)
$−ΔE¯=∂F2∂t¯=WωRF,$
(B18)
$ϕcp=∂F2∂W=ωRFt¯−hsR.$
Then, the new Hamiltonian $H2$ is described as
(B19)
$H2≃p˜x2+p˜y22ps+ps2(1+WωRFEs)(Kxx˜2+Kyy˜2) +eΦc(x˜−WhpsRD,y˜,−cβsϕcpωRF+Dp˜xps−dDdsx˜)βsγs2c −psEs(x˜−ωRFWpsβscD)Fx−KxDpsx˜W2ωRF2c2+12KxD2βsW3ωRF3ps2c3 +(1γs2−Dρ)h22R2psW2−eVrfωRFδp(s)cos(ϕcp+φs),$
where Eq. (B2) is used.
By extracting the Hamiltonian $H3,L$ for the synchrotron oscillation, we obtain
(B20)
$H3,L≡−ηh22R2psW2+eVrfcosφs4πRωRFϕcp2=−ηh22R2psW2−Esc2βs42ηω0RωRF2(νs0R)2ϕcp2,$
where the synchrotron tune $νs0$ is given by
(B21)
$νs0=1βs(−ηheVrfcosφs2πEs)1/2.$
Accordingly, Eq. (B19) is rewritten as
(B22)
$H2≃H3,L+p˜x2+p˜y22ps+ps2(1+WωRFEs)(Kxx˜2+Kyy˜2) +eΦc(x˜−WhpsRD,y˜,−cβsϕcpωRF+Dp˜xps−dDdsx˜)βsγs2c −psEs(x˜−ωRFWpsβscD)Fx−KxDpsx˜W2ωRF2c2+12KxD2βsW3ωRF3ps2c3 +(1γs2−Dρ)h22R2psW2−eVrfωRFδp(s)cos(ϕcp+φs) +ηh22R2psW2−eVrfcosφs4πRωRFϕcp2.$
Before describing the Hamiltonian in terms of action-angle variables, let us continue to make the canonical transformations from $(x˜,p˜x),(y˜,p˜y)$ to $(x¯¯,p¯¯x),(y¯¯,p¯¯y)$, respectively, which are generated by the function $F3$:
(B23)
$F3(x¯¯,p˜x,y¯¯,p˜y)=−x¯¯p˜xps−y¯¯p˜yps.$
The canonical transformations are expressed as
(B24)
$p¯¯x=p˜xps, x˜=x¯¯ps,$
(B25)
$p¯¯y=p˜yps, y˜=y¯¯ps.$
The new Hamiltonian $H3$ is divided as
(B26)
$H3=H3,L+H3,T+ΔH3,T+ΔH3,L,$
where $H3,L$ is given by Eq. (B20), and
(B27)
$H3,T≡p¯¯x2+p¯¯y22+12(Kxx¯¯2+Kyy¯¯2)−psEs(x¯¯ps−ωRFWpsβscD)Fx,$
(B28)
$ΔH3,T=WωRF2Es(Kxx¯¯2+Kyy¯¯2)+eΦc(x¯¯ps−WhpsRD,y¯¯ps,−cβsϕcpωRF+Dp¯¯xps−dDdsx¯¯ps)βsγs2c −KxDpsx¯¯psW2ωRF2c2+12KxD2βsW3ωRF3ps2c3,$
(B29)
$ΔH3,L=(1γs2−Dρ)h22R2psW2−eVrfωRFδp(s)cos(ϕcp+φs)+ηh22R2psW2−eVrfcosφs4πRωRFϕcp2.$
For the longitudinal motion, let us consider the generating function
(B30)
$F(ϕcp,ϕL,s)=−cpsβsνs02h2|η|ω0ϕcp2tan(ϕL−π2) ,$
which gives
(B31)
$ϕcp=(2JLh2|η|ω0cpsβsνs0)1/2sinϕL,$
(B32)
$W=(2JLνs0cpsβs|η|h2ω0)1/2cosϕL.$
Consequently, the Hamiltonian is written as
(B33)
$H3,L+ΔH3,L=−|η|ηνs0RJL+|η|ηνs0RJLsin2ϕL −eVrfωRFδp(s)cos((2JLh2|η|ω0cpsβsνs0)1/2sinϕL+φs) .$
To extract the Twiss parameters dependence from the transverse variables $(x¯¯,p¯¯x)$ and $(y¯¯,p¯¯y)$, we consider the canonical transformations generated by the function $F4$:
(B34)
$F4(x¯¯,ψx,y¯¯,ψy,s)=−x¯¯22βx(s)[tan(ψx+ϕx(s)−QxRs)+αx(s)] −y¯¯22βy(s)[tan(ψy+ϕy(s)−QyRs)+αy(s)],$
(B35)
$ϕx(s)=∫sdsβx(s),$
(B36)
$ϕy(s)=∫sdsβy(s).$
We obtain the canonical transformations from $(x¯¯,p¯¯x),(y¯¯,p¯¯y)$ to $(Jx,ψx),(Jy,ψy)$, respectively, as
(B37)
$p¯¯x=∂F4∂x¯¯=−x¯¯βx(s)[tan(ψx+ϕx(s)−QxRs)+αx(s)] ,$
(B38)
$Jx=−∂F4∂ψx=x¯¯22βx(s)cos2(ψx+ϕx(s)−QxRs),$
(B39)
$p¯¯y=−y¯¯βy(s)[tan(ψy+ϕy(s)−QyRs)+αy(s)] ,$
(B40)
$Jy=y¯¯22βy(s)cos2(ψy+ϕy(s)−QyRs),$
where the Twiss parameters satisfy
(B41)
$d2ds2βi+Kiβi−1(βi)3=0,$
(B42)
$αi=−12dβids,$
(B43)
$βiγs,i−αi2=1,$
and $i$ denotes $x$ or $y$.
Thus, the new Hamiltonian $H4$ is expressed as
(B44)
$H4≃H4,0+ΔH4,$
where
(B45)
$H4,0=QxJxR+QyJyR−η|η|νs0JLR −2βsc((βx(s)Jx2ps)1/2cos(ψx+ϕx(s)−QxRs)−D(s)βs(ω0νs0JL2Es|η|)1/2cosϕL)Fx,$
(B46)
$ΔH4=Kxβx(s)Jx(2JLνs0βs2ω0|η|Es)1/2cosϕLcos2(ψx+ϕx(s)−QxRs) +Kyβy(s)Jy(2JLνs0βs2ω0|η|Es)1/2cosϕLcos2(ψy+ϕy(s)−QyRs)+eΦc(X,Y,Z)βsγs2c −KxD(2βx(s)JxcβsEs)1/22JLω0νs0βs|η|cos(ψx+ϕx(s)−QxRs)cos2ϕL +Kx2D2ω03Esc(JLνs0|η|ω0)3/2cos3ϕL +|η|ηνs0RJLsin2ϕL−eVrfωRFδp(s)cos((2JLh2|η|ω0cpsβsνs0)1/2sinϕL+φs),$
(B47)
$X=(2βx(s)Jxps)1/2cos(ψx+ϕx(s)−QxRs)−hpsRD(2JLνs0cpsβs|η|h2ω0)1/2cosϕL,$
(B48)
$Y=(2βy(s)Jyps)1/2cos(ψy+ϕy(s)−QyRs) ,$
(B49)
$Z=cβsωRF(2JLh2|η|ω0cpsβsνs0)1/2sinϕL +Dps(2Jxβx(s))1/2[αx(s)cos(ψx+ϕx(s)−QxRs)+sin(ψx+ϕx(s)−QxRs)] +dDds(2βx(s)Jxps)1/2cos(ψx+ϕx(s)−QxRs) .$
The application of the canonical perturbation theory (see, e.g., Ref. [36]) for the Hamiltonian and neglecting the higher-order terms lead to the new Hamiltonian $H$:
(B50)
$H≃QxJx+QyJy+JLνs0+Ux+Y′,$
with its independent variable $θ=s/R$, where
(B51)
$Ux=−R2βsc((βx(s)Jx2ps)1/2cos(ψx+ϕx(s)−QxRs)−D(s)βs(ω0νs0JL2Es|η|)1/2cosϕL)Fx,$
(B52)
$Y′=eR8π3βsγs2c∫02πdψx∫02πdψy∫02πdϕL ×Φc[(2βx(s)Jxps)1/2cosψx−D(s)R(2JLνs0cβs|η|psω0)1/2cosϕL,(2βy(s)Jyps)1/2cosψy, ch(2JL|η|ω0Esνs0)1/2sinϕL+D(s)ps(2Jxβx(s))1/2(αx(s)cosψx+sinψx) +dDds(2βx(s)Jxps)1/2cosψx] ,$
$Ux$ and $Y′$ are the effect of the horizontal wake and the space charge forces, respectively.
Here, we consider a rather nonrelativistic condition, namely, a long bunch beam in the ring with the conditions
(B53)
$(2βx(s)Jxps)1/2≪|D|R(2JLνs0cβs|η|psω0)1/2,$
(B54)
$|Dps(2Jxβx(s))1/2(αx(s)+1)+dDds(2βx(s)Jxps)1/2|≪ch(2JL|η|ω0Esνs0)1/2.$
In this case, the function $Y′$ is approximated as
(B55)
$Y′=Ycoh,0′(JL)+Y′coh,2(JL)(βx(s)Jxps+βy(s)Jyps) +Y′coh,4(JL)(3βx2(s)Jx22+3βy2(s)Jy22+2βx(s)βy(s)JxJy) ,$
where
(B56)
$Ycoh,0′(JL)=eRZ0eNb2π2βsγsσx2∫0adρ′ρ′exp(−ρ′22σx2)∫0∞dλexp(−λ2γs2σz22) ×[K0(λρ′)−K0(λa)I0(λa)I0(λρ′)]J0[γsλch(2JL|η|ω0Esνs0)1/2] ,$
(B57)
$Y′coh,2(JL)=eRZ0eNb2π2σx2βsγs∫0adρ′ρ′exp(−ρ′22σx2)∫0∞dλ(λ24+ρ′24σx4−12σx2) ×exp(−λ2γs2σz22)[K0(λρ′)−K0(λa)I0(λa)I0(λρ′)]J0[γsλch(2JL|η|ω0Esνs0)1/2] +eRZ0eNb4σx4π2βsγs∫0adρ′ρ′2exp(−ρ′22σx2)∫0∞dλλexp(−λ2γs2σz22) ×[K1(λρ′)−K1(λa)I1(λa)I1(λρ′)]J0[γsλch(2JL|η|ω0Esνs0)1/2] −eRZ0eNb8π2βsγs2σx2σz(π2)1/2exp(−c2JL|η|2σz2h2ω0Esνs0)I0(c2JL|η|2σz2h2ω0Esνs0),$
(B58)
$Y′coh,4(JL)=eRZ0eNb16π2σx2βsγsps2∫0adρ′ρ′exp(−ρ′22σx2)J0[γsλch(2JL|η|ω0Esνs0)1/2] ×∫0∞dλ(λ48+ρ′2λ22σx4+ρ′48σx8−λ2σx2−ρ′2σx6+1σx4)exp(−λ2γs2σz22) ×[K0(λρ′)−K0(λa)I0(λa)I0(λρ′)] +eRZ0eNb8π2σx4βsγsps2∫0adρ′exp(−ρ′22σx2)J0[γsλch(2JL|η|ω0Esνs0)1/2] ×∫0∞dλ(ρ′4λ4σx4+ρ′2λ34−ρ′2λσx2)exp(−λ2γs2σz22)[K1(λρ′)−K1(λa)I1(λa)I1(λρ′)] +eRZ0eNb64π2σx6βsγsps2∫0adρ′exp(−ρ′22σx2)ρ′3∫0∞dλλ2exp(−λ2γs2σz22) ×[K2(λρ′)−K2(λa)I2(λa)I2(λρ′)]J0[γsλch(2JL|η|ω0Esνs0)1/2] +eRZ0eNbβsγs2π2σx2ps2(π2)1/2exp(−c2JL|η|2h2ω0Esνs0σz2) ×[(−1128γs2σz3+164σx2σz+c2JL|η|128γs2σz5h2ω0Esνs0) ×I0(c2JL|η|2h2ω0Esνs0σz2)−c2JL|η|I1(c2JL|η|2h2ω0Esνs0σz2)128γs2σz5h2ω0Esνs0].$
Finally, they are simplified to Eqs. (7), (8), and (9) as in the text, under a typical parameter region.
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1After this paper had been submitted on 22 June 2016, the authors attended the 57th ICFA Advanced Beam Dynamics Workshop on High-Intensity and High-Brightness Hadron Beams (HB2016: https://hb2016.esss.se/) and found that A. Burov had submitted a document entitled “Coupled-beam and coupled-bunch instabilities” to http://arxiv.org/pdf/1606.07430v1.pdf on 27 June 2016. He discusses the space charge effect on coupled-bunch-type instabilities by another approach.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
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# Weekday captions with the TikZ library “calendar”
I use the TikZ library calendar to produce a week list. The code looks like this:
\documentclass{article}
\usepackage{pgf,tikz}
\usetikzlibrary{calendar}
\begin{document}
\tikz\calendar[dates=2011-02-01 to 2011-02-last,week list];
\end{document}
However, I want to add abbreviated captions for the weekdays (M, T, W, ...) in the first line of the week list:
M T W T F S S
1 2 3 4 5 6
7 8 9 10 11 12 13
14 15 16 17 18 19 20
21 22 23 24 25 26 27
28
I couldn't find any option to the \calendar macro which produces this effect. Is there any other solution to achieve my goal?
-
There's no way of doing this with a single option, but the following code is a way of getting the weekday headings in a way that keeps all options of the week list calendar. It adds the style day letter headings (which in this form only makes sense to use with the week list style). The style of the headings can be set with the option day headings.
\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{calendar}
\makeatletter%
execute before day scope={ \ifdate{day of month=1}{%
\pgfmathsetlength{\pgf@ya}{\tikz@lib@cal@yshift}%
\pgfmathsetlength\pgf@xa{\tikz@lib@cal@xshift}%
\pgftransformyshift{-\pgf@ya}
\foreach \d/\l in {0/M,1/T,2/W,3/T,4/F,5/S,6/S} {
\pgf@xa=\d\pgf@xa%
\pgftransformxshift{\pgf@xa}%
\pgftransformyshift{\pgf@ya}%
}
}{}%
}%
]
\makeatother%
\begin{document}
\tikz\calendar[dates=2011-02-01 to 2011-04-last,
week list,
month label above centered,
day xshift = 0.8cm,
BTW: A slash is missing between \d and \l (it should be \foreach \d/\l in {0/M,1/T,2/W,3/T,4/F,5/S,6/S} {, otherwise, I get an error message). – diabonas Feb 4 '11 at 12:39
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# Primary fuel
Primary fuels or primary energy sources are dense sources of primary energy found as natural resources. Primary fuels are fuels that are found in nature and can be extracted, captured, cleaned, or graded without any sort of energy conversion or transformation process. This means that all processing and collecting of the fuel is done before the fuel is converted into heat or mechanical work.[1] These primary fuels tend to be non-renewable, and some of the most commonly known primary fuels are fossil fuels. Energy harvested from primary fuels tends to make up most of a country's total primary energy supply or TPES.
Primary fuels such as coal should be distinguished from flows like wind and solar power. Although they are all sources of primary energy, primary fuels are "consumed". In contrast, a flow such as wind doesn't get "consumed"—rather, it makes more sense to think of flows as being "utilized". It is also important to distinguish primary fuels from secondary fuels like gasoline. Secondary fuels are derived from primary fuels through some sort of distillation process. Gasoline, for example, is created after distilling a primary fuel, oil.
The distinction between primary fuels and energy currencies like electricity is also important. Primary fuels are sources of primary energy, and this energy—once obtained from the fuel itself—is transformed into forms that are easier to use, transport, or store.[2] Essentially, energy currencies are ways of transporting the energy obtained from primary fuels.
## Types of Primary Fuels
As mentioned above, most of the primary fuels used currently are non-renewable. However, one major renewable primary fuel is from biomass sources. Other examples of primary fuel include:[3]
Primary fuels such as these can be obtained from the ground using several different mining techniques. While coal is mined more "traditionally" using strip mines or deep mines, certain sources of natural gas such as shale gas or tight gas are obtained by fracturing large stones with poor permeability. Overall, there are many different ways to obtain these vital resources.
## Interactive Data Simulation
The data simulation below shows the energy production of countries worldwide based on their energy source. Note that some of the energy sources are from flows, such as hydro, wind, and solar. Production from the fuels such as oil, coal, peat, natural gas, and nuclear (uranium and thorium) are all energy produced from primary fuels.
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# What is the binding energy of one mole of 234Th90 if the mass defect is 1.908 g/mol?
May 3, 2017
Well, $\text{energy"="mass} \times {c}^{2}$, where $c = \text{speed of light.}$
#### Explanation:
You know that masses are conserved in chemical reactions. However, this is a PHYSICAL not a chemical reaction, where nuclear processes occur with the release of a vast amount of energy. So try it out...........
We know that $1 \cdot g \equiv 1 \times {10}^{-} 3 \cdot k g$.
${N}_{A} = 6.022 \times {10}^{23} \cdot m o {l}^{-} 1$
$c = 3.00 \times {10}^{8} \cdot m \cdot {s}^{-} 1$.
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# General FAQs
Performance Breakdowns on higher Core Numbers
In case you would like to optimize the performance of your application or you observe performance breakdowns on higher core numbers please test the following scenario:
The variable PSP_ONDEMAND influences the creation of MPI connections. If you change the value of this variable into PSP_ONDEMAND=1 within your batch script, then the connections will be created dynamically when they are used the first time. We observed a performance increase of several applications with these dynamic connections.
Our recommendation is to perform a test run with PSP_ONDEMAND=1 and to compare the results to the runs without this specification.
Attention: If you have all-to-all communication in your application, PSP_ONDEMAND=1 might not be possible, see Using dynamic memory allocation for MPI connections.
Regular and -mt Variants of the ParaStation Modules on Juropa
Question:
Can someone help clarify the expected use of the regular and -mt variants of the parastation modules on juropa:
module whatis parastation parastation/gcc: Parastation library for parallel computing (GCC). parastation/intel: Parastation library for parallel computing (Intel Compiler). parastation/gcc-mt: Parastation library for multi-threaded parallel computing (GCC). parastation/intel-mt: Parastation library for multi-threaded parallel computing (Intel Compiler).
Is the expectation that the "-mt" variants should be exclusively/preferentially used for hybrid/mixed-mode applications combining OpenMP+MPI (as seems to be suggested by the module descriptions) and/or also for pure MPI applications using non-blocking communication (where internal MPI threads are used to improve performance)?
How can I activate Turbo Mode on the Nehalem processors of Juropa/HPC-FF?
Turbo Mode makes it possible to automatically overclock the cores under certain conditions (see: Intel Turbo Boost Technology). The standard frequency of 2.933 GHz can be increased to a maximum value of roughly 3.2 GHz.
The following command enables Turbo Mode on the cores of the reserved compute node:
msub -l nodes=1:ppn=8:turbomode
Information about the actually applied clock frequency can be obtained from the following file of the corresponding node:
/sys/devices/system/cpu/cpu?/cpufreq/cpuinfo_max_freq
The default version of ParaStation-MPI installed on JUROPA/HPC-FF does not support MPI_THREAD_MULTIPLE, i.e. it is not possible that multiple threads may call MPI, with no restriction. This functionality is given by a special version of ParaStation-MPI that can be used through the invokation of the corresponding module:
In order to avoid conflicts, ensure that at the relevant time only one MPI version is loaded.
My scp file transfer crashes with time limit. What can I do?
File transfer with scp (ssh secure copy) may consume significant amounts of CPU time due to the inherent data encryption/decryption. In order to allow for the transfer of big files to and from JUROPA/HPC-FF the CPU time limit has been specifically increased on the GPFS nodes.
CPU limits
GPFS nodes21600 sec
For this reason, it is strongly recommended to use the GPFS nodes for file transfer instead of the Login nodes.
Example:
scp <userID>@juropagpfs.fz-juelich.de:<source file> <destination file>
For what do I need the module tool?
On JUROPA/HPC-FF, general purpose applications and libraries are made available to users through the use of the module command. The user's environment in the current shell will be updated so that the software under consideration can be used. To get an overview of the modules available on JUROPA/HPC-FF type module avail on the command line. Further useful commands are:
CommandDescription
module load <module>Enables the use of the corresponding software package
module listPrints out a list of loaded modules
module help <module>Gives some information about the package under consideration
module unload <module>Opposite of module load <module>. Some software packages provoke conflicts if several versions are installed at the same time, so it might make sense to unload versions that are not needed for the moment
module show <module>Shows information about the location of the software and variables that will be set by invoking this <module>
I have deleted some critical files by mistake, is it possible to recover them?
Files in your home directory are backed up on tape. To recover the deleted files, please do the following:
1. Log on to one of the Juropa-GPFS-Nodes:
ssh -X <userid>@juropagpfs.fz-juelich.de
2. Start the backup recovery routine:
3. On the prompt, select home as the desired target:
home
4. A graphical panel will pop up. Select the function restore from the Backup panel.
5. A new window will pop up, showing the Juropa file hierarchy. Open the tree File Level and herein your home directory:
e.g.: jhome12 and then hmz29 and hmz298.
6. You can now further refine the choice of data you want to restore and finally checkmark all files and/or subtrees you need to restore.
7. Finally select the button Restore and decide whether to restore into the original place or to a new location.
Please note that the restore may take a couple of minutes, since the data has to be retrieved from magnetic tape.
How can I include Fortran subroutines in C programs?
In order to use some general Fortran subroutines in C programs you have to include the corresponding libraries.
-lifcore -lifport
PSIlogger: Timeout: Not all clients joined the first pmi barrier ...
mpiexec along with -x can be used for the export of all environment variables to the processes spawned by mpiexec. Unfortunately, this strategy might provoke the error given in the headline in dependence of the amount of variables which are exported. Instead, it is recommended to export the needed environment variables by the option --exports. An example can be found here: Quick Introduction
ipo: warning #11010: file format not recognized ...
Object files generated by the Intel compilers using the option -ipo contain additional information for the compiler/linker in order to perform code optimizations. The message ipo: warning #11010: file format not recognized for ..., possible linker script occurs if the gnu command ar is used to build static libraries from such object files. To avoid this warning, please use the Intel tool
xiar
Important: If you ignore this warning, the compiler cannot find the corresponding object files and finally the linker will abort with an unresolved symbol error.
Intel Compiler 12.0.3: ld: cannot find -lmkl_lapack
Starting with Intel Compilers 12.0.3 the LAPACK routines are no longer in a separate library mkl_lapack but in mkl_intel_lp64. If your Makefile contains -lmkl_lapack you will get the error message ld: cannot find -lmkl_lapack. You can just omit the -lmkl_lapack and linking will work as expected.
Error:Connecting ... failed : Invalid exchange - Protocol driver not attached
The following error messages during batch job execution may hint to an out-of-memory condition:
Error:Connecting 10.1.22.23:51388 to 10.1.17.42:50020 (rank 1476 to 4058) failed : Invalid exchangeError:Connecting 10.1.21.49:59181 to 10.1.16.50:56717 (rank 1940 to 4520) failed : Protocol driver not attached
The given IP addresses, port and rank numbers may vary from case to case.
In order to solve the problem, try one of the solutions given in chapter Memory Optimisation:
How to generate and upload ssh keys?
In order to access the JSC computer systems you need to generate a SSH key pair. This pair consists of a public and a private part.
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# ergodic
share
## Examplesergodic's examples
• Is it possible to decompose an invariant measure into ergodic. measures in such a way that the study of the former reduces to the One can easily show that ergodic measures are extreme points of. — “A Guide to the Ergodic Decomposition Theorem: Ergodic”, cimat.mx
• In this volume we develop the beginnings of ergodic theory and dynamical view on ergodic theory, with different kinds of examples, may be found. — “Ergodic Theory”,
• of determining K and Γ so that action of Γ on K is ergodic if and only if Γ ergodically but no element of which is ergodic. If K is a compact connected finite. — “Distal actions and ergodic actions on compact groups”, nyjm.albany.edu
• Ergodic theory. A discrete dynamical system consists of a structure, X , and an map T. from X to X : Think of the underlying set of X as the set of states of a system. If x is a state, T x gives the state after one unit of time. In ergodic theory, X is assumed to be a finite measure space (X, B, µ). — “Computability in ergodic theory”, andrew.cmu.edu
• Definition of word from the Merriam-Webster Online Dictionary with audio pronunciations, thesaurus, Word of the Day, and word games. Definition of ERGODIC. 1 : of or relating to a process in which every sequence or sizable sample is equally representative of the whole (as in regard to a statistical. — “Ergodic - Definition and More from the Free Merriam-Webster”, merriam-
• Definition of ergodic in the Online Dictionary. Meaning of ergodic. Pronunciation of ergodic. Translations of ergodic. ergodic synonyms, ergodic antonyms. Information about ergodic in the free online English dictionary and encyclopedia. ergodic. — “ergodic - definition of ergodic by the Free Online Dictionary”,
• ergodic theory ( ər′gädik ′thēərē ) ( mathematics ) The study of measure-preserving transformations. ( statistical mechanics ) Mathematical theory. — “Ergodic theory: Definition from ”,
• An ergodic dynamical system is one in which, with respect to some probability distribution, all invariant sets either have measure 0 or measure 1. (Sometimes non-ergodic systems can be decomposed into a number of components, each of which is separately ergodic. — “Ergodic Theory”, cscs.umich.edu
• A central aspect of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. Two of the most important examples are ergodic theorems of Birkhoff and von Neumann. — “Ergodic theory - Wikipedia, the free encyclopedia”,
• Let be an H-invariant probability measure on which is ergodic with respect to H (i.e. all H-invariant sets either have full measure or zero measure). Then is homogeneous in the sense that there exists a closed connected subgroup and a closed orbit such that is L-invariant and supported on Lx. — “254A – ergodic theory " What's new”,
• We say that $T$ is ergodic if all the subsets $A \in \mathfrak{B}$ such that $T^{-1}(A)=A$ have measure $0$ or $1 The transformation$T$is ergodic precisely when$T\$ cannot be decomposed into simpler transformations. — “PlanetMath: ergodic”,
• 1.1 What is ergodic theory and how it came about. Dynamical systems and ergodic theory. Ergodic theory is a part of the theory of 1.2 The abstract setup of ergodic theory. 3. Poincar'e's Recurrence. — “Lecture Notes on Ergodic Theory”, math.psu.edu
• Computational ergodic theory, by Geon Ho Choe, Springer, Berlin, Heidelberg, ergodic, but in many cases it is not. In the discussion of recurrence properties there is so far no mention. — “Computational ergodic theory, by Geon Ho Choe, Springer”,
• Ergodic theory, metric theory of dynamical systems. The branch of the theory of dynamical systems that studies systems with an invariant measure and related problems. 1) In the "abstract" or "general" part of ergodic theory one examines measurable dynamical systems. — “Springer Online Reference Works”,
• 2.6.3 The Multiplicative Ergodic Theorem for Invertible Cocycles 61 1.1 What is ergodic theory and how it came about. Dynamical systems and ergodic theory. Ergodic theory is a part of the theory of. — “Lecture Notes on Ergodic Theory”, wisdom.weizmann.ac.il
• Ergodic theory has connections to many areas of mathematics, but primarily to the area Due to the recent development of the subject and the requisite background, ergodic theory. — “May 1, 2003 ERGODIC THEORY”, math.utah.edu
• We outline the ergodic theory background needed to un- derstand these results, with an emphasis on recent developments in ergodic. theory and the relation to recent developments in additive combinatorics. These notes are based on four lectures given during the School on Additive. — “Ergodic Methods in Additive Combinatorics”, math.northwestern.edu
• Ergodic theory has fundamental applications in. probability theory, starting from areas Definition 6 A dynamical system is called ergodic if it has no nontrivial in. — “Ergodic Theory”, users.ece.utexas.edu
• on the set EA(G) of isomorphism classes of ergodic actions of G. such that the following holds: for any continuous field of ergodic. actions of G over a locally compact Hausdorff space T the map. T EA(G) sending each t in T to the isomorphism class of the. — “COMPACT QUANTUM METRIC SPACES AND ERGODIC ACTIONS OF COMPACT”, math.buffalo.edu
• A collection of systems forms an ergodic ensemble if the modes of behavior found in any one system from time to time resemble its behavior at other temporal periods and if the behavior of any other system when chosen at random also is like the one system. — “ERGODIC”, pespmc1.vub.ac.be
• ergodic (comparative more ergodic, superlative most ergodic) (mathematics, physics) Of or related to certain systems that, given enough time, will eventually return to previously experienced state. (statistics, engineering) Of or relating to. — “ergodic - Wiktionary”,
• Global aspects of ergodic group actions. Introduction. In this talk, I will discuss some aspects of recent work concerning. the global structure of the space of measure preserving actions and. their associated cohomology. This is part of a forthcoming book,. — “Global aspects of ergodic group actions”, math.ucla.edu
## Blogs & Forumblogs and forums about ergodic
• “Para evitar problemas con usuarios (o intrusos) que consuman demasiados recursos la mejor Primero, hemos de editar los servicios con la configuración PAM”
— Limitar los recursos de un usuario en GNU/Linux " Computación, ergodic.ugr.es
• “forum.universal- - Welcome to UDI Forum. forum.universal- Forum Viewing profile :: ergodic. Avatar. All about ergodic. Advanced. Joined: 19 Mar 2009. Total”
— forum.universal- :: Viewing profile, forum.universal-
• “New Indicator: Ergodic Indicator (Page 1) - Technical Indicators - Forex Forum - Forex Software and Strategies”
— New Indicator: Ergodic Indicator (Page 1) - Technical,
• “Welcome to Max Raginsky's new blog, The Information Structuralist, which in two weeks has already produced more useful content than this poor blog does in several months. Posted by Anand Sarwate under Uncategorized | Tags: blog”
— blog " An Ergodic Walk,
• “Ergodic TVI custom indicator downloaded from this forum comes up blank on screen - MQL4 forum”
• “YF on An Ergodic Walk Blog. October 17, 2010. Another YF review': http:// UT Blog: House Seats. Favorite Links. Actors' Equity Association”
— YF on An Ergodic Walk Blog " Mo`olelo Blog,
• “That's why this blog has been inactive for the past three months. But now that the But this isn't a blog entry about his music, his level of fame, or”
— Shailesh's Ergodic Life,
• “Ergodic CCI”
Ergodic CCI,
• “[Archive] Request for Ergodic indicator Suggestions And Feedback NinjaTrader Support Forum > Suggestions and Feedback > Suggestions And Feedback > Request for Ergodic indicator”
— Request for Ergodic indicator [Archive] - NinjaTrader Support,
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## Postdoctoral Research Positions in Mathematics (Algebraic Number Theory and Its Applications to Communications and Storage)
Organization:
Aalto University, School of Science, Department of mathematics and systems analysis
Email:
camilla.hollanti$aalto.fi Job Description: The Algebra and Combinatorics research group in the Department of Mathematics and Systems analysis is presently seeking to hire 1-3 postdoctoral researchers in the field of algebraic number theory and its applications to communications and storage. For more information on the department's research fields in general and the department itself, please see www.mathsys.aalto.fi/en The positions are fixed term for the maximum period of two years, starting at the latest by September 1st, 2012. Candidates are required to hold a PhD degree before the start of the contract in order to qualify. Successful candidates will carry out research in collaboration with the Algebra and Combinatorics research group active at the department, with some teaching included. The salary is competitive, determined based on experience and qualifications, including occupational health and social security benefits. The application should include at least -CV -Cover letter justifying why the candidate is suitable for working in this research group -Short description of previous research and teaching experience The applications should be in English, addressed to 'Camilla Hollanti and the research group', and e-mailed to camilla.hollanti$aalto.fi by
Tuesday, March 20th 2012.
HR Coordinator Janna Ahtiainen, (in recruitment process questions and practical issues)
or Camilla Hollanti (research areas, and departmental information). Our e-mails follow the form “firstname.lastname@aalto.fi”.
Job Categories:
Universities and colleges
Postdoctoral fellowships
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QUESTION
# Homework help
)In a certain year, when she was a high school senior, Idonna scored 670 on the mathematics part of the SAT. The distribution of SAT math scores in that year was Normal with mean 515 and standard deviation 113. Jonathan took the ACT and scored 21 on the mathematics portion. ACT math scores for that year were Normally distributed with mean 20.9 and standard deviation 4.7. Find the standardized scores for both students. (Round your answers to two decimal places.)
Idonna =
z = (x-µ)/sd
z =(670-515)/113 = 1.37
Jonathan =
z = (x-µ)/sd
(21-20.9)/4.7 = 0.02
2)Almost all medical schools in the United States require students to take the Medical College Admission Test (MCAT). The exam is composed of three multiple-choice sections (Physical Sciences, Verbal Reasoning, and Biological Sciences). The score on each section is converted to a 15-point scale so that the total score has a maximum value of 45. The total scores follow a Normal distribution, and in a certain year the mean was 25.2 with a standard deviation of 6.9. There is little change in the distribution of scores from year to year. (Round your answers to four decimal places.)
(a) What proportion of students taking the MCAT had a score over 32
For the proportion of student who scored above 32 is
X> 32 which corresponds with Z > (32- 25.2)/6.9 = 0.99
from a standard normal distribution the probability of a z value of less or equal to 0.99 is 0.8389
so P(X>32) =1-0.8389= 0.1611
so the PROPORTION which scored above 32 is
0.1611
(b) What proportion had a score between 22 and 25?
Z (22) = (22- 25.2)/6.9= -0.46
Z(25) = (25- 25.2)/6.9 = -0.02
we are checking the probability of lying between
-0.02> Z > -0.46
=.4920 – 3228 =0.1692
3)The total scores on the Medical College Admission Test (MCAT) follow a Normal distribution with mean 24.8 and standard deviation 6.9.
(a) What are the median and the first and third quartiles of the MCAT scores? (Enter your median to one decimal place. Round your quartiles to the nearest whole number.)
median=
The median is 50% point of the data which is a probability 0.5
From standard normal distribution it corresponds to a z of 0.00
0.0 =( x –24.8)/6.9 x = 24.8
Again a normal distribution is symmetrical with mean at the center
Q1= this is the 0.25 of the left tail
The z value for this probability is -0.67
So z= (x- 24.8)/6.9 =-0.67
-4.623 = x-24.8
X = -4.23+24.8 =20
Q3= this is the 0.75 of the data
The z value for this probability is 0.67
Z =(x- 24.8)/6.9 =0.67
4.623 +24.8=29
What is the interquartile range? (Round your answer to the nearest whole number.)
The interquartile range is upper quarter – lower quater
29-20 =9
(b) Give the interval that contains the central 80% of the MCAT scores. (Round your answers to the nearest whole number.)
This can be obtained by taking 90th perventile – 10th percentile
For the 90th we find z corresponding to probability 0.9
This is 1.28 from an std normal table
1.28=( x -24.8)/6.9
8.832 =x-24
X= 32.832 =33
For 0.10 corresponds to -1.29
-1.29 =( x-24.8)/6.9
-8.901 =x-24.8
X= 15.899 = 16
The interval is 16 -32
4)It appears that people who are mildly obese are less active than leaner people. One study looked at the average number of minutes per day that people spend standing or walking. Among mildly obese people, minutes of activity varied according to the
N(37868)
distribution. Minutes of activity for lean people had the
N(526106)
distribution. Within what limits do the active minutes for 95% of the people in each group fall? Use the 68-95-99.7 rule. Within what limits do the active minutes for 95% of the people in the mildly obese group fall?
Mean =378 sd =68
It lies between 2.5th quartile and 97.5th quartile
The z value for 0.975 is 1.96
1.96= (x-378)/68
133.28= x-378
X= 133.28+ 378 = 511.28
For 2.5th quartile
Z =-1.96
-133.28 =x-378
X= 244.72
( 244.72 ) to ( 511 ) min
Within what limits do the active minutes for 95% of the people in the lean group fall?
It lies between 2.5th quartile and 97.5th quartile
Mean =526,sd =106
The z value for 0.975 is 1.96
1.96= (x-526)/106
207.76= x-526
X= 207.76+ 526 = 733.76
For 2.5th quartile
Z =-1.96
-1.96= (x-526)/106
-207.76= x-526
X= -207.76+ 526 = 318.24
( 318 ) to ( 734 ) min
5)The common fruit fly Drosophila melanogaster is the most studied organism in genetic research because it is small, is easy to grow, and reproduces rapidly. The length of the thorax (where the wings and legs attach) in a population of male fruit flies is approximately Normal with mean 0.810 millimeters (mm) and standard deviation 0.083 mm. (Round your answers to four decimal places.)
(a) What proportion of flies have thorax lengths less than 0.71 mm?
Mean =0.810 ,sd=0.083
The z corresponding to length 0.71 = 0.55
Z =(x-u)/sd
=(0.71-0.810)/0.083
=-1.2048193
From the table it is
0.1151
(b) What proportion have thorax lengths greater than 0.91 mm?
The z corresponding to probability 0.09 is -1.34
Z= ( 0.91-0.810)/0.083
=1.2048193
From the table
=1-0.8849 =
=0.1151
(c) What proportion have thorax lengths between 0.71 mm and 0.91 mm?
Proportion corresponding to 0.91
= P(0.91) –P(0.71)= 0.8849-0.1151
= 0.7698
6)In a study of exercise, a large group of male runners walk on a treadmill for 6 minutes. Their heart rates in beats per minute at the end vary from runner to runner according to the
N(10612.6)
distribution. The heart rates for male nonrunners after the same exercise have the
N(13917)
(a) What percent of the runners have heart rates above 143
Z =(143-106)/12.6
=2.9365079
From an std normal table we have
0.9984
Prop Above 143 is 1-0.9984
=0.0016
(b) What percent of the nonrunners have heart rates above 143?
Z =(143-139)/17
=0.24
From an std normal table we have
0.5948
Prop Above 143 is 1-0.5948
=0.4052
Files: eee.docx
• @
• 10 orders completed
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# Prove that: $2^a+3^b<3a+4b$
Let be $a, b$ in $(0,1)$ such that $a+b>1$. I need to prove that:
$$2^a+3^b<3a+4b$$
I'm looking for an elementary proof that doesn't resort to the calculus tools.
-
Are you allow to use the fact that $2^a + 3^b$ is convex? Then you just need to check at the end-points $(a,b) \in \{ (0,1), (1,0), (1,1)\}$. – Willie Wong Jun 15 '12 at 12:36
@Willie Wong: yes. This idea came to me too late. – Chris's sis Jun 15 '12 at 12:41
From the graph of the function $f(x)=2^x$ we see that on interval $(0,1)$ it is bellow the line $y=x+1$ joining the points $(0,f(0))=(0,1)$ and $(1,f(1))=(1,2)$. Thus we have $$2^a<1+a$$ for $a\in(0,1)$.
Using similar argument for $3^x$ we get $$3^b<1+2b$$ for $b\in(0,1)$.
Adding the two inequalities together and using $1<a+b$ we obtain $$2^a+3^b<2+a+2b<2(a+b)+a+2b=3a+4b.$$
-
nice proof! Thanks! – Chris's sis Jun 15 '12 at 12:41
Using Bernoulli's Inequality, $2^a \leq 1 + a$ and $3^b \leq 1 + 2b$. Therefore, $$2^a + 3^b \leq 2 + a + 2b < 2(a + b) + a + 2b = 3a + 4b$$
The problem is that both proofs use calculus in a hidden way ;) Proving Bernoulli for $a \in (0,1)$ requires calculus, and so does the concavity of $2^x$.... – N. S. Jun 15 '12 at 14:18
That is true, but I think convexity of $2^x$ can be established without calculus. $2^{\lambda x + (1 - \lambda)y} = (2^x)^{\lambda} (2^y)^{1 - \lambda} \leq \lambda 2^x + (1 - \lambda) 2^y$ where the last step comes from the generalized AM-GM inequality. – TenaliRaman Jun 15 '12 at 16:54
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# How Do You Calculate Wavelength?
Wavelength is calculated by formula Lambda (Wavelength) = c/f where c= Speed of Light is Vaccum (3*10&8m/s) & f= frequency of the wave.
How to Calculate Wavelength
Electromagnetic waves are wavy disturbances known as the electric force field that repeat over a distance (or wavelength). Examples of electromagnetic waves include transmissions from light, microwaves, X-rays, television, radio, etc. Electromagnetic... More »
Difficulty: Moderate
Source: www.ehow.com
Q&A Related to "How Do You Calculate Wavelength"
In order to calculate wavelength, you first want to determine the speed of light in meters per seconds, which is 299,792,458 m/s. Then you want to find out what the frequency of the http://answers.ask.com/Science/Physics/how_to_calc...
1. Find out the frequency of the wave you are trying to measure. Whether you're dealing with microwaves or light rays, frequency is often indicated in hertz, megahertz or kilohertz. http://www.ehow.com/how_5146929_calculate-waveleng...
1 Know the formula for calculating wavelength. To find the wavelength of a wave, you just have to divide the wave's speed by its frequency. The formula for calculating wavelength http://www.wikihow.com/Calculate-Wavelength
frequency = Speed of light/ wavelength. http://wiki.answers.com/Q/How_to_calculate_the_wav...
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# Using colcon to build packages¶
This is a brief tutorial of how to create and build a ROS 2 workspace with colcon. It is a practical tutorial and not designed to replace the core documentation.
## Background¶
colcon is an iteration on the ROS build tools catkin_make, catkin_make_isolated, catkin_tools and ament_tools. For more information on the design of colcon see this document.
The source code can be found in the colcon GitHub organization.
## Prerequisites¶
### Install colcon¶
sudo apt install python3-colcon-common-extensions
python3 -m pip install colcon-common-extensions
pip install -U colcon-common-extensions
### Install ROS 2¶
To build the samples, you will need to install ROS 2.
Attention
If installing from Debian packages, this tutorial requires the desktop installation.
## Basics¶
A ROS workspace is a directory with a particular structure. Commonly there is a src subdirectory. Inside that subdirectory is where the source code of ROS packages will be located. Typically the directory starts otherwise empty.
colcon does out of source builds. By default it will create the following directories as peers of the src directory:
• The build directory will be where intermediate files are stored. For each package a subfolder will be created in which e.g. CMake is being invoked.
• The install directory is where each package will be installed to. By default each package will be installed into a separate subdirectory.
• The log directory contains various logging information about each colcon invocation.
Note
Compared to catkin there is no devel directory.
### Create a workspace¶
First, create a directory (ros2_example_ws) to contain our workspace:
mkdir -p ~/ros2_example_ws/src
cd ~/ros2_example_ws
mkdir -p ~/ros2_example_ws/src
cd ~/ros2_example_ws
md \dev\ros2_example_ws\src
cd \dev\ros2_example_ws
At this point the workspace contains a single empty directory src:
.
└── src
1 directory, 0 files
Let’s clone the examples repository into the src directory of the workspace:
git clone https://github.com/ros2/examples src/examples
Attention
It is recommended to checkout a branch that is compatible with the version of ROS that was installed (e.g. crystal).
cd ~/ros2_example_ws/src/examples/
git checkout \$ROS_DISTRO
cd ~/ros2_example_ws
Now the workspace should have the source code to the ROS 2 examples:
.
└── src
└── examples
├── CONTRIBUTING.md
├── rclcpp
├── rclpy
4 directories, 3 files
### Source an underlay¶
It is important that we have sourced the environment for an existing ROS 2 installation that will provide our workspace with the necessary build dependencies for the example packages. This is achieved by sourcing the setup script provided by a binary installation or a source installation, ie. another colcon workspace (see Installation). We call this environment an underlay.
Our workspace, ros2_examples_ws, will be an overlay on top of the existing ROS 2 installation. In general, it is recommended to use an overlay when you plan to iterate on a small number of packages, rather than putting all of your packages into the same workspace.
### Build the workspace¶
Attention
To build packages on Windows you need to be in a Visual Studio environment, see Building the ROS 2 Code for more details.
In the root of the workspace, run colcon build. Since build types such as ament_cmake do not support the concept of the devel space and require the package to be installed, colcon supports the option --symlink-install. This allows the installed files to be changed by changing the files in the source space (e.g. Python files or other not compiled resourced) for faster iteration.
colcon build --symlink-install
After the build is finished, we should see the build, install, and log directories:
.
├── build
├── install
├── log
└── src
4 directories, 0 files
### Run tests¶
To run tests for the packages we just built, run the following:
colcon test
### Source the environment¶
When colcon has completed building successfully, the output will be in the install directory. Before you can use any of the installed executables or libraries, you will need to add them to your path and library paths. colcon will have generated bash/bat files in the install directory to help setup the environment. These files will add all of the required elements to your path and library paths as well as provide any bash or shell commands exported by packages.
. install/setup.bash
. install/setup.bash
call install\setup.bat
### Try a demo¶
With the environment sourced we can run executables built by colcon. Let’s run a subscriber node from the examples:
ros2 run examples_rclcpp_minimal_subscriber subscriber_member_function
In another terminal, let’s run a publisher node (don’t forget to source the setup script):
ros2 run examples_rclcpp_minimal_publisher publisher_member_function
You should see messages from the publisher and subscriber with numbers incrementing.
colcon uses the package.xml specification defined in REP 149 (format 2 is also supported).
colcon supports multiple build types. The recommended build types are ament_cmake and ament_python. Also supported are pure cmake packages.
An example of an ament_python build is the ament_index_python package , where the setup.py is the primary entry point for building.
A package such as demo_nodes_cpp uses the ament_cmake build type, and uses CMake as the build tool.
For convenience, you can use the tool ros2 pkg create to create a new package based on a template.
Note
For catkin users, this is the equivalent of catkin_create_package.
## Tips¶
• If you do not want to build a specific package place an empty file named COLCON_IGNORE in the directory and it will not be indexed.
• If you want to avoid configuring and building tests in CMake packages you can pass: --cmake-args -DBUILD_TESTING=0.
• If you want to run a single particular test from a package:
colcon test --packages-select YOUR_PKG_NAME --ctest-args -R YOUR_TEST_IN_PKG
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### Home > CCAA8 > Chapter 9 Unit 10 > Lesson CCA: 9.3.1 > Problem9-71
9-71.
Algeria has decided to take out an advertisement in the U.N. newspaper, Liberty Daily. The newspaper charges a base fee of $\1200$ for an ad. There is an additional fee of $\300$ for every inch in height. If Algeria is willing to spend any amount up to (and including) $\2700$, what choices does the country have for the height of the ad?
Let $x$ represent the extra inches in height.
Write an inequality using the information in the problem.
$300x+1200\le2700$
Find the boundary point.
Test numbers on both sides of the boundary point.
$x\le5$
Algeria can order an advertisement up to $5$ inches high.
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X}zŁ fɖ߂ S O100 100 ŐV50
read.cgi ver 05.04.02 2018/11/22 Walang Kapalit
FOX DSO(Dynamic Shared Object)
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• # C01 Automorphisms and embeddings of manifolds
This project concerns the homotopical properties of spaces of smooth and topological automorphisms of manifolds, their classifying spaces and spaces of smooth and topological embeddings of manifolds. Known characteristic classes for manifold bundles will play an important role. It is conceivable that new ones will be constructed. The action of automorphisms and embeddings on finite subsets of manifolds, more precisely on the configuration category of a manifold, will be exploited.
• # Project Leaders & Staff
Prof. Dr. Johannes Ebert
Prof. Dr. Michael Weiss
Staff
Étienne Batelier
Leon Hendrian
Lukas Stöveken
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# how does heat energy start to speed up a gas molecule?
If it was possible to place a single gas molecule in a cell and freeze it to near absolute zero. What would the molecule do as it thawed out?
Would it translate the heat energy into it's electrons and nucleus into vibrations and remain where it was? Would these vibrations cause it to start bouncing off the floor until it gets enough speed to convert the heat into kinetic energy?
Do gas molecules need convection to start speeding them up ( like the atmosphere), or do they translate heat into kinetic energy straight away all by themselves?
-
I don't think the definition of heat can be used down to the molecular level, as it is a macroscopic measure for the average vibrational energy of the different molecules. In the same way, you cannot define freezing for a single molecule as it describes a phase change, which you can only describe using a bunch of molecules.
Suppose you have a set-up that is capable of extracting all energy to "freeze" it, then there is also no vibration. Putting "heat" into the system is the same thing as adding vibrations to the molecule. So this comes from outside the molecule, not from the inside.
Convection is caused by the net displacement (flow) of molecules from one location to the other. Suppose you have a bunch of molecules, which are not of the same temperature, than diffusion will be the mechanism of heat transfer to surrounding molecules.
So it all comes down to: what is heat precisely, and that is a difficult to relate from macroscopic to molecular scales.
-
The question of molecular motion in response to "heat" is answered by statistical mechanics. If you have a single molecule at a temperature T, all that means is that the probability of any state of the molecule with energy E is as likely as $e^{-E\over T}$.
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# Clustering synthetic control data
## Introduction
This example will demonstrate clustering of time series data, specifically control charts. Control charts are tools used to determine whether a manufacturing or business process is in a state of statistical control. Such control charts are generated / simulated repeatedly at equal time intervals. A simulated dataset is available for use in UCI machine learning repository.
A time series of control charts needs to be clustered into their close knit groups. The data set we use is synthetic and is meant to resemble real world information in an anonymized format. It contains six different classes: Normal, Cyclic, Increasing trend, Decreasing trend, Upward shift, Downward shift. In this example we will use Mahout to cluster the data into corresponding class buckets.
For the sake of simplicity, we won’t use a cluster in this example, but instead show you the commands to run the clustering examples locally with Hadoop.
## Setup
We need to do some initial setup before we are able to run the example.
1. Start out by downloading the dataset to be clustered from the UCI Machine Learning Repository: http://archive.ics.uci.edu/ml/databases/synthetic_control/synthetic_control.data.
3. Unpack the release binary and switch to the mahout-distribution-0.x folder
4. Make sure that the JAVA_HOME environment variable points to your local java installation
5. Create a folder called testdata in the current directory and copy the dataset into this folder.
## Clustering Examples
Depending on the clustering algorithm you want to run, the following commands can be used:
bin/mahout org.apache.mahout.clustering.syntheticcontrol.canopy.Job
bin/mahout org.apache.mahout.clustering.syntheticcontrol.kmeans.Job
bin/mahout org.apache.mahout.clustering.syntheticcontrol.fuzzykmeans.Job
The clustering output will be produced in the output directory. The output data points are in vector format. In order to read/analyze the output, you can use the clusterdump utility provided by Mahout.
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# tm_layout
0th
Percentile
##### Layout of cartographic maps
This element specifies the map layout. The main function tm_layout controls title, margins, aspect ratio, colors, frame, legend, among many other things. The function tm_legend is a shortcut to access all legend. arguments without this prefix. The other functions are wrappers for two purposes: tm_format specifies position related layout settings such as margins, and tm_style specifies general styling related layout settings such as colors and font. Typically, the former functions are shape dependent, and the latter functions are shape independent. See details for predefined styles and formats. With tmap.style, a default style can be specified. Multiple tm_layout elements (or wrapper functions) can be stacked: called arguments will be overwritten.
##### Usage
tm_layout(title, scale, title.size, bg.color, aes.color, aes.palette,
attr.color, sepia.intensity, saturation, frame, frame.lwd,
frame.double.line, asp, outer.margins, inner.margins, between.margin,
outer.bg.color, fontface, fontfamily, compass.type, earth.boundary,
earth.boundary.color, earth.boundary.lwd, earth.datum, space.color,
legend.show, legend.only, legend.outside, legend.outside.position,
legend.outside.size, legend.position, legend.stack, legend.just,
legend.width, legend.height, legend.hist.height, legend.hist.width,
legend.title.color, legend.title.size, legend.title.fontface,
legend.title.fontfamily, legend.text.color, legend.text.size,
legend.text.fontface, legend.text.fontfamily, legend.hist.size,
legend.format, legend.frame, legend.frame.lwd, legend.bg.color,
legend.bg.alpha, legend.hist.bg.color, legend.hist.bg.alpha,
title.snap.to.legend, title.position, title.color, title.fontface,
title.fontfamily, title.bg.color, title.bg.alpha, panel.show,
panel.labels, panel.label.size, panel.label.color, panel.label.fontface,
panel.label.fontfamily, panel.label.bg.color, panel.label.height,
panel.label.rot, main.title, main.title.size, main.title.color,
main.title.fontface, main.title.fontfamily, main.title.position,
attr.outside, attr.outside.position, attr.outside.size, attr.position,
attr.just, design.mode)tm_legend(...)tm_style(style, ...)tm_format(format, ...)
##### Arguments
title
Global title of the map. For small multiples, multiple titles can be specified. The title is drawn inside the map. Alternatively, use panel.labels to print the map as a panel, with the title inside the panel header (especially useful for small multiples). Another alternative is the main.title which prints a title above the map. Titles for the legend items are specified at the layer functions (e.g. tm_fill).
scale
numeric value that serves as the global scale parameter. All font sizes, symbol sizes, border widths, and line widths are controlled by this value. Each of these elements can be scaled independently with the scale, lwd, or size arguments provided by the tmap-elements.
title.size
Relative size of the title
bg.color
Background color. By default it is "white". A recommended alternative for choropleths is light grey (e.g., "grey85").
aes.color
Default color values for the aesthetics layers. Should be a named vector with the names chosen from: fill, borders, symbols, dots, lines, text, na. Use "#00000000" for transparency.
aes.palette
Default color palettes for the aesthetics. It takes a list of three items: seq for sequential palettes, div for diverging palettes, and cat for categorical palettes. By default, Color Brewer palettes (see (see tmaptools::palette_explorer())) are used. It is also possible provide a vector of colors for any of these items.
attr.color
Default color value for map attributes
sepia.intensity
Number between 0 and 1 that defines the amount of sepia effect, which gives the map a brown/yellowish flavour. By default this effect is disabled (sepia.intensity=0). All colored used in the map are adjusted with this effect.
saturation
Number that determines how much saturation (also known as chroma) is used: saturation=0 is greyscale and saturation=1 is normal. A number larger than 1 results in very saturated maps. All colored used in the map are adjusted with this effect. Hacking tip: use a negative number.
frame
Either a boolean that determines whether a frame is drawn, or a color value that specifies the color of the frame.
frame.lwd
width of the frame
frame.double.line
draw a double frame line border?
asp
Aspect ratio. The aspect ratio of the map (width/height). If NA, it is determined by the bounding box (see argument bbox of tm_shape), the outer.margins, and the inner.margins. If 0, then the aspect ratio is adjusted to the aspect ratio of the device.
outer.margins
Relative margins between device and frame. Vector of four values specifying the bottom, left, top, and right margin. Values are between 0 and 1. When facets are created, the outer margins are the margins between the outer panels and the device borders (see also between.margin)
inner.margins
Relative margins inside the frame. Vector of four values specifying the bottom, left, top, and right margin. Values are between 0 and 1. By default, 0 for each side if master shape is a raster, otherwise 0.02.
between.margin
Margin between facets (small multiples) in number of text line heights. The height of a text line is automatically scaled down based on the number of facets.
outer.bg.color
Background color outside the frame.
fontface
global font face for the text in the map. It can also be set locally per element (see e.g. title.fontface).
fontfamily
global font family for the text in the map. It can also be set locally per (see e.g. title.fontfamily).
compass.type
type of compass, one of: "arrow", "4star", "8star", "radar", "rose". Of course, only applicable if a compass is shown. The compass type can also be set within tm_compass.
earth.boundary
Logical that determines whether the boundaries of the earth are shown or an object that specifies the boundaries. This object can be a vector of size four, a 2 by 2 matrix (bounding box), or an extent object. By default, the boundaries are c(-180, -90, 180, 90). Useful for projected world maps. Often, it is useful to crop both poles (e.g., with c(-180, -88, 180, 88)).
earth.boundary.color
Color of the earth boundary.
earth.boundary.lwd
Line width of the earth boundary.
earth.datum
Geodetic datum to determine the earth boundary. By default "WGS84", other frequently used datums are "NAD83" and "NAD27". Any other PROJ.4 character string can be used.
space.color
Color of the space, i.e. the region inside the frame, and outside the earth boundary.
legend.show
Logical that determines whether the legend is shown.
legend.only
logical. Only draw the legend (without map)? Particularly useful for small multiples with a common legend.
legend.outside
Logical that determines whether the legend is plot outside of the map/facets. Especially useful when using facets that have a common legend (i.e. with free.scales=FALSE).
legend.outside.position
Character that determines the outside position of the legend. Only applicable when legend.outside=TRUE. One of: "right", "left", "top", or "bottom".
legend.outside.size
Numeric value that determines the relative size of the legend, when legend.outside=TRUE. If the first value of legend.outside.position is "top" or "bottom", then it is the width of the legend, else it is the height of the legend. Note that the actual height or width of the legend is determined by the content of the legend (and the used font sizes). This argument specifies the upperbound of the width or height.
legend.position
Position of the legend. Vector of two values, specifying the x and y coordinates. Either this vector contains "left", "LEFT", "center", "right", or "RIGHT" for the first value and "top", "TOP", "center", "bottom", or "BOTTOM" for the second value, or this vector contains two numeric values between 0 and 1 that specifies the x and y coordinates of the left bottom corner of the legend. The uppercase values correspond to the position without margins (so tighter to the frame). By default, it is automatically placed in the corner with most space based on the (first) shape object. If legend.outside=TRUE, this argument specifies the legend position within the outside panel.
legend.stack
Value that determines how different legends are stacked: "vertical" or "horizontal". To stack items within a same legend, look at "legend.is.portrait" in the specific layer calls.
legend.just
Justification of the legend relative to the point coordinates. The first value specifies horizontal and the second value vertical justification. Possible values are: "left" , "right", "center", "bottom", and "top". Numeric values of 0 specify left/bottom alignment and 1 right/top alignment. This option is only used, if legend.position is specified by numeric coordinates.
legend.width
width of the legend. If it is a negative number, it will be the exact legend width. If it is a positive number (by default), it will be the maximum legend width; the actual legend width will be decreased automatically based on the legend content and font sizes.
legend.height
height of the legend. If it is a negative number, it will be the exact legend height. If it is a positive number (by default), it will be the maximum legend height; the actual legend height will be decreased automatically based on the legend content and font sizes.
legend.hist.height
height of the histogram. This height is initial. If the total legend is downscaled to legend.height, the histogram is downscaled as well.
legend.hist.width
width of the histogram. By default, it is equal to the legend.width.
legend.title.color
color of the legend titles
legend.title.size
Relative font size for the legend title
legend.title.fontface
font face for the legend title. By default, set to the global parameter fontface.
legend.title.fontfamily
font family for the legend title. By default, set to the global parameter fontfamily.
legend.text.color
color of the legend text
legend.text.size
Relative font size for the legend text elements
legend.text.fontface
font face for the legend text labels. By default, set to the global parameter fontface.
legend.text.fontfamily
font family for the legend text labels. By default, set to the global parameter fontfamily.
legend.hist.size
Relative font size for the choropleth histogram
legend.format
list of formatting options for the legend numbers. Only applicable for layer functions (such as tm_fill) where labels is undefined. Parameters are:
fun
Function to specify the labels. It should take a numeric vector, and should return a character vector of the same size. By default it is not specified. If specified, the list items scientific, format, and digits (see below) are not used.
scientific
Should the labels be formatted scientifically? If so, square brackets are used, and the format of the numbers is "g". Otherwise, format="f", and text.separator, text.less.than, text.or.more, and big.num.abbr are used. Also, the numbers are automatically rounded to millions or billions if applicable.
format
By default, "f", i.e. the standard notation xxx.xxx, is used. If scientific=TRUE then "g", which means that numbers are formatted scientifically, i.e. n.dddE+nn if needed to save space.
digits
Number of digits after the decimal point if format="f", and the number of significant digits otherwise.
big.num.abbr
Vector that defines whether and which abbrevations are used for large numbers. It is a named numeric vector, where the name indicated the abbreviation, and the number the magnitude (in terms on numbers of zero). Numbers are only abbrevation when they are large enough. Set it to NA to disable abbrevations. The default is c("mln" = 6, "bln" = 9). For layers where style is set to log10 or log10_pretty, the default is NA.
text.separator
Character string to use to separate numbers in the legend (default: "to").
text.less.than
Character value(s) to use to translate "Less than". When a character vector of length 2 is specified, one for each word, these words are aligned when text.to.columns = TRUE
text.or.more
Character value(s) to use to translate "or more". When a character vector of length 2 is specified, one for each word, these words are aligned when text.to.columns = TRUE
text.align
Value that determines how the numbers are aligned, "left", "center" or "right"
. By default "left" for legends in portrait format (legend.is.protrait = TRUE), and "center" otherwise.
text.to.columns
Logical that determines whether the text is aligned to three columns (from, text.separator, to). By default FALSE.
text.align
Value that determines how the numbers are aligned, "left", "center" or "right"
. By default "left" for legends in portrait format (legend.is.protrait = TRUE), and "center" otherwise.
text.to.columns
Logical that determines whether the text is aligned to three columns (from, text.separator, to). By default FALSE.
...
Other arguments passed on to formatC
legend.frame
either a logical that determines whether the legend is placed inside a frame, or a color that directly specifies the frame border color.
legend.frame.lwd
line width of the legend frame (applicable if legend.frame is TRUE or a color)
legend.bg.color
Background color of the legend. Use TRUE to match with the overall background color bg.color.
legend.bg.alpha
Transparency number between 0 (totally transparent) and 1 (not transparent). By default, the alpha value of the legend.bg.color is used (normally 1).
legend.hist.bg.color
Background color of the histogram
legend.hist.bg.alpha
Transparency number between 0 (totally transparent) and 1 (not transparent). By default, the alpha value of the legend.hist.bg.color is used (normally 1).
title.snap.to.legend
Logical that determines whether the title is part of the legend. By default FALSE, unless the legend is drawn outside the map (see legend.outside).
title.position
Position of the title. Vector of two values, specifying the x and y coordinates. Either this vector contains "left", "LEFT", "center", "right", or "RIGHT" for the first value and "top", "TOP", "center", "bottom", or "BOTTOM" for the second value, or this vector contains two numeric values between 0 and 1 that specifies the x and y coordinates of the tile. The uppercase values correspond to the position without margins (so tighter to the frame). By default the title is placed on top of the legend (determined by legend.position).
title.color
color of the title
title.fontface
font face for the title. By default, set to the global parameter fontface.
title.fontfamily
font family for the title. By default, set to the global parameter fontfamily.
title.bg.color
background color of the title. Use TRUE to match with the overall background color bg.color. By default, it is TRUE if legend.frame is TRUE or a color.
title.bg.alpha
Transparency number between 0 (totally transparent) and 1 (not transparent). By default, the alpha value of the title.bg.color is used (normally 1).
panel.show
Logical that determines if the map(s) are shown as panels. If TRUE, the title will be placed in the panel header instead of inside the map. By default, it is TRUE when small multiples are created with the by variable. (See tm_facets)
panel.labels
Panel labels. Only applicable when panel.show is TRUE. For cross tables facets, it should be a list containing the row names in the first, and column names in the second item.
panel.label.size
Relative font size of the panel labels
panel.label.color
Font color of the panel labels
panel.label.fontface
font face for the panel labels. By default, set to the global parameter fontface.
panel.label.fontfamily
font family for the panel labels. By default, set to the global parameter fontfamily.
panel.label.bg.color
Background color of the panel labels
panel.label.height
Height of the labels in number of text line heights.
panel.label.rot
Rotation angles of the panel labels. Vector of two values: the first is the rotation angle (in degrees) of the row panels, which are only used in cross-table facets (when tm_facets's by is specified with two variables). The second is the rotation angle of the column panels.
main.title
Title that is printed above the map (or small multiples). When multiple pages are generated (see along argument of tm_facets), a vector can be provided. By default, the main title is only printed when this along argument is specified.
main.title.size
Size of the main title
main.title.color
Color of the main title
main.title.fontface
font face for the main title. By default, set to the global parameter fontface.
main.title.fontfamily
font family for the main title. By default, set to the global parameter fontfamily.
main.title.position
Position of the main title. Either a numeric value between 0 (left) and 1 (right), or a character value: "left", "center", or "right".
attr.outside
Logical that determines whether the attributes are plot outside of the map/facets.
attr.outside.position
Character that determines the outside position of the attributes: "top" or "bottom". Only applicable when attr.outside=TRUE. If the legend is also drawn outside (with legend.outside=TRUE) and on the same side of the map (e.g. also "top" or "bottom"), the attributes are placed between the map and the legend. This can be changed by setting attr.outside.position to "TOP" or "BOTTOM": in this case, the attributes are placed above respectively below the legend.
attr.outside.size
Numeric value that determines the relative height of the attribute viewport, when attr.outside=TRUE.
attr.position
Position of the map attributes, which are tm_credits, tm_scale_bar, tm_compass, and tm_minimap. Vector of two values, specifying the x and y coordinates. The first value is "left", "LEFT", "center", "right", or "RIGHT", and the second value "top", "TOP", "center", "bottom", or "BOTTOM". The uppercase values correspond to the position without margins (so tighter to the frame). Positions can also be set separately in the map attribute functions. If attr.outside=TRUE, this argument specifies the position of the attributes within the outside panel.
attr.just
Justification of the attributes relative to the point coordinates. The first value specifies horizontal and the second value vertical justification. Possible values are: "left" , "right", "center", "bottom", and "top". Numeric values of 0 specify left/bottom alignment and 1 right/top alignment. This option is only used, if attr.position is specified by numeric coordinates. It can also be specified per attribute function.
design.mode
Logical that enables the design mode. If TRUE, inner and outer margins, legend position, aspect ratio are explicitly shown. Also, feedback text in the console is given.
...
other arguments from tm_layout
style
name of the style
format
name of the format
##### Details
Predefined styles:
"white" White background, commonly used colors (default) "gray"/"grey" Grey background, useful to highlight sequential palettes (e.g. in choropleths) "natural" Emulation of natural view: blue waters and green land "bw" Greyscale, obviously useful for greyscale printing "classic" Classic styled maps (recommended) "cobalt" Inspired by latex beamer style cobalt "albatross" Inspired by latex beamer style albatross "beaver" Inspired by latex beamer style beaver --------------------------- ---------------------------------------------------------------------------------------------------
Predefined formats
"World" Format specified for world maps "World_wide" Format specified for world maps with more space for the legend "NLD" Format specified for maps of the Netherlands "NLD_wide" Format specified for maps of the Netherlands with more space for the legend --------------------------- ---------------------------------------------------------------------------------------------------
##### References
Tennekes, M., 2018, tmap: Thematic Maps in R, Journal of Statistical Software, 84(6), 1-39, DOI
vignette("tmap-getstarted")
• tm_layout
• tm_legend
• tm_style
• tm_format
##### Examples
# NOT RUN {
data(World, land)
tm_shape(World) +
tm_fill("pop_est_dens", style="kmeans", title="Population density") +
tm_style("albatross", frame.lwd=10) + tm_format("World", title="The World")
# }
# NOT RUN {
tm_shape(land) +
tm_raster("elevation", breaks=c(-Inf, 250, 500, 1000, 1500, 2000, 2500, 3000, 4000, Inf),
palette = terrain.colors(9), title="Elevation", midpoint = NA) +
tm_shape(World, is.master=TRUE) +
tm_borders("grey20") +
tm_grid(projection="longlat", labels.size = .5) +
tm_text("name", size="AREA") +
tm_compass(position = c(.65, .15), color.light = "grey90") +
tm_credits("Eckert IV projection", position = c("right", "BOTTOM")) +
tm_style("classic") +
tm_layout(bg.color="lightblue",
inner.margins=c(.04,.03, .02, .01),
earth.boundary = TRUE,
space.color="grey90") +
tm_legend(position = c("left", "bottom"),
frame = TRUE,
bg.color="lightblue")
# }
# NOT RUN {
tm_shape(World, projection="robin") +
tm_polygons("HPI", palette="div", n=7,
title = "Happy Planet Index") +
tm_credits("Winkel Tripel projection", position = c("right", "BOTTOM")) +
tm_style("natural", earth.boundary = c(-180, -87, 180, 87), inner.margins = .05) +
tm_legend(position=c("left", "bottom"), bg.color="grey95", frame=TRUE)
# Example to illustrate the type of titles
tm_shape(World) +
tm_polygons(c("income_grp", "economy"), title = c("Legend Title 1", "Legend Title 2")) +
tm_layout(main.title = "Main Title",
main.title.position = "center",
main.title.color = "blue",
title = c("Title 1", "Title 2"),
title.color = "red",
panel.labels = c("Panel Label 1", "Panel Label 2"),
panel.label.color = "purple",
legend.text.color = "brown")
# }
# NOT RUN {
# global option tmap.style demo
# get current style
current.style <- tmap_style()
qtm(World, fill = "economy", format = "World")
tmap_style("col_blind")
qtm(World, fill = "economy", format = "World")
tmap_style("cobalt")
qtm(World, fill = "economy", format = "World")
# set to current style
tmap_style(current.style)
# }
# NOT RUN {
# TIP: check out these examples in view mode, enabled with tmap_mode("view")
# }
Documentation reproduced from package tmap, version 2.3-1, License: GPL-3
### Community examples
Looks like there are no examples yet.
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# Is there any means by which to severely discourage breaking a line in a would-be box?
I have a small section of text—no more than 15 or 30 characters—that is split up into a few words. I'd like to add a penalty to breaking in this section, but I don't want to completely bar it and risk a truly awful paragraph. How can I effect this?
I have some text and \DiscourageBreak{some text that \emph{shouldn't} be broken.}
I looked at a list of penalties available on this site, but I didn't see one that seemed appropriate.
Not robust, in the sense that it can't go in the argument to another command; the space is made active and defined to issue a high penalty and a space. A space following a penalty can't be taken as a line break point, only the penalty can, so this discourages breaks. Probably you want to disallow hyphenation, which can be easily done by adding \language=\l@nohyphenation in the definition of \DiscourageBreak.
\documentclass{article}
\makeatletter
\newcommand{\DiscourageBreak}{%
\begingroup\obeyspaces\sean@penalizespace
\sean@discouragebreak
}
\newcommand{\sean@discouragebreak}[1]{#1\endgroup}
\newcommand{\sean@penalizespace}{%
\begingroup\lccode\~=\ %
\lowercase{\endgroup\def~}{\penalty9000 \space}%
}
\makeatother
\begin{document}
I have some text and some other text and again.
I have some text and \DiscourageBreak{some text that \emph{shouldn't} be broken.}
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Online Calculators
# Percentage Increase Calculator
Percentage Increase Calculator
= 6.72478% increase
## Calculator Use
The Percentage Increase Calculator finds the increase from one value to another in terms of a percentage.
Enter starting value and final value to find percentage increase.
## How to Calculate Percentage Increase
• Subtract final value minus starting value
• Divide that amount by the absolute value of the starting value
• Multiply by 100 to get percent increase
• If the percentage is negative, it means there was a decrease and not an increase.
## Percentage Increase Formula
You can use the percentage increase formula for any percent increase calculation:
$$\text{Percentage Increase} = \\ \dfrac{\text{Final Value} - \text{Starting Value}}{\left|\text{Starting Value}\right|} \times 100$$
## Example Problem: Percentage Increase
Last year your favorite jeans cost $36 per pair. This year they cost$45 per pair. What is the percentage increase in the price of these jeans from last year to this year?
Percentage Increase = [ (Final Value - Starting Value) / |Starting Value| ] × 100
45 - 36 = 9
9 / 36 = 0.25
0.25 × 100 = 25%
So the price of your favorite jeans increased by 25% from last year to this year.
### Related Calculators
Use the Percentage Decrease Calculator to find the percent decrease from one value to another.
Use the Percent Difference Calculator when you are comparing two values and want to find the percentage difference between them.
The Percent Change Calculator finds the change between two numbers as a percentage. It is similar to finding percentage increase or percentage decrease but it doesn't label the change as an increase or a decrease.
Cite this content, page or calculator as:
Furey, Edward "Percentage Increase Calculator"; CalculatorSoup, https://www.calculatorsoup.com - Online Calculators
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# If A/B test results are not significant, how does increasing sample size affect things?
If I run an A/B with the recommended sample size (using this for example), and at the end my results are not significant, what can be done at this point? If someone requests to continue running the test (i.e. increase the sample size), what implications does this have?
I guess this will cause a "peeking" problem, but how do you quantify this if someone asks you to, say, double the sample size? And if I do double the sample size, what is the likeihood of getting significant results, given that the recommended sample size did not produce significant results?
So a specific example
1. I run a test with a recommended 10K impressions on each variation
2. At the end of this test, the p-value is above 5% so there is not significant results
3. Someone suggests to continue running the test to see if it does produce significant results. Is this a bad or good idea? And how do you motivate the answer?
• If you're interested in what looking twice does to the significance level, you should look into methods of sequential design – Cliff AB Jan 7 '16 at 20:13
What is the cost of running the additional time?
What is the cost of a type I error (finding significance when there really is not a difference)?
What is the cost of a type II error (not finding a difference that is there)?
How would the decision be made if you don't run any longer?
Yes, you are looking at a case of the peeking problem, or multiple comparisons, but the consequences range from dire to irrelevant depending on the answers to the questions above.
For example, if there is really not much cost to running longer (just your time sometime in the future to run a second test) and if you don't run longer then the decision will be made by flipping a coin, then getting more data at worst is a slow coin flip and will not really hurt.
On the other hand, if running longer deters you from finding a better solution, or could lead to a more expensive solution when the cheaper is just as good, then the cost of getting more data is more severe.
One way to quantify the effect would be to simulate data. Simulate data where the null is true at the final sample size, then analyze just the first part and the whole data. Repeat this a bunch of times to see how often the 1st test is not significant and the 2nd is. Also include any known costs. You can also redo this with the null false at various differences to see those effects as well.
When you first chose your sample size using the calculator, you entered a "minimum detectable effect". If this level that you entered is the smallest effect that would be relevant for you (which would have probably been the best way to run the test, if not cost prohibitive), then you should stop, since it is highly likely that the true difference, if there is any, is below this minimum detectable effect.
If, however, you would care about an effect size smaller than the minimum detectable effect you entered at first, it may be worthwhile to continue. Adjusting the power and minimum detectable effect sliders on that calculator will give you a better sample size needed for your application.
As you mentioned, this does lead to a "peeking" problem, which means your p-values will be biased downward. To see this, imagine that you were looking at the results after every impression and calculating a p-value. It is essentially a statistical guarantee that you will eventually get a "significant" p-value eventually, even if there is no true effect. The more often you check, the more biased your p-values will be when you are finished. To counter this, if you indeed decide that you would like to be able to detect a smaller effect than your original test could have, you should choose a smaller value for the significance level $\alpha$ than you would otherwise, since a low p-value does not hold as much meaning when you make decisions of whether or not to continue the test based on earlier p-values.
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# Neutral technical progress
Recently I learnt that Cobb-Douglas production function has elasticity of substitution equal to 1, therefore it has neutral technical progress. Then Leontief production function has zero elasticity of substitution. Does that imply that it cannot have neutral technical progress? On the other hand, factors of production in Leontief cannot be substituted for one another so I thought it can never have labour or capital saving technical progress.
There are different notions of neutrality of technical progress in macroeocnomics.
You can have Hicks-neutral technical progress - that is a technical progress that increases the marginal productivity of all factors of production by the same proportion at the same capital-labor ratio. An example of Hicks-neutral production function would be one where $$Y^*= F(AK, A L) = A F( K,L)$$ and consequently $$\frac{\partial Y / \partial K}{\partial Y / \partial K}= \frac{AF_k}{AF_L}= \frac{F_K}{F_L}$$. Also note the technical progress would be neutral when its neither capital or labor saving, according to Hick's definition in his the theory of wages, labor saving technology would be one that improves the marginal product of labor relatively more and capital saving that which improves the marginal product of capital relative to labor.
I dont recall ever seeing Leontief production function that would be assumed to be Hicks-neutral in research papers but I think it should be possible to have special cases such as $$F = \min{ \{A K, A L}\}$$ where $$K$$ and $$L$$ have to be consumed exactly in the same proportion you could say that the technology is Hicks-neutral as it is neither labor nor capital saving, also the $$K$$ and $$L$$ ratio would have to remain constant as they are perfect complements and here we would need one unit of capital for every one unit of labor.
Whether the production function is Hicks-neutral depends also on factor elasticity of substitution as well as product elasticity of demand but that is more in sense that the same production function can be or not be Hicks-netural depending on the elasticity of substitution it has.
Then you can also have Harrod-neutral technical progress. In this definition the technological change is neutral as long as capital-output ratio and the rate of profit stays constant. An example of such function would be a function where technology is labor augmenting like $$F(K, AL)$$.
Next, you can also have Solow-neutral technical progress which is defined as technological change that leaves the relative input shares unchanged for a given labour-output ratio. With example being capital augmenting technology $$F(AK,L)$$.
Moreover, you can even have Leontief-neutral technical change. This occurs when the augmentation effect of technology would be different for both factors of production so you would have $$F(A_1 K, A_2L)$$ where $$A_1\neq A_2$$.
None of the above definitions seem to exclude Leontief production function per se.
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# Is it possible for the phase of electric charge to change over large general relativistic distances?
Jackson provides examples of how magnetic charge and electric charge form together to create complex charge, \begin{align} \rho = \rho_e+i\rho_m \end{align} which gives rise to the complex faraday field \begin{align} \boldsymbol{F}=\boldsymbol{E}+i\boldsymbol{B} \end{align} I assume magnetic charge has not been observed on a local scale, however is it possible to bend space such that two charges are observed to have different phases over vast cosmic distances. For example \begin{align} \rho = \rho_e e^{i r/\lambda} \end{align}
If every electromagnetic charge has the same ratio of magnetic and electric charge then you can rotate that complex charge $\rho=\rho_e+i\rho_m$ (and a duality rotation for the fields) to get a purely real (electric) charge with the normal maxwell equations.
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Intermediate Example of the Principle of Weak Mathematical Induction
Intermediate Example of the Principle of Weak Mathematical Induction
We will now look at an intermediate example of the principle of weak mathematical induction.
Example 1
Prove that $n^2 < 2^n$ for each positive integer $n \geq 5$.
Let $P(n)$ be the statement that $n^2 < 2^n$.
Base Step: The statement $P(5)$ says that $5^2 = 25 < 2^5 = 32$ which is true.
Induction Step: Suppose that for some $k \geq 5$, $P(k)$ is true, that is, $k^2 < 2^k$. We want to show that $P(k+1)$ is true, i.e., $(k+1)^2 < 2^{k+1}$. We have that:
(1)
\begin{align} \quad (k+1)^2 = k^2 + 2k + 1 \overset{I.H.} < 2^k + (2k + 1) \quad (*) \end{align}
To complete this proof we need to do another induction. Let $S(n)$ be the statement that $2n + 1 < 2^n$ for each $n \geq 5$ (this is actually true for each $n \geq 3$ but this is not needed in the proof above).
Base Step: The statement $S(5)$ says that $2(5) + 1 = 11 < 2^5 = 32$ which is true.
Induction Step: Suppose that for some $k \geq 5$, $S(k)$ is true, that is, $2k + 1 < 2^k$. We want to show that $P(k + 1)$ is true, i.e., $2(k + 1) + 1 < 2^{k+1}$. We have that:
(2)
\begin{align} \quad 2(k + 1) + 1 = 2k +1 + 2 \overset{I.H.}< 2k + 2 < 2^k + 2^k = 2^{k+1} \end{align}
So by the principle of mathematical induction, $S(n)$ is true for all $n \geq 5$.
Going back up to $(*)$ we have that:
(3)
\begin{align} \quad (k+1)^2 < 2^k + 2^k = 2^{k+1} \end{align}
So by the principle of mathematical induction, $P(n)$ is true for all $n \geq 5$.
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License
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# Parallel Random Number Generators
This is the second of a multi-part series about the MATLAB random number generators. If you ask for help rng, you will get lots of information, including the fact that there are three modern generators.
'twister' Mersenne Twister
'combRecursive' Combined Multiple Recursive
'multFibonacci' Multiplicative Lagged Fibonacci
My previous post was about twister. Today's post is about the other two, 'combRecursive' and 'multFibonacci', which are designed for parallel computing.
### Contents
#### Parallel computing demo
I frequently use the card game Blackjack to demonstrate parallel computing. At the same time I can demonstrate the random number generators. I regard Blackjack as a financial instrument, not unlike the stock of a publicly traded company. Simulating the size of an investment as a function of time is a typical application of the Monte Carlo technique.
Begin by opening a pool of workers.
parpool;
Starting parallel pool (parpool) using the 'local' profile ... connected to 2 workers.
Four players each play twenty thousand hands of Blackjack.
p = 4;
n = 20000;
Collect the results in this array.
B = zeros(n,p);
A parallel for loop executes a Blackjack program that knows nothing about parallel computing.
parfor k = 1:p
B(:,k) = cumsum(blackjacksim(n));
end
Plot the results.
plot(B)
title(['Blackjack , $', int2str(sum(B(end,:)))]) xlabel('Number of hands') ylabel('Stake, ($)')
axis([0 n -2500 2500])
#### Parallel streams
The card dealing calls the random number generator. It is essential that the different parallel workers have different, independent streams of random numbers. The default MATLAB generator twister does not offer this feature. Simply starting twister with different seeds on different workers does not provide statistically independent streams. So we turn to the other generators. To see which one is in use here, run a small spmd, "single program, multiple data" block.
spmd
r = rng
s = r.State'
format short
x = rand(1,7)
end
Lab 1:
r =
Type: 'combRecursive'
Seed: 0
State: [12x1 uint32]
s =
Columns 1 through 6
1720035765 2052922678 1637499698 3048064580 1173461082 2391850890
Columns 7 through 12
1862757735 2368998908 1385613640 1660833332 146924518 3104031825
x =
0.8789 0.6969 0.0409 0.4609 0.7528 0.2871 0.5241
Lab 2:
r =
Type: 'combRecursive'
Seed: 0
State: [12x1 uint32]
s =
Columns 1 through 6
323405913 3817048408 3712601073 1070773748 1552739185 3267875480
Columns 7 through 12
1594297407 2533167732 3377045245 3413340742 2651847732 1248925296
x =
0.1072 0.3194 0.1048 0.6623 0.0878 0.3692 0.8035
We see that we have two workers, that they are both using the combRecursive generator, that they have the same seed, but different states, so they are generating different random numbers.
#### combRecursive
Also known as mrg32k3a. A 32-bit combined multiple recursive generator (CMRG), due to Pierre L'Ecuyer, at the Universite de Montreal, and his colleagues, described in the papers referenced below. This generator is similar to the CMRG implemented in the RngStreams package. It has a period of $2^{127}$, and supports up to $2^{63}$ parallel streams, via sequence splitting, and $2^{51}$ substreams each of length $2^{76}$. Here is a link to the C source code. combmrg2.c
The state of the backbone generator is a 2-by-3 array S that evolves at each step according to the linear recurrence expressed succinctly in MATLAB by
m1 = 2^32 - 209;
m2 = 2^32 - 22853;
x1 = mod(1403580*S(1,2)-810728*S(1,3),m1);
x2 = mod(527612*S(2,1)-1370589*S(2,3),m2);
S = [x1 S(1,1) S(1,2))
x2 S(2,1) S(2,2)];
A single precision random real u is then produced by
z = mod(x1-x2,m1);
if z > 0, u = z/(m1+1); else, u = m1/(m1+1); end
The important feature of this generator is that it is possible to create different initial states for each worker in a parallel pool so that the resulting streams of random numbers are statistically independent.
#### multFibonacci
Also known as mlfg6331_64. A 64-bit multiplicative lagged Fibonacci generator (MLFG), developed by Michael Mascagni and Ashok Srinivasan at Florida State University. This generator, which has lags $l=63$ and $k=31$, is similar to the MLFG implemented in the SPRNG package. It has a period of approximately $2^{124}$. It supports up to $2^{61}$ parallel streams, via parameterization, and $2^{51}$ substreams each of length $2^{72}$.
The state of this generator is a length 63 vector of 64-bit integers S. The recurrence relation is
$$x_n = x_{n-k} \times x_{n-l} (mod 2^{64})$$
Each random double precision value is created using one 64-bit integer from the generator; the possible values are all multiples of $2^{-53}$ strictly within the interval (0,1).
Again, the important feature of this generator is that it is possible to create different initial states for each worker in a parallel pool so that the resulting streams of random numbers are statistically independent.
#### Which one?
Which one should you use? Most of the time, stick with the default and you'll be OK. You will get 'twister' in serial computations and 'combRecursive' on the workers in a parallel pool. You can use
rng('shuffle')
at the beginning of a session if you want different sequences of random numbers in different sessions. Otherwise, don't worry about setting the generator or the seed.
If you want to experiment, you can use rng to try different generators and different starting seeds on your computation. If you find a problem where it makes a significant difference, please let us know.
#### Thanks
Thanks again to Peter Perkins.
#### References
Pierre L'Ecuyer, R. Simard, E. J. Chen, and W. D. Kelton. "An Objected-Oriented Random-Number Package with Many Long Streams and Substreams." Operations Research, 50(6):1073-1075. 2002. <http://www.iro.umontreal.ca/~lecuyer/myftp/papers/streams00.pdf>
Pierre L'Ecuyer. "Good Parameters and Implementations for Combined Multiple Recursive Random Number Generators." Operations Research 47(1):159-164. 1999. <http://dx.doi.org/10.1287/opre.47.1.159>
Michael Mascagni and Ashok Srinivasan. "Parameterizing Parallel Multiplicative Lagged-Fibonacci Generators." Parallel Computing, 30: 899-916. 2004. <http://www.cs.fsu.edu/~asriniva/papers/mlfg.ps>
Michael Mascagni and Ashok Srinivasan. "SPRNG: A Scalable Library for Pseudorandom Number Generation." ACM Transactions on Mathematical Software, Vol 26 436-461. 2000. <http://www.cs.fsu.edu/~asriniva/papers/sprngacm.ps>
Published with MATLAB® R2014b
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Zhong2010
Effects of attachment preferences on coevolution of opinions and networks
L.-X. Zhong, F. Ren, T. Qiu, J.-R. Xu, B.-H. Chen, and C.-F. Liu
Physica A, 389(13), 2557-2565, 2010.
In the coevolution of network structures and opinion formation, we investigate the effects of a mixed population with distinctive relinking preferences on both the convergence time and the network structures. It has been found that a heterogeneous network structure is easier to be reached with more high-degree-preferential (HDP) nodes. There exists high correlation between the convergence time and the network heterogeneity. The heterogeneous degree distribution caused by preferential attachment accelerates the convergence to a consensus state and the shortened convergence time inhibits the occurrence of the following disquieting situation that occurs in a continuously evolving network: with preferential attachment and long-time evolvement, most of the nodes would become separated and only a few leaders would have immediate neighbors. Analytical calculations based on mean field theory reveal that both the transition point $p_{tr}$and the consensus time $\tau$ depend upon the standard deviation of the degree distribution $\sigma_{d}$. $p_{tr}$ increases while $\tau$ decreases with the rise of $\sigma_{d}$. Functions of $p_{tr} = \langle k \rangle / \left( \langle k \rangle + 1\right)$ and $\tau = a\left(N\right)K / \left( \sigma_{D}^2 - K\right)$ are found. Theoretical analyses are in accordance with simulation data.
This paper in Physica A
page revision: 3, last edited: 10 Sep 2010 13:57
Materials on this page may be the property of the authors and/or journals named.
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# ECNI guide to the public sector equality and disability duties
A short guide to the public sector equality and disability duties was published by the Equality Commission for Northern Ireland in October 2014.
This guidance note aims to provide an overview of some key aspects of the public sector equality and disability duties. The guidance focuses on the concepts of ‘due regard’ and ‘regard’ and their implementation within the context of a public authority equality scheme and disability action plan.
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# Anthropic reasoning in everyday life
Thought experiment from a past post:
A stranger comes up to you and offers to play the following game with you: “I will roll a pair of dice. If they land snake eyes (i.e. they both land 1), you give me one dollar. Otherwise, if they land anything else, I give you a dollar.”
Do you play this game?
[…]
Now imagine that the stranger is playing the game in the following way: First they find one person and offer to play the game with them. If the dice land snake eyes, then they collect a dollar and stop playing the game. Otherwise, they find ten new people and offer to play the game with them. Same as before: snake eyes, the stranger collects \$1 from each and stops playing, otherwise he moves on to 100 new people. Et cetera forever.
When we include this additional information about the other games the stranger is playing, then the thought experiment becomes identical in form to the dice killer thought experiment. Thus updating on the anthropic information that you have been kidnapped gives a 90% chance of snake-eyes, which means you have a 90% chance of losing a dollar and only a 10% chance of gaining a dollar. Apparently you should now not take the offer!
This seems a little weird. Shouldn’t it be irrelevant if the game if being offered to other people? To an anthropic reasoner, the answer is a resounding no. It matters who else is, or might be, playing the game, because it gives us additional information about our place in the population of game-players.
Thus far this is nothing new. But now we take one more step: Just because you don’t know the spatiotemporal distribution of game offers doesn’t mean that you can ignore it!
So far the strange implications of anthropic reasoning have been mostly confined to bizarre thought experiments that don’t seem too relevant to the real world. But the implication of this line of reasoning is that anthropic calculations bleed out into ordinary scenarios. If there is some anthropically relevant information that would affect your probabilities, then you need to consider the probability that this information
In other words, if somebody comes up to you and makes you the offer described above, you can’t just calculate the expected value of the game and make your decision. Instead, you have to consider all possible distributions of game offers, calculate the probability of each, and average over the implied probabilities! This is no small order.
For instance, suppose that you have a 50% credence that the game is being offered only one time to one person: you. The other 50% is given to the “dice killer” scenario: that the game is offered in rounds to a group that decuples in size each round, and that this continues until the dice finally land snake-eyes. Presumably you then have to average over the expected value of playing the game for each scenario.
$EV_1 = - \1 \cdot \frac{35}{36} + \1 \cdot \frac{1}{36} = \ \frac{34}{36} \approx \0.94 \\~\\ EV_2 = \1 \cdot 0.1 + - \1 \cdot 0.9 = - \ 0.80 \\~\\ EV = 0.50 \cdot EV_1 + 0.50 \cdot EV_2 \approx \ .07$
In this case, the calculation wasn’t too bad. But that’s because it was highly idealized. In general, representing your knowledge of the possible distributions of games offered seems quite difficult. But the more crucial point is that it is apparently not enough to go about your daily life calculating the expected value of the decisions facing you. You have to also consider who else might be facing the same decisions, and how this influences your chances of winning.
Can anybody think of a real-life example where these considerations change the sign of the expected value calculation?
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## A Reduction Result for Planar Location Problems with Polygonal Barriers
• In this paper we consider the problem of locating one new facility in the plane with respect to a given set of existing facility where a set of polygonal barriers restricts traveling. This non-convex optimization problem can be reduced to a finite set of convex subproblems if the objective function is a convex function of the travel distances between the new and the existing facilities (like e.g. the Median and Center objective functions). An exact Algorithm and a heuristic solution procedure based on this reduction result are developed.
Author: Kathrin Klamroth urn:nbn:de:hbz:386-kluedo-4854 Report in Wirtschaftsmathematik (WIMA Report) (42) Preprint English 1999 1999 Technische Universität Kaiserslautern 2000/04/03 location; non-convex optimization Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011
$Rev: 13581$
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# HSC Physics Module 7 Practice Questions with Solutions
Can you solve these 10 Must Know HSC Physics Module 7 Exam questions? Test your exam readiness with our Module 7 The Nature of Light questions.
## Test your exam readiness with these 10 Must Know Physics Module 7 questions
HSC Physics Module 7 The Nature of Light is considered by many students as the most interesting module in the Physics course. It explores the development of quantum mechanics and the “strangeness” of the Special Theory of Relativity.
Students find Special Relativity fascinating but difficult to comprehend.
## What type of questions are commonly asked in HSC Physics Module 7?
11 commonly asked question types from the HSC Physics Module 7 Syllabus are listed below:
NESA Content Block Question Type Electromagnetic Spectrum Describing the production and propagation of electromagnetic waves and relate these processes qualitatively to the predictions made by Maxwell’s electromagnetic theory.Investigating how the spectra of stars can provide information on:surface temperaturerotational and translational velocitydensitychemical composition Light: Wave Model Analysing qualitatively the diffraction of lightAnalysing quantitatively the interference of light using double slit apparatus and diffraction gratings $d\sin\theta=m\lambda$.Analysing quantitatively the plane polarisation of light using Malus’ Law. Light: Quantum Model Analysing the photoelectric effect as it occurs in metallic elements by applying the law of conservation of energy and the photon model of light. Light and Special Relativity Analysing and evaluating the evidence confirming or denying Einstein’s two postulates:the speed of light in a vacuum is an absolute constantall inertial frames of reference are equivalentAnalysing quantitatively situations in which time dilation and length contraction are observed.Investigating the evidence for time dilation and length contraction such as:observations of cosmic-origin muons at the Earth’s surfaceatomic clocks (Hafele–Keating experiment)evidence from particle acceleratorsevidence from cosmological studiesDescribing the consequences and applications of relativistic momentumUsing Einstein’s mass–energy equivalence relationship $E=mc^2$ to calculate the energy released by processes in which mass is converted to energy:production of energy by the sunparticle–antiparticle interactions, eg positron–electron annihilationcombustion of conventional fuel
Source: NESA website
## 10 Must Know Questions for HSC Physics Module 7
### Question 1 (5 marks)
A student conducts an experiment to investigate Malus’ law. Their light source is a laser that emits light polarised at an unknown angle. To conduct this experiment the student uses a detector with a light sensor and angle sensor.
The light sensor measures the intensity $I$ of incident light. In front of the light sensor there is a polariser, which can be rotated or removed. The angle sensor measures the angle $\theta$ of the polariser.
A schematic diagram of the apparatus is shown below, with the polariser currently in place and vertical at $\theta = 0 \degree$.
The student records the following data from conducting their experiment:
Angle Sensor Reading (degrees) Light Sensor Reading $(Wm ^{-2})$ Angle $\theta$ between laser polarisation and polariser (degrees) 0 3.75 10 4.41 30 5.00 40 4.85 70 2.94 90 1.25
(a) Describe a procedure that would be suitable to investigate Malus’ Law using this equipment. 2 (b) The intensity of the laser is $5.00 \ Wm^{-2}$. Complete the third column of the table above by finding the angle between the laser polarisation and the polariser. 1 (c) Explain why this experiment verifies Malus’ Law. 2
See Question 1 Solution
### Question 2 (4 marks)
An experimental setup to demonstrate Young’s double slit experiment is shown below.
Monochromatic light is passed through a double slit and the interference pattern is projected onto a screen. The distance between the central bright fringe and the bright fringes adjacent to it is $10 \ cm$.
(a) The student makes an adjustment that resulted in the new distance between the central bright fringe and the adjacent bright fringes increasing to 30 cm. Outline TWO possible changes the student could have made. 2 (b) At the time Young’s double-slit experiment was conducted, there were two competing models of the nature of light. Explain how Young’s experiment supported one of the models. 2
See Question 2 Solution
### Question 3 (3 marks)
The diagram below shows an AC voltage connected to the wires of an antenna. This is a common technique for producing radio waves.
(a) Explain how the apparatus shown produces radio waves. 2 (b) With reference to a classical theory of light, outline Maxwell’s contribution to our understanding of light that was supported by the discovery of radio waves. 3
See Question 3 Solution
### Question 4 (7 marks)
A plot of some experimental blackbody radiation spectra at different temperatures is shown in the diagram below, together with a spectrum predicted by classical theory.
(a) Explain how Planck accounted for the discrepancy between the experimental results and the predictions of classical theory. 3 (b) Describe results of two experiments that support Planck’s hypothesis regarding black body radiation. 3 (c) Using Planck’s energy equation, determine the energy (in joules) of a released quantum of red light if it has a wavelength of $700 \ nm$. Express your answer to three significant figures. 1
See Question 4 Solution
### Question 5 (3 marks)
Spectroscopy can be used to determine a large amount of information about objects in the galaxy. The following spectrum of a star was recorded.
Explain what information about the star can be determined from this spectrum.
See Question 5 Solution.
### Question 6 (6 marks)
Light is incident on a Zirconium surface in a vacuum. The graph below shows the variation of the stopping voltage $V_{s}$ of the electrons emitted from the surface with the frequency $f$ of the incident light.
(a) From the graph, determine the work function of Zirconium. 2 (b) Explain how the graph of stopping voltage variation with incident light frequency provides evidence supporting the particle model of light. 3
See Question 6 Solution
### Question 7 (3 marks)
The principle of relativity was proposed by Galileo Galilei in 1632.
(a) State Galileo’s principle of relativity. 1 (b) Describe the consequences of Galileo’s principle of relativity on observers in both inertial and non-inertial frames of reference. 2
See Question 7 Solution.
### Question 8 (6 marks)
A spacecraft leaves Earth at a speed of $0.80 \ c$ as measured by an observer on Earth. It heads towards, and continues beyond, a distant planet in the star system Gliese 3325. The planet is 30 light years (ly) away from Earth as measured by an observer on Earth.
When the spacecraft leaves Earth, Emmet, one of the astronauts in the spacecraft, is 20 years old.
(a) Calculate the time taken for the journey to the planet as measured by an observer on Earth. 2 (b) Calculate the distance between the Earth and the planet, as measured by Emmet. 2 (c) As the spacecraft goes past the planet, Emmet sends a radio signal to Earth. Calculate, as measured by the spacecraft observers, the time it takes for the signal to arrive at Earth. 2
See Question 8 Solution
### Question 9 (5 marks)
A deuteron (which consists of 1 proton and 1 neutron) is accelerated from rest to a speed of $0.8 \ c$ in a particle accelerator.
• Mass of a proton: $1.673 \times 10^{-27} \ kg$
• Mass of a neutron: $1.675 \times 10^{-27} \ kg$
(a) Calculate the accelerating potential $V$ in volts required to achieve this velocity. 3 (b) Calculate the work required to accelerate the deuteron from $0.8 \ c$ to $0.9 \ c$. 2
See Question 9 Solution
### Question 10 (6 marks)
Around 1850 a French scientist Leon Foucault performed an experiment to determine whether light travels faster in air or in water. His experimental setup is shown below.
In Foucault’s experiment a beam of light is reflected off mirror R to mirror M, back to mirror R, and then to position 1 on a screen, as shown in the left diagram.
Mirror R is then spun clockwise at a high rate, as shown in the middle diagram. Due to the rotation of mirror R during the flight time of the beam of light, the returning beam is reflected to a displaced position on the screen, position 2.
In the third phase of the experiment a 3 m long tube of water is placed between mirrors M and R, in the path of the beam of light, as shown in the diagram on the right.
(a) Outline the results of his experiment. 2 (b) Which early theories of light did the results of this experiment support or refute? Justify your answer by outlining each of the theories. 4
See Question 10 Solution
## Solutions to HSC Physics Module 7 Practice Questions
Detailed, step-by-step solutions to the Module 5 Advanced Mechanics questions are provided below.
Marking practice exams is just as important as answering the questions
### Question 1 Solution
Part (a):
1. Direct the laser beam into the detector. Measure the laser intensity in the absence of the polariser to obtain the control variable $I_{max}$.
2. Put the polariser in place and rotate it to specific angles, recording the measured laser intensity for each angle of the polariser. Use at least 5 different values of $\theta$.
3. Analyse the data, specifically by plotting a graph of $\frac{I}{I_{max}}$ vs $cos^2 \theta$.
4. Malus’ Law is $I = I_{max} cos^2 \theta$, which will be verified if the graph produced in Step 3 exhibits a gradient of 1.
Part (b):
Given the laser intensity is $I_{max} = 5.00 \ Wm^{-2}$, the ratio $\frac{I}{I_{max}}$ can be found from the intensity measurements recorded in the experiment.
Assuming Malus’ Law, $\theta$ can be found as follows:
\begin{aligned} \dfrac{I}{I_{max}} &= \cos^2 \theta \\\\ \cos \theta &= \sqrt{\dfrac{I}{I_{max}}} \\\\ \therefore \theta &= \cos^{-1} \bigg(\sqrt{\dfrac{I}{I_{max}}}\bigg) \end{aligned}
Calculating $(\theta)$ for each measured intensity yields:
Angle $\theta$ between laser polarisation and polariser (degrees) 30 20 0 10 40 60
Part (c):
Step 1: Plot a graph of $\dfrac{I}{I_{max}} \ vs \ \cos^2 \theta$. Draw a line of best fit through your data.
Step 2: Calculate the gradient of the line of best fit.
The gradient of the line of best fit is calculated from $m = \frac{rise}{run}$:
\begin{aligned} m &= \dfrac{rise}{run} \\\\ &= \dfrac{0.6-0.4}{0.6-0.4} \\\\ &= \dfrac{0.2}{0.2} \\\\ &= 1 \end{aligned}
Therefore, $y = x$ and hence $\dfrac{I}{I_{max}} = \cos^2 \theta$. Rearranging this equation, $I = I_{max} \cos^2 \theta$. This is Malus’ Law, which is thus verified by this experiment.
Back to Question 1
### Question 2 Solution
Part (a):
For the bright fringe adjacent to the central bright fringe, $m = 1$, in the relationship:
$d\sin \theta = m \lambda$
For a small $\theta$, $\sin \theta = \tan \theta= \frac{h}{L}$:
\begin{aligned}\therefore \frac{dh}{L} &= \lambda \\\\ h &= \frac{ \lambda L}{d} \end{aligned}
The separation on the screen of the central bright fringe and the adjacent bright fringes can be increased from $10 \ cm$ to $30 \ cm$ by:
• Moving the screen to three times the initial distance, $3L$.
• Increasing the wavelength of light to three times the initial wavelength, $3\lambda$.
• Decreasing the slit spacing to one third of the initial spacing, $\frac{1}{3}d$.
Note that there are THREE possible changes that could be made, of which the question requires TWO, so choose two from the above list.
Part (b):
Young’s double slit experiment demonstrated the occurrence of interference effects with light, which produced the pattern of bright and dark fringes on the screen. The wave model of light was able to explain the interference effect, while the particle model could not provide an explanation. Hence, Young’s double slit experiment supported the wave model of light.
Back to Question 2
### Question 3 Solution
Part (a):
Radio waves are electromagnetic waves. The AC powered antenna setup includes a dipole – the two separate metal rods. The application of AC causes charge to oscillate between the two rods of the dipole, resulting in changing electric and magnetic fields around the dipole antenna.
These changes in electric and magnetic field propagate sideways away from the antenna at the speed of light, as a transverse wave. This is an electromagnetic wave. The frequency of the wave is determined by the frequency of AC, so radio frequency AC produces radio waves.
Part (b):
Maxwell unified known, empirical laws of electricity and magnetism into the theory of electromagnetism, as expressed by Maxwell’s equations, which describe all electromagnetic phenomena. Maxwell’s equations can be combined to derive the wave equation, which predicts electromagnetic waves travel at a speed of $v = 3 \times 10^8 \ ms^{-1}$ in a vacuum. This is the known speed of light.
The theory also predicts that electromagnetic waves are transverse. The agreement between the speed of the hypothesised electromagnetic waves and the known speed of light led Maxwell to propose that light must be an electromagnetic oscillation, i.e. an electromagnetic wave.
Back to Question 3
### Question 4 Solution
Part (a):
When approaching questions about Planck’s explanation of black body radiation, students should be aware of the following context:
Classical theory predicts that atoms can exist at any energy and oscillate at any frequency, and that there is no preferred frequency for emitted radiation. Hence all frequencies are emitted equally. Given c = fλ, hence λ = c/f, therefore shorter wavelengths should dominate at any temperature.
Planck proposed that the intensity and hence energy of radiation emitted by a black body could only be increased by fixed amounts proportional to the frequency of radiation, given by $E = hf$. This required atoms to have discrete (or quantised) energy levels, and hence the atomic energy changes associated with radiation emission must be discrete changes. (1 mark)
At a given temperature one such discrete energy change would be most probable, hence occur most often, giving rise to the peak wavelength in the spectrum – a wavelength of light emitted more than any other and thus having the highest intensity. (1 mark)
Furthermore, atoms lack sufficiently large discrete energy changes to able to produce high frequency (short wavelength) radiation such as X-rays and gamma rays. Consequently, these wavelengths are not observed at any temperature. (1 mark)
Part (b):
Classical theory could not explain the photoelectric effect, while the concept of quantised light energy (photons with energy $E = hf$) was able to fully explain the phenomenon. The maximum kinetic energy of the ejected electrons was found to be $K_{max} = hf - \phi$, where the photon energy $hf$ must exceed the work function $\phi$ of the metal, else electrons are not ejected.
Classical theory could also not explain the emission line spectra from gases in discharge tubes. The specific wavelengths of emission (while no other wavelengths were emitted) constitute direct evidence of discrete (i.e. quantised) energy levels in atoms.
Part (c):
Planck’s energy equation is $E = hf$. This can be re-written as $E = \dfrac{hc}{\lambda}$ using the equation $v = c = f \lambda$.
By substituting the value for wavelength into this equation, we can calculate the energy of a quantum of red light:
\begin{aligned} E &= \dfrac{hc}{\lambda} \\\\ &= \dfrac{ 6.626 \times 10^{-34} \times 3 \times 10^8 }{ 700 \times 10^{-9} } \\\\ &= 2.84 \times 10^{-19} \ J \ (3 \ s.f.) \end{aligned}
Back to Question 4
### Question 5 Solution
The surface temperature $T$ of the star can be determined from the peak wavelength $\lambda_{max}$ of the black body spectrum by using Wien’s Law: $\lambda_{max} = \frac{2.898 \times 10^{-3}}{T}$.
The chemical composition of the stellar atmosphere is determined from the presence of spectral lines. Laboratory spectra of atoms/molecules can be used to identify those elements/compounds in the spectrum of the star. The strengths (depths) of the absorption lines are proportional to the concentrations of the atoms/molecules, indicating how much of each chemical is in the star’s atmosphere.
Students could also discuss information gained from the broadening of spectral lines:
The rotational velocity of the star can be determined from the extent of broadening of spectral lines (whole line) while density can be determined from the extent of pressure broadening (wings of line).
Back to Question 5
### Question 6 Solution
Part (a):
The magnitude of the stopping voltage in volts is equal to the magnitude of the maximum kinetic energy in electron volts. Therefore, a graph of stopping voltage vs frequency can be interpreted as a graph of maximum kinetic energy vs frequency. Hence, the y-intercept of the graph represents the work function $\phi$, in electron volts.
By extrapolating the graph back to the y-axis, the work function of Zirconium is found to be $4 \ eV$.
Part (b):
From the graph, a relationship between maximum kinetic energy and frequency of incident light can be obtained.
The gradient of the graph is equal to Planck’s constant $h$.
\begin{aligned} y &= mx + c \\ K_{max}& =gradient \times f - \phi \\ K_{max}& =hf - \phi \end{aligned}
From the photoelectric effect equation, we can infer:
1. the maximum kinetic energy $K_{max}$ of ejected photoelectrons is directly proportional to the frequency $f$ of light.
2. the energy absorbed by an electron is equal to the energy carried by a photon $hf$.
This supports the particle model of light, which stated that the energy of the photon is determined by its frequency by $E = hf$ and there is a one-to-one interaction between an electron and a photon.
Back to Question 6
### Question 7 Solution
Part (a):
Galileo’s principle of relativity states that the mechanical laws of physics are the same for every observer moving uniformly with constant speed in a straight line (i.e. for every observer in an inertial frame).
Part (b):
Galileo’s principle of relativity states that there is no physical way, from within an inertial frame, to differentiate between a body moving at a constant speed (one inertial frame of reference) and a body at rest (another inertial frame of reference). This means that constant motion cannot be detected in an inertial frame of reference.
However, acceleration can be detected in non-inertial frames of reference. This means that the principle of relativity holds true for any frames of reference that are not accelerating (i.e. inertial frames of reference).
Back to Question 7
### Question 8 Solution
Part (a):
Using the formula, $t = \frac{d}{v}$, we can calculate the time taken for the journey to the planet.
$t = \frac{30 \ \text{c years}}{0.80 \ c } = 37.5 \ years$
1 light year can be written as 1 c year.
Part (b):
Using the formula for length contraction, we can calculate the distance between the Earth and the planet as measured by Emmet on the spacecraft:
\begin{aligned} l &= l_0 \times \sqrt{1-\frac{v^2}{c^2}} \\\\ &= 30 \times \sqrt{1-\frac{0.8c^2}{c^2}} \\\\ &= 30 \times \sqrt{1-0.64} \\\\ &= 18 \ light \ years \end{aligned}
Part (c):
Let the time taken for the signal to arrive at Earth be denoted by $T$. This means that the signal reaches Earth after travelling a distance of $cT$.
However, this distance is also equal to 18 light years (from part (b)) plus the distance travelled by the Earth in time $T$, i.e:
$d = 18 \text{c years} + 0.80 cT$
By rearranging this equation, we can calculate the time taken for the signal to arrive at earth:
\begin{aligned} cT &= 18c + 0.80 cT \\\\ \therefore 0.20 T &= 18 \ \text{c years} \\\\ T &= 90 \ years \end{aligned}
Back to Question 8
### Question 9 Solution
Part (a):
\begin{aligned} E &= m_0 c^2 \\\\ &= (1.673 \times 10^{-27} + 1.675 \times 10^{-27})c^2 \\\\ &= 3.348 \times 10^{-27} \times (3 \times 10^8) ^2 \\\\ &= 3.0132 \times 10^{-10} \ J \end{aligned}
\begin{aligned} E_{0.8 \ c} &= m_v c^2 \\\\ &= \frac{ ( m_0 c^2 ) }{ \sqrt{ 1 - \frac{ v^2 }{ c^2 } } } \\\\ &= \frac{ ( 3.0132 \times 10^{-10} ) }{ \sqrt{ 1 - (0.8)^2 } } \\\\ &= 5.022 \times 10^{-10} \ J \end{aligned}
\begin{aligned} K &= m_{\nu} c^2 - m_0 c^2 \\ \Delta K&=qV \\ E_{0.8 \ c} - E_{rest}&= qV \\ V&= \frac{E_{0.8c} - E_{rest}}{q} \\ &=\frac{5.022 \times 10^{-10} -3.0132 \times 10^{-10}}{1.602 \times 10^{-19}} \\ &= 1.254 \ GV \end{aligned}
Part (b):
Step 1: Calculate the total energy of the deuteron when it is travelling at $0.9 \ c$.
\begin{aligned} E_{0.9c} &= m_v c^2 \\\\ &= \frac{ ( m_0 c^2 ) }{ \sqrt{ 1 - \frac{ v^2 }{ c^2 } } } \\\\ &= \frac{ ( 3.0132 \times 10^{-10} ) }{ \sqrt{ 1 - (0.9)^2 } } \\\\ &= 6.912 \times 10^{-10} \ J \end{aligned}
Step 2: Calculate the work required to increase the energy of the deuteron from the previous value at $v = 0.8 \ c$ to the new value at $v = 0.9 \ c$.
\begin{aligned} W &= E_{0.9 c} - E_{0.8 c} \\\\ &= 6.912 \times 10^{-10} - 5.022 \times 10^{-10} \\\\ &= 1.890 \times 10^{-10} \ J \end{aligned}
Back to Question 9
### Question 10 Solution
Part (a):
Light travelling through only air reached the screen at position 2. Light travelling through air and the tube of water reached the screen at a more displaced position on the screen, position 3.
Part (b):
Foucault’s experimental results indicated that light travels slower in water than in air. The presence of the tube of water increased the flight time of the beam of light (by slowing the light down), which allowed the spinning mirror to rotate more and reflect the returning beam of light to a more displaced position on the screen.
At the time there were two major and competing models of light: Huygens’ wave model and Newton’s corpuscle model. For light entering a more dense medium:
• Huygens’ model predicted that waves of light will decrease in velocity.
• Newton’s model predicted that corpuscles of light will increase in velocity.
Hence Foucault’s result refuted Newton’s corpuscle model and supported Huygens’ wave model.
Back to Question 10
## Access our library of HSC Physics Module 7 Exam Questions
Test your understanding of any HSC Physics Module 7 concepts in just 10 minutes with Learnable’s customisable quizzes with over 500+ questions for each module. Instant feedback provides immediate adjustments on your misconceptions. Try Learnable for free now.
### Written by DJ Kim
DJ is the founder of Learnable and has a passionate interest in education and technology. He is also the author of Physics resources on Learnable.
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# Change of variable integral
Consider the following integral $$\int_{\Omega} f(y,My-z_2)\, g(z_1,z_2,y) ~ dz_1\, dz_2\, dy$$
$f(y,My-z_2) = 1$ (a constant function for each value of $y$ and $z_2$)
$g(z_1,z_2,y) = z_2$
The integration region $\Omega$ is defined as $\{(z_1, z_2, y) ~|~ z_1 = 0; 0 \leq z_2 \leq 1 - My; 0 \leq y \leq 1 \}$. I wish to calculate this integral, but I am facing some problems, due to the fact that $\Omega$ is defined in such a way that $z_1 = 0$.
If is a triple integral, is zero because the domain $\Omega$ has measure zero.
• Suppose I eliminate variable $z1$ and compute the following double integral: Consider the following integral $$\int_{\Omega} f(y,My-z_2)\, g(z_2,y) ~ dz_2\, dy$$ where $f(y,My-z_2) = 1$ and $g(z_2,y) = z_2$. The integration region $\Omega$ is defined as $\{(z_2, y) ~|~ 0 \leq z_2 \leq 1 - My; 0 \leq y \leq 1 \}$. What about this? – Statistics Optimization Real E Nov 11 '14 at 11:58
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Publication
Title
Circular motion analysis of time-varying bioimpedance
Author
Abstract
This paper presents a step forward towards the analysis of a linear periodically time-varying (PTV) bioimpedance ${{Z}_{\text{PTV}}}(\,j\omega ,t)$ , which is an important subclass of a linear time-varying (LTV) bioimpedance. Similarly to the Fourier coefficients of a periodic signal, a PTV impedance can be decomposed into frequency dependent impedance phasors, \${{Z}_{r}}(\,j\omega ){{\text{e}}
Language
English
Source (journal)
Physiological measurement / Institute of Physical Sciences in Medicine. - Bristol, 1980, currens
Publication
Bristol : IOP, 2015
ISSN
0967-3334
Volume/pages
36:11(2015), p. 2353-2367
ISI
000367844400009
Full text (Publisher's DOI)
Full text (open access)
Full text (publisher's version - intranet only)
UAntwerpen
Faculty/Department Research group Publication type Subject Affiliation Publications with a UAntwerp address
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×
## Discrete Probability
How often will a die come up "4"? How likely is it to rain tomorrow? Probability is one of the most powerful frameworks for modeling the world around us.
# Probability Misconceptions
Suppose you flip a fair coin (i.e., a coin that lands Heads with probability 1/2 and Tails with probability 1/2) 4 times. Which of these outcomes is most likely?
Suppose you flip a fair coin (i.e., a coin that lands Heads with probability 1/2 and Tails with probability 1/2) 4 times. Which of these sequences of Heads (H) and Tails (T) is most likely?
Suppose you flip a fair coin (i.e., a coin that lands Heads with probability 1/2 and Tails with probability 1/2) 10 times. Which of these sequences of Heads (H) and Tails (T) is most likely?
Suppose you flip a fair coin (i.e., a coin that lands Heads with probability 1/2 and Tails with probability 1/2) 100 times. Which is more likely:
If you flip a fair coin (i.e., a coin that lands Heads with probability 1/2 and Tails with probability 1/2) 10 times, which is more likely?
×
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## Thursday, March 15, 2012
### Power Laws
Many non linear phenomena where shown to obey a power law. However, this paper shows that we could have inferred them improperly. The paper, I realize, is quite famous, since it gained over than 1000 citations according to Google Scholar (but I did not know it). ...
I gave it a look and it is actually very interesting. In fact, my good all power laws (e.g. Rigon et al., 1996; Maritan et al., 1996; Rinaldo et al., 1998; Convertino et al., 2007) were a little different, since my exponents were all less than 1 (not greater than 1, as those treated in the paper): but I think that most of the result can be appropriately adapted. There are also these weblog pages where there are further comments and a lot of information (maybe too much ?) by one of the authors.
For general information about statistics give a look here.
#### 1 comment:
1. The contents has provided meaningful information thanks for sharing info
Pub licence
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## What’s Wrong With Modern Macro? Part 5 Filtering Away All Our Problems
Part 5 in a series of posts on modern macroeconomics. In this post, I will describe some of the issues with the Hodrick-Prescott (HP) filter, a tool used by many macro models to isolate fluctuations at business cycle frequencies.
A plot of real GDP in the United States since 1947 looks like this
Although some recessions are easily visible (like 2009), the business cycle is not especially easy to see. The growing trend of economic growth over time dominates more frequent business cycle fluctuations. While the trend is important, if we want to isolate more frequent business cycle fluctuations, we need some way to remove the trend. The HP Filter is one method of doing that. Unlike the trends in my last post, which simply fit a line to the data, the HP filter allows the trend to change over time. The exact equation is a bit complex, but here’s the link to the Wikipedia page if you’d like to see it. Plotting the HP trend against actual GDP looks like this
By removing this trend, we can isolate deviations from trend. Usually we also want to take a log of the data in order to be able to interpret deviations in percentages. Doing that, we end up with
The last picture is what the RBC model (and most other DSGE models) try to explain. By eliminating the trend, the HP filter focuses exclusively on short term fluctuations. This shift in focus may be an interesting exercise, but it eliminates much of what make business cycles important and interesting. Look at the HP trend in recent years. Although we do see a sharp drop marking the 08-09 recession, the trend quickly adjusts so that we don’t see the slow recovery at all in the HP filtered data. The Great Recession is actually one of the smaller movements by this measure. But what causes the change in the trend? The model has no answer to this question. The RBC model and other DSGE models that explain HP filtered data cannot hope to explain long periods of slow growth because they begin by filtering them away.
Let’s go back one more time to these graphs
Notice that these graphs show the fit of the model to HP filtered data. Here’s what they look like when the filter is removed
Not so good. And it gets worse. Remember we have already seen in the last post that TFP itself was highly correlated with each of these variables. If we remove the TFP trend, the picture changes to
Yeah. So much for that great fit. Roger Farmer describes the deceit of the RBC model brilliantly in a blog post called Real Business Cycle and the High School Olympics. He argues that when early RBC modelers noticed a conflict between their model and the data, they didn’t change the model. They changed the data. They couldn’t clear the olympic bar of explaining business cycles, so they lowered the bar, instead explaining only the “wiggles.” But, as Farmer contends, the important question is in explaining those trends that the HP filter assumes away.
Maybe the most convincing reason that something is wrong with this method of measuring business cycles is that measurements of the cost of business cycles using this metric are tiny. Based on a calculation by Lucas, Ayse Imrohoroglu explains that under the assumptions of the model, an individual would need to be given $28.96 per year to be sufficiently compensated for the risk of business cycles. In other words, completely eliminating recessions is worth about as much as a couple decent meals. Obviously to anyone who has lived in a real economy this number is completely ridiculous, but when business cycles are relegated to the small deviations from a moving trend, there isn’t much to be gained from eliminating the wiggles. There are more technical reasons to be wary of the HP filter that I don’t want to get into here. A recent paper called “Why You Should Never Use the Hodrick-Prescott Filter” by James Hamilton, a prominent econometrician, goes into many of the problems with the HP filter in detail. He opens by saying “The HP filter produces series with spurious dynamic relations that have no basis in the underlying data-generating process.” If you don’t trust me, at least trust him. TFP doesn’t measure productivity. HP filtered data doesn’t capture business cycles. So the RBC model is 0/2. It doesn’t get much better from here. ## What’s Wrong With Modern Macro? Part 4 How Did a "Measure of our Ignorance" Become the Cause of Business Cycles? Part 4 in a series of posts on modern macroeconomics. Parts 1, 2, and 3 documented the revolution that transformed Keynesian economics into DSGE economics. This post begins my critique of that revolution, beginning with a discussion of its reliance on total factor productivity (TFP) shocks. In my last post I mentioned that the real business cycle model implies technology shocks are the primary driver of business cycles. I didn’t, however, describe what these technology shocks actually are. To do that, I need to bring in the work of another Nobel Prize winning economist: Robert Solow. ### The Solow Residual and Total Factor Productivity Previous posts in this series have focused on business cycles, which encompass one half of macroeconomic theory. The other half looks at economic growth over a longer period. In a pair of papers written in 1956 and 1957, Solow revolutionized growth theory. His work attempted to quantitatively decompose the sources of growth. Challenging the beliefs of many economists at the time, he concluded that changes in capital and labor were relatively unimportant. The remainder, which has since been called the “Solow Residual” was attributed to “total factor productivity” (TFP) and interpreted as changes in technology. Concurrent research by Moses Abramovitz confirmed Solow’s findings, giving TFP 90% of the credit for economic growth in the US from 1870 to 1950. In the RBC model, the size of technology shocks is found by subtracting the contributions of labor and capital from total output. What remains is TFP. ### “A Measure of Our Ignorance” Because TFP is nothing more than a residual, it’s a bit of a stretch to call it technical change. As Abramovitz put it, it really is just “a measure of our ignorance,” capturing the leftover effects in reality that have not been captured by the simple production function assumed. He sums up the problem in a later paper summarizing his research Standard growth accounting is based on the notion that the several proximate sources of growth that it identifies operate independently of one another. The implication of this assumption is that the contributions attributable to each can be added up. And if the contribution of every substantial source other than technological progress has been estimated, whatever of growth is left over – that is, not accounted for by the sum of the measured sources – is the presumptive contribution of technological progress Moses Abramovitz (1993) – The Search for the Sources of Growth: Areas of Ignorance, Old and New p. 220 In other words, only if we have correctly specified the production function to include all of the factors that determine total output can we define what is left as technological progress. So what is this comprehensive production function that is supposed to encompass all of these factors? Often, it is simply $Y=AK^\alpha L^{1-\alpha}$ Y represents total output in the economy. All labor hours, regardless of the skill of the worker or the type of work done, are stuffed into L. Similarly, K covers all types of capital, treating diverse capital goods as a single homogeneous blob. Everything that doesn’t fit into either of these becomes part of A. This simple function, known as the Cobb-Douglas function, is used in various economic applications. Empirically, the Cobb-Douglas function matches some important features in the data, notably the constant shares of income accrued to both capital and labor. Unfortunately, it also appears to fit any data that has this feature, as Anwar Shaikh humorously points out by fitting it to fake economic data that is made to spell out the word HUMBUG. Shaikh concludes that the fit of the Cobb-Douglas function is a mathematical trick rather than a proper description of the fundamentals of production. Is it close enough to consider its residual an accurate measure of technical change? I have some doubts. There are also more fundamental problems with aggregate production functions that will need to wait for a later post. ### Does TFP Measure Technological Progress or Something Else? The technical change interpretation of the Solow residual runs into serious trouble if there are other variables correlated with it that are not directly related to productivity, but that also affect output. Robert Hall tests three variables that possibly fit this criteria (military spending, oil prices, and the political party of the president), and finds that all three affect the residual, casting doubt on the technology interpretation. Hall cites five possible reasons for the discrepancy between Solow’s interpretation and reality, but they are somewhat technical. If you are interested, take a look at the paper linked above. The main takeaway should be that the simple idealized production function is a bit (or a lot) too simple. It cuts out too many of the features of reality that are essential to the workings of a real economy. ### TFP is Nothing More Than a Noisy Measure of GDP The criticisms of the production function above are concerning, but subject to debate. We cannot say for sure whether the Cobb-Douglas, constant returns to scale formulation is close enough to reality to be useful. But there is a more powerful reason to doubt the TFP series. In my last post, I put up these graphs that appear to show a close relationship between the RBC model and the data. Here’s another graph from the same paper This one is not coming from a model at all. It simply plots TFP as measured as the residual described above against output. And yet the fit is still pretty good. In fact, looking at correlations with all of the variables shows that TFP alone “explains” the data about as well as the full model What’s going on here? Basically, the table above shows that the fit of the RBC model only works so well because TFP is already so close to the data. For all its talk of microfoundations and the importance of including the optimizing behavior of agents in a model, the simple RBC framework does little more than attempt to explain changes in GDP using a noisy measure of GDP itself. Kevin Hoover and Kevin Salyer make this point in a revealing paper where they claim that TFP is nothing more than “colored noise.” To defend this claim, they construct a fake “Solow residual” by creating a false data series that shares similar statistical properties to the true data, but whose coefficients come from a random number generator rather than a production function. Constructed in this way, the new residual certainly does not have anything to do technology shocks, but feeding this false residual into the RBC model still provides an excellent fit to the data. Hoover and Salyer conclude The relationship of actual output and model output cannot indicate that the model has captured a deep economic relationship; for there is no such relationship to capture. Rather, it shows that we are seeing a complicated version of a regression fallacy: output is regressed on a noisy version of itself, so it is no wonder that a significant relationship is found. That the model can establish such a relationship on simulated data demonstrates that it can do so with any data that are similar in the relevant dimensions. That it has done so for actual data hardly seems subject to further doubt. Hoover and Salyer (1998) – Technology Shocks or Coloured Noise? Why real-business-cycle models cannot explain actual business cycles, p.316 Already the RBC framework appears to be on shaky ground, but I’m just getting started (my plan for this series seems to be constantly expanding – there’s even more wrong with macro than I originally thought). My next post will be a brief discussion of the filtering method used in many DSGE applications. I will follow that with an argument that the theoretical justification for using an aggregate production function (Cobb-Douglas or otherwise) is extremely weak. At some point I will also address rational expectations, the representative agent assumption, and why the newer DSGE models that attempt to fix some of the problems of the RBC model also fail. ## What’s Wrong with Modern Macro? Part 3 Real Business Cycle and the Birth of DSGE Models Part 3 in a series of posts on modern macroeconomics. Part 1 looked at Keynesian economics and part 2 described the reasons for its death. In this post I will explain dynamic stochastic general equilibrium (DSGE) models, which began with the real business cycle (RBC) model introduced by Kydland and Prescott and have since become the dominant framework of modern macroeconomics. “What I am going to describe for you is a revolution in macroeconomics, a transformation in methodology that has reshaped how we conduct our science.” That’s how Ed Prescott began his Nobel Prize lecture after being awarded the prize in 2004. While he could probably benefit from some of Hayek’s humility, it’s hard to deny the truth in the statement. Lucas and Friedman may have demonstrated the failures of Keynesian models, but it wasn’t until Kydland and Prescott that a viable alternative emerged. Their 1982 paper, “Time to Build and Aggregate Fluctuations,” took the ideas of microfoundations and rational expectations and applied them to a flexible model that allowed for quantitative assessment. In the years that followed, their work formed the foundation for almost all macroeconomic research. ### Real Business Cycle The basic setup of a real business cycle (RBC) model is surprisingly simple. There is one firm that produces one good for consumption by one consumer. Production depends on two inputs, labor and capital, as well as the level of technology. The consumer chooses how much to work, how much to consume, and how much to save based on its preferences, the current wage, and interest rates. Their savings are added to the capital stock, which, combined with their choice of labor, determines how much the firm is able to produce. There is no money, no government, no entrepreneurs. There is no unemployment (only optimal reductions in hours worked), no inflation (because there is no money), and no stock market (the one consumer owns the one firm). There are essentially none of the features that most economists before 1980 as well as non-economists today would consider critically important for the study of macroeconomics. So how are business cycles generated in an RBC model? Exclusively through shocks to the level of technology (if that seems strange it’s probably even worse than you expect – stay tuned for part 4). When consumers and firms see changes in the level of technology, their optimal choices change which then causes total output, the number of hours worked, and the level of consumption and investment to fluctuate as well. Somewhat shockingly, when the parameters are calibrated to match the data, this simple model does a good job capturing many of the features of measured business cycles. The following graphs (from Uhlig 2003) demonstrate a big reason for the influence of the RBC model. Looking at those graphs, you might wonder why there is anything left for macroeconomists to do. Business cycles have been solved! However, as I will argue in part 4, the perceived closeness of model and data is largely an illusion. There are, in my opinion, fundamental issues with the RBC framework that render it essentially meaningless in terms of furthering our understanding of real business cycles. ### The Birth of Dynamic Stochastic General Equilibrium Models Although many economists would point to the contribution of the RBC model in explaining business cycles on its own, most would agree that its greater significance came from the research agenda it inspired. Kydland and Prescott’s article was one of the first of what would come to be called Dynamic Stochastic General Equilibrium (DSGE) models. They are dynamic because they study how a system changes over time and stochastic because they introduce random shocks. General equilibrium refers to the fact that the agents in the model are constantly maximizing (consumers maximizing utility and firms maximizing profits) and markets always clear (prices are set such that supply and demand are equal in each market in all time periods). Due in part to the criticisms I will outline in part 4, DSGE models have evolved from the simple RBC framework to include many of the features that were lost in the transition from Keynes to Lucas and Prescott. Much of the research agenda in the last 30 years has aimed to resurrect Keynes’s main insights in microfounded models using modern mathematical language. As a result, they have come to be known as “New Keynesian” models. Thanks to the flexibility of the DSGE setup, adding additional frictions like sticky prices and wages, government spending, and monetary policy was relatively simple and has enabled DSGE models to become sufficiently close to reality to be used as guides for policymakers. I will argue in future posts that despite this progress, even the most advanced NK models fall short both empirically and theoretically. ## What’s Wrong With Modern Macro? Part 1 Before Modern Macro - Keynesian Economics Part 1 in a series of posts on modern macroeconomics. This post focuses on Keynesian economics in order to set the stage for my explanation of modern macro, which will begin in part 2. If you’ve never taken a macroeconomics class, you almost certainly have no idea what macroeconomists do. Even if you have an undergraduate degree in economics, your odds of understanding modern macro probably don’t improve much (they didn’t for me at least. I had no idea what I was getting into when I entered grad school). The gap between what is taught in undergraduate macroeconomics classes and the research that is actually done by professional macroeconomists is perhaps larger than in any other field. Therefore, for those of you who made the excellent choice not to subject yourself to the horrors of a first year graduate macroeconomics sequence, I will attempt to explain in plain English (as much as possible), what modern macro is and why I think it could be better. But before getting to modern macro itself, it is important to understand what came before. Keep in mind throughout these posts that the pretense of knowledge is quite strong here. For a much better exposition that is still somewhat readable for anyone with a basic economic background, Michael De Vroey has a comprehensive book on the history of macroeconomics. I’m working through it now and it’s very good. I highly recommend it to anyone who is interested in what I say in this series of posts. ### Keynesian Economics Although Keynes was not the first to think about business cycles, unemployment, and other macroeconomic topics, it wouldn’t be too much of an exaggeration to say that macroeconomics as a field didn’t truly appear until Keynes published his General Theory in 1936. I admit I have not read the original book (but it’s on my list). My summary here will therefore be based on my undergraduate macro courses, which I think capture the spirit (but probably not the nuance) of Keynes. Keynesian economics begins by breaking aggregate spending (GDP) into four pieces. Private spending consists of consumption (spending by households on goods and services) and investment (spending by firms on capital). Government spending on goods and services makes up the rest of domestic spending. Finally, net exports (exports minus imports) is added to account for foreign expenditures. In a Keynesian equilibrium, spending is equal to income. Consumption is assumed to be a fraction of total income, which means that any increase in spending (like an increase in government spending) will cause an increase in consumption as well. An important implication of this setup is that increases in spending increase total income by more than the initial increase (called the multiplier effect). Assume that the government decides to build a new road that costs$1 million. This increase in expenditure immediately increases GDP by $1 million, but it also adds$1 million to the income of the people involved in building the road. Let’s say that all of these people spend 3/4 of their income and save the rest. Then consumption also increases by $750,000, which then becomes other people’s incomes, adding another$562,500, and the process continues. Some algebra shows that the initial increase of $1 million leads to an increase in GDP of$4 million. Similar results occur if the initial change came from investment or changes in taxes.
The multiplier effect also works in the other direction. If businesses start to feel pessimistic about the future, they might cut back on investment. Their beliefs then become self-fulfilling as the reduction in investment causes a reduction in consumption and aggregate spending. Although the productive resources in the economy have not changed, output falls and some of these resources become underutilized. A recession occurs not because of a change in economic fundamentals, but because people’s perceptions changed for some unknown reason – Keynes’s famous “animal spirits.” Through this mechanism, workers may not be able to find a job even if they would be willing to work at the prevailing wage rate, a phenomenon known as involuntary unemployment. In most theories prior to Keynes, involuntary unemployment was impossible because the wage rate would simply adjust to clear the market.
Keynes’s theory also opened the door for government intervention in the economy. If investment falls and causes unemployment, the government can replace the lost spending by increasing its own expenditure. By increasing spending during recessions and decreasing it during booms, the government can theoretically smooth the business cycle.
### IS-LM
The above description is Keynes at its most basic. I haven’t said anything about monetary policy or interest rates yet, but both of these were essential to Keynes’s analysis. Unfortunately, although The General Theory was a monumental achievement for its time and probably the most rigorous analysis of the economy that had been written, it is not exactly the most readable or even coherent theory. To capture Keynes’s ideas in a more tractable framework, J.R. Hicks and other economists developed the IS-LM model.
I don’t want to give a full derivation of the IS-LM model here, but the basic idea is to model the relationship between interest rates and income. The IS (Investment-Savings) curve plots all of the points where the goods market is in equilibrium. Here we assume that investment depends negatively on interest rates (if interest rates are high, firms would rather put their money in a bank then invest in new projects). A higher interest rate then lowers investment and decreases total income through the same multiplier effect outlined above. Therefore we end up with a negative relationship between interest rates and income.
The LM (Liquidity Preference-Money) curve plots all of the points where the money market is in equilibrium. Here we assume that the money supply is fixed. Money demand depends negatively on interest rates (since a higher interest rate means you would rather keep money in the bank than in your wallet) and positively on income (more cash is needed to buy stuff). Together these imply that a higher level of income results in a lower interest rate required to clear the money market. An equilibrium in the IS-LM model comes when both the money market and the goods market are in equilibrium (the point where the two lines cross).
The above probably doesn’t make much sense if you haven’t seen it before. All you really need to know is that an increase in government spending or investment shifts the IS curve right, which increases both income and interest rates. If the central bank increases the money supply, the LM curve shifts right, increasing income and decreasing interest rates. Policymakers then have two powerful options to combat economic downturns.
In the decades following Keynes and Hicks, the IS-LM model grew to include hundreds or thousands of equations that economists attempted to estimate econometrically, but the basic features remained in place. However, in the 1970s, the Keynesian model came under attack due to both empirical and theoretical failures. Part 2 will deal with these failures and the attempts to solve them.
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# Creating a Graph Data Structure
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Hi guys i am wanting to create a graph data structure in c++ and link the edges to the nodes. so the graph looks like: I want to create this graph so i can then implement a A* pathfinding algorithm. I have never created a graph data structure before so any help would be muchly appreciated.
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A waypoint (i.e. nodes of the graph) is suitable to provide information for each point of interest in your dungeon. Obviously points of interest are most of those marked by the green dots in your image. Necessary data for each waypoint is what waypoints can be reached directly.
Another data perhaps necessary is the grid location of the waypoint. Looking furthur, informations of the length of the way (i.e. edges of the graphs) from one waypoint to the next may be of interest if the shortest way should be found. The length can be calculated from the grid location stored with waypoints. More advanced information for the way would be an additional walkability value for determining a maximum speed if using that way.
Until now waypoints like 14 and 21 need not necessarily be given. However, they become of interest if used e.g. to steer animations, since they allow to determine a change in direction of movement. However, for a pure path finding they are not needed.
Another aspect is whether ways may also be unidirectional, i.e. they lead from waypoint A to waypoint B but not the other way (e.g. a trap door).
Whether waypoints and/or ways are explicitely modeled by an own class depends on the amount of information inherent, and the amount of redundancy a solution will have. One possibility would be the following:
typedef enum Direction_t { NORTH, EAST, SOUTH, WEST}class Waypoint {public: Waypoint* next(Direction_t inDirection) { return _waypoints[inDirection]; } float costs(Direction_t inDirection) const { float result = INFINITY; Waypoint* neighbour = next(inDirection); if(neighbour) { // the following assumes pure horizontal or vertical direction result = _walkability[inDirection] * ( ::abs(_x-neighbour->_x) + ::abs(_y-neighbour->_y) ); } return result; }private: // points to the next Waypoint indexed by a Direction_t; nil if no Waypoint in that direction Waypoint* _neighbours[4]; // a value for each neighboured Waypoint float _walkability[4]; // grid location int32_t _x, _y;};
This (incomplete) solution allows to have unidirectional ways since each Waypoint stores only how to reach its possible neighbours, but not the way back. If a way back would be possible than the particular neighboured Waypoint must have set its _waypoints[oppositeDirection] to the original Waypoint.
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A more abstracted version, allowing for non-NESW connections, would be:
struct Waypoint { struct Connection { Waypoint* target; float value }; std::vector<Connection> neighbours; // grid location; could be a point int32_t _x, _y;};
... i.e. a waypoint is just a list of connections to other waypoints and a location.
You're probably also going to want to read the connection information in from a (text) file.
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I made a graph/map data struct in one of my classes at school and it worked out quite well.
Create a Vertex Class
Each Vertex has a list of connection lines
Each connection line has a Vertex it connects to
Then all you have to do is traverse through these vertecies to find a path.
I recommend you make your graph be templated. This way you can have whatever data stored at each vertex and whatever data stored at each connection.
Luck,
J
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# Understanding $V=K\otimes_{\mathbb{F}_p}\mathbb{F}_{p^n}$ where $K$ is the algebraic closure of $\mathbb{F}_p$ [duplicate]
So, let $K$ be an algebraic closure of $\mathbb{F}_{p}$ and consider the space $V=K\otimes_{\mathbb{F}_p}\mathbb{F}_{p^n}$ as a $K$ vector space. It seems pretty straight forward to show that this vector space is $n$ dimensional since for $\{\alpha_1,\ldots\alpha_n\}$ linearly independent elements of $\mathbb{F}_{p^n}$ (considered as itself a vector space over $\mathbb{F}_p$) we have the linearly independant tensors $1\otimes\alpha_i$. Then I think it is clear that any elementary tensor is a $K$ linear sum of these and so these must span $V$.
Anyway, what I don't understand is in what way this vector space is practically different from simply considering $\mathbb{F}_{p^n}$ as a $\mathbb{F}_p$ vector space. Studying for my algebra qualifying exam, I worked on a problem that asked to calculate the basis for $V$ for which the Froebenius automorphism is in Jordan canonical form. I don't see how this is different from doing it for $\mathbb{F}_{p^n}$. Although, we know a priori that the form exists since the underlying field is complete algebraically closed, but isn't that true for $\mathbb{F}_{p^n}$?
Thanks.
## marked as duplicate by Community♦May 24 '17 at 20:34
• The required roots of unity are not necessarily available in $\Bbb{F}_{p^n}$. To see this consider the case $p=2,n=3$. The Frobenius automorphism is of order three, and it is diagonalizable in $V$, because the primitive third roots of unity are in $K$. However, we don't have those roots of unity in $\Bbb{F}_8$, because $3\nmid(8-1)$. – Jyrki Lahtonen May 24 '17 at 20:31
• I don't think this is a duplicate, $\overline{\mathbb{F}}\otimes_{\mathbb{F}_p}\mathbb{F}_{p^n}$, the Frobenius acting on it and what is useful for is a different question – reuns May 24 '17 at 20:36
• @ JyrkiLahtonen Wait I didn't understand what really means "the Frobenius $T$ in Jordan normal form". We want a basis such that $T(\alpha_i) = m_i \alpha_i$ with $m \in \mathbb{F}_p$ ? – reuns May 24 '17 at 20:48
• The minimal polynomial of $F$ is $T^n-1$. So if $\gcd(n,p)=1$ the eigenvalues of $F$ are $n$th roots of unity. Those exist in $K$, but not necessarily in $\Bbb{F}_{p^n}$. Therefore $F$ is diagonalizable over $K$ but not necessarily over the prime field, or even over $\Bbb{F}_{p^n}$. – Jyrki Lahtonen May 24 '17 at 20:49
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6.11. Bidirectional Recurrent Neural Networks¶
All the recurrent neural network models so far discussed have assumed that the current time step is determined by the series of earlier time steps. Therefore, they all pass information through hidden states in a forward direction. Sometimes, however, the current time step can be determined by later time steps. For example, when we write a statement, we may modify the words at the beginning of the statement based on the words at the end. Bidirectional recurrent neural networks add a hidden layer that passes information in a backward direction to more flexibly process such information. Figure 6.12 demonstrates the architecture of a bidirectional recurrent neural network with a single hidden layer.
Now, we will look at the specifics of such a network. For a given time step $$t$$, the mini-batch input is $$\boldsymbol{X}_t \in \mathbb{R}^{n \times d}$$ (number of examples: $$n$$, number of inputs: $$d$$) and the hidden layer activation function is $$\phi$$. In the bidirectional architecture: We assume the forward hidden state for this time step is $$\overrightarrow{\boldsymbol{H}}_t \in \mathbb{R}^{n \times h}$$ (number of forward hidden units: $$h$$) and the backward hidden state is $$\overleftarrow{\boldsymbol{H}}_t \in \mathbb{R}^{n \times h}$$ (number of backward hidden units: $$h$$). Thus, we can compute the forward and backward hidden states:
\begin{split}\begin{aligned} \overrightarrow{\boldsymbol{H}}_t &= \phi(\boldsymbol{X}_t \boldsymbol{W}_{xh}^{(f)} + \overrightarrow{\boldsymbol{H}}_{t-1} \boldsymbol{W}_{hh}^{(f)} + \boldsymbol{b}_h^{(f)}),\\ \overleftarrow{\boldsymbol{H}}_t &= \phi(\boldsymbol{X}_t \boldsymbol{W}_{xh}^{(b)} + \overleftarrow{\boldsymbol{H}}_{t+1} \boldsymbol{W}_{hh}^{(b)} + \boldsymbol{b}_h^{(b)}), \end{aligned}\end{split}
Here, the weight parameters $$\boldsymbol{W}_{xh}^{(f)} \in \mathbb{R}^{d \times h}, \boldsymbol{W}_{hh}^{(f)} \in \mathbb{R}^{h \times h}, \boldsymbol{W}_{xh}^{(b)} \in \mathbb{R}^{d \times h}, and \boldsymbol{W}_{hh}^{(b)} \in \mathbb{R}^{h \times h}$$ and bias parameters $$\boldsymbol{b}_h^{(f)} \in \mathbb{R}^{1 \times h} and \boldsymbol{b}_h^{(b)} \in \mathbb{R}^{1 \times h}$$ are all model parameters.
Then we concatenate the forward and backward hidden states $$\overrightarrow{\boldsymbol{H}}_t$$ and $$\overleftarrow{\boldsymbol{H}}_t$$ to obtain the hidden state $$\boldsymbol{H}_t \in \mathbb{R}^{n \times 2h}$$ and input it to the output layer. The output layer computes the output $$\boldsymbol{O}_t \in \mathbb{R}^{n \times q}$$ (number of outputs: $$q$$):
$\boldsymbol{O}_t = \boldsymbol{H}_t \boldsymbol{W}_{hq} + \boldsymbol{b}_q,$
Here, the weight parameter $$\boldsymbol{W}_{hq} \in \mathbb{R}^{2h \times q}$$ and bias parameter $$\boldsymbol{b}_q \in \mathbb{R}^{1 \times q}$$ are the model parameters of the output layer. The two directions can have different numbers of hidden units.
6.11.1. Summary¶
• In bidirectional recurrent neural networks, the hidden state for each time step is simultaneously determined by the subseries before and after this time step (including the input for the current time step).
6.11.2. Problems¶
• If the different directions use a different number of hidden units, how will the shape of $$\boldsymbol{H}_t$$ change?
• Referring to figures 6.11 and 6.12, design a bidirectional recurrent neural network with multiple hidden layers.
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## Encyclopedia > Wikipedia talk:Image use policy
Article Content
# Wikipedia talk:Image use policy
For older talk see:
international What is the current policy (recommendations) for making images usable for other Wikipedias ? Leaving out descriptions, lettering, captions etc that would help immensely other Wikipedias use them.
Is it at all possible to have editable images apart from SVG (which we don't support yet) ?
See Wikipedia:Embassy for in a way successful story of sharing an image between international Wikipedias.
Kpjas 07:30 14 Jun 2003 (UTC)
it's a good idea to avoid text in an image if possible. If text is essential, keep a source version without the text, as someone from another language site might want to make their own version -- see for example :Image:Europeanunion-med.png -- Tarquin 07:55 14 Jun 2003 (UTC)
The value of the width (here 270) should be slightly larger than the actual width of the thumbnail.
This is really just a hack, and not recommended for a number of obvious reasons. padding is better, but margin-lefg is even better because it places the spacing only where it is needed. Which I think looks better. -- Egil 13:43 May 5, 2003 (UTC)
PS: I also have left out the <small> for captions, since it decreases readability. The Wikipedia:Manual of Style recommends straight italics, and at least to my taste that makes the image caption stand out to a sufficient degree. -- Egil
Floating images Moved from Wikipedia:Village pump on Wednesday, June 4th, 02003.
A few weeks ago, somebody discovered a set of HTML code that makes aligning works in most versions of most browser. But I can't find it. --Menchi 12:52 1 Jun 2003 (UTC)
You mean a floating frame? that's <div style="float:right;">{image}</div>. To make it work in all browsers, you can make an aligned table by doing <table align=right><tr><td>{image}</table> LittleDan 18:07 1 Jun 2003 (UTC)
Yes, thanks. --Menchi 04:48 2 Jun 2003 (UTC)
See wikipedia:image use policy for markup suggestions.
Size issues Why so small? I have a diagram that I made that's 362px wide, and there's really no way to shrink it without either rendering the text labels illegible or having them take over the whole image. I don't think it's too big from a page-layout perspective, even for a 640-px-wide screen. And don't even bring up the size issue; it's 4.53 KB. -Smack 02:28 8 Jun 2003 (UTC)
You can have images wider than 300 px in articles - just center them and don't have text flow around them. --mav 05:31 8 Jun 2003 (UTC)
Linking to http://burnallgifs.org/#software (http://burnallgifs.org/#software) for the request to convert .gifs to .pngs may be useful. People need to be warned against using .mngs though.
Are there any examples of two images overlayed with CSS, perhaps a photo covered with labels, as suggested in the policy?
If that's possible, (i don't know much CSS), would it be alright to use a GIF for the overlay, as IE5 doesn't support PNG transparency. Thanks Tristanb 03:26 16 Jun 2003 (UTC)
IE5 supports transparency in indexed PNGs in exactly the same way as in GIFs (which only comes in indexed flavour anyway), so I don't see the problem. branko
I disagree with the recommended thumbnail sizes, 150 or 200 seems unnecessarily small. Even if that's the only available size, a graphics editor can upsize to, say, 300 pixels, with trivial loss of quality. I find 300 pixels the ideal width (with 750 pixels width for any bigger version) for a landscape format pic, and 250 pixels width (with 500 pixels width for any larger version) for a portrait format pic. They allow text to flow down the side of the image even on an 800 by 600 screen. (I do not think we should make our pic sizes suit a 640 by 480 screen). Am I allowed to change the advice accordingly?
Note: the 500 and 750 values are chosen so that the reader does not have scroll his screen much, I find having to scroll a lot to view all of a pic is irritating.
Adrian Pingstone 14:06 16 Jun 2003 (UTC)
I've been going towards 300px in practice too, although I noticed at poppy mallow that a 300px image makes the taxobox too wide, I'll probably replace with a smaller image. I've been doing large images between 600-750px, poorer images don't always do well with the 750px size. Stan 14:18 16 Jun 2003 (UTC)
Additional idea: if you're going to make a smaller one for the taxobox, instead of calling it "Image:Poppy_mallow.small.jpg" or "Image:Poppy_mallow.thmb.jpg" or something like that, make three versions: the full size, a small size in case it's ever wanted in an article body, and a "Image:Poppy_mallow.taxo.jpg" or something like that, sized for the taxobox. I wouldn't want to see the taxobox-sized and full-sized versions being the only options. -- John Owens 23:27 16 Jun 2003 (UTC)
Some people browse the web from mobile devices with limited screen resolutions. Some people are still stuck with 640x480 or 800x600 screens. Some people browse the web with non-maximised windows. The current recommendation for floating images is 150->250... I suppose I could live with increasing that to 200->300, but it might be quite controversial. Wait a bit before updating it.
For non-floating images, I'd recommend 450->600. Larger than 600 will pagewiden on an 800x600 screen, which is bad. Martin 14:42 16 Jun 2003 (UTC)
And some people don't have web browsing capabilities at all! How can we cater to them?
There's going to be a cut-off line somewhere, no matter what.
We cater to people without internet access or web browsing capabilities by licensing our content under the GFDL, so a friend can give them a CD with a TomeRaider download on it, for example, or print off a copy. What, that was a rhetorical question? ;-)
If an image is too large it completely screws up the layout for people with small browser windows. If an image is too small... it's not actually a major problem, especially when we provide a link to a larger version anyway. Hence, I'd rather bias in the direction of too small rather than too big... Martin
I was going to add an unrelated suggestion (v.i.), but even before I came to the Talk: page here, I looked at the Wikipedia: page, saw rule of thumb #4, and thought, "I'm going to add a suggestion that the range be upped to 300." Now I see Adrian beat me to it, but I'll certainly throw in my support. -- John Owens 23:27 16 Jun 2003 (UTC)
In this thumbnail-making integer ratio suggestion, it should clarify whether that means aspect ratio or sizing ratio. I'm pretty sure it means sizing ratio, right? And what's the better term for that? -- John Owens 11:50 17 Jun 2003 (UTC)
Moved from the village pump:
Can some kind soul show me how to put a sequence of four images down the right hand side of the page so that they form one unit and no text can sneak in between them. I'm referring to Sistine Chapel that I illustrated. I reckon the page would look nicer if I had all four pics together and there would be no problems with pics overlapping in different browsers. Thanks.
Adrian Pingstone 09:01 14 Jun 2003 (UTC)
LittleDan told me something like this a few weeks ago:
<table align=right>
<tr><td>
<div style="float:right;">
[[image:___Name___ | __Description_]] <br>
<small>''More detailed explanation<br>
[[media:___Name___ | Larger version]]''</small>
<br><br>
[[image:___Name2___ | __Description_]] <br>
<small>''More detailed explanation<br>
[[media:___Name2___ | Larger version]]''</small>
<br><br>
[[image:___Name3___ | __Description_]] <br>
<small>''More detailed explanation<br>
[[media:___Name3___ | Larger version]]''</small>
<br><br>
[[image:___Name4___ | __Description_]] <br>
<small>''More detailed explanation<br>
[[media:___Name4___ | Larger version]]''</small>
</div>
</td></tr>
</table>
--Menchi 09:08 14 Jun 2003 (UTC)
It's easier than that, no need for tables. I've done the first two -- Tarquin
Thanks, Menchi for your info. I'm sorry you had to do so much typing but I'll use Tarquins method (Tarquin, thanks).
Adrian Pingstone 09:46 14 Jun 2003 (UTC)
Not much typing, just copy-and-paste. If Tarquin's method is simpler, the better. :-) --Menchi 09:57 14 Jun 2003 (UTC)
OK, the code is now in place in Sistine Chapel to put four (or whatever) pics in a column, which are then effectively one pic. Thanks to Menchi and Tarquin for their replies.
Adrian Pingstone 11:22 22 Jun 2003 (UTC)
Thanks to wapcaplet for the section on markup.
In the interests of scalability and uniformity, it might be better to always use the version of the <div> tag with the text-align attribute. -Smack 17:09 20 Jun 2003 (UTC)
I suggest a system to use the same images from a language wikipedia (i.e. wikipedia in french, in another languages, i.e. in spanish wikipedia), easily.
Why does my uploaded image not show up on my watchlist? I really would like to know if someone is modifying something regarding my image. Is there a reason for it that it is not automatically on the Watchlist? Thanks, Fantasy 09:56 28 Jun 2003 (UTC)
Are you referring to the image or the image description page? Martin
I think the "image description page".Fantasy
Strange - I thought it did. :-( Well, you can go and fix that yourself (Wikipedia software is open source) or beg a developer on wikipedia:village pump, I guess. It's definately a good idea :) Martin 18:09 28 Jun 2003 (UTC)
A "watch this image" box for each upload would be very nice to have. I've uploaded hundreds of my own images and have yet to put them all on my watch list. --mav
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