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# Goals for today's section¶
• Go over general problem for w5 homework -- the underlying models of data we are considering, and the big picture behind the calculation of evidences for the models.
## What's the general problem?¶
For ten different loci, we want to know if the genotype at that locus is linked to the phenotype.
The phenotype for fly $k$ is $f_k$, a song frequency measured in Hz.
At a particular locus, we also have the genotype of the $k$th fly, $g_k$.
Our null model is that the genotype does not explain the phenotype:
$$f_k = c + \epsilon, ~~\epsilon \sim N(0, \sigma)$$
Here, $c$ is some constant song frequency, and $\varepsilon$ is a Gaussian noise variable with standard deviation $\sigma$.
Our QTL model:
$$f_k = a + b g_k + \epsilon, ~~\epsilon \sim N(0, \sigma)$$
What does this mean?
• if fly $k$ has $g_k = 0$, then its phenotype has mean frequency $a$ with Gaussian noise with standard deviation $\sigma$ (a $N(a, \sigma)$ process),
• whereas if $g_k = 1$, then it's $N(a+b, \sigma)$
Let's visualize this for different $a, b, c$:
Given our data, we want to compare two hypotheses, so let's compute our old friend:
\begin{aligned} \frac{P(QTL \mid D)}{P(NQTL \mid D)} &= \frac{P(D \mid QTL)}{P(D \mid NQTL)} \frac{P(QTL)}{P(NQTL)} \\ &= \frac{P(D \mid QTL)}{P(D \mid NQTL)} ~\mathrm{(assuming~models~equally~ likely)} \\ \end{aligned}
Let's go through $P(D \mid NQTL)$.
In the NQTL model, what we assume is that phenotypes are governed by
$$f_k = c + \epsilon, ~~\epsilon \sim N(0, \sigma)$$
So, a frequency $f_k$ is assumed to come from a $N(c, \sigma)$ distribution.
The pdf of a Gaussian $N(\alpha, \beta)$ is:
$$f(x; \alpha, \beta) = \frac{1}{\sqrt{2\pi}\beta}e^{-\frac{(x-\alpha)^2}{2\beta^2}}$$
That's exactly what we use to write the likelihood of a single datapoint:
$$P(f_k \mid c, \sigma, NQTL) = \frac{1}{\sqrt{2\pi}\sigma}e^{\frac{-(f_k - c)^2}{2\sigma^2}}$$
And for the full data, with $N$ flies: \begin{aligned} P(D \mid c, \sigma, NQTL) &= \prod_{k=1}^N P(f_k \mid c, \sigma, NQTL) \\ &= \prod_{k=1}^N \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(f_k - c)^2}{2\sigma^2}} \\ &= \frac{1}{(\sqrt{2\pi}\sigma)^N}\prod_{k=1}^N e^{-\frac{(f_k - c)^2}{2\sigma^2}} \\ &= \frac{1}{(\sqrt{2\pi}\sigma)^N} e^{\frac{-\sum_k (f_k - c)^2}{2\sigma^2}} \\ \end{aligned}
The $f_k$'s are fixed and given to us. We can compute the term above if we set a particular $c$ and $\sigma$. But, what $c$ should we use? We will marginalize over different $c$. For our homework, it is OK to assume a single $\sigma$ (though in principle, you could leave this as a parameter and marginalize over different $\sigma$'s too).
$$P(D \mid NQTL) = \int_{-\infty}^\infty dc~ P(D \mid c, \sigma, NQTL)P(c \mid NQTL)$$
Similar to previous homeworks, we will consider uniform priors on parameters like $c$ and call $\sigma_c = c_+ -c_-$, where $c_+$ and $c_-$ are endpoints for $c$ that are arbitrary but don't cut off meaningful values.
$$P(D \mid NQTL) = \frac{1}{\sigma_c}\int_{-\infty}^\infty dc~ P(D \mid c, \sigma, NQTL)$$
We need to compute the above integral in order to get the evidence for the NQTL model. What follows in the lecture notes is a simplification of that integral by doing a Taylor expansion around the peak, very similar to how we did the Laplace approximation of using Taylor expansions around a peak of a posterior to find an approximating Gaussian.
Here, we consider the log likelihood:
\begin{aligned} L(c) &= \log P(D \mid c, \sigma, NQTL) \\ &\approx L(c^*) + \frac{1}{2} \frac{\partial^2 L}{\partial c^2} (c - c^*)^2 \end{aligned}
We are expanding around an optimal $c^*$, for which the first partial derivative of $L$ in $c$ is zero. Hence, there are no first partial derivatives in the above Taylor expansion. We got this $c^*$ by the following:
$$\frac{\partial L}{\partial c} = \frac{1}{\sigma^2} \sum_k (f_k - c) = 0 \implies c^* = \frac{1}{N} \sum_k f_k = \bar{f}$$
There is another key result from lecture notes, $\frac{\partial^2 L}{\partial c^2} = -\frac{N}{\sigma^2}$, which we will now plug in to the above expression:
\begin{aligned} L(c) &= \log P(D \mid c, \sigma, NQTL) \\ &\approx L(c^*) + \frac{1}{2} \frac{\partial^2 L}{\partial c^2} (c - c^*)^2\\ &\approx L(c^*) - \frac{N}{2\sigma^2} (c - c^*)^2 \end{aligned}
Going back to non-logs:
\begin{aligned} P(D \mid c, \sigma, NQTL) &= \exp\big(\log P(D \mid c, \sigma, NQTL)\big)\\ &\approx \exp\big( L(c^*) - \frac{N}{2\sigma^2} (c - c^*)^2 \big)\\ &\approx e^{L(c^*)}e^{-\frac{N}{2\sigma^2} (c - c^*)^2}\\ &\approx P(D \mid c^*, \sigma, NQTL)e^{-\frac{N}{2\sigma^2} (c - c^*)^2}\\ \end{aligned}
Plug this into our integral:
\begin{aligned} P(D \mid NQTL) &= \frac{1}{\sigma_c}\int_{-\infty}^\infty dc~ P(D \mid c, \sigma, NQTL) \\ &\approx \frac{1}{\sigma_c}\int_{-\infty}^\infty dc~ P(D \mid c^*, \sigma, NQTL)e^{-\frac{N}{2\sigma^2} (c - c^*)^2}\\ &\approx \frac{1}{\sigma_c} P(D \mid c^*, \sigma, NQTL) \int_{-\infty}^\infty dc~ e^{-\frac{N}{2\sigma^2} (c - c^*)^2}\\ \end{aligned}
That integral is simple to integrate (it is a form of a Gaussian integral), so we arrive at a simple expression:
$$P(D\mid NQTL) \approx P(D\mid c^\ast, \sigma, NQTL) \times \sqrt{\frac{2\pi\sigma^2}{N}}\frac{1}{\sigma_c}$$
Here, $P(D \mid c^*, \sigma, NQTL)$ is simply the likelihood of the data, given a particular $c=c^* = \frac{1}{N} \sum_k f_k$ (which makes sense, right? In the null model, we assume that genotype has no effect on phenotype, and that each observed phenotype is generated by a gaussian with mean frequency $c$... so if we wanted to estimate $c$, the best we can do is take the mean of the frequencies $f$).
Let's sum up. So what did we do here? We
• assumed that we had a known $\sigma$ (something you can set by looking at the data -- what would be a reasonable choice? You can set this as a variable in your code and see how your results change for different $\sigma$'s if you're interested),
• used a Taylor expansion around the optimal $c$ to approximate $P(D \mid c, \sigma, NQTL)$
• doing that approximation gave us a new integral with an easy analytical answer, giving us a one-line calculation for the integral we wanted to compute in the first place (we don't need to do integration ourselves! just need to evaluate the above expression)
For the QTL model, the lecture notes go through a similar process. But there, we have two parameters, $a$ and $b$, and we incorporate genotype information, $g$.
We will do the same tricks though!
\begin{aligned} P(D\mid QTL) &= \int_{-\infty}^{\infty} da \int_{-\infty}^{\infty} db ~P(D\mid a, b, \sigma, QTL) P(a,b\mid QTL) \\ &= \frac{1}{\sigma_a\sigma_b}\int_{-\infty}^{\infty} da \int_{-\infty}^{\infty} db ~P(D\mid a, b, \sigma, QTL) \\ &\approx \frac{1}{\sigma_a\sigma_b} P(D\mid a^*, b^*, \sigma, QTL)\int_{-\infty}^{\infty} da \int_{-\infty}^{\infty} db ~e^{-\frac{1}{2}(a-a^*, b-b^*)Q{a-a^\ast\choose b-b^\ast}} \\ &\approx \frac{1}{\sigma_a\sigma_b} P(D\mid a^*, b^*, \sigma, QTL)\frac{2\pi}{\sqrt{\det Q}} \\ \end{aligned}
Above, I introduce quantities like the matrix $Q$, and the optimal $a^*, b^*$ that are defined in lecture notes. The key point is that the we again arrived at a closed form solution of the integral using terms that can be quickly computed using the genotypes $g$ and frequencies $f$ for the data that we're interested in.
Finally, we arrive at our ratio of probabilities:
$$\frac{P(QTL\mid D)}{P(NQTL\mid D)} = \frac{P(D\mid a^\ast, b^\ast, \sigma, QTL)}{P(D\mid c^\ast, \sigma, NQTL)} \times \sqrt{\frac{2\pi/N}{\overline{gg} - \bar{g} \bar{g}}} \times \sigma \times \frac{\sigma_c}{\sigma_a\sigma_b}$$
So really this is the term you need to compute for your homework, after all the beautiful simplification shown in full in the lecture notes. Terms like $\overline{gg}$ are defined there. If this term is much greater than 1, then the QTL model is more plausible. If it's close to 0, NQTL is more plausible.
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| English | 简体中文 |
# 1189. Maximum Number of Balloons
## Description
Given a string text, you want to use the characters of text to form as many instances of the word "balloon" as possible.
You can use each character in text at most once. Return the maximum number of instances that can be formed.
Example 1:
Input: text = "nlaebolko"
Output: 1
Example 2:
Input: text = "loonbalxballpoon"
Output: 2
Example 3:
Input: text = "leetcode"
Output: 0
Constraints:
• 1 <= text.length <= 104
• text consists of lower case English letters only.
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# Replace the output of a module containing local variables by their actual value
I want to construct a function as the output of a Module, but I obtain the local variables as part of the output. This is a problem when I try to construct many of these functions in a parallel table, because all these local variables appear to die when the parallel computations end.
Minimal example:
If I write
Module[{i = RandomInteger[{0, 100}]},
Function[x, x + i]
]
I obtain
Function[x$$, x$$ + i$197286] Now the problem: If I write t = ParallelTable[ Module[{i = RandomInteger[{0, 100}]}, Function[x, x + i] ], 4 ]; Through[t[5]] I obtain {5 + i$$19726, 5 + i$$21792, 5 + i$$20597, 5 + i$$19520} and the variables like i$19724 have no value.
If I use Table instead of ParallelTable I get what I want
{60, 13, 62, 13}
I found a workaround by doing
Module[{i = RandomInteger[{0, 100}]},
With[{ei = i}, Function[x, x + ei]]
]
obtaining
Function[x$$, x$$ + 15]
which is EXACTLY what I want. I prefer to have concrete values inside the functions, even if I don't use parallelization.
But I have to introduce a new variable. And this is a problem if there are many of these constants inside the function. Is there a way to solve this problem without using a new variable?
• Function has the attribute "HoldAll". That is why i is not replaced by its value. You can force the replacement by writing: With[{i = i}, Function[x, x + i]] instead of: Function[x, x + i] – Daniel Huber Dec 19 '20 at 19:29
• Is that "good practice" ? the syntax coloring gets all red. – julian haddad Dec 21 '20 at 2:44
• @julianhaddad I think it's fairly standard. If you don't like the coloring, you can always rename the inner variable to something else, e.g. With[{j=i},Function[x,x+j]] – Lukas Lang Dec 24 '20 at 9:18
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# Time series forecasting with hour data, prediction for next 24 hours
i'm a newbie in Time Series Analysis. I have a 2 year pandas dataframe about water consumptions in hour granularity (24 records for day, 365 days).
Water_consumptions
Data
2017-01-01 00:00:00 315.546173
2017-01-01 01:00:00 322.469203
2017-01-01 02:00:00 305.497974
2017-01-01 03:00:00 291.905637
2017-01-01 04:00:00 268.990071
2017-01-01 05:00:00 267.545479
...
I would like to predict day water consumptions (the next 24 records) based on this two years. Which kind of model is most accurate for this task?
I've read about Sarimax and Recurrent Neural Network (LSTM) as a possibility. Are there other possibilities?
My series has also trending and seasonal component. Have my series to be stationarized? Why? in which way i have to use trending, seasonal and residuals in my model after my series was stationarized? i think i can't remove annual seasonality with 2 only years of data:
Plotting before stationarity:
Plotting after stationarity, except for annual seasonality:
Thanks
You very probably have : intra-daily, intra-weekly and intra-yearly. Therefore, my first choice would be models that explicitly address these. Examples of such models are and . Both are available in the forecast package for R (beware: they take a long time to fit), but I am not aware of any implementations for Python.
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## 6C. 19 part C
Acidity $K_{a}$
Basicity $K_{b}$
The Conjugate Seesaw $K_{a}\times K_{b}=K_{w}$
Mansi_1D
Posts: 50
Joined: Fri Aug 02, 2019 12:15 am
### 6C. 19 part C
For number 19 part c in Focus 6C, it asks which of the two pairs, HBrO2 and HClO2 is a stronger acid. The corrected answer is HClO2 and I wanted to why HClO2 is stronger because Br and Cl are in the same group so wouldn't you look at the bond strength/ atom size to determine acid strength? Br is larger than Cl so wouldn't it be easier to remove a proton from HBrO2 than HClO2?
Brian Tangsombatvisit 1C
Posts: 119
Joined: Sat Aug 17, 2019 12:15 am
### Re: 6C. 19 part C
In both HBrO2 and HClO2, the H+ atom is bonded to an O, not Cl and Br. If the H+ were bonded to the Cl or Br, then you would be correct in that the H-Br is a weaker bond than H-Cl, making it a stronger acid. However, this is not the case. To determine the acid that is stronger, we would have to look instead at the resulting stability of the anion after the acid donates its proton since both acids have the same O-H bond. HClO- would be a more stable anion than HBrO- since Cl is more electronegative, pulling more of the electron density away from the negatively charged O once it donates the proton. Because of this, HClO2 is the stronger acid.
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## 6.2 Cost-Volume-Profit Analysis for Multiple-Product and Service Companies
### Learning Objective
1. Perform cost-volume-profit analysis for multiple-product and service companies.
Question: Although the previous section illustrated cost-volume-profit (CVP) analysis for companies with a single product easily measured in units, most companies have more than one product or perhaps offer services not easily measured in units. Suppose you are the manager of a company called Kayaks-For-Fun that produces two kayak models, River and Sea. What information is needed to calculate the break-even point for this company?
Answer: The following information is required to find the break-even point:
### Review Problem 6.2
International Printer Machines (IPM) builds three computer printer models: Inkjet, Laser, and Color Laser. Information for these three products is as follows:
Inkjet Laser Color Laser Total Selling price per unit $250$400 $1,600 Variable cost per unit$100 $150$ 800 Expected unit sales (annual) 12,000 6,000 2,000 20,000 Sales mix 60 percent 30 percent 10 percent 100 percent
Total annual fixed costs are $5,000,000. Assume the sales mix remains the same at all levels of sales. 1. 1. How many printers in total must be sold to break even? 2. How many units of each printer must be sold to break even? 2. 1. How many printers in total must be sold to earn an annual profit of$1,000,000?
2. How many units of each printer must be sold to earn an annual profit of $1,000,000? Solution to Review Problem 6.2 Note: All solutions are rounded. 1. 1. IPM must sell 20,408 printers to break even: 2. As calculated previously, 20,408 printers must be sold to break even. Using the sales mix provided, the following number of units of each printer must be sold to break even: 2. 1. IPM must sell 24,490 printers to earn$1,000,000 in profit:
2. As calculated previously, 24,490 printers must be sold to earn $1,000,000 in profit. Using the sales mix provided, the following number of units for each printer must be sold to earn$1,000,000 in profit:
## Finding the Break-Even Point and Target Profit in Sales Dollars for Multiple-Product and Service Companies
A restaurant like Applebee’s, which serves chicken, steak, seafood, appetizers, and beverages, would find it difficult to measure a “unit” of product. Such companies need a different approach to finding the break-even point. Figure 6.4 "Type of Good or Service Determines Whether to Calculate Break-Even Point and Target Profit Points in Units or Sales Dollars" illustrates this point by contrasting a company that has similar products easily measured in units (kayaks) with a company that has unique products (meals at a restaurant) not easily measured in units.
## Break-Even Point in Sales Dollars and the Weighted Average Contribution Margin Ratio
Question: For companies that have unique products not easily measured in units, how do we find the break-even point?
Answer: Rather than measuring the break-even point in units, a more practical approach for these types of companies is to find the break-even point in sales dollars. We can use the formula that follows to find the break-even point in sales dollars for organizations with multiple products or services. Note that this formula is similar to the one used to find the break-even point in sales dollars for an organization with one product, except that the contribution margin ratio now becomes the weighted average contribution margin ratio.
### Key Equation
For example, suppose Amy’s Accounting Service has three departments—tax, audit, and consulting—that provide services to the company’s clients. Figure 6.5 "Income Statement for Amy’s Accounting Service" shows the company’s income statement for the year. Amy, the owner, would like to know what sales are required to break even. Note that fixed costs are known in total, but Amy does not allocate fixed costs to each department.
Figure 6.5 Income Statement for Amy’s Accounting Service
The contribution margin ratio differs for each department:
Tax 70 percent (= $70,000 ÷$100,000) Audit 20 percent (= $30,000 ÷$150,000) Consulting 50 percent (= $125,000 ÷$250,000)
Question: We have the contribution margin ratio for each department, but we need it for the company as a whole. How do we find the contribution margin ratio for all of the departments in the company combined?
Answer: The contribution margin ratio for the company as a whole is the weighted average contribution margin ratioThe total contribution margin divided by total sales.. We calculate it by dividing the total contribution margin by total sales. For Amy’s Accounting Service, the weighted average contribution margin ratio is 45 percent (= $225,000 ÷$500,000). For every dollar increase in sales, the company will generate an additional 45 cents ($0.45) in profit. This assumes that the sales mix remains the same at all levels of sales. (The sales mix here is measured in sales dollars for each department as a proportion of total sales dollars.) Now that you know the weighted average contribution margin ratio for Amy’s Accounting Service, it is possible to calculate the break-even point in sales dollars: Amy’s Accounting Service must achieve$266,667 in sales to break even.The weighted average contribution margin ratio can also be found by multiplying each department’s contribution margin ratio by its proportion of total sales. The resulting weighted average contribution margin ratios for all departments are then added. The calculation for Amy’s Accounting Service is as follows:45 percent weighted average contribution margin ratio = (tax has 20 percent of total sales × 70 percent contribution margin ratio) + (audit has 30 percent of total sales × 20 percent contribution margin ratio) + (consulting has 50 percent of total sales × 50 percent contribution margin ratio)Thus 45 percent = 14 percent + 6 percent + 25 percent.
## Target Profit in Sales Dollars
Question: How do we find the target profit in sales dollars for companies with products not easily measured in units?
Answer: Finding the target profit in sales dollars for a company with multiple products or services is similar to finding the break-even point in sales dollars except that profit is no longer set to zero. Instead, profit is set to the target profit the company would like to achieve.
### Key Equation
For example, assume Amy’s Accounting Service would like to know sales dollars required to make $250,000 in annual profit. Simply set the target profit to$250,000 and run the calculation:
Amy’s Accounting Service must achieve $822,222 in sales to earn$250,000 in profit.
## Important Assumptions
Question: Several assumptions are required to perform break-even and target profit calculations for companies with multiple products or services. What are these important assumptions?
Answer: These assumptions are as follows:
• Costs can be separated into fixed and variable components.
• Contribution margin ratio remains constant for each product, segment, or department.
• Sales mix remains constant with changes in total sales.
These assumptions simplify the CVP model and enable accountants to perform CVP analysis quickly and easily. However, these assumptions may not be realistic, particularly if significant changes are made to the organization’s operations. When performing CVP analysis, it is important to consider the accuracy of these simplifying assumptions. It is always possible to design a more accurate and complex CVP model. But the benefits of obtaining more accurate data from a complex CVP model must outweigh the costs of developing such a model.
## Margin of Safety
Question: Managers often like to know how close expected sales are to the break-even point. As defined earlier, the excess of projected sales over the break-even point is called the margin of safety. How is the margin of safety calculated for multiple-product and service organizations?
Answer: Let’s return to Amy’s Accounting Service and assume that Amy expects annual sales of $822,222, which results in expected profit of$250,000. Given a break-even point of $266,667, the margin of safety in sales dollars is calculated as follows: Thus sales revenue can drop by$555,555 per year before the company begins to incur a loss.
### Key Takeaways
• The key formula used to calculate the break-even or target profit point in units for a company with multiple products is as follows. Simply set the target profit to $0 for break-even calculations, or to the appropriate profit dollar amount for target profit calculations. • The formula used to find the break-even point or target profit in sales dollars for companies with multiple products or service is as follows. Simply set the “Target Profit” to$0 for break-even calculations, or to the appropriate profit dollar amount for target profit calculations:
### Review Problem 6.3
Ott Landscape Incorporated provides landscape maintenance services for three types of clients: commercial, residential, and sports fields. Financial projections for this coming year for the three segments are as follows:
Assume the sales mix remains the same at all levels of sales.
1. How much must Ott Landscape have in total sales dollars to break even?
2. How much must Ott Landscape have in total sales dollars to earn an annual profit of $1,500,000? 3. What is the margin of safety, assuming projected sales are$5,000,000 as shown previously?
Solution to Review Problem 6.3
1. Sales of $1,000,000 are required to break even: *Weighted average contribution margin ratio =$1,000,000 ÷ $5,000,000 = 20 percent or 0.20. 2. Sales of$8,500,000 are required to make a profit of $1,500,000: 3. The margin of safety is$4,000,000 in sales:
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The information given below is only for those who are still able to use AscoGraph on simple Antescofo score. Ascograph is no longer supported.
## AscoGraph is deprecated¶
AscoGraph is discontinued since 2016. It was based on graphic libraries that has not survived the evolution of compilers and software.
The available version is the last version compiled in 2016. Even if the binary is sometimes still executable (depending on the machine architecture and the OS version), the syntax recognized by AscoGraph is restricted to Antescofo 0.3. This means that scores that make use of new features fail to be loaded by Ascograph. In any case, its use is deprecated.
There are some tools that can be used to achieve the functionalities of Ascograph.
### Edition¶
To translate a MusicXML score or a midi file into a list of musical events in the Antescofo format, the translation MusicXML/Midi $\rightarrow$ antescofo available as a web service is working.
For editing an Antescofo score, you can use Sublime with a dedicated theme. The theme enables the highlighting of Antescofo score but also specific command with a keystroke.
For example option-P send the selected area for evaluation to a running instance of Antescofo. It is very useful for “live coding” or for the work in studio. See :
Check also useful tutorials in our Forum.
### Edition and Automatic Filewatch in the Max Environment¶
Max users can also use the Antescofo Max Object Autowatch attribute. This means that if the loaded score is modified elsewhere, it'll be automatically reloaded in the Max object. Obviously, you'd want to turn this attribute off during live concerts! Autowatch is currently not available for PureData objects.
### Monitoring¶
To control the score following during a performance, a possibility is to export the score to a bach.roll object in the bach computer-aided composition library using the @bach_score function in the Antescofo library.
During the performance, the position produced by the listener can be used to animate thae cursor in the bach.roll. See a patch produced by Serge Lemouton and also this post: Ascograph for Windows is ready?
## Using AscoGraph¶
Ascograph is a graphical tool that can be used to edit and then to control a running instance of Antescofo through OSC messages.
Ascograph and Antescofo are two independent applications but the coupling between Ascograph and an Antescofo instance running in MAX appears transparent for the user: a double-click on the Antescofo object launches Ascograph, saving a file under Ascograph will reload the file under the Antescofo object, loading a file under the Antescofo object will open it under Ascograph, etc.
Ascograph was available in the same bundle as Antescofo on the IRCAM Forum. It was far from being stable but has proven useful for authoring/visualizing complex scores and for monitoring your live performances.
### The Ascograph interface¶
As you can see below (open the screenshoot in another tab to have a larger view), the Ascograph interface contains 3 parts :
• the piano roll, where you can see the musician's score and the label at the top
• the action view, where you can see the graphical representation of the electronic score and where you can manually edit the curves.
• the text editor, where you can edit the musican's score and the electronic score.
At the top of the ascograph window, there are five buttons that correspond respectively to the Antescofo functions previous event, stop, play, start and next event that you can launch directly in your patch.
The play button simply sequences from the beginning of the score to the end, using given event timing and internal tempi, and undertaking actions wherever available.
The Play string function (Menu/Transport/Play string) only plays the selected part of your score. This enables to test on the fly Antescofo fragment (only actions are performed, musical events are ignored) and even allows a limited form of live-coding.
The text Editor menu contains all the functions relevant to the text editor. Included is the display mode selector, where you can toggle between integrated, floating (default) and hidden. The View menu retains the functions pertaining to the visual editor.
### Color Scheme¶
AscoGraph comes with a new color scheme for both the text and visual editor. By default, we provide dark backgrounds for both. The philosophy behind this design choice is the fact that many of us use AscoGraph during live concerts and bright backgrounds create too much light on your screens which make things annoying for you and your audience who are most of the time in the dark.
Many users however prefer working on white background while typing. You can easily Import a new Color Scheme for AscoGraph's Text Editor from the Text editor Menu. You can design your own color scheme by doing an Export Color Scheme from the Text Editor menu, modify the XML file and import it back. Users interested in classical white background text editor can import Larry Nelson's Color Scheme in particular.
### Interaction Between Visual and Text Editors in AscoGraph¶
Clicking on a Musical Event or Action in the Visual Editor will bring the text editor to the corresponding text. Conversely, Right/Ctrl-click on an event in the text editor takes you to the corresponding place in the visual editor.
Vertical mouse-wheel (or double-finger on pad) gesture allows you to browse over the text-editor, as well as horizontal mouse-wheel (double-finger) gesture on the visual editor.
### Shortcuts¶
Holding CMD + mouse scroll: zoom in/out (visual & text editors) CMD + <number> switches the text editor between the open files (main .asco score and any @INSERT-ed files). You can also switch using the tabs at the top of the editor.
Ascograph is also very useful to visualize and edit the curves constructs.
You can Edit Antescofo Curves easily from the Visual Editor. Applying graphical changes will automatically create the corresponding text into the right place. See Nadir B's very userful YouTube Tutorials.
To start, you can create a curve after a note with the menu "create/actions/curve". After saving the score, if you see the action view, below the piano roll, you can see your curve.
If you click on the arrow at the right of the curve band, the curve is moved on the piano roll, to see the superpositions and you can add or move the curve's points. You can also choose the type of the interpolation between each points.
### AscoGraph Automatic Filewatch: Using Another Text Editor¶
Starting with AscoGraph version 0.25, there is an Automatic Filewatch integrated into the AscoGraph editor. This means that you can now use any of your favourite text editors in addition to AscoGraph. The moment you save the Antescofo Score outside, it'll be automatically reloaded (and visualized) in the AscoGraph window without any intervention (and the Antescofo object will reload the score automatically in the Max environment).
So you can use in parallel the Piano Roll and action view and still use your favorite text editor.
The addition of the Automatic Filewatch feature just makes score editing with AscoGraph more coherent with other editors as they all integrate automatic filewatch as well.
Max users can also use the Antescofo Max Object Autowatch attribute. This means that if the loaded score is modified elsewhere, it'll be automatically reloaded in the Max object. Obviously, you'd want to turn this attribute off during live concerts! Autowatch is currently not available for PureData objects but upcoming.
For the people that prefer use their own beloved text editor, we have created some syntax highlightings for Sublime Text, TextWrangler and... emacs ! You have just to download them in our Forum.
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# Should we take a stand on qbit vs. qubit?
I noticed recently that I had edited a posting of an otherwise informed answer solely to replace "q-bit" with "qubit", but then I note that I don't bother to correct the unhyphenated "qbit" to "qubit" when I see it in other similar questions and answers.
Should we have a preferred policy to encourage use of "qubit" in much the same way that we lightly encourage "quantum advantage" or "quantum computational supremacy" over "quantum supremacy"?
Nonetheless there appears to be less than 100 postings that use qbit, while the vast majority, >4k, use qubit, so it might be a don't care in the grand scheme of things. Most of the high-rep users on this site use "qubit" I believe, so I'll go out on a limb and say that "qubit" is preferred but no one will make a stink about "qbit". Maybe some monstrosities like "Q bit" or "quibit" would trigger an edit.
• "qbit" is definitely a monstruosity on par with "q-bit" and such in my book. I edit it anytime I see it, nor I think I'd be able to force myself not to do it even if I wanted to. I do often wonder where it even comes from though. Is it just people than never saw it spelled and go by the way it's pronounced, or is there actually some place in the internet that uses it? – glS Jan 20 at 22:58
• Apparently in David Mermin's book Quantum Computer Science? See here. David Bacon took issue with Mermin's choice here. We actively and happily and gleefully edit |0> to $\vert 0\rangle$ - that of course is fingernails on the chalkboard - but it happens so often that it's hard to keep up. – Mark S Jan 20 at 23:38
• ah, I didn't know that, I guess that explains why it crops out every so often then. Still, I think it's quite uncontroversial to say that "qubit" is the widespread spelling, used in the overwhelming majority of papers (...right?). Regarding editing posts though... I don't think there's much we can do about it. Most people using these spellings won't read, nor probably care, about we decide here on meta. The only solution that comes to mind is to set up some bot to correct these types of minor spelling mistakes in posts... but I don't think such bots are well accepted in the SE network – glS Jan 21 at 0:02
• Thanks! All of this is probably fun and silly shibboleth talk that doesn't advance QCSE's implied mission to inform and instruct the public on what is known and unknown about the capabilities and limitations of a quantum computer, but it still amazes that "qubit" had not yet been coined by the time of Shor's prime factorization paper. How much easier it is to think about quantum algorithms once "qubit" is internalized! OK Shor's algorithm is still pretty tough for new learners but imagine learning it, or as for Shor, discovering it, without the word! – Mark S Jan 21 at 0:11
• agree this is probably not so significant in itself but maybe theres something bigger to pursue here. eg a dictionary of standard QC terminology. it would also be interesting to analyze the origins and nuances of slightly different terminology. re q-bit, or qbit, think they are not terrible. – vzn Jan 21 at 21:51
• @glS I think the more natural question is why most people with certain language background (mostly western languages, I guess) feel uncomfortable when they see a "q" without a "u" right after it. (Includes me!) – Norbert Schuch Jan 23 at 15:40
• @Mark "QCSE's implied mission to inform and instruct the public on what is known and unknown about the capabilities and limitations of a quantum computer" -- Is that the mission of qc.se? – Norbert Schuch Jan 23 at 15:42
• @NorbertSchuch mh, that might indeed be a big reason why many find the "qbit" spelling "ugly". Unsurprisingly, it seems to be due to the latin/greek influences in many (most?) European languages. – glS Jan 23 at 17:49
• @Mark Well, for sure there are SE sites which whose target audience are experts of various levels rather than the interested public. Defining the scope if up to the members of qc.se. There have been meta discussions a while ago, and there were diverging opinions IIRC, but given how young this site is, they are potentially outdated. – Norbert Schuch Jan 23 at 18:51
• in some discussions of quantum communication that "qbit" shows up as the simplest construction that follows the same pattern as "cbit", "ebit" and "cobit" and is actually a very natural choice in that context – forky40 Jan 26 at 1:51
• I am mostly used to 'qubit', but I don't think we should force everyone to spell it a specific way. The main reasons are (1) as Mark S mentioned, there are legit references that use 'qbit' (2) as mentioned by NorbertSchuch and gIS, there may be cultural reasons why some prefer 'qubit' to 'qbit' or vice-versa, (3) as forky mentioned, there may be legitimate reasons that 'qbit' follows naturally and (4) it's definitely nice to have words for quantum qubits as mentioned by Mark S but nobody is really confusing the term 'qbit' with anything else, so there's no upside in policing the word. – Rajiv Krishnakumar Mar 8 at 9:25
Just for the sake of having it written as an answer, I'll sum up what was said in the comments of the question.
I think it's safe to say that most people here strongly prefer the "qubit" spelling over any other. This is also the spelling adopted in the overwhelming majority of scientific writing on the subject, as far as I can tell, albeit with some notable exceptions such as David Mermin’s book Quantum Computer Science. This was also discussed in some blog posts, e.g. here and here.
I've personally always edited the other spellings any time I've noticed them, and I'd encourage other people to do the same. It is indeed often quite an annoying janitorial task, but we can hardly do anything about it, as most people using alternative spellings are unlikely to read meta posts anyway. Individual repeated offenders can be asked to use the correct spelling of course, but there's always going to be new people coming in writing "qbit" etc.
I think "qubit" should be used instead of "qbit", "q-bit", or "Q bit" always, unless one of the other spellings becomes more popular overall (in the whole world, not just here on Stack Exchange, and I don't think one of those spellings will become more popular than "qubit" anyway).
• If you see "qbit" on a new post, feel free to edit it if you have enough reputation to make edits without having to bother reviewers with such a "trifle" matter.
• If you see "qbit" on an old post, feel free to edit it on an "isolated basis" if you have enough reputation to make edits without having to bother reviewers with such a "trifle" matter.
• However, if you're thinking of going through 100 old posts and editing just "qbit" to "qubit", then what you'll be doing is bumping up 100 old posts to the top of the site's list of questions, and this will make the recent questions (often by new users who are just about to get their first experience of whether or not this is a good place to ask a quantum computing question) get buried.
• Buried questions lack visibility and because of that might never get answered.
• If people's first impression of the site is that their question doesn't get answers, they are less likely to come back, and that is not exactly what we want here.
So please feel free to edit the spelling if you have enough reputation to make edits unilaterally, as long as it's on new questions or an isolated old question (not on 100 old questions at the same time, for example).
• The last parenthetical can probably be generalized to avoid block-editing a bunch of questions for other similar issues. Richard Jozsa's name is spelled with the "z" first; however about 20% (20 of 94) of all posts that mention Jozsa or Deutsch-Jozsa spell as Josza with "s" first (many posts by me embarrassingly!) Unlike qbit vs. qubit, however, there's just no controversy about whether we should take a stand on spelling his name correctly (we should). – Mark S yesterday
• @MarkS Interesting. I hadn't noticed there's been so many misspellings of Jozsa! We should take a stand on it, but I think the right way to do it is to catch it when it first happens, and perhaps do 2-3 edits at a time sporadically to correct the rest of them. It seems that it was spelled josza only 20 times and I've just corrected it on the most recent 3 of these posts (two of these were your posts!). Maybe next week we can do another 3, and at 3/week we'll get them all fixed soon enough :) – user1271772 yesterday
• I suppose one way to work around the issue of accidentally burying new questions is to edit affected new questions after the bulk edit is complete. It is generally not a problem to find something about a new question to improve (typos, spelling, capitalization, terminology, mathjax, grammar, punctuation, removing unnecessary text like "Thanks!" etc). Is there a reason this should not be done? – Adam Zalcman yesterday
• @AdamZalcman That's also doable, and a good idea. I think the issue though is that: how do you choose which questions to unbury? There's 17 more posts that use the mis-spelling "Josza", and correcting the spellings would push about 50 questions into lower visibility. Also, even more than 50 questions will be affected in the "active questions" page. I suppose 50 or 50+ questions to unbury, is quite a bit much! – user1271772 yesterday
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H Chapter Contents
H Chapter Introduction
NAG Library Manual
# NAG Library Routine DocumentH05AAF
Note: before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.
## 1 Purpose
Given a set of $m$ features and a scoring mechanism for any subset of those features, H05AAF selects the best $n$ subsets of size $p$ using a reverse communication branch and bound algorithm.
## 2 Specification
SUBROUTINE H05AAF ( IREVCM, MINCR, M, IP, NBEST, DROP, LZ, Z, LA, A, BSCORE, BZ, MINCNT, GAMMA, ACC, ICOMM, LICOMM, RCOMM, LRCOMM, IFAIL)
INTEGER IREVCM, MINCR, M, IP, NBEST, DROP, LZ, Z(M-IP), LA, A(max(NBEST,M)), BZ(M-IP,NBEST), MINCNT, ICOMM(LICOMM), LICOMM, LRCOMM, IFAIL REAL (KIND=nag_wp) BSCORE(max(NBEST,M)), GAMMA, ACC(2), RCOMM(LRCOMM)
## 3 Description
Given $\Omega =\left\{{x}_{i}:i\in ℤ,1\le i\le m\right\}$, a set of $m$ unique features and a scoring mechanism $f\left(S\right)$ defined for all $S\subseteq \Omega$ then H05AAF is designed to find ${S}_{o1}\subseteq \Omega ,\left|{S}_{o1}\right|=p$, an optimal subset of size $p$. Here $\left|{S}_{o1}\right|$ denotes the cardinality of ${S}_{o1}$, the number of elements in the set.
The definition of the optimal subset depends on the properties of the scoring mechanism, if
$fSi ≤ fSj , for all Sj ⊆ Ω and Si ⊆ Sj$ (1)
then the optimal subset is defined as one of the solutions to
$maximize S⊆Ω fS subject to S = p$
else if
$f Si ≥ fSj , for all Sj ⊆ Ω and Si ⊆ Sj$ (2)
then the optimal subset is defined as one of the solutions to
$minimize S ⊆ Ω fS subject to S = p .$
If neither of these properties hold then H05AAF cannot be used.
As well as returning the optimal subset, ${S}_{o1}$, H05AAF can return the best $n$ solutions of size $p$. If ${S}_{o\mathit{i}}$ denotes the $\mathit{i}$th best subset, for $\mathit{i}=1,2,\dots ,n-1$, then the $\left(i+1\right)$th best subset is defined as the solution to either
$maximize S ⊆ Ω - Soj : j∈ℤ , 1≤j≤i fS subject to S = p$
or
$minimize S ⊆ Ω - Soj : j∈ℤ,1≤j≤i fS subject to S = p$
depending on the properties of $f$.
The solutions are found using a branch and bound method, where each node of the tree is a subset of $\Omega$. Assuming that (1) holds then a particular node, defined by subset ${S}_{i}$, can be trimmed from the tree if $f\left({S}_{i}\right)<\stackrel{^}{f}\left({S}_{on}\right)$ where $\stackrel{^}{f}\left({S}_{on}\right)$ is the $n$th highest score we have observed so far for a subset of size $p$, i.e., our current best guess of the score for the $n$th best subset. In addition, because of (1) we can also drop all nodes defined by any subset ${S}_{j}$ where ${S}_{j}\subseteq {S}_{i}$, thus avoiding the need to enumerate the whole tree. Similar short cuts can be taken if (2) holds. A full description of this branch and bound algorithm can be found in Ridout (1988).
Rather than calculate the score at a given node of the tree H05AAF utilizes the fast branch and bound algorithm of Somol et al. (2004), and attempts to estimate the score where possible. For each feature, ${x}_{i}$, two values are stored, a count ${c}_{i}$ and ${\stackrel{^}{\mu }}_{i}$, an estimate of the contribution of that feature. An initial value of zero is used for both ${c}_{i}$ and ${\stackrel{^}{\mu }}_{i}$. At any stage of the algorithm where both $f\left(S\right)$ and $f\left(S-\left\{{x}_{i}\right\}\right)$ have been calculated (as opposed to estimated), the estimated contribution of the feature ${x}_{i}$ is updated to
$ciμ^i + f S - f S - xj ci+1$
and ${c}_{i}$ is incremented by $1$, therefore at each stage ${\stackrel{^}{\mu }}_{i}$ is the mean contribution of ${x}_{i}$ observed so far and ${c}_{i}$ is the number of observations used to calculate that mean.
As long as ${c}_{i}\ge k$, for the user-supplied constant $k$, then rather than calculating $f\left(S-\left\{{x}_{i}\right\}\right)$ this routine estimates it using $\stackrel{^}{f}\left(S-\left\{{x}_{i}\right\}\right)=f\left(S\right)-\gamma {\stackrel{^}{\mu }}_{i}$ or $\stackrel{^}{f}\left(S\right)-\gamma {\stackrel{^}{\mu }}_{i}$ if $f\left(S\right)$ has been estimated, where $\gamma$ is a user-supplied scaling factor. An estimated score is never used to trim a node or returned as the optimal score.
Setting $k=0$ in this routine will cause the algorithm to always calculate the scores, returning to the branch and bound algorithm of Ridout (1988). In most cases it is preferable to use the fast branch and bound algorithm, by setting $k>0$, unless the score function is iterative in nature, i.e., $f\left(S\right)$ must have been calculated before $f\left(S-\left\{{x}_{i}\right\}\right)$ can be calculated.
## 4 References
Narendra P M and Fukunaga K (1977) A branch and bound algorithm for feature subset selection IEEE Transactions on Computers 9 917–922
Ridout M S (1988) Algorithm AS 233: An improved branch and bound algorithm for feature subset selection Journal of the Royal Statistics Society, Series C (Applied Statistics) (Volume 37) 1 139–147
Somol P, Pudil P and Kittler J (2004) Fast branch and bound algorithms for optimal feature selection IEEE Transactions on Pattern Analysis and Machine Intelligence (Volume 26) 7 900–912
## 5 Parameters
Note: this routine uses reverse communication. Its use involves an initial entry, intermediate exits and re-entries, and a final exit, as indicated by the parameter IREVCM. Between intermediate exits and re-entries, all parameters other than BSCORE must remain unchanged.
1: IREVCM – INTEGERInput/Output
On initial entry: must be set to $0$.
On intermediate exit: ${\mathbf{IREVCM}}=1$ and before re-entry the scores associated with LA subsets must be calculated and returned in BSCORE.
The LA subsets are constructed as follows:
${\mathbf{DROP}}=1$
The $j$th subset is constructed by dropping the features specified in the first LZ elements of Z and the single feature given in ${\mathbf{A}}\left(j\right)$ from the full set of features, $\Omega$. The subset will therefore contain ${\mathbf{M}}-{\mathbf{LZ}}-1$ features.
${\mathbf{DROP}}=0$
The $j$th subset is constructed by adding the features specified in the first LZ elements of Z and the single feature specified in ${\mathbf{A}}\left(j\right)$ to the empty set, $\varnothing$. The subset will therefore contain ${\mathbf{LZ}}+1$ features.
In both cases the individual features are referenced by the integers $1$ to M with $1$ indicating the first feature, $2$ the second, etc., for some arbitrary ordering of the features. The same ordering must be used in all calls to H05AAF.
If ${\mathbf{LA}}=0$, the score for a single subset should be returned. This subset is constructed by adding or removing only those features specified in the first LZ elements of Z.
If ${\mathbf{LZ}}=0$, this subset will either be $\Omega$ or $\varnothing$.
The score associated with the $j$th subset must be returned in ${\mathbf{BSCORE}}\left(j\right)$.
On intermediate re-entry: IREVCM must remain unchanged.
On final exit: ${\mathbf{IREVCM}}=0$, and the algorithm has terminated.
Constraint: ${\mathbf{IREVCM}}=0$ or $1$.
2: MINCR – INTEGERInput
On entry: flag indicating whether the scoring function $f$ is increasing or decreasing.
${\mathbf{MINCR}}=1$
$f\left({S}_{i}\right)\le f\left({S}_{j}\right)$, i.e., the subsets with the largest score will be selected.
${\mathbf{MINCR}}=0$
$f\left({S}_{i}\right)\ge f\left({S}_{j}\right)$, i.e., the subsets with the smallest score will be selected.
For all ${S}_{j}\subseteq \Omega$ and ${S}_{i}\subseteq {S}_{j}$.
Constraint: ${\mathbf{MINCR}}=0$ or $1$.
3: M – INTEGERInput
On entry: $m$, the number of features in the full feature set.
Constraint: ${\mathbf{M}}\ge 2$.
4: IP – INTEGERInput
On entry: $p$, the number of features in the subset of interest.
Constraint: $1\le {\mathbf{IP}}\le {\mathbf{M}}$.
5: NBEST – INTEGERInput
On entry: $n$, the maximum number of best subsets required. The actual number of subsets returned is given by LA on final exit. If on final exit ${\mathbf{LA}}\ne {\mathbf{NBEST}}$ then ${\mathbf{IFAIL}}={\mathbf{53}}$ is returned.
Constraint: ${\mathbf{NBEST}}\ge 1$.
6: DROP – INTEGERInput/Output
On initial entry: DROP need not be set.
On intermediate exit: flag indicating whether the intermediate subsets should be constructed by dropping features from the full set (${\mathbf{DROP}}=1$) or adding features to the empty set (${\mathbf{DROP}}=0$). See IREVCM for details.
On intermediate re-entry: DROP must remain unchanged.
On final exit: is undefined.
7: LZ – INTEGERInput/Output
On initial entry: LZ need not be set.
On intermediate exit: the number of features stored in Z.
On intermediate re-entry: LZ must remain unchanged.
On final exit: is undefined.
8: Z(${\mathbf{M}}-{\mathbf{IP}}$) – INTEGER arrayInput/Output
On initial entry: Z need not be set.
On intermediate exit: ${\mathbf{Z}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{LZ}}$, contains the list of features which, along with those specified in A, define the subsets whose score is required. See IREVCM for additional details.
On intermediate re-entry: Z must remain unchanged.
On final exit: is undefined.
9: LA – INTEGERInput/Output
On initial entry: LA need not be set.
On intermediate exit: if ${\mathbf{LA}}>0$, the number of subsets for which a score must be returned.
If ${\mathbf{LA}}=0$, the score for a single subset should be returned. See IREVCM for additional details.
On intermediate re-entry: LA must remain unchanged.
On final exit: the number of best subsets returned.
10: A($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{NBEST}},{\mathbf{M}}\right)$) – INTEGER arrayInput/Output
On initial entry: A need not be set.
On intermediate exit: ${\mathbf{A}}\left(\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{LA}}$, contains the list of features which, along with those specified in Z, define the subsets whose score is required. See IREVCM for additional details.
On intermediate re-entry: A must remain unchanged.
On final exit: is undefined.
11: BSCORE($\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{NBEST}},{\mathbf{M}}\right)$) – REAL (KIND=nag_wp) arrayInput/Output
On initial entry: BSCORE need not be set.
On intermediate exit: is undefined.
On intermediate re-entry: ${\mathbf{BSCORE}}\left(j\right)$ must hold the score for the $j$th subset as described in IREVCM.
On final exit: holds the score for the LA best subsets returned in BZ.
12: BZ(${\mathbf{M}}-{\mathbf{IP}}$,NBEST) – INTEGER arrayInput/Output
On initial entry: BZ need not be set.
On intermediate exit: is used for storage betweeen calls to H05AAF.
On intermediate re-entry: BZ must remain unchanged.
On final exit: the $j$th best subset is constructed by dropping the features specified in ${\mathbf{BZ}}\left(\mathit{i},\mathit{j}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{M}}-{\mathbf{IP}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{LA}}$, from the set of all features, $\Omega$. The score for the $j$th best subset is given in ${\mathbf{BSCORE}}\left(j\right)$.
13: MINCNT – INTEGERInput
On entry: $k$, the minimum number of times the effect of each feature, ${x}_{i}$, must have been observed before $f\left(S-\left\{{x}_{i}\right\}\right)$ is estimated from $f\left(S\right)$ as opposed to being calculated directly.
If $k=0$ then $f\left(S-\left\{{x}_{i}\right\}\right)$ is never estimated. If ${\mathbf{MINCNT}}<0$ then $k$ is set to $1$.
14: GAMMA – REAL (KIND=nag_wp)Input
On entry: $\gamma$, the scaling factor used when estimating scores. If ${\mathbf{GAMMA}}<0$ then $\gamma$ is set to $1$.
15: ACC($2$) – REAL (KIND=nag_wp) arrayInput
On entry: a measure of the accuracy of the scoring function, $f$.
Letting ${a}_{i}={\epsilon }_{1}\left|f\left({S}_{i}\right)\right|+{\epsilon }_{2}$, then when confirming whether the scoring function is strictly increasing or decreasing (as described in MINCR), or when assessing whether a node defined by subset ${S}_{i}$ can be trimmed, then any values in the range $f\left({S}_{i}\right)±{a}_{i}$ are treated as being numerically equivalent.
If $0\le {\mathbf{ACC}}\left(1\right)\le 1$ then ${\epsilon }_{1}={\mathbf{ACC}}\left(1\right)$, otherwise ${\epsilon }_{1}=0$.
If ${\mathbf{ACC}}\left(2\right)\ge 0$ then ${\epsilon }_{2}={\mathbf{ACC}}\left(2\right)$, otherwise ${\epsilon }_{2}=0$.
In most situations setting both ${\epsilon }_{1}$ and ${\epsilon }_{2}$ to zero should be sufficient. Using a nonzero value, when one is not required, can significantly increase the number of subsets that need to be evaluated.
16: ICOMM(LICOMM) – INTEGER arrayCommunication Array
On initial entry: ICOMM need not be set.
On intermediate exit: is used for storage between calls to H05AAF.
On intermediate re-entry: ICOMM must remain unchanged.
On final exit: is not defined. The first two elements, ${\mathbf{ICOMM}}\left(1\right)$ and ${\mathbf{ICOMM}}\left(2\right)$ contain the minimum required value for LICOMM and LRCOMM respectively.
17: LICOMM – INTEGERInput
On entry: the length of the array ICOMM. If LICOMM is too small and ${\mathbf{LICOMM}}\ge 2$ then ${\mathbf{IFAIL}}={\mathbf{172}}$ is returned and the minimum value for LICOMM and LRCOMM are given by ${\mathbf{ICOMM}}\left(1\right)$ and ${\mathbf{ICOMM}}\left(2\right)$ respectively.
Constraints:
• if ${\mathbf{MINCNT}}=0\phantom{\rule{0ex}{0ex}}$, ${\mathbf{LICOMM}}\ge 2×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{NBEST}},{\mathbf{M}}\right)+{\mathbf{M}}\left({\mathbf{M}}+2\right)+\left({\mathbf{M}}+1\right)×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}}-{\mathbf{IP}},1\right)+27$;
• otherwise ${\mathbf{LICOMM}}\ge 2×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{NBEST}},{\mathbf{M}}\right)+{\mathbf{M}}\left({\mathbf{M}}+3\right)+\left(2{\mathbf{M}}+1\right)×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}}-{\mathbf{IP}},1\right)+25$.
18: RCOMM(LRCOMM) – REAL (KIND=nag_wp) arrayCommunication Array
On initial entry: RCOMM need not be set.
On intermediate exit: is used for storage between calls to H05AAF.
On intermediate re-entry: RCOMM must remain unchanged.
On final exit: is not defined.
19: LRCOMM – INTEGERInput
On entry: the length of the array RCOMM. If LRCOMM is too small and ${\mathbf{LICOMM}}\ge 2$ then ${\mathbf{IFAIL}}={\mathbf{172}}$ is returned and the minimum value for LICOMM and LRCOMM are given by ${\mathbf{ICOMM}}\left(1\right)$ and ${\mathbf{ICOMM}}\left(2\right)$ respectively.
Constraints:
• if ${\mathbf{MINCNT}}=0$, ${\mathbf{LRCOMM}}\ge 9+{\mathbf{NBEST}}+{\mathbf{M}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}}-{\mathbf{IP}},1\right)$;
• otherwise ${\mathbf{LRCOMM}}\ge 8+{\mathbf{M}}+{\mathbf{NBEST}}+{\mathbf{M}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{M}}-{\mathbf{IP}},1\right)$.
20: IFAIL – INTEGERInput/Output
On initial entry: IFAIL must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{ or }1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, because for this routine the values of the output parameters may be useful even if ${\mathbf{IFAIL}}\ne {\mathbf{0}}$ on exit, the recommended value is $-1$. When the value $-\mathbf{1}\text{ or }1$ is used it is essential to test the value of IFAIL on exit.
On final exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).
## 6 Error Indicators and Warnings
If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=11$
On entry, ${\mathbf{IREVCM}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{IREVCM}}=0$ or $1$.
${\mathbf{IFAIL}}=21$
On entry, ${\mathbf{MINCR}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{MINCR}}=0$ or $1$.
${\mathbf{IFAIL}}=22$
MINCR has changed between calls.
On intermediate entry, ${\mathbf{MINCR}}=⟨\mathit{\text{value}}⟩$.
On initial entry, ${\mathbf{MINCR}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=31$
On entry, ${\mathbf{M}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{M}}\ge 2$.
${\mathbf{IFAIL}}=32$
M has changed between calls.
On intermediate entry, ${\mathbf{M}}=⟨\mathit{\text{value}}⟩$.
On initial entry, ${\mathbf{M}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=41$
On entry, ${\mathbf{IP}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{M}}=⟨\mathit{\text{value}}⟩$.
Constraint: $1\le {\mathbf{IP}}\le {\mathbf{M}}$. On entry, ${\mathbf{IP}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{M}}=⟨\mathit{\text{value}}⟩$.
Constraint: $1\le {\mathbf{IP}}\le {\mathbf{M}}$.
${\mathbf{IFAIL}}=42$
IP has changed between calls.
On intermediate entry, ${\mathbf{IP}}=⟨\mathit{\text{value}}⟩$.
On initial entry, ${\mathbf{IP}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=51$
On entry, ${\mathbf{NBEST}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{NBEST}}\ge 1$.
${\mathbf{IFAIL}}=52$
NBEST has changed between calls.
On intermediate entry, ${\mathbf{NBEST}}=⟨\mathit{\text{value}}⟩$.
On initial entry, ${\mathbf{NBEST}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=53$
On entry, ${\mathbf{NBEST}}=⟨\mathit{\text{value}}⟩$.
But only $⟨\mathit{\text{value}}⟩$ best subsets could be calculated.
${\mathbf{IFAIL}}=61$
DROP has changed between calls.
On intermediate entry, ${\mathbf{DROP}}=⟨\mathit{\text{value}}⟩$.
On initial entry, ${\mathbf{DROP}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=71$
LZ has changed between calls.
On entry, ${\mathbf{LZ}}=⟨\mathit{\text{value}}⟩$.
On previous exit, ${\mathbf{LZ}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=91$
LA has changed between calls.
On entry, ${\mathbf{LA}}=⟨\mathit{\text{value}}⟩$.
On previous exit, ${\mathbf{LA}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=111$
${\mathbf{BSCORE}}\left(⟨\mathit{\text{value}}⟩\right)=⟨\mathit{\text{value}}⟩$, which is inconsistent with the score for the parent node. Score for the parent node is $⟨\mathit{\text{value}}⟩$. ${\mathbf{BSCORE}}\left(⟨\mathit{\text{value}}⟩\right)=⟨\mathit{\text{value}}⟩$, which is inconsistent with the score for the parent node. Score for the parent node is $⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=131$
MINCNT has changed between calls.
On intermediate entry, ${\mathbf{MINCNT}}=⟨\mathit{\text{value}}⟩$.
On initial entry, ${\mathbf{MINCNT}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=141$
GAMMA has changed between calls.
On intermediate entry, ${\mathbf{GAMMA}}=⟨\mathit{\text{value}}⟩$.
On initial entry, ${\mathbf{GAMMA}}=⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=151$
${\mathbf{ACC}}\left(1\right)$ has changed between calls.
On intermediate entry, ${\mathbf{ACC}}\left(1\right)=⟨\mathit{\text{value}}⟩$.
On initial entry, ${\mathbf{ACC}}\left(1\right)=⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=152$
${\mathbf{ACC}}\left(2\right)$ has changed between calls.
On intermediate entry, ${\mathbf{ACC}}\left(2\right)=⟨\mathit{\text{value}}⟩$.
On initial entry, ${\mathbf{ACC}}\left(2\right)=⟨\mathit{\text{value}}⟩$.
${\mathbf{IFAIL}}=161$
ICOMM has been corrupted between calls. ICOMM has been corrupted between calls.
${\mathbf{IFAIL}}=171$
On entry, ${\mathbf{LICOMM}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{LRCOMM}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{LICOMM}}\ge ⟨\mathit{\text{value}}⟩$, ${\mathbf{LRCOMM}}\ge ⟨\mathit{\text{value}}⟩$.
ICOMM is too small to return the required array sizes.
${\mathbf{IFAIL}}=172$
On entry, ${\mathbf{LICOMM}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{LRCOMM}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{LICOMM}}\ge ⟨\mathit{\text{value}}⟩$, ${\mathbf{LRCOMM}}\ge ⟨\mathit{\text{value}}⟩$.
The minimum required values for LICOMM and LRCOMM are returned in ${\mathbf{ICOMM}}\left(1\right)$ and ${\mathbf{ICOMM}}\left(2\right)$ respectively. On entry, ${\mathbf{LICOMM}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{LRCOMM}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{LICOMM}}\ge ⟨\mathit{\text{value}}⟩$, ${\mathbf{LRCOMM}}\ge ⟨\mathit{\text{value}}⟩$.
The minimum required values for LICOMM and LRCOMM are returned in ${\mathbf{ICOMM}}\left(1\right)$ and ${\mathbf{ICOMM}}\left(2\right)$ respectively.
${\mathbf{IFAIL}}=181$
RCOMM has been corrupted between calls.
${\mathbf{IFAIL}}=-999$
Dynamic memory allocation failed.
## 7 Accuracy
The subsets returned by H05AAF are guaranteed to be optimal up to the accuracy of your calculated scores.
The maximum number of unique subsets of size $p$ from a set of $m$ features is $N=\frac{m!}{\left(m-p\right)!p!}$. The efficiency of the branch and bound algorithm implemented in H05AAF comes from evaluating subsets at internal nodes of the tree, that is subsets with more than $p$ features, and where possible trimming branches of the tree based on the scores at these internal nodes as described in Narendra and Fukunaga (1977). Because of this it is possible, in some circumstances, for more than $N$ subsets to be evaluated. This will tend to happen when most of the features have a similar effect on the subset score.
If multiple optimal subsets exist with the same score, and NBEST is too small to return them all, then the choice of which of these optimal subsets is returned is arbitrary.
## 9 Example
This example finds the three linear regression models, with five variables, that have the smallest residual sums of squares when fitted to a supplied dataset. The data used in this example was simulated.
### 9.1 Program Text
Program Text (h05aafe.f90)
### 9.2 Program Data
Program Data (h05aafe.d)
### 9.3 Program Results
Program Results (h05aafe.r)
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Chapter 2.3, Problem 84E
### Applied Calculus for the Manageria...
10th Edition
Soo T. Tan
ISBN: 9781285464640
Chapter
Section
### Applied Calculus for the Manageria...
10th Edition
Soo T. Tan
ISBN: 9781285464640
Textbook Problem
# Charter Revenue The owner of a luxury motor yacht that sails among the 4000 Greek islands charges $600/person per day if exactly 20 people sign up for the cruise. However, if more than 20 people sign up for the cruise (up to the maximum capacity of 90), the fare for all the passengers is reduced by$4/person for each additional passenger. Assume that at least 20 people sign up for the cruise, and let x denote the number of passengers above 20. a. Find a function R giving the revenue per day realized from the charter. b. What is the revenue per day if 60 people sign up for the cruise? c. What is the revenue per day if 80 people sign up for the cruise?
(a)
To determine
The function R that gives the revenue per day.
Explanation
Given information:
The luxury motor yacht charges $600/person per day if exactly 20 people sign up for the cruise. If more than 20 person sign up for the cruise, the fare for all person is reduced by$4/person.
Calculation:
Let x be the number of passengers above 20, then the total number of passengers becomes 20+x .
For each additional passengers, the fare decreased by $4/person and the fair for one person is$600/person, so the fair per person for (20+x) passenger becomes (6004x)
(b)
To determine
The revenue per day if 60 people sign up for the cruise.
(c)
To determine
The revenue per day if 80 people sign up for the cruise.
### Still sussing out bartleby?
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# Chemical Partition of the Radiative Decay Rate of Luminescence of Europium Complexes
## Abstract
The spontaneous emission coefficient, Arad, a global molecular property, is one of the most important quantities related to the luminescence of complexes of lanthanide ions. In this work, by suitable algebraic transformations of the matrices involved, we introduce a partition that allows us to compute, for the first time, the individual effects of each ligand on Arad, a property of the molecule as a whole. Such a chemical partition thus opens possibilities for the comprehension of the role of each of the ligands and their interactions on the luminescence of europium coordination compounds. As an example, we applied the chemical partition to the case of repeating non-ionic ligand ternary complexes of europium(III) with DBM, TTA and BTFA, showing that it allowed us to correctly order, in an a priori manner, the non-obvious pair combinations of non-ionic ligands that led to mixed-ligand compounds with larger values of Arad.
## Introduction
Almost 40 years ago, systems containing lanthanide ions started attracting the interest of many research groups around the world1,2 due to their very peculiar luminescence properties. Such properties make these ions, such as trivalent europium, essential for the development of photo and electroluminescent devices3,4. Since intra-ion f-f transitions are Laporte forbidden, the light absorption capability of the lanthanide ions is poor and therefore luminescence cannot be generated from direct excitation. However, when the lanthanide ion is surrounded by coordinating ligands who can effectively absorb radiation and subsequently transfer the energy, in an intramolecular manner, to the coordinated lanthanide ion – a process known as the antenna effect5 – then something entirely different happens. Asymmetries in the ligand field, asymmetries due to different spatial arrangement of the ligands around the lanthanide ion, asymmetries due to mixed ligands coordination6, asymmetries due to thermal vibrations, etc, all make the intra-ion f-f decay less forbidden, and, as a result, strong luminescence ensues. Nevertheless, this strong luminescence is still a result of an almost forbidden intra-ion f-f transition, which therefore both exhibits long lifetimes and high color purity. Indeed, the exhibited luminescence is essentially monochromatic. Moreover, since the lanthanide trication 4f orbitals belong to an inner shell which is shielded by the outer 5s25p6 shell, this luminescence displays a high degree of insensitivity to environmental quenching and is of value for applications in sensors7, displays3, fluoroimmunoasssays8, fluorescence microscopy9, etc.
The rapid technological development of the last decades10,11 has contributed to escalating the interest in lanthanide complexes. So much so, that, currently, there are about one thousand articles published every year on the subject12. However, less than 5% of the studies involving lanthanide ions make use of theoretical tools12. Even though, it is possible to verify an increase in the use of theoretical tools assisting the experimental studies in their quest to design increasingly more efficient luminescent systems13,14,15,16. Hence, the extant theoretical tools and techniques are slowly becoming increasingly popular, such as the Sparkle Models17,18,19,20,21 and RM1 for lanthanides22,23,24,25, which are fully available in the MOPAC software26, as well as the new lanthanide luminescence software package LUMPAC (www.lumpac.pro.br)12, the first and only software dedicated to the study of the luminescence properties of systems containing europium ions.
Recently, we showed that the charge factors of the simple overlap model and the polarizabilities of Judd-Ofelt theory can be uniquely modeled by perturbation theory on the semiempirical electronic wavefunction of the complex27. Consequently, the terms of the intensity parameters related to dynamic coupling and electric dipole mechanisms are made unique, leading to a unique set of intensity parameters per complex.
Luminescence of europium complexes is a phenomenon entirely dependent on the chemical nature of the ligands and on the fine details of their geometric arrangements, as they are coordinated to the central metal ion. Nevertheless, luminescence is still a property of the complex as a whole. Hence, the emission spectrum of the complex conceals a profusion of ligand contributions to the luminescence phenomenon. For example, by examining the luminescence properties, it is not always entirely clear which ligand should replace another, in a given compound, in order to design a novel and more luminescent complex.
In the present article, we are advancing a novel formalism for a partition of the radiative decay rate of luminescence of europium complexes into ligand contributions. Such a chemical partition scheme is shown to be general and applicable to any europium complex. Finally, we exemplify the usage of this novel chemical partition for the choice of the best couple of non-ionic ligands for the design of mixed ligand complexes with boosted luminescence.
## Intensity parameters from experiment
Luminescence in europium complexes is a process by which ultraviolet light is converted into visible light, mostly in the red-orange region of the spectrum. The first step of this light conversion is the absorption of the ultra-violet light by high-absorbance ligands. Then, the absorbed energy is transferred to the central trivalent europium ion that, as a result, is placed in its 5D0 excited state. From that excited state, two types of processes occur: radiative decays to the low-lying 7FJ states and nonradiative decays to the ground state. This final step of the luminescence phenomenon is expressed mathematically in terms of the emission efficiency η = Arad/(Arad + Anrad), where Arad and Anrad are, respectively, the radiative and nonradiative total decay rates.
The radiative decay rate Arad is the sum of all radiative decay rates from the 5D0 excited state to each of the 7FJ states, with J = 0,1,2,3,4,5,6, whereas the non-radiative rate, Anrad, is actually a bundle of non-radiative processes. Since we are not interested in non-radiative decays, there is no need to identify any of their terms.
The experimental radiative decay rate, , is the sum of the radiative decay rates of each of the possible transitions 5D0 → 7FJ with J ranging from 0–6:
The transition 5D0 → 7F1 is governed by a magnetic dipole mechanism and is therefore insensitive to electric dipole contributions. This transition is thus regarded as insensitive to the essentially static electric fields produced by the ligands and can be determined through the expression28 , where n is the refractive index of the medium and is the barycenter, the weighted mean of the frequencies in cm−1, corresponding the 5D0 → 7F1 transition. From this value, we can now compute the other radiative decay rates, with J from 0–6:
where are the energies of the barycenters of the respective transitions; and S[5D0 → 7FJ] are the areas under the spectra corresponding to the respective transitions. Finally, the experimental intensity parameters can be calculated from28:
where is Planck-Dirac constant, e is the fundamental electric charge, is the frequency of the transition in wavenumbers, χ is the Lorentz local-field correction term given by χ = n(n2 + 2)2/9 and are the square reduced matrix elements whose values are 0.0032, 0.0023 and 0.0002 for λ = 2, 4 and 6 and J = λ in the case of europium29.
Another way of obtaining the is from absorption spectra30. In such a procedure, the are obtained by parameterization so that the calculated oscillator strengths match the experimental ones from the absorption spectra and the parameterization error propagates to the intensity parameters. Besides, further errors may derive from uncertainties in the choice of theoretical transitions to relate to the experimental ones. Indeed, when Judd-Ofelt parameters are estimated from absorption spectra in this manner, errors are of the order of 10–20%30 and that is why, in this article, we chose to evaluate them from emission spectra instead.
## Theoretical intensity parameters
The theoretical radiative decay rate for the forced electric dipole and magnetic dipole governed transitions,, is given by, where, for europium,
and is equal to
where e is the elementary charge; 2J + 1 is the degeneracy of the initial state, in this case 5D0 and therefore J = 0. Transitions 5D0 → 7FJ with J = 0, 3 and 5 depend on contributions from both electric and magnetic dipole mechanisms and thus are not as easy to calculate. Fortunately, their intensities are very low and, thus, they can be safely disregarded. Transition 5D0 → 7F1 is the only one which does not have an electric dipole contribution, therefore, is not sensitive to the presence of the ligands around the europium ion and is relatively small: its magnetic dipole strength is theoretically evaluated as being Smd = 96 × 10–42 esu2 cm2 31. Therefore, for the purpose of this work, we define , restricted to the chemically interesting transitions 5D0 → 7FJ with J = 2, 4 and 6. Their corresponding theoretical intensity parameters Ωλ (λ = 2, 4, 6) emerge from the Judd-Ofelt theory32,33, depend on the coordination interaction between the lanthanide cation and the ligands and are given by the following expression:
The Bλ,t,p terms in eq. (6) are given by the following expression:
where corresponds to the forced electric dipole contribution and corresponds to the dynamic coupling contribution, respectively given by:
where ΔE is a constant, approximately given by the energy difference between the barycenters of the ground 4f N and first opposite parity excited state of configuration 4f (N−1)5d of the europium ion; = 2.567541 × 10−17cm2 and = 1.58188 × 10−33 cm4 are radial integrals, pre-defined for the europium ion34; θ(t,λ) are numerical factors for a given lanthanide, estimated by Malta et al.35 from Hartree-Fock calculations of the radial integrals as being: θ(1,2) = −0.17; θ(3,2) = 0.345; θ(3,4) = 0.18, θ(5,4) = −0.24, θ(5,6) = −0.24, θ(7,6) = 0.24; γtp are the odd-rank ligand field parameters; (1 − σλ) is a shielding factor due to the filled 5 s and 5p sub-shells of the lanthanide ion35, with σ2, σ4 and σ6 being, respectively, 0.6, 0.139 and 0.100 for Eu(III), as previously calculated by Malta and Silva36; is a Racah tensor operator of rank λ whose values for λ = 2,4,6, are −1.3660, 1.128 and −1.270, respectively, for any lanthanide; is also a sum over coordinating atoms which further reflects the chemical environment; finally, δt,λ+1 is a Kronecker delta symbol. All these pre-defined parameters are taken as constants for all europium complexes.
The odd rank ligand field parameters are, in turn, given by:
where the term , according to the Simple Overlap Model (SOM)37,38, introduces a correction to the crystal field parameters of the point charge electrostatic model (PCEM)39, , such that , which confers a degree of covalency to the point charge model through the inclusion of parameter ρ, since PCEM only treats the metal-ligand atom bonds as a purely electrostatic phenomenon; gi is the charge factor associated to the lanthanide-ligand atom bond; is the lanthanide-ligand atom bond distance; and are complex conjugate spherical harmonics.
The other odd-rank parameter , which further reflects the chemical environment is given by:
where αi is the polarizability associated to the lanthanide-ligand atom bond.
In a recent article27, we introduced a protocol to model the charge factors gj of the simple overlap model by electron densities and the polarizabilities αi of Judd-Ofelt theory by superdelocalizabilities, all obtained by perturbation theory on the semiempirical electronic wavefunction of the complex. A fitting of the theoretical intensity parameters is then carried out, which reproduces the experimentally obtained using only three adjustable constants: Q, D and C, which must obey the acceptance criterion D/C > 1 leading to a unique adjustment. Whenever D/C ≤ 1, it was shown that the presumed geometry of the coordination polyhedron is not seemingly compatible with the experimental intensity parameters and requires improvement, either via calculation by another theoretical model, such as another Sparkle Model17,18,19,20,21 or RM1 model for lanthanides22, or via X-ray crystallographic measurements, etc. The importance of this previous work is that all derived quantities become also uniquely determined for a given complex geometry27, including the chemical partition that is being advanced in this article.
## Results and Discussion
### Partitioning Arad into ligand terms
According to the theory, . The first term, , with even J, is mainly driven by electric dipole transitions. The second term, , with odd J, is mainly driven by magnetic dipole transitions. Besides, recall that the strengths of the transitions 5D0 → 7FJ with J = 0, 3 and 5 are set at zero because of their low values. Likewise, the magnetic dipole driven transition 5D0 → 7F1 is not sensitive to the ligands and, as a result, is not directly relevant from a chemical point of view. Therefore, the partition will focus on the electric dipole driven transitions with J = 2, 4, 6. Accordingly, we will partition a subset of Arad, we call Arad′, defined by:
the most significant term being the decay rate of the so-called hypersensitive transition 5D0 → 7F2 , which is highly susceptible to the presence of the ligands.
Now, let us turn to compute the terms from eqs (8, 9, 10, 11).
which can be rewritten as,
Accordingly, are obtained by sums over all coordinating atoms of the ligands of the product of a term, , defined below, which depends only on the lanthanide ion and on the particular coordinating bond as previously described, with a complex conjugate spherical harmonic.
Therefore,
and we can define an auxiliary matrix, , which is a function of only the coordinating atoms of the complex as
Note that all diagonal elements . Besides, is an Hermitian matrix because *. Therefore, its eigenvalues are all real numbers. Moreover, we will show that is a positive semi-definite matrix and therefore all its eigenvalues are, not only real, but also equal to or greater than zero. Q is said to be positive semi-definite if is non-negative for every non-zero column vector t of n real numbers. Here tT denotes the transpose of t.
As such,
Replacing Q by its expression,
That is, . Therefore, Q is positive semi-definite.
The terms of eq. (16) can now be computed in terms of the directly coordinating atoms as:
then,
Likewise, we can define the efficacy of luminescence matrix as
where νλ is the frequency (in cm−1) corresponding to the energy gap between the initial 5D0 and final 7FJ states.
Note that is a real symmetric positive semi-definite matrix, since is also a real symmetric positive semi-definite matrix and the coefficients multiplying in eq. (22) are all positive. The diagonal terms are coordinating atom contributions and the off diagonal terms with , are the coordinated atom pair contributions to Arad[5D0 → 7FJ], and, ultimately, indirect contributions to the emission efficiency. As a result,
From the point of view of coordination chemistry, it is more useful to aggregate all contributions from each ligand in the complex and define the ligand contributions to the radiative decay rate by matrix , in terms of summations over the coordinating atoms k of ligand L as:
Likewise, we can define the ligand pair contribution to the radiative decay rate by matrix , which is a measure of how well the ligands interact to enhance the radiative decay rate, in terms of summations over the coordinating atoms k and m, of ligands L and L′, respectively, as:
Note that the ligand-pair contributions do not contain any atomic contributions, since, by being different ligands, L and L’ do not share any directly coordinating atom.
Finally,
We will use the efficacies of luminescence, or simply efficacies and in order to interpret, from a chemical point of view, the various influences of the ligands and of their atoms, together with their pair influences, directly on . The elements of both efficacy matrices A and Al are partial decay rates (see eqs (23) and (26)), being expressed in units of a decay rate, usually s−1. So, in order to obtain the total decay rate in s−1, just add all elements of either matrix, that is, their grand sums or defined as and , where is a column vector with all elements equal to unity.
### The chemical partition of Arad′
The elements of the efficacy matrices A and Al, are often negative, which require an interpretation, which may sometimes be useful, but is certainly less chemically intuitive, of the partition of . For example, if a given complex displays a very low luminescence, that is an close to zero, it is possible that the contribution Al from one of its ligands be +800 s−1 and the Al of another ligand be −800 s−1. Such a situation in which one contribution annihilates the other, renders the role of each of the ligands on the luminescence phenomenon somewhat indiscernible, especially when they are chemically identical.
A chemically more intuitive partition would require only coordinated atom or ligand contributions, always positive and hence, whenever is zero, they all should be zero.
In order to define such a partition, we start by recognizing that matrix Q is Hermitian and positive semi-definite, and, as a consequence, matrix is also Hermitian and positive semi-definite. That implies that their eigenvalues are not only all real, but they are all greater than or equal to zero. Now, define the orthonormal eigenvectors of as U1, U2, …, Un. Let , where denotes an inner product. Then, . Let the eigenvalues associated to the eigenvectors U1Un be, respectively, λ1, … λn. Thus, . In fact,
Let,
Observe that
So, we define the relative contribution of the coordinated atom j to the eigenvalue λi as
We will now obtain a nicer expression for this relative contribution. Since , we have , thus . Finally, . Thus, is the relative contribution of the coordinated atom j to eigenvalue λi because these relative contributions are non-negative and sum 1 and thus can be viewed as proportions as we intended. We can now define the vectors with the ligand contributions to the eigenvectors:
In a similar manner, we can define the absolute contribution of coordinated atom j to as times the relative contribution of coordinated atom j to the eigenvalue λi, i.e . Finally, we can define the absolute contribution of coordinated atom j to Su(A), Λj, as the sum of the contributions of coordinated atom j over all values , with i = 1…n. Let . Then, the contribution of coordinated atom j to Su(A), Λj, is given by
due to the definition above, , with .
The set of coefficients Λj thus constitutes a partition of in n terms, all positive, each corresponding to each of the n coordinated atoms. These terms reflect how each coordinated atom contributed to make the 5D0 → 7FJ less forbidden.
As before, we can aggregate all contributions from each ligand in the complex and define the ligand contributions to in terms of summations over the coordinating atoms k of ligand L as:
Of course,
In summary, the chemical partitions we introduce in this article, , , , , Λj and ΛL, are from now on available, ready to be interpreted from a myriad of chemical perspectives depending on the system of interest, subject to the creativity of the researcher.
### LUMPAC chemical partition implementation
The luminescent software package LUMPAC12, since 2013 freely available from http://www.lumpac.pro.br/, is the first state of the art complete software to treat europium luminescence from a theoretical point of view. Recently, the unique adjustment of theoretical intensity parameters developed by our group27 was implemented in LUMPAC. Now, the chemical partition being advanced in this article has also been implemented and is already available to all users of LUMPAC.
From eqs (10) and (11), clearly geometry makes a profound impact on the calculation of the theoretical intensity parameters Ω2, Ω4 and Ω6, and, by extension makes an impact on our partition scheme. Therefore, users must first determine the most stable geometry of the complex of interest via either RM122 or any of the Sparkle Models17,18,19,20,21 in such a manner as to satisfy the binary outcome acceptance conditions for the unique adjustment of theoretical intensity parameters as described in the “LUMPAC implementation” section of ref. 27. Once the adjustment is considered accepted, calculation of the chemical partition follows in a seamless manner.
The partition is first computed per directly coordinated atom and then subsequently aggregated per ligand by summing up the terms of the directly coordinated atoms of each ligand. It has been proven useful to further aggregate the terms of the ligands into terms for classes of ligands, such as the terms of all ionic ligands and those of all non-ionic ligands.
### Interpretation of the chemical partition ligand terms
Interpretation of the ligand terms requires an understanding of the fact that, according to Laporte rule, the electronic f-f transitions in lanthanide complexes should be forbidden in centrosymmetric molecules, since they conserve parity with respect to the inversion center where the metal is located. In this sense, luminescence happens because the centrosymmetry can be broken by ligands coordinating the lanthanide ion. Since luminescence happens in the europium ion, not at the ligands, the ligand terms of the chemical partition cannot possibly be regarded as ligand contributions to , but rather as measures of the relaxation of the forbidding character of the 5D0 → 7FJ transitions (J = 2, 4, 6), conferred by each of the respective ligands to the europium ion. Note that each ligand term of the chemical partition is defined within the distinctive chemical ambiance of the particular complex and cannot be expected to be transferable from complex to complex.
### Sensitivity of the chemical partition to complex geometry
We now turn to exemplify aspects of our partition scheme by first studying three specific complexes of the general formula Eu(β-diketonate)3(TPPO)2, where TPPO is the non-ionic ligand triphenylphosphine oxide and β-diketonate stands for one of the ionic ligands TTA, 1-(2-thenoyl),3,3,3-trifluoroacetone, BTFA, 4,4,4-trifluoro-1-phenyl-2,4-butanedione, or DBM, 1,3-diphenylpropane-1,3-dione.
A complex of this general formula Eu(β-diketonate)3(TPPO)2 may display two possible ligand arrangements: both TPPOs are either adjacent or opposite to each other. All structural data have been computed by either RM122, or, in a single case, by the Sparkle/PM318 method. The choice of method followed the QDC acceptance criterion27 defined in a previous article on the unique adjustment of theoretical intensity parameters27. We will now examine the impact of these different geometric arrangements on the chemical partition of . However, since the emission spectra, Arad, Ω2 and Ω4 have been measured for these complexes in the opposite TPPO configuration6,40, we will use these same values to arrive at the chemical partition for each of the two possible geometrical arrangements for enlightening purposes only. Tables S1 and S2 of the Supplementary Information contain information on the adjustments of the theoretical intensity parameters, unique for each of the geometrical arrangements and the partition results, aggregated by ligand, for all three complexes considered.
Figure 1 shows the chemical partition of the radiative decay rate for each of the ligands coordinated to the metal ion for both cases of adjacent and opposite non-ionic ligands for all three TPPO complexes considered.
Figure 1 evidences the chemical nature of the partition because, now, has been sliced into ligand terms that depend on the chemical nature of the ligands, as well as on their collective arrangements around the europium ion.
In an environment with three other identical ionic ligands, adjacent non-ionic ligands are less centrosymmetric than opposite ones. So, one would expect that adjacent same ligands should contribute more to the relaxation of the forbidding character of the 5D0 → 7FJ transitions (J = 2,4,6) than opposite ones. That is indeed the case for Eu(TTA)3(TPPO)2, where the sum of terms of the adjacent TPPOs is 382 s−1, whereas for opposite TPPOs it is 122 s−1. Equivalent numbers for Eu(BTFA)3(TPPO)2 are 422 s−1 and 160 s−1 and for Eu(DBM)3(TPPO)2, they are 96 s−1 and 102 s−1, a more balanced situation which arises seemingly due to the more symmetric and bulky nature of DBM.
Conversely, the β-diketonate ligands are more centrosymmetric when the TPPOs are adjacent (two of them tend to occupy opposite axial-like positions) and therefore they should contribute less to the relaxation of the forbidding character of the 5D0 → 7FJ transitions (J = 2, 4, 6). On the other hand, the β-diketonate ligands are less centrosymmetric when the TPPOs are opposite, because in this case they tend to occupy planar trigonal-like positions, in which case they should contribute more to the relaxation of the forbidding character of the 5D0 → 7FJ transitions (J = 2, 4, 6). For Eu(TTA)3(TPPO)2, the sum of the three β-diketonate terms for opposite TPPOs is 636 s−1, whereas for adjacent TPPOs it is 374 s−1. Equivalent numbers for Eu(BTFA)3(TPPO)2 are 635 s−1 and 454 s−1. For Eu(DBM)3(TPPO)2 the numbers are 190 s−1 and 233 s−1, which we again attribute to the bulky nature of DBM which is in itself the most symmetric of the β-diketonates used and makes the whole complex overall more centrosymmetric, severely reducing the value of Arad to 335 s−1 when compared to the other two, which average 858 s−1.
### Applications of the chemical partition
Complexes Eu(TTA)3(TPPO)2 and Eu(BTFA)3(TPPO)2 had their geometries determined by crystallography and deposited in the Cambridge Structural Database, CSD41,42,43, with refcodes SABHIM and WIFWIR, respectively. In both cases, the non-ionic ligands appear opposite to each other. Recently, a theoretical determination of the thermodynamic properties of Eu(DBM)3(TPPO)2 also indicated that the opposite TPPO configuration should be the preferred one40. This study further extended the analysis for other non-ionic ligands and predicted that Eu(DBM)3(DBSO)2 and Eu(DBM)3(PTSO)2 should also display opposite non-ionic ligand configurations, where DBSO is dibenzyl sulfoxide and PTSO is p-tolyl sulfoxide. As a consequence, in the present article we assume that all complexes of the general formula Eu(β-diketonate)3(L)2 where L is a non-ionic ligand, will adopt opposite non-ionic ligand configurations. As before, all structural data have been computed by either RM122, or, in a single case, by the Sparkle/PM318 method - the choice of method followed the QDC acceptance criterion27.
Table 1 presents luminescence results for 9 different complexes of the general formula Eu(β-diketonate)3(L)2, radiative decay rates, both experimental () and calculated, (), the latter one partitioned by ligands and summed up into ionic ligand () and non-ionic ligand () terms. Please remember that () will always be larger than () because refers to all 5D0 → 7FJ transitions, while only adds up those with J = 2,4,6.
Examination of the average values in Table 1 reveals that the contribution of the non-ionic ligands to the triggering of luminescence decay by the excited europium trivalent ion is much smaller, in average 120 s−1, than the corresponding contribution of the ionic ligands, 544 s−1. That could lead to a misunderstanding that the non-ionic ligands, in these cases, are not too relevant to the luminescence phenomenon. However, their modest contribution to () is seemingly due to that fact that they are opposite to each other, and, therefore, in a symmetric configuration with respect to the europium ion, which does not help to relax the Laporte’s rule.
Recently, our group introduced a simple strategy to boost three important luminescence properties of complexes of the general formula Eu(β-diketonate)3(L)2: the quantum yield, Φ, the emission efficiency η and , which was mathematically translated into the following conjecture6:
where P stands for either Φ, η, or and L and L′ are different non-ionic ligands. This conjecture states that mixed non-ionic ligand complexes, Eu(β-diketonate)3(L,L′), should display larger luminescence properties when compared to the average of the same properties for repeating ligand complexes, Eu(β-diketonate)3(L)2 and Eu(β-diketonate)3(L′)2. This conjecture has already been proven experimentally for all combinations of the non-ionic ligands DBSO, TPPO and PTSO, for all ternary europium complexes of TTA, BTFA and DBM.
Table 2 displays , , and for the complexes with mixed non-ionic ligands. The role of the non-ionic ligands in the mixed-ligand complexes becomes clearer. Indeed, now the contribution of both different non-ionic ligands, to the triggering of luminescence decay, becomes accentuated to an average of 302 s−1, up from the average of 120 s−1 in the repeating non-ionic ligand complexes of Table 1. Once again, this behavior can be rationalized in terms of symmetry: the different non-ionic ligands are opposite to each other, rendering the situation considerably more asymmetric, thus much more capable of triggering the luminescent decay of the excited europium ion. On the other hand, the role of the ionic ligands remains, in average, unaffected. Indeed, the average of for the mixed non-ionic ligand complexes is 552 s−1, whereas in the repeating non-ionic ligand complexes it is 544 s−1, thus reinforcing the protagonist role of the non-ionic ligands in the luminescence boost.
So far, we have argued the usefulness of our chemical partition in terms of averages across complexes, which helped us understand some global aspects of the luminescence phenomenon.
We now turn to examine the usefulness of the chemical partition per complex. The conjecture, eq. (36), indicates that in order to obtain a more luminescent complex, of the type Eu(β-diketonate)3(L)2, one should synthesize complexes of the type Eu(β-diketonate)3(L,L′). However, the conjecture does not provide a hint of which pair combination of non-ionic ligands L,L′ one must choose to obtain the mixed non-ionic ligand complex of maximum . Actually, this is by no means trivial as exemplified by the case of Eu(DBM)3(L)2. Which pair of non-ionic ligands L,L′ will lead to the complex Eu(DBM)3(L,L′) with the largest ? If one naively decides to choose in terms of of Table 1, by picking the non-ionic ligands of the two complexes with the highest values of : Eu(DBM)3(DBSO)2 and Eu(DBM)3(PTSO)2, one would synthesize complex Eu(DBM)3(DBSO,PTSO) and would be left with the mixed non-ionic DBM ternary complex with the smallest . On the other hand, for the case of TTA ternary complexes, such a choice would lead to the correct complex, that is, to Eu(TTA)3(DBSO,TPPO). Clearly, choosing mixed non-ionic ligand combinations from of repeating non-ionic ligand complexes is not correct, as extraneous factors seem to be playing a role in the different ligands synergy, factors that, as will be shown below, the chemical partition seems to unveil.
Since what one needs to choose is one pair of non-ionic ligands which will lead to the mixed ligand complex with the highest value of , one must therefore simply look, instead, at the non-ionic ligand term in the corresponding repeating ligand complexes in Table 1. Thus, for the before mentioned case of DBM ternary complexes, one would choose the two non-ionic ligands L and L′, whose corresponding repeating non-ionic ligand complexes display the largest values of : Eu(DBM)3(TPPO)2 and Eu(DBM)3(DBSO)2, respectively, 102 s−1 and 59 s−1, leading to complex Eu(DBM)3(DBSO,TPPO). Indeed, this complex has the largest of 652 s−1. Furthermore, one can use the same reasoning to arrive at the DBM ternary complex with the next larger , where two possibilities exist: Eu(DBM)3(PTSO,TPPO) and Eu(DBM)3(DBSO,PTSO). The corresponding pair of largest that are left are 102 s−1 for Eu(DBM)3(TPPO)2 and 29 s−1 for Eu(DBM)3(PTSO)2, leading to complex Eu(DBM)3(PTSO,TPPO), which, indeed, is the next best complex, with an of 572 s−1. Finally, the last one left is complex Eu(DBM)3(DBSO,PTSO), with an of 540 s−1.
Undoubtedly, for the case of ternary complexes of DBM, usage of the chemical partition allowed us to perfectly order, in an a priori way, the pair combinations of non-ionic ligands in terms of their, a single outcome out of six possibilities.
Let us now apply the same strategy to the other ternary complexes of TTA and BTFA. The overall situation is present in pictorial format in Fig. 2, which shows that choosing non-ionic ligands by their terms in non-ionic repeating ligand complexes also perfectly orders all mixed non-ionic ligand complexes in terms of their values for the remaining cases of TTA and BTFA ternary complexes.
Overall, the chemical partition predicted, in an a priori manner, a single joint outcome out of 216 possibilities, which, if it were by chance, would have a probability of only 0.46% of occurrence.
## Conclusions
For the first time, a molecular global property related to luminescence, the radiative decay rate of europium complexes, is partitioned into ligand contributions. Such a novel approach gives rise to possible chemical interpretations of the effect of each ligand and their interactions on the luminescence phenomenon, allowing for the design of cutting edge compounds with enhanced brightness upon ultra-violet illumination.
As a demonstration of an application made only possible with the chemical partition, we address the case of repeating non-ionic ligand ternary complexes of europium(III) with DBM, TTA and BTFA. We show that the chemical partition allows us to perfectly order, in an a priori way, the non-obvious pair combinations of non-ionic ligands that led to mixed-ligand compounds with larger values of Arad.
Our new chemical partition has been implemented in the LUMPAC software, which is freely available12.
How to cite this article: Lima, N. B. D. et al. Chemical Partition of the Radiative Decay Rate of Luminescence of Europium Complexes. Sci. Rep. 6, 21204; doi: 10.1038/srep21204 (2016).
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## Acknowledgements
The authors appreciate the financial support of the following Brazilian Agencies and Institutes: FACEPE (Pronex), CNPq, FAPITEC/SE and INCT/INAMI.
## Author information
Authors
### Contributions
A.M.S. and N.B.D.L. identified the need for a chemical partition of the radiative decay rate of luminescence. A.M.S. derived the theory formalism. J.D.L.D. and R.O.F. coded the necessary modifications in LUMPAC to implement the chemical partition. N.B.D.L., A.M.S. and S.M.C.G. arrived at the application of the chemical partition to guide the choice of ligands for the design of mixed-ligand complexes displaying boosted luminescence. R.O.F. wrote the introductions of the article and of the Supplementary Information. All authors wrote and reviewed the manuscript.
## Ethics declarations
### Competing interests
The authors declare no competing financial interests.
## Rights and permissions
Reprints and Permissions
Lima, N., Dutra, J., Gonçalves, S. et al. Chemical Partition of the Radiative Decay Rate of Luminescence of Europium Complexes. Sci Rep 6, 21204 (2016). https://doi.org/10.1038/srep21204
• Accepted:
• Published:
• ### Competition Between Different Decay Modes of Lanthanides
• H. C. Manjunatha
• , L. Seenappa
• , P. S. Damodara Gupta
• , K. N. Sridhar
• & B. Chinnappa Reddy
Brazilian Journal of Physics (2021)
• ### Quantum Yield and Photoluminescence Intensity Enhancement Effects of a Diphosphine Dioxide Ligand on a 6-Coordinate Eu(III)-β-Diketonate Complex with Low Luminescence
• Hiroki Iwanaga
• & Fumihiko Aiga
ACS Omega (2021)
• ### Energy transfer mechanism in luminescence Eu(III) and Tb(III) complexes of coumarin-3-carboxylic acid: A theoretical study
• Ivelina Georgieva
• , Tsvetan Zahariev
• , Natasha Trendafilova
• & Hans Lischka
Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy (2020)
• ### Effects of alkyl groups in diphosphine dioxide-ligand for Eu(III)-β-diketonate complexes on photoluminescence properties
• Hiroki Iwanaga
Chemical Physics Letters (2019)
• ### Energy Transfer from Antenna Ligand to Europium(III) Followed Using Ultrafast Optical and X-ray Spectroscopy
• Michael W. Mara
• , David S. Tatum
• , Anne-Marie March
• , Gilles Doumy
• , Evan G. Moore
• & Kenneth N. Raymond
Journal of the American Chemical Society (2019)
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[SOLVED]Convergence 2
dwsmith
Well-known member
$\sum\limits_{n = 2}^{\infty}n^p\left(\frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}}\right)$
where p is any fixed real number.
If this was just the telescoping series or the p-series, this wouldn't be a problem.
CaptainBlack
Well-known member
$\sum\limits_{n = 2}^{\infty}n^p\left(\frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}}\right)$
where p is any fixed real number.
If this was just the telescoping series or the p-series, this wouldn't be a problem.
$\left( \frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}} \right) \sim \frac{n^{-3/2}}{2}$
CB
dwsmith
Well-known member
$\left( \frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}} \right) \sim \frac{n^{-3/2}}{2}$
CB
How did you come up with that though?
chisigma
Well-known member
$\displaystyle \frac{1}{\sqrt{n-1}}- \frac{1}{\sqrt{n}}= \frac{\sqrt{n}- \sqrt{n-1}} {\sqrt{n^{2}-n}}= \frac{1}{\sqrt{n^{2}-n}\ (\sqrt{n}+\sqrt{n+1})} = \frac{1}{\sqrt{n^{3}-n^{2}} + \sqrt{n^{3}-2\ n^{2} + n}} \sim \frac{1}{2} n^{-\frac{3}{2}}$
Kind regards
$\chi$ $\sigma$
CaptainBlack
Well-known member
How did you come up with that though?
\begin{aligned}\frac{1}{\sqrt{n-1}}-\frac{1}{\sqrt{n}}&=\frac{1}{\sqrt{n}\sqrt{1-1/n}}-\frac{1}{\sqrt{n}}\\&= \frac{1}{\sqrt{n}}\left(1+(-1/2)(-n)^{-1}+O(n^{-2})\right)-\frac{1}{\sqrt{n}}\\ & = \dots \end{aligned}
CB
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# Don’t Let This Slip Through Your Fingers Like Little Grains of Sand
(0)
Tuition given in the topic of Miss Loi's Temple aka Joss Sticks Tuition Centre from the desk of at 1:24 am (Singapore time)
Updated on
Update: Seems like this tree is filling up nicely. We’ll hold all of you to your wishes! 😀
It’s Autumn, where trees in many parts of the world shed their leaves in preparation for the long, dark winter.
Nearer to home however, a solitary tree offers its shed branches to the men and women of The Temple to “sow” their O Level wishes, hopeful that they will bear fruit in January.
You have your future
And you hold it in our hands
Don’t let it slip through your fingers
Like little grains of sand …
(inspired by lyrics from some obscure dance track discovered on Youtube :P)
Do Your Very Best & Good Luck for O Level 2014!
### Revision Exercise
To show that you have understood what Miss Loi just taught you from the centre, you must:
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SERVING THE QUANTITATIVE FINANCE COMMUNITY
• 1
• 2
rwang716
Topic Author
Posts: 12
Joined: July 18th, 2019, 6:18 am
### Explicit Formula for computing IV
Working on the fast computation of BS implied volatility. I have read some papers regarding the explicit formula ofr BS IV. Like the Li(2005) paper: A new formula for computing implied volatility. However, I met a very serious problem in their formulas because the terms in the square root bracket can goes to negative is the option price is small enough, the simple formula by Corrado and Miller(1996) also suffers from the same problem. How can I solve this problem?
BTW, the explicit formula only serves as the initial guess for the subsequent numerical methods.
I don't know how to insert image for the formula, sorry.
Cuchulainn
Posts: 62126
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:
### Re: Explicit Formula for computing IV
There was a long-running discussion on this a while back
https://forum.wilmott.com/viewtopic.php?f=34&t=97812&p=732891&hilit=all+for+iv#p732891
The most robust was least squares AFAIR. The posts have unfortunately been become almost unreadable.
Since this is a nonlinear problem I would be surprised if there is an explicit formula. It would be the first in history.
terms in the square root bracket can goes to negative is the option price is small enough,
Can happen. It means the (implicit) assumptions/constraints on which the algorithm is based no longer hold.
rwang716
Topic Author
Posts: 12
Joined: July 18th, 2019, 6:18 am
### Re: Explicit Formula for computing IV
There are some approximation IV formulas that are 99.9% accurate. However, the problem is that those formula only works for near-ATM options, for options that are, e.g., 25% ITM/OTM, the formula becomes nonsense. So I just wonder if there are some approximation formulas that does not have this kind of issue.
FaridMoussaoui
Posts: 507
Joined: June 20th, 2008, 10:05 am
Location: Genève, Genf, Ginevra, Geneva
### Re: Explicit Formula for computing IV
Isnt' Li formula (15) designed for the deep I(O)TM calls?
rwang716
Topic Author
Posts: 12
Joined: July 18th, 2019, 6:18 am
### Re: Explicit Formula for computing IV
Do you mean Li(2005) formula?
I don't think so. I have tried this formula and it produces imaginary numbers.
Try the input: S = 2800, K = 3050, T = 1/12, r = 0.02, divi = r, call price = 15.36.
The result is a negative value in the square root bracket.
Cuchulainn
Posts: 62126
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:
### Re: Explicit Formula for computing IV
I tried c = 15.3559 and get iv = 0.10979904 using both Differential Evolution (DE) and Brent's method in Boost C++.
A benign problem, numerically.
//
The approach taken in LI 2005 (Taylor series, >>) does not appeal to me at all.
I've lost count of the number of time that 'explicit formula' is mentioned in the text..
Last edited by Cuchulainn on July 29th, 2019, 7:18 pm, edited 1 time in total.
FaridMoussaoui
Posts: 507
Joined: June 20th, 2008, 10:05 am
Location: Genève, Genf, Ginevra, Geneva
### Re: Explicit Formula for computing IV
May be because he treated only two cases (as shown in the appendix derivation) for the "asymptotic" behaviour of the IV.
You can have a look to Quantlib function "blackFormulaImpliedStdDevApproximation" for approximations of IV.
rwang716
Topic Author
Posts: 12
Joined: July 18th, 2019, 6:18 am
### Re: Explicit Formula for computing IV
As far as the paper I have read, this is an inevitable problem for all formulas that stem from Taylor expansion. All of these formulas works for a small range of option prices and moneyness, which makes them much less useful in practice.
I guess I might just have to read a lot more papers to find the solution.
Last edited by rwang716 on July 30th, 2019, 12:08 am, edited 1 time in total.
rwang716
Topic Author
Posts: 12
Joined: July 18th, 2019, 6:18 am
### Re: Explicit Formula for computing IV
I tried c = 15.3559 and get iv = 0.10979904 using both Differential Evolution (DE) and Brent's method in Boost C++.
A benign problem, numerically.
//
The approach taken in LI 2005 (Taylor series, >>) does not appeal to me at all.
I've lost count of the number of time that 'explicit formula' is mentioned in the text..
The iteration methods for computing IV is definitely applicable in all cases, and accurate. However, since in my cases we are looking for the extreme speed-to reduce the time consumption to less than or equal to 1 BS formula time. This makes most numerical solvers methods inappropriate for us.
Alan
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Location: California
Contact:
### Re: Explicit Formula for computing IV
Please justify the need for extreme speed. What is the application?
rwang716
Topic Author
Posts: 12
Joined: July 18th, 2019, 6:18 am
### Re: Explicit Formula for computing IV
I have pretty much solved the problem myself. Anyway, thanks guys for the help.
BTW, by extreme speed, I mean I hope the time for computing the BS implied vol is less than 0.02us, hopefully 0.01us. My current speed is 0.022us on my laptop but I think with further code optimization I can reduce it to 0.015us.
Cuchulainn
Posts: 62126
Joined: July 16th, 2004, 7:38 am
Location: Amsterdam
Contact:
### Re: Explicit Formula for computing IV
I tried c = 15.3559 and get iv = 0.10979904 using both Differential Evolution (DE) and Brent's method in Boost C++.
A benign problem, numerically.
//
The approach taken in LI 2005 (Taylor series, >>) does not appeal to me at all.
I've lost count of the number of time that 'explicit formula' is mentioned in the text..
The iteration methods for computing IV is definitely applicable in all cases, and accurate. However, since in my cases we are looking for the extreme speed-to reduce the time consumption to less than or equal to 1 BS formula time. This makes most numerical solvers methods inappropriate for us.
Prove it! Sounds a bit waffly...
haslipg
Posts: 5
Joined: July 11th, 2013, 9:03 am
### Re: Explicit Formula for computing IV
I used found this method very reliable https://www.ssrn.com/abstract=2908494
Cuchulainn
Posts: 62126
Joined: July 16th, 2004, 7:38 am
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Contact:
### Re: Explicit Formula for computing IV
I used found this method very reliable https://www.ssrn.com/abstract=2908494
Looks good. I am just wondering about multiple calls to exp() and testing for y < 0 etc. AFAIR if-else tests in loops are an issue?
Cuchulainn
Posts: 62126
Joined: July 16th, 2004, 7:38 am
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Contact:
### Re: Explicit Formula for computing IV
Another choice is a bisection/Newton combi (reliability/speed) that I find useful if the nonlinear function is unimodal:
1. We can pre-compute the number of iterations (and hence number of function evaluations!) needed for a given tolerance (e.g. 1/10?) with Bisection. The value will be the seed for Newton. Espen Haug in his book how to compute the initial interval/bracket for the solution.
https://en.wikipedia.org/wiki/Bisection_method
2. Since the seed is in the neighbourhood of the true solution, then 2-3 iterations of Newton is enough. all things being equal.
3. More upmarket, we might need to compute the 'initial bracket'.
https://en.wikipedia.org/wiki/Kantorovich_theorem
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# If in a non-right-angled triangle ABC, tan A and tan B are rational and the vertices A and B are also rational points, then
16 views
If in a non-right-angled triangle ABC, tan A and tan B are rational and the vertices A and B are also rational points, then the correct statements is/are
(A) tan C must be rational
(B) C must be a rational point
(C) tan C may be irrational
(D) C may be an irrational point
+1 vote
by (53.5k points)
selected
Correct option (A) , (B)
tan A + tan B + tan C = tan A tan B tan C
⇒ tan C = tan A + tan B/tan A tan B -1
⇒ tan C must be rational
Since the slope of line AB is rational and tan A is also rational, therefore slope of line AC is also rational. Similarly, slope of line BC is also rational.
⇒ C must be a rational point.
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## Browse
### Recent Submissions
Now showing 1 - 5 of 428
• Publication
Interesting Issues Relating to the Succession inLien of the Heirs Disinherited by the de cujusbefore the Death of the de cujus and theSuccession in Lien of the Heirs Excluded fromthe Succession after the Death of the de cujus
(Chulalongkorn University Printing House, 2008)
University of the Thai Chamber of Commerce. Journal Editorial Office
When a person dies, his estate devolves on the heirs by statutory rights, called çstatutoryheirsé, or on the heirs who are entitled by will, called çlegateesé, as the case may be. Butif such heirs behave in an unworthy way vis-a$-vis the de cujus, and are still entitled tothe succession, it would appear to be unjust. Hence, the stipulations of the law thatsuch heirs shall be disinherited ipso jure on account of their unworthy behaviorsvis-a$-vis the de cujus, by being excluded from the succession under two circumstances,i.e. exclusion from the succession for having diverted or concealed the property or forunworthy behaviors vis-a\$-vis the de cujus. The exclusion from the succession could occurboth before and after the death of the de cujus. If the exclusion from the successionoccurs before the death of the de cujus, there are provisions of the law that entitle thesuccession in lien of such heirs by the çdescendenté of the excluded heirs. But if theexclusion from the succession occurs after the death of the de cujus, the çdescendentéof the excluded heirs could also succeed by their right of succession in lien of theexcluded heirs. In the view of the author, there could be a problem of interpretation ofthe law if it is asserted that the çdescendenté of the excluded heirs could succeed bytheir right of succession in lien of the excluded heirs.
6 63
• Publication
Application of the Culture of Thai Literary Art: A Teaching Strategy for Creative Writing
(Chulalongkorn University Printing House, 2008)
University of the Thai Chamber of Commerce. Journal Editorial Office
This paper aims to demonstrate the use of traditional poetry in teaching creative writing.To attract new generation studentsû interest in the beauty of poetic language, as well as toenable them to write creatively, teachers may make use of some traditional forms of poetryfrom Thai literary art and apply them to todayûs social context. The use of Phamee(a poetic riddle) and Klon Nirat (a traveling poem) as teaching media are the strategiesthat make the convention of poetic writing relevant to contemporary contents.
2 52
• Publication
Chemical Composition and Physico-ChemicalProperties of Chinese Water Chestnut(Eleocharisdulcis Trin.) Flour
(Chulalongkorn University Printing House, 2008)
University of the Thai Chamber of Commerce. Journal Editorial Office
Chinese water chestnut (Eleocharis dulcis Trin.) is one of the most popular ingredientsof various Asian cuisines. It is traditionally used as peeled corms. However, the flourprepared from the corms also finds its applications in some Chinese dishes. The objectiveof this study was to investigate the chemical compositions and some physicochemicalproperties of Chinese water chestnut flour. The flour samples of different particle sizes(60-, 80- and 100-mesh) were investigated for their microscopic structure, pH, color(CIE L* a* and b*), swelling power and water solubility. The thermodynamic properties ofthe flour samples were monitored using a Differential Scanning Calorimeter. The flour pastingbehavior was investigated using a Rapid Visco Analyzer (RVA) and, finally, the productionyield of the flour was calculated. It was found that Chinese water chestnut flour sampleswere high in amylose, especially the 100-mesh sample (32.75% amylose). The ScanningElectron Microscopy (SEM) showed that the flour sample with larger particle size wasmore likely to attach and form clusters when compared to that of smaller particle size.The pH values of Chinese water chestnut flour samples were in the range of 5.62-5.66.Chinese water chestnut flour was light yellow while the flour with larger particle size hadlower L* value than that with smaller particle size. Swelling power and water solubility of theflour samples increased with increasing temperature. Moreover, the flour sample with higherstarch content required more energy for gelatinization than that with lower starch content.The 60-mesh flour produced a gel with higher stability upon cooling than the 80- and 100-mesh samples, respectively. The production yield of Chinese water chestnut flour was65.41% on a dry weight basis.
21 418
• Publication
Management of Thai Firms in the New Millennium for Sustainability
(Chulalongkorn University Printing House, 2008)
University of the Thai Chamber of Commerce. Journal Editorial Office
This paper proposes to incorporate His Majesty King Bhumipolûs philosophy ofSufficiency Economy into a new model for doing business as a logical approach to a sustainable and successful business endeavour. The concept of Sufficiency Economy was initially formulated for rural development, mainly in the agricultural sector; but it has gained currency and recognized as a valuablemodel applicable to individuals, groups, business firms, to countries and eveninternational communities.
• Publication
Activity-Based Costing for Improving Calculationof Product Cost of a Printing Company
(Chulalongkorn University Printing House, 2008)
University of the Thai Chamber of Commerce. Journal Editorial Office
Indirect cost allocation to product cost is considered a main problem of calculatingproduct cost. In the past, indirect cost was allocated by a method that resulted in theproduct cost being significantly wrong. Activity-based costing (ABC) is a costingtechnique based on an analysis of the resources actually used to produce a product orprovide a service. This method will calculate the cost of the product more accurately andeffectively. Any enterprise can apply this method. This article shows activity-basedcosting for improving the calculation of product costing of a small printing company.
12 98
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### An HMM example
We now describe a simple problem that we will analyze using the HMM tools. Consider the HMM with the following parameters:
The output of the HMM is a time series with a 16-sample step size (i.e. the state is allowed to change every 16 output samples). The output is Gaussian with mean and variance depending on the state as follows:
State Mean Var 1 0 1 2 0 4 3 2 1
For each 16-sample segment, the sample mean and standard deviation are computed. This constitutes a 2-dimensional feature vector that is the observation space of the HMM.
Baggenstoss 2017-05-19
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# From a circular disc of radius R and mass 9M , a small disc of radius R/3 is removed from the disc. The moment of inertia ….
Q: Free a circular disc of radius R and mass 9M , a small disc of radius R/3 is removed from the disc. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through O is
(a) 4 MR2
(b) $\frac{40}{9}MR^2$
(c) $10 MR^2$
(d) $\frac{37}{9}MR^2$
Ans: (a)
Sol: Mass of removed disc , $M’ = \frac{9 M}{\pi R^2} \times \pi (\frac{R}{3})^2 = M$
Moment of Inertia of complete circular disc about O ,
$\displaystyle I = \frac{9 M R^2}{2}$
Moment of Inertia of removed disc about O will be ,
$\displaystyle I’ = (\frac{M(R/3)^2}{2}) + M (\frac{2R}{3})^2$ (Applying Parallel Axes Theorem)
$\displaystyle I’ = \frac{M R^2}{18} + \frac{4 M R^2}{9}$
$\displaystyle I’ = \frac{M R^2}{2}$
Required Moment of Inertia ,
$\displaystyle I_{req} = I – I’$
$\displaystyle I_{req} = \frac{9 M R^2}{2} – \frac{M R^2}{2} = 4 M R^2$
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# Fake Medallion – PlaidCTF 2021
by goulov and ice and mandlebro
Points: 420 (4 solves)
Given: Medallions are tokens for games at our carnival, selling at $1000 each. Our system makes sure that you cannot forge your own. Can you win some money at the carnival? fake.medallion.pwni.ng 15215, 15214, 15213, 15212, 15211, 15210 fake-medallion.535079809a4644a52ec5721be7e49868cd8e386c844851a58da53b5a345ebc13.tgz # tldr 1. The protocol resembles the Quantum Money scheme, where we are given 30 random qubits (a medallion) from $$\{\left\vert 0\right>, \left\vert 1\right>, \left\vert +\right>, \left\vert -\right>\}$$, and are asked to clone them and sell 15 medallions. 2. The bases and values of the qubits are generated with the Python Mersenne Twister PRNG with the outputs truncated, which is insecure. But recovering the internal state still requires us to clone the 30 qubits once, to get enough bits. 3. Construct a circuit to clone a qubit with high enough fidelity $$(5/6)$$, such that we can successfully duplicate the money once, allowing us to break the PRNG and predict with certainty how the next medallions will be prepared, giving us infinite money and winning the game. # breaking the truncated Mersenne Twister Assume we have enough money to exchange medallions 624 times using the med_for_med option, which costs$1. Everytime we exchange a medallion for another medallion, the server leaks the full quantum state of the previous medallion:
...
return {'msg': f"New medallion {new_id} created. " +
"Your old one was " +
old_medallion + #HERE
". That one is now invalid."}
The state will look something like 011-1100++-000++--0+0-0++--11-, which we can decode by inverting the get_medallion function:
def get_res(qubits):
'''
Inverse of get_medallion function
'''
bits = ''
bases = ''
for i in qubits:
if i == '0':
bits += '0'
bases += '0'
elif i == '1':
bits += '1'
bases += '0'
elif i == '+':
bits += '0'
bases += '1'
elif i == '-':
bits += '1'
bases += '1'
return bits[::-1], bases[::-1]
This will give us the random numbers bits and bases that were generated for that medallion. It’s now well known in the CTF world that if you have 624 consecutive outputs from the Python builtin random generator, you can learn its internal state and predict all “random numbers” that may come next. What may not be so well known is that even if those outputs are partially truncated, you can still find the internal state, given enough additional numbers. To our knowledge, there aren’t many tools online that let you crack Python’s random with truncated output.
So we went ahead and modeled the Mersenne Twister algorithm (MT19937) as a Z3 program:
def symbolic_twist(self, MT, n=624, \
for i in range(n):
xA = LShR(x, 1)
xB = If(x & 1 == 0, xA, xA ^ a)
MT[i] = MT[(i + m) % n] ^ xB
return MT
And implemented a program that given partial outputs from getrandbits, recovers a valid internal state using Z3 SMT solver. You can find it here.
ut = Untwister()
for i in range(624):
out = interact(p, {'option': 'med_for_med', 'med_id': med_id})
med_id = out['msg'].split(' ')[2]
old_med = out['msg'].split(' ')[-6][:-1]
bits, bases = get_res(old_med)
ut.submit(bases + '??') #2 unknown bits because getrandbits(30) truncated 2 lsbs
ut.submit(bits + '??')
#Obtain cloned random
r = ut.get_random()
bases = r.getrandbits(NUM_QUBITS) #predict bases
bits = r.getrandbits(NUM_QUBITS) #predict bits
So, we have to exchange a medallion for another medallion 624 times, but we only have $23 after buying the first medallion. If only we could sell a medallion twice… # (almost) cloning a qubit From here on we’ll assume you have a little knowledge of Quantum Physics/Computation. The protocol in this challenge looks heavily based on the Quantum Money protocol, which holds on the fact that arbitrary quantum states cannot be perfectly copied (known as the No-cloning theorem) to devise a test which will fail if one tries to duplicate the money (i.e. clone the qubits). Intuitively what’s happening is: some entity, let’s call it the bank, creates the money (i.e. prepares the qubits) and keeps a record of its properties (i.e. its state). Then, if you try to clone the money (read qubits) you’re going to need to somehow “mess with it”, and by doing so, you’ll change its properties and you’ll be caught when the bank tests the qubits. So what is this test? Basically, the test checks if the properties of the money are still the same as when it was created. These properties are the basis and eigenvalue of each qubit. Here, the bases are the computational basis (or Z-basis) and the X-basis, and the eigenvalues are $$\{-1,1\}$$, which result in four possibilities for the qubits, $$\{\left|0\right>$$, $$\left|1\right>$$, $$\left|+\right>$$, $$\left|-\right>\}$$. Now, to check if the properties are unaltered, the bank measures each qubit in the basis it was prepared, and compares the eigenvalue with the original one, passing if it matches and aborting if it doesn’t. For this it uses the fact that these two bases are mutually unbiased bases, which implies that: • if a qubit is measured in the same basis that it was prepared, then its outcome will be the same eigenvalue as when it was prepared; • if a qubit is measured in another basis then all outcomes will have an equal probability of occurring. Notice that this algorithm is probabilistic, and even if a wrong basis is used, the measured eigenvalue will still be correct half the times, meaning that there is actually a (3/4) probability of yielding the correct eigenvalue if the measurement is preformed in a random basis. Meaning that for $$n$$ qubits, the probability to randomly pass the test exponentially drops to $$(3/4)^n$$. Coming back to the challenge… The challenge uses IBM Qiskit framework to simulate the quantum computations. The server, which is playing the bank, uses Python’s random modules to sample twice 30 bits, first 30 bases (bitstring bases, bit 0 for the computational basis and bit 1 for the X-basis) and then 30 eigenvalues (bitstring bits, bit 0 for eigenvalue 1 and bit 1 for eigenvalue -1), and prepares the 30 qubits accordingly. These 30 qubits make a medallion, which is the unit of currency of this challenge, and is associated with an ID, that is the AES encryption of the eigenvalues and bases of all 30 qubits. In order to sell a medallion with some ID, the test must pass for the bases and eigenvalues the bank recovers by decrypting the ID (only the bank knows the key). For each money-qubit (q_0) we have access to two auxiliary qubits (ancillas) (q_1, q_2) which the server does not touch. The system is initialized in the separable state $$\left|000\right>$$, and it is prepared by applying to q_0 the NOT gate (X-gate) if the eigenvalue-bit a was 1, and applying the Hadamard gate if the basis-bit b was 1. Again, the q_0 state will be: $$\left|0\right>$$ if a=0, b=0, $$\left|1\right>$$ if a=1, b=0, $$\left|+\right>$$ if a=0, b=1, $$\left|-\right>$$ if a=1, b=1, and it will be separable from the ancilla qubits of which we have total control. We’ll be presenting the circuits (for 1 qubit) one step at a time; this is what the server does before handing the medallion to us: ┌──────────────────────────────┐┌───┐┌───┐ » q_0: ┤0 ├┤ Xᵃ├┤ Hᵇ├─» │ │└───┘└───┘ » q_1: ┤1 initialize(1,0,0,0,0,0,0,0) ├───────────» │ │ » q_2: ┤2 ├───────────» └──────────────────────────────┘ » c0: 1/═══════════════════════════════════════════» » After, the server allows us to perform any arbitrary unitary operation on the entire system, i.e. q_0 and the two ancillary qubits q_1, q_2, or a measurement. Indeed, we know that we won’t be able to clone q_0 to an ancilla because of the No-cloning theorem. However, this doesn’t mean that we cannot make a good enough copy that allows us to win the challenge with high-enough probability. An interesting page on wikipedia about quantum cloning tells us that an arbitrary qubit can be cloned into two identical copies with a fidelity of $$5/6$$ and the circuit (also below) is found in the references (for instance here). So, we get two copies (one in q_0 and another in q_1) which resemble the original state on q_0, and which will pass the test with probability $$5/6$$ each. « ┌───┐┌───┐ » «q_0: ───────────────────────────────────────────────────■────■──┤ X ├┤ X ├─» « ┌──────────┐ ┌───┐┌───────────┐┌─┴─┐ │ └─┬─┘└─┬─┘ » «q_1: ─┤RY(1.1071)├──■───────────────┤ X ├┤RY(0.46365)├┤ X ├──┼────■────┼───» « └──────────┘┌─┴─┐┌───────────┐└─┬─┘└───────────┘└───┘┌─┴─┐ │ » «q_2: ─────────────┤ X ├┤RY(0.72973)├──■────────────────────┤ X ├───────■───» « └───┘└───────────┘ └───┘ » «c0: 1/══════════════════════════════════════════════════════════════════════» « » When we sell our only medallion, the bank measures the 30 qubits in the basis they were prepared in (according to the decryption of ID), and aborts if any measurement gives the wrong eigenvalue. To test our circuit, we use the Aer.get_backend('qasm_simulator') Qiskit backend and run it 1000 times getting back the measurements {'0': 835, '1': 165} for a $$\left|+\right>$$ prepared state which matches the expected $$5/6 = 0.8(3)$$ probability (remember 0 means eigenvalue 1), double-checking the other possible states yields identical results. Be warned that we will fail some times, i.e. only with probability $$(5/6)^{30} ≈ 0.0042$$ we will be able to sell this slightly modified medallion. « ┌───┐┌─┐┌─────────────────┐ » «q_0: ──┤ Hᵇ├┤M├┤ initialize(1,0) ├──» « └───┘└╥┘└─────────────────┘ » «q_1: ────────╫──────────────────────» « ║ » «q_2: ────────╫──────────────────────» « ║ » «c0: 1/════════╩══════════════════════» « 0 » Now we have no medallion left… No problem, we ask to sell the same medallion ID (which correspond to a medallion with 30 qubits with the same bases and eigenvalues as before) and we just swap our ancilla q_1, holding the almost clone of the first medallion, to the q_0 register using the SWAP gate, and sell this copy instead! « » «q_0: ───X───» « │ » «q_1: ───X───» « » «q_2: ───────» « » «c0: 1/═══════» « » Once more, we’ll fail a few times. We can simulate the circuit again and check that ({'0': 832, '1': 168}) we are getting away cheating with probability $$5/6$$ per qubit again. « ┌───┐┌─┐ «q_0: ──┤ Hᵇ├┤M├ « └───┘└╥┘ «q_1: ────────╫─ « ║ «q_2: ────────╫─ « ║ «c0: 1/════════╩═ « 0 Assembling everything (and compiling the circuit into only one unitary to save time communicating with the server using Aer.get_backend('unitary_simulator')), we can now cheat the bank by selling two clones of the same medallion with probability $${(5/6)^{30}}^2 ≈ 1.775\cdot 10^{-5}$$. This means that in 1 in around 60k tries we should be able to cheat the bank twice, for q_0 and q_1. This challenge requires solving a proof-of-work before being able to try, but nothing that spawning a few hundred connections simultaneously doesn’t do in a few hours. # reconstructing the state We sold our cloned medallion twice, and now have$1998 to add to our previous $23, so we have enough money to buy a medallion and exchange it 624 times to have enough consecutive outputs from the Python random generator and recover the Mersenne Twister state. However, we still need$15213 to get the flag. Since we know the internal state of the PRNG, we know exactly what are the bases and eigenvalues of the qubits the upcoming medallion, which we buy. Therefore, we simply sell this same medallion 15 times by constructing its quantum state using the available unitary operations (NOT and Hadamard gates), so all the qubits will be in a correctly verifiable state with probability 1.
def state_reconstruct(c, bases, bits):
''' Copied from challenge '''
for qubit in range(NUM_QUBITS):
measure(c, 0, qubit=qubit)
if bases % 2:
if bits % 2:
# This gives |->
xgate(c, 0, qubit=qubit)
else:
# This gives |+>
else:
if bits % 2:
# This gives |1>
xgate(c, 0, qubit=qubit)
else:
# This gives |0>, cuz why not
xgate(c, 0, qubit=qubit)
xgate(c, 0, qubit=qubit)
bases //= 2
bits //= 2
for _ in range(15):
assert sell_medallion(p, med_id) #Always selling the same medallion
state_reconstruct(p, bases, bits) #We can reset the state because we know the bases and bits
# conclusion
Finally, and after way more attempts than expected (more than 100k instead of 60k), we receive the flag and become one of the only four teams to solve this very interesting challenge, which combined knowledge on multiple fields of modern cryptography.
{'msg': 'Here you go. ', 'flag': "We shan't begin to fathom how you cheated at our raffle game. To attempt to appease you, here is a flag: PCTF{1_D1d_nO7_L13_7O_YoU_R422l3_R34lLy_15_4_5c4M}", 'curr_money': 16382}
The solution code can be found in sol.py.
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# Find all $c\in\mathbb Z^+$ for which $\exists a,b\in\mathbb Z^+, a\neq b$ with $a+c\mid ab$ and $b+c\mid ab$
Find all $c\in\mathbb Z^+$ for which $\exists a,b\in\mathbb Z^+, a\neq b$ with $\begin{cases}a+c\mid ab\\b+c\mid ab\end{cases}$
For those $c$, prove only finitely many $(a,b)$ exist.
My attempt:
Let $(a+c,b+c)=d$. Then $\color{RoyalBlue}{d\mid ab}$ and $\text{lcm}(a+c,b+c)\mid ab\iff \frac{(a+c)(b+c)}{d}\mid ab$.
So $d\neq 1$, because $(a+c)(b+c)\mid ab\,\Rightarrow ab\ge (a+c)(b+c)$, impossible.
$a\equiv -c\equiv b\,\Rightarrow\, a\equiv b$ mod $d$. Then $\color{RoyalBlue}{ab\equiv 0}\equiv a^2\equiv b^2\,\Rightarrow\, d\mid a^2,b^2$.
So $p\mid d\mid a+c,b+c,a^2,b^2\,\Rightarrow\, p\mid a,b,c$, and $(a,b,c)\ge 2$.
• How do you deduce $a^2 \equiv b^2 \equiv 0 \pmod{d}$ in your reasoning? – A.P. May 2 '15 at 14:19
• @A.P. $a\equiv b\implies ab\equiv a^2$ – Hagen von Eitzen May 2 '15 at 14:20
Let $$C=\{\,c\in\mathbb Z^+\mid \exists a,b\in\mathbb Z^+\colon a\ne b\land a+c| ab\land b+c| ab\,\}$$ be the set we are looking for. For $c\ge 2$ we can let $$\tag{\star} a=c^3+c^2-c,\qquad b=c(a-1)=c(c+1)(c^2-1)$$Then certainly $0<a<b$ and we have $$ab = c^2(c^2+c-1)(c+1)(c^2-1)=(a+c)(c^2+c-1)(c^2-1)$$ and $$ab=ac(a-1)=(b+c)(a-1).$$ This shows $c\in C$ for $c\ge 2$. On the other hand $1\notin C$ for if $a+1\mid ab$ then $\gcd(a+1,a)=1$ implies $a+1\mid b$; likewise $b+1\mid ab$ implies $b+1\mid a$. Specifically $b\ge a+1$ and $a\ge b+1$, which is absurd. Therefore we have $$C=\mathbb Z^+\setminus\{1\}.$$
Remains to show that for each $c\in C$ there are only finitely many $(a,b)$. Empirically, it seems that the choice for $a,b$ made in $(\star)$ is the maximal possible choice. Showing that $\max\{a,b\}>c(c+1)(c^2-1)$ is really impossible would of course show finiteness.
• How do my findings imply $c\mid a,b$? I could see that if $c$ were prime, because $p\mid a,b,c\,\Rightarrow\, p\mid c\,\Rightarrow\,p=c$, but you later said we never used the assumption that $c$ is prime. – user236182 May 2 '15 at 15:45
• @user236182 Alright, we did use it there, but with hindsight, once we have found a way to compute suitable $a,b$ from given $c$, the proof that $a+c\mid ab$ and $b+c\mid ab$ does not use that $c$ is prime. - Maybe I should have dropped that paragraph altogether and started right away with the display formula. But that would then feel like unmotivated and out of the blue, wouldn't it? – Hagen von Eitzen May 2 '15 at 15:57
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Challenges
# Find good coalitions
+3
−0
As you might know, there were elections in Germany, and now the parties have to form a government coalition. Let's help them with it!
A good coalition has the strict majority of seats (that is, more seats than the opposition). Moreover you don't want more parties than necessary in a coalition, that is, if you still have a majority after removing one of the parties, it's not a good coalition.
You are given a list of parties and their number of seats. Your task is to output a list of good coalitions, where each coalition is given by a list of its members. For this task, a coalition may also consist of a single party.
Input and output may be in any suitable form (e.g. they need not be lists, but can be sets, dictionaries, XML, …). Note that the order of the parties in each coalition does not matter, nor does the order of coalitions. For example, the output [["nano", "vi"], ["nano", Emacs"], ["vi", "Emacs"]] is equivalent to [["vi", Emacs"], ["Emacs", "nano"], ["nano", "vi"]].
This is code-golf, the shortest code wins.
Test cases:
{"nano": 25}
→ [["nano"]]
{"vi": 20, "Emacs": 21}
→ [["Emacs"]]
{"vi": 20, "Emacs": 20}
→ [["vi", "Emacs"]]
{"nano": 20, "vi": 15, "Emacs": 12}
→ [["nano", "vi"], ["nano", Emacs"], ["vi", "Emacs"]]
{"nano": 25, "vi": 14, "Emacs": 8, "gedit": 6}
→ [["nano", "vi"], ["nano", "Emacs"], ["nano", "gedit"], ["vi", "Emacs", "gedit"]]
{"nano": 16, "vi": 14, "Emacs": 8, "gedit": 6}
→ [["nano", "vi"], ["nano", "Emacs"], ["vi", "Emacs", "gedit"]]
{"nano": 10, "vi": 10, "Emacs": 5, "gedit": 5}
→ [["nano", "vi"], ["nano", "Emacs", "gedit"], ["vi", "Emacs", "gedit]]
Why does this post require moderator attention?
Why should this post be closed?
+3
−0
import Control.Monad
f a=map(map fst)$filter(\c->let{g=map snd;e=sum(g a)div2;f=g c;d=sum f}in d>e&&all((<=e).(d-))f)$filterM(pure[1<0..])a
Try it online!
Explanation: we generate all possible coalitions and then filter out the bad ones. filterM(pure[1<0..])a is the powerset of input.
Why does this post require moderator attention?
+2
−0
# Python 3, 191 bytes
lambda p:[c for c in chain(*[C(p,r)for r in range(len(p)+1)])if all(all(i==(sum(p.values())/2>=sum(p[n]for n in d))for d in C(c,len(c)-i))for i in[0,1])]
from itertools import*;C=combinations
Try it online!
Yay for quadruple nested list compressions...
## Explanation
First we take the powerset:
lambda p:[c for c in chain(*[C(p,r)for r in range(len(p)+1)])
Then we filter.
if all(all(i==(sum(p.values())/2>=sum(p[n]for n in d))for d in C(c,len(c)-i))
for i in[0,1])]
from itertools import*;C=combinations # Imports and aliases
There's a neat little trick: we want to have the set sum greater than s/2 (the average), while having the n-1 sized subsets' sums less than or equal to s/2. By realizing that the full set is equal to the n sized subset, we can combine both of them:
n subset > s/2 and n-1 subset <= s/2
is equivalent to
(n subset <= s/2) == 0 and (n-1 subset <= s/2) == 1
is equivalent to
(n-i subset <= s/2) == i) for i in [0,1]
Why does this post require moderator attention?
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# Small question about derivative
how to derive $\int_0^1 G(t,s) e(s)ds$ with respect to $t$
Where $G(t,s)$ is a Green function and $e:(0,1)\rightarrow \mathbb{R}$ continuous and $e\in L(0,1)$
Please help me
Thank you
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add comment
## 1 Answer
You can commute the derivative and the integral operators in this case. See here.
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$\frac{d}{dt}(\int_0^1 G(t,s)e(s) ds)=\int_0^1 \frac{\partial G(t,s)}{dt} e(s)ds$ it's right ? – Vrouvrou May 16 at 7:20
yes, you are correct – julian fernandez May 16 at 7:39
add comment
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# Joining the definitions of entropy
$\int \frac{Q_{rev}}{T} = \Delta(k_B\ln\Omega)=\Delta S$
Could anyone give some definite proof for this?
I was able to prove that the two definitions of change in entropy are equivalent for an isothermal process carried out on a gas (by quantizing space and then limiting the quantization to infinity), but my proof makes the absolute entropy of the gas infinite. If the process is not isothermal, the particle's velocities come into the picture and I don't know how to deal with that. I tried making various assumptions (quantizing time, etc), but it didn't work. I know that once I prove it for another process, it will be proven for any process carried out on ideal gases(as I can write any process as the combination of isothermal and another process).
Could someone please nudge me in the right direction/give a proof?
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In what framework do you work? If you know define the temperature as $\frac{1}{T}=\frac{\partial S}{\partial U}$, then you can just solve the first law of thermodynamics $\delta Q=PV+dU(S,V)$ for $dS$. – NikolajK Jan 31 '12 at 14:35
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# Lab 2-2: Pluralizer¶
## Lab Requirements and Specifications¶
Two horses. Five monkeys. Twenty flies. One cat. Nine lives. Despite the English language being one of the most commonly used languages around the world, it is also quite complicated to learn if you are not a native speaker. Taking a noun and converting it to its plural form has a dozen rules and even more special cases.
In this lab, you will be creating a program that will take in two inputs: a non-negative number, and a singular noun. If the number entered is 1, then you would just print out “1 horse”. However, if the number is 0 or greater than 1, you would print out the number and the pluralized version of the word - like “0 horses” or “4 horses”.
Your program will be handling the following rules to pluralize nouns:
If it ends in... Rule Example
-fe Replace “fe” with “ves” knife, knives
-y Replace “y” with “ies” family, families
-sh, -ch Add “es” to the end bush, bushes
-us Replace “us” with “i” cactus, cacti
-ay, -oy, -ey, -uy Add “s” to the end
guy, guys
boy, boys
key, keys
day, days
(The above rule is included because it is a special case that conflicts with the first “y” rule.)
All other cases Add “s” to the end cat, cats
Note that in order to handle cases where the word ends in “y” correctly, you will need to take some care. It is important that you order your conditions so that your code will check for the special case endings of “ay”, “oy”, “ey”, and “uy” before simply checking whether a word ends in “y”.
You should name your file FILN_pluralizer.py, where FILN is your first initial and last name, all lowercase, no space.
Although it is a good idea to come up with some of your own test cases to realize the limitations your own code for this lab (and other labs) the following are some inputs you can use to verify that your program works as intended. I would also recommend using the examples in the table above.
Input (#) Input (word) Result 0 dog 0 dogs 1 city 1 city 5 fish 5 fishes 33 fly 33 flies 8 life 8 lives 20 cactus 20 cacti
Note that some words covered by special cases will look incorrect if used in this program. Examples: bus should be buses, not bi / foot should be feet, not foots, etc. It’s important to realize that for now, you want to make sure that the words follow your rules, regardless of whether it is actually an English word or not.
The following space is provided in case you want to test code out or write it in the browser:
## Taking it Further¶
You can extend the functionality of your program by including all possible rules and special cases. Performing a google search on “plural nouns rules” will provide numerous resources on the various rules and exceptions in making a noun plural. See how many you can incorporate, then challenge your peers or teachers to stump your program!
Next Section - Lesson 03-01: Categories of Errors
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# zbMATH — the first resource for mathematics
A shrinking projection method for generalized mixed equilibrium problems, variational inclusion problems and a finite family of quasi-nonexpansive mappings. (English) Zbl 1206.47077
Summary: The purpose of this paper is to consider a shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of fixed points of a finite family of quasi-nonexpansive mappings, and the set of solutions of variational inclusion problems. Then, we prove a strong convergence theorem of the iterative sequence generated by the shrinking projection method under some suitable conditions in a real Hilbert space. Our results improve and extend recent results announced by Peng et al. (2008), Takahashi et al. (2008), S. Takahashi and W. Takahashi (2008), and many others.
##### MSC:
47J25 Iterative procedures involving nonlinear operators 47J22 Variational and other types of inclusions 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
Full Text:
##### References:
[1] Brézis, H, Opérateur maximaux monotones, No. 5, (1973), Amsterdam, The Netherlands [2] Lemaire, B, Which fixed point does the iteration method select?, No. 452, 154-167, (1997), Berlin, Germany · Zbl 0882.65042 [3] Blum, E; Oettli, W, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63, 123-145, (1994) · Zbl 0888.49007 [4] Chadli, O; Schaible, S; Yao, JC, Regularized equilibrium problems with application to noncoercive hemivariational inequalities, Journal of Optimization Theory and Applications, 121, 571-596, (2004) · Zbl 1107.91067 [5] Chadli, O; Wong, NC; Yao, JC, Equilibrium problems with applications to eigenvalue problems, Journal of Optimization Theory and Applications, 117, 245-266, (2003) · Zbl 1141.49306 [6] Konnov, IV; Schaible, S; Yao, JC, Combined relaxation method for mixed equilibrium problems, Journal of Optimization Theory and Applications, 126, 309-322, (2005) · Zbl 1110.49028 [7] Moudafi, A; Théra, M, Proximal and dynamical approaches to equilibrium problems, No. 477, 187-201, (1999), Berlin, Germany · Zbl 0944.65080 [8] Zeng, L-C; Wu, S-Y; Yao, J-C, Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems, Taiwanese Journal of Mathematics, 10, 1497-1514, (2006) · Zbl 1121.49005 [9] Ceng, LC; Sahu, DR; Yao, JC, Implicit iterative algorithms for asymptotically nonexpansive mappings in the intermediate sense and Lipschitz-continuous monotone mappings, Journal of Computational and Applied Mathematics, 233, 2902-2915, (2010) · Zbl 1188.65076 [10] Ceng, L-C; Yao, J-C, A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem, Nonlinear Analysis: Theory, Methods & Applications, 72, 1922-1937, (2010) · Zbl 1179.49003 [11] Ceng, LC; Petruşel, A; Yao, JC, Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings, Journal of Optimization Theory and Applications, 143, 37-58, (2009) · Zbl 1188.90256 [12] Cianciaruso, F; Marino, G; Muglia, L; Yao, Y, A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem, No. 2010, 19, (2010) · Zbl 1203.47043 [13] Chadli, O; Liu, Z; Yao, JC, Applications of equilibrium problems to a class of noncoercive variational inequalities, Journal of Optimization Theory and Applications, 132, 89-110, (2007) · Zbl 1158.35050 [14] Cholamjiak, P; Suantai, S, A new hybrid algorithm for variational inclusions, generalized equilibrium problems, and a finite family of quasi-nonexpansive mappings, No. 2009, 20, (2009) · Zbl 1186.47060 [15] Jaiboon, C; Kumam, P, A general iterative method for addressing mixed equilibrium problems and optimization problems, Nonlinear Analysis, Theory, Methods and Applications, 73, 1180-1202, (2010) · Zbl 1205.49011 [16] Jaiboon, C; Kumam, P, Strong convergence for generalized equilibrium problems, fixed point problems and relaxed cocoercive variational inequalities, No. 2010, 43, (2010) · Zbl 1187.47048 [17] Jaiboon, C; Chantarangsi, W; Kumam, P, A convergence theorem based on a hybrid relaxed extragradient method for generalized equilibrium problems and fixed point problems of a finite family of nonexpansive mappings, Nonlinear Analysis: Hybrid Systems, 4, 199-215, (2010) · Zbl 1179.49011 [18] Kumam, P; Jaiboon, C, A new hybrid iterative method for mixed equilibrium problems and variational inequality problem for relaxed cocoercive mappings with application to optimization problems, Nonlinear Analysis: Hybrid Systems, 3, 510-530, (2009) · Zbl 1221.49010 [19] Peng, J-W; Yao, J-C, A new hybrid-extragradient method for generalized mixed equilibrium problems, fixed point problems and variational inequality problems, Taiwanese Journal of Mathematics, 12, 1401-1432, (2008) · Zbl 1185.47079 [20] Peng, J-W; Yao, J-C, Some new iterative algorithms for generalized mixed equilibrium problems with strict pseudo-contractions and monotone mappings, Taiwanese Journal of Mathematics, 13, 1537-1582, (2009) · Zbl 1193.47068 [21] Yao, Y; Liou, Y-C; Yao, J-C, A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems, No. 2008, 15, (2008) · Zbl 1203.47087 [22] Yao, Y; Liou, Y-C; Wu, Y-J, An extragradient method for mixed equilibrium problems and fixed point problems, No. 2009, 15, (2009) · Zbl 1203.47086 [23] Takahashi, W; Takeuchi, Y; Kubota, R, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications, 341, 276-286, (2008) · Zbl 1134.47052 [24] Takahashi, S; Takahashi, W, Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space, Nonlinear Analysis: Theory, Methods & Applications, 69, 1025-1033, (2008) · Zbl 1142.47350 [25] Zhang, S-S; Lee, JHW; Chan, CK, Algorithms of common solutions to quasi variational inclusion and fixed point problems, Applied Mathematics and Mechanics. English Edition, 29, 571-581, (2008) · Zbl 1196.47047 [26] Peng, J-W; Wang, Y; Shyu, DS; Yao, J-C, Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems, No. 2008, 15, (2008) · Zbl 1161.65050 [27] Kumam, P; Katchang, P, A general iterative method of fixed points for mixed equilibrium problems and variational inclusion problems, No. 2010, 25, (2010) · Zbl 1189.47066 [28] Acedo, GL; Xu, H-K, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Analysis: Theory, Methods & Applications, 67, 2258-2271, (2007) · Zbl 1133.47050 [29] Opial, Z, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bulletin of the American Mathematical Society, 73, 591-597, (1967) · Zbl 0179.19902 [30] Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan; 2000. · Zbl 0997.47002 [31] Peng, J-W; Liou, YC; Yao, JC, An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions, No. 2009, 21, (2009) · Zbl 1163.91463 [32] Atsushiba, S; Takahashi, W, Strong convergence theorems for a finite family of nonexpansive mappings and applications, Indian Journal of Mathematics, 41, 435-453, (1999) · Zbl 1055.47514
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.
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# Re: Integral involving square root of e^x
1. Mar 30, 2008
### Schrodinger's Dog
[SOLVED] Re: Integral involving square root of e^x
1. The problem statement, all variables and given/known data
$$\int \sqrt{1-e^{-x}}$$
2. Relevant equations
Sub rule.
3. The attempt at a solution
I realised that it's fairly obvious I can use u=e^-x/2 to give $\sqrt {1-u^2}$
but I'm kind of looking at the answers and I'm not seeing how I can get to them from here?
The answer I have from my maths program looks a bit awkward? Not sure how they got here?
$$2(\sqrt{1-e^x})-2\tanh^{-1}(\sqrt{1-e^x})+c$$
Are there any other useful subs anyone can think of, or tips for this one.
Not a homework question as such but I thought this was the best place for it.
2. Mar 30, 2008
### cristo
Staff Emeritus
Well, remember that the actual interview you are trying to determine is
$$\int \sqrt{1-e^{-x}}dx$$. Performing the substitution $u=e^{-x/2}$ gives $du=-\frac{1}{2}e^{-x/2}\cdot dx$, yielding$$-2\int\frac{\sqrt{1-u^2}}{u}\cdot du$$. You can integrate this by another subsitution, u=cosy, say.
Last edited: Mar 30, 2008
3. Mar 30, 2008
### Schrodinger's Dog
Thanks Christo I'm still not sure here. I'm sure I'm just being thick but could you spell it out, I'm not seeing how I'm going to end up with something even remotely like that answer, even with your sub? It's probably the hyperbolic function, I've got to admit they weren't heavily dealt with in the course I did. I can see where the answer turns into what it is with the sub cos y but the arctanh, where does that come from exactly?
Last edited: Mar 30, 2008
4. Mar 30, 2008
### Schrodinger's Dog
Thanks Christo I'm still not sure here. I'm sure I'm just being thick but could you spell it out, I'm not seeing how I'm going to end up like that answer, even with your sub? It's probably the hyperbolic function, I've got to admit they weren't heavily dealt with in the course I did. I can see where the answer turns into what it is with the sub cos y but the arctanh, where does that come from exactly? Is it some clever trig identity I'm missing
I can see where you get
$$2(1-e^x)-2tanh^{-1}(1-e^x)+c$$
from
$$-2\int\frac{\sqrt{1-u^2}}{u}\cdot du$$ but why the arctanh?
Last edited: Mar 30, 2008
5. Mar 30, 2008
### cristo
Staff Emeritus
Hmm, I don't really know to be honest. My substitution doesn't really work, since the fraction has a "u" in the denominator and not a "u^2," which is what I must have assumed.
I guess we'll have to wait for someone else to come along, as I can't see how to integrate that by hand, at the moment.
6. Mar 30, 2008
### tiny-tim
Hi Schrodinger's Dog!
Put u = sechv, du = -sechvtanhvdv.
Then ∫(√(1 - u^2))/udu = -∫(tanh^2(v)sechv/sechv)dy
= ∫(sech^2(v) - 1)dv
= tanhv - v
= √(1 - u^2)) - tanh^-1√(1 - u^2)
= √(1 - e^-x) - tanh^-1√(1 - e^-x).
(hmm … might have been quicker to start by putting e^(-x/2) = sechv )
7. Mar 30, 2008
### Schrodinger's Dog
Thanks tiny tim, when I get a spare moment I'll go through that thoroughly.
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makeCPO creates a Feature Operation CPOConstructor, i.e. a constructor for a CPO that will operate on feature columns. makeCPOTargetOp creates a Target Operation CPOConstructor, which creates CPOs that operate on the target column. makeCPORetrafoless creates a Retrafoless CPOConstructor, which creates CPOs that may operate on both feature and target columns, but have no retrafo operation. See OperatingType for further details on the distinction of these. makeCPOExtendedTrafo creates a Feature Operation CPOConstructor that has slightly more flexibility in its data transformation behaviour than makeCPO (but is otherwise identical). makeCPOExtendedTargetOp creates a Target Operation CPOConstructor that has slightly more flexibility in its data transformation behaviour than makeCPOTargetOp but is otherwise identical.
See example section for some simple custom CPO.
makeCPO(cpo.name, par.set = makeParamSet(), par.vals = NULL,
dataformat = c("df.features", "split", "df.all", "task", "factor",
"ordered", "numeric"), dataformat.factor.with.ordered = TRUE,
export.params = TRUE, fix.factors = FALSE,
properties.data = c("numerics", "factors", "ordered", "missings"),
properties.adding = character(0), properties.needed = character(0),
properties.target = c("cluster", "classif", "multilabel", "regr",
"surv", "oneclass", "twoclass", "multiclass"), packages = character(0),
cpo.train, cpo.retrafo)
makeCPOExtendedTrafo(cpo.name, par.set = makeParamSet(),
par.vals = NULL, dataformat = c("df.features", "split", "df.all",
dataformat.factor.with.ordered = TRUE, export.params = TRUE,
fix.factors = FALSE, properties.data = c("numerics", "factors",
properties.needed = character(0), properties.target = c("cluster",
"classif", "multilabel", "regr", "surv", "oneclass", "twoclass",
"multiclass"), packages = character(0), cpo.trafo, cpo.retrafo)
makeCPORetrafoless(cpo.name, par.set = makeParamSet(), par.vals = NULL,
dataformat.factor.with.ordered = TRUE, export.params = TRUE,
fix.factors = FALSE, properties.data = c("numerics", "factors",
properties.needed = character(0), properties.target = c("cluster",
"classif", "multilabel", "regr", "surv", "oneclass", "twoclass",
"multiclass"), packages = character(0), cpo.trafo)
makeCPOTargetOp(cpo.name, par.set = makeParamSet(), par.vals = NULL,
dataformat = c("df.features", "split", "df.all", "task", "factor",
"ordered", "numeric"), dataformat.factor.with.ordered = TRUE,
export.params = TRUE, fix.factors = FALSE,
properties.data = c("numerics", "factors", "ordered", "missings"),
properties.adding = character(0), properties.needed = character(0),
properties.target = "cluster", task.type.out = NULL,
predict.type.map = c(response = "response"), packages = character(0),
constant.invert = FALSE, cpo.train, cpo.retrafo, cpo.train.invert,
cpo.invert)
makeCPOExtendedTargetOp(cpo.name, par.set = makeParamSet(),
par.vals = NULL, dataformat = c("df.features", "split", "df.all",
dataformat.factor.with.ordered = TRUE, export.params = TRUE,
fix.factors = FALSE, properties.data = c("numerics", "factors",
properties.needed = character(0), properties.target = "cluster",
task.type.out = NULL, predict.type.map = c(response = "response"),
packages = character(0), constant.invert = FALSE, cpo.trafo,
cpo.retrafo, cpo.invert)
## Arguments
cpo.name
[character(1)]
The name of the resulting CPOConstructor / CPO. This is used for identification in output, and as the default id.
par.set
[ParamSet]
Optional parameter set, for configuration of CPOs during construction or by hyperparameters. Default is an empty ParamSet. It is recommended to use pSS to construct this, as it greatly reduces the verbosity of creating a ParamSet and makes it more readable.
par.vals
[list | NULL]
Named list of default parameter values for the CPO. These are used instead of the parameter default values in par.set, if not NULL. It is preferred to use ParamSet default values, and not par.vals. Default is NULL.
dataformat
[character(1)]
Indicate what format the data should be as seen by the cpo.train and cpo.retrafo function. The following table shows what values of dataformat lead to what is given to cpo.train and cpo.retrafo as data and target parameter value. (Note that for Feature Operating CPOs, cpo.retrafo has no target argument.) Possibilities are:
dataformat data target “df.all” data.frame with target cols target colnames “df.features” data.frame without target data.frame of target “task” full Task target colnames “split” list of data.frames by type data.frame of target [type] data.frame of [type] feats only data.frame of target
[type] can be any one of “factor”, “numeric”, “ordered”; if these are given, only a subset of the total data present is seen by the CPO.
Note that makeCPORetrafoless accepts only “task” and “df.all”.
For dataformat == "split", cpo.train and cpo.retrafo get a list with entries “factor”, “numeric”, “other”, and, if dataformat.factor.with.ordered is FALSE, “ordered”.
If the CPO is a Feature Operation CPO, then the return value of the cpo.retrafo function must be in the same format as the one requested. E.g. if dataformat is “split”, the return value must be a named list with entries $numeric, $factor, and $other. The types of the returned data may be arbitrary: In the given example, the $factor slot of the returned list may contain numeric data. (Note however that if data is returned that has a type not already present in the data, properties.needed must specify this.)
For Feature Operating CPOs, if dataformat is either “df.all” or “task”, the target column(s) in the returned value of the retrafo function must be identical with the target column(s) given as input.
If dataformat is “split”, the $numeric slot of the value returned by the cpo.retrafo function may also be a matrix. If dataformat is “numeric”, the returned object may also be a matrix. Default is “df.features” for all functions except makeCPORetrafoless, for which it is “df.all”. dataformat.factor.with.ordered [logical(1)] Whether to treat ordered typed features as factor typed features. This affects how dataformat is handled, for which it only has an effect if dataformat is “split” or “factor”. If dataformat is “ordered”, this must be FALSE. It also affects how strictly data fed to a CPORetrafo object is checked for adherence to the data format of data given to the generating CPO. Default is TRUE. export.params [logical(1) | character] Indicates which CPO parameters are exported by default. Exported parameters can be changed after construction using setHyperPars, but exporting too many parameters may lead to messy parameter sets if many CPOs are combined using composeCPO or %>>%. The exported parameters can be set during construction, but export.params determines the default exported parameters. If this is a logical(1), TRUE exports all parameters, FALSE to exports no parameters. It may also be a character, indicating the names of parameters to be exported. Default is TRUE. fix.factors [logical(1)] Whether to constrain factor levels of new data to the levels of training data, for each factorial or ordered column. If new data contains factors that were not present in training data, the values are set to NA. Default is FALSE. properties.data [character] The kind if data that the CPO will be able to handle. This can be one or more of: “numerics”, “factors”, “ordered”, “missings”. There should be a bias towards including properties. If a property is absent, the preproc operator will reject the data. If an operation e.g. only works on numeric columns that have no missings (like PCA), it is recommended to give all properties, ignore the columns that are not numeric (using dataformat = "numeric"), and giving an error when there are missings in the numeric columns (since missings in factorial features are not a problem). Defaults to the maximal set. properties.adding [character] Can be one or many of the same values as properties.data for Feature Operation CPOs, and one or many of the same values as properties.target for Target Operation CPOs. These properties get added to a Learner (or CPO) coming after / behind this CPO. When a CPO imputes missing values, for example, this should be “missings”. This must be a subset of “properties.data” or “properties.target”. Note that this may not contain a Task-type property, even if the CPO is a Target Operation CPO that performs conversion. Property names may be postfixed with “.sometimes”, to indicate that adherence should not be checked internally. This distinction is made by not putting them in the $adding.min slot of the getCPOProperties return value when get.internal = TRUE.
Default is character(0).
properties.needed
[character]
Can be one or many of the same values as properties.data for Feature Operation CPOs, and one or many of the same values as properties.target. These properties are required from a Learner (or CPO) coming after / behind this CPO. E.g., when a CPO converts factors to numerics, this should be “numerics” (and properties.adding should be “factors”).
Note that this may not contain a Task-type property, even if the CPO is a Target Operation CPO that performs conversion.
Property names may be postfixed with “.sometimes”, to indicate that adherence should not be checked internally. This distinction is made by not putting them in the $needed slot of properties. They can still be found in the $needed.max slot of the getCPOProperties return value when get.internal = TRUE.
Default is character(0).
properties.target
[character]
For Feature Operation CPOs, this can be one or more of “cluster”, “classif”, “multilabel”, “regr”, “surv”, “oneclass”, “twoclass”, “multiclass”. Just as properties.data, it indicates what kind of data a CPO can work with. To handle data given as data.frame, the “cluster” property is needed. Default is the maximal set.
For Target Operation CPOs, this must contain exactly one of “cluster”, “classif”, “multilabel”, “regr”, “surv”. This indicates the type of Task the CPO can work on. If the input is a data.frame, it is treated as a “cluster” type Task. If the properties.target contains “classif”, the value must then also contain one or more of “oneclass”, “twoclass”, or “multiclass”. Default is “cluster”.
packages
[character]
Package(s) that should be loaded when the CPO is constructed. This gives the user an error if a package required for the CPO is not available on his system, or can not be loaded. Default is character(0).
cpo.train
[function | NULL]
This is a function which must have the parameters data and target, as well as the parameters specified in par.set. (Alternatively, the function may have only some of these arguments and a dotdotdot argument). It is called whenever a CPO is applied to a data set to prepare for transformation of the training and prediction data. Note that this function is only used in Feature Operating CPOs created with makeCPO, and in Target Operating CPOs created with makeCPOExtendedTargetOp.
The behaviour of this function differs slightly in Feature Operation and Target Operation CPOs.
For Feature Operation CPOs, if cpo.retrafo is NULL, this is a constructor function which must return a “retrafo” function which will then modify (possibly new unseen) data. This retrafo function must have exactly one argument--the (new) data--and return the modified data. The format of the argument, and of the return value of the retrafo function, depends on the value of the dataformat parameter, see documentation there.
If cpo.retrafo is not NULL, this is a function which must return a control object. This control object returned by cpo.train will then be given as the control argument of the cpo.retrafo function, along with (possibly new unseen) data to manipulate.
For Target Operation CPOs, if cpo.retrafo is NULL, cpo.train.invert (or cpo.invert if constant.invert is TRUE) must likewise be NULL. In that case cpo.train's return value is ignored and it must define, within its namespace, two functions cpo.retrafo and cpo.train.invert (or cpo.invert if constant.invert is TRUE) which will take the place of the respective functions. cpo.retrafo must take the parameters data and target, and return the modified target target (or data, depending on dataformat) data. cpo.train.invert must take a data and control argument and return either a modified control object, or a cpo.invert function. cpo.invert must have a target and predict.type argument and return the modified target data.
If cpo.retrafo is not NULL, cpo.train.invert (or cpo.invert if constant.invert is TRUE) must likewise be non-NULL. In that case, cpo.train must return a control object. This control object will then be given as the control argument of both cpo.retrafo and cpo.train.invert (or the control.invert argument of cpo.invert if constant.invert is TRUE).
This parameter may be NULL, resulting in a so-called stateless CPO. For Target Operation CPOs created with makeCPOTargetOp, constant.invert must be TRUE in this case. A stateless CPO does the same transformation for initial CPO application and subsequent prediction data transformation (e.g. taking the logarithm of numerical columns). Note that cpo.retrafo and cpo.invert should not have a control argument in a stateless CPO.
cpo.retrafo
[function | NULL]
This is a function which must have the parameters data, target (Target Operation CPOs only) and control, as well as the parameters specified in par.set. (Alternatively, the function may have only some of these arguments and a dotdotdot argument). In Feature Operation CPOs created with makeCPO, if cpo.train is NULL, the control argument must be absent.
This function gets called during the “retransformation” step where prediction data is given to the CPORetrafo object before it is given to a fitted machine learning model for prediction. In makeCPO Featore Operation CPOs and makeCPOTargetOp Target Operation CPOs, this is also called during the first trafo step, where the CPO object is applied to training data.
In Feature Operation CPOs, this function receives the data to be transformed and must return the transformed data in the same format as it received them. The format of data is the same as the format in cpo.train and cpo.trafo, with the exception that if dataformat is “task” or “df.all”, the behaviour here is as if “df.split” had been given.
In Target Operation CPOs created with makeCPOTargetOp, this function receives the data and target to be transformed and must return the transformed target. The input format of these parameters depends on dataformat. If dataformat is “task” or “df.all”, the returned value must be the modified Task / data.frame with the feature columns not modified. Otherwise, the target values to be modified are in the target parameter, and the return value must be a data.frame of the modified target values only.
In Target Operation CPOs created with makeCPOExtendedTargetOp, this function is called during the retrafo step, and it must create a control.invert object in its environment to be used in the inversion step, as well as return the modified target data.The format of the data given to cpo.retrafo in Target Operation CPOs created with makeCPOExtendedTargetOp is the same as in other functions, with the exception that, if dataformat is “df.all” or “task”, the full data.frame or Task will be given as the target parameter, while the data parameter will behave as if dataformat “df.split”. Depending on what object the CPORetrafo object was applied to, the target argument may be NULL; in that case NULL must also be returned by the function.
If cpo.invert is NULL, cpo.retrafo should create a cpo.invert function in its environment instead of creating the control object; this function should then take the target and predict.type arguments. If constant.invert is TRUE, this function does not need to define the control.invert or cpo.invert variables, they are instead taken from cpo.trafo.
cpo.trafo
[function]
This is a function which must have the parameters data and target, as well as the parameters specified in par.set. (Alternatively, the function may have only some of these arguments and a dotdotdot argument). It is called whenever a CPO is applied to a data set to transform the training data, and (except for Retrafoless CPOs) to collect a control object used by other transformation functions. Note that this function is not used in makeCPO.
This functions primary task is to transform the given data when the CPO gets applied to training data. For Target Operating CPOs (created with makeCPOExtendedTargetOp(!)), it must return the complete transformed target column(s), unless dataformat is “df.all” (in which case the complete, modified, data.frame must be returned) or “task” (in which case the complete, modified, Task must be returned). It must furthermore create the control objects for cpo.retrafo and cpo.invert, or create these functins themselves, and save them in its function environment (see below). For Retrafoless CPOs (created with makeCPORetrafoless) and Feature Operation CPOs (created with makeCPOExtendedTrafo(!)), it must return the data in the same format as received it in its data argument (depending on dataformat). If dataformat is a df.all or task, this means the target column(s) contained in the data.frame or Task returned must not be modified.
For CPOs that are not Retrafoless, a unit of information to be carried over to the retrafo step needs to be created inside the cpo.trafo function. This unit of information is a variable that must be defined inside the environment of the cpo.trafo function and will be retrieved by the CPO framework.
If cpo.retrafo is not NULL the unit is an object named “control” that will be passed on as the control argument to the cpo.retrafo function. If cpo.retrafo is NULL, the unit is a function, called “cpo.retrafo”, that will be used instead of the cpo.retrafo function passed over to makeCPOExtendedTargetOp / makeCPOExtendedTrafo. It must behave the same as the function it replaces, but has only the data (and target, for Target Operation CPOs) argument.
For Target Operation CPOs created with makeCPOExtendedTargetOp, another unit of information to be used by cpo.invert must be used. The options here are similar to cpo.retrafo: Either a control object, named control.invert, is created, or the cpo.invert function itself is given (and cpo.invert in the makeCPOExtendedTargetOp call is set to NULL), with the target and predict.type arguments.
[character(1) | NULL]
If Task conversion is to take place, this is the output task that the data should be converted to. Note that the CPO framework takes care of the conversion if dataformat is not “task”, but the target column needs to have the proper format for that.
If this is NULL, Tasks will not be converted. Default is NULL.
predict.type.map
[character | list]
This becomes the CPO's predict.type, explained in detail in PredictType.
In short, the predict.type.map is a character vector, or a list of character(1), with names according to the predict types predict can request in its predict.type argument when the created CPO was used as part of a CPOLearner to create the model under consideration. The values of predict.type.map are the predict.type that will be requested from the underlying Learner for prediction.
predict.type.map thus determines the format that the target parameter of cpo.invert can take: It is the format according to predict.type.map[predict.type], where predict.type is the respective cpo.invert parameter.
constant.invert
[logical(1)]
Whether the cpo.invert step should not have information from the previous cpo.retrafo or cpo.train.invert step in Target Operation CPOs (makeCPOTargetOp or makeCPOExtendedTargetOp).
For makeCPOTargetOp, if this is TRUE, the cpo.train.invert argument must be NULL. If cpo.retrafo and cpo.invert are given, the same control object is given to both of them. Otherwise, if cpo.retrafo and cpo.invert are NULL, the cpo.train function must return NULL and define a cpo.retrafo and cpo.invert function in its namespace (see cpo.train documentation for more details). If constant.invert is FALSE, cpo.train may either return a control object that will then be given to cpo.train.invert, or define a cpo.retrafo and cpo.train.invert function in its namespace.
For makeCPOExtendedTargetOp, if this is TRUE, cpo.retrafo does not need to generate a control.invert object. The control.invert object created in cpo.trafo will then always be given to cpo.invert for all data sets.
Default is FALSE.
cpo.train.invert
This is a function which must have the parameters data, and control, as well as the parameters specified in par.set. (Alternatively, the function may have only some of these arguments and a dotdotdot argument).
This function receives the feature columns given for prediction, and must return a control object that will be passed on to the cpo.invert function, or it must return a function that will be treated as the cpo.invert function if the cpo.invert argument is NULL. In the latter case, the returned function takes exactly two arguments (the prediction column to be inverted, and predict.type), and otherwise behaves identically to cpo.invert.
If constant.invert is TRUE, this must be NULL.
cpo.invert
[function | NULL]
This is a function which must have the parameters target (a data.frame containing the columns of a prediction made), control.invert, and predict.type, as well as the parameters specified in par.set. (Alternatively, the function may have only some of these arguments and a dotdotdot argument).
The predict.type requested by the predict or invert call is given as a character(1) in the predict.type argument. Note that this is not necessarily the predict.type of the prediction made and given as target argument, depending on the value of predict.type.map (see there).
This function performs the inversion for a Target Operation CPO. It takes a control object, which summarizes information from the training and retrafo step, and the prediction as returned by a machine learning model, and undoes the operation done to the target column in the cpo.trafo function.
For example, if the trafo step consisted of taking the logarithm of a regression target, the cpo.invert function could return the exponentiated prediction values by taking the exp of the only column in the target data.frame and returning the result of that. This kind of operation does not need the cpo.retrafo step and should have skip.retrafo set to TRUE.
As a more elaborate example, a CPO could train a model on the training data and set the target values to the residues of that trained model. The cpo.retrafo function would then make predictions with that model on the new prediction data and save the result to the control object. The cpo.invert function would then add these predictions to the predictions given to it in the target argument to “invert” the antecedent subtraction of model predictions from target values when taking the residues.
## Value
[CPOConstructor]. A Constructor for CPOs.
## CPO Internals
The mlrCPO package offers a powerful framework for handling the tasks necessary for preprocessing, so that the user, when creating custom CPOs, can focus on the actual data transformations to perform. It is, however, useful to understand what it is that the framework does, and how the process can be influenced by the user during CPO definition or application. Aspects of preprocessing that the user needs to influence are:
Operating Type
The core of preprocessing is the actual transformation being performed. In the most general sense, there are three points in a machine learning pipeline that preprocessing can influence.
1. Transformation of training data before model fitting, done in mlr using train. In the CPO framework (when not using a CPOLearner which makes all of these steps transparent to the user), this is done by a CPO.
2. transformation of new validation or prediction data that is given to the fitted model for prediction, done using predict. This is done by a CPORetrafo retrieved using retrafo from the result of step 1.
3. transformation of the predictions made to invert the transformation of the target values done in step 1, which is done using the CPOInverter retrieved using inverter from the result of step 2.
The framework poses restrictions on primitive (i.e. not compound using composeCPO) CPOs to simplify internal operation: A CPO may be one of three OperatingTypes (see there). The Feature Operation CPO does not transform target columns and hence only needs to be involved in steps 1 and 2. The Target Operation CPO only transforms target columns, and therefore mostly concerns itself with steps 1 and 3. A Retrafoless CPO may change both feature and target columns, but may not perform a retrafo or inverter operation (and is therefore only concerned with step 1). Note that this is effectively a restriction on what kind of transformation a Retrafoless CPO may perform: it must not be a transformation of the data or target space, it may only act or subtract points within this space. The Operating Type of a CPO is ultimately dependent on the function that was used to create the CPOConstructor: makeCPO / makeCPOExtendedTrafo, makeCPOTargetOp / makeCPOExtendedTargetOp, or makeCPORetrafoless.
Data Transformation
At the core of a CPO is the modification of data it performs. For Feature Operation CPOs, the transformation of each row, during training and prediction, should happen in the same way, and it may only depend on the entirety of the training data--i.e. the value of a data row in a prediction data set may not influence the transformation of a different prediction data row. Furthermore, if a data row occurs in both training and prediction data, its transformation result should ideally be the same. This property is ensured by makeCPO by splitting the transformation into two functions: One function that collects all relevant information from the training data (called cpo.train), and one that transforms given data, using this collected information and (potentially new, unseen) data to be transformed (called cpo.retrafo). The cpo.retrafo function should handle all data as if it were prediction data and unrelated to the data given to cpo.train. Internally, when a CPO gets applied to a data set using applyCPO, the cpo.train function is called, and the resulting control object is used for a subsequent cpo.retrafo call which transforms the data. Before the result is given back from the applyCPO call, the control object is used to create a CPORetrafo object, which is attached to the result as attribute. Target Operating CPOs additionally create and add a CPOInverter object. When a CPORetrafo is then applied to new prediction data, the control object previously returned by cpo.train is given, combined with this new data, to another cpo.retrafo call that performs the new transformation. makeCPOExtendedTrafo gives more flexibility by having calling only the cpo.trafo in the training step, which both creates a control object and modifies the data. This can increase performance if the underlying operation creates a control object and the transformed data in one step, as for example PCA does. Note that the requirement that the same row in training and prediction data should result in the same transformation result still stands. The cpo.trafo function returns the transformed data and creates a local variable with the control information, which the CPO framework will access.
Inversion
If a CPO performs transformations of the target column, the predictions made by a following machine learning process should ideally have this transformation undone, so that if the process makes a prediction that coincides with a target value after the transformation, the whole pipeline should return a prediction that equals to the target value before this transformation. This is done by the cpo.invert function given to makeCPOTargetOp. It has access to information from both the preceding training and prediction steps. During the training step, cpo.train createas a control object that is not only given to cpo.retrafo, but also to cpo.train.invert. This latter function is called before the prediction step, whenever new data is fed to the machine learning process. It takes the new data and the old control object and transforms it to a new control.invert object to include information about the prediction data. This object is then given to cpo.invert. It is possible to have Target Operation CPOs that do not require information from the retrafo step. This is specified by setting constant.invert to TRUE. It has the advantage that the same CPOInverter can be used for inversion of predictions made with any new data. Otherwise, a new CPOInverter object must be obtained for each new data set after the retrafo step (using the inverter function on the retrafo result). Having constant.invert set to TRUE results in hybrid retrafo / inverter objects: The CPORetrafo object can then also be used for inversions. When defining a constant.invert Target Operating CPO, no cpo.train.invert function is given, and the same control object is given to both cpo.retrafo and cpo.invert. makeCPOExtendedTargetOp gives more flexibility and allows more efficient implementation of Target Operating CPOs at cost of more complexity. With this method, a cpo.trafo function is given that is executed during the first training step; It must return the transformed target column, as well as a control and control.invert object. The cpo.retrafo function not only transforms the target, but must also create a new control.invert object (unless constant.invert is TRUE). The semantics of cpo.invert is identical with the basic makeCPOTargetOp.
cpo.train-cpo.retrafo information transfer
One possibility to transfer information from cpo.train to cpo.retrafo is to have cpo.train return a control object (a list) that is then given to cpo.retrafo. The CPO is then called an object based CPO. Another possibility is to not give the cpo.retrafo argument (set it to NULL in the makeCPO call) and have cpo.train instead return a function instead. This function is then used as the cpo.retrafo function, and should have access to all relevant information about the training data as a closure. This is called functional CPO. To save memory, the actual data (including target) given to cpo.train is removed from the environment of its return value in this case (i.e. the environment of the cpo.retrafo function). This means the cpo.retrafo function may not reference a “data” variable. There are similar possibilities of functional information transfer for other types of CPOs: cpo.trafo in makeCPOExtendedTargetOp may create a cpo.retrafo function instead of a control object. cpo.train in makeCPOTargetOp has the option of creating a cpo.retrafo and cpo.train.invert (cpo.invert if constant.invert is TRUE) function (and returning NULL) instead of returning a control object. Similarly, cpo.train.invert may return a cpo.invert function instead of a control.invert object. In makeCPOExtendedTargetOp, cpo.trafo may create a cpo.retrafo or a cpo.invert function, each optionally instead of a control or control.invert object (one or both may be functional). cpo.retrafo similarly may create a cpo.invert function instead of giving a control.invert object. Functional information transfer may be more parsimonious and elegant than control object information transfer.
Hyperparameters
The action performed by a CPO may be influenced using hyperparameters, during its construction as well as afterwards (then using setHyperPars). Hyperparameters must be specified as a ParamSet and given as argument par.set. Default values for each parameter may be specified in this ParamSet or optionally as another argument par.vals. Hyperparameters given are made part of the CPOConstructor function and can thus be given during construction. Parameter default values function as the default values for the CPOConstructor function parameters (which are thus made optional function parameters of the CPOConstructor function). The CPO framework handles storage and changing of hyperparameter values. When the cpo.train and cpo.retrafo functions are called to transform data, the hyperparameter values are given to them as arguments, so cpo.train and cpo.retrafo functions must be able to accept these parameters, either directly, or with a ... argument. Note that with functional CPOs, the cpo.retrafo function does not take hyperparameter arguments (and instead can usually refer to them by its environment). Hyperparameters may be exported (or not), thus making them available for setHyperPars. Not exporting a parameter has advantage that it does not clutter the ParamSet of a big CPO or CPOLearner pipeline with many hyperparameters. Which hyperparameters are exported is chosen during the constructing call of a CPOConstructor, but the default exported hyperparameters can be chosen with the export.params parameter.
Properties
Similarly to Learners, CPOs may specify what kind of data they are and are not able to handle. This is done by specifying .properties.* arguments. The names of possible properties are the same as possible LearnerProperties, but since CPOs mostly concern themselves with data, only the properties indicating column and task types are relevant. For each CPO one must specify
1. which kind of data does the CPO handle,
2. which kind of data must the CPO or Learner be able to handle that comes after the given CPO, and
3. which kind of data handling capability does the given CPO add to a following CPO or Learner if coming before it in a pipeline.
The specification of (1) is done with properties.data and properties.target, (2) is specified using properties.needed, and (3) is specified using properties.adding. Internally, properties.data and properties.target are concatenated and treated as one vector, they are specified separately in makeCPO etc. for convenience reasons. See CPOProperties for details. The CPO framework checks the cpo.retrafo etc. functions for adherence to these properties, so it e.g. throws an error if a cpo.retrafo function adds missing values to some data but didn't declare “missings” in properties.needed. It may be desirable to have this internal checking happen to a laxer standard than the property checking when composing CPOs (e.g. when a CPO adds missings only with certain hyperparameters, one may still want to compose this CPO to another one that can't handle missings). Therefore it is possible to postfix listed properties with “.sometimes”. The internal CPO checking will ignore these when listed in properties.adding (it uses the ‘minimal’ set of adding properties, adding.min), and it will not declare them externally when listed in properties.needed (but keeps them internally in the ‘maximal’ set of needed properties, needed.max). The adding.min and needed.max can be retrieved using getCPOProperties with get.internal = TRUE.
Data Format
Different CPOs may want to change different aspects of the data, e.g. they may only care about numeric columns, they may or may not care about the target column values, sometimes they might need the actual task used as input. The CPO framework offers to present the data in a specified formats to the cpo.train, cpo.retrafo and other functions, to reduce the need for boilerplate data subsetting on the user's part. The format is requested using the dataformat and dataformat.factor.with.ordered parameter. A cpo.retrafo function is expected to return data in the same format as it requested, so if it requested a Task, it must return one, while if it only requested the feature data.frame, a data.frame must be returned.
Target Operation CPOs can be used for conversion between Tasks. For this, the type.out value must be given. Task conversion works with all values of dataformat and is handled by the CPO framework. The cpo.trafo function must take care to return the target data in a proper format (see above). Note that for conversion, not only does the Task type need to be changed during cpo.trafo, but also the prediction format (see above) needs to change.
Fix Factors
Some preprocessing for factorial columns needs the factor levels to be the same during training and prediction. This is usually not guarranteed by mlr, so the framework offers to do this if the fix.factors flag is set.
ID
To prevent parameter name clashes when CPOs are concatenated, the parameters are prefixed with the CPOs id. The ID can be set during CPO construction, but will default to the CPOs name if not given. The name is set using the cpo.name parameter.
Packages
Whenever a CPO needs certain packages to be installed to work, it can specify these in the packages parameter. The framework will check for the availability of the packages and throw an error if not found during construction. This means that loading a CPO from a savefile will omit this check, but in most cases it is a sufficient measure to make the user aware of missing packages in time.
Target Column Format
Different Task types have the target in a different formats. They are listed here for reference. Target data is in this format when given to the target argument of some functions, and must be returned in this format by cpo.trafo in Target Operation CPOs. Target values are always in the format of a data.frame, even when only one column.
Task type target format “classif” one column of factor “cluster” data.frame with zero columns. “multilabel” several columns of logical “regr” one column of numeric “surv” two columns of numeric
When inverting, the format of the target argument, as well as the return value of, the cpo.invert function depends on the Task type as well as the predict.type. The requested return value predict.type is given to the cpo.invert function as a parameter, the predict.type of the target parameter depends on this and the predict.type.map (see PredictType). The format of the prediction, depending on the task type and predict.type, is:
Task type predict.type target format “classif” “response” factor “classif” “prob” matrix with nclass cols “cluster” “response” integer cluster index “cluster” “prob” matrix with nclustr cols “multilabel” “response” logical matrix “multilabel” “prob” matrix with nclass cols “regr” “response” numeric “regr” “se” 2-col matrix “surv” “response” numeric “surv” “prob” [NOT YET SUPPORTED]
All matrix formats are numeric, unless otherwise stated.
In the place of all cpo.* arguments, it is possible to make a headless function definition, consisting only of the function body. This function body must always begin with a ‘{’. For example, instead of cpo.retrafo = function(data, control) data[-1], it is possible to use cpo.retrafo = function(data, control) { data[-1] }. The necessary function head is then added automatically by the CPO framework. This will always contain the necessary parameters (e.g. “data”, “target”, hyperparameters as defined in par.set) in the names as required. This can declutter the definition of a CPOConstructor and is recommended if the CPO consists of few lines.
Note that if this is used when writing an R package, inside a function, this may lead to the automatic R correctness checker to print warnings.
Other CPOConstructor related: CPOConstructor, getCPOClass, getCPOConstructor, getCPOName, identicalCPO, print.CPOConstructor
Other CPO lifecycle related: CPOConstructor, CPOLearner, CPOTrained, CPO, NULLCPO, %>>%, attachCPO, composeCPO, getCPOClass, getCPOConstructor, getCPOTrainedCPO, identicalCPO
## Examples
# an example constant feature remover CPO
constFeatRem = makeCPO("constFeatRem",
dataformat = "df.features",
cpo.train = function(data, target) {
names(Filter(function(x) { # names of columns to keep
length(unique(x)) > 1
}, data))
}, cpo.retrafo = function(data, control) {
data[control]
})
# alternatively:
constFeatRem = makeCPO("constFeatRem",
dataformat = "df.features",
cpo.train = function(data, target) {
cols.keep = names(Filter(function(x) {
length(unique(x)) > 1
}, data))
# the following function will do both the trafo and retrafo
result = function(data) {
data[cols.keep]
}
result
}, cpo.retrafo = NULL)
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{}
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If a shunt 1//10 of the coil r...
# If a shunt 1//10 of the coil resistance is applied to a moving coil galvanometer, its sensitivity becomes
Updated On: 27-06-2022
Text Solution
10 fold 11 fold(1)/(10) fold(1)/(11) fold
Solution : (I_(g))/(I) = (S)/(S + G) = ((G //10))/((G//10) + G) = (1)/(11) <br> Initially, alpha_(1) = theta//I_(g) (i) <br> Finally, after the shunt is used, <br> alpha_(f) = theta//I(ii) <br> :. (alpha_(f))/(alpha_(I)) = (theta//I)/(theta//I_(g)) = (I_(g))/(I) = (1)/(11) <br> So current sensitivity becomes 1//11 - fold.
Step by step solution by experts to help you in doubt clearance & scoring excellent marks in exams.
Transcript
hello students our question is efficient 1 upon 10 of the coil resistance is applied to a moving coil galvanometer it sensitivity becomes so solution for this will be sensitivity become solution will be let this is our galvanometer ok connected with the central distance electric current is my current in this branch will be IG endangers branch will be I minus I ok now we calculate
igf-i which is equals to as upon as plus ok so I ji apan I will be equal to 10 upon Ji Bhai ok so from here Ayyappan I will be equal to one upon 11 Nau initially initially Alpha I will be equals to eat upon Eyes finally Alpha to
sorry Alpha Alpha final will be closed state of an I take the rate of Alpha Alpha AP it will be question theta upon upon heater upon I from here we get IG account I so if we put the value of Alpha upon Alpha ap is equal to 1 upon I and the value of igf-i is 1111 so this means current so this means current sensitivity witty becomes one upon 11 Pol ABCD option
option for our current correct option
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{}
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## Tag: lie-algebras
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24 Follow-up to Steinberg's problem (12) in his 1966 ICM talk? 2016-07-18T20:24:07.893
21 How are these two ways of thinking about the cross product related? 2010-07-30T07:13:55.153
21 Pythagorean 5-tuples 2011-04-24T09:14:09.013
21 When is a finite dimensional real or complex Lie Group not a matrix group 2011-05-07T14:02:23.740
21 Why the BGG category O? 2011-05-13T20:26:15.770
20 "isotropic" subspaces of a simple Lie algebra 2009-10-25T17:50:38.683
20 Formal Geometry 2009-11-19T03:16:14.307
20 Automorphism group of real orthogonal Lie groups 2016-04-09T12:43:40.137
20 Curious fact about number of roots of $\mathfrak{sl}_n$ 2017-05-13T01:29:50.093
19 Number of triples of roots (of a simply-laced root system) which sum to zero 2011-05-26T23:07:33.007
18 What is the Zariski closure of the space of semisimple Lie algebras? 2009-12-25T01:06:33.500
18 Outer automorphisms of simple Lie Algebras 2010-02-09T03:31:21.120
18 Lie algebra automorphisms and detecting knot orientation by Vassiliev invariants 2010-08-27T02:20:28.893
18 On a drawing in Dixmier's Enveloping Algebras 2015-03-23T16:20:41.833
17 Generators of the cohomology of a Lie algebra 2010-12-17T16:35:47.267
17 homotopy type of connected Lie groups 2011-01-24T16:49:45.873
17 Invariants for the exceptional complex simple Lie algebra $F_4$ 2012-05-06T18:10:59.720
17 Is homology finitely generated as an algebra? 2014-10-03T03:24:28.203
17 Does there exist something like an $H_3$ and $H_4$ (icosahedral) Lie algebra or algebraic group? 2016-08-17T13:25:27.640
16 Introduction to W-Algebras/Why W-algebras? 2010-01-01T22:09:25.353
16 Matrix representation for $F_4$ 2010-02-26T04:07:34.233
16 Combinatorial identity involving the Coxeter numbers of root systems 2010-09-16T16:47:55.000
16 How do I stop worrying about root systems and decomposition theorems (for reductive groups)? 2010-10-20T20:48:28.903
16 Gap in an argument in Fulton & Harris? 2011-02-06T20:17:32.260
16 (Dis)similarity between groups and Lie algebras 2011-06-20T04:56:22.563
16 Is a retract of a free object free? 2012-03-09T17:17:56.073
16 Motivation of Virasoro algebra 2012-12-26T23:37:33.753
16 Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group? 2016-01-07T11:00:03.163
16 Are the unipotent and nilpotent varieties isomorphic in bad characteristics? 2017-05-02T14:14:32.003
15 Modern reference for maximal connected subgroups of compact Lie groups 2011-04-01T19:42:17.397
15 Why do sl(2) and so(3) correspond to different points on the Vogel plane? 2011-05-04T19:19:00.707
15 Lie algebras vs. graph complexes 2013-12-15T21:22:11.343
15 factorization of the regular representation of the symmetric group 2016-01-02T22:57:07.053
14 Folding by Automorphisms 2009-11-03T02:11:23.030
14 Is every finite-dimensional Lie algebra the Lie algebra of a closed linear Lie group? 2009-12-01T23:43:20.270
14 Understanding moment maps and Lie brackets 2010-02-06T20:03:23.447
14 What is the physical meaning of a Lie algebra symmetry? 2010-05-17T05:19:22.533
14 Is every G-invariant function on a Lie algebra a trace? 2010-05-21T00:06:41.840
14 Basis-free definition of Casimir element? 2011-01-20T06:53:22.040
14 Proof for which primes H*G has torsion 2012-06-14T02:16:35.763
14 What is a Homotopy between $L_\infty$-algebra morphisms 2013-08-11T16:52:53.747
14 "Special" meanders 2013-11-03T09:50:16.393
14 Associators, Grothendieck-Teichmüller group and monoidal categories 2014-01-02T21:49:13.203
14 Examples of representations of quantum groups 2017-10-05T17:54:39.417
13 What does the nilpotent cone represent? 2010-01-08T17:41:47.973
13 What is the recent development of D-module and representation theory of Kac-Moody algebra? 2010-02-09T12:32:45.973
13 Polynomial invariants of the exceptional Weyl groups 2010-09-03T11:35:08.780
13 Deformations of semisimple Lie algebras 2010-11-26T18:42:08.527
13 Torsion for Lie algebras and Lie groups 2011-06-04T03:37:20.340
13 So, did Poincaré prove PBW or not? 2011-09-08T19:12:23.647
13 Ternary "Lie structure" 2012-05-10T16:21:14.160
13 About the intrinsic definition of the Weyl group of complex semisimple Lie algebras 2012-08-02T02:10:29.030
13 Deligne's 1996 note on exceptional Lie groups 2013-01-10T13:57:14.327
13 decompositions for exceptional Lie algebras $E_6$, $E_7$ and $E_8$ 2013-01-17T16:58:32.247
13 Characteristic subgroup of nilpotent group that is not invariant under powering 2013-03-12T07:16:48.117
13 Why are affine Lie algebras called affine? 2013-05-19T00:24:05.467
13 Is this error in this paper of Langlands fixable? 2013-10-09T21:13:31.880
13 Intuition for the Cartan connection and "rolling without slipping" in Cartan geometry 2016-01-28T23:39:35.727
13 Free groups and free restricted Lie algebras 2017-02-03T19:49:58.383
13 Constructing $E_8$ from its branching to $A_8$ 2017-06-13T14:22:05.727
13 Can one define quantized universal enveloping algebras in a basis-free way? 2017-07-19T19:28:54.367
13 Is there a simple system that has $\text{SU}(3)$ symmetry? 2017-10-30T20:39:57.340
12 Matrices into path algebras 2009-10-26T22:16:44.997
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## Cryptology ePrint Archive: Report 2010/597
A New Class of Bent--Negabent Boolean Functions
Abstract: In this paper we develop a technique of constructing bent--negabent Boolean functions by using complete mapping polynomials. Using this technique we demonstrate that for each $\ell \ge 2$ there exits bent--negabent functions on $n = 12\ell$ variables with algebraic degree $\frac{n}{4}+1 = 3\ell + 1$. It is also demonstrated that there exist bent--negabent functions on $8$ variables with algebraic degrees $2$, $3$ and $4$.
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Looking for Algebra worksheets?
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##### Browse Questions
You can create printable tests and worksheets from these Grade 6 Function and Algebra Concepts questions! Select one or more questions using the checkboxes above each question. Then click the add selected questions to a test button before moving to another page.
1 2 3 4 ... 48
Grade 6 Linear Equations CCSS: 6.EE.B.6
Solve: $x-4=13$
1. $9$
2. $17$
3. $\frac{13}{4}$
4. $7$
Which one shows:
Ten less than a number is fifteen.
1. 10 - x =15
2. x - 15 = 10
3. x - 10 = 15
4. 15 - 10 = x
Which phase best describes the number line?
1. any number less than zero
2. any number greater than zero
3. any number less than or equal to zero
4. any number greater than or equal to zero
The solution of which inequality is shown by the number line?
1. $x > -5$
2. $x <-5$
3. $x <= -5$
4. $x >= -5$
The solution of which inequality is shown by the number line?
1. $x > -5$
2. $x >= -5$
3. $x <-5$
4. $x <= -5$
The solution of which inequality is shown by the number line?
1. $x < 0$
2. $x <= 0$
3. $x > 0$
4. $x >= 0$
The solution of which inequality is shown by the number line?
1. $x < 0$
2. $x <= 0$
3. $x > 0$
4. $x >= 0$
The solution of which inequality is shown by the number line?
1. $x >= 4$
2. $x > 4$
3. $x < 4$
4. $x <= 4$
The solution of which inequality is shown by the number line?
1. $x <= -1$
2. $x <-1$
3. $x >= -1$
4. $x > -1$
1. $x > -1$
2. $x >= -1$
3. $x <-1$
4. $x <= -1$
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# A nontrivial everywhere continuous function with uncountably many roots?
This is my first post on SE, forgive any blunders.
I am looking for an example of a function $f:\mathbb{R} \to \mathbb{R}$ which is continuous everywhere but has uncountably many roots ($x$ such that $f(x) = 0$). I am not looking for trivial examples such as $f = 0$ for all $x$. This is not a homework problem. I'd prefer a nudge in the right direction rather than an explicit example.
Thanks!
Edit: Thanks all! I've constructed my example with your help.
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The roots of a continuous function is always a closed subset of $\mathbb{R}$ : $\{0\}$ is closed, thus $f^{-1}(\{0\})$ is closed too.
If you have a closed set $S$, you can define a function $f : x \mapsto d(x,S)$, which is continuous and whose set of roots is exactly $S$ : you can make a continuous function have any closed set as its set of roots.
Therefore you only have to look for closed sets that are uncountable.
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This is the best answer, since it says exactly which sets appear as zero sets of continuous functions (with an explicit construction). Thus although the OP did not specify what "nontrivial" means -- but it must mean something: e.g. it is probably not very exciting to have a function which is constantly zero on some nontrivial interval -- with this answer in hand it doesn't really matter: one knows the general case. – Pete L. Clark May 22 '11 at 17:49
@Pete: By nontrivial, I meant exactly what you said. If a function is constantly zero on some continuous interval of $\mathbb{R}$, of course it has uncountably many roots, but this is not interesting (I thought I had made this clear by stating I was not interested in, eg, $f(x) = 0$ for all $x$). Thanks for the great answer, chandok! – barf May 22 '11 at 21:47
I guess the Cantor set would do as an uncountable closed set. – gary Jun 30 '11 at 21:12
This could be interesting for you.
-
Andre Henriques's answer has an impressive 86 upvotes! – Grumpy Parsnip May 22 '11 at 1:56
Even though you seem to be happy with the given answers, I can't resist pointing out the following construction which I already mentioned in this thread. If this example is already contained in one of the links provided in the other answer, I apologize for the duplication:
Choose a space-filling curve $c: \mathbb{R} \to \mathbb{R}^2$ and compose it with the projection $p$ to one of the coordinate axes. This gives you an example of a continuous and surjective function $f = p \circ c: \mathbb{R} \to \mathbb{R}$ all of whose pre-images are uncountable.
-
Why not $f(x)=x+|x|$? Looks quite nontrivial to me.
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I think he is looking for an example that is not 0 on a dense interval, otherwise the problem is pretty boring. – Listing May 22 '11 at 8:49
OK, but I only wanted to notice, that he should specify what kind of a "nontrivial" example he is looking for. – t22 May 22 '11 at 9:00
If $f(x)$ is the function then $E = \{ x \colon f(x) = 0 \}$ is a closed set. Do you know of a nontrivial, by your standards, closed set with uncountable many points? The complement of $E$ is open. Is there a way to describe the complement of $E$ so that it would be easy to construct the remainder of the function in such a way that the function was never $0$ on the complement of $E$.
-
If you just want a continuous function, then let $f(x) = 0$ over an interval say $[a,b]$ where $a<b$ and over the rest of the real line find functions $g(x)$ and $h(x)$ such that $g(a) = 0$ and $h(b) = 0$ and define $f(x) = g(x), \forall x \leq a$ and $f(x) = h(x), \forall x \geq b$.
You could also have infinitely smooth functions with zero on an uncountable set as for instance $$f(x)=\begin{cases}e^{-1/x}&x>0\newline 0&x\leq 0\end{cases}$$
And if you want an infinitely smooth function with a compact support
$$f(x) = \begin{cases} \exp(- \frac{1}{1 - x^2}), \quad |x| < 1 \\ 0 , \quad |x| \geq 1 \end{cases}$$
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I just thought I'd mention that a Brownian motion has this property (almost surely).
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# A question about dense subsets of the real line
1. Feb 19, 2009
### AxiomOfChoice
Consider the closed interval $$A = [a,b]\subset \mathbb{R}$$. Are the only dense subsets of $$A$$ the set of all rational numbers in $$A$$ and the set of all irrational numbers in $$A$$? Something tells me that there's got to be more than that, but I can't think of any examples.
2. Feb 19, 2009
### HallsofIvy
Staff Emeritus
Well, obviously, the set of, say, rational numbers with some irrational numbers included would be dense, the set of all irrational numbers with some rational numbers included would be dense. Or you could partition the real numbers into any number of disjoint subsets, take rational numbers in some of the subsets and irrational numbers in the others.
3. Feb 20, 2009
### maze
You could basically throw out a bunch of rational or irrational numbers, so long as you don't throw out too many of them. For example, Q\{1} is still dense. You could probably even get away with throwing out countably infinite rational numbers from Q and be OK - throw out "every other" rational number (they are countable so you can order them).
You could also construct other countable dense subsets by adding an irrational number to all rational numbers. eg, {q+pi: q rational}
That is an interesting question if there are other countable dense subsets that are fundamentally different from Q in a meaningful way. I don't know the answer to that. Probably yes?
4. Feb 20, 2009
### lurflurf
What does that mean?
would
Z+sqrt(2)Z={a+sqrt(2)b |a,b integers}
Z[1/2]={a/2^b|a,b integers}
Sin(Z)={sin(a)|a an integer}
qualify?
5. Feb 20, 2009
### maze
I don't understand? Those aren't dense. Do you mean rationals Q instead of integers Z?
6. Feb 21, 2009
### lurflurf
What definition of dense are you using? Try something like if A is dense in B, for any b in B and epsilon in (0,infinity), there exist a in A such that d(a,b)<epsilon. That is A contains points arbitrarily close to any point in B.
Example sin(Z) is dense in [-1,1]
since sine is a continuous function with period 2pi our result follows from the fact that
Z+2piZ is dense in R
which follows from the fact that
aZ+bZ is dense in R when a is rational (and not zero) and b is irrational
7. Feb 21, 2009
### HallsofIvy
Staff Emeritus
AxiomofChoice said in his first post that the set of irrationals is dense in the reals. That is about as "fundamentally different from Q in a meaningful way" as you can get.
You seem to be asking for sets of real numbers that do no involve rational or irrational numbers. There are no such sets- rational and irrational numbers are all we have!
8. Feb 21, 2009
### slider142
Hmm, how about changing the sieve. What about algebraic vs. transcendental numbers. Are transcendental numbers dense on the real line? :tongue2:
Or going the other way, are undefinable numbers dense in the reals? By Cantor's slash, they're a "pretty large" subset.
9. Feb 21, 2009
### maze
Well I mean I was looking for countable dense subsets, not any dense subset. But lurf's example of Sin(Z) repeated each interval seems like a pretty good example to me.
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# Creating a decent PDF/A document
Let's say I have a simple document having some sections and Unicode and/or math in their titles. How to produce a decent PDF output? By that I mean:
• It contains correct bookmarks.
• It is a PDF/A document.
I have produced the following example. (I know that it may not be a good idea to have that much math and Unicode in titles, but to some extent it's ok, and the source below is just an example.)
\begin{filecontents*}{\jobname.xmpdata}
\Title{Properties of ℝ^ω}
\Author{Foo Barovič\sep Baz Pequeño}
\end{filecontents*}
\documentclass[a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage[nomath]{lmodern}
\usepackage{amssymb}
\usepackage[a-2u]{pdfx}
\pdfstringdefDisableCommands{
\def\omega{ω}
}
\begin{document}
\title{Properties of $\mathbb{R}^\omega$}
\author{Foo Barovič, Baz Pequeño}
\date{}
\maketitle
\section{\texorpdfstring
{Basic properties of $\mathbb{R}^\omega$}
{Basic properties of ℝ\textasciicircumω}
}
Here we shall talk about the basic properties.
\subsection{\texorpdfstring
{$\mathbb{R}^\omega$ vs.\ $\bigcup_{n \in \omega} \mathbb{R}^n$ according to Pequeño}
{ℝ\textasciicircumω vs. ⋃\_\{n ∈ ω\} ℝ\textasciicircum n according to Pequeño}
}
\section{\texorpdfstring
{Advanced properties of $\mathbb{R}^\omega$}
• The duplicities in titles. It is possible to map a Unicode character to a macro via the inputenc package. For PDF bookmarks the situation is opposite: it is possible to map macros to Unicode characters using \pdfstringdefDisableCommands. This works well for \omega or \bigcup, but what to do with superscripts, subscripts, groups, $, and \mathbb{R}? Is there a way to declare that ^, _, {, } should be taken literally, $ should be ignored, and \mathbb{R} means ℝ? Then, \texorpdfstring won't be needed.
• The fact that filecontents* does not rewrite existing document. I know this can be changed by filecontents package, but it still emits a warning, and it cannot be loaded before \documentclass, while generating of xmpdata is recommended before \documentclass.
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Language Constructs¶
Proof scripts are textual representations of rule applications, settings changes and strategy invocations (in the case of KeY as underlying verification system also referred to as macros).
Variables and Types¶
We need to distinguish between: logic, program, and script variables.
• logic variable: Occur on sequents, bounded by a quantifier or in an update
• program variable: declared and used in Java programs. They are constants on the sequent.
• script variable: declared and assignable within a script
Proof Script Language has script variables.
A script variable has a name, a type and a value. Variables are declared by
var0 : type;
var1 : type := value;
var2 := value;
Both statements declare a variable, in the latter case (var1 and var2) we directly assign a value, in the first form var0 receives a default value.
Types and Literals¶
We have following types: INT, TERM<Sort>, String.
• INT represents integer of arbitrary size.
42
-134
• TERM<S> represents a term of sort S in KeY. S can be any sort given by KeY. If the sort is ommitied, then S=Any.
f(x)
g(a)
imp(p,q)
• STRING
"i am a string"
Special Variables¶
To expose settings of the underlying prover to the user we include special variables:
• MAX_STEPS : amount denotes the maximum number of proof steps the underlying prover is allowed to perform
Proof Commands¶
command argName="argument" "positional argument" ;
Every command is terminated with a semicolon. There are named arguments in the form argName="argument" and unnamed argument without name. Single '...' and double quotes "..." can both be used.
Single-line comments are start with //.
KeY Rules/Taclets¶
All KeY rules can be used as proof command. The following command structure is used to apply single KeY rules onto the sequent of a selected goal node. If no argument is following the proof command the taclet corresponding to this command has to match at most once on the sequent.
If more terms or formulas to which a proof command is applicable exist, arguments have to be given that indicate the where to apply the rule to.
A rule command has the following syntax:
RULENAME [on=TERM]?
[formula=TERM]
[occ=INT]
[inst_*=TERM]
with:
• TERM specific sub-term
• TOP_LEVEL_FORMULA: specific top level formula
• OCCURENCE_IN_SEQUENT: Number of occurence in the sequent
• maxSteps the number of steps KEY should at most use until it terminateds teh proof search
If a rule has schema variables which must be instantiated manually, such instantiations can be provided as arguments. A schema variable named sv can be instantiated by setting the argument sv="..." or by setting inst_sv="..." (the latter works also for conflict cases like inst_occ="...").
Examples¶
• andRight;
Applicable iff there is only one matching spot on the sequent
• eqSymm formula="a=b";
This command changes the sequent a=b ==> c=d to b=a ==> c=d Using only eqSymm; alone would have been ambiguous.
• eqSymm formula="a=b->c=d" occ=2;
This command changes sequent a=b->c=d ==> to a=b->d=c ==>. The occurrence number is needed since there are two possible applications on the formula
• eqSymm formula="a=b->c=d" on="c=d";
This command changes the sequent "a=b->c=d ==>" to "a=b->d=c ==>". It is simialr to the example above, but here the option to specify a subterm instead of an occurrence number is used.
• cut cutFormula="x > y";
This command is almost the same as cut \x>y
Macro-Commands¶
In the KeY system macro commands are proof strategies tailored to specific proof tasks. The available macro commands can be found using the command help. Using them in a script is similar to using rule commands:
MACRONAME (PARAMETERS)?
Often used macro commands are:
• symbex : performs symbolic execution
• auto: invokes the automatic strategy of key
• heap_simp: performs heap simplification
• autopilot: full autopilot
• autopilot_prep: preparation only autopilot
• split_prop: propositional expansion w/ splits
• nosplit_prop: propositional expansion w/o splits
• simp_upd: update simplification
• simp_heap: heap simplification
Example:
auto;
Selectors
As nited before proof commands are implemented as single goal statements. Some proof commands split a proof into more than one goal. To allow to apply proof commands in proof state with more than one proof goal, the language allows for a selector statement cases. Such a cases-statement has the following structure:
cases {
case MATCHER:
STATEMENTS
[default:
STATEMENTS]?
}
Control Flow Statements
The script language allows different statements for control-flow. Control-Flow statements define blocks, therefor it is neccessary to use curly braces after a control-flow statement.
• foreach {STATEMENTS}
• theOnly {STATEMENTS}
• repeat {STATEMENTS}
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Fourier Transforms, Momentum and Position
1. Jun 9, 2009
r16
In quantum mechanics, why does the Fourier transform
$$f(x) = \int_{-\infty}^\infty F(k) e^{ikx}dk$$
represent position and
$$F(k) = \int_{-\infty}^\infty f(x) e^{-ikx} dx$$
represent momentum?
2. Jun 9, 2009
malawi_glenn
you must have something given to you, e.g. given that f(x) is the position wavefunction, then it's fourier transform is the momenum wavefunction. It just follows since p = - i hbar d/dx
<x'|p|p'> = -i h d/dx' <x'|p'>
p'<x'|p'> = -i h d/dx' <x'|p'>
solve this differential equation
<x'|p'> = N exp { i p' x' /h }
now look at
psi_a(x') = <x'|a> = (insert completeness realtion ) = integral dp' <x'|p'><p'|a>
psi_a(x') = N integral dp' exp { i p' x' /h } phi_a(p')
determine the normalization N using the delta function:
delta(x' - x'') = N^2 integral dp' exp { i p'( x' - x'') /h }
we get N = 1/sqrt(2 pi h)
3. Jun 10, 2009
r16
I'm only starting out in Quantum Mechanics (chapter 2 of the griffiths book) and I am not familiar with the notation
I'm sure ill get to it later on in the book. Until then, could you explain it?
4. Jun 10, 2009
malawi_glenn
you might want to look up "bra - ket" notation or "dirac notation" in Griffiths book, then I can explain if you don't understand, but basically:
p|p'> = "p operator on p-eigenstate with momentum p' " = p'|p'> (I denote operator with the letter and eigenvalues with prime"
since p = -i h d/dx, we can do the same operation on the right hand side, but with <x'|p'> as just an arbitrary function of x' and p'
The basic idea is that, without getting too much into math behind it;
|a'> is a vector in hilbert space, it denotes the state with quantun number a'
a|a'> = a'|a>
ok?
these are called "kets"
now the dual vector, called "bra":
<a'|
we can "think" of this as the ket's beeing column vectors and bra's as row vectors:
<a''|a'> is then a number
ok, this was assuming that a',a'' are discrete quantum numbers
now, for x and p, which are continuous, we can use the same notation, but we can not imagine/represent them as discrete vectors as we do in introductory linear algebra.
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## Precalculus (6th Edition) Blitzer
polynomial; $3x^2+2x-5$
The exponents of the variables are whole numbers therefore the given algebraic expression is a polynomial. Arrange the terms in descending degree to obtain $3x^2+2x-5.$
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# Modular arithmetic and equivalence classes
I completely understand the concept of modulus arithmetic and grasp all the main ideas. The only thing I don’t understand is equivalence classes.
The formal definition is:
Another interpretation is that modular arithmetic deals with all the integers, but divide them into N equivalence classes, each of the form $\{i + kN \mid k \in\mathbb{Z}\}$ for some $i$ between $0$ and $N-1$.
Could someone explain this in layman’s terms with an example possibly? I have absolutely no idea what equivalence classes are.
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Do you know what a relation on a set is. I think that'd help me write an answer. BTW, Is this answer of mine that deals with the the equivalence classes modulo $3$ help you? – user21436 Mar 11 '12 at 23:11
Well, it's too late in here. I'll write an answer tommorrow if this is still on the Unanswered list. – user21436 Mar 11 '12 at 23:12
@KannappanSampath Sorry, but I do not know what a relation on a set is. – user26649 Mar 11 '12 at 23:13
Your definition appears to be from p.26 of Algorithms, by S. Dasgupta, C. H. Papadimitriou, and U. V. Vazirani. Is your knowledge of modular arithmetic based exclusively on the (too) short sketch there (section 1.2)? The more you reveal about your level of knowledge, the more likely you'll receive answers at your level. – Bill Dubuque Mar 11 '12 at 23:40
The notion of an equivalence class always arises when dealing with something called an "equivalence relation." So to explain the former, we'll first discuss the latter.
An equivalence relation is (speaking broadly) a generalized notion of equality. For example, if we are conducting a survey, and we are only concerned with gender, we might say that two people are "equivalent" if they have the same gender. Similarly, in the case of modular arithmetic, we say that two numbers are equivalent if they have the same remainder when divided by some number $n$. So, if we are working mod $4$, we say that $1 \equiv 5$. That does not mean that $1=5$, just that they are equivalent for our purposes.
More formally, an equivalence relationship needs to be reflexive (everything should be equivalent to itself), symmetric (if $A$ is equivalent to $B$, $B$ is equivalent to $A$) and transitive (if $A$ is equivalent to $B$, and $B$ is equivalent to $C$, $A$ is equivalent to $C$). Thus, this generalized notion of equality behaves very similarly to the traditional "$=$".
When we construct an equivalence relationship, one can think of it as putting on a pair of coarse glasses, so that we can no longer distinguish between "equivalent" objects. Thus, mod $4$, it appears to us that $1,5,9,\cdots$ are all the same thing. This entire family of objects that are equivalent is called an equivalence class. When we are working mod $4$, there will be $4$ equivalence classes, one corresponding to all things with remainder $0$, another for all things with remainder $1$, etc. Moreover, when you add two things with remainder $2$, you get something with remainder $0$. Thus, we can say that when working mod $4$, there are only four objects: families of "equivalent" numbers, and we do arithmetic with these families.
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Thank you, I understood it perfectly. – user26649 Mar 11 '12 at 23:26
Given $N$, we will make $N$ "boxes", labeled $0$, $1$, $2,\ldots,N-1$. We will divide all the integers among the boxes as follows: given an integer $a$, divide $a$ by $N$ with remainder; put $a$ in the box corresponding to its remainder.
So, for example, if $N=11$, then we have boxes labeled $0,1,\ldots,10$; in the box labeled $0$ we have all multiples of $11$; in the box labeled $1$ we have all numbers of the form $11k + 1$, with $k\in\mathbb{Z}$; and so on: \begin{align*} {}[0] &= \{\ldots, -22, -11, 0, 11, 22, 33,\ldots\} = \{0+11k\mid k\in\mathbb{Z}\};\\ {}[1] &= \{\ldots, -21, -10, 1, 12, 23, 34, \ldots\} = \{1+11k\mid k\in\mathbb{Z}\};\\ {}[2] &= \{\ldots, -20, -9, 2, 13, 24, 35,\ldots\} = \{2+11k\mid k\in\mathbb{Z}\};\\ &\vdots\\ {}[10] &= \{\ldots,-12, -1, 10, 21, 32, 43,\ldots\} = \{10+11k\mid k\in\mathbb{Z}\}. \end{align*}
Each of this boxes is an "equivalence class": we consider two integers to be "equivalent [modulo $N$]" if they are in the same box. Note that every integer is in some box, and no integer is in more than one box; the boxes partition the integers.
We can then define addition of boxes as follows: to add the box $[a]$ with the box $[b]$, take any number $x\in[a]$, any number $y\in[b]$, add them like you add integers normally, and then find the one and only one box $[c]$ such that $x+y\in[c]$. We define $[a]+[b]$ to be the box $[c]$.
This definition requires that you show that it is "well-defined": the definition of $[a]+[b]$ seems to depend on the numbers $x$ and $y$ that are chosen; we need to show that if you pick a different $x'\in[a]$ and a different $y'\in[b]$, then $x'+y'$ will be in the same box as $x+y$. This can be done directly, or as a consequence of structure that these boxes have and is derived from the general algebraic theory of "congruences".
In a sense, we are dealing with all integers now, because the boxes include all the integers. We can then define a relation on the integers by saying that "$a$ is congruent to $b$ modulo $N$" if and only if $a$ and $b$ are in the same box. We write this $a\equiv b\pmod{N}$. Then what I discussed above says that if $x\equiv x'\pmod{N}$ and $y\equiv y'\pmod{N}$, then $x+y\equiv x'+y'\pmod{N}$.
You can define modular arithmetic as regular arithmetic in terms of the boxes.
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Am instructive example - one which the OP surely knows - is fraction arithmetic (rationals). Alas, I don't have the time now to elaborate, but perhaps someone else does. – Bill Dubuque Mar 11 '12 at 23:57
To understand equivalence classes, you need to understand equivalence relations.
Let $A$ be any old set. A relation is defined to be any subset $R\subset A\times A$; if $(a,b)\in R$ we would usually write $aRb$. This definition seems kind of silly, and it kind of is, so let's make more assumptions on $R$. We say $R$ is an equivalence relation if it satisfies three axioms:
1) for every $a\in A,\ aRa$ (reflexivity)
2) for every $a,b\in A$, if $aRb$ then $bRa$ (symmetry)
3) for every $a,b,c\in A$, if $aRb$ and $bRc$ then $aRc$ (transitivity)
You can verify that the relation on $\mathbb{Z}$ defined by "$a\equiv_m b$ iff $m|(a-b)$" is an equivalence relation for any $m$
The special thing about equivalence relations is that they partition your set into equivalence classes . If $a\in A$ and $R$ is an equivalence relation on $A$, define the equivalence class $[a]=\{b\in A\ |\ aRb \}$. For example, if $A=\mathbb{Z}$ and $R$ is the familiar relation $\equiv_m$, then the equivalence class of $1$ is simply all integers $a$ such that $a\equiv 1\pmod{m}$, i.e. $[a]=\{1+km\ |\ k\in\mathbb{Z} \}$
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In very layman's terminology, you can think of an analogy of students in a class(room).
The three conditions for an Equivalence relation can be imagined as such:
Reflexive: Every student is in the same class as himself.
Symmetric: If student A is in the same class as student B, then student B is in the same class as student A.
Transitive: If student A is in the same class as student B, and student B is in the same class as student C, then student A is in the same class as student C.
In the context of modular arithmetic (e.g. modulo 6), one can see that the three conditions hold.
Then the integers in a certain class all have the the same reminder (modulo 6).
So for example, 1,7,13 are all in the same class, denoted by [1], as they all have remainder 1 when divided by 6.
Hope this analogy helps.
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# Configuring VMware vRealize Operations Manager Object Groups
There are two sections I will cover in this post: ‘Group Types’ and ‘Object groups’. An example of when you might want to consider creating a group type …lets say you have multiple data centers, a group type could be used as a way to group all objects types of that data center into one folder. In other worlds: The group type for the data center in Texas would be used as a sorting container for the group objects such as data stores, vCenters, hosts, virtual machines, etc.. and keep them separated from the data center in New York.
The way you can do this is by clicking on the Content icon, selecting group types and then clicking the green plus sign to add a new group type.
Next you can click on the environment icon (blue globe), Environment Overview, click the green plus icon to create a new object group, and then in the group type drop down, you can select the group type you just created. As far as policy selection goes, the built in VMware policies are a great place to start. You can easily update this selection later when you create a custom policy. I would recommend checking ‘Keep group membership up to date’.
Define membership criteria section. This is where the water can get muddy as you will have more than one way to target your desired environment objects. In the drop down menu ‘Select the Object Type that matches all of the following criteria’, the object types selection can grow in number depending how many additional adapter Management PAKs are installed on the vROps appliance. This selection will also be important because of the way vROps alerts off from the management packs.
– An example would be alerts on Host Systems. One would assume you would select the vCenter Adapter and then select Host Systems, however if you have the vCloud Director Adapter Management PAK installed for example, that PAK also has metrics for Host Systems that vROps will alert from, you would also need to select Host systems under that solution to target those alerts and systems as well.
For this example, we will use the Host System under the vCenter Adapter. There are different ways to target the systems, for this example I will showcase using Object name, contains option. This option allows us to target several systems IF they have a common name like so:
You also have the option to target systems based on their Relationship. In this example we have clusters of hosts group under the name of MGMT, so I chose Relationship, Child of, contains, MGMT – to target all systems in that cluster like so:
There is a Preview button in the lower left corner which you can use to see if your desired systems are picked up by the membership criteria you selected.
You can also target multiple system names by using the Add another criteria set option like so:
Depending on the size of the environment, I’ve noticed that it helps to make a selection in the ‘in navigation tree’ drop down as well
When you have the desired systems targeted in the group, click OK. Groups are subject to the metrics interval collection, so the groups page will show grey question marks next to the groups until the next interval has passed. Any policies that were applied to these custom groups will not change the alerts page until that metrics collection has occurred.
The added benefit to having custom groups is that vROps will also show you the group health, and if you click on the group name you will be brought to a custom interface with alerts only for that group of targeted systems.
In my next post I will go over the creating of policies and how to apply them to object groups.
# Nightly Automated Script To Gather VM and Host Information, and Output To A CSV
Admittedly this was my first attempt at creating a Powershell script, but thought I would share my journey. We wanted a way to track the location of customer VMs running on our ESXi hosts, and their power state in the event of a host failure. Ideally this would be automated to run multiple times during the day, and the output would be saved to a csv on a network share.
We had discovered a bug in vCenter 5.5 where if the ESXi 5.5 host was in a disconnected state, and an attempt was made to reconnect it using the vSphere client without knowing that the host was actually network isolated, HA would not restart the VMs on another host as expected. We would later test this in a lab to find that if we had not used the reconnect option, HA would restart the VMs as expected on other hosts. We again tested this scenario in vCenter 6 update 2, and the bug was not present.
So the first powercli one-liner I came up with was the following:
> get-vm | select VMhost, Name, @{N="IP Address";E={@($_.guest.IPaddress[0])}}, PowerState | where {$_.PowerState -ne "PoweredOff"} | sort VMhost
I wanted a list of powered on VMs, their IPs, what host they were running on, and I wanted to sort the output by the host name. Knowing I was on the right track, now I wanted to be able to connect to multiple data centers, and have each data center’s output saved to a CSV on a network share. I didn’t really need to hang on to the CSVs in that network share for more than seven days, so I also wanted to build in logic so that it would essentially cleanup after itself.
That script looks something like this:
#Initial variables
$vCenter = @()$sites = @("vcenter01","vcenter02","vcenter03")
#get array of sites and establishes connections to vCenters
foreach ($site in$sites) {
$vCenter =$site + "domain.net"
Connect-VIServer $vCenter #get list of not equal to powered off VMs, there IP and which hosts they're running on get-vm | select VMhost, Name, @{N="IP Address";E={@($_.guest.IPaddress[0])}}, PowerState | where {$_.PowerState -ne "PoweredOff"} | sort VMhost | Export-Csv -Path "c:\path\to\output\$site $((Get-Date).ToString('MM-dd-yyyy_hhmm')).csv" -NotypeInformation -useculture #Disconnect from vCenters Disconnect-VIServer -Force -Confirm:$false -Server $vCenter } #Cleanup old csv after 7 days.$limit = (Get-Date).AddDays(-7)
$path = "c:\path\to\output\" Get-ChildItem -Path$path -Recurse -Force | Where-Object { !$_.PSIsContainer -and$_.CreationTime -lt \$limit } | Remove-Item -Force
Something I did not know until after running this in a large production environment, is that the get-vm call is heavy and not very efficient. When I ran this script in my lab it took less than 15 seconds to run. However in a production environment, connecting to data centers all over the globe, it took over 40 minutes to run.
A colleague of mine who had automation experience, exposed me to another cmdlet called get-view, and said it would be much faster to run as it was more efficient gathering the data needed. So I rewrote my script and now it looks like:
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The new code took less than a couple of minutes to run in my production environment. I have a Windows VM deployed that’s running the VMware poweractions fling, and it also runs some scheduled scripts. This script would be running from that server, so I added an additional function to the script that creates a WIN event entry so it could be tracked from a syslog server.
So the final script can be downloaded here. *Disclaimer – test this in a lab first as the code will need to be updated to suit your needs.
# Add The vROps License, Configuring LDAP, and The SMTP For vRealize Operations Manager (vROps)
If you’ve been following my previous posts, I discussed what vRealize Operations Manager is, how to get the appliance deployed, how to get the master replica, data nodes and remote collectors attached to the cluster, and finally how to get data collection started.
Now it’s time to license vRealize Operations Manager. This can be achieved by logging into the appliance via: < https ://vrops-appliance/ui/login.action >. Next go into the Administration section, and there you’ll see Licensing. Click the green plus sign to add the vROps license.
About seven down from Licensing on the left hand column, you will see Authentication Sources. This is where you configure LDAP.
Again click the green plus sign to configure the LDAP source.
Once the credentials have been added, test the connection and then if everything checks out click OK.
Lastly lets get the SMTP service configured, about three down from Authentication sources you’ll find outbound settings. Click the green plus to add a new smtp.
Once you have the SMTP information added, test the connection, and if everything checks out click save.
So now you should have a functioning licensed vrops instance. In future posts I will cover creating object groups, policies, and configuring some alert emails.
# Configuring VMware vRealize Operations Manager Adapters For Data Collection
If you’ve followed my recent blog post on Installing vRealize Operations Manager (vROps) Appliance, you are now ready to configure the built in vSphere adapter to start data collection.
Depending on how big your environment is, and IF you have remote collectors deployed, you may want to consider configuring collector groups. A Collector group allows you to group multiple remote collectors within the same data center, and the idea is that this would allow for resiliency for those remote collectors, that way when you have the vROps adapters pointed to the collector group instead of the individual remote collector, if one of the remote collectors went down the other would essentially pick up the slack and continue collecting from that data center, so there would be no data loss. You can also create a collector group for a single remote collector for ease of expansion later if you want to add that data collection resiliency.
Go ahead and get logged into the appliance using the regular UI <https//vrops-appliance/ui/login.action>. From here click Administration. If you just need to configure the vSphere adapter for data collection, you can skip ahead to Section 2. Otherwise lets continue in section 1, and configure the collector groups.
Section 1
Click on Collector Groups
You can see that I already have collector groups created for my remote data centers, but if you were to create new, just click the green plus sign
Give the collector group a name, and then in the lower window select the corresponding remote collector. Then rinse-wash-and-repeat until you have the collector groups configured. Click Save when finished. Now lets move on to Section 2.
Section 2
From the Administration area, click on Solutions
Now because this is your new deployment, you would only have Operating Systems / Remote Service Monitoring and VMware vSphere. For the purpose on this post I will only cover configuring the VMware vSphere adapter. Click it to select it, and then click the gears to the right of the green plus sign.
Here just fill out the display name, the vCenter Server it will be connecting to, the credentials, and if you click the drop down arrow next to Advanced Settings, you will see the Collectors/Groups drop down menu. Expand that if you have created the custom collectors in Section 1, and select the desired group. Otherwise vROps will use the Default collector group, which is fine if you only have one data center, otherwise I recommend at least selecting a remote collector here if you do not have a collector group configured. This basically puts the load onto the remote collectors for data collection, and allows the cluster to focus on processing all of that lovely data. Click Test Connection to verify connectivity, and then click save. Then rinse-wash-and-repeat until you have all vCenters collecting. Close when finished.
Important to note that vROps by default will collect data every five minutes, and currently that is the lowest setting possible. You can monitor the status of your solutions or adapters here. Once they start collecting their statuses will change to green.
If you’d like to add additional solutions otherwise known as “Management PAKs”, head on over to VMware’s Solution Exchange . I currently work for a cloud provider running NSX, so I also have the NSX and vCloud Director Management PAKs installed. From the same solutions page, instead of clicking on the gears, click the green plus sign and add the additional solutions to your environment. This would also be used when you are updating solutions to newer versions. Currently there is no system to update you when a newer version is available.
Go to Global Settings on the Administration page, where you can configure the object history, or data retention policy, along with a few other settings.
Finally, Go back to home by clicking the house icon. By now the Health Risk and Efficiency badges should all be colored. Ideally green, but your results may vary. This is the final indication that vROps is collecting.
# Sizing and Installing The VMware vRealize Operations (vROps) Appliance
VMware has a sizing guide that will aid you in determining how many appliances you need to deploy. If you have multiple data centers, and somewhere north of 200 hosts, and more than 5,000 VMs, I’d recommend at least starting out with two servers configured as Large deployments. Once you get the built in vSphere adapter collecting for each environment, you can run an audit on the environment using vROps to get the raw numbers, and expand the cluster accordingly. Come prepared. Walk through your environments and get a list of how many hosts, data stores, vCenters, and get a rough count of the virtual machines deployed.
KB2093783 has more details on the sizing, and I strongly urge you to visit the KB, as there are links to the latest releases of vROps, and each KB has a sizing guide attachment at the bottom, where you can input the information you collected from your environment to get a more accurate size.
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Appliance Manual Installation
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Architectural Note
• Before proceeding be sure you have:
• The appropriate host resources
• The appropriate storage
• IP addresses assigned and entered into DNS
• The appropriate ports opened between data centers listed in VMWare’s documentation
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Once you have the latest edition of the vROps appliance ovf downloaded, and after consulting the documentation, use either the vSphere client or web, and deploy the OVF template. I’ll skip through browsing for, verifying the details of, accepting the licence agreement for, and naming the appliance.
So now you’ve come to the OVF deployment step where you must select the size of your appliance. No matter the size, the remainder of the deployment is the same, but for this example I will deploy an appliance as Large.
You can deploy the appliance in several sizing configurations depending on the size of your environment and those are: Extra Small, Small, Medium and Large.
• Extra Small = 2 vCPUs and 8GBs of memory
• Small = 4 vCPUs and 16GBs of memory
• Large= 16 vCPUs and 48GBs of memory
You can also choose to deploy a remote collector and they come in two sizes:
• Standard = 2 vCPUs and 4GBs of memory
• Large = 4 vCPUs and 16GBs of memory
You will notice that with each selection, VMware has given a definition of what it entails. Choose the one that best suits your needs. Click next
Storage dialog
• Depending on the size of your environment, vrops VMs can get to over a terabyte in size each
• Architectural Note – If adding a master replica node to your vROps cluster, I’d recommend keeping the Master and Master Replicas on separate XIVs, or whatever you use to serve up storage to your environment.
Disk Format dialog
• The default is Lazy Zeroed, and that’s how my environments have been deployed. I’d strongly advise not using thin provision for this appliance.
Network Mapping dialog
• Select the appropriate destination network like a management network, where it can capture traffic from your hosts, VMs, vCenters and datastores.
Properties dialog
• Here you can set the Timezone for the appliance, and choose whether to use IPv6
• Once you’ve filled out the network information, click next
Configuration Verification dialog
• Read it carefully to be sure there were no fat fingers at play. Click finish when ready.
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Before you proceed in turning on the appliance, you may want to take the opportunity now and expand its disk. This can be done a couple of ways. You can expand the existing Hard Disk 2, however keep in mind that the current file system can only see disks under 2TB. Any disk space allocated over 2TB the appliance wont be able to see. For my production environment, I increased disk 2 to 1TB in size, and then added 500GB disks as more storage was needed. Also keep in mind the amount of data you are going to be retaining. My appliances are configured for 6 months, but this can be changed as needs change. We’ll go over this later in another post. The cool thing about this appliance is that as you increase the size of disk 2, or add additional storage, the appliance during the power-on process, expands the data partition automatically.
Power up the appliance, open a console to it in vCenter to watch it boot up, and go through some scripted configurations.
• To get logged in, press ALT + F1 keys. Enter root for the user, leave the password blank and hit enter. Now you will be prompted to input the current password, so leave it blank and hit enter. Now enter a new password, hit enter and enter the new password once more for verification.
• Now depending on how locked down your environment is, you may not be able to but I always ping out to 8.8.8.8 along with hitting a few internal servers to verify network settings.
• Also unless you really enjoy VMware’s console, I’d recommend running a couple commands to turn on SSH, so any future administrative tasks can be performed with a putty session.
• The first command is: # chkconfig sshd on
• This enables the sshd service at system boot
• The second command is: # service sshd start
• This turns on the sshd service so you can connect to the box with a putty session.
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Using Microsoft Edge, Firefox or Chrome, browse to < https ://vrops-appliance-name/ >. This will redirect you to the Getting started page where you can choose Three options:
Express Installation, where you can set the admin password and that’s pretty much it.
New Installation gives you a few more options to configure, like which NTP server(s) you want to use, and a TLS/SSL certificate you’ve created specifically for this system (or just use the built-in one).
Expand An Existing Installation – this option would be used for additional data nodes or remote collectors as you’ll have the option to pick under “node type”.
For this installation we will select New Installation. As a rule of thumb and for better appliance performance, I’d use the NTP servers on your network that vCenter and the ESXi hosts are using to keep time in check. Once you’ve made it though the wizard click finish.
It shouldn’t take too long for the master appliance to setup and take you to a log in screen.
You’re not done yet however. You still have to configure your cluster if you have additional data nodes, and remote collectors to add. If you have a master replica, data nodes, or remote collector, get them connected to the master. Each will have their own web UI < https ://vrops-appliance-name/ >, only this time you can use the Expand An Existing Installation Option. You will also need to log into the admin section for some of this <https ://vrops-appliance-name/admin/login.action>
Lets get the master replica added first. When you use the expand an existing cluster option, you’ll need to add it as a data node. Then wait for the cluster to expand to it.
Then click the finish adding new nodes button.
To enable HA, you’ll notice in the center of the screen there is a High Availability option, but it is disabled. Go ahead an click enable
Now select the data node that will be the master replica, make sure enable high availability is checked, and click OK. This part will take a little while, and the cluster services will be restarted. After word the High Availability status will be enabled.
Add any remaining data nodes and remote collectors using the Expand An Existing Installation Option.
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Architectural Note
• I’d recommend going into vCenter and adding an anti-affinity rule to keep the master and master replica on separate hosts
• If you’ve deployed vROps to its own host cluster, I’d recommend turning down vSphere DRS to conservative. The appliances are usually pretty busy in an active environment, and having one vmotion on you can cause cluster performance degradation, and will throw some interesting alarms within vROps. It will recover on its own, but better to avoid when possible.
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Next up – You”ll need to configure the built in vSphere adapter so you can start collecting data. I’ll have more on that in my next post.
Recent Post: What Is VMware’s vRealize Operations Manager?
# What Is VMware’s vRealize Operations Manager?
Formally Known as vCenter Operations (vCOps), vRealize Operations Manager (vROps) really has become the center of the vSphere universe. vROps is an appliance that sits in your environment collecting system metrics from vCenter, virtual servers and ESXi. It acts as the single pane of glass to the virtual environment, allowing the administrator to track and mitigate resource contention, along with performance and capacity constraints. vROps will also “learn” about the environment, and given a couple of months, the data collection can be used to perform future calculations to determine things like when more capacity is required based on growth and resource consumption, and that’s pretty cool. Data collection is not just limited to VMware products however, you can also install additional management PAKs from VMware’s Solutions Exchange, and there are always more being added. Which brings us to the next topic: Sizing your vROps deployment.
Unlike vCOps which consisted of two virtual machines within a vApp, vROps is a single virtual appliance, that can be expanded and clustered for additional compute resources. A single appliance or node can be deployed and looks like the following in figure 1:
Figure 1 – A single vROps node
Now the cool thing about vROps is that it has the built in functionality of clustering two appliances together as a Master and Master Replica, giving you resiliency in case of a failure. You can also add additional nodes to the cluster known as Data nodes, that will allow you to collect and process even more metrics. It should go without saying, but as you add more Management PAKs from VMware’s solution exchange, keep in mind that you may need to add additional data nodes. It’s also important to note that the master and master replica servers can also be referred to as data nodes, and that is important because since vRealize Operations Manager 6.x, you can have a total of 16 data nodes in a cluster. That means you can have an HA pair, and 14 additional data nodes. You can deploy the appliance in several sizing configurations depending on the size of your environment and those are: Extra Small, Small, Medium and Large.
• Extra Small = 2 vCPUs and 8GBs of memory
• Small = 4 vCPUs and 16GBs of memory
• Large= 16 vCPUs and 48GBs of memory
You can also deploy additional appliances known as a Remote Collector, and since vRealize Operations Manager 6.1.x, you can have a total of 50 remote collectors, allowing the collection from 120,000 objects and 30,000,000 individual metrics. Now that’s a lot of data! Now these remote collectors come in different size configurations as well.
• Standard = 2 vCPUs and 4GBs of memory
• Large = 4 vCPUs and 16GBs of memory
In figure 2, this is what a clustered installation would look like and where remote collectors fit in.
Figure 2 – A vROps cluster
As you can see, the main cluster or the master, replica and data nodes all share a database and analytics processing engine, but the remote collector does not. It’s goal is simply to act as a vacuum, to collect and push the metrics collected from those remote data centers back to the main cluster for processing and storage.
All together this makes for a fantastic resource for troubleshooting and metrics data retention for historical data. I will caution that the vROps appliance requires a lot of CPU and memory depending on your environments configuration, and you should be sure to have ample resources supporting it. To get the most from this appliance, I’d also recommend at least one dedicated engineer to vROps, as there is a great deal of information to be had, and much to configure and maintain.
A Final Word
As someone who has been responsible for several large deployments, I can tell you this appliance has come a long way from its former days as VMware vCenter Operations Manager, and the developers dedicated to this platform are hard at work making it even better as it becomes the center of the software defined datacenter universe, within the VMware stack.
There are excellent blogs over at VMware that dives deeper into this appliance and it’s capabilities. For more information visit their site via blogs.vmware.com
# An Engineer’s Guide To The Galaxy Using KeePass
If you’re like me, and have multiple cloud environments, and multiple servers to manage, the task can be quite daunting. There are many paid for utilities out there that can help you with this task, but I’ve found that a utility called: KeePass Password Safe, does the job flawlessly if you are willing to do some custom configurations. KeePass is not just for IT engineers, it’s free and open source so anyone can use it. It really is the modern day Swiss Army knife for all geeks alike.
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The Benefits
• The database is encrypted using the most secure and best known algorithms AES and Twofish.
• It is password protected, so assuming you are following password best practices, the KeePass database with your environment variables will remain secure if it is misplaced.
• You can use KeePass from your local box, plus drop the same database file onto a jump server within your environment, to easily setup a secondary base of operations for yourself.
• IT IS FREE. Open source too (OSI certified).
You Know You Want It
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Now, there are many platforms that support KeePass, but this post will focus on Windows, as the majority of legacy IT departments are not too keen on running Linux or OSX in their environments, although those two platforms are quickly gaining traction in the modern era of hyper converged infrastructure.
The two key features I will focus on in this post are the abilities to use KeePass to open SSH and RDP sessions. Assuming you already have KeePass installed, go into the Tools menu and then click Options
Now go to the Integration tab in the options window, and click the URL Overrides button
We will be creating two custom URL Override entries one for SSH (putty session) and the other for RDP (Microsoft Remote Desktop)
Click the Add button to get started:
Assuming you installed Putty to its default directory, you need to tell KeePass where to find the executable. You can call Scheme whatever you wish, but for simplicity ssh was chosen for this example.
Scheme: ssh
URL override: cmd://”C:\Program Files (x86)\PuTTY\PuTTY.exe” -ssh {URL:RMVSCM}
Click OK when finished.
Now for RDP sessions we will need to string together several commands in order to get the desired result. Here we are calling MSTSC (RDP) through the command prompt, configuring a timeout, and passing through credentials. You can call Scheme whatever you wish, but for simplicity rdp was chosen for this example.
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# UR: Making the Most of Radio Peaks in Gamma Ray Burst Afterglows
The Undergraduate Research series is where we feature the research that you’re doing. If you are an undergraduate that took part in an REU or similar astro research project and would like to share this on Astrobites, please check out our submission page for more details. We would also love to hear about your more general research experience!
Ruby Duncan
George Washington University
This guest post was written by Ruby Duncan. Ruby is an undergraduate senior pursuing a degree in Astronomy and Astrophysics at George Washington University. She’s been working on research projects focusing on gamma-ray burst afterglows since the summer of 2018, though this work will be her capstone research project before graduating.
Gamma-ray bursts (GRBs) are caused by extremely energetic jets arising from cataclysmic events, which can produce as much energy in a few seconds as our Sun will in its entire lifetime (10 billion years!). These jets emit short bursts of gamma rays, followed by an afterglow of photons that range from X-rays all the way down to radio waves. Observations of GRBs and their afterglows across the entire electromagnetic spectrum can be used to study particle acceleration under extreme conditions.
There are several physical parameters that can inform us about a burst and its environment. These include the energy of the shock wave at the front of the jet, the density of the medium surrounding the burst, and the energy in the associated radiation-producing electrons. These parameters can be constrained by modeling the energy emitted by the GRB across the entire electromagnetic spectrum. This modeling is typically done using multiwavelength observations, but those observations are not always available. My work focuses on determining GRB parameters from restricted parts of the spectrum so that a robust picture of bursts can be developed even from limited observations.
The afterglow photons that are seen at radio wavelengths are especially useful. Peaks seen in the light curves from radio afterglow emission can be used to constrain the fraction of the shock energy that resides in electrons, $\epsilon_{e}$. This diagnostic tool, developed by Beniamini & van der Horst (2017) , depends on a relationship between the observables of a GRB, such as its redshift and observed energy, as well as the time and flux of the peak seen in the radio light curve. We have expanded upon their work by including peaks of radio spectra (frequency versus flux) along with radio light curves (flux versus time) and also by adding 15 more GRBs to the initial sample of 36. This allowed for a systematic check of the modeling approach, as well as enabling us to further constrain the distribution of $\epsilon_{e}$. We further expanded this work by exploring other parameters of particle acceleration that could potentially be constrained using our determined values of electron energy ($\epsilon_{e}$).
Astrobite edited by: Tarini Konchady
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# 15.7: Electrolytes and Nonelectrolytes
Millions of people in the world jog for exercise. For the most part, jogging can be a healthy way to stay fit. However, problems can also develop for those who jog in the heat. Excessive sweating can lead to electrolyte loss that could be life-threatening. Early symptoms of electrolyte deficiency can include nausea, fatigue, and dizziness. If not treated, individuals can experience muscle weakness and increased heart rate (which could lead to a heart attack). Many sports drinks can be consumed to restore electrolytes quickly in the body.
### Electrolytes and Nonelectrolytes
An electrolyte is a compound that conducts an electric current when it is in an aqueous solution or melted. In order to conduct a current, a substance must contain mobile ions that can move from one electrode to the other. All ionic compounds are electrolytes. When ionic compounds dissolve, they break apart into ions which are then able to conduct a current (conductivity). Even insoluble ionic compounds such as $$\ce{CaCO_3}$$ are electrolytes because they can conduct a current in the molten (melted) state.
nonelectrolyte is a compound that does not conduct an electric current in either aqueous solution or in the molten state. Many molecular compounds, such as sugar or ethanol, are nonelectrolytes. When these compounds dissolve in water, they do not produce ions. The figure below illustrates the difference between an electrolyte and a nonelectrolyte.
Figure 15.7.1: Conductivity apparatus.
### Roles of Electrolytes in the Body
Several electrolytes play important roles in the body. Here are a few significant electrolytes:
1. Calcium - in bones and teeth. Also important for muscle contraction, blood clotting, and nerve function.
2. Sodium - found outside the cell. Mainly involved in water balance as well as nerve signaling.
3. Potassium - major cation inside the cell. Important for proper functioning of heart, muscles, kidneys, and nerves.
4. Magnesium - in bone and cells. Involved in muscle, bone, nervous system, and takes part in many biochemical reactions.
### Summary
• Electrolytes conduct electric current when in solution or melted.
• Nonelectrolytes do not conduct electric current when in solution or melted.
• Some electrolytes play important roles in the body.
### Contributors
• CK-12 Foundation by Sharon Bewick, Richard Parsons, Therese Forsythe, Shonna Robinson, and Jean Dupon.
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• ### Neural network an1alysis of sleep stages enables efficient diagnosis of narcolepsy(1710.02094)
Feb. 28, 2019 cs.NE
Analysis of sleep for the diagnosis of sleep disorders such as Type-1 Narcolepsy (T1N) currently requires visual inspection of polysomnography records by trained scoring technicians. Here, we used neural networks in approximately 3,000 normal and abnormal sleep recordings to automate sleep stage scoring, producing a hypnodensity graph - a probability distribution conveying more information than classical hypnograms. Accuracy of sleep stage scoring was validated in 70 subjects assessed by six scorers. The best model performed better than any individual scorer (87% versus consensus). It also reliably scores sleep down to 5 instead of 30 second scoring epochs. A T1N marker based on unusual sleep-stage overlaps achieved a specificity of 96% and a sensitivity of 91%, validated in independent datasets. Addition of HLA-DQB1*06:02 typing increased specificity to 99%. Our method can reduce time spent in sleep clinics and automates T1N diagnosis. It also opens the possibility of diagnosing T1N using home sleep studies.
• ### On Estimation of Isotonic Piecewise Constant Signals(1705.06386)
Aug. 2, 2019 math.ST, stat.TH
Consider a sequence of real data points $X_1,\ldots, X_n$ with underlying means $\theta^*_1,\dots,\theta^*_n$. This paper starts from studying the setting that $\theta^*_i$ is both piecewise constant and monotone as a function of the index $i$. For this, we establish the exact minimax rate of estimating such monotone functions, and thus give a non-trivial answer to an open problem in the shape-constrained analysis literature. The minimax rate involves an interesting iterated logarithmic dependence on the dimension, a phenomenon that is revealed through characterizing the interplay between the isotonic shape constraint and model selection complexity. We then develop a penalized least-squares procedure for estimating the vector $\theta^*=(\theta^*_1,\dots,\theta^*_n)^T$. This estimator is shown to achieve the derived minimax rate adaptively. For the proposed estimator, we further allow the model to be misspecified and derive oracle inequalities with the optimal rates, and show there exists a computationally efficient algorithm to compute the exact solution.
• ### On inference validity of weighted U-statistics under data heterogeneity(1804.00034)
March 30, 2018 math.ST, stat.TH
Motivated by challenges on studying a new correlation measurement being popularized in evaluating online ranking algorithms' performance, this manuscript explores the validity of uncertainty assessment for weighted U-statistics. Without any commonly adopted assumption, we verify Efron's bootstrap and a new resampling procedure's inference validity. Specifically, in its full generality, our theory allows both kernels and weights asymmetric and data points not identically distributed, which are all new issues that historically have not been addressed. For achieving strict generalization, for example, we have to carefully control the order of the "degenerate" term in U-statistics which are no longer degenerate under the empirical measure for non-i.i.d. data. Our result applies to the motivating task, giving the region at which solid statistical inference can be made.
• ### An Extreme-Value Approach for Testing the Equality of Large U-Statistic Based Correlation Matrices(1502.03211)
March 30, 2018 math.ST, stat.TH, stat.ML
There has been an increasing interest in testing the equality of large Pearson's correlation matrices. However, in many applications it is more important to test the equality of large rank-based correlation matrices since they are more robust to outliers and nonlinearity. Unlike the Pearson's case, testing the equality of large rank-based statistics has not been well explored and requires us to develop new methods and theory. In this paper, we provide a framework for testing the equality of two large U-statistic based correlation matrices, which include the rank-based correlation matrices as special cases. Our approach exploits extreme value statistics and the Jackknife estimator for uncertainty assessment and is valid under a fully nonparametric model. Theoretically, we develop a theory for testing the equality of U-statistic based correlation matrices. We then apply this theory to study the problem of testing large Kendall's tau correlation matrices and demonstrate its optimality. For proving this optimality, a novel construction of least favourable distributions is developed for the correlation matrix comparison.
• ### Pairwise Difference Estimation of High Dimensional Partially Linear Model(1705.08930)
Jan. 12, 2018 math.ST, stat.TH, stat.ME
This paper proposes a regularized pairwise difference approach for estimating the linear component coefficient in a partially linear model, with consistency and exact rates of convergence obtained in high dimensions under mild scaling requirements. Our analysis reveals interesting features such as (i) the bandwidth parameter automatically adapts to the model and is actually tuning-insensitive; and (ii) the procedure could even maintain fast rate of convergence for $\alpha$-H\"older class of $\alpha\leq1/2$. Simulation studies show the advantage of the proposed method, and application of our approach to a brain imaging data reveals some biological patterns which fail to be recovered using competing methods.
• ### A Provable Smoothing Approach for High Dimensional Generalized Regression with Applications in Genomics(1509.07158)
July 21, 2017 stat.ME
In many applications, linear models fit the data poorly. This article studies an appealing alternative, the generalized regression model. This model only assumes that there exists an unknown monotonically increasing link function connecting the response $Y$ to a single index $X^T\beta^*$ of explanatory variables $X\in\mathbb{R}^d$. The generalized regression model is flexible and covers many widely used statistical models. It fits the data generating mechanisms well in many real problems, which makes it useful in a variety of applications where regression models are regularly employed. In low dimensions, rank-based M-estimators are recommended to deal with the generalized regression model, giving root-$n$ consistent estimators of $\beta^*$. Applications of these estimators to high dimensional data, however, are questionable. This article studies, both theoretically and practically, a simple yet powerful smoothing approach to handle the high dimensional generalized regression model. Theoretically, a family of smoothing functions is provided, and the amount of smoothing necessary for efficient inference is carefully calculated. Practically, our study is motivated by an important and challenging scientific problem: decoding gene regulation by predicting transcription factors that bind to cis-regulatory elements. Applying our proposed method to this problem shows substantial improvement over the state-of-the-art alternative in real data.
• ### Distribution-Free Tests of Independence in High Dimensions(1410.4179)
July 21, 2017 math.ST, stat.TH
We consider the testing of mutual independence among all entries in a $d$-dimensional random vector based on $n$ independent observations. We study two families of distribution-free test statistics, which include Kendall's tau and Spearman's rho as important examples. We show that under the null hypothesis the test statistics of these two families converge weakly to Gumbel distributions, and propose tests that control the type I error in the high-dimensional setting where $d>n$. We further show that the two tests are rate-optimal in terms of power against sparse alternatives, and outperform competitors in simulations, especially when $d$ is large.
• ### On Gaussian Comparison Inequality and Its Application to Spectral Analysis of Large Random Matrices(1607.01853)
May 26, 2017 math.ST, stat.TH, stat.ME
Recently, Chernozhukov, Chetverikov, and Kato [Ann. Statist. 42 (2014) 1564--1597] developed a new Gaussian comparison inequality for approximating the suprema of empirical processes. This paper exploits this technique to devise sharp inference on spectra of large random matrices. In particular, we show that two long-standing problems in random matrix theory can be solved: (i) simple bootstrap inference on sample eigenvalues when true eigenvalues are tied; (ii) conducting two-sample Roy's covariance test in high dimensions. To establish the asymptotic results, a generalized $\epsilon$-net argument regarding the matrix rescaled spectral norm and several new empirical process bounds are developed and of independent interest.
• ### Testing the science/technology relationship by analysis of patent citations of scientific papers after decomposition of both science and technology(1705.00258)
April 30, 2017 physics.soc-ph, cs.DL
The relationship of scientific knowledge development to technological development is widely recognized as one of the most important and complex aspects of technological evolution. This paper adds to our understanding of the relationship through use of a more rigorous structure for differentiating among technologies based upon technological domains (defined as consisting of the artifacts over time that fulfill a specific generic function using a specific body of technical knowledge).
• ### An Exponential Inequality for U-Statistics under Mixing Conditions(1609.06821)
Nov. 15, 2016 math.PR, math.ST, stat.TH
The family of U-statistics plays a fundamental role in statistics. This paper proves a novel exponential inequality for U-statistics under the time series setting. Explicit mixing conditions are given for guaranteeing fast convergence, the bound proves to be analogous to the one under independence, and extension to non-stationary time series is straightforward. The proof relies on a novel decomposition of U-statistics via exploiting the temporal correlatedness structure. Such results are of interest in many fields where high dimensional time series data are present. In particular, applications to high dimensional time series inference are discussed.
• ### ECA: High Dimensional Elliptical Component Analysis in non-Gaussian Distributions(1310.3561)
Oct. 3, 2016 stat.ML
We present a robust alternative to principal component analysis (PCA) --- called elliptical component analysis (ECA) --- for analyzing high dimensional, elliptically distributed data. ECA estimates the eigenspace of the covariance matrix of the elliptical data. To cope with heavy-tailed elliptical distributions, a multivariate rank statistic is exploited. At the model-level, we consider two settings: either that the leading eigenvectors of the covariance matrix are non-sparse or that they are sparse. Methodologically, we propose ECA procedures for both non-sparse and sparse settings. Theoretically, we provide both non-asymptotic and asymptotic analyses quantifying the theoretical performances of ECA. In the non-sparse setting, we show that ECA's performance is highly related to the effective rank of the covariance matrix. In the sparse setting, the results are twofold: (i) We show that the sparse ECA estimator based on a combinatoric program attains the optimal rate of convergence; (ii) Based on some recent developments in estimating sparse leading eigenvectors, we show that a computationally efficient sparse ECA estimator attains the optimal rate of convergence under a suboptimal scaling.
• ### Statistical analysis of latent generalized correlation matrix estimation in transelliptical distribution(1305.6916)
Sept. 28, 2016 stat.ML
Correlation matrices play a key role in many multivariate methods (e.g., graphical model estimation and factor analysis). The current state-of-the-art in estimating large correlation matrices focuses on the use of Pearson's sample correlation matrix. Although Pearson's sample correlation matrix enjoys various good properties under Gaussian models, it is not an effective estimator when facing heavy-tailed distributions. As a robust alternative, Han and Liu [J. Am. Stat. Assoc. 109 (2015) 275-287] advocated the use of a transformed version of the Kendall's tau sample correlation matrix in estimating high dimensional latent generalized correlation matrix under the transelliptical distribution family (or elliptical copula). The transelliptical family assumes that after unspecified marginal monotone transformations, the data follow an elliptical distribution. In this paper, we study the theoretical properties of the Kendall's tau sample correlation matrix and its transformed version proposed in Han and Liu [J. Am. Stat. Assoc. 109 (2015) 275-287] for estimating the population Kendall's tau correlation matrix and the latent Pearson's correlation matrix under both spectral and restricted spectral norms. With regard to the spectral norm, we highlight the role of "effective rank" in quantifying the rate of convergence. With regard to the restricted spectral norm, we for the first time present a "sign sub-Gaussian condition" which is sufficient to guarantee that the rank-based correlation matrix estimator attains the fast rate of convergence. In both cases, we do not need any moment condition.
• ### Robust Inference of Risks of Large Portfolios(1501.02382)
Jan. 10, 2015 math.ST, stat.TH, q-fin.PM
We propose a bootstrap-based robust high-confidence level upper bound (Robust H-CLUB) for assessing the risks of large portfolios. The proposed approach exploits rank-based and quantile-based estimators, and can be viewed as a robust extension of the H-CLUB method (Fan et al., 2015). Such an extension allows us to handle possibly misspecified models and heavy-tailed data. Under mixing conditions, we analyze the proposed approach and demonstrate its advantage over the H-CLUB. We further provide thorough numerical results to back up the developed theory. We also apply the proposed method to analyze a stock market dataset.
• ### Challenges of Big Data Analysis(1308.1479)
Dec. 15, 2014 stat.ML
Big Data bring new opportunities to modern society and challenges to data scientists. On one hand, Big Data hold great promises for discovering subtle population patterns and heterogeneities that are not possible with small-scale data. On the other hand, the massive sample size and high dimensionality of Big Data introduce unique computational and statistical challenges, including scalability and storage bottleneck, noise accumulation, spurious correlation, incidental endogeneity, and measurement errors. These challenges are distinguished and require new computational and statistical paradigm. This article give overviews on the salient features of Big Data and how these features impact on paradigm change on statistical and computational methods as well as computing architectures. We also provide various new perspectives on the Big Data analysis and computation. In particular, we emphasis on the viability of the sparsest solution in high-confidence set and point out that exogeneous assumptions in most statistical methods for Big Data can not be validated due to incidental endogeneity. They can lead to wrong statistical inferences and consequently wrong scientific conclusions.
• ### A Direct Estimation of High Dimensional Stationary Vector Autoregressions(1307.0293)
Oct. 29, 2014 stat.ML
The vector autoregressive (VAR) model is a powerful tool in modeling complex time series and has been exploited in many fields. However, fitting high dimensional VAR model poses some unique challenges: On one hand, the dimensionality, caused by modeling a large number of time series and higher order autoregressive processes, is usually much higher than the time series length; On the other hand, the temporal dependence structure in the VAR model gives rise to extra theoretical challenges. In high dimensions, one popular approach is to assume the transition matrix is sparse and fit the VAR model using the "least squares" method with a lasso-type penalty. In this manuscript, we propose an alternative way in estimating the VAR model. The main idea is, via exploiting the temporal dependence structure, to formulate the estimating problem into a linear program. There is instant advantage for the proposed approach over the lasso-type estimators: The estimation equation can be decomposed into multiple sub-equations and accordingly can be efficiently solved in a parallel fashion. In addition, our method brings new theoretical insights into the VAR model analysis. So far the theoretical results developed in high dimensions (e.g., Song and Bickel (2011) and Kock and Callot (2012)) mainly pose assumptions on the design matrix of the formulated regression problems. Such conditions are indirect about the transition matrices and not transparent. In contrast, our results show that the operator norm of the transition matrices plays an important role in estimation accuracy. We provide explicit rates of convergence for both estimation and prediction. In addition, we provide thorough experiments on both synthetic and real-world equity data to show that there are empirical advantages of our method over the lasso-type estimators in both parameter estimation and forecasting.
• ### On the Impact of Dimension Reduction on Graphical Structures(1404.7547)
Oct. 13, 2014 stat.ME
Statisticians and quantitative neuroscientists have actively promoted the use of independence relationships for investigating brain networks, genomic networks, and other measurement technologies. Estimation of these graphs depends on two steps. First is a feature extraction by summarizing measurements within a parcellation, regional or set definition to create nodes. Secondly, these summaries are then used to create a graph representing relationships of interest. In this manuscript we study the impact of dimension reduction on graphs that describe different notions of relations among a set of random variables. We are particularly interested in undirected graphs that capture the random variables' independence and conditional independence relations. A dimension reduction procedure can be any mapping from high dimensional spaces to low dimensional spaces. We exploit a general framework for modeling the raw data and advocate that in estimating the undirected graphs, any acceptable dimension reduction procedure should be a graph-homotopic mapping, i.e., the graphical structure of the data after dimension reduction should inherit the main characteristics of the graphical structure of the raw data. We show that, in terms of inferring undirected graphs that characterize the conditional independence relations among random variables, many dimension reduction procedures, such as the mean, median, or principal components, cannot be theoretically guaranteed to be a graph-homotopic mapping. The implications of this work are broad. In the most charitable setting for researchers, where the correct node definition is known, graphical relationships can be contaminated merely via the dimension reduction. The manuscript ends with a concrete example, characterizing a subset of graphical structures such that the dimension reduction procedure using the principal components can be a graph-homotopic mapping.
• ### Joint Estimation of Multiple Graphical Models from High Dimensional Time Series(1311.0219)
Oct. 8, 2014 stat.ML
In this manuscript we consider the problem of jointly estimating multiple graphical models in high dimensions. We assume that the data are collected from n subjects, each of which consists of T possibly dependent observations. The graphical models of subjects vary, but are assumed to change smoothly corresponding to a measure of closeness between subjects. We propose a kernel based method for jointly estimating all graphical models. Theoretically, under a double asymptotic framework, where both (T,n) and the dimension d can increase, we provide the explicit rate of convergence in parameter estimation. It characterizes the strength one can borrow across different individuals and impact of data dependence on parameter estimation. Empirically, experiments on both synthetic and real resting state functional magnetic resonance imaging (rs-fMRI) data illustrate the effectiveness of the proposed method.
• ### High Dimensional Semiparametric Scale-Invariant Principal Component Analysis(1402.4507)
Feb. 18, 2014 stat.ML
We propose a new high dimensional semiparametric principal component analysis (PCA) method, named Copula Component Analysis (COCA). The semiparametric model assumes that, after unspecified marginally monotone transformations, the distributions are multivariate Gaussian. COCA improves upon PCA and sparse PCA in three aspects: (i) It is robust to modeling assumptions; (ii) It is robust to outliers and data contamination; (iii) It is scale-invariant and yields more interpretable results. We prove that the COCA estimators obtain fast estimation rates and are feature selection consistent when the dimension is nearly exponentially large relative to the sample size. Careful experiments confirm that COCA outperforms sparse PCA on both synthetic and real-world datasets.
• ### Sparse Median Graphs Estimation in a High Dimensional Semiparametric Model(1310.3223)
Oct. 11, 2013 stat.AP
In this manuscript a unified framework for conducting inference on complex aggregated data in high dimensional settings is proposed. The data are assumed to be a collection of multiple non-Gaussian realizations with underlying undirected graphical structures. Utilizing the concept of median graphs in summarizing the commonality across these graphical structures, a novel semiparametric approach to modeling such complex aggregated data is provided along with robust estimation of the median graph, which is assumed to be sparse. The estimator is proved to be consistent in graph recovery and an upper bound on the rate of convergence is given. Experiments on both synthetic and real datasets are conducted to illustrate the empirical usefulness of the proposed models and methods.
• ### Sparse Principal Component Analysis for High Dimensional Vector Autoregressive Models(1307.0164)
June 30, 2013 stat.ML
We study sparse principal component analysis for high dimensional vector autoregressive time series under a doubly asymptotic framework, which allows the dimension $d$ to scale with the series length $T$. We treat the transition matrix of time series as a nuisance parameter and directly apply sparse principal component analysis on multivariate time series as if the data are independent. We provide explicit non-asymptotic rates of convergence for leading eigenvector estimation and extend this result to principal subspace estimation. Our analysis illustrates that the spectral norm of the transition matrix plays an essential role in determining the final rates. We also characterize sufficient conditions under which sparse principal component analysis attains the optimal parametric rate. Our theoretical results are backed up by thorough numerical studies.
• ### Sterile Neutrino Search Using China Advanced Research Reactor(1303.0607)
June 18, 2013 hep-ex, physics.ins-det
We study the feasibility of a sterile neutrino search at the China Advanced Research Reactor by measuring $\bar {\nu}_e$ survival probability with a baseline of less than 15 m. Both hydrogen and deuteron have been considered as potential targets. The sensitivity to sterile-to-regular neutrino mixing is investigated under the "3(active)+1(sterile)" framework. We find that the mixing parameter $\sin^2(2\theta_{14})$ can be severely constrained by such measurement if the mass square difference $\Delta m_{14}^2$ is of the order of $\sim$1 eV$^2$.
• ### High Dimensional Semiparametric Gaussian Copula Graphical Models(1202.2169)
July 27, 2012 stat.ML
In this paper, we propose a semiparametric approach, named nonparanormal skeptic, for efficiently and robustly estimating high dimensional undirected graphical models. To achieve modeling flexibility, we consider Gaussian Copula graphical models (or the nonparanormal) as proposed by Liu et al. (2009). To achieve estimation robustness, we exploit nonparametric rank-based correlation coefficient estimators, including Spearman's rho and Kendall's tau. In high dimensional settings, we prove that the nonparanormal skeptic achieves the optimal parametric rate of convergence in both graph and parameter estimation. This celebrating result suggests that the Gaussian copula graphical models can be used as a safe replacement of the popular Gaussian graphical models, even when the data are truly Gaussian. Besides theoretical analysis, we also conduct thorough numerical simulations to compare different estimators for their graph recovery performance under both ideal and noisy settings. The proposed methods are then applied on a large-scale genomic dataset to illustrate their empirical usefulness. The R language software package huge implementing the proposed methods is available on the Comprehensive R Archive Network: http://cran. r-project.org/.
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# Mixed model with lmer: Variance of residuals should give the same as level 1 variance?
I expected that the variance of residuals from a mixed model computed by, for example, lmer should give the same as the residual variance from the summary output.
set.seed(123)
# -- Data simulation
group <- rep(letters[1:10], each=10)
groupx <- rep(rnorm(10,0,5), each=10)
x0 <- 1:10
y0 <- rep(3*x0,10)
y <- y0+groupx + rnorm(100,0,3)
data0 <- data.frame(grp=group,x=x0,y=y)
# -- library(lattice)
xyplot(y ~ x|grp, data=data0)
# -- Model
f0 <- lmer(y ~ x + (1|grp), data= data0, REML=FALSE)
f0s <- summary(f0)
f0s
# -- Compare level 1 variance (Residual):
as.data.frame(VarCorr(f0))[,c("grp","vcov")]
# -- with variance of residuals:
var(resid(f0))
In my example I got:
> as.data.frame(VarCorr(f0))[,c("grp","vcov")]
grp vcov
1 grp 20.621959
2 Residual 7.090458
> var(resid(f0))
[1] 6.469677
I expected that var(resid(f0)) should be exactly the same as the variance of Residuals from summary. Could someone gives me a hint what is wrong in my argumentation?
UPDATE 1: According to the formula in link
$\hat{\sigma}^2_{\alpha}=E\left[\frac{1}{10}\sum_{s=1}^{10} \alpha_s^2\right]= \frac{1}{10}\sum_{s=1}^{10}\hat{ \alpha }_s^2 +\frac{1}{10}\sum_{s=1}^{48}var(\hat{ \alpha }_s)$
I tried to compute the variance of the random variable:
> 1/10 * sum((se.ranef(f0)$grp)^2) + var(ranef(f0)$grp)
(Intercept)
(Intercept) 22.83712
which seems quite different compared to 20.621959 in the summary above.
With method REML I got the following results:
f1 <- lmer(y ~ x + (1|grp), data= data0, REML=TRUE)
f1s <- summary(f1)
as.data.frame(VarCorr(f1))[,c("grp","vcov")]
1/10 * sum((se.ranef(f1)$grp)^2) + var(ranef(f1)$grp)
> f1 <- lmer(y ~ x + (1|grp), data= data0, REML=TRUE)
> f1s <- summary(f1)
> as.data.frame(VarCorr(f1))[,c("grp","vcov")]
grp vcov
1 grp 22.984104
2 Residual 7.170126
> 1/10 * sum((se.ranef(f1)$grp)^2) + var(ranef(f1)$grp)
(Intercept)
(Intercept) 22.984
Obviously, the formula is correct when method REML is used.
With
var(resid(f0))*99/90
I got a SD for residuals of 7.116645 which is near to 7.09. But how to justify a degree of freedom of 90? Or does it makes no sense to speak about df in mixed models?
• Does this answer you question? stats.stackexchange.com/questions/69882/… – Steve Sep 16 '14 at 13:42
• @Steve Thanks a lot for the link. Supposing that the statement at the link is also valid for EML (and not only for REML) it does answer my question. – giordano Sep 17 '14 at 10:59
• I think so.. I played around a bit with different simulations and formulas and I think that's the issue. If you have a very big sample you see that the estimator and the var of residuals are pretty much the same, which makes some sense. – Steve Sep 17 '14 at 11:24
• @Steve Oviously, the formula is valid for REML but not EML. – giordano May 1 '16 at 17:43
You might want to have a look at Interpretation of various output of "lmer" function in R.
Basically the estimated residual variance $\hat{\sigma}$ is an estimation for a population parameter $\sigma$. Whereas the estimated variance of your residual is an estimation based on a subset of your population. You will encounter the same thing in simple linear models:
set.seed(123)
x <- rnorm(50)
y <- 1+2*x + rnorm(50, sd = 2)
mod <- lm(y~x)
summary(mod)\$sigma #1.828483
sd(resid(mod)) #1.809729
• Thank you very much for this hint. I can compute the sgima from the residuals: sqrt(sum(resid(mod)2/(50−2))) or sd(resid(mod))∗sqrt(49/48) gives both 1.828483. But I cannot reproduce the standard deviation of the random effect and the residuals in lmer (see update 1). – giordano 20 hours ago – giordano May 2 '16 at 14:08
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# Sequences and Series Grade 11 Revision
Quiz
Materials
Consider the following quadratic pattern: $$8 ; 18 ; 30 ; 44 ; …$$
0
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# statistics
posted by Kabelo
How large a sample should be taken if the population mean is to be estimated with 99% confidence to within \$75? The population has a standard deviation of \$900.
1. MsPi_3.14159265
Thanks to Dr Bob for presenting this solution earlier : )
mu = bar x +/- 3(sigma/sqrt N)
mu is the population mean.
bar x is the average.
3 gives the 99% confidence interval.
sigma is standard deviation. N is the number of sample. Solve for N.
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# Table of content as right side growing tree
\documentclass{article}
\usepackage{tikz}
\usepackage{tikz-qtree}
\begin{document}
\begin{tikzpicture}
\tikzset{grow'=right,level distance=32pt}
\tikzset{execute at begin node=\strut}
\tikzset{every tree node/.style={anchor=base west}}
\Tree [.S [.Introduction ] [.Taxonomy
] ]
\end{tikzpicture}
\section{Introduction}
\section{Taxonomy}
\section{User Association}
\end{document}
• How much have you got so far? Please help us help you and add a minimal working example (MWE) that illustrates your problem. Reproducing the problem and finding out what the issue is will be much easier when we see compilable code, starting with \documentclass{...} and ending with \end{document}. – DG' Apr 17 '17 at 8:31
• I tried using qtree, however I don't know how to add table of contents in qtree, thus, I thought MWE is no use here. – Mithun Apr 17 '17 at 16:06
• A MWE is always useful. And almost mandatory in your case, since we don't know anything about your document (i.e. documentclass, packages, etc.) – DG' Apr 17 '17 at 16:10
• Ok, I'll add some that might be helpful. – Mithun Apr 17 '17 at 16:11
• @DG' , I added, please kindly check. – Mithun Apr 17 '17 at 16:43
Definitely shouldn't have tried to answer this ....
Proof of concept:
\documentclass{article}
\usepackage[edges]{forest}
\forestset{%
subsection/.style={%
delay n={%
>O{n} %
}{%
for preceding siblings={do dynamics},
temptoksa/.option=name,
if nodewalk valid={previous}{%
for previous={%
if section={append/.register=temptoksa}{}%
}%
}{%
replace by={[, coordinate, append]},
},
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declare boolean={section}{0},
}
\makeatletter
\newcommand\foresttoc{%
\begingroup
\xdef\foresttoctoc{}%
\renewcommand\contentsline[3]{%
\let\tempb\foresttoctoc
\xdef\foresttoctoc{\tempb[{##2}, ##1]}%
}%
\renewcommand\numberline[1]{Sec.~##1 }%
\bracketset{action character=@}%
\@starttoc{toc}%
\begin{forest}
before typesetting nodes={
forked edges,
for tree={
grow'=0,
anchor=parent,
tier/.option=level,
},
where level=1{fork sep'=0pt}{},
},
[, coordinate, @+\foresttoctoc ]
\end{forest}%
\endgroup
}
\makeatother
\begin{document}
\foresttoc
% \tableofcontents
\subsection{Awkward}
\section{User Association}
\section{Introduction}
\section{Taxonomy}
\section{User Association}
• @Mithun Please see edited version above. A regular ToC is now permitted, but no longer required. (But we still use the .toc file and apparatus to collect the information etc.) – cfr Apr 21 '17 at 0:26
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# damp
Natural frequency and damping ratio
## Syntax
``damp(sys)``
``````[wn,zeta] = damp(sys)``````
``````[wn,zeta,p] = damp(sys)``````
## Description
example
````damp(sys)` displays the damping ratio, natural frequency, and time constant of the poles of the linear model `sys`. For a discrete-time model, the table also includes the magnitude of each pole. The poles are sorted in increasing order of frequency values.```
example
``````[wn,zeta] = damp(sys)``` returns the natural frequencies `wn`, and damping ratios `zeta` of the poles of `sys`.```
example
``````[wn,zeta,p] = damp(sys)``` also returns the poles `p` of `sys`.```
## Examples
collapse all
For this example, consider the following continuous-time transfer function:
`$sys\left(s\right)=\frac{2{s}^{2}+5s+1}{{s}^{3}+2s-3}.$`
Create the continuous-time transfer function.
`sys = tf([2,5,1],[1,0,2,-3]);`
Display the natural frequencies, damping ratios, time constants, and poles of `sys`.
`damp(sys)`
``` Pole Damping Frequency Time Constant (rad/seconds) (seconds) 1.00e+00 -1.00e+00 1.00e+00 -1.00e+00 -5.00e-01 + 1.66e+00i 2.89e-01 1.73e+00 2.00e+00 -5.00e-01 - 1.66e+00i 2.89e-01 1.73e+00 2.00e+00 ```
The poles of `sys` contain an unstable pole and a pair of complex conjugates that lie int he left-half of the s-plane. The corresponding damping ratio for the unstable pole is -1, which is called a driving force instead of a damping force since it increases the oscillations of the system, driving the system to instability.
For this example, consider the following discrete-time transfer function with a sample time of 0.01 seconds:
$sys\left(z\right)=\frac{5{z}^{2}+3z+1}{{z}^{3}+6{z}^{2}+4z+4}$.
Create the discrete-time transfer function.
`sys = tf([5 3 1],[1 6 4 4],0.01)`
```sys = 5 z^2 + 3 z + 1 --------------------- z^3 + 6 z^2 + 4 z + 4 Sample time: 0.01 seconds Discrete-time transfer function. ```
Display information about the poles of `sys` using the `damp` command.
`damp(sys)`
``` Pole Magnitude Damping Frequency Time Constant (rad/seconds) (seconds) -3.02e-01 + 8.06e-01i 8.61e-01 7.74e-02 1.93e+02 6.68e-02 -3.02e-01 - 8.06e-01i 8.61e-01 7.74e-02 1.93e+02 6.68e-02 -5.40e+00 5.40e+00 -4.73e-01 3.57e+02 -5.93e-03 ```
The `Magnitude` column displays the discrete-time pole magnitudes. The `Damping`, `Frequency`, and `Time Constant` columns display values calculated using the equivalent continuous-time poles.
For this example, create a discrete-time zero-pole-gain model with two outputs and one input. Use sample time of 0.1 seconds.
`sys = zpk({0;-0.5},{0.3;[0.1+1i,0.1-1i]},[1;2],0.1)`
```sys = From input to output... z 1: ------- (z-0.3) 2 (z+0.5) 2: ------------------- (z^2 - 0.2z + 1.01) Sample time: 0.1 seconds Discrete-time zero/pole/gain model. ```
Compute the natural frequency and damping ratio of the zero-pole-gain model `sys`.
`[wn,zeta] = damp(sys)`
```wn = 3×1 12.0397 14.7114 14.7114 ```
```zeta = 3×1 1.0000 -0.0034 -0.0034 ```
Each entry in `wn` and `zeta` corresponds to combined number of I/Os in `sys`. `zeta` is ordered in increasing order of natural frequency values in `wn`.
For this example, compute the natural frequencies, damping ratio and poles of the following state-space model:
`$A=\left[\begin{array}{cc}-2& -1\\ 1& -2\end{array}\right],\phantom{\rule{1em}{0ex}}B=\left[\begin{array}{cc}1& 1\\ 2& -1\end{array}\right],\phantom{\rule{1em}{0ex}}C=\left[\begin{array}{cc}1& 0\end{array}\right],\phantom{\rule{1em}{0ex}}D=\left[\phantom{\rule{0.1em}{0ex}}\begin{array}{cc}0& 1\end{array}\right].$`
Create the state-space model using the state-space matrices.
```A = [-2 -1;1 -2]; B = [1 1;2 -1]; C = [1 0]; D = [0 1]; sys = ss(A,B,C,D);```
Use `damp` to compute the natural frequencies, damping ratio and poles of `sys`.
`[wn,zeta,p] = damp(sys)`
```wn = 2×1 2.2361 2.2361 ```
```zeta = 2×1 0.8944 0.8944 ```
```p = 2×1 complex -2.0000 + 1.0000i -2.0000 - 1.0000i ```
The poles of `sys` are complex conjugates lying in the left half of the s-plane. The corresponding damping ratio is less than 1. Hence, `sys` is an underdamped system.
## Input Arguments
collapse all
Linear dynamic system, specified as a SISO, or MIMO dynamic system model. Dynamic systems that you can use include:
• Continuous-time or discrete-time numeric LTI models, such as `tf` (Control System Toolbox), `zpk` (Control System Toolbox), or `ss` (Control System Toolbox) models.
• Generalized or uncertain LTI models such as `genss` (Control System Toolbox) or `uss` (Robust Control Toolbox) models. (Using uncertain models requires Robust Control Toolbox™ software.)
`damp` assumes
• current values of the tunable components for tunable control design blocks.
• nominal model values for uncertain control design blocks.
## Output Arguments
collapse all
Natural frequency of each pole of `sys`, returned as a vector sorted in ascending order of frequency values. Frequencies are expressed in units of the reciprocal of the `TimeUnit` property of `sys`.
If `sys` is a discrete-time model with specified sample time, `wn` contains the natural frequencies of the equivalent continuous-time poles. If the sample time is not specified, then `damp` assumes a sample time value of 1 and calculates `wn` accordingly. For more information, see Algorithms.
Damping ratios of each pole, returned as a vector sorted in the same order as `wn`.
If `sys` is a discrete-time model with specified sample time, `zeta` contains the damping ratios of the equivalent continuous-time poles. If the sample time is not specified, then `damp` assumes a sample time value of 1 and calculates `zeta` accordingly. For more information, see Algorithms.
Poles of the dynamic system model, returned as a vector sorted in the same order as `wn`. `p` is the same as the output of `pole(sys)`, except for the order. For more information on poles, see `pole`.
## Algorithms
`damp` computes the natural frequency, time constant, and damping ratio of the system poles as defined in the following table:
Continuous TimeDiscrete Time with Sample Time Ts
Pole Location
`$s$`
`$z$`
Equivalent Continuous-Time Pole
`$s=\frac{ln\left(z\right)}{{T}_{s}}$`
Natural Frequency
`${\omega }_{n}=|s|$`
`${\omega }_{n}=|s|=|\frac{ln\left(z\right)}{{T}_{s}}|$`
Damping Ratio
`$\zeta =-cos\left(\angle s\right)$`
`$\begin{array}{lll}\zeta \hfill & =-cos\left(\angle s\right)\hfill & =-cos\left(\angle ln\left(z\right)\right)\hfill \end{array}$`
Time Constant
`$\tau =\frac{1}{{\omega }_{n}\zeta }$`
`$\tau =\frac{1}{{\omega }_{n}\zeta }$`
If the sample time is not specified, then `damp` assumes a sample time value of 1 and calculates `zeta` accordingly.
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Waiting for answer This question has not been answered yet. You can hire a professional tutor to get the answer.
QUESTION
# How is the kinetic theory of matter related to gases?
The kinetic molecular theory tells us how gas molecules are supposed to behave. In doing this, it makes a few assumptions about how gases behave. Let's have a look at them and see if they're right:
• Gas particles are infinitely small. In the real world, atoms and molecules are not infinitely small, making this incorrect.
• Intermolecular forces can be ignored. Because all gas particles experience intermolecular forces of one kind or another, this is also incorrect.
• Gas particles undergo completely elastic collisions (i.e. collisions that don't result in the loss of energy). Perfectly elastic collisions don't happen because some energy isn't transferred, making this wrong.
• The kinetic energy of gas particles is directly proportional to the temperature in Kelvin. As far as I know, this is correct.
OK.
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# Definition of geometric monodromy
Consider a polynomial $$f \in \mathbb C[x_1,\dots ,x_n]$$. An atypical value of $$f$$ is a complex number about which $$f:\mathbb C^n\to \mathbb C$$ is not a topological fiber bundle. Writing $$\mathrm{Atyp}(f)$$ for the atypical values of $$f$$ we thus have a fiber bundle $$\mathbb C^n\setminus f^{-1}(\mathrm{Atyp}(f))\to \mathbb C\setminus \mathrm{Atyp}(f).$$
A book I'm reading now says the following.
The fundamental group $$\pi_1(\mathbb C\setminus \mathrm{Atyp}(f))$$ acts therefore on the generic fiber $$f^{-1}(t_0)$$ up to isotopy. The image of this action is called the geometric monodromy group. The pullback of $$f$$ along an admissible loop (image doesn't hit atypical values) is a fiber bundle which yields an automorphism of the fiber, called the geometric monodromy.
I am confused by this. We have for any Hurewicz fibration $$E\to B$$ the transport functor $$\pi_1(B)\to \mathsf{hTop}$$ which lands in the homotopy category. This is the only meaning I can think of for "acts up to isotopy". On the other hand, this does not give an automorphism of the fiber in the topological category.
Question. Is the geometric monodromy defined as the homotopy type of the continuous map between fibers given by the homotopy lifting property? Is the geometric monodromy group(oid) given by the image $$\pi_1(B)$$ in $$\mathsf{hTop}$$?
• Take every closed curve $\gamma(r), r \in [a,b], \gamma(a) = t_0$ and every lift $\phi(r) \in \text{Isom}(f^{-1}(t_0),f^{-1}(\gamma(r)), \phi(a)=\text{Id} \in \text{Aut}(f^{-1}(t_0))$ and make a group from all the obtained $\phi(b) \in \text{Aut}(f^{-1}(t_0))$, it will contain the automorphisms isotopic to the identity plus some other isotopy classes. Jun 23 '19 at 1:43
As you say, a Hurewicz fibration defines a map from $$\pi_1(B)$$ to the automorphisms of the fiber in the homotopy category.
But a topological fiber bundle has more structure than a Hurewicz fibration. In fact, a topological fiber bundle defines a map from $$\pi_1(B)$$ to the mapping class group of the fiber (i.e. the component group of the group of topological automorphisms of the fiber). This is what "acts up to isotopy" means.
To see this, maybe one way is to observe that the set of pairs of a point in $$B$$ and an isomorphism between the fiber over that point and $$F$$ is an an $$\operatorname{Aut}(F)$$-torsor, and modding out by the identity component gives a $$\pi_0(\operatorname{Aut}(F))$$-torsor, which gives a homomorphism $$\pi_1(B) \to \pi_0(\operatorname{Aut}(F))$$.
• @Arrow In this context, the geometric monodromy is the homomorphism from $\pi_1(B)$ to $\pi_0(\operatorname{Aut}(F))$, and the geometric monodromy group is its image. (In other contexts, these terms will mean slightly different things). Jun 23 '19 at 21:02
• @Arrow With regards to a reference, I don't know one, and wasn't able to find one with Google. But I don't really understand what needs a reference. The second paragraph makes claims, which the third paragraph proves. I can add a little more detail: To check this set is a torsor, we can work locally over an open set $U$ where the fiber bundle trivializes. Then we must check that the set of pairs of a point in $U$ and an isomorphism between $F$ and $F$ is an $\operatorname{Aut}(F)$-torsor, which is obvious. Jun 23 '19 at 21:13
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[1] The equal kid points to the next character in the word. 1 You signed in with another tab or window. We were delighted to find that the trees also lead to practical computer programs. M Like other prefix trees, a ternary search tree can be used as an associative map structure with the ability for incremental string search. Dr. Dobb's is part of the Informa Tech Division of Informa PLC. + Say that the external node being added onto is node A. This is very good for doing search and are many many applications for the same in the real world. For every node, all nodes down the left child have smaller values, while all nodes down the right child have greater values. Some programmers might feel more comfortable with the iterative version of the search function (which has only one argument): This is substantially faster on some compilers, and we'll use it in later experiments. 1 Defines big text 1 Self-balancing BSTs are flexible data structures, in that it's easy to extend them to efficiently record additional information or perform new operations. +2h = 2h+1−1 nodes. , = Tarjan describe theoretical balancing algorithms for ternary search trees in "Self-Adjusting Binary Search Trees" (Journal of the ACM, July 1985). [1] Additionally, hash maps do not allow for many of the uses of ternary search trees, such as near-neighbor lookups. This Dr. Dobb's article Ternary Search Trees By Jon Bentley and Bob Sedgewick provides the following guidance: You can build a completely balanced tree by inserting the median element of the input set, then recursively inserting all lesser elements and greater elements. If, however, the key's first character is equal to the node's value then the insertion procedure is called on the equal kid and the key's first character is pruned away. 3 Due to this, the worst case time-complexity of operations such as search, insertion and deletion is as the height of a 2-3 tree is . We described the theory of ternary search trees at a symposium in 1997 (see "Fast Algorithms for Sorting and Searching Strings," by J.L. ⌈ + This recursive method is continually called on nodes of the tree given a key which gets progressively shorter by pruning characters off the front of the key. Each node in a ternary search tree is represented by the following structure: The value stored at the node is splitchar, and the three pointers represent the three children. {\displaystyle n\leq 2^{h+1}-1}. Informa PLC is registered in England and Wales with company number 8860726 whose registered and head office is 5 Howick Place, London, SW1P 1WG. Access is fast, but information about relative order is lost. Contact UsAbout UsRefund PolicyPrivacy PolicyServices DisclaimerTerms and Conditions, Accenture For example, when the items are inserted in sorted key order, the tree degenerates into a linked list with n nodes. O ( Ternary search trees combine attributes of binary search trees and digital search tries. log When we want to enter a new node, then we will rotate the tree to make sure that the tree is always balanced. 1 log {\displaystyle \lfloor \log _{2}n\rfloor } 2 h [1] Like binary search trees and other data structures, ternary search trees can become degenerate depending on the order of the keys. italic text, Defines an anchor = Clampett sketched a primitive version of the structure. [1] Therefore, most self-balanced BST algorithms keep the height within a constant factor of this lower bound. D.D. When adding a new node, we should always rotate the tree to make sure that the tree’s properties are always maintained. You could use hash tables, which sprinkle the strings throughout an array. Any node in the data structure can be reached by starting at root node and repeatedly following references to either the left, mid or right child. Digital search tries to store strings character by character. When you have to store a set of strings, what data structure should you use? If this method reaches a node that has not been created, it creates the node and assigns it the character value of the first character in the key. + h + To find the word "ON", we compare it to "IN" and take the right branch. Defines computer code text ≤ In a tree representing words of lowercase letters, each node has 26-way branching (though most branches are empty, and not shown in Figure 2). Nodes can be inserted into ternary trees in between three other nodes or added after an external node. Dr. Dobb's Journal is devoted to mobile programming. As a final case, if the first character of the string is equal to the character of the current node then the function returns the node if there are no more characters in the key. Although a certain overhead is involved, it may be justified in the long run by ensuring fast execution of later operations. It follows that for any tree with n nodes and height h: n We help students to prepare for placements with the best study material, online classes, Sectional Statistics for better focus and Success stories & tips by Toppers on PrepInsta. they're used to log you in. ⌋ A generalization of the binary search tree is the B-Tree, which works for fanouts anywhere from 2 and up. Data structures implementing this type of tree include: Self-balancing binary search trees can be used in a natural way to construct and maintain ordered lists, such as priority queues. The picture below is a binary search tree that represents 12 two-letter words. The running time of ternary search trees varies significantly with the input. Even though this tree is not a fully balanced tree, the searching operation only takes O (log n) time. ( The root of the tree is declared to be Tptr root. {\displaystyle {\frac {3^{h+1}-1}{2}}} 2 + The following code illustrates that shorter is not always cleaner, simpler, faster, or better: These tags can be used alone and don't need an ending tag. [1], This article is about rooted trees with three children per node. h h View, Insert and remove happen at O(log n), Generally, we apply it when we are going operations like lookup, example searching in your phone’s contacts directory, This another example of a tree that is auto-balancing, The name is given because either a node will be painted as red or black, according to red-black tree properties. We use optional third-party analytics cookies to understand how you use GitHub.com so we can build better products. ( 3 For example, if binary tree sort is implemented with a self-balanced BST, we have a very simple-to-describe yet asymptotically optimal O(n log n) sorting algorithm. Leon Balogun Sister, Kamen Rider Grease New World Kissasian, Is There A Problem With Telus Optik Tv, Fly Fishing Deadwood Reservoir Idaho, Homer App Vs Abc Mouse, Fablehaven Book 1 Theme, Happy Anderson John Goodman,
balanced ternary search tree 2020
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# Which way is correct and why?
## Homework Statement:
This week's question
A race car driver travels x km at 40km/h and y km at 50 km/h. The total time taken is 2.5 hrs. If the time to travel 6x km at 30km/h and 4y km at 50km/h is 14 hrs. Find x and y
## Relevant Equations:
Speed = distance /time
Ave speed = total distance/ total time
Ave speed= (speed 1 + speed 2 ) /2
I get two different answers even the area of the graph.
I think there is something wrong with the equation I constructed for average speed part.
I know the method 1 is correct. But I'd like to know why the average speed cannot be used. And why I get two different areas for the graph
Thank you
## Answers and Replies
Related Introductory Physics Homework Help News on Phys.org
haruspex
Science Advisor
Homework Helper
Gold Member
Ave speed= (speed 1 + speed 2 ) /2
False.
False.
Why? An average can be the total / the number of things added to make the total
Could you explain a bit more
You need to take into account the times travelled at each speed. Useful sites for help in caculating average speeds are inly a simple search away.
haruspex
Science Advisor
Homework Helper
Gold Member
Why? An average can be the total / the number of things added to make the total
Could you explain a bit more
It is a matter of definition.
Average speed is defined to be (total distance travelled )/( total time taken). This does not give the same number as (v1+v2)/2 unless the two times happen to be the same.
It is a matter of definition.
Average speed is defined to be (total distance travelled )/( total time taken). This does not give the same number as (v1+v2)/2 unless the two times happen to be the same.
I understand very well
Could you give me an example to two situations where it works and it doesn't. I just want to wrap my head around it
haruspex
Science Advisor
Homework Helper
Gold Member
I understand very well
Could you give me an example to two situations where it works and it doesn't. I just want to wrap my head around it
As an analogy, consider a group of four people, one of height x, one of height y and two of height z. There are three heights, so in one sense the average height is (x+y+z)/3, but we find it more useful to define the average height of the four people as (x+y+2z)/4.
The number of people of a given height in the analogy corresponds to the time spent at a given speed.
lioric
As an analogy, consider a group of four people, one of height x, one of height y and two of height z. There are three heights, so in one sense the average height is (x+y+z)/3, but we find it more useful to define the average height of the four people as (x+y+2z)/4.
I actually meant an analogy where average speed = v1+v2/2 and average speed not equal to v1+v2/2
haruspex
Science Advisor
Homework Helper
Gold Member
I actually meant an analogy where average speed = v1+v2/2 and average speed not equal to v1+v2/2
Oh, you mean an example of where the answers are different?
If I walk at a steady 5kph for 59 minutes then stand still for one minute, is it sensible to say that my average speed is 2.5kph?
lioric
Oh, you mean an example of where the answers are different?
If I walk at a steady 5kph for 59 minutes then stand still for one minute, is it sensible to say that my average speed is 2.5kph?
Hmmm that is true. Then an a case where someone walks for 5kph for 59mins and 6kph for 15 mins, can we say the average speed is 5.5kph?
PeroK
Science Advisor
Homework Helper
Gold Member
Hmmm that is true. Then an a case where someone walks for 5kph for 59mins and 6kph for 15 mins, can we say the average speed is 5.5kph?
The average of the numbers ##5## and ##6## is ##5.5##. But that's not how average speed is defined.
symbolipoint
Homework Helper
Education Advisor
Gold Member
Ioric (lioric?)
The system of equations shown in the left-side rectangle-contained spot of the image of your paper are the correct ones for the system. You have two linear equation in two variables. The rest of YOUR work is difficult to follow. I am guessing by this time you have your solution straightened-out. Otherwise, you have the correct system. Just simplify and use Elimination Method to find x and y.
The simplified system I find is this:
5x+4y=500
AND
5x+2y=350
Simple use of Elimination Method will quickly give you value for y.
The average of the numbers ##5## and ##6## is ##5.5##. But that's not how average speed is defined.
This what I'm trying to clarify. In which situations does the summing and dividing of speeds not fit the average speed
PeroK
Science Advisor
Homework Helper
Gold Member
Hmmm that is true. Then an a case where someone walks for 5kph for 59mins and 6kph for 15 mins, can we say the average speed is 5.5kph?
To see this, suppose you have the following data:
5km/h for 15 mins
5km/h for 15 mins
6 km/h for 15 mins
5 km/h for 15 mins
Would the average speed be 5.5km/h? Why not?
PeroK
Science Advisor
Homework Helper
Gold Member
This what I'm trying to clarify. In which situations does the summing and dividing of speeds not fit the average speed
The average speed is defined to be the total distance divided by the total time.
Ioric (lioric?)
The system of equations shown in the left-side rectangle-contained spot of the image of your paper are the correct ones for the system. You have two linear equation in two variables. The rest of YOUR work is difficult to follow. I am guessing by this time you have your solution straightened-out. Otherwise, you have the correct system. Just simplify and use Elimination Method to find x and y.
The simplified system I find is this:
Simple use of Elimination Method will quickly give you value for y.
Yes it's Lioric.
I know the solution to this problem. There is no doubt about that. X is 40 y is 75
But I'm just trying to make sense why the average speed version doesn't work. In my gut I sort of know why it cannot be that average speed is not V1+v2 /2. But I'd like to be able to explain it too. I just cannot find the words
To see this, suppose you have the following data:
5km/h for 15 mins
5km/h for 15 mins
6 km/h for 15 mins
5 km/h for 15 mins
Would the average speed be 5.5km/h? Why not?
For one the unit is per hour.
PeroK
Science Advisor
Homework Helper
Gold Member
For one the unit is per hour.
What is the average speed in the case I mentioned?
What is the average speed in the case I mentioned?
5+5+5+6/1
Since 4x15 min is 1 hour
So 21
PeroK
Science Advisor
Homework Helper
Gold Member
5+5+5+6/1
Since 4x15 min is 1 hour
So 21
Well that's just absurd. I can't help you if you are going to give daft answers.
Well that's just absurd. I can't help you if you are going to give daft answers.
Sorry
Well that was total distance over total time. I miss took the speed for distance. My bad. Very sorry
5+5+5+6 /4
Which is 5.25
Then the other is
1.25+1.25+.1.25+1.5/1
=5.25
1.25 is the distance walked at 5kph for 15min
And 1.5 is 6kpm for 15min
PeroK
Science Advisor
Homework Helper
Gold Member
Sorry
Well that was total distance over total time. I miss took the speed for distance. My bad. Very sorry
5+5+5+6 /4
Which is 5.25
It has to be 5.25km/h. The key is that the durations must be the same. You can't treat 5km/h for 1 minute with the same weight as 6 km/h for 59 mins. Because the second speed is sustained for 59 times longer.
The answer to your question is that the average of 5km/h and 6km/h is only 5.5 km/h when the two speed are sustained for the same duration.
You can see that from the definition of total distance/total time. Which you need to understand.
robphy
Science Advisor
Homework Helper
Gold Member
"Average velocity" and "Average speed" are really time-weighted averages (not straight averages)
Here's average velocity... I just pasted the equations here. For details, read my posts.
https://www.physicsforums.com/threads/a-question-about-average-velocity.41732/post-303642 (calculus) from what should be the definition... then special case of constant acceleration
\begin{align*} v_{avg} &\equiv\frac{\int_{t_1}^{t_2} v\ dt}{\int_{t_1}^{t_2}\ dt}=\frac{\Delta x}{\Delta t}\\ &=\frac{\int_{t_1}^{t_2} (at)\ dt}{\int_{t_1}^{t_2}\ dt}\\ &=\frac{a\frac{1}{2}(t_2^2 - t_1^2)}{t_2-t_1}\\ &=\frac{1}{2}a(t_2+t_1)\\ &=\frac{1}{2}(v_2+v_1) \end{align*}
https://www.physicsforums.com/threa...anics-kinematics-comments.813456/post-5106894 (algebra) from what should be the definition... then special case of constant acceleration
\begin{align*}
\vec v_{avg}
&\equiv \frac{\vec v_1\Delta t_1+\vec v_2\Delta t_2+\vec v_3\Delta t_3}{\Delta t_1+\Delta t_2+\Delta t_3} \\
&=
\frac{\Delta \vec x_1+\Delta\vec x_2+\Delta \vec x_3}{\Delta t_1+\Delta t_2+\Delta t_3}\\
&=
\frac{\Delta \vec x_{total}}{\Delta t_{total}}\\
\end{align*}
\begin{align*} \vec v_{avg} &\equiv \frac{\Delta \vec x_{total}}{\Delta t_{total}}\\ &\stackrel{\scriptsize\rm const \ a }{=}\frac{\frac{1}{2}\vec a(\Delta t)^2 +\vec v_i\Delta t}{\Delta t}\\ &\stackrel{\scriptsize\rm const \ a }{=}\frac{1}{2}\vec a(\Delta t) +\vec v_i\\ &\stackrel{\scriptsize\rm const \ a }{=}\frac{1}{2}(\vec v_f -\vec v_i) +\vec v_i \\ &\stackrel{\scriptsize\rm const \ a }{=} \frac{1}{2}(\vec v_f+ \vec v_i) \end{align*}
archaic
It has to be 5.25km/h. The key is that the durations must be the same. You can't treat 5km/h for 1 minute with the same weight as 6 km/h for 59 mins. Because the second speed is sustained for 59 times longer.
The answer to your question is that the average of 5km/h and 6km/h is only 5.5 km/h when the two speed are sustained for the same duration.
You can see that from the definition of total distance/total time. Which you need to understand.
Yep totally
It was what was going in my head the but couldn't explain it. Thank you.
So to sum it up
If the speeds are of the same duration, then we can get a mean value out of it.
But if it's from different durations then the mean value cannot be equal to Ave speed
symbolipoint
Homework Helper
Education Advisor
Gold Member
Yes it's Lioric.
I know the solution to this problem. There is no doubt about that. X is 40 y is 75
But I'm just trying to make sense why the average speed version doesn't work. In my gut I sort of know why it cannot be that average speed is not V1+v2 /2. But I'd like to be able to explain it too. I just cannot find the words
lioric
I could not find any reason to look specifically for any average speed in analyzing and solving the problem. I did not actually finish trying to solve it; maybe in a further step, an expression for average speed may appear, or maybe not. What do you find when you take from the simplified system and solve for x and for y? Does an expression for average speed show?
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## Gallium reactors
How useful are reactors cooled by gallium alloys?
Ga has cross section for neutrons of 2,9 barns. Somewhat high - but K has 2,1 barns, and NaK is mostly (76 %) K.
Pure Ga melts under +30 Celsius. The reactor may freeze - but it is much easier to melt than Pb/Bi eutectic (+125 Celsius).
Ga melting point can be further lowered by alloying. While In, Cd and Hg are neutron poisons, the other low melting metals are reasonable - Pb, Bi, also Sn (0,62 barns) and Zn (1,1 barns).
Would a reactor with Ga-Sn coolant be convenient to handle? Ga/Sn melt is nowhere as reactive as Na/K melt, and also not poisonous until irradiated....
PhysOrg.com science news on PhysOrg.com >> King Richard III found in 'untidy lozenge-shaped grave'>> Google Drive sports new view and scan enhancements>> Researcher admits mistakes in stem cell study
Admin Gallium's price has been on the order of $650/kg but could be has high as$750 to $970/kg as compared to sodium at around$150/kg. In a reactor, liquid metal embrittlement with fuel and structural materials is an issue. Liquid metals are usually used for fast reactors.
Clementine managed to work with mercury coolant - despite its huge neutron cross-section. How does the quoted price of gallium compare with the price of highly enriched uranium 235, or plutonium 239? These are presumably priceless because no one wants to sell them....
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# Human fingers are very sensitive, detecting vibrations with amplitudes as low as 2.4 X 10^-5 m. Consider a sound wave with a wavelength exactly 1000 times greater than the lowest amplitude detectable by fingers. The speed of sound in air is 377 m/s. What is this wave’s frequency?
Human fingers are very sensitive, detecting vibrations with amplitudes as low as $2.4×{10}^{-5}$ m. Consider a sound wave with a wavelength exactly 1000 times greater than the lowest amplitude detectable by fingers. The speed of sound in air is 377 m/s. What is this wave’s frequency?
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tiltat9h
Wavelength $=1000×$ Lowest amplitude
$\lambda =1000×2.4×{10}^{-5}m=2.4×{10}^{-2}m$
Speed, wavelength and frequency are related as
$v=f\lambda$
$f=\frac{v}{\lambda }=\frac{377m/s}{2.4×{10}^{-2}m}=15708.3Hz$
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User:Waqjan/Fluid Mechanics Draft
Introduction
Fluid mechanics is the study of how fluids flow.
A fluid is simply a liquid or a gas, or more precisely, a substance that deforms continuously when subjected to a tangential or shear stress, however small the shear stress may be. This means that some materials, such as amorphous solids, colloidal suspensions and gelatinous materials, while not being completely liquids or gases, are fluids so the general theories of fluid mechanics can be applied to them. However, these substances are not commonly studied at introductory fluid mechanics.
Fluid mechanics is a subdivision of continuum mechanics. This means that fluids will be considered as being able to be infinitely divisible, and not as collections of atoms or molecules.
Fluid Properties
In addition to the properties like mass, velocity, and pressure usually considered in physical problems, the following are the basic properties of a fluid:
Density
The density of a fluid is defined as the mass per unit volume of the fluid.
${\displaystyle \rho ={\frac {m}{V}}\ }$
Viscosity
Viscosity (represented by μ) is a material property, unique to fluids, that measures the fluid's resistance to flow. Though this is a property of the fluid, its effect is understood only when the fluid is in motion. When different elements move with different velocities, then the each element tries to drag its neighboring elements along with it. Thus shear stress can be identified between fluid elements of different velocities.
Velocity gradient in laminar shear flow
The relationship between the shear stress and the velocity field was studied by Isaac Newton and he proposed that the shear stresses are directly proportional to the velocity gradient. ${\displaystyle \tau =\mu {\frac {\partial u}{\partial y}}}$
The constant of proportionality is called the coefficient of dynamic viscosity.
Another coefficient, known as the kinematic viscosity is defined as the ratio of dynamic viscosity and density. ${\displaystyle \nu =\mu /\rho }$
Reynolds Number
There are several dimensionless parameters that are important in fluid dynamics. Reynolds number (after Osborne Reynolds, 1842-1912) is an important parameter in the study of fluid flows. Physically it is the ratio between inertial and viscous forces. The value of Reynolds number determines the kind of flow of the fluid.
${\displaystyle Re={\frac {\rho VL}{\mu }}={\frac {VL}{\nu }}}$
where ρ(rho) is the density, μ(mu) is the viscosity, V is the velocity of the flow, and L is the dimension representing length for the flow. Additionally, we define a parameter ν(nu) as the kinematic viscosity.
Low Re indicates creeping flow, medium Re is laminar flow, and high Re indicates turbulent flow.
Reynolds number can also be transformed to take account of different flow conditions. For example the reynolds number for flow within a pipe is expressed as
${\displaystyle Re={\frac {\rho ud}{\mu }}}$
where u is the average fluid velocity within the pipe and d is the inside diameter of the pipe.
Application of dynamic forces (and the Reynolds number) to the real world: sky-diving, where friction forces equal the falling body's weight. (jjam)
Pathlines and Streamlines
The path which a fluid element traces out in space is called a pathline. For steady non fluctuating flows where a pathline is followed continuously by a number of fluid elements , the pathline is called streamline. A streamline is the imaginary line whose tangent gives the velocity of flow at all times if the flow is steady, however in an unsteady flow, the streamline is constantly changing and thus the tangent gives the velocity of an element at an instant of time. A common practice in analysis is taking some of the walls of a control volume to be along streamlines. Since there is no flow perpendicular to streamlines, only the flow across the other boundaries need be considered.
Hydrostatics
The pressure distribution in a fluid under gravity is given by the relation dp/dz = −ρg where dz is the change in the direction of the gravitational field (usually in the vertical direction). Note that it is quite straightforward to get the relations for arbitrary fields too, for instance, the pseudo field due to rotation.
The pressure in a fluid acts equally in all directions. When it comes in contact with a surface, the force due to pressure acts normal to the surface. The force on a small area dA is given by p dA where the force is in the direction normal to dA. The total force on the area A is given by the vector sum of all these infinitesimal forces.
Control Volume Analysis
A fluid dynamic system can be analysed using a control volume, which is an imaginary surface enclosing a volume of interest. The control volume can be fixed or moving, and it can be rigid or deformable. Thus, we will have to write the most general case of the laws of mechanics to deal with control volumes.
The first equation we can write is the conservation of mass over time. Consider a system where mass flow is given by dm/dt, where m is the mass of the system. We have,
${\displaystyle {\dot {m}}=\int _{CS}\rho \left(\mathbf {V} \cdot \mathbf {n} \right)dA}$
${\displaystyle \int _{CS}\rho \left(\mathbf {V} \cdot \mathbf {n} \right)dA=0}$
And for incompressible flow, we have
${\displaystyle \int _{CS}\left(\mathbf {V} \cdot \mathbf {n} \right)dA=0}$
If we consider flow through a tube, we have, for steady flow,
${\displaystyle \rho _{1}A_{1}V_{1}=\rho _{2}A_{2}V_{2}}$
and for incompressible steady flow, A1V1 = A2V2.
Law of conservation of momentum as applied to a control volume states that
${\displaystyle \sum F={\frac {d}{dt}}\left(\int _{CV}\mathbf {V} \mathbf {\rho } \right)+\int _{CS}\mathbf {V} \mathbf {\rho } \left(\mathbf {V} \cdot \mathbf {n} \right)dA}$
where V is the velocity vector and n is the unit vector normal to the control surface at that point.
Law of Conservation of Energy (First Law of Thermodynamics)
${\displaystyle {\frac {d\mathbf {Q} }{dt}}+{\frac {d\mathbf {W} }{dt}}={\frac {d}{dt}}\left(\int _{CV}e\mathbf {\rho } \right)+\int _{CS}e\mathbf {\rho } \left(\mathbf {V} \cdot \mathbf {n} \right)dA}$
where e is the energy per unit mass.
Bernoulli's Equation
Bernoulli's equation considers frictionless flow along a streamline.
For steady, incompressible flow along a streamline, we have
${\displaystyle {\frac {p}{\rho }}+{\frac {V^{2}}{2}}+gz=constant}$
We see that Bernoulli's equation is just the law of conservation of energy without the heat transfer and work.
It may seem that Bernoulli's equation can only be applied in a very limited set of situations, as it requires ideal conditions. However, since the equation applies to streamlines, we can consider a streamline near the area of interest where it is satisfied, and it might still give good results, i.e., you don't need a control volume for the actual analysis (although one is used in the derivation of the equation).
${\displaystyle {\frac {p}{\rho g}}+{\frac {V^{2}}{2g}}+z=constant}$
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auctex
[Top][All Lists]
## RE: [AUCTeX] New version of Emacs/AUCTeX bundle for Windows available
From: Adam Johnson Subject: RE: [AUCTeX] New version of Emacs/AUCTeX bundle for Windows available Date: Wed, 06 Jun 2007 10:12:37 -0500
There is a new version of the Emacs/AUCTeX bundle for Windows
available.
* Update to Emacs pretest 22.0.990.
Not sure what is the reason, but the Emacs is very easy to crash in this bundle (i.e., being exceptionally closed). The previous version is more robust than this version somehow.
I have used many such bundles, and some versions were more easy to crash than others.
p.s., whether was the new RefTeX also included in this bundle?
Thanks.
--
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## The Annals of Statistics
### Large Sample Behaviour of the Product-Limit Estimator on the Whole Line
Richard Gill
#### Abstract
Weak convergence results are proved for the product-limit estimator on the whole line. Applications are given to confidence band construction, estimation of mean lifetime, and to the theory of $q$-functions. The results are obtained using stochastic calculus and in probability linear bounds for empirical processes.
#### Article information
Source
Ann. Statist., Volume 11, Number 1 (1983), 49-58.
Dates
First available in Project Euclid: 12 April 2007
https://projecteuclid.org/euclid.aos/1176346055
Digital Object Identifier
doi:10.1214/aos/1176346055
Mathematical Reviews number (MathSciNet)
MR684862
Zentralblatt MATH identifier
0518.62039
JSTOR
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Sidebar
seminars:comb:abstract.201112nye
Computational Techniques for Magic Square Enumeration
Abstract for the Combinatorics Seminar 2011 December 6
An n×n magic square has n2 distinct positive integers such that row, column, and diagonal sums are equal. Counting the number of magic squares for a given magic sum is difficult. Constraint programming offers a simple way to enumerate them directly, listing all magic squares. I'll show how to do this for 3×3 squares.
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[Madera Canyon, AZ with wife Emily, Oct. 2016]
Gordon C. Brown
Department of Mathematics
University of Oklahoma
Norman, OK 73019-3103
gbrown [at] math [dot] ou [dot] edu
Welcome
I'm a PhD candidate and teaching assistant in the
math department at the University of Oklahoma.
I received my BS in math from UT Austin in 2012,
and my MA in math from OU in 2014. I'm orginally
from Dallas, TX.
My CV is here. I will graduate in December 2017.
Research
[At right is the Specht polytope P(3,1).]
My research interests lie in representation theory
and combinatorics. I'm particularly interested in
modules over Lie superalgebras and their quantum
groups. The prefix "super" refers to a $\mathbb{Z}_2$-grading
on these algebras, giving them an extra layer of
complexity than in the classical, non-super case.
At the moment I'm studying the type Q Lie
superalgebra $\mathfrak{q}_n$ and its quantum group $\operatorname{U}_q(\mathfrak{q}_n)$.
One of the two so-called "strange" Lie
superalgebras, $\mathfrak{q}_n$ resists investigation by some
of the methods that apply to the other Lie
superalgebras. There are, however, underlying
combinatorics that play a central role and are
fascinating all on their own. In fact whenever
possible I view morphisms diagrammatically.
I'm also very interested in the modular
representation theory of the symmetric groups,
and the areas of its modern day study - cyclotomic
KLR algebras, Kac-Moody 2-categories, etc.
[At right are the first 5 levels of the Young lattice.]
Teaching
For Fall 2017 I'm teaching Math 2924-012,
Differential and Integral Calculus II. Enrolled
students can find all course information and
materials on Canvas.
Please see my CV for my full teaching history.
As of Spring 2017 I've taught over 500 students!
Service
• Mentor in the OU Directed Reading Program.
[Same as the DRP at University of Maryland.]
• LGBTQ and Diversity Faculty Ally at OU.
• Co-organizer of CTS, GAS, and TORAS.
• Learning Specialist in the OU Math Center.
• Co-administrator of WeBWorK software at OU.
• Notes and Talks
• Young symmetrizers, Fall 2016 GAS talk.
• Quiver notes, for my mentee Christy Strauss.
• Intro to categorification, Spring 2016 GAS talk.
• Category $\mathscr{O}$ and AS functors, Fall 2015 GAS talk.
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# A FREE ℤp-ACTION AND THE SEIBERG-WITTEN INVARIANTS
• Nakamura, Nobuhiro
• Published : 2002.01.01
• 60 6
#### Abstract
We consider the situation that ${\mathbb{Z}_p}\;=\;{\mathbb{Z}/p\mathbb{Z}}$ acts freely on a closed oriented 4-manifold X with ${b_2}^{+}\;{\geq}\;2$. In this situation, we study the relation between the Seiberg-Witten invariants of X and those of the quotient manifold $X/{\mathbb{Z}}_p$. We prove that the invariants of X are equal to those of $X/{\mathbb{Z}}_p$ modulo p.
#### Keywords
4-manifold;Seiberg-Witten invariants;group action
#### References
1. M. F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes: I, Ann. of Math. 86, 374-407. https://doi.org/10.2307/1970694
2. S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-Manifold, Oxford, 1990
3. M. Furuta, A remark on a fixed point of finite group action on $S^4$, Topology 28 (1989), 35-38 https://doi.org/10.1016/0040-9383(89)90030-X
4. D. Kotschick, J. W. Morgan, and C. H. Taubes, Four-manifold without symplectic structures but with nontrivial Seiberg- Witt invariants, Math. Res. Lett. 2 (1995), 119-124. https://doi.org/10.4310/MRL.1995.v2.n2.a1
5. J. W. Morgan, The Seiberg- Witten equations and application to the topology of smooth four-manifolds, Mathematical Notes, Princeton Univ, Press, 1996
6. M. Ue, A note on Donaldson and Seiberg- Witten invariants for some reducible 4-manifolds (preprint)
7. M. Ue, Exotic group actions in dimension four and Seiberg- Witten Theory (preprint) https://doi.org/10.3792/pjaa.74.68
8. E. Witten, Monopoles and 4-manifolds, Math. Res. Lett. 1 (1994), 769-796 https://doi.org/10.4310/MRL.1994.v1.n6.a13
#### Cited by
1. $$G$$ G -monopole invariants on some connected sums of 4-manifolds vol.178, pp.1, 2015, https://doi.org/10.1007/s10711-015-0044-1
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## Introduction
The Convention on International Trade in Endangered Species of Wild Fauna and Flora (CITES) is a binding and multilateral environmental agreement with the objective of prohibiting trade of wildlife species in immediate danger of extinction (Appendix I listing) and preventing additional threatened species reaching this critical stage by regulating trade so that it is traceable, legal, and sustainable (Appendix II listing). CITES is the first line of defense against illegal wildlife trafficking, with 183 parties (i.e. signatory nations), and lists over 35,000 animal and plant species1. The vast majority of these species (96%) are listed in Appendix II and exporting parties have the obligation to document that traded specimens are traceable through the supply chain, were legally obtained, and that trade it is not detrimental for the survival of the species2. Importing parties are required to monitor imports and certify that incoming specimens are accompanied by the appropriate documentation2. Failure to properly implement CITES for any listed species can eventually lead to international trade sanctions being placed on offending parties2.
Sharks are a group of marine fish many of which are threatened by overexploitation to satisfy demand for internationally traded products, primarily fins (for use in luxury Asian soup dishes) and meat3,4. Twelve shark species have been listed in CITES Appendix II to date. The first round of shark listings took place from 2001–2004, with the whale shark Rhincodon typus (2001), basking shark Cetorhinus maximus (2001), and great white shark Carcharodon carcharias (2004). These species all share the characteristics of being iconic, extremely large bodied (>4.0 m total length at maturity), and some products from these species (e.g., fins, dressed carcasses) can be readily identified by their very large sizes (except when procured from juvenile stages) or other morphological features (e.g., jaws and teeth for great whites). These species also share the characteristic that Appendix II listing is not as stringent as the domestic protection these species receive in many jurisdictions (i.e., landing and trade prohibition4,5). Products of these species have generally not been detected in post-listing studies of major shark fin and meat markets, suggesting they are rare in trade6,7,8,9,10.
The second round of shark listings took place from 2013–2016, where the porbeagle shark Lamna nasus (2013), scalloped hammerhead Sphyrna lewini (2013), great hammerhead S. mokarran (2013), smooth hammerhead S. zygaena (2013), the oceanic whitetip shark Carcharhinus longimanus (2013), silky shark C. falciformis (2016), bigeye thresher shark Alopias superciliosus (2016), pelagic thresher shark A. pelagicus (2016), and common thresher shark A. vulpinus (2016) were added to Appendix II. This group of species all share the characteristics of being smaller at maturity than white, whale, or basking sharks, and being much more common in trade both pre and post-CITES listing9,10. The silky, scalloped hammerhead, and smooth hammerhead represent the second, fourth, and fifth most common species in the contemporary fin trade9,10. Some of the dried unprocessed fins (e.g., first dorsal, pectoral) and dressed carcasses of these sharks can be identified using morphological characters11,12. Trade in meat, other processed products, and some, less distinctive fin types (e.g., lower caudal) are more challenging to detect without the aid of genetic testing. The Secretariat of CITES recently recommended that parties “share experiences with, and knowledge of, forensic means to efficiently, reliably and cost- effectively identify shark products in trade” (SC69 Doc. 50), including genetic methods.
It is evident that enforcing CITES regulations is a major challenge, in part due to limited availability of rapid, reliable, and cost-effective tools to identify traded products3, especially for the species listed from 2013–201610. Wildlife forensics tools, such as DNA barcoding13,14,15 and PCR-based species-specific assays16,17,18, are the most reliable and widely applicable species identification approaches available to assess and detect illicit activities associated with wildlife trade19,20,21. However, the application of these tools to detect CITES listed species is difficult when inspections must be conducted in a short time frame (e.g. less than 24 hours as is the legal requirement in some locations, such as the United States of America, Hong Kong Special Administrative Region of China), with limited scope for randomly sampling unknown products, transferring tissue samples to a laboratory, and going through all of the steps required for DNA barcoding (i.e., DNA isolation, PCR, product purification, sequencing19) or using available species-specific PCR tests (i.e., DNA isolation, testing for CITES listed species contained in up to four separate multiplex PCR assays, gel electrophoresis19). This is especially problematic given that individual shipments of shark products can be quite large (i.e., 102–103 kg20). Real-time PCR (rtPCR) techniques are an alternative tool that reduce the number of steps to species diagnosis and offers the advantage over DNA barcoding of being able to be conducted outside of the laboratory. Typically, rtPCR techniques have been developed for the quantification of nucleic acids21, and for the detection of pathogen microorganisms22,23,24. This technique uses target-specific primers and fluorescent dyes (e.g. SYBR Green, TaqMan probes) to detect if PCR amplification occurred due to the presence of the targeted nucleic acid template, eliminating the need for sequencing and agarose gels, and enabling a reliable identification of targeted species in real time. In addition, the last stage of rtPCR protocols usually generates melt curves and melt curve temperatures, that can further be used for the identification of the target species/locus25,26. Melt curves are generated by the dissociation of double-stranded DNA during heating, and melt curve temperatures typically vary with the GC/AT ratio of the generated amplicon and its size26.
Here, we present a rtPCR protocol capable of rapidly detecting the presence of nine of the twelve CITES-listed shark species in a single multiplex PCR that could be used as a field-based tool to detect the illegal exportation/importation of these species (i.e. cases of import/export without the correct documentation). The multiplex includes all of the commercially important sharks listed from 2013–2016, with the exception of the oceanic whitetip shark. While the test itself does not discriminate which of the species is present (with a few exceptions based on melt curve analysis) it rapidly and cost effectively (~4 hours for 95 samples; 0.94 USD per sample) provides sufficient information to justify holding shipments based on the illegal presence of CITES listed species. This would provide time for more rigorous forensic testing to verify species-of-origin to meet typical evidentiary standards required for prosecution. We also present three field tests conducted in Hong Kong to inspect the contents of a shark fin consignment in collaboration with local authorities that provide proof-of-concept of the portability of this approach. ## Methods Previously published species-specific primers for CITES-listed species16,17,27,28 were used as a starting point for our multiplex rtPCR assay. The original published studies validated these primers against large, geographically broad samples of the target species, and demonstrated that the primer failed to amplify a large number of non-target species including all or most of the target species’ close relatives (i.e. congeners) through agarose gel electrophoresis18,29.Originally, published primers for hammerhead and thresher shark species were tested on 56 species17,28, great white shark primers on 68 species16, and porbeagle and silky shark primers on 34 species27. Initially, all published primers for these CITES listed species (Table 1) were tested individually with one universal reverse primer against 48 target and non-target species comprising six orders and twelve families (Table 2) to ensure complete specificity of each individual primer when adapted to SYBR Green chemistry. Each 20 μL reaction included, 10 μL of PowerUp SYBR Green Master Mix (Applied Biosystems), 5 μL of primer mix (equal mixture of primers from 10 μM stock solutions), and 5 μL of extracted DNA. DNA extractions were either conducted from dried unprocessed fin, meat, or wet unprocessed fins using Chelex DNA extraction protocols30,31, where a small sample (2 mm2) was added to a PCR tube containing 200 μL of 10% Chelex solution and heated for 20 minutes at 60 °C and for 25 minutes at 99 °C, followed by a brief centrifugation and storage at 4 °C. A no-template negative control was included in every extraction and rtPCR batch to control for reagent contamination. Species-specific amplifications were tested against a wide range of annealing temperatures (60°–70 °C) on the QuantStudio5 System (Thermofisher) to determine the most stringent temperature that could be used to amplify the target species and an annealing temperature of 69 °C was selected for further trials. Early trials showed that the silky shark primer, described by27, did not amplify well in a rtPCR format. It also frequently amplified genomic DNA from the spinner shark (C. brevipinna). Thus, we designed a novel species-specific primer for silky sharks (Table 1). We aligned all reference sequences for the internal transcribed spacer 2 (ITS2) deposited in GeneBank by27 in SeaView v432, and designed a putative species-specific primer based on nucleotide differences in this locus. The ITS2 locus was chosen as the target locus for silky sharks because (i) it is highly conserved within shark species, (ii) contains enough nucleotide polymorphisms to achieve species-specific diagnosis and sequence differences between distantly related species are sufficiently large to preclude meaningful alignments, and (iii) all other species-specific primers in our multiplex were designed for this particular locus, thus allowing us to use the same universal reverse primer to amplify the novel primer16,17,27,28. This novel silky shark primer was individually tested for amplification performance in the rtPCR format with silky shark tissues collected over a wide geographic area (Fig. 1) and against all test samples in Table 2. Once we had validated this silky primer, all species-specific primers and one universal reverse primer were combined into a single multiplex of 20 μL including 10 μL of PowerUp SYBR Green Master Mix (Applied Biosystems), 5 μL of primer mix (equal mixture of all primers from 10 μM stock solutions), and 5 μL of extracted DNA. This final multiplex was tested against all test species (Table 2) to detect possible primer interaction that could obstruct the amplification. The thermal cycle profile is shown in Supplementary Fig. S1. Our expected result was that genomic DNA from the nine CITES listed species would amplify and enable SYBR green to fluoresce, while DNA from any other species would not. We also tested our multiplex rtPCR protocol against processed shark fin samples collected in the Hong Kong retail markets (Table 2) in order to assess its applicability on highly degraded tissue samples. Only one of the nine species present in the multiplex was not tested on processed products (i.e., great white shark), since no samples for this species have been found after more than four years of extensive sampling in Hong Kong and mainland China (Cardeñosa et al. unpublished data). DNA extraction methods, rtPCR protocols, and thermal profiles followed the ones described above. Three field tests were conducted in Hong Kong in collaboration with the Agriculture Fisheries and Conservation Department (AFCD). During these field tests the QuantStudio5 System, reagents, and plastic consumables were transported to local inspection facilities as a mobile lab. We used the following workflow during our field tests in Hong Kong. After AFCD personnel visually inspected the fins, tissue samples were taken and the QuantStudio5 System was used for Chelex extraction of DNA (~45 minutes), 5 μL of which was then added to the reaction plate. Two wells were used for a positive control using DNA from a previously known scalloped hammerhead sample and a negative control. Once the primer mix and SYBR Green were added, 35 PCR cycles were run with a posterior melt curve analysis (Supplementary Fig. S1). At the conclusion of the PCR and melt curve stages, the melt curve analysis was used to determine which species were present. The field tests involved (i) a container that had sacks of small, unprocessed fins (i.e., <10 cm across the base), (ii) an undeclared shipment of dried seafood that, among other products, included processed shark fins, and (iii) a container that was primarily carrying large unprocessed primary fins (i.e., first dorsal, pectorals, caudal). In the first two cases (i.e., small fins, processed fins) no positive visual identifications were made and we tested our rtPCR protocol against haphazardly selected fins. For the third case (i.e., large unprocessed primary fins), we tested our rtPCR protocol against fins identified as being from CITES listed species based on visual identification protocols (N = 21) and haphazardly selected fins that were thought not to be from these species (N = 35). Since Hong Kong authorities had not yet started to enforce the 2016 shark CITES listings at the time, the species-specific primers to detect the silky and thresher sharks were removed from the multiplex for these specific law-enforcement scenarios. For all of these cases a portion of the mitochondrial COI gene was later amplified and sequenced from unprocessed fins following the protocols by31, and from processed fins following the protocols by15 to assess concordance with the multiplex results. Sequences were compared to BOLD (FISH-BOL) and BLAST (GenBank) databases to identify them to the lowest taxonomic category possible. ## Results Our putative silky shark specific primer performed well individually and in multiplex format, amplifying silky sharks from all over the world (Fig. 1). The final multiplex rtPCR assay containing this primer and the previously published primers for eight additional species correctly detected when any of these CITES-listed species were present and failed to amplify when using genomic DNA templates from all non-target species (Fig. 2). The multiplex proved to be highly reliable in this context, with a false positive rate of 0% with the test samples used (Table 2). There were no differences in the amplification profiles when primers were run individually or as a multiplex protocol. Melt curve temperatures for CITES-listed species occurred between 89° and 91 °C (Fig. 3), making the instantaneous post-PCR identification to species difficult. However, particular melt curve presented consistent shapes that were distinctive for certain species, enabling unambiguous post-PCR identification of S. lewini and A. superciliosus (Fig. 3). Our protocol was also able to consistently detect seven of the eight species tested on processed shark fin samples, with the exception of A. superciliosus, due to its large amplicon (i.e., around 1000 bps). The rtPCR protocol achieved positive identification of fins from CITES listed species (N = 24) across all three field tests in Hong Kong (Fig. 4). All positive results from the rtPCR were subsequently corroborated with DNA barcoding (i.e., there were no false positives). There were four false negative results out of a total of 118 fins tested from CITES listed species, involving one processed fin and three large unprocessed fins. Three of these not only failed with rtPCR but also with DNA barcoding and were only identified as CITES listed species using morphological guides. The barcoding analysis for the first case identified 77% of the samples as milk shark (Rhizoprionodon acutus [55%]), spinner shark (C. brevipinna [16%]), silky shark (C. falciformis [3%]) and scalloped hammerhead (S. lewini [3%]), the latter of which tested positive with the rtPCR. For the second case, species identification was achieved in 92.9% of the tested samples and were identified as blue shark (P. glauca [23.8%]), brownbanded bamboo shark (Chiloscyllium punctatum [21.4%]), bigeye thresher shark (A. superciliosus [19.1%]), scalloped hammerhead (S. lewini [7.1%, all but one of which had tested positive with the rtPCR]), white-spotted guitarfish (Rhinchobatus australiae [7.1%]), spottail shark (C. sorrah [4.8%]), whitecheek shark (C. dussumieri [4.8%]), blacktip shark (C. limbatus [2.4%]), and salmon shark (L. ditropis [2.4%]). Barcoding analysis identified 71.4% of the samples from the third case. All fins that tested positive with rtPCR were CITES-listed smooth hammerhead (S. zygaena [16.1%]), great hammerhead (S. mokarran [8.9%]), and scalloped hammerhead (S. lewini [7.1%]). All of them were correctly visually identified as being from CITES listed hammerheads by AFCD personnel. Three large fins marked as CITES-listed species during the visual inspection failed to amplify with both the rtPCR protocol and the barcoding analysis. Fins that had negative test results with rt-PCR later proved to be non-CITES listed tiger shark (Galeocerdo cuvier [7.1%]), blacktip shark (C. limbatus [3.6%]), snaggletooth shark (Hemipristis elongata [3.6%]), bull shark (C. leucas [1.8%]), silvertip shark (C. albimarginatus [1.8%]), and graceful shark (C. amblyrhynchoides [1.8%]). Barcoding also verified CITES listed oceanic whitetip (C. longimanus [7.1%]) and silky sharks (C. falciformis [1.8%]) that were not part of the multiplex. ## Discussion Here, we present an easy-to-use, reliable, and to our knowledge, the fastest molecular protocol to date to detect the majority of CITES-listed shark species that are relatively common in the international trade (at least from the perspective of fins9). The multiplex used previously published and extensively tested PCR primers, adapting them to rtPCR format with SYBR Green chemistry. The only exception was the silky shark primer presented in27, which did not work well in this format. The novel silky primer we designed produced a smaller amplicon, promoting more efficient PCR and reliable detection. It amplified 96 known silky shark samples collected from all ocean basins where they occur and failed to amplify any other species, including 22 of 28 described congeners. We did not test the silky primer against all Carcharhinus species and none of the species that we lacked tissue samples for had a sequence in the National Center of Biotechnology Information (NCBI) GeneBank (http://www.ncbi.nlm.nih.gov/genbank/) that would enable us to check for primer mismatches that could prevent amplification. However, the untested Carcharhinus species (n = 6) are phylogenetically more distant to the silky shark than others that we have examined and presented negative results33 (Table 2). Therefore, it is unlikely that the novel primer amplifies other untested Carcharhinus species, especially at an annealing temperature of 69 °C. We also tested against all of the Carcharhinus species that are common in the fin trade, with the untested species occurring only rarely or not occurring at all9. Across all species our rtPCR multiplex generated 0% false positives during the testing phase and field trials. It is likely that the false positive rate of this multiplex will be zero or near zero because of the extensive testing of the published primers16,17,27,28 and the testing we conducted of the novel silky primer. Overall, we contend that the most probable source of false positives is contamination, which can be minimized by rinsing tissues with clean water prior to sampling and by using negative controls. A positive result from this protocol is robust evidence of the presence of one of these CITES listed species, which should be sufficient probable cause to at least hold the product for DNA barcoding that is currently widely accepted in an evidentiary context. False negative results are unavoidable and no method, genetic or morphological, is entirely immune to them. With our rtPCR assay the frequency of false negative results could vary between individual runs according to the frequency of samples with low DNA quality (e.g., when tested on processed products) or the presence of PCR inhibitors. When the rtPCR multiplex was tested against processed shark fin samples, the bigeye thresher was the only CITES listed species included in the multiplex that failed to amplify, possibly due to the resulting large amplicon (i.e., >1000 bp)15. If false negatives are considered to be a major problem in law enforcement contexts we suggest that all test samples could be run twice: once with the rtPCR multiplex described and once with a protocol expected to amplify all shark products (e.g., a universal shark mini-barcode assay34 or universal ITS2 primers27). Samples that fail to amplify using both approaches could therefore be deemed inconclusive rather than registering as false negatives. This protocol identifies nine out of twelve CITES-listed shark species, only omitting whale, basking, and oceanic whitetip sharks. The whale and basking sharks are relatively rare in the fin9,10 and meat trade and many of their fins are visually identifiable given their large size and other morphological characters12. The oceanic whitetip is more common in the contemporary shark fin trade9,10 but there are no published primers for this species. Examining the sequences for this species available on GenBank such as ITS2 and cytochrome oxidase I (COI) revealed very high sequence similarity (98%) between this and the dusky and Galapagos sharks (C. obscurus, C. galapagensis), which makes it extremely difficult to design a specific primer (see Shivji et al.27 who noted similar issues). While future efforts will explore other loci, the fins of oceanic whitetips have unique morphological characteristics (e.g. rounded apices with mottled white markings) that make them easily recognizable for law enforcement officers and inspectors12 and reduce the need for them to be in the present multiplex to monitor the fin trade. We conclude that this assay is currently ready-to-use for monitoring the fin trade given that the omitted species all have visually identifiable fins. It is ready-to-use for monitoring the trade of meat and other products, with the caveat that it does not amplify three CITES listed species, two of which are likely to be extremely rare7,10. Future work should aim to develop specific primers for these species that could be integrated into the present multiplex, although this may require surveys of additional loci. The largest obstacle to routine screening of shark products is cost and time given large shipment volumes typical of the shark trade. The QuantStudio5 System and other rtPCR thermal cyclers have a relatively small footprint and are sufficiently portable, together with the laptop and all necessary reagents, pipettes, and disposable plastics, for this protocol to be deployed in ports or customs inspection areas as shown during our first field tests. The only on-site requisite is a source of electricity and cover from the elements. The speed at which this multiplex can detect these species is a key advance. We suggest the following workflow for sample identification using this multiplex, which can be completed in approximately 4 hours for 94 samples. After taking tissue samples (up to 94 for one run), the QuantStudio5 System can be used for Chelex extraction of DNA (~45 minutes), 5 μL of which is then added to all but one well of a 96 well reaction plate. Once the primer mix and SYBR Green are added, 35 PCR cycles should be run and positive amplifications (i.e., presence of CITES listed species) detected in wells that reach a conservative ΔRn value (i.e. the magnitude of the fluorescent signal generated by the PCR reaction) of 1.00 (Fig. 2), since no non-target species ever crossed that threshold during our testing with the optimized protocol. This threshold is normally reached in 60–70 minutes. We suggest reserving two wells, on a 96 well plate, for a positive control that includes DNA from a known CITES listed shark species in the multiplex that has previously amplified, and a negative control without a template. If the positive control sample fails to amplify it suggests the reagents are compromised. At the conclusion of PCR, the melt curve can be analyzed to determine which species is present (when possible; see Fig. 3), although the presence of any CITES listed species without the proper documentation is illegal and allows for the shipment to be seized and inspected in more detail for prosecution. Given this workflow, the concentrations of reagents and consumables used, the cost to screen one sample with the rtPCR multiplex was0.94 USD.
Combining this multiplex with morphological identification guides12 enabled highly efficient and cost effective screening of imports at the border for the most common CITES listed sharks in trade from a wide range of potential fin products. The rtPCR protocol also proved to enhance the capacity of law-enforcement officers in Hong Kong to detect illicit trade. First, it achieved detection of CITES-listed species fins that otherwise went undetected because they were outside of the scope of morphological identification guides that are part of the AFCD inspection protocol (e.g., small fins and processed fins in the first and second cases respectively). In the third case it gave them near real time genetic confirmation of suspected CITES-listed shark fins that were flagged after their initial visual inspection, with at least 85.7% of their visual identifications that were tested proving to be accurate. The remainder failed to amplify with the rtPCR protocol and the barcoding analysis, probably due to degraded DNA or PCR inhibitors present in the extraction. Overall, our results highlight the need to implement this new rtPCR protocol to detect the CITES listed species in shipments of fins from small sharks, small secondary fin types (e.g., pelvic, anal, second dorsal fins), and processed fins that have been previously gone uninspected due to the lack of morphological identification protocols. All three cases contained CITES-listed shark fins that were exported without the required permits, aligning with recent evidence of CITES-listed species remaining among the most common in the contemporary shark fin trade in Hong Kong10. Also, the barcoding analysis of the COI locus allowed the confirmation of the CITES species detection by our protocol in all three cases.
The rtPCR approach described here can aid law enforcement officers in major shark trade hubs around the world meet their CITES requirements and can also serve as a model for other monitoring applications for sharks. For example, the International Commission for the Conservation of Atlantic Tunas (ICCAT) prohibits the retention, transhipment, landing, storing, or trade of the three hammerheads and bigeye thresher sharks35, which could be monitored and enforced by member states in port using a modified five-primer version of our multiplex assay. Domestic and regional prohibitions on other species await either the development of a specific multiplex or design of species-specific primers. Usually, rtPCR protocols for quantification purposes (e.g. gene expression21), or using TaqMan probes, are recommended for amplicon sizes of less that 200 bps. We demonstrated that the SYBR Green chemistry allows for amplification and detection of fragments as long as 1000 bps (Table 1), increasing the scope for finding an appropriate region to design many species-specific primers within a locus (e.g., ITS2) and tailor rtPCR assays for sharks according to the specific legal application. Given the widespread detection of illegal trade of meat and fins from sharks and their relatives8,10,36,37,38,39,40 it is imperative that we deploy all available tools, including portable rtPCR field assays, for better monitoring and enforcement of laws intended to protect them.
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# Running rfgb¶
Object and their relationships are a natural way to think about the world. In this example, we have some facts about the world which we want to learn from. More specifically, we have a table of people and their relationships.
Name Gender Child Sibling James Male [Harry] Lily Female [Harry] Petunia Harry Male Arthur Male [Ron, Fred] Molly Female [Ron, Fred] Ron Male [Fred] Fred Male [Ron]
Assume that the goal is to learn father(Y,X). We want to learn logical rules representing that domain object X is the father of Y (both of which are people in this case), given that you know information about their gender, children, and siblings.
## From Tables to First-order Predicate Logic¶
Once we have a high-level idea of what these relationships look like, the next step is to convert this into predicate logic format. This format is standard for most Prolog-based systems.
A few assumptions we will make about our data:
1. ‘Name’ is an identifier.
2. ‘Gender’ is male or female in this case, so we can make it a true/false value.
3. ‘Child’ and ‘Sibling’ are binary relationships encoding a relationship between two people (e.g. childof(lily, harry) denotes that ‘harry’ is the childof ‘lily’).
The target we want to learn is father(x,y). To learn this rule, rfgb learns a decision tree that most effectively splits the positive and negative examples. This example is fairly small so a small number of trees should suffice, but for more complicated problem more may be needed to learn a robust model.
Positive Examples:
father(harrypotter,jamespotter).
father(ginnyweasley,arthurweasley).
father(ronweasley,arthurweasley).
father(fredweasley,arthurweasley).
...
Negative Examples:
father(harrypotter,mollyweasley).
father(georgeweasley,jamespotter).
father(harrypotter,arthurweasley).
father(harrypotter,lilypotter).
father(ginnyweasley,harrypotter).
father(mollyweasley,arthurweasley).
father(fredweasley,georgeweasley).
father(georgeweasley,fredweasley).
father(harrypotter,ronweasley).
father(georgeweasley,harrypotter).
father(mollyweasley,lilypotter).
...
Facts:
male(jamespotter).
male(harrypotter).
male(arthurweasley).
male(ronweasley).
male(fredweasley).
male(georgeweasley).
siblingof(ronweasley,fredweasley).
siblingof(ronweasley,georgeweasley).
siblingof(ronweasley,ginnyweasley).
siblingof(fredweasley,ronweasley).
siblingof(fredweasley,georgeweasley).
siblingof(fredweasley,ginnyweasley).
siblingof(georgeweasley,ronweasley).
siblingof(georgeweasley,fredweasley).
siblingof(georgeweasley,ginnyweasley).
siblingof(ginnyweasley,ronweasley).
siblingof(ginnyweasley,fredweasley).
siblingof(ginnyweasley,georgeweasley).
childof(jamespotter,harrypotter).
childof(lilypotter,harrypotter).
childof(arthurweasley,ronweasley).
childof(mollyweasley,ronweasley).
childof(arthurweasley,fredweasley).
childof(mollyweasley,fredweasley).
childof(arthurweasley,georgeweasley).
childof(mollyweasley,georgeweasley).
childof(arthurweasley,ginnyweasley).
childof(mollyweasley,ginnyweasley).
...
## Training a Model¶
There is one more piece we still need: background knowledge about the world.
// Parameters
setParam: maxTreeDepth=3.
setParam: nodeSize=1.
setParam: numOfClauses=8.
// Modes
mode: male(+name).
mode: childof(+name,+name).
mode: siblingof(+name,-name).
mode: father(+name,+name).
Begin training:
python -m rfgb --help
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# Variance of a hermitian operator
Take an hermitian operator $$O$$ such that $$O|\psi\rangle = x|\psi\rangle$$. The variance of an operator $$O$$ is defined as $$(\Delta O)^2 = \langle{O^2}\rangle - \langle{O}\rangle^2.$$ Let's consider the first term, I would write it as
$$\langle{O^2}\rangle = \langle\psi|O O|\psi\rangle = x\langle\psi|O|\psi\rangle = x^2\langle\psi|\psi\rangle = x^2.$$ But then for the second term I get the same result $$\langle{O}\rangle^2 = \langle\psi| O|\psi\rangle^2 = (x\langle\psi|\psi\rangle)^2 = x^2.$$ Therefore I end up with $$(\Delta O)^2 = 0$$ which is obviously wrong. What's the problem here?
The only idea I have is that $$O^2|\psi\rangle \neq O(O|\psi\rangle)$$, but i cannot understand why... What am I doing wrong?
As you have written things, the variance is indeed $$0$$ because $$\vert\psi\rangle$$ is an eigenstate of $$O$$: thankfully this is so as it means the outcome with eigenvalue $$x$$ is not uncertain and we can use the eigenvalue $$x$$ to label the state.
• @user85231 The coherent state is an eigenstate of $\hat a$, which is not hermitian. The variance is not $0$ in this case. See physics.stackexchange.com/questions/158849/… – ZeroTheHero Mar 11 '19 at 18:21
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# Surface area of revolution around the x-axis and y-axis
## Formulas to find the surface area of revolution
We can use integrals to find the surface area of the three-dimensional figure that’s created when we take a function and rotate it around an axis and over a certain interval.
The formulas we use to find surface area of revolution are different depending on the form of the original function and the axis of rotation.
Hi! I'm krista.
1. When the function is in the form ???y=f(x)??? and you’re rotating around the ???y???-axis, the interval is ???a\leq{x}\leq{b}??? and the formula is
???S=\int^b_a2\pi{x}\sqrt{1+\left(\frac{dy}{dx}\right)^2}\ dx???
2. When the function is in the form ???y=f(x)??? and you’re rotating around the ???x???-axis, the interval is ???a\leq{x}\leq{b}??? and the formula is
???S=\int^b_a2\pi{y}\sqrt{1+\left(\frac{dy}{dx}\right)^2}\ dx???
3. When the function is in the form ???x=g(y)??? and you’re rotating around the ???y???-axis, the interval is ???c\leq{y}\leq{d}??? and the formula is
???S=\int^d_c2\pi{x}\sqrt{1+\left(\frac{dx}{dy}\right)^2}\ dy???
4. When the function is in the form ???x=g(y)??? and you’re rotating around the ???x???-axis, the interval is ???c\leq{y}\leq{d}??? and the formula is
???S=\int^d_c2\pi{y}\sqrt{1+\left(\frac{dx}{dy}\right)^2}\ dy???
## Finding surface area of the rotation around the x-axis over an interval
Example
Find the area of the surface generated by rotating the function about the ???x???-axis over ???0\leq{x}\leq3???.
???y=x^3???
Since the equation is in the form ???y=f(x)???, and we’re rotating around the ???x???-axis, we’ll use the formula
???S=\int^b_a2\pi{y}\sqrt{1+\left(\frac{dy}{dx}\right)^2}\ dx???
We’ll calculate ???dy/dx??? and then substitute it back into the equation.
???\frac{dy}{dx}=3x^2???
???S=\int^3_02\pi{x^3}\sqrt{1+\left(3x^2\right)^2}\ dx???
???S=\int^3_02\pi{x^3}\sqrt{1+9x^4}\ dx???
Using u-substitution and setting ???u=1+9x^4??? and ???du=36x^3\ dx???, we calculate
???x=\left(\frac{u-1}{9}\right)^{\frac14}???
???dx=\frac{1}{36x^3}\ du???
???dx=\frac{1}{36\left[\left(\frac{u-1}{9}\right)^{\frac{1}{4}}\right]^3}\ du???
The formulas we use to find surface area of revolution are different depending on the form of the original function and the axis of rotation.
Plugging these values back into the integral, we get
???S=\int^3_02\pi{\left[\left(\frac{u-1}{9}\right)^{\frac{1}{4}}\right]^3}\sqrt{u}\frac{1}{36\left[\left(\frac{u-1}{9}\right)^{\frac{1}{4}}\right]^3}\ du???
???S=\int^3_02\pi{\left(\frac{u-1}{9}\right)^{\frac{3}{4}}}\sqrt{u}\frac{1}{36\left(\frac{u-1}{9}\right)^{\frac{3}{4}}}\ du???
???S=\frac{\pi}{18}\int^3_0{\left(\frac{u-1}{9}\right)^{\frac{3}{4}}}\sqrt{u}\frac{1}{\left(\frac{u-1}{9}\right)^{\frac{3}{4}}}\ du???
???S=\frac{\pi}{18}\int^3_0\sqrt{u}\ du???
Integrate.
???S=\left(\frac{\pi}{18}\right)\left(\frac{2}{3}u^{\frac{3}{2}}\right)\bigg|^3_0???
???S=\frac{\pi}{27}u^{\frac{3}{2}}\bigg|^3_0???
We’ll plug back in for ???u???, remembering that ???u=1+9x^4???, and then evaluate over the interval.
???S=\frac{\pi}{27}\left(1+9x^4\right)^{\frac{3}{2}}\bigg|^3_0???
???S=\frac{\pi}{27}\left[1+9(3)^4\right]^{\frac{3}{2}}-\left[\frac{\pi}{27}\left(1+9(0)^4\right)^{\frac{3}{2}}\right]???
???S=2,294.8??? square units
The surface area obtained by rotating ???y=x^3??? around the ???x???-axis over the interval ???0\leq{x}\leq3??? is ???S=2,294.8???.
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Question
# Answer the following true or false. If false, explain why. a. We can reduce alpha (alpha) by pushing the critical regions further into the tails of th
Modeling data distributions
Answer the following true or false. If false, explain why. a. We can reduce $$\alpha$$ (alpha) by pushing the critical regions further into the tails of the distribution. b. A decrease in the probability of the type II error always results in an increase in the probability of the type I error, provided that the sample size n does not change. c. The p-value of a left-sided hypothesis test is always half of a two-sided hypothesis test.
a. The quantity, $$\alpha$$ is also known as the probability of Type-I error. It is the probability of rejecting the null hypothesis when it is true. In a hypothesis testing problem, the probability assigned to the critical region is alpha, which is proportional to the area of the critical region. This critical region is the region, which lies in the extremities of the sampling distribution of the statistic, and is the region that marks the extreme or unlikely values of the distribution. If one pushes the critical regions further into the tails, the probability assigned to it will become smaller, thus reducing alpha Hence, the statement is True. b. The probability of Type-II error is the probability of failing to reject the null hypothesis when it is false. In reality, if the data and all the conditions attached to the hypothesis test are kept unchanged, there is only one way to decrease the probability of Type-II error, which is, by increasing the probability of Type-I error. It is not possible to simultaneously reduce both errors keeping all other factors in the study exactly identical. In a study, the only way to decrease both types of errors simultaneously, is by increasing the sample size. Hence, the statement is True. c. The p-value of a left-sided hypothesis test will be equal to half of that of a two-sided test, only in case of a symmetric sampling distribution of the test statistic (example: t, Normal distributions). For a symmetric distribution, the probability to the right of a certain value is equal to the probability to the left of the value which is at the mirror-image location of that value. However, this is not true for any distribution that is not symmetric (example: $$F, \chi^{2}$$ distributions). Hence, the statement is False.
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# Getting Record entries
I have a class Record which keeps entries of strings(implemented using vector of strings). I want to keep the data member private to the class. To get entries, I have defined a member function get_entry which accepts a reference to vector of strings to store results.
Below is the code.
#include <iostream>
#include <vector>
#include <string>
class Record {
public:
{
static int seq = 1;
entries.push_back(std::to_string(seq++) + " "+ entry);
}
void get_entry(std::vector<std::string>& res) const
{
for (auto &v: entries)
res.push_back(v);
}
private:
std::vector<std::string> entries;
};
int main()
{
Record name;
std::vector<std::string> res;
name.get_entry(res);
for (auto &v : res)
std::cout << v << "\n";
return 0;
}
Are there other better ways to get entries from such a class? I don't want to make entries data member public and access it directly.
The obvious downside of your current approach is that you do a lot potentially unnecessary copying. One common solution is to return (const) iterators to the entries, i.e.,:
class Record
{
public:
// ...
auto cbegin() const { return entries.cbegin(); }
auto cend() const { return entries.cend(); }
// Perhaps similar methods for getting non-const iterators ...
};
This solution allows you to use e.g., standard algorithms on the entries, but you can't manipulate the underlying range from the outside (due to const-ness).
To support your earlier use case, you can still do std::vector<std::string> res(r.cbegin(), r.cend()); if you wish, where r is an object of type Record.
Just return entries by value. It will make a deep copy automatically:
std::vector<std::string> get_entries() const
{
return entries;
}
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# C++ Mandelbulb-set generator
This mandelbulb-set generator creates a .xyz file which contains all the points that are in the set which lie on a 3D grid. It does what it is supposed to.
However, the number of calculations done is proportional to the cube of the resolution*, which means that the script takes very long if the resolution is in the order of 500 or 1000.
* The resolution is the amount of points that each axis is divided into. A resolution of 5 means that the x, y and z axis are each divided by 5. Therefore, there are $$\5^3 = 125\$$ points that are calculated.
Here is the entire code:
#include <iostream>
#include <cmath>
#include <fstream>
using namespace std;
/* Mandelbulb: https://en.wikipedia.org/wiki/Mandelbulb
*
* The sequence v(n+1) = v(n)^power + c is iterated several times for every point in 3d space. v(0) and c are equal to that point.
* If the length of v(n) does not exceed 'maxlength' (i.e. it does not diverge) then the point is an element of the mandelbulb set
* and saved to the file.
*/
int main()
{
//settings:
int iter = 5;
int resolution = 50;
double power = 8.0;
double maxlength = 2.0;
double rangemin = -1.75;
double rangemax = 1.75;
//Points:
double xpos, ypos, zpos; //positions to cycle through
double xpoint, ypoint, zpoint; //point to be calculated at that position
double cx, cy, cz; //C-point
double r, phi, theta; //spherical coordinates
//calculated variables:
double div = (rangemax - rangemin) / double(resolution);
//other variables:
double length = 0;
double density = 0; //Named 'density' since the 3d-volume has density=1 if the point is in the set and density=0 if point is not in the set
double progress, progdiv = 100.0 / pow((double(resolution) + 1), 3.0);
//User interface:
cout << endl;
cout << "+--------------------+" << endl;
cout << "|Mandelbulb generator|" << endl;
cout << "+--------------------+" << endl << endl;
cout << "Standard settings:" << endl;
cout << "Iterations: " << iter << endl;
cout << "Power: " << power << endl;
cout << "Maxlength: " << maxlength << endl;
cout << "rangemin: " << rangemin << endl;
cout << "rangemax: " << rangemax << endl;
cout << "Resolution: " << resolution << endl;
cout << endl;
//create output file:
ofstream file ("Mandelbulb.xyz");
file << div << endl; //first line contains distance between points for other scripts to read
//x,y,z-loop:
for(xpos = rangemin; xpos <= rangemax; xpos += div){
for(ypos = rangemin; ypos <= rangemax; ypos += div){
for(zpos = rangemin; zpos <= rangemax; zpos += div){
//Display progress in console:
progress += progdiv;
cout << " \r";
cout << progress << "%\r";
cout.flush();
//reset for next point:
xpoint = xpos;
ypoint = ypos;
zpoint = zpos;
cx = xpos;
cy = ypos;
cz = zpos;
//Sequence loop:
for(int i = 0; i <= iter; i++)
{
r = sqrt(xpoint*xpoint + ypoint*ypoint + zpoint*zpoint);
phi = atan(ypoint/xpoint);
theta = acos(zpoint/r);
xpoint = pow(r, power) * sin(power * theta) * cos(power * phi) + cx;
ypoint = pow(r, power) * sin(power * theta) * sin(power * phi) + cy;
zpoint = pow(r, power) * cos(power * theta) + cz;
length = r;
if(length >= maxlength)
{
density = 0.0;
break;
}
density = 1.0;
}
if(density == 1.0)
{
file << xpos << " " << ypos << " " << zpos << endl;
}
}//zpos loop end
}//ypos loop end
}//xpos loop end
cout << endl << "Done." << endl;
return 0;
}
I hope that the comments make clear what happens, otherwise please ask. Again, the goal is to make it execute as fast as possible. I am a beginner at C++ and might not see some obvious bottlenecks. I'd like to avoid using external libraries.
• Parallelism is the answer here. Generating a fractal is an embarrassingly parallel problem. Split the space into k threads and have each of them calculate the subspace autonomously. If you take this approach to the limit then you would use GPUs, and have a thread calculate a single point. I wrote an article some time ago on fractals on CUDA (Julia set in this case) davidespataro.it/cuda-julia-set-fractals. – Davide Spataro Dec 3 '19 at 21:22
• You should definitely avoid flush() and endl. They will slow your program down since they force it to write to the file/stdout right now. For this it has to wait on the system. Without flushing, it can buffer the writing for later (it will certainly do it at some point) and then can write more data in a single write, which minimizes waiting. See also here – n314159 Dec 3 '19 at 21:40
I probably can't say much about the algorithm but I see many things which can help you improve your use of C++ and programming in general.
• ### Don't use using namespace std
Why? You pollute the whole std namespace into your program which could lead to name clashes. More about it here.
• ### Use const and constexpr
Consider these declarations:
//settings:
int iter = 5;
int resolution = 50;
double power = 8.0;
double maxlength = 2.0;
double rangemin = -1.75;
double rangemax = 1.75;
In your program you can now happily modify these parameters by accident. Since they should be constants say so with the keyword const:
const int iter = 5;
const int resolution = 50;
const double power = 8.0;
const double maxlength = 2.0;
const double rangemin = -1.75;
const double rangemax = 1.75;
If C++11 is available, even better: make them constexpr:
constexpr int iter = 5;
constexpr int resolution = 50;
constexpr double power = 8.0;
constexpr double maxlength = 2.0;
constexpr double rangemin = -1.75;
constexpr double rangemax = 1.75;
This makes these variables true constants. That means they eat up no memory. They are just aliases for the numbers.
• ### Use auto whenever possible
You don't have to mention types explicitly. Instead use auto. So we get:
constexpr auto iter = 5;
constexpr auto resolution = 50;
constexpr auto power = 8.0;
constexpr auto maxlength = 2.0;
constexpr auto rangemin = -1.75;
constexpr auto rangemax = 1.75;
• ### omit return 0 in main()
In C++, return 0 gets automatically inserted into the main function if you don't write it. So for this special case it is not necessary to write the return.
• ### Avoid declaring several variables on one line
double progress, progdiv = 100.0 / pow((double(resolution) + 1), 3.0);
//...
//Display progress in console:
progress += progdiv;
My compiler shows a warning in the last line:
variable 'progress' is uninitalized when used here
That means progress can have any value here and gets added here.
So better write two lines and initialize:
auto progress = 0.0;
auto progdiv = 100.0 / pow((double(resolution) + 1), 3.0);
• ### Avoid std::endl
This is really a very common beginners' trap. Even some books use std::endl all the time but it is wrong.
First of all, std::endl is nothing but this:
'\n' + std::flush
So it gives you a new line (What you want) and the performs a expensive flush of the output sequence which is in 99.9% of the cases not necessary. So stick to \n.
• ### Declare variables as late as possible:
This:
//reset for next point:
xpoint = xpos;
ypoint = ypos;
zpoint = zpos;
cx = xpos;
cy = ypos;
cz = zpos;
Should be:
//reset for next point:
auto xpoint = xpos;
auto ypoint = ypos;
auto zpoint = zpos;
auto cx = xpos;
auto cy = ypos;
auto cz = zpos;
Unlike in old C Standard, you can declare variables as late as possible. They should be declared when they are needed. This avoids mistakes.
Same goes for this:
auto r = sqrt(xpoint*xpoint + ypoint*ypoint + zpoint*zpoint);
auto phi = atan(ypoint/xpoint);
auto theta = acos(zpoint/r);
• ### Limit your line length
Consider this nice line:
//other variables:
double length = 0.0;
double density = 0.0; //Named 'density' since the 3d-volume has density=1 if the point is in the set and density=0 if point is not in the set
You should limit your line length to a value (80 or 100 are common) so you can read all the code without scrolling to the right. With this limit you can also open two code pages next to each other. More about it here.
Most IDEs support a visible line length marker. In mine it looks like this:
So better like this:
double length = 0.0;
//Named 'density' since the 3d-volume has density=1 if the point is in the
//set and density=0 if point is not in the set
double density = 0.0;
• ### Avoid long functions
Your main function is very long and it's hard to grasp from looking at it what is going on. Prefer shorter functions which only do one thing. The main does several things (user input, calculation etc). Split that into smaller functions. And the smaller functions into even smaller functions. This way you also have the option to run the algorithm without any user input, maybe for unit tests.
Refactored, the code looks like this:
#include <iostream>
#include <cmath>
#include <fstream>
/* Mandelbulb: https://en.wikipedia.org/wiki/Mandelbulb
*
* The sequence v(n+1) = v(n)^power + c is iterated several times for every
* point in 3d space. v(0) and c are equal to that point.
* If the length of v(n) does not exceed 'maxlength' (i.e. it does not diverge)
* then the point is an element of the mandelbulb set
* and saved to the file.
*/
struct Parameters{
const int iter;
const int resolution;
const double power;
const double maxLength;
const double rangemin;
const double rangemax;
};
void printParameters(std::ostream &os, const Parameters ¶meters);
void calculation(
std::ostream &calcOutput,
std::ostream &progressOutput,
bool progressOutputOn,
const Parameters ¶meters);
bool sequenceLoop(double xpos, double ypos, double zpos, int iter,
double power, double maxLength);
bool sequence(double &xpoint, double &ypoint, double &zpoint, double power,
double maxLength, double cx, double cy, double cz);
void printParameters(std::ostream &os, const Parameters ¶meters)
{
os << '\n'
<< "+--------------------+" << '\n'
<< "|Mandelbulb generator|" << '\n'
<< "+--------------------+" << '\n'
<<'\n'
<< "Standard settings:" << '\n'
<< "Iterations: " << parameters.iter << '\n'
<< "Power: " << parameters.power << '\n'
<< "Maxlength: " << parameters.maxLength << '\n'
<< "rangemin: " << parameters.rangemin << '\n'
<< "rangemax: " << parameters.rangemax << '\n'
<< "Resolution: " << parameters.resolution << '\n'
<<'\n';
}
void calculation(
std::ostream &calcOutput,
std::ostream &progressOutput,
bool progressOutputOn,
const Parameters ¶meters)
{
auto div = (parameters.rangemax - parameters.rangemin)
/ double(parameters.resolution);
//first line contains distance between points for other scripts to read
calcOutput << div << '\n';
auto progress = 0.0;
auto progdiv = 100.0 / pow((double(parameters.resolution) + 1), 3.0);
//x,y,z-loop:
for(auto xpos = parameters.rangemin;
xpos <= parameters.rangemax; xpos += div) {
for(auto ypos = parameters.rangemin;
ypos <= parameters.rangemax; ypos += div) {
for(auto zpos = parameters.rangemin;
zpos <= parameters.rangemax; zpos += div) {
if(progressOutputOn) {
//Display progress in console:
progress += progdiv;
progressOutput << " \r"
<< progress << "%\r";
progressOutput.flush();
}
if(sequenceLoop(xpos, ypos, zpos, parameters.iter,
parameters.power, parameters.maxLength))
{
calcOutput << xpos << " " << ypos << " " << zpos << '\n';
}
}
}
}
if(progressOutputOn) {
progressOutput << '\n'
<< "Done." << '\n';
}
}
bool sequenceLoop(
double xpos,
double ypos,
double zpos,
double power,
int iter,
double maxLength)
{
auto xpoint = xpos;
auto ypoint = ypos;
auto zpoint = zpos;
for(auto i = 0; i <= iter; i++)
{
if(!sequence(xpoint, ypoint, zpoint, power, maxLength,
xpos, ypos, zpos)) {
return false;
}
}
return true;
}
bool sequence(
double &xpoint,
double &ypoint,
double &zpoint,
double power,
double maxLength,
double cx,
double cy,
double cz)
{
auto r = sqrt(xpoint*xpoint + ypoint*ypoint + zpoint*zpoint);
auto phi = atan(ypoint/xpoint);
auto theta = acos(zpoint/r);
xpoint = pow(r, power) * sin(power * theta) * cos(power * phi) + cx;
ypoint = pow(r, power) * sin(power * theta) * sin(power * phi) + cy;
zpoint = pow(r, power) * cos(power * theta) + cz;
return r < maxLength;
}
int main()
{
constexpr Parameters parameters{ 5, 50, 8.0, 2.0, -1.75, 1.75 };
printParameters(std::cout, parameters);
constexpr auto outputFileName = "Mandelbulb.xyz";
std::ofstream ofs (outputFileName);
calculation(ofs, std::cout, true, parameters);
}
I hope this gives some idea how to make the code more readable. I think it can be still improved further.
For a bit of speedup, I suggest to turn off the progress echoing in the console with progressOutputOn = false for the method calculation.
As mentioned in the comments, maybe you can split up the calculation into several threads. For example:
xpoint = pow(r, power) * sin(power * theta) * cos(power * phi) + cx;
ypoint = pow(r, power) * sin(power * theta) * sin(power * phi) + cy;
zpoint = pow(r, power) * cos(power * theta) + cz;
These 3 points get calculated one after each other. But they are independent of each other, so probably they can be excecuted in parallel. I sould try that out with std::async.
That's my first thought but maybe even the whole algorithm needs to be set up differently to have it ready for parallel computation.
Now that you have the code cleaned up, you can measure the time between the calculations to see how the speed is (also with other parameters).
|
{}
|
# VC dimension, Rademacher complexity, and growth function
March 15, 2018 | by
We define the VC dimension, the Rademacher complexity, and the growth function of a class of functions.
Let $X$ be the set of inputs. The general learning question we consider is the following: we want to learn a target function $f : X \to \left\{0,1\right\}$ through a number of samples $(x_i,f(x_i))_{i \in [1,m]}$. The function $f$ is unknown, but we assume some background knowledge in the form of a set of hypotheses $H$, which are functions $h : X \to \left\{0,1\right\}$.
We make a strong assumption called realizability: $f \in H.$ This assumption can be lifted to obtain more general results, which we will not do here.
We would like to quantify how hard it is to learn a function from $H$. More precisely, we are interested in the PAC-learnability of $H$; see below for a definition. There are two approaches:
• a combinatorial approach, called the VC dimension,
• a probabilistic approach, called the Rademacher complexity.
## PAC learning
We let $D$ denote distributions over the inputs, and write $x \sim D$ to mean that $x$ is sampled according to $D$. If the sample set has size $m$ we assume that the inputs are identically independently sampled according to $D$, written $S \sim D^m$, where $S = (x_i)_{i \in [1,m]}$. The distribution $D$ is unknown, in particular we do not want the learning algorithm to depend on it.
The first notion we need is the loss function: how good is a hypothesis $h$ if the target function is $f$?
• The zero-one loss is $L_f(h,x) = 1 \text{ if } h(x) \neq f(x), \text{ and } 0 \text{ otherwise}$.
• The quality of $h$ is measured by $L_{D,f}(h) = E_{x \sim D}[L_f(h,x)]$.
• The empirical quality of $h$ against the samples $S$ is $L_{S,f}(h) = \frac{1}{m} \sum_{i = 1}^m L_f(h,x_i)$.
Definition: We say that $H$ is PAC-learnable if
for all $\varepsilon > 0$ (precision),
for all $\delta > 0$ (confidence),
there exists an algorithm $A$,
there exists $m$ a number of samples,
for all objective function $f \in H$,
for all distributions $D$ over $X$:
$P_{S \sim D^m} ( L_D(h,f) \le \varepsilon ) \ge 1 - \delta,$ where we let $h$ denote the hypothesis picked by the algorithm $A$ when receiving the samples $S = (x_i,f(x_i))_{i \in [1,n]}$.
## VC dimension
We say that a subset $Y$ of $X$ is shattered by $H$ if by considering the restrictions of the functions in $H$ to $Y$ we obtain all functions $Y \to \left\{0,1 \right\}$. Formally, let $H_{\mid Y}$ denote $\left\{ h_{\mid Y} : Y \to \left\{0,1\right\} \mid h \in H \right\}$. The subset $Y$ is shattered by $H$ if $\text{Card} (H_{\mid Y}) = 2^Y$.
Intuitively, $Y$ being shattered by $H$ means that to learn how the function $f$ behaves on $Y$ one needs the values of $f$ on each element of $Y$. Hence the if $H$ shatters large sets, it is difficult to learn a function from $H$.
Definition: The VC dimension of $H$ is the size of the largest set $Y$ which is shattered by $H$.
We unravel the definition: the VC dimension of $H$ is the size of the largest set $Y$ for which all possible functions are realised.
The beauty here is that this is a purely combinatorial definition, which abstracts away probabilities and in particular the probabilistic distribution on the inputs. We will see that it anyway says something about the PAC-learnability of $H$.
The Rademacher complexity quantifies how a set of functions fits random noise. To define it for $H$, we consider the loss functions induced by $H$ and $f$.
Hence the question is how much do the loss functions induced by $H$ and $f$ fit random noise. A random noise is $\sigma = (\sigma_i)_{i \in [1,m]}$ where each $\sigma_i \in \left\{-1,+1\right\}$ is drawn uniformly and independently at random. To measure how well $h$ correlates with random noise we use the quantity
$\frac{1}{m} \sum_{i = 1}^m \sigma_i L_f(h,x_i)$.
Hence the function $h$ which best fits random noise maximises the above quantity.
The empirical Rademacher complexity of $H$ and $f$ against the samples $S$ is $E_{\sigma} \left[ \sup_{h \in H} \frac{1}{m} \sum_{i = 1}^m \sigma_i L_f(h,x_i) \right]$
Definition: The Rademacher complexity $R_H(m)$ of $H$ is $\sup_{f \in H} E_{S \sim D^m} \left[ E_{\sigma} \left[ \sup_{h \in H} \frac{1}{m} \sum_{i = 1}^m \sigma_i L_f(h,x_i) \right] \right]$.
## Growth function
The notion of growth function will be a bridge between VC dimension and Rademacher complexity.
Definition: The grow function $\tau_H$ of $H$ associates with $m$ the number $\max_{Y \subseteq X, |Y| = m} |H_{\mid Y}|$.
Remark that by definition, if $H$ has VC dimension $d$, then for $m \le d$, we have $\tau_H(m) = 2^m$.
The following theorem relates the Rademacher complexity to the growth function:
Massart’s Theorem: $R_H(m) \le \sqrt{\frac{2 \log(\pi_H(m))}{m}}$.
The proof of analytical and relies on Hoeffding’s inequality. See p39/40 of the Foundations of Machine Learning book for a proof.
|
{}
|
# Bibliotek
Musik » Christina Aguilera »
## Something's Got a Hold on Me
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Something's Got a Hold on Me 3:04 23 apr 2014, 09:35
Something's Got a Hold on Me 3:04 7 apr 2014, 19:03
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Something's Got a Hold on Me 3:04 28 mar 2014, 01:26
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Something's Got a Hold on Me 3:04 14 mar 2014, 18:13
Something's Got a Hold on Me 3:04 7 mar 2014, 08:56
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Something's Got a Hold on Me 3:04 27 jan 2014, 20:20
Something's Got a Hold on Me 3:04 22 jan 2014, 09:13
Something's Got a Hold on Me 3:04 15 jan 2014, 11:46
Something's Got a Hold on Me 3:04 14 dec 2013, 22:52
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Something's Got a Hold on Me 3:04 1 dec 2013, 09:44
Something's Got a Hold on Me 3:04 11 nov 2013, 21:29
Something's Got a Hold on Me 3:04 9 nov 2013, 12:48
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Something's Got a Hold on Me 3:04 3 okt 2013, 22:34
Something's Got a Hold on Me 3:04 19 sep 2013, 12:22
Something's Got a Hold on Me 3:04 6 sep 2013, 11:29
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Something's Got a Hold on Me 3:04 13 aug 2013, 00:28
Something's Got a Hold on Me 3:04 5 aug 2013, 05:58
Something's Got a Hold on Me 3:04 29 jul 2013, 02:47
Something's Got a Hold on Me 3:04 18 jul 2013, 22:22
Something's Got a Hold on Me 3:04 25 jan 2013, 23:56
Something's Got a Hold on Me 3:04 18 okt 2012, 22:34
Something's Got a Hold on Me 3:04 7 sep 2012, 19:25
Something's Got a Hold on Me 3:04 16 aug 2012, 00:49
Something's Got a Hold on Me 3:04 12 jun 2012, 20:45
Something's Got a Hold on Me 3:04 2 jun 2012, 12:33
Something's Got a Hold on Me 3:04 15 maj 2012, 14:56
Something's Got a Hold on Me 3:04 22 apr 2012, 21:48
Something's Got a Hold on Me 3:04 22 apr 2012, 21:13
Something's Got a Hold on Me 3:04 8 apr 2012, 15:53
Something's Got a Hold on Me 3:04 27 mar 2012, 13:48
Something's Got a Hold on Me 3:04 25 mar 2012, 15:38
Something's Got a Hold on Me 3:04 24 mar 2012, 04:01
Something's Got a Hold on Me 3:04 5 mar 2012, 02:10
Something's Got a Hold on Me 3:04 19 feb 2012, 22:03
Something's Got a Hold on Me 3:04 15 feb 2012, 23:12
Something's Got a Hold on Me 3:04 21 dec 2011, 21:40
Something's Got a Hold on Me 3:04 14 dec 2011, 23:00
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Topic
# space flight
12,484 media by topicpage 1 of 125
## S05-43-1547 - STS-005 - Earth observations taken during STS-5 mission
The original finding aid described this as: Description: Earth observations taken during the STS-5 mission from the space shuttle Columbia. Many of the views on this roll are framed by a portion of the orbiter... More
## S08-49-1745 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations (framed by the aft flight deck window) taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERV... More
## S08-50-1800 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERVATIONS (FROM SPACE) STS-8 Categories:... More
## S08-50-1835 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERVATIONS (FROM SPACE) STS-8 Categories:... More
## S08-50-1837 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERVATIONS (FROM SPACE) STS-8 Categories:... More
## S08-50-1838 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERVATIONS (FROM SPACE) STS-8 Categories:... More
## S08-50-1840 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. An active 5, 500 foot high volcano on Adonara Island in Indonesia ... More
## S08-50-1841 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERVATIONS (FROM SPACE) STS-8 Categories:... More
## S08-50-1849 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERVATIONS (FROM SPACE) STS-8 Categories:... More
## S08-50-1851 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERVATIONS (FROM SPACE) STS-8 Categories:... More
## S08-50-1860 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERVATIONS (FROM SPACE) STS-8 Categories:... More
## S08-50-1875 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERVATIONS (FROM SPACE) STS-8 Categories:... More
## S08-50-1891 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERVATIONS (FROM SPACE) STS-8 Categories:... More
## S08-37-1902 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERVATIONS (FROM SPACE) STS-8 Categories:... More
## S05-46-1943 - STS-005 - Earth observations taken during STS-5 mission
The original finding aid described this as: Description: Earth observations taken during the STS-5 mission from the space shuttle Columbia. Subject Terms: EARTH OBSERVATIONS (FROM SPACE), STS-5, COLUMBIA (ORB... More
## S08-37-1943 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERVATIONS (FROM SPACE) STS-8 Categories:... More
## S08-37-1957 - STS-008 - Earth observations taken during STS-8 mission
The original finding aid described this as: Description: Earth observations taken during the STS-8 mission from the space shuttle Challenger. Subject Terms: EARTH OBSERVATIONS (FROM SPACE) STS-8 Categories:... More
## S05-46-1957 - STS-005 - Earth observations taken during STS-5 mission
The original finding aid described this as: Description: Earth observations taken during the STS-5 mission from the space shuttle Columbia. Subject Terms: EARTH OBSERVATIONS (FROM SPACE), STS-5, COLUMBIA (ORB... More
## Photograph of John H. Glenn, Jr., Virgil L. Grissom and Alan B. Shepar...
Original caption: The three Project Mercury astronauts training for the nation's first manned space flight are (from left) John H. Glenn, Jr., Virgil I. Grissom and Alan B. Shepard, Jr. The forthcoming sub-orb... More
## Photograph of Virgil L. Grissom Preparing for Centrifuge Tests
Original caption: Astronaut Virgil I. Grissom, 35, prepares for centrifuge tests at the U.S. Navy's center at Johnsville, Pennsylvania during April of this year. These tests were conducted to determine the rea... More
## Mercury 1A Mission - Clouds, Earth observations
The original caption reads: View of clouds taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observations ... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - lighting fixture
The original caption reads: Underexposed (almost totally dark) frame with lighting fixture visible in the center taken prior to the Mercury Redstone 1A flight. A clock face is visible in the top left of the pho... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Clouds, Earth observations
The original caption reads: View of clouds taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observations ... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - lighting fixture
The original caption reads: Underexposed (almost totally dark) frame with lighting fixture visible in the center taken prior to the Mercury Redstone 1A flight. A clock face is visible in the top left of the pho... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - underexposed, NASA Mercury program
The original caption reads: Underexposed (almost totally dark) frame taken during to the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Cate... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - lighting fixture
The original caption reads: Underexposed (almost totally dark) frame with lighting fixture visible in the center taken prior to the Mercury Redstone 1A flight. A clock face is visible in the top left of the pho... More
## Mercury 1A Mission - lighting fixture
The original caption reads: Underexposed (almost totally dark) frame with lighting fixture visible in the center taken prior to the Mercury Redstone 1A flight. A clock face is visible in the top left of the pho... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Clouds, Earth observations
The original caption reads: View of clouds taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observations ... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Views
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - lighting fixture
The original caption reads: Underexposed (almost totally dark) frame with lighting fixture visible in the center taken prior to the Mercury Redstone 1A flight. A clock face is visible in the top left of the pho... More
## Mercury 1A Mission - underexposed, NASA Mercury program
The original caption reads: Underexposed (almost totally dark) frame taken during to the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Cate... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - underexposed, NASA Mercury program
The original caption reads: Underexposed (almost totally dark) frame taken during to the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Cate... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - underexposed, NASA Mercury program
The original caption reads: Underexposed (almost totally dark) frame taken during to the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Cate... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
## Mercury 1A Mission - Clouds, Earth observations
The original caption reads: View of clouds taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observations ... More
## Mercury 1A Mission - Earth Observations
The original caption reads: Earth Observations taken during the Mercury Redstone 1A flight. A clock face is visible in the top left of the photo. Film type was 70mm Dupont Cronar. Categories: Earth Observatio... More
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## NARA & DVIDS Public Domain Archive
The objects in this collection are from The U.S. National Archives and Defense Visual Information Distribution Service. The U.S. National Archives and Records Administration (NARA) was established in 1934 by President Franklin Roosevelt. NARA keeps those Federal records that are judged to have continuing value—about 2 to 5 percent of those generated in any given year. There are approximately 10 billion pages of textual records; 12 million maps, charts, and architectural and engineering drawings; 25 million still photographs and graphics; 24 million aerial photographs; 300,000 reels of motion picture film; 400,000 video and sound recordings; and 133 terabytes of electronic data. The Defense Visual Information Distribution Service provides a connection between world media and the American military personnel serving at home and abroad. All of these materials are preserved because they are important to the workings of Government, have long-term research worth, or provide information of value to citizens.
Disclaimer: A work of the U.S. National Archives and DVIDS is "a work prepared by an officer or employee" of the federal government "as part of that person's official duties." In general, under section 105 of the Copyright Act, such works are not entitled to domestic copyright protection under U.S. law and are therefore in the public domain. This website is developed as a part of the world's largest public domain archive, PICRYL.com, and not developed or endorsed by the U.S. National Archives or DVIDS. https://www.picryl.com
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Lab 10: Graphs and dot files
• Due before 23:59 Wednesday, November 9
• Submit using the submit program as 301 lab 10.
This will be covered in more detail in the notes for Unit 9, but here is an overview of how we store a graph with an adjacency list data structure:
• Every vertex is identified by some label. (Usually, these labels are just strings.)
• Every vertex is associated with a list that contains the labels of its neighbors, i.e. the other vertices which are connected to that vertex by some edge.
• There is a map that associates each vertex's label to its list of neighbors.
So the main representation is a map of strings to lists of strings. For example, the adjacency list of the following graph:
would be the following map of strings to lists of strings:
{'A': ['B', 'C'],
'B': ['A', 'C', 'D'],
'C': ['A', 'B'],
'D': ['B'] }
2 DOT Files
A common file format for encoding graph information is called a DOT file. We're going to write a class to read in a .dot file of an undirected, unweighted graph, store the information in an adjacency list, and then run some common operations on that graph.
Here's an example dot file. The first line names the graph. Following that is a list of all edges in the graph. You may assume the vertices are "named" using a single word, but they may not be single letters like this. You can also assume the graph is undirected and unweighted, and that there is one edge specified per line, like in the example file.
3 Assignment: Write a Graph class
In a file called graph.py, you need to write a class Graph with the following methods:
• A constuctor which accepts a file name, stores the graph name, and builds the adjacency list. You can assume a competent user is calling this, and the file is, indeed, a .dot file. (Ex: g=Graph('example.dot') )
• isEdge(self, vertexA, vertexB): Returns True if the two vertices are adjacent, and False if they are not. (Ex: Given our example.dot, g.isEdge('B','A') would return True, while g.isEdge('D','A') would return False.) Because the graph is undirected, the order of the arguments shouldn't matter.
• neighbors(self, vertex): Returns a list of all vertices adjacent to vertex.
• __str__(self): Returns a string which re-creates the .dot file from the adjacency list. It doesn't need to be in the same order as it was when it came in, but it should contain the same information. You may not just store all the text and regurgitate it for this method. Take note, the description should be returned as a single string, not printed.
This lab will be the basis of our next project and lab, so understanding it carries some extra importance. If you have questions or run into trouble, just ask for help!
Note: if you're stuck on how to read in the dot file, have a look at the built-in methods of file objects and, more importantly, the built-in methods of strings.
4 Visualizing .dot files
If you have a .dot file, you can use some built-in Linux tools to visualize it. For example, given your example.dot, run fdp -Tpng example.dot > example.png. You can then look at example.png to see your graph. Run man dot for the full specifications.
5 Directed Graphs
The rest of this lab is optional for your enrichment and enjoyment. Be sure to work on this only after getting everything else perfect. You should still submit this work, but make sure it doesn't break any of the parts you already had working correctly!
.dot files can also handle directed graphs, though right now, your program probably can't. Fix that! If A -> B;, then isEdge('A','B') should return True, but isEdge('B','A') should return False.
Arrows always point right.
Directed graphs are labelled as such: in the first line, instead of "graph graphName {" it would say "digraph graphName {". In digraphs, ALL edges are directed (A -- B; is not allowed). In regular graphs, NO edges are directed.
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Overview
DirectEffects is an R package to estimate controlled direct effects (CDEs), which are the effect of a treatment fixing a set of downstream mediators to particular values. As of now, the package supports sequential g-estimation and a two-stage matching approach called telescope matching. For more information on how CDEs can be useful for applied research and a brief introduction to sequential g-estimation, see Acharya, Blackwell, and Sen (2016). For more on the telescope matching procedure, see Blackwell and Strezhnev (2022).
Installation
You can install DirectEffects via CRAN for the current stable version or via GitHub for the development version.
# Installing from CRAN
install.packages("DirectEffects")
# Installing development version from Github:
# install.packages("devtools")
devtools::install_github("mattblackwell/DirectEffects", build_vignettes = TRUE)
Usage
The main functions for estimating CDEs in DirectEffects are:
DirectEffects also provides diagnostics for these two approaches, including sensitivity analyses and balance checks.
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# finding the probability for coin toss and expected number of flips required, is this the right answer for this problem?
A coin having probability $$p$$ of coming up head is to be successively flipped until the first head appears.
1. find the probability that the first head occurs in odd number of tosses ?
2. find the expected number of flips required for the first head ?
the solution for number 1 $$p(1 + (1-p)^2 + (1-p)^4 + \ldots) = p\displaystyle\sum_{i=0}^\infty ((1-p)^2)^i = \frac{p}{1-(1-p)^2} = \frac{1}{2-p}.$$
the solution for number 2
$$\sum\limits_{i=1}^\infty i p (1-p)^{i-1}=\frac{1}{p}~~~~~~~$$
is this the right answer for this problem and thank you
## 2 Answers
Both are correct.
As a way to avoid the use of infinite series:
For the first note that we either win on the first roll, lose on the second, or reset. Thus, letting $$\Psi$$ denote the answer, we have $$\Psi=p+(1-p)p\times 0+(1-p)^2\Psi\implies \left(1-(1-p)^2\right)\Psi=p\implies \Psi=\frac p{1-(1-p^2)}$$
For the second note the recursive relation $$E=p\times 1 +(1-p)\times (E+1)\implies pE=1\implies E=\frac 1p$$
• thank you very much for confirming the answer and for explaining this more clear way to obtain it – green life Dec 16 '18 at 19:28
Let $$X =$$ number of tosses until a head appears.
Then $$X$$ ~ Geometric$$(\frac{1}{2})$$.
## Part 1
The probability we wish to calculate is,
$$\Pr(odd) = \sum_{k=0}^{\infty}{\Pr(X = 2k + 1)} = \sum_{k=0}^{\infty}{p(1 - p)^{2k}} = p\sum_{k=0}^{\infty}{((1-p)^{2})^{k}} = \frac{p}{1 - (1-p)^{2}} = \frac{1}{2-p}$$
## Part 2
Let $$Y =$$ heads was seen on the first flip.
Then $$Y$$ ~ Bernoulli$$(p)$$.
$$E[X] = E[E[X|Y]] = \Pr(Y = 1)E[X|Y=1] + \Pr(Y=0)E[X| Y=0] = p(1) + (1-p)(1 + E[X])$$
This then reduces to the equation, $$E[X] = 1 + (1-p)E[X]$$ And then solving for $$E[X]$$ we get $$E[X] = \frac{1}{p}$$
## Remarks
My calculations agree with yours, so I would conclude that your answers are correct.
• thank you very much for confirming the answer and for the explanation – green life Dec 16 '18 at 19:42
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# Question
You work for Microsoft Corporation (ticker: MSFT), and you are considering whether to develop a new software product. The risk of the investment is the same as the risk of the company.
a. Using the data in Table 13.1 and in the table above, calculate the cost of capital using the FFC factor specification if the current risk-free rate is 3% per year.
b. Microsoft’s CAPM beta over the same time period was 0.84. What cost of capital would you estimate using theCAPM?
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# Oh! Matryoshka!
Given polynomial $f(x) = a_0 + a_1 x+a_2 x^2 + ... + a_{n-1}x^{n-1}+a_n x^n$, we wish to evaluate integral
$\int \frac{f(x)}{(x-a)^p}\;dx, \quad p \in N^+\quad\quad\quad(1)$
When $p = 1$,
$\int \frac{f(x)}{x-a} \;dx= \int \frac{f(x)-f(a)+f(a)}{x-a}\;dx$
$= \int \frac{f(x)-f(a)}{x-a}\;dx + \int \frac{f(a)}{x-a}\;dx$
$=\int \frac{f(x)-f(a)}{x-a}\;dx + f(a)\cdot \log(x-a)$.
Since
$f(x) = a_0 + a_1x + a_2x^2 + ... + a_{n-1}x^{n-1} + a_n x^n$
and
$f(a) = a_0 + a_1 a + a_2 a^2 + ... + a_{n-1}a^{n-1} + a_n a^n$
It follows that
$f(x)-f(a) = a_1(x-a) + a_2(x^2-a^2) + ... + a_{n-1}(x^{n-1}-a^{n-1}) + a_n (x^n-a^n)$.
That is
$f(x)-f(a) = \sum\limits_{k=1}^{n}a_k(x^k-a^k)$
By the fact (see “Every dog has its day“) that
$x^k-a^k =(x-a)\sum\limits_{i=1}^{k}x^{k-i}a^{i-1}$,
we have
$f(x)-f(a) = \sum\limits_{k=1}^{n}a_k(x-a)\sum\limits_{i=1}^{k}x^{k-i}a^{i-1}=(x-a)\sum\limits_{k=1}^{n}(a_k\sum\limits_{i=1}^{k}x^{k-i}a^{i-1})$
or,
$\frac{f(x)-f(a)}{x-a}= \sum\limits_{k=1}^{n}(a_k\sum\limits_{i=1}^{k}x^{k-i}a^{i-1})\quad\quad\quad(2)$
Hence,
$\int\frac{f(x)}{x-a}\;dx = \int \sum\limits_{k=1}^{n}(a_k \sum\limits_{i=1}^{k}x^{k-i}a^{i-1})\;dx + f(a)\log(x-a)$
$=\sum\limits_{k=1}^{n}(a_k \sum\limits_{i=1}^{k}\int x^{k-i}a^{i-1}\; dx)+ f(a)\log(x-a)$
i.e.,
$\int \frac{f(x)}{x-a} = \sum\limits_{k=1}^{n}(a_k\sum\limits_{i=1}^{k}\frac{x^{k-i+1}}{k-i+1}a^{i-1})+ f(a)\log(x-a)$
Let us now consider the case when $p>1$:
$\int \frac{f(x)}{(x-a)^p}\; dx$
$=\int \frac{f(x)-f(a)+f(a)}{(x-a)^p}\;dx$
$=\int \frac{f(x)-f(a)}{(x-a)^p} + \frac{f(a)}{(x-a)^p}\;dx$
$=\int \frac{f(x)-f(a)}{(x-a)}\cdot\frac{1}{(x-a)^{p-1}} + \frac{f(a)}{(x-a)^p}\;dx$
$= \int \frac{f(x)-f(a)}{x-a}\cdot\frac{1}{(x-a)^{p-1}}\;dx + \int\frac{f(a)}{(x-a)^p}\; dx$
$\overset{(2)}{=}\int \frac{g(x)}{(x-a)^{p-1}}\;dx + \frac{f(a)(x-a)^{1-p}}{1-p}$
where
$g(x) = \frac{f(x)-f(a)}{x-a}=\sum\limits_{k=1}^{n}(a_k\sum\limits_{i=1}^{k}x^{k-i}a^{i-1})$, a polynomial of order $n-1$.
What emerges from the two cases of $p$ is a recursive algorithm for evaluating (1):
Given polynomial $f(x) = \sum\limits_{k=0}^{n} a_k x^k$,
$\int \frac{f(x)}{(x-a)^p} \;dx, \; p \in N^+= \begin{cases}p=1: \sum\limits_{k=1}^{n}(a_k\sum\limits_{i=1}^{k}\frac{x^{k-i+1}}{k-i+1}a^{i-1})+ f(a)\log(x-a) \\p>1: \int \frac{g(x)}{(x-a)^{p-1}}\;dx + \frac{f(a)(x-a)^{1-p}}{1-p}, \\ \quad\quad\quad g(x) = \frac{f(x)-f(a)}{x-a}=\sum\limits_{k=1}^{n}(a_k\sum\limits_{i=1}^{k}x^{k-i}a^{i-1}). \end{cases}$
Exercise-1 Optimize the above recursive algorithm (hint: examine how it handles the case when $f(x)=0$)
# Integration of Trigonometric Expressions
We will introduce an algorithm for obtaining indefinite integrals such as
$\int \frac{(1+\sin(x))}{\sin(x)(1+\cos(x))}\;dx$
or, in general, integral of the form
$\int R(\sin(x), \cos(x))\;dx\quad\quad\quad(1)$
where $R$ is any rational function $R(p, q)$, with $p=\sin(x), q=\cos(x)$.
Let
$t = \tan(\frac{x}{2})\quad\quad(2)$
Solving (2) for $x$, we have
$x = 2\cdot\arctan(t)\quad\quad\quad(3)$
which provides
$\frac{dx}{dt} = \frac{2}{1+t^2}\quad\quad\quad(4)$
and,
$\sin(x) =2\sin(\frac{x}{2})\cos(\frac{x}{2})\overset{\cos^(\frac{x}{2})+\sin^2(\frac{x}{2})=1}{=}\frac{2\sin(\frac{x}{2})\cos(\frac{x}{2})}{\cos^2(\frac{x}{2})+\sin^2(\frac{x}{2})}=\frac{2\frac{\sin(\frac{x}{2})}{\cos(\frac{x}{2})}}{1+\frac{\sin^2(\frac{x}{2})}{\cos^2(\frac{x}{2})}}=\frac{2\tan(\frac{x}{2})}{1+\tan^2(\frac{x}{2})}$
yields
$\sin(x) = \frac{2 t}{1+t^2}\quad\quad\quad(5)$
Similarly,
$\cos(x) = \cos^2(\frac{x}{2})-\sin^2(\frac{x}{2})=\frac{\cos^2(\frac{x}{2})-\sin^2(\frac{x}{2})}{\cos^2(\frac{x}{2})+\sin^2(\frac{x}{2})}=\frac{1+\frac{\sin^2(\frac{x}{2})}{\cos^2(\frac{x}{2})}}{1+\frac{\sin^2(\frac{x}{2})}{\cos^2(\frac{x}{2})}}=\frac{1-\tan^2(\frac{x}{2})}{1+\tan^2(\frac{x}{2})}$
gives
$\cos(x)=\frac{1-t^2}{1+t^2}\quad\quad\quad(6)$
We also have (see “Finding Indefinite Integrals” )
$\int f(x)\;dx \overset{x=\phi(t)}{=} \int f(\phi(t))\cdot\frac{d\phi(t)}{dt}\;dt$.
Hence
$\int R(\cos(x), \sin(x))\;dx \overset{(2), (4), (5), (6)}{=} \int R(\frac{1-t^2}{1+t^2}, \frac{2t}{1+t^2})\cdot\frac{2}{1+t^2}\;dt$,
and (1) is reduced to an integral of rational functions in $t$.
Example-1 Evaluate $\int \csc(x)\;dx$.
Solution: $\csc(x) = \frac{1}{\sin(x)}\implies \int \csc(x)\;dx = \int \frac{1}{\sin(x)}\;dx$
$= \int \frac{1}{\frac{2t}{1+t^2}}\cdot\frac{2}{1+t^2}\;dt=\int\frac{1}{t}\;dt = \log(t) = \log(\tan(\frac{x}{2}))$.
Example-2 Evaluate $\int \sec(x)\;dx$.
Solution: $\sec(x) = \frac{1}{\cos(x)}\implies \int \sec(x)\; dx =\int \frac{1}{\cos(x)}\;dx$
$= \int \frac{1}{\frac{1-t^2}{1+t^2}}\cdot \frac{2}{1+t^2}\; dt=\int \frac{2}{1-t^2}\;dt=\int \frac{2}{(1+t)(1-t)}\;dt=\int \frac{1}{1+t} + \frac{1}{1-t}\;dt$
$=\int \frac{1}{1+t}\;dt - \int \frac{-1}{1-t}\;dt$
$=\log(1+t) -\log(1-t) =\log\frac{1+t}{1-t}=\log(\frac{1+\tan(\frac{x}{2})}{1-\tan(\frac{x}{2})})$.
According to CAS (see Fig. 1),
Fig. 1
However, the two results are equivalent as a CAS-aided verification (see Fig. 2) confirms their difference is a constant (see Corollary 2 in “Sprint to FTC“).
Fig. 2
Exercise-1 According to CAS,
Show that it is equivalent to the result obtained in Example-1
Exercise-2 Try
$\int \frac{1}{\sin(x)+1}\;dx$
$\int \frac{1}{\sin(x)+\cos(x)}\;dx$
$\int \frac{1}{(2+\cos(x))\sin(x)}\;dx$
$\int \frac{1}{5+4\sin(x)}\;dx$
$\int \frac{1}{2\sin(x)-\cos(x)+5}\;dx$
and of course,
$\int \frac{1+\sin(x)}{\sin(x)(1+\cos(x))}\;dx$
# Deriving Lagrange's multiplier
We wish to consider a special type of optimization problem:
Find the maximum (or minimum) of a function $f(x,y)$ subject to the condition $g(x,y)=0\quad\quad(1)$
If it is possible to solve $g(x)=0$ for $y$ so that it is expressed explicitly as $y=\psi(x)$, by substituting $y$ in (1), it becomes
Find the maximum (or minimum) of a single variable function $f(x, \psi(x))$.
In the case that $y$ can not be obtained from solving $g(x,y)=0$, we re-state the problem as:
Find the maximum (or minimum) of a single variable function $z=f(x,y)$ where $y$ is a function of $x$, implicitly defined by $g(x, y)=0\quad\quad\quad(2)$
Following the traditional procedure of finding the maximum (or minimum) of a single variable function, we differentiate $z$ with respect to $x$:
$\frac{dz}{dx} = f_x(x,y) + f_y(x,y)\cdot \frac{dy}{dx}\quad\quad\quad(3)$
Similarly,
$g_x(x,y) + g_y(x,y)\cdot \frac{dy}{dx}=0\quad\quad\quad(4)$
By grouping $g(x, y)=0$ and (3), we have
$\begin{cases} \frac{dz}{dx}= f_x(x, y)+g_x(x, y)\cdot \frac{dy}{dx}\\ g(x,y) = 0\end{cases}\quad\quad\quad(5)$
The fact that $\frac{dz}{dx}= 0$ at any stationary point $x^*$ means for all $(x^*, y^*)$ where $g(x^*, y^*)=0$,
$\begin{cases} f_x(x^*, y^*)+g_x(x^*, y^*)\cdot \frac{dy}{dx}\vert_{x=x^*}=0 \\ g(x^*,y^*) = 0\end{cases}\quad\quad\quad(6)$
If $g_y(x^*,y^*) \ne 0$ then from (4),
$\frac{dy}{dx}\vert_{x=x^*} = \frac{-g_x(x^*, y^*)}{g_y(x^*, y^*)}$
Substitute it into (6),
$\begin{cases} f_x(x^*, y^*)+f_y(x^*, y^*)\cdot (\frac{-g_x(x^*, y^*)}{g_y(x^*, y^*)})=f_x(x^*, y^*)+g_x(x^*, y^*)\cdot (\frac{-f_y(x^*, y^*)}{g_y(x^*, y^*)})\\ g(x^*,y^*) = 0\end{cases}\quad\quad\quad(7)$
Let $\lambda = \frac{-f_y(x^*, y^*)}{g_y(x^*, y^*)}$, we have
$f_y(x^*, y^*) + \lambda g_y(x^*, y^*) =0\quad\quad\quad(8)$
Combining (7) and (8) gives
$\begin{cases} f_x(x^*, y^*)+\lambda g_x(x^*, y^*) = 0 \\ f_y(x^*, y^*)+\lambda g_y(x^*, y^*)=0 \\ g(x^*, y^*) = 0\end{cases}$
It follows that to find the stionary points of $z$, we solve
$\begin{cases} f_x(x, y)+\lambda g_x(x, y) = 0 \\ f_y(x, y)+\lambda g_y(x, y)=0 \\ g(x, y) = 0\end{cases}\quad\quad\quad(9)$
for $x, y$ and $\lambda$.
This is known as the method of Lagrange’s multiplier.
Let $F(x,y,\lambda) = f(x,y) + \lambda g(x,y)$.
Since
$F_x(x,y,\lambda) = f_x(x,y) + \lambda g_x(x,y)$,
$F_y(x,y,\lambda)=f_y(x,y) + \lambda g_y(x,y)$,
$F_{\lambda}(x,y,\lambda) = g(x, y)$,
(9) is equivalent to
$\begin{cases} F_x(x, y, \lambda)=0 \\ F_y(x,y,\lambda)=0 \\ F_{\lambda}(x, y) = 0\end{cases}\quad\quad\quad(10)$
Let’s look at some examples.
Example-1 Find the minimum of $f(x, y) = x^2+y^2$ subject to the condition that $x+y=4$
Let $F(x, y, \lambda) = x^2+y^2+\lambda(x+y-4)$.
Solving
$\begin{cases}F_x=2x-\lambda=0 \\ F_y = 2y-\lambda = 0 \\ F_{\lambda} = x+y-4=0\end{cases}$
for $x, y, \lambda$ gives $x=y=2, \lambda=4$.
When $x=2, y=2, x^2+y^2=2^2+2^2=8$.
$\forall (x, y) \ne (2, 2), x+y=4$, we have
$(x-2)^2 + (y-2)^2 > 0$.
That is,
$x^2-4x+4 + y^2-4y+4 = x^2+y^2-4(x+y)+8 \overset{x+y=4}{=} x^2+y^2-16+8>0$.
Hence,
$x^2+y^2>8, (x,y) \ne (2,2)$.
The target function $x^2+y^2$ with constraint $x+y=4$ indeed attains its global minimum at $(x, y) = (2, 2)$.
I first encountered this problem during junior high school and solved it:
$(x-y)^2 \ge 0 \implies x^2+y^2 \ge 2xy$
$x+y=4\implies (x+y)^2=16\implies x^2+2xy +y^2=16\implies 2xy=16-(x^2+y^2)$
$x^2+y^2 \ge 16-(x^2+y^2) \implies x^2+y^2 \ge 8\implies z_{min} = 8$.
I solved it again in high school when quadratic equation is discussed:
$x+y=4 \implies y =4-x$
$z=x^2+y^2 \implies z = x^2+(4-x)^2 \implies 2x^2-8x+16-z=0$
$\Delta = 64-4 \cdot 2\cdot (16-z) \ge 0 \implies z \ge 8\implies z_{min} = 8$
In my freshman calculus class, I solved it yet again:
$x+y=4 \implies y=4-x$
$z = x^2+(4-x)^2$
$\frac{dz}{dx} = 2x+2(4-x)(-1)=2x-8+2x=4x-8$
$\frac{dz}{dx} =0 \implies x=2$
$\frac{d^2 z}{dx^2} = 4 > 0 \implies x=2, z_{min}=2^2+(4-2)^2=8$
Example-2 Find the shortest distance from the point $(1,0)$ to the parabola $y^2=4x$.
We minimize $f = (x-1)^2+y^2$ where $y^2=4x$.
If we eliminate $y^2$ in $f$, then $f = (x-1)^2+4x$. Solving $\frac{df}{dx} = 2x+2=0$ gives $x=-1$, Clearly, this is not valid for it would suggest that $y^2=-4$ from $y^2=4x$, an absurdity.
By Lagrange’s method, we solve
$\begin{cases} 2(x-1)-4\lambda=0 \\2y\lambda+2y = 0 \\y^2-4x=0\end{cases}$
Fig. 1
The only valid solution is $x=0, y=0, k=-\frac{1}{2}$. At $(x, y) = (0, 0), f=(0-1)^2+0^2=1$. It is the global minimum:
$\forall (x, y) \ne (0, 0), y^2=4x \implies x>0$.
$(x-1)^2+y^2 \overset{y^2=4x}{=}(x-1)^2+4x=x^2-2x+1+4x=x^2+2x+1\overset{x>0}{>}1=f(0,0)$
Example-3 Find the shortest distance from the point $(a, b)$ to the line $Ax+By+C=0$.
We want find a point $(x_0, y_0)$ on the line $Ax+By+C=0$ so that the distance between $(a, b)$ and $(x_0, y_0)$ is minimal.
To this end, we minimize $(x_0-a)^2+(y_0-b)^2$ where $Ax_0+By_0+C=0$ (see Fig. 2)
Fig. 2
We found that
$x_0=\frac{aB^2-bAB-AC}{A^2+B^2}, y_0=\frac{bA^2-aAB-BC}{A^2+B^2}$
and the distance between $(a, b)$ and $(x_0, y_0)$ is
$\frac{|Aa+Bb+C|}{\sqrt{A^2+B^2}}\quad\quad\quad(11)$
To show that (11) is the minimal distance, $\forall (x, y) \ne (x_0, y_0), Ax+By+C=0$.
Let $d_1 = x-x_0, d_2=y-y_0$, we have
$x = x_0 + d_1, y=y_0 + d_2, d_1 \ne 0, d_2 \ne 0$.
Since $Ax+By+C=0$,
$A(x_0+d_1)+B(y_0+d_2)+C=Ax_0+Ad_1+By_0+Bd_2+C=0$
That is
$Ax_0+By_0+C+Ad_1+Bd_2=0$.
By the fact that $Ax_0+By_0+C=0$, we have
$Ad_1 + Bd_2 =0\quad\quad\quad(12)$
Compute $(x-a)^2+(y-b)^2 - ((x_0-a)^2+(y_0-b)^2)$ (see Fig. 3)
Fig. 3
yields
$\boxed{-\frac{2d_2BC}{B^2+A^2}-\frac{2d_1AC}{B^2+A^2}}+[\frac{d_2^2B^2}{B^2+A^2}]-\underline{\frac{2bd_2B^2}{B^2+A^2}}+(\frac{d_1^2B^2}{B^2+A^2})-\frac{2ad_2AB}{B^2+A^2}-\underline{\frac{2bd_1AB}{B^2+A^2}}+[\frac{d_2A^2}{B^2+A^2}]+(\frac{d_1^2A^2}{B^2+A^2})-\frac{2ad_1A^2}{B^2+A^2}$
After some rearrangement and factoring, it becomes
$\frac{-2C}{A^2+B^2}(Ad_1+Bd_2)+\frac{-2B}{A^2+B^2}(Ad_1+Bd_2)+\frac{-2A}{A^2+b^2}(Ad_1+Bd_2) + d_1^2+d_2^2$
By (12), it reduces to
$d_1^2 + d_2^2$.
This is clearly a positive quantity. Therefore,
$\forall (x, y) \ne (x_0, y_0), Ax+By+C=0 \implies (x-a)^2+(y-b)^2> (x_0-1)^2+(y_0-b)^2$
# Introducing Operator Delta
The $r^{th}$ order finite difference of function$f(x)$ is defined by
$\Delta^r f(x) = \begin{cases} f(x+1)-f(x), r=1\\ \Delta(\Delta^{r-1}f(x)), r > 1\end{cases}$
From this definition, we have
$\Delta f(x) = \Delta^1 f(x) = f(x+1)-f(x)$
and,
$\Delta^2 f(x) = \Delta (\Delta^{2-1} f(x))$
$= \Delta (\Delta f(x))$
$= \Delta( f(x+1)-f(x))$
$= (f(x+2)-f(x+1)) - (f(x+1)-f(x))$
$= f(x+2)-2f(x)+f(x+1)$
as well as
$\Delta^3 f(x) = \Delta (\Delta^2 f(x))$
$= \Delta (f(x+2)-2f(x)+f(x+1))$
$= (f(x+3)-2f(x+1)+f(x+2)) - (f(x+2)-2f(x)+f(x+1))$
$= f(x+3)-3f(x+2)+3f(x+1)-f(x)$
The function shown below generates $\Delta^r f(x), r:1\rightarrow 5$ (see Fig. 1).
delta_(g, n) := block(
local(f),
define(f[1](x),
g(x+1)-g(x)),
for i : 2 thru n do (
define(f[i](x),
f[i-1](x+1)-f[i-1](x))
),
return(f[n])
);
Fig. 1
Compare to the result of expanding $(f(x)-1)^r=\sum\limits_{i=0}^r(-1)^i \binom{r}{i} f(x)^{r-i}, r:1\rightarrow 5$ (see Fig. 2)
Fig. 2
It seems that
$\Delta^r f(x) = \sum\limits_{i=0}^r(-1)^i \binom{r}{i} f(x+r-i)\quad\quad\quad(1)$
Lets prove it!
We have already shown that (1) is true for $r= 1, 2, 3$.
Assuming (1) is true when $r=k-1 \ge 4$:
$\Delta^{k-1} f(x) = \sum\limits_{i=0}^{k-1}(-1)^i \binom{r}{i} f(x+k-1-i)\quad\quad\quad(2)$
When $r=k$,
$\Delta^k f(x) = \Delta(\Delta^{k-1} f(x))$
$\overset{(2)}{=}\Delta (\sum\limits_{i=0}^{k-1}(-1)^i \binom{k-1}{i}f(x+k-1-i))$
$=\sum\limits_{i=0}^{k-1}(-1)^i \binom{k-1}{i}f(x+1+k-1-i)-\sum\limits_{i=0}^{k-1}(-1)^i \binom{k-1}{i}f(x+k-1-i)$
$=(-1)^0 \binom{k-1}{0}f(x+k-0)$
$+\sum\limits_{i=1}^{k-1}(-1)^i \binom{k-1}{i}f(x+k-i)-\sum\limits_{i=0}^{k-2}(-1)^i \binom{k-1}{i}f(x+k-1-i)$
$-(-1)^{k-1}\binom{k-1}{k-1}f(x+k-1-(k-1))$
$\overset{\binom{k-1}{0} = \binom{k-1}{k-1}=1}{=}$
$f(x+k)+ \sum\limits_{i=1}^{k-1}(-1)^i \binom{k-1}{i}f(x+k-i)-\sum\limits_{i=0}^{k-2}(-1)^i \binom{k-1}{i}f(x+k-1-i) -(-1)^{k-1}f(x)$
$=f(x+k)+ \sum\limits_{i=1}^{k-1}(-1)^i \binom{k-1}{i}f(x+k-i)+\sum\limits_{i=0}^{k-2}(-1)^{i+1}\binom{k-1}{i}f(x+k-1-i) -(-1)^{k-1}f(x)$
$\overset{j=i+1, i:0 \rightarrow k-2\implies j:1 \rightarrow k-1}{=}$
$f(x+k)+ \sum\limits_{i=1}^{k-1}(-1)^i\binom{k-1}{i}f(x+k-i) + \sum\limits_{j=1}^{k-1}(-1)^j \binom{k-1}{j-1}f(x+k-j)-(-1)^{k-1}f(x)$
$= f(x+k)+ \sum\limits_{i=1}^{k-1}(-1)^i\binom{k-1}{i}f(x+k-i) + \sum\limits_{i=1}^{k-1}(-1)^i\binom{k-1}{i-1}f(x+k-i)+(-1)^k f(x)$
$= f(x+k) + \sum\limits_{i=1}^{k-1}(-1)^i f(x+k-i) (\binom{k-1}{i} + \binom{k-1}{i-1})+(-1)^k f(x)$
$\overset{\binom{k-1}{i} + \binom{k-1}{i-1}=\binom{k}{i}}{=}$
$f(x+k)+ \sum\limits_{i=1}^{k-1}(-1)^i \binom{k}{i} f(x+k-i)+(-1)^k f(x)$
$= (-1)^0 \binom{k}{0}f(x+k-0)+\sum\limits_{i=1}^{k-1}(-1)^i \binom{k}{i} f(x+k-i)+(-1)^k \binom{k}{k} f(x+k-k)$
$= \sum\limits_{i=0}^{k}(-1)^i \binom{k}{i} f(x+k-i)$
# A pair of non-identical twins
A complex number $x + i y$ can be plotted in a complex plain where the $x$ coordinate is the real axis and the $y$ coordinate the imaginary.
Let’s consider the following iteration:
$z_{n+1} = z_{n}^2 + c\quad\quad\quad(1)$
where $z, c$ are complex numbers.
If (1) are started at $z_0 = 0$ for various values of $c$ and plotted in c-space, we have the Mandelbrot set:
When $c$ is held fixed and points generated by (1) are plotted in z-space, the result is the Julia set:
# Constructing the tangent line of circle without calculus
The tangent line of a circle can be defined as a line that intersects the circle at one point only.
Put a circle in the rectangular coordinate system.
Let $(x_0, y_0)$ be a point on a circle. The tangent line at $(x_0, y_0)$ is a line intersects the circle at $(x_0, y_0)$ only.
Let’s first find a function $y=kx+m$ that represents the line.
From circle’s equation $x^2+y^2=r^2$, we have
$y^2=r^2-x^2$
Since the line intersects the circle at $(x_0, y_0)$ only,
$r^2-x^2=(kx+m)^2$
has only one solution.
That means
$k^2x^2+x^2+2kmx+m^2-r^2 =0$
has only one solution. i.e., its discriminant
$(2km)^2-4(k^2+1)(m^2-r^2)=0\quad\quad\quad(1)$
By the definition of slope,
$kx+m-y_0 = k(x-x_0)$.
It follows that
$m =y_0-kx_0\quad\quad\quad(2)$
Substitute (2) into (1) and solve for $k$ gives
$k = \frac{-x_0}{y_0}\quad\quad\quad(3)$
The slope of line connecting $(0, 0)$ and $(x_0, y_0)$ where $x_0 \neq 0$ is $\frac{y_0}{x_0}$.
Since $\frac{-x_0}{y_0}\cdot \frac{y_0}{x_0} = -1$, the tangent line is perpendicular to the line connecting $(0, 0)$ and $(x_0, y_0)$.
Substitute (3) into $y = k x +m$, we have
$y=-\frac{x_0}{y_0} x + m\quad\quad\quad(4)$.
The fact that the line intersects the circle at $(x_0, y_0)$ means
$y_0 = -\frac{x_0^2}{y_0} + m$
or
$y_0^2=-x_0^2+ my_0$.
Hence,
$m =\frac{x_0^2+y_0^2}{y_0} = \frac{r^2}{y_0}$.
It follows that by (4),
$x_0 x +y_0 y = r^2\quad\quad\quad(5)$
(5) is derived under the assumption that $y_0 \neq 0$. However, by letting $y_0 =0$ in (5), we obtain two tangent lines that can not be expressed in the form of $y=kx+m$:
$x=-r, x=r$
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### What the foundations of quantum computer science teach us about chemistry
Jarrod Ryan McClean, Nicholas Rubin, Joonho Lee, Matthew Harrigan, Thomas E O'Brien, William J. Huggins, Ryan Babbush, Hsin-Yuan (Robert) Huang · arXiv:2106.03997 (2021)
With the rapid development of quantum technology, one of the leading applications that has been identified is the simulation of chemistry. Interestingly, even before full scale quantum computers are available, quantum computer science has exhibited a remarkable string of results that directly impact what is possible in chemical simulation, even with a quantum computer. Some of these results even impact our understanding of chemistry in the real world. In this perspective, we take the position that direct chemical simulation is best understood as a digital experiment. While on one hand this clarifies the power of quantum computers to extend our reach, it also shows us the limitations of taking such an approach too directly. Leveraging results that quantum computers cannot outpace the physical world, we build to the controversial stance that some chemical problems are best viewed as problems for which no algorithm can deliver their solution in general, known in computer science as undecidable problems. This has implications for the predictive power of thermodynamic models and topics like the ergodic hypothesis. However, we argue that this perspective is not defeatist, but rather helps shed light on the success of existing chemical models like transition state theory, molecular orbital theory, and thermodynamics as models that benefit from data. We contextualize recent results showing that data-augmented models are more powerful rote simulation. These results help us appreciate the success of traditional chemical theory and anticipate new models learned from experimental data. Not only can quantum computers provide data for such models, but they can extend the class and power of models that utilize data in fundamental ways. These discussions culminate in speculation on new ways for quantum computing and chemistry to interact and our perspective on the eventual roles of quantum computers in the future of chemistry.
### Unbiasing Fermionic Quantum Monte Carlo with a Quantum Computer
William J. Huggins, Bryan O'Gorman, Nicholas Rubin, David Reichman, Ryan Babbush, Joonho Lee · arXiv:2106.16235 (2021)
Many-electron problems pose some of the greatest challenges in computational science, with important applications across many fields of modern science. Fermionic quantum Monte Carlo (QMC) methods are among the most powerful approaches to these problems. However, they can be severely biased when controlling the fermionic sign problem using constraints, as is necessary for scalability. Here we propose an approach that combines constrained QMC with quantum computing tools to reduce such biases. We experimentally implement our scheme using up to 16 qubits in order to unbias constrained QMC calculations performed on chemical systems with as many as 120 orbitals. These experiments represent the largest chemistry simulations performed on quantum computers (more than doubling the size of prior electron correlation calculations), while obtaining accuracy competitive with state-of-the-art classical methods. Our results demonstrate a new paradigm of hybrid quantum-classical algorithm, surpassing the popular variational quantum eigensolver in terms of potential towards the first practical quantum advantage in ground state many-electron calculations.
### Variational Quantum Algorithms
Marco Cerezo, Andrew Arrasmith, Ryan Babbush, Simon Benjamin, Suguro Endo, Keisuke Fujii, Jarrod Ryan McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, Patrick Coles · Nature Reviews Physics (2021)
Applications such as simulating large quantum systems or solving large-scale linear algebra problems are immensely challenging for classical computers due their extremely high computational cost. Quantum computers promise to unlock these applications, although fault-tolerant quantum computers will likely not be available for several years. Currently available quantum devices have serious constraints, including limited qubit numbers and noise processes that limit circuit depth. Variational Quantum Algorithms (VQAs), which employ a classical optimizer to train a parametrized quantum circuit, have emerged as a leading strategy to address these constraints. VQAs have now been proposed for essentially all applications that researchers have envisioned for quantum computers, and they appear to the best hope for obtaining quantum advantage. Nevertheless, challenges remain including the trainability, accuracy, and efficiency of VQAs. In this review article we present an overview of the field of VQAs. Furthermore, we discuss strategies to overcome their challenges as well as the exciting prospects for using them as a means to obtain quantum advantage.
### Power of data in quantum machine learning
Hsin-Yuan (Robert) Huang, Michael Blythe Broughton, Masoud Mohseni, Ryan Babbush, Sergio Boixo, Hartmut Neven, Jarrod Ryan McClean · Nature Communications, vol. 12 (2021), pp. 2631
The use of quantum computing for machine learning is among the most exciting prospective applications of quantum technologies. However, machine learning tasks where data is provided can be considerably different than commonly studied computational tasks. In this work, we show that some problems that are classically hard to compute can be easily predicted by classical machines learning from data. Using rigorous prediction error bounds as a foundation, we develop a methodology for assessing potential quantum advantage in learning tasks. The bounds are tight asymptotically and empirically predictive for a wide range of learning models. These constructions explain numerical results showing that with the help of data, classical machine learning models can be competitive with quantum models even if they are tailored to quantum problems. We then propose a projected quantum model that provides a simple and rigorous quantum speed-up for a learning problem in the fault-tolerant regime. For near-term implementations, we demonstrate a significant prediction advantage over some classical models on engineered data sets designed to demonstrate a maximal quantum advantage in one of the largest numerical tests for gate-based quantum machine learning to date, up to 30 qubits.
### Low-Depth Mechanisms for Quantum Optimization
Jarrod Ryan McClean, Matthew P Harrigan, Masoud Mohseni, Nicholas Rubin, Zhang Jiang, Sergio Boixo, Vadim Smelyanskiy, Ryan Babbush, Hartmut Neven · PRX Quantum, vol. 3 (2021), pp. 030312
One of the major application areas of interest for both near-term and fault-tolerant quantum computers is the optimization of classical objective functions. In this work, we develop intuitive constructions for a large class of these algorithms based on connections to simple dynamics of quantum systems, quantum walks, and classical continuous relaxations. We focus on developing a language and tools connected with kinetic energy on a graph for understanding the physical mechanisms of success and failure to guide algorithmic improvement. This physical language, in combination with uniqueness results related to unitarity, allow us to identify some potential pitfalls from kinetic energy fundamentally opposing the goal of optimization. This is connected to effects from wavefunction confinement, phase randomization, and shadow defects lurking in the objective far away from the ideal solution. As an example, we explore the surprising deficiency of many quantum methods in solving uncoupled spin problems and how this is both predictive of performance on some more complex systems while immediately suggesting simple resolutions. Further examination of canonical problems like the Hamming ramp or bush of implications show that entanglement can be strictly detrimental to performance results from the underlying mechanism of solution in approaches like QAOA. Kinetic energy and graph Laplacian perspectives provide new insights to common initialization and optimal solutions in QAOA as well as new methods for more effective layerwise training. Connections to classical methods of continuous extensions, homotopy methods, and iterated rounding suggest new directions for research in quantum optimization. Throughout, we unveil many pitfalls and mechanisms in quantum optimization using a physical perspective, which aim to spur the development of novel quantum optimization algorithms and refinements.
Ryan Babbush, Jarrod Ryan McClean, Michael Newman, Craig Michael Gidney, Sergio Boixo, Hartmut Neven · PRX Quantum, vol. 2 (2021), pp. 010103
In this perspective we discuss conditions under which it would be possible for a modest fault-tolerant quantum computer to realize a runtime advantage by executing a quantum algorithm with only a small polynomial speedup over the best classical alternative. The challenge is that the computation must finish within a reasonable amount of time while being difficult enough that the small quantum scaling advantage would compensate for the large constant factor overheads associated with error correction. We compute several examples of such runtimes using state-of-the-art surface code constructions under a variety of assumptions. We conclude that quadratic speedups will not enable quantum advantage on early generations of such fault-tolerant devices unless there is a significant improvement in how we realize quantum error correction. While this conclusion persists even if we were to increase the rate of logical gates in the surface code by more than an order of magnitude, we also repeat this analysis for speedups by other polynomial degrees and find that quartic speedups look significantly more practical.
### Microwaves in Quantum Computing
Joseph Bardin, Daniel Slichter, David Reilly · IEEE Journal of Microwaves, vol. 1 (2021)
The growing field of quantum computing relies on a broad range of microwave technologies, and has spurred development of microwave devices and methods in new operating regimes. Here we review the use of microwave signals and systems in quantum computing, with specific reference to three leading quantum computing platforms: trapped atomic ion qubits, spin qubits in semiconductors, and superconducting qubits. We highlight some key results and progress in quantum computing achieved through the use of microwave systems, and discuss how quantum computing applications have pushed the frontiers of microwave technology in some areas. We also describe open microwave engineering challenges for the construction of large-scale, fault-tolerant quantum computers.
### Removing leakage-induced correlated errors in superconducting quantum error correction
Matt McEwen, Dvir Kafri, Jimmy Chen, Juan Atalaya, Kevin Satzinger, Chris Quintana, Paul Victor Klimov, Daniel Sank, Craig Michael Gidney, Austin Fowler, Frank Carlton Arute, Kunal Arya, Bob Benjamin Buckley, Brian Burkett, Nicholas Bushnell, Benjamin Chiaro, Roberto Collins, Sean Demura, Andrew Dunsworth, Catherine Erickson, Brooks Riley Foxen, Marissa Giustina, Trent Huang, Sabrina Hong, Evan Jeffrey, Seon Kim, Kostyantyn Kechedzhi, Fedor Kostritsa, Pavel Laptev, Anthony Megrant, Xiao Mi, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Alexandru Paler, Nick Redd, Pedram Roushan, Ted White, Jamie Yao, Ping Yeh, Adam Jozef Zalcman, Yu Chen, Vadim Smelyanskiy, John Martinis, Hartmut Neven, J. Kelly, Alexander Korotkov, Andre Gregory Petukhov, Rami Barends · Nature Communications, vol. 12 (2021), pp. 1761
Quantum computing becomes scalable through error correction, but logical error rates only decrease with system size when physical errors are sufficiently uncorrelated. During computation, the unused high energy states of the qubits can become excited. In weakly nonlinear qubits, such as the superconducting transmon, these leakage states are long-lived and mobile, opening a path to errors that are correlated in space and time. The effects of leakage and its mitigation during quantum error correction remain an open question. Here, we report a reset protocol that returns a qubit to the ground state from all relevant higher level states. It requires no additional hardware and combines speed, fidelity, and resilience to noise. We test its performance with the bit-flip stabilizer code, a simplified version of the surface code scheme for quantum error correction. We investigate the accumulation and dynamics of leakage during the stabilizer codes. Using this protocol, we find lower rates of logical errors, and an improved scaling and stability of error suppression with qubits. This demonstration provides a key step on the path towards scalable quantum computing.
### Fault-Tolerant Quantum Simulations of Chemistry in First Quantization
Yuan Su, Dominic W. Berry, Nathan Wiebe, Nicholas Rubin, Ryan Babbush · arXiv:2105.12767 (2021)
Quantum simulations of chemistry in first quantization offer important advantages over approaches in second quantization including faster convergence to the continuum limit and the opportunity for practical simulations outside the Born-Oppenheimer approximation. However, as all prior work on quantum simulation in first quantization has been limited to asymptotic analysis, it has been impossible to compare the resources required for these approaches to those for more commonly studied algorithms in second quantization. Here, we analyze and optimize the resources required to implement two first quantized quantum algorithms for chemistry from Babbush et al that realize block encodings for the qubitization and interaction picture frameworks of Low et al. The two algorithms we study enable simulation with gate complexities O(η^{8/3} N^{1/3} t+η^{4/3} N^{2/3} t) and O(η^{8/3} N^{1/3} t) where η is the number of electrons, N is the number of plane wave basis functions, and t is the duration of time-evolution (t is inverse to target precision when the goal is to estimate energies). In addition to providing the first explicit circuits and constant factors for any first quantized simulation and introducing improvements which reduce circuit complexity by about a thousandfold over naive implementations for modest sized systems, we also describe new algorithms that asymptotically achieve the same scaling in a real space representation. We assess the resources required to simulate various molecules and materials and conclude that the qubitized algorithm will often be more practical than the interaction picture algorithm. We demonstrate that our qubitized algorithm often requires much less surface code spacetime volume for simulating millions of plane waves than the best second quantized algorithms require for simulating hundreds of Gaussian orbitals.
### Quantum Approximate Optimization of Non-Planar Graph Problems on a Planar Superconducting Processor
Matthew Harrigan, Kevin Jeffery Sung, Kevin Satzinger, Matthew Neeley, Frank Carlton Arute, Kunal Arya, Juan Atalaya, Joseph Bardin, Rami Barends, Sergio Boixo, Michael Blythe Broughton, Bob Benjamin Buckley, David A Buell, Brian Burkett, Nicholas Bushnell, Jimmy Chen, Yu Chen, Ben Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Austin Fowler, Brooks Riley Foxen, Craig Michael Gidney, Marissa Giustina, Rob Graff, Steve Habegger, Sabrina Hong, Lev Ioffe, Sergei Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul Klimov, Alexander Korotkov, Fedor Kostritsa, Dave Landhuis, Pavel Laptev, Martin Leib, Mike Lindmark, Orion Martin, John Martinis, Jarrod Ryan McClean, Matthew McEwen, Anthony Megrant, Xiao Mi, Masoud Mohseni, Wojtek Mruczkiewicz, Josh Mutus, Ofer Naaman, Charles Neill, Florian Neukart, Murphy Yuezhen Niu, Thomas E O'Brien, Bryan O'Gorman, A.G. Petukhov, Harry Putterman, Chris Quintana, Pedram Roushan, Nicholas Rubin, Daniel Sank, Andrea Skolik, Vadim Smelyanskiy, Doug Strain, Michael Streif, Marco Szalay, Amit Vainsencher, Ted White, Jamie Yao, Adam Zalcman, Leo Zhou, Hartmut Neven, Dave Bacon, Erik Lucero, Edward Farhi, Ryan Babbush · Nature Physics (2021)
Faster algorithms for combinatorial optimization could prove transformative for diverse areas such as logistics, finance and machine learning. Accordingly, the possibility of quantum enhanced optimization has driven much interest in quantum technologies. Here we demonstrate the application of the Google Sycamore superconducting qubit quantum processor to combinatorial optimization problems with the quantum approximate optimization algorithm (QAOA). Like past QAOA experiments, we study performance for problems defined on the planar connectivity graph native to our hardware; however, we also apply the QAOA to the Sherrington–Kirkpatrick model and MaxCut, non-native problems that require extensive compilation to implement. For hardware-native problems, which are classically efficient to solve on average, we obtain an approximation ratio that is independent of problem size and observe that performance increases with circuit depth. For problems requiring compilation, performance decreases with problem size. Circuits involving several thousand gates still present an advantage over random guessing but not over some efficient classical algorithms. Our results suggest that it will be challenging to scale near-term implementations of the QAOA for problems on non-native graphs. As these graphs are closer to real-world instances, we suggest more emphasis should be placed on such problems when using the QAOA to benchmark quantum processors.
### Tuning Quantum Information Scrambling on a 53-Qubit Processor
Adam Jozef Zalcman, Alan Derk, Alan Ho, Alex Opremcak, Alexander Korotkov, Alexandre Bourassa, Andre Gregory Petukhov, Andreas Bengtsson, Andrew Dunsworth, Anthony Megrant, Austin Fowler, Bálint Pató, Benjamin Chiaro, Benjamin Villalonga, Brian Burkett, Brooks Riley Foxen, Catherine Erickson, Charles Neill, Chris Quintana, Cody Jones, Craig Michael Gidney, Daniel Eppens, Daniel Sank, Dave Landhuis, David A Buell, Doug Strain, Dvir Kafri, Edward Farhi, Eric Ostby, Erik Lucero, Evan Jeffrey, Fedor Kostritsa, Frank Carlton Arute, Hartmut Neven, Igor Aleiner, Jamie Yao, Jarrod Ryan McClean, Jeffrey Marshall, Jeremy Patterson Hilton, Jimmy Chen, Jonathan Arthur Gross, Joseph Bardin, Josh Mutus, Juan Atalaya, Julian Kelly, Kevin Miao, Kevin Satzinger, Kostyantyn Kechedzhi, Kunal Arya, Marco Szalay, Marissa Giustina, Masoud Mohseni, Matt McEwen, Matt Trevithick, Matthew Neeley, Matthew P Harrigan, Michael Blythe Broughton, Michael Newman, Murphy Yuezhen Niu, Nicholas Bushnell, Nicholas Redd, Nicholas Rubin, Ofer Naaman, Orion Martin, Paul Victor Klimov, Pavel Laptev, Pedram Roushan, Ping Yeh, Rami Barends, Roberto Collins, Ryan Babbush, Sabrina Hong, Salvatore Mandra, Sean Demura, Sean Harrington, Seon Kim, Sergei Isakov, Sergio Boixo, Ted White, Thomas E O'Brien, Trent Huang, Trevor Mccourt, Vadim Smelyanskiy, Vladimir Shvarts, William Courtney, Wojtek Mruczkiewicz, Xiao Mi, Yu Chen, Zhang Jiang · arXiv (2021)
As entanglement in a quantum system grows, initially localized quantum information is spread into the exponentially many degrees of freedom of the entire system. This process, known as quantum scrambling, is computationally intensive to study classically and lies at the heart of several modern physics conundrums. Here, we characterize scrambling of different quantum circuits on a 53-qubit programmable quantum processor by measuring their out-of-time-order correlators (OTOCs). We observe that the spatiotemporal spread of OTOCs, as well as their circuit-to-circuit fluctuation, unravel in detail the time-scale and extent of quantum scrambling. Comparison with numerical results indicates a high OTOC measurement accuracy despite the large size of the quantum system. Our work establishes OTOC as an experimental tool to diagnose quantum scrambling at the threshold of being classically inaccessible.
### Exponential suppression of bit or phase flip errors with repetitive quantum error correction
Adam Jozef Zalcman, Alan Derk, Alan Ho, Alex Opremcak, Alexander Korotkov, Alexandre Bourassa, Andre Gregory Petukhov, Andreas Bengtsson, Andrew Dunsworth, Anthony Megrant, Austin Fowler, Bálint Pató, Benjamin Chiaro, Benjamin Villalonga, Brian Burkett, Brooks Riley Foxen, Catherine Erickson, Charles Neill, Chris Quintana, Cody Jones, Craig Michael Gidney, Daniel Eppens, Daniel Sank, Dave Landhuis, David A Buell, Doug Strain, Dvir Kafri, Edward Farhi, Eric Ostby, Erik Lucero, Evan Jeffrey, Fedor Kostritsa, Frank Carlton Arute, Hartmut Neven, Igor Aleiner, Jamie Yao, Jarrod Ryan McClean, Jeremy Patterson Hilton, Jimmy Chen, Jonathan Arthur Gross, Joseph Bardin, Josh Mutus, Juan Atalaya, Julian Kelly, Kevin Miao, Kevin Satzinger, Kostyantyn Kechedzhi, Kunal Arya, Marco Szalay, Marissa Giustina, Masoud Mohseni, Matt McEwen, Matt Trevithick, Matthew Neeley, Matthew P Harrigan, Michael Broughton, Michael Newman, Murphy Yuezhen Niu, Nicholas Bushnell, Nicholas Redd, Nicholas Rubin, Ofer Naaman, Orion Martin, Paul Victor Klimov, Pavel Laptev, Pedram Roushan, Ping Yeh, Rami Barends, Roberto Collins, Ryan Babbush, Sabrina Hong, Sean Demura, Sean Harrington, Seon Kim, Sergei Isakov, Sergio Boixo, Ted White, Thomas E O'Brien, Trent Huang, Trevor Mccourt, Vadim Smelyanskiy, Vladimir Shvarts, William Courtney, Wojtek Mruczkiewicz, Xiao Mi, Yu Chen, Zhang Jiang · Nature (2021)
Realizing the potential of quantum computing will require achieving sufficiently low logical error rates. Many applications call for error rates below 10^-15, but state-of-the-art quantum platforms typically have physical error rates near 10^-3. Quantum error correction (QEC) promises to bridge this divide by distributing quantum logical information across many physical qubits so that errors can be corrected. Logical errors are then exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold. QEC also requires that the errors are local, and that performance is maintained over many rounds of error correction, a major outstanding experimental challenge. Here, we implement 1D repetition codes embedded in a 2D grid of superconducting qubits which demonstrate exponential suppression of bit or phase-flip errors, reducing logical error per round by more than 100x when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analyzing error correlations with high precision, and characterize the locality of errors in a device performing QEC for the first time. Finally, we perform error detection using a small 2D surface code logical qubit on the same device, and show that the results from both 1D and 2D codes agree with numerical simulations using a simple depolarizing error model. These findings demonstrate that superconducting qubits are on a viable path towards fault tolerant quantum computing.
### Efficient and Noise Resilient Measurements for Quantum Chemistry on Near-Term Quantum Computers
William Huggins, Jarrod Ryan McClean, Nicholas Rubin, Zhang Jiang, Nathan Wiebe, K. Birgitta Whaley, Ryan Babbush · Nature Quantum Information, vol. 7 (2021)
Variational algorithms are a promising paradigm for utilizing near-term quantum devices for modeling electronic states of molecular systems. However, previous bounds on the measurement time required have suggested that the application of these techniques to larger molecules might be infeasible. We present a measurement strategy based on a low-rank factorization of the two-electron integral tensor. Our approach provides a cubic reduction in term groupings over prior state-of-the-art and enables measurement times three orders of magnitude smaller than those suggested by commonly referenced bounds for the largest systems we consider. Although our technique requires execution of a linear-depth circuit prior to measurement, this is compensated for by eliminating challenges associated with sampling nonlocal Jordan–Wigner transformed operators in the presence of measurement error, while enabling a powerful form of error mitigation based on efficient postselection. We numerically characterize these benefits with noisy quantum circuit simulations for ground-state energies of strongly correlated electronic systems.
### Even More Efficient Quantum Computations of Chemistry through Tensor Hyper-Contraction
Joonho Lee, Dominic W. Berry, Craig Michael Gidney, William J. Huggins, Jarrod Ryan McClean, Nathan Wiebe, Ryan Babbush · PRX Quantum, vol. 2 (2021), pp. 030305
We describe quantum circuits with only $\widetilde{\cal O}(N)$ Toffoli complexity that block encode the spectra of quantum chemistry Hamiltonians in a basis of $N$ arbitrary (e.g., molecular) orbitals. With ${\cal O}(\lambda / \epsilon)$ repetitions of these circuits one can use phase estimation to sample in the molecular eigenbasis, where $\lambda$ is the 1-norm of Hamiltonian coefficients and $\epsilon$ is the target precision. This is the lowest complexity that has been shown for quantum computations of chemistry within an arbitrary basis. Furthermore, up to logarithmic factors, this matches the scaling of the most efficient prior block encodings that can only work with orthogonal basis functions diagonalizing the Coloumb operator (e.g., the plane wave dual basis). Our key insight is to factorize the Hamiltonian using a method known as tensor hypercontraction (THC) and then to transform the Coulomb operator into an isospectral diagonal form with a non-orthogonal basis defined by the THC factors. We then use qubitization to simulate the non-orthogonal THC Hamiltonian, in a fashion that avoids most complications of the non-orthogonal basis. We also reanalyze and reduce the cost of several of the best prior algorithms for these simulations in order to facilitate a clear comparison to the present work. In addition to having lower asymptotic scaling spacetime volume, compilation of our algorithm for challenging finite-sized molecules such as FeMoCo reveals that our method requires the least fault-tolerant resources of any known approach. By laying out and optimizing the surface code resources required of our approach we show that FeMoCo can be simulated using about four million physical qubits and under four days of runtime, assuming $1 \, \mu {\rm s}$ cycle times and physical gate error rates no worse than $0.1\%$.
### \textttp$^\dagger$q: A tool for prototyping many-body methods for quantum chemistry
Nicholas Rubin, Albert Eugene DePrince III · arXiv:2106.06850 (2021)
\texttt{p$^\dagger$q} is a C++ accelerated Python library designed to generate equations for many-body quantum chemistry methods and to realize proof-of-concept implementations of these equations for rapid prototyping. Central to this library is a simple interface to define strings of second-quantized creation and annihilation operators and to bring these strings to normal order with respect to either the true vacuum state or the Fermi vacuum. Tensor contractions over fully-contracted strings can then be evaluated using standard Python functions ({\em e.g.}, Numpy's einsum). Given one- and two-electron integrals these features allow for the rapid implementation and assessment of a wide array of many-body quantum quantum chemistry methods.
### Low Rank Representations for Quantum Simulations of Electronic Structure
Mario Motta, Erika Ye, Jarrod Ryan McClean, Zhendong Li, Austin Minnich, Ryan Babbush, Garnet Kin-Lic Chan · NPJ Quantum Information, vol. 7 (2021)
The quantum simulation of quantum chemistry is a promising application of quantum computers. However, for N molecular orbitals, the O(N^4) gate complexity of performing Hamiltonian and unitary Coupled Cluster Trotter steps makes simulation based on such primitives challenging. We substantially reduce the gate complexity of such primitives through a two-step low-rank factorization of the Hamiltonian and cluster operator, accompanied by truncation of small terms. Using truncations that incur errors below chemical accuracy, we are able to perform Trotter steps of the arbitrary basis electronic structure Hamiltonian with O(N^3) gate complexity in small simulations, which reduces to O(N^2 log N) gate complexity in the asymptotic regime, while our unitary Coupled Cluster Trotter step has O(N^3) gate complexity as a function of increasing basis size for a given molecule. In the case of the Hamiltonian Trotter step, these circuits have O(N^2) depth on a linearly connected array, an improvement over the O(N^3) scaling assuming no truncation. As a practical example, we show that a chemically accurate Hamiltonian Trotter step for a 50 qubit molecular simulation can be carried out in the molecular orbital basis with as few as 4,000 layers of parallel nearest-neighbor two-qubit gates, consisting of fewer than 100,000 non-Clifford rotations. We also apply our algorithm to iron-sulfur clusters relevant for elucidating the mode of action of metalloenzymes.
### Error Mitigation via Verified Phase Estimation
Thomas E O'Brien, Stefano Polla, Nicholas Rubin, Bill Huggins, Sam Connor McArdle, Sergio Boixo, Jarrod Ryan McClean, Ryan Babbush · PRX Quantum, vol. 2 (2021)
We present a novel error mitigation technique for quantum phase estimation. By post-selecting the system register to be in the starting state, we convert all single errors prior to final measurement to a time-dependent decay (up to on average exponentially small corrections), which may be accurately corrected for at the cost of additional measurement. This error migitation can be built into phase estimation techniques that do not require control qubits. By separating the observable of interest into a linear combination of fast-forwardable Hamiltonians and measuring those components individually, we can convert this decay into a constant offset. Using this technique, we demonstrate the estimation of expectation values on numerical simulations of moderately-sized quantum circuits with multiple orders of magnitude improvement over unmitigated estimation at near-term error rates. We further combine verified phase estimation with the optimization step in a variational algorithm to provide additional mitigation of control error. In many cases, our results demonstrate a clear signature that the verification technique can mitigate against any single error occurring in an instance of a quantum computation: the error $\epsilon$ in the expectation value estimation scales with $p^2$, where $p$ is the probability of an error occurring at any point in the circuit. Further numerics indicate that our scheme remains robust in the presence of sampling noise, though different classical post-processing methods may lead to up to an order of magnitude error increase in the final energy estimates.
### The Fermionic Quantum Emulator
Nicholas Rubin, Toru Shiozaki, Kyle Throssell, Garnet Kin-Lic Chan, Ryan Babbush · arXiv:2104.13944 (2021)
The fermionic quantum emulator (FQE) is a collection of protocols for emulating quantum dynamics of fermions efficiently taking advantage of common symmetries present in chemical, materials, and condensed-matter systems. The library is fully integrated with the OpenFermion software package and serves as the simulation backend. The FQE reduces memory footprint by exploiting number and spin symmetry along with custom evolution routines for sparse and dense Hamiltonians, allowing us to study significantly larger quantum circuits at modest computational cost when compared against qubit state vector simulators. This release paper outlines the technical details of the simulation methods and key technical advantages.
### Analog/Mixed-Signal Integrated Circuits for Quantum Computing
Joseph Bardin · Proceedings of the 2020 IEEE BiCMOS and Compound Semiconductor Integrated Circuits and Technology Symposium (BCICTS)
Future error-corrected quantum computers will require analog/mixed-signal control systems capable of providing a million or more high-precision analog waveforms. In this article, we begin by introducing the reader to superconducting transmon qubit technology and then explain how the state of a quantum processor implemented in this technology is controlled. By example, we then show how one can implement one part of the quantum control system as a cryogenic integrated circuit and review experimental results from the characterization of such an IC. The article concludes with a discussion of analog/mixed-signal design challenges that must be overcome in order to build the system required to control a million-qubit quantum computer.
### Fermionic partial tomography via classical shadows
Akimasa Miyake, Andrew Zhao, Nicholas Rubin · arXiv:2010.16094 (2020)
We propose a tomographic protocol for estimating any $k$-body reduced density matrix ($k$-RDM) of an $n$-mode fermionic state, a ubiquitous step in near-term quantum algorithms for simulating many-body physics, chemistry, and materials. Our approach extends the framework of classical shadows, a randomized approach to learning a collection of quantum state properties, to the fermionic setting. Our sampling protocol uses randomized measurement settings generated by a discrete group of fermionic Gaussian unitaries, implementable with linear-depth circuits. We prove that estimating all $k$-RDM elements to additive precision $\varepsilon$ requires on the order of $\binom{n}{k} k^{3/2} \log(n) / \varepsilon^2$ repeated state preparations, which is optimal up to the logarithmic factor. Furthermore, numerical calculations show that our protocol offers a substantial improvement in constant overheads for $k \geq 2$, as compared to prior deterministic strategies. We also adapt our method to particle-number symmetry, wherein the additional circuit depth may be halved at the cost of roughly 2--5 times more repetitions.
### A Low-Power CMOS Quantum Controller for Transmon Qubits
Joseph Bardin · Proceedings of the 2020 IEEE International Electron Devices Meeting (IEDM)
The development of large-scale quantum computers will require the implementation of scalable control and measurement electronics, which may have to operate at a temperature as low as 4 K. In this paper, we review a first-generation control IC that has been implemented for use with transmon qubits. We first explain the requirements which such an IC must meet and then present the circuit design. We then describe a series of quantum control experiments. The paper concludes with a discussion of future work.
### The Snake Optimizer for Learning Quantum Processor Control Parameters
Paul Victor Klimov, Julian Kelly, John M. Martinis, Hartmut Neven · arXiv:2006.04594 (2020)
High performance quantum computing requires a calibration system that learns optimal control parameters much faster than system drift. In some cases, the learning procedure requires solving complex optimization problems that are non-convex, high-dimensional, highly constrained, and have astronomical search spaces. Such problems pose an obstacle for scalability since traditional global optimizers are often too inefficient and slow for even small-scale processors comprising tens of qubits. In this whitepaper, we introduce the Snake optimizer for efficiently and quickly solving such optimization problems by leveraging concepts in artificial intelligence, dynamic programming, and graph optimization. In practice, the Snake has been applied to optimize the frequencies at which quantum logic gates are implemented in frequency-tunable superconducting qubits. This application enabled state-of-the-art system performance on a 53 qubit quantum processor, serving as a key component of demonstrating quantum supremacy. Furthermore, since the Snake optimizer scales favorably with qubit number, is amenable to local re-optimization, and is parallelizable, it shows promise for optimizing much larger quantum processors.
### Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization
Yuval Sanders, Dominic W. Berry, Pedro C. S. Costa, Louis W. Tessler, Nathan Wiebe, Craig Michael Gidney, Hartmut Neven, Ryan Babbush · PRX Quantum, vol. 1 (2020), pp. 020312
Here we explore which heuristic quantum algorithms for combinatorial optimization might be most practical to try out on a small fault-tolerant quantum computer. We compile circuits for several variants of quantum-accelerated simulated annealing including those using qubitization or Szegedy walks to quantize classical Markov chains and those simulating spectral-gap-amplified Hamiltonians encoding a Gibbs state. We also optimize fault-tolerant realizations of the adiabatic algorithm, quantum-enhanced population transfer, the quantum approximate optimization algorithm, and other approaches. Many of these methods are bottlenecked by calls to the same subroutines; thus, optimized circuits for those primitives should be of interest regardless of which heuristic is most effective in practice. We compile these bottlenecks for several families of optimization problems and report for how long and for what size systems one can perform these heuristics in the surface code given a range of resource budgets. Our results discourage the notion that any quantum optimization heuristic realizing only a quadratic speedup achieves an advantage over classical algorithms on modest superconducting qubit surface code processors without significant improvements in the implementation of the surface code. For instance, under quantum-favorable assumptions (e.g., that the quantum algorithm requires exactly quadratically fewer steps), our analysis suggests that quantum-accelerated simulated annealing requires roughly a day and a million physical qubits to optimize spin glasses that could be solved by classical simulated annealing in about 4 CPU-minutes.
### Quantum Computing: An Introduction for Microwave Engineers
Joseph Bardin, Daniel Sank, Ofer Naaman, Evan Jeffrey · IEEE Microwave Magazine, vol. 21 (2020), pp. 24-44
This paper is a tutorial on quantum computing designed for hardware engineers. We begin by explaining the type of computation that is carried out on a universal quantum computer then describe the requirement for hardware used to build a such system. We then explain how each of these requirements can be met using superconducting qubit technology. Connections are made throughout to familiar microwave concepts. The paper ends with a discussion of what is required of a fault-tolerant quantum computer and how the expertise of the microwave engineering community will be required to engineer such a system.
### Investigating Quantum Approximate Optimization Algorithms under Bang-bang Protocols
Daniel Liang, Li Li, Stefan Leichenauer · Phys. Rev. Research, vol. 2 (2020), pp. 033402
The quantum approximate optimization algorithm (QAOA) is widely seen as a possible usage of noisy intermediate-scale quantum (NISQ) devices. We analyze the algorithm as a bang-bang protocol with fixed total time and a randomized greedy optimization scheme. We investigate the performance of bang-bang QAOA on MAX-2-SAT, finding the appearance of phase transitions with respect to the total time. As the total time increases, the optimal bang-bang protocol experiences a number of jumps and plateaus in performance, which match up with an increasing number of switches in the standard QAOA formulation. At large times, it becomes more difficult to find a globally optimal bang-bang protocol and performances suffer. We also investigate the effects of changing the initial conditions of the randomized optimization algorithm and see that better local optima can be found by using an adiabatic initialization.
### Improved Fault-Tolerant Quantum Simulation of Condensed-Phase Correlated Electrons via Trotterization
Ian Kivlichan, Craig Michael Gidney, Dominic Berry, Jarrod Ryan McClean, Wei Sun, Zhang Jiang, Nicholas Rubin, Austin Fowler, Alán Aspuru-Guzik, Hartmut Neven, Ryan Babbush · Quantum, vol. 4 (2020), pp. 296
Recent work has deployed linear combinations of unitaries techniques to significantly reduce the cost of performing fault-tolerant quantum simulations of correlated electron models. Here, we show that one can sometimes improve over those results with optimized implementations of Trotter-Suzuki-based product formulas. We show that low-order Trotter methods perform surprisingly well when used with phase estimation to compute relative precision quantities (e.g. energy per unit cell), as is often the goal for condensed-phase (e.g. solid-state) systems. In this context, simulations of the Hubbard model and plane wave electronic structure models with $N < 10^5$ fermionic modes can be performed with roughly O(1) and O(N^2) T complexities. We also perform numerics that reveal tradeoffs between the error of a Trotter step and Trotter step gate complexity across various implementations; e.g., we show that split-operator techniques have less Trotter error than popular alternatives. By compiling to surface code fault-tolerant gates using lattice surgery and assuming error rates of one part in a thousand, we show that one can error-correct quantum simulations of interesting, classically intractable instances with only a few hundred thousand physical qubits.
### Quantum Optimization with a Novel Gibbs Objective Function and Ansatz Architecture Search
Li Li, Minjie Fan, Marc Coram, Patrick Riley, Stefan Leichenauer · Phys. Rev. Research, vol. 2 (2020), pp. 023074
The quantum approximate optimization algorithm (QAOA) is a standard method for combinatorial optimization with a gate-based quantum computer. The QAOA consists of a particular ansatz for the quantum circuit architecture, together with a prescription for choosing the variational parameters of the circuit. We propose modifications to both. First, we define the Gibbs objective function and show that it is superior to the energy expectation value for use as an objective function in tuning the variational parameters. Second, we describe an ansatz architecture search (AAS) algorithm for searching the discrete space of quantum circuit architectures near the QAOA to find a better ansatz. Applying these modifications for a complete graph Ising model results in a 244.7% median relative improvement in the probability of finding a low-energy state while using 33.3% fewer two-qubit gates. For Ising models on a 2d grid we similarly find 44.4% median improvement in the probability with a 20.8% reduction in the number of two-qubit gates. This opens a new research field of quantum circuit architecture design for quantum optimization algorithms.
### Using Models to Improve Optimizers for Variational Quantum Algorithms
Kevin Jeffery Sung, Jiahao Yao, Matthew P Harrigan, Nicholas Rubin, Zhang Jiang, Lin Lin, Ryan Babbush, Jarrod Ryan McClean · Quantum Science and Technology, vol. 5 (2020), pp. 044008
Variational quantum algorithms are a leading candidate for early applications on noisy intermediate-scale quantum computers. These algorithms depend on a classical optimization outer-loop that minimizes some function of a parameterized quantum circuit. In practice, finite sampling error and gate errors make this a stochastic optimization with unique challenges that must be addressed at the level of the optimizer. The sharp trade-off between precision and sampling time in conjunction with experimental constraints necessitates the development of new optimization strategies to minimize overall wall clock time in this setting. We introduce an optimization method and numerically compare its performance with common methods in use today. The method is a simple surrogate model-based algorithm designed to improve reuse of collected data. It does so by estimating the gradient using a least-squares quadratic fit of sampled function values within a moving trusted region. To make fair comparisons between optimization methods, we develop experimentally relevant cost models designed to balance efficiency in testing and accuracy with respect to cloud quantum computing systems. The results here underscore the need to both use relevant cost models and optimize hyperparameters of existing optimization methods for competitive performance. We compare tuned methods using cost models presented by superconducting devices accessed through cloud computing platforms. The method introduced here has several practical advantages in realistic experimental settings, and has been used successfully in a separately published experiment on Google's Sycamore device.
### Accurately computing electronic properties of materials using eigenenergies
Adam Jozef Zalcman, Alan Derk, Alan Ho, Alex Opremcak, Alexander Korotkov, Andre Gregory Petukhov, Andreas Bengtsson, Andrew Dunsworth, Anthony Megrant, Austin Fowler, Bálint Pató, Benjamin Chiaro, Benjamin Villalonga, Bob Benjamin Buckley, Brian Burkett, Brooks Riley Foxen, Catherine Erickson, Charles Neill, Chris Quintana, Cody Jones, Craig Michael Gidney, Daniel Eppens, Daniel Sank, Dave Landhuis, David A Buell, Doug Strain, Dvir Kafri, Edward Farhi, Eric Ostby, Erik Lucero, Evan Jeffrey, Fedor Kostritsa, Frank Carlton Arute, Hartmut Neven, Igor Aleiner, Jamie Yao, Jarrod Ryan McClean, Jeremy Patterson Hilton, Jimmy Chen, Jonathan Arthur Gross, Joseph Bardin, Josh Mutus, Juan Atalaya, Juan Campero, Julian Kelly, Kevin Miao, Kevin Satzinger, Kostyantyn Kechedzhi, Kunal Arya, Lev Ioffe, Marco Szalay, Marissa Giustina, Masoud Mohseni, Matt Jacob-Mitos, Matt McEwen, Matt Trevithick, Matthew Neeley, Matthew P Harrigan, Michael Blythe Broughton, Michael Newman, Murphy Yuezhen Niu, Nicholas Bushnell, Nicholas Redd, Nicholas Rubin, Ofer Naaman, Orion Martin, Paul Victor Klimov, Pavel Laptev, Pedram Roushan, Ping Yeh, Rami Barends, Roberto Collins, Ryan Babbush, Sabrina Hong, Sean Demura, Sean Harrington, Seon Kim, Sergei Isakov, Sergio Boixo, Ted White, Thomas E O'Brien, Trent Huang, Trevor Mccourt, Vadim Smelyanskiy, Vladimir Shvarts, William Courtney, William J. Huggins, Wojtek Mruczkiewicz, Xiao Mi, Yu Chen, Zhang Jiang · arXiv preprint arXiv:2012.00921 (2020)
A promising approach to study quantum materials is to simulate them on an engineered quantum platform. However, achieving the accuracy needed to outperform classical methods has been an outstanding challenge. Here, using superconducting qubits, we provide an experimental blueprint for a programmable and accurate quantum matter simulator and demonstrate how to probe fundamental electronic properties. We illustrate the underlying method by reconstructing the single-particle band-structure of a one-dimensional wire. We demonstrate nearly complete mitigation of decoherence and readout errors and arrive at an accuracy in measuring energy eigenvalues of this wire with an error of ~0.01 radians, whereas typical energy scales are of order 1 radian. Insight into this unprecedented algorithm fidelity is gained by highlighting robust properties of a Fourier transform, including the ability to resolve eigenenergies with a statistical uncertainty of 1e-4 radians. Furthermore, we synthesize magnetic flux and disordered local potentials, two key tenets of a condensed-matter system. When sweeping the magnetic flux, we observe avoided level crossings in the spectrum, a detailed fingerprint of the spatial distribution of local disorder. Combining these methods, we reconstruct electronic properties of the eigenstates where we observe persistent currents and a strong suppression of conductance with added disorder. Our work describes an accurate method for quantum simulation and paves the way to study novel quantum materials with superconducting qubits.
### TensorFlow Quantum: A Software Framework for Quantum Machine Learning
Michael Broughton, Guillaume Verdon, Trevor McCourt, Antonio J. Martinez, Jae Hyeon Yoo, Sergei V. Isakov, Philip Massey, Ramin Halavati, Murphy Yuezhen Niu, Alexander Zlokapa, Evan Peters, Owen Lockwood, Andrea Skolik, Sofiene Jerbi, Vedran Djunko, Martin Leib, Michael Streif, David Von Dollen, Hongxiang Chen, Chuxiang Cao, Roeland Wiersema, Hsin-Yuan Huang, Jarrod R. McClean, Ryan Babbush, Sergio Boixo, Dave Bacon, Alan K. Ho, Hartmut Neven, Masoud Mohseni · (2020)
We introduce TensorFlow Quantum (TFQ), an open source library for the rapid prototyping of hybrid quantum-classical models for classical or quantum data. This framework offers high-level abstractions for the design and training of both discriminative and generative quantum models under TensorFlow and supports high-performance quantum circuit simulators. We provide an overview of the software architecture and building blocks through several examples and review the theory of hybrid quantum-classical neural networks. We illustrate TFQ functionalities via several basic applications including supervised learning for quantum classification, quantum control, simulating noisy quantum circuits, and quantum approximate optimization. Moreover, we demonstrate how one can apply TFQ to tackle advanced quantum learning tasks including meta-learning, layerwise learning, Hamiltonian learning, sampling thermal states, variational quantum eigensolvers, classification of quantum phase transitions, generative adversarial networks, and reinforcement learning. We hope this framework provides the necessary tools for the quantum computing and machine learning research communities to explore models of both natural and artificial quantum systems, and ultimately discover new quantum algorithms which could potentially yield a quantum advantage.
### Learnability and Complexity of Quantum Samples
Murphy Yuezhen Niu, Andrew Mingbo Dai, Li Li, Augustus Odena, Zhengli Zhao, Vadim Smelyanskiy, Hartmut Neven, Sergio Boixo · arxiv (2020)
Given a quantum circuit, a quantum computer can sample the output distribution exponentially faster in the number of bits than classical computers. A similar exponential separation has yet to be established in generative models through quantum sample learning: given samples from an n-qubit computation, can we learn the underlying quantum distribution using models with training parameters that scale polynomial in n under a fixed training time? We study four kinds of generative models: Deep Boltzmann machine (DBM), Generative Adversarial Networks (GANs), Long Short-Term Memory (LSTM) and Autoregressive GAN, on learning quantum data set generated by deep random circuits. We demonstrate the leading performance of LSTM in learning quantum samples, and thus the autoregressive structure present in the underlying quantum distribution from random quantum circuits. Both numerical experiments and a theoretical proof in the case of the DBM show exponentially growing complexity of learning-agent parameters required for achieving a fixed accuracy as n increases. Finally, we establish a connection between learnability and the complexity of generative models by benchmarking learnability against different sets of samples drawn from probability distributions of variable degrees of complexities in their quantum and classical representations.
### Nearly Optimal Measurement Scheduling for Partial Tomography of Quantum States
Xavier Bonet, Ryan Babbush, Thomas O'Brien · Physical Review X, vol. 10 (2020), pp. 031064
Many applications of quantum simulation require to prepare and then characterize quantum states by performing an efficient partial tomography to estimate observables corresponding to k-body reduced density matrices (k-RDMs). For instance, variational algorithms for the quantum simulation of chemistry usually require that one measure the fermionic 2-RDM. While such marginals provide a tractable description of quantum states from which many important properties can be computed, their determination often requires a prohibitively large number of circuit repetitions. Here we describe a method by which all elements of k-RDMs acting on N qubits can be sampled with a number of circuits scaling as O(3^k log^{k-1} N), an exponential improvement in N over prior art. Next, we show that if one is able to implement a linear depth circuit on a linear array prior to measurement, then one can sample all elements of the fermionic 2-RDM using only O(N^2) circuits. We prove that this result is asymptotically optimal. Finally, we demonstrate that one can estimate the expectation value of any linear combination of fermionic 2-RDM elements using O(N^4 / w) circuits, each with only O(w) gates on a linear array where w < N is a free parameter. We expect these results will improve the viability of many proposals for near-term quantum simulation.
### Hartree-Fock on a Superconducting Qubit Quantum Computer
Frank Carlton Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph Bardin, Rami Barends, Sergio Boixo, Michael Blythe Broughton, Bob Benjamin Buckley, David A Buell, Brian Burkett, Nicholas Bushnell, Yu Chen, Jimmy Chen, Benjamin Chiaro, Roberto Collins, William Courtney, Sean Demura, Andrew Dunsworth, Edward Farhi, Austin Fowler, Brooks Riley Foxen, Craig Michael Gidney, Marissa Giustina, Rob Graff, Steve Habegger, Matthew P Harrigan, Alan Ho, Sabrina Hong, Trent Huang, William J. Huggins, Lev Ioffe, Sergei Isakov, Evan Jeffrey, Zhang Jiang, Cody Jones, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Seon Kim, Paul Klimov, Alexander Korotkov, Fedor Kostritsa, Dave Landhuis, Pavel Laptev, Mike Lindmark, Erik Lucero, Orion Martin, John Martinis, Jarrod Ryan McClean, Matthew McEwen, Anthony Megrant, Xiao Mi, Masoud Mohseni, Wojtek Mruczkiewicz, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Hartmut Neven, Murphy Yuezhen Niu, Thomas E O'Brien, Eric Ostby, Andre Gregory Petukhov, Harry Putterman, Chris Quintana, Pedram Roushan, Nicholas Rubin, Daniel Sank, Kevin Satzinger, Vadim Smelyanskiy, Doug Strain, Kevin Jeffery Sung, Marco Szalay, Tyler Y. Takeshita, Amit Vainsencher, Ted White, Nathan Wiebe, Jamie Yao, Ping Yeh, Adam Zalcman · Science, vol. 369 (2020), pp. 6507
As the search continues for useful applications of noisy intermediate scale quantum devices, variational simulations of fermionic systems remain one of the most promising directions. Here, we perform a series of quantum simulations of chemistry which involve twice the number of qubits and more than ten times the number of gates as the largest prior experiments. We model the binding energy of ${\rm H}_6$, ${\rm H}_8$, ${\rm H}_{10}$ and ${\rm H}_{12}$ chains as well as the isomerization of diazene. We also demonstrate error-mitigation strategies based on $N$-representability which dramatically improve the effective fidelity of our experiments. Our parameterized ansatz circuits realize the Givens rotation approach to free fermion evolution, which we variationally optimize to prepare the Hartree-Fock wavefunction. This ubiquitous algorithmic primitive corresponds to a rotation of the orbital basis and is required by many proposals for correlated simulations of molecules and Hubbard models. Because free fermion evolutions are classically tractable to simulate, yet still generate highly entangled states over the computational basis, we use these experiments to benchmark the performance of our hardware while establishing a foundation for scaling up more complex correlated quantum simulations of chemistry.
### Increasing the Representation Accuracy of Quantum Simulations of Chemistry without Extra Quantum Resources
Tyler Takeshita, Nicholas Rubin, Zhang Jiang, Eunseok Lee, Ryan Babbush, Jarrod Ryan McClean · Physical Review X, vol. 10 (2020), pp. 011004
Proposals for near-term experiments in quantum chemistry on quantum computers leverage the ability to target a subset of degrees of freedom containing the essential quantum behavior, sometimes called the active space. This approximation allows one to treat more difficult problems using fewer qubits and lower gate depths than would otherwise be possible. However, while this approximation captures many important qualitative features, it may leave the results wanting in terms of absolute accuracy (basis error) of the representation. In traditional approaches, increasing this accuracy requires increasing the number of qubits and an appropriate increase in circuit depth as well. Here we introduce a technique requiring no additional qubits or circuit depth that is able to remove much of this approximation in favor of additional measurements. The technique is constructed and analyzed theoretically, and some numerical proof of concept calculations are shown. As an example, we show how to achieve the accuracy of a 20 qubit representation using only 4 qubits and a modest number of additional measurements for a simple hydrogen molecule. We close with an outlook on the impact this technique may have on both near-term and fault-tolerant quantum simulations.
### Creating and manipulating a Laughlin-type ν=1/3 fractional quantum Hall state on a quantum computer with linear depth circuits
Armin Rahmani, Kevin J. Sung, Harald Putterman, Pedram Roushan, Pouyan Ghaemi, Zhang Jiang · PRX Quantum, vol. 1 (2020), pp. 020309
Here we present an efficient quantum algorithm to generate an equivalent many-body state to Laughlin’s ν= 1/3 fractional quantum Hall state on a digitized quantum computer. Our algorithm only uses quantum gates acting on neighboring qubits in a quasi one-dimensional setting, and its circuit depth is linear in the number of qubits, i.e., the number of Landau levels in the second quantized picture. We identify correlation functions that serve as signatures of the Laughlin state and discuss how to obtain them on a quantum computer. We also discuss a generalization of the algorithm for creating quasiparticles in the Laughlin state. This paves the way for several important studies, including quantum simulation of non-equilibrium dynamics and braiding of quasiparticles in quantum Hall states.
### Demonstrating a Continuous Set of Two-qubit Gates for Near-term Quantum Algorithms
Brooks Riley Foxen, Charles Neill, Andrew Dunsworth, Pedram Roushan, Ben Chiaro, Anthony Megrant, Julian Kelly, Jimmy Chen, Kevin Satzinger, Rami Barends, Frank Carlton Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph Bardin, Sergio Boixo, David A Buell, Brian Burkett, Yu Chen, Roberto Collins, Edward Farhi, Austin Fowler, Craig Michael Gidney, Marissa Giustina, Rob Graff, Matthew P Harrigan, Trent Huang, Sergei Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Paul Klimov, Alexander Korotkov, Fedor Kostritsa, Dave Landhuis, Erik Lucero, Jarrod McClean, Matthew McEwen, Xiao Mi, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Murphy Yuezhen Niu, Chris Quintana, Nicholas Rubin, Daniel Sank, Vadim Smelyanskiy, Amit Vainsencher, Ted White, Adam Zalcman, Jamie Yao, Hartmut Neven, John Martinis · arXiv:2001.08343 (2020)
Quantum algorithms offer a dramatic speedup for computational problems in machine learning, material science, and chemistry. However, any near-term realizations of these algorithms will need to be heavily optimized to fit within the finite resources offered by existing noisy quantum hardware. Here, taking advantage of the strong adjustable coupling of gmon qubits, we demonstrate a continuous two qubit gate set that can provide a 5x reduction in circuit depth. We implement two gate families: an iSWAP-like gate to attain an arbitrary swap angle, $\theta$, and a CPHASE gate that generates an arbitrary conditional phase, $\phi$. Using one of each of these gates, we can perform an arbitrary two qubit gate within the excitation-preserving subspace allowing for a complete implementation of the so-called Fermionic Simulation, or fSim, gate set. We benchmark the fidelity of the iSWAP-like and CPHASE gate families as well as 525 other fSim gates spread evenly across the entire fSim($\theta$, $\phi$) parameter space achieving purity-limited average two qubit Pauli error of $3.8 \times 10^{-3}$ per fSim gate.
### Decoding Quantum Errors Using Subspace Expansions
Jarrod Ryan McClean, Zhang Jiang, Nicholas Rubin, Ryan Babbush, Hartmut Neven · Nature Communications, vol. 11 (2020), pp. 636
With the rapid developments in quantum hardware comes a push towards the first practical applications on these devices. While fully fault-tolerant quantum computers may still be years away, one may ask if there exist intermediate forms of error correction or mitigation that might enable practical applications before then. In this work, we consider the idea of post-processing error decoders using existing quantum codes, which are capable of mitigating errors on encoded logical qubits using classical post-processing with no complicated syndrome measurements or additional qubits beyond those used for the logical qubits. This greatly simplifies the experimental exploration of quantum codes on near-term devices, removing the need for locality of syndromes or fast feed-forward, allowing one to study performance aspects of codes on real devices. We provide a general construction equipped with a simple stochastic sampling scheme that does not depend explicitly on a number of terms that we extend to approximate projectors within a subspace. This theory then allows one to generalize to the correction of some logical errors in the code space, correction of some physical unencoded Hamiltonians without engineered symmetries, and corrections derived from approximate symmetries. In this work, we develop the theory of the method and demonstrate it on a simple example with the perfect [[5,1,3]] code, which exhibits a pseudo-threshold of p≈0.50 under a single qubit depolarizing channel applied to all qubits. We also provide a demonstration under the application of a logical operation and performance on an unencoded hydrogen molecule, which exhibits a significant improvement over the entire range of possible errors incurred under a depolarizing channel.
### Discontinuous Galerkin Discretization for Quantum Simulation of Chemistry
Jarrod Ryan McClean, Fabian M. Faulstich, Qinyi Zhu, Bryan O'Gorman, Yiheng Qiu, Steven White, Ryan Babbush, Lin Lin · New Journal of Physics, vol. 22 (2020)
Methods for electronic structure based on Gaussian and molecular orbital discretizations offer a well established, compact representation that forms much of the foundation of correlated quantum chemistry calculations on both classical and quantum computers. Despite their ability to describe essential physics with relatively few basis functions, these representations can suffer from a quartic growth of the number of integrals. Recent results have shown that, for some quantum and classical algorithms, moving to representations with diagonal two-body operators can result in dramatically lower asymptotic costs, even if the number of functions required increases significantly. We introduce a way to interpolate between the two regimes in a systematic and controllable manner, such that the number of functions is minimized while maintaining a block diagonal structure of the two-body operator and desirable properties of an original, primitive basis. Techniques are analyzed for leveraging the structure of this new representation on quantum computers. Empirical results for hydrogen chains suggest a scaling improvement from O(N^4.5) in molecular orbital representations to O(N^2.6) in our representation for quantum evolution in a fault-tolerant setting, and exhibit a constant factor crossover at 15 to 20 atoms. Moreover, we test these methods using modern density matrix renormalization group methods classically, and achieve excellent accuracy with respect to the complete basis set limit with a speedup of 1-2 orders of magnitude with respect to using the primitive or Gaussian basis sets alone. These results suggest our representation provides significant cost reductions while maintaining accuracy relative to molecular orbital or strictly diagonal approaches for modest-sized systems in both classical and quantum computation for correlated systems.
### Materials loss measurements using superconducting microwave resonators
Corey Rae Harrington McRae, Haozhi Wang, Jiansong Gao, Michael Vissers, Teresa Brecht, Andrew Dunsworth, David Pappas, Josh Mutus · Review of Scientific Instruments (invited) (2020)
The performance of superconducting circuits for quantum computing is limited by materials losses in general, and two-level system (TLS) losses in particular, at single photon powers and millikelvin temperatures. The identification of low loss fabrication techniques, materials, and thin film dielectrics is critical to achieving scalable architectures for superconducting quantum computing. Superconducting microwave resonators provide a convenient qubit proxy for assessing loss performance and studying loss mechanisms such as TLS loss, non-equilibrium quasiparticles, and magnetic flux vortices. In this review article, we provide an overview of considerations for designing accurate resonator experiments to characterize loss including applicable types of loss, cryogenic setup, device design, and methods for extracting material and interface loss tangents, summarizing techniques that have been evolving for over two decades. Results from measurements of a variety of materials and processes are also summarized. Lastly, we present recommendations for the reporting of loss data from superconducting microwave resonators to facilitate materials comparisons across the field.
### Virtual Distillation for Quantum Error Mitigation
William J. Huggins, Sam Connor McArdle, Thomas E O'Brien, Joonho Lee, Nicholas Rubin, Sergio Boixo, Birgitta Whaley, Ryan Babbush, Jarrod Ryan McClean · arXiv:2011.07064 (2020)
Contemporary quantum computers have relatively high levels of noise, making it difficult to use them to perform useful calculations, even with a large number of qubits. Quantum error correction is expected to eventually enable fault-tolerant quantum computation at large scales, but until then it will be necessary to use alternative strategies to mitigate the impact of errors. We propose a near-term friendly strategy to mitigate errors by entangling and measuring $$M$$ copies of a noisy state $$\rho$$. This enables us to estimate expectation values with respect to a state with dramatically reduced error, $$\rho^M/ \tr(\rho^M)$$, without explicitly preparing it, hence the name virtual distillation''. As $$M$$ increases, this state approaches the closest pure state to $$\rho$$, exponentially quickly. We analyze the effectiveness of virtual distillation and find that it is governed in many regimes by the behaviour of this pure state (corresponding to the dominant eigenvector of $$\rho$$). We numerically demonstrate that virtual distillation is capable of suppressing errors by multiple orders of magnitude and explain how this effect is enhanced as the system size grows. Finally, we show that this technique can improve the convergence of randomized quantum algorithms, even in the absence of device noise.
### Optimal fermion-to-qubit mapping via ternary trees with applications to reduced quantum states learning
Zhang Jiang, Amir Kalev, Wojtek Mruczkiewicz, Hartmut Neven · Quantum, vol. 4 (2020), pp. 276
We introduce a fermion-to-qubit mapping using ternary trees. The mapping has a simple structure where any single Majorana operator on an n-mode fermionic system is mapped to a multi-qubit Pauli operator acting nontrivially on log_3 (2n+1) qubits. We prove that the ternary-tree mapping is optimal in the sense that it is impossible to construct a Pauli operator in any fermion-to-qubit mapping which acts nontrivially on less than log_3 (2n+1) qubits. We apply this mapping to the problem of learning k-fermion reduced density matrix (RDM); a problem relevant in various quantum simulation applications. We show that using this mapping one can determine the elements of all k-fermion RDMs, to precision ε, by repeating a single quantum circuit for ~ (2n+1) k / ε^2 times. This result is based on a method we develop here that allows one to determine the elements of all k-qubit RDMs, to precision ε, by repeating a single quantum circuit for ~ 3k /ε^2 times, independent of the system size. This method improves over existing ones for determining qubit RDMs.
### XY-mixers: analytical and numerical results for QAOA
Eleanor Rieffel, Jason M. Dominy, Nicholas Rubin, Zhihui Wang · Phys. Rev. A, vol. 101 (2020), pp. 012320
The Quantum Alternating Operator Ansatz (QAOA) is a promising gate-model meta-heuristic for combinatorial optimization. Extending the algorithm to include hard constraints presents an implementation challenge for near-term quantum resources. This work explores strategies for enforcing hard constraints by using XY-hamiltonians as the mixer. Despite the complexity of the XY-Hamiltonian mixer, we demonstrate that for problems represented through one-hot-encoding, certain classes of the mixer Hamiltonian can be implemented without Trotter error in depth $O(\kappa)$ where $\kappa$ is the number of assignable colors. We also specify general strategies for implementing QAOA circuits on all-to-all connected graphs and linearly connected graphs inspired by fermionic simulation techniques. Performance is validated on graph coloring problems that are known to be challenging for a given classical algorithm. The general strategy of using the XY-mixers is validated numerically demonstrating a significant improvement over the general X-mixer.
### Quantum Simulation of the Sachdev-Ye-Kitaev Model by Asymmetric Qubitization
Ryan Babbush, Dominic W. Berry, Hartmut Neven · Physical Review A Rapid Communication, vol. 99 (2019), 040301(R)
We show that one can quantum simulate the dynamics of a Sachdev-Ye-Kitaev model with $N$ Majorana modes for time $t$ to precision $\epsilon$ with gate complexity ${\cal O}(N^{7/2} t + N^{5/2} \log(1 / \epsilon) / \log\log(1/\epsilon))$. In addition to scaling sublinearly in the number of Hamiltonian terms, this gate complexity represents an exponential improvement in $1/\epsilon$ and large polynomial improvement in $N$ and $t$ over prior state-of-the-art algorithms which scale as ${\cal O}(N^{10} t^2 / \epsilon)$. Our approach involves a variant of the qubitization technique in which we encode the Hamiltonian $H$ as an asymmetric projection of a signal oracle $U$ onto two different signal states prepared by distinct state oracles, $A\ket{0} \mapsto \ket{A}$ and $B\ket{0} \mapsto \ket{B}$, such that $H = \bra{B} U \ket{A}$. Our strategy for applying this method to the Sachdev-Ye-Kitaev model involves realizing $B$ using only Hadamard gates and realizing $A$ as a random quantum circuit.
### Learning to learn with quantum neural networks via classical neural networks
Guillaume Verdon, Michael Broughton, Jarrod Ryan McClean, Kevin Jeffery Sung, Ryan Babbush, Zhang Jiang, Hartmut Neven, Masoud Mohseni · arXiv:1907.05415 (2019)
Quantum Neural Networks (QNNs) are a promising variational learning paradigm with applications to near-term quantum processors, however they still face some significant challenges. One such challenge is finding good parameter initialization heuristics that ensure rapid and consistent convergence to local minima of the parameterized quantum circuit landscape. In this work, we train classical neural networks to assist in the quantum learning process, also know as meta-learning, to rapidly find approximate optima in the parameter landscape for several classes of quantum variational algorithms. Specifically, we train classical recurrent neural networks to find approximately optimal parameters within a small number of queries of the cost function for the Quantum Approximate Optimization Algorithm (QAOA) for MaxCut, QAOA for Sherrington-Kirkpatrick Ising model, and for a Variational Quantum Eigensolver for the Hubbard model. By initializing other optimizers at parameter values suggested by the classical neural network, we demonstrate a significant improvement in the total number of optimization iterations required to reach a given accuracy. We further demonstrate that the optimization strategies learned by the neural network generalize well across a range of problem instance sizes. This opens up the possibility of training on small, classically simulatable problem instances, in order to initialize larger, classically intractably simulatable problem instances on quantum devices, thereby significantly reducing the number of required quantum-classical optimization iterations.
### Design and Characterization of a 28-nm Bulk-CMOS Cryogenic Quantum Controller Dissipating Less than 2 mW at 3 K
Joseph Bardin, Evan Jeffrey, Erik Lucero, Trent Huang, Sayan Das, Daniel Sank, Ofer Naaman, Anthony Megrant, Rami Barends, Ted White, Marissa Giustina, Kevin Satzinger, Kunal Arya, Pedram Roushan, Ben Chiaro, Julian Kelly, Zijun Chen, Brian Burkett, Yu Chen, Andrew Dunsworth, Austin Fowler, Brooks Foxen, Craig Michael Gidney, Rob Graff, Paul Klimov, Josh Mutus, Matthew McEwen, Matthew Neeley, Charles Neill, Chris Quintana, Amit Vainsencher, Hartmut Neven, John Martinis · IEEE Journal of Solid State Circuits, vol. 54(11) (2019), pp. 3043 - 3060
Implementation of an error corrected quantum computer is believed to require a quantum processor with on the order of a million or more physical qubits and, in order to run such a processor, a quantum control system of similar scale will be required. Such a controller will need to be integrated within the cryogenic system and in close proximity with the quantum processor in order to make such a system practical. Here, we present a prototype cryogenic CMOS quantum controller designed in a 28-nm bulk CMOS process and optimized to implement a 4-bit XY gate instruction set for transmon qubits. After introducing the transmon qubit, including a discussion of how it is controlled, design considerations are discussed, with an emphasis on error rates and scalability. The circuit design is then discussed. Cryogenic performance of the underlying technology is presented and the results of several quantum control experiments carried out using the integrated controller are described. The paper ends with a comparison to the state of the art. It has been shown that the quantum control IC achieves comparable performance with a conventional rack mount control system while dissipating less than 2mW of total AC and DC power and requiring a digital data stream of less than 500 Mb/s.
### A 28nm Bulk-CMOS 4-to-8GHz <2mW Cryogenic Pulse Modulator for Scalable Quantum Computing
Joseph Bardin, Evan Jeffrey, Erik Lucero, Trent Huang, Ofer Naaman, Rami Barends, Ted White, Marissa Giustina, Daniel Sank, Pedram Roushan, Kunal Arya, Ben Chiaro, Julian Kelly, Jimmy Chen, Brian Burkett, Yu Chen, Andrew Dunsworth, Austin Fowler, Brooks Foxen, Craig Michael Gidney, Rob Graff, Paul Klimov, Josh Mutus, Matthew McEwen, Anthony Megrant, Matthew Neeley, Charles Neill, Chris Quintana, Amit Vainsencher, Hartmut Neven, John Martinis · Proceedings of the 2019 International Solid State Circuits Conference, IEEE, pp. 456-458
Future quantum computing systems will require cryogenic integrated circuits to control and measure millions of qubits. In this paper, we report design and measurement of a prototype cryogenic CMOS integrated circuit that has been optimized for the control of transmon qubits. The circuit has been integrated into a quantum measurement setup and its performance has been validated through multiple quantum control experiments.
### Quantum Supremacy using a Programmable Superconducting Processor
Frank Arute, Kunal Arya, Ryan Babbush, Dave Bacon, Joseph Bardin, Rami Barends, Rupak Biswas, Sergio Boixo, Fernando Brandao, David Buell, Brian Burkett, Yu Chen, Jimmy Chen, Ben Chiaro, Roberto Collins, William Courtney, Andrew Dunsworth, Edward Farhi, Brooks Foxen, Austin Fowler, Craig Michael Gidney, Marissa Giustina, Rob Graff, Keith Guerin, Steve Habegger, Matthew Harrigan, Michael Hartmann, Alan Ho, Markus Rudolf Hoffmann, Trent Huang, Travis Humble, Sergei Isakov, Evan Jeffrey, Zhang Jiang, Dvir Kafri, Kostyantyn Kechedzhi, Julian Kelly, Paul Klimov, Sergey Knysh, Alexander Korotkov, Fedor Kostritsa, Dave Landhuis, Mike Lindmark, Erik Lucero, Dmitry Lyakh, Salvatore Mandrà, Jarrod Ryan McClean, Matthew McEwen, Anthony Megrant, Xiao Mi, Kristel Michielsen, Masoud Mohseni, Josh Mutus, Ofer Naaman, Matthew Neeley, Charles Neill, Murphy Yuezhen Niu, Eric Ostby, Andre Petukhov, John Platt, Chris Quintana, Eleanor G. Rieffel, Pedram Roushan, Nicholas Rubin, Daniel Sank, Kevin J. Satzinger, Vadim Smelyanskiy, Kevin Jeffery Sung, Matt Trevithick, Amit Vainsencher, Benjamin Villalonga, Ted White, Z. Jamie Yao, Ping Yeh, Adam Zalcman, Hartmut Neven, John Martinis · Nature, vol. 574 (2019), 505–510
The promise of quantum computers is that certain computational tasks might be executed exponentially faster on a quantum processor than on a classical processor. A fundamental challenge is to build a high-fidelity processor capable of running quantum algorithms in an exponentially large computational space. Here we report the use of a processor with programmable superconducting qubits to create quantum states on 53 qubits, corresponding to a computational state-space of dimension 2^53 (about 10^16). Measurements from repeated experiments sample the resulting probability distribution, which we verify using classical simulations. Our Sycamore processor takes about 200 seconds to sample one instance of a quantum circuit a million times-our benchmarks currently indicate that the equivalent task for a state-of-the-art classical supercomputer would take approximately 10,000 years. This dramatic increase in speed compared to all known classical algorithms is an experimental realization of quantum supremacy for this specific computational task, heralding a much-anticipated computing paradigm.
### Qubitization of Arbitrary Basis Quantum Chemistry Leveraging Sparsity and Low Rank Factorization
Dominic W. Berry, Craig Michael Gidney, Mario Motta, Jarrod Ryan McClean, Ryan Babbush · Quantum, vol. 3 (2019), pp. 208
Recent work has dramatically reduced the gate complexity required to quantum simulate chemistry by using linear combinations of unitaries based methods to exploit structure in the plane wave basis Coulomb operator. Here, we show that one can achieve similar scaling even for arbitrary basis sets (which can be hundreds of times more compact than plane waves) by using qubitized quantum walks in a fashion that takes advantage of structure in the Coulomb operator, either by directly exploiting sparseness, or via a low rank tensor factorization. We provide circuits for several variants of our algorithm (which all improve over the scaling of prior methods) including one with O(N^{3/2} λ) T complexity, where N is number of orbitals and λ is the 1-norm of the chemistry Hamiltonian. We deploy our algorithms to simulate the FeMoco molecule (relevant to Nitrogen fixation) and obtain circuits requiring almost one thousand times less surface code spacetime volume than prior quantum algorithms for this system, despite us using a larger and more accurate active space.
### Quantum Simulation of Chemistry with Sublinear Scaling in Basis Size
Ryan Babbush, Dominic W. Berry, Jarrod Ryan McClean, Hartmut Neven · NPJ Quantum Information, vol. 5 (2019)
We present a quantum algorithm for simulating quantum chemistry with gate complexity Õ(η^{8/3}N^{1/3}),where η is the number of electrons and N is the number of plane wave orbitals. In comparison, the most efficient prior algorithms for simulating electronic structure using plane waves (which are at least as efficient as algorithms using any other basis) have complexity Õ(η^{2/3}N^{8/3}). We achieve our scaling in first quantization by performing simulation in the rotating frame of the kinetic operator using interaction picture techniques. Our algorithm is far more efficient than all prior approaches when N ≫ η, as is needed to suppress discretization error when representing molecules in the plane wave basis, or when simulating without the Born-Oppenheimer approximation.
### Majorana Loop Stabilizer Codes for Error Mitigation in Fermionic Quantum Simulations
Zhang Jiang, Jarrod Ryan McClean, Ryan Babbush, Hartmut Neven · Physical Review Applied, vol. 12 (2019), pp. 064041
Fermion-to-qubit mappings that preserve geometric locality are especially useful for simulating lattice fermion models (e.g., the Hubbard model) on a quantum computer. They avoid the overhead associated with geometric nonlocal parity terms in mappings such as the Jordan-Wigner transformation and the Bravyi-Kitaev transformation. As a result, they often provide quantum circuits with lower depth and gate complexity. In such encodings, fermionic states are encoded in the common +1 eigenspace of a set of stabilizers, akin to stabilizer quantum error-correcting codes. Here, we discuss several known geometric locality-preserving mappings and their abilities to correct and detect single-qubit errors. We introduce a geometric locality-preserving map, whose stabilizers correspond to products of Majorana operators on closed paths of the fermionic hopping graph. We show that our code, which we refer to as the Majorana loop stabilizer code (MLSC) can correct all single-qubit errors on a two-dimensional square lattice, while previous geometric locality-preserving codes can only detect single-qubit errors on the same lattice. Compared to existing codes, the MLSC maps the relevant fermionic operators to lower-weight qubit operators despite having higher code distance. Going beyond lattice models, we demonstrate that the MLSC is compatible with state-of-the-art algorithms for simulating quantum chemistry, and can offer those simulations the same error-correction properties without additional asymptotic overhead. These properties make the MLSC a promising candidate for error-mitigated quantum simulations of fermions on near-term devices
### Diabatic gates for frequency-tunable superconducting qubits
Rami Barends, Chris Quintana, A.G. Petukhov, Yu Chen, Dvir Kafri, Kostyantyn Kechedzhi, Roberto Collins, Ofer Naaman, Sergio Boixo, Frank Carlton Arute, Kunal Arya, David A Buell, Brian Burkett, Jimmy Chen, Ben Chiaro, Andrew Dunsworth, Brooks Foxen, Austin Fowler, Craig Michael Gidney, Marissa Giustina, Rob Graff, Trent Huang, Evan Jeffrey, J. Kelly, Paul Victor Klimov, Fedor Kostritsa, Dave Landhuis, Erik Lucero, Matt McEwen, Anthony Megrant, Xiao Mi, Josh Mutus, Matthew Neeley, Charles Neill, Eric Ostby, Pedram Roushan, Daniel Sank, Amit Vainsencher, Ted White, Jamie Yao, Ping Yeh, Adam Jozef Zalcman, Hartmut Neven, Vadim Smelyanskiy, John Martinis · Physical Review Letters, vol. 123 (2019), pp. 210501
We demonstrate diabatic two-qubit gates with Pauli error rates down to 4.3(2)*10^{-3} in as fast as 18 ns using frequency-tunable superconducting qubits. This is achieved by synchronizing the entangling parameters with minima in the leakage channel. The synchronization shows a landscape in gate parameter space that agrees with model predictions and facilitates robust tune-up. We test both iSWAP-like and CPHASE gates with cross-entropy benchmarking. The presented approach can be extended to multibody operations as well.
### Low-Depth Quantum Simulation of Materials
Ryan Babbush, Nathan Wiebe, Jarrod McClean, James McClain, Hartmut Neven, Garnet Chan · Physical Review X, vol. 8 (2018), pp. 011044
Quantum simulation of the electronic structure problem is one of the most researched applications of quantum computing. The majority of quantum algorithms for this problem encode the wavefunction using $N$ molecular orbitals, leading to Hamiltonians with ${\cal O}(N^4)$ second-quantized terms. To avoid this overhead, we introduce basis functions which diagonalize the periodized Coulomb operator, providing Hamiltonians for condensed phase systems with $N^2$ second-quantized terms. Using this representation we can implement single Trotter steps of the Hamiltonians with gate depth of ${\cal O}(N)$ on a planar lattice of qubits -- a quartic improvement over prior methods. Special properties of our basis allow us to apply Trotter based simulations with planar circuit depth in $\widetilde{\cal O}(N^{7/2} / \epsilon^{1/2})$ and Taylor series methods with circuit size $\widetilde{\cal O}(N^{11/3})$, where $\epsilon$ is target precision. Variational algorithms also require significantly fewer measurements to find the mean energy using our representation, ameliorating a primary challenge of that approach. We conclude with a proposal to simulate the uniform electron gas (jellium) using a linear depth variational ansatz realizable on near-term quantum devices with planar connectivity. From these results we identify simulation of low-density jellium as an ideal first target for demonstrating quantum supremacy in electronic structure.
### Strategies for Quantum Computing Molecular Energies Using the Unitary Coupled Cluster Ansatz
Jhonathan Romero Fontalvo, Ryan Babbush, Jarrod McClean, Cornelius Hempel, Peter J. Love, Alán Aspuru-Guzik · Quantum Science and Technology, vol. 4 (2018), pp. 14008
The variational quantum eigensolver (VQE) algorithm combines the ability of quantum computers to efficiently compute expectations values with a classical optimization routine in order to approximate ground state energies of quantum systems. In this paper, we study the application of VQE to the simulation of molecular energies using the unitary coupled cluster (UCC) ansatz. We introduce new strategies to reduce the circuit depth for the implementation of UCC and improve the optimization of the wavefunction based on efficient classical approximations of the cluster amplitudes. Additionally, we propose a method to compute the energy gradient within the VQE approach. We illustrate our methodology with numerical simulations for a system of four hydrogen atoms that exhibit strong correlation and show that the cost of the circuit depth and execution time of VQE using a UCC ansatz can be reduced without introducing significant loss of accuracy in the final wavefunctions and energies.
### Barren Plateaus in Quantum Neural Network Training Landscapes
Jarrod McClean, Sergio Boixo, Vadim Smelyanskiy, Ryan Babbush, Hartmut Neven · Nature Communications, vol. 9 (2018), pp. 4812
Many experimental proposals for noisy intermediate scale quantum devices involve training a parameterized quantum circuit with a classical optimization loop. Such hybrid quantum-classical algorithms are popular for applications in quantum simulation, optimization, and machine learning. Due to its simplicity and hardware efficiency, random circuits are often proposed as initial guesses for exploring the space of quantum states. We show that the exponential dimension of Hilbert space and the gradient estimation complexity make this choice unsuitable for hybrid quantum-classical algorithms run on more than a few qubits. Specifically, we show that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is non-zero to some fixed precision is exponentially small as a function of the number of qubits. We argue that this is related to the 2-design characteristic of random circuits, and that solutions to this problem must be studied.
### Quantum Chemistry Calculations on a Trapped-Ion Quantum Simulator
Cornelius Hempel, Christine Maier, Jhonathan Romero, Jarrod McClean, Thomas Monz, Heng Shen, Petar Jurcevic, Ben Lanyon, Peter J. Love, Ryan Babbush, Alán Aspuru-Guzik, Rainer Blatt, Christian Roos · Physical Review X, vol. 8 (2018), pp. 031022
Quantum-classical hybrid algorithms are emerging as promising candidates for near-term practical applications of quantum information processors in a wide variety of fields ranging from chemistry to physics and materials science. We report on the experimental implementation of such an algorithm to solve a quantum chemistry problem, using a digital quantum simulator based on trapped ions. Specifically, we implement the variational quantum eigensolver algorithm to calculate the molecular ground-state energies of two simple molecules and experimentally demonstrate and compare different encoding methods using up to four qubits. Furthermore, we discuss the impact of measurement noise as well as mitigation strategies and indicate the potential for adaptive implementations focused on reaching chemical accuracy, which may serve as a cross-platform benchmark for multiqubit quantum simulators.
### Postponing the Orthogonality Catastrophe: Efficient State Preparation for Electronic Structure Simulations on Quantum Devices
Norman Tubman, Carlos Mejuto Zaera, Jeffrey Epstein, Diptarka Hait, Daniel Levine, William Huggins, Zhang Jiang, Jarrod Ryan McClean, Ryan Babbush, Martin Head-Gordon, K. Birgitta Whaley · arXiv:1809.05523 (2018)
Despite significant work on resource estimation for quantum simulation of electronic systems, the challenge of preparing states with sufficient ground state support has so far been largely neglected. In this work we investigate this issue in several systems of interest, including organic molecules, transition metal complexes, the uniform electron gas, Hubbard models, and quantum impurity models arising from embedding formalisms such as dynamical mean-field theory. Our approach uses a state-of-the-art classical technique for high-fidelity ground state approximation. We find that easy-to-prepare single Slater determinants such as the Hartree-Fock state often have surprisingly robust support on the ground state for many applications of interest. For the most difficult systems, single-determinant reference states may be insufficient, but low-complexity reference states may suffice. For this we introduce a method for preparation of multi-determinant states on quantum computers.
### A flexible high-performance simulator for the verification and benchmarking of quantum circuits implemented on real hardware
Benjamin Villalonga, Sergio Boixo, Bron Nelson, Christopher Henze, Eleanor Rieffel, Rupak Biswas, Salvatore Mandra · arXiv:1811.09599 (2018)
Here we present a flexible tensor network based simulator for quantum circuits on different topologies, including the Google Bristlecone QPU. Our simulator can compute both exact amplitudes, a task essential for the verification of the quantum hardware, as well as low-fidelity amplitudes to mimic Noisy Intermediate-Scale Quantum (NISQ) devices. While our simulator can be used to compute amplitudes of arbitrary quantum circuits, we focus on random quantum circuits (RQCs) [Boixo et al., Nature Physics 14] in the range of sizes expected for supremacy experiments. Our simulator enables the simulation of sampling on quantum circuits that were out of reach for previous approaches. For instance, our simulator is able to output single amplitudes with depth 1+32+1 for the full Google Bristlecone QPU in less than (f · 4200) hours on a single core, where 0 < f ≤ 1 is the target fidelity, on 2 × 20-core Intel Xeon Gold 6148 processors (Skylake). We also estimate that computing 106 amplitudes (with fidelity 0.50%) needed to sample from the full Google Bristlecone QPU with depth (1+32+1) would require about 3.5 days using the NASA Pleiades and Electra supercomputers combined. In addition, we discuss the hardness of the classical simulation of RQCs, as well as give evidence for the higher complexity in the simulation of Google’s Bristlecone topology as compared to other two-dimensional grids with the same number of qubits. Our analysis is supported by extensive simulations on NASA HPC clusters Pleiades (27th in the November 2018 TOP500 list) and Electra (33rd in the November 2018 TOP500 list). For the most computationally demanding simulation we had, namely the simulation of a 60-qubit sub-lattice of Bristlecone, the two HPC clusters combined reached a peak of 20 PFLOPS (single precision), that is 64% of their maximum achievable performance. To date, this numerical computation is the largest in terms of sustained PFLOPS and number of nodes utilized ever run on NASA HPC clusters.
### Application of Fermionic Marginal Constraints to Hybrid Quantum Algorithms
Nicholas Rubin, Jarrod McClean, Ryan Babbush · New Journal of Physics, vol. 20 (2018), pp. 053020
Many quantum algorithms, including recently proposed hybrid classical/quantum algorithms, make use of restricted tomography of the quantum state that measures the reduced density matrices, or marginals, of the full state. The most straightforward approach to this algorithmic step estimates each component of the marginal independently without making use of the algebraic and geometric structure of the marginals. Within the field of quantum chemistry, this structure is termed the fermionic $n$-representability conditions, and is supported by a vast amount of literature on both theoretical and practical results related to their approximations. In this work, we introduce these conditions in the language of quantum computation, and utilize them to develop several techniques to accelerate and improve practical applications for quantum chemistry on quantum computers. As a general result, we demonstrate how these marginals concentrate to diagonal quantities when measured on random quantum states. We also show that one can use fermionic $n$-representability conditions to reduce the total number of measurements required by more than an order of magnitude for medium sized systems in chemistry. As a practical demonstration, we simulate an efficient restoration of the physicality of energy curves for the dilation of a four qubit diatomic hydrogen system in the presence of three distinct one qubit error channels, providing evidence these techniques are useful for pre-fault tolerant quantum chemistry experiments.
### Fluctuations of Energy-Relaxation Times in Superconducting Qubits
Paul Klimov, Julian Kelly, Jimmy Chen, Matthew Neeley, Anthony Megrant, Brian Burkett, Rami Barends, Kunal Arya, Ben Chiaro, Yu Chen, Andrew Dunsworth, Austin Fowler, Brooks Foxen, Craig Michael Gidney, Marissa Giustina, Rob Graff, Trent Huang, Evan Jeffrey, Erik Lucero, Josh Mutus, Ofer Naaman, Charles Neill, Chris Quintana, Pedram Roushan, Daniel Sank, Amit Vainsencher, Jim Wenner, Ted White, Sergio Boixo, Ryan Babbush, Vadim Smelyanskiy, Hartmut Neven, John Martinis · Physical Review Letters, vol. 121 (2018), pp. 090502
Superconducting qubits are an attractive platform for quantum computing since they have demonstrated high-fidelity quantum gates and extensibility to modest system sizes. Nonetheless, an outstanding challenge is stabilizing their energy-relaxation times, which can fluctuate unpredictably in frequency and time. Here, we use qubits as spectral and temporal probes of individual two-level-system defects to provide direct evidence that they are responsible for the largest fluctuations. This research lays the foundation for stabilizing qubit performance through calibration, design and fabrication.
### Quantum Simulation of Electronic Structure with Linear Depth and Connectivity
Ian D. Kivlichan, Jarrod McClean, Nathan Wiebe, Craig Michael Gidney, Alán Aspuru-Guzik, Garnet Kin-Lic Chan, Ryan Babbush · Physical Review Letters, vol. 120 (2018), pp. 110501
As physical implementations of quantum architectures emerge, it is increasingly important to consider the cost of algorithms for practical connectivities between qubits. We show that by using an arrangement of gates that we term the fermionic swap network, we can simulate a Trotter step of the electronic structure Hamiltonian in exactly N depth and with N^2/2 two-qubit entangling gates, and prepare arbitrary Slater determinants in at most N/2 depth, all assuming only a minimal, linearly connected architecture. We conjecture that no explicit Trotter step of the electronic structure Hamiltonian is possible with fewer entangling gates, even with arbitrary connectivities. These results represent significant practical improvements on the cost of all current proposed algorithms for both variational and phase estimation based simulation of quantum chemistry.
### Physical qubit calibration on a directed acyclic graph
Hartmut Neven, John Martinis, Julian Kelly, Matthew Neeley, Peter James Joyce O'Malley · arXiv, possibly npj Quantum Information (2018)
High-fidelity control of qubits requires precisely tuned control parameters. Typically, these param- eters are found through a series of bootstrapped calibration experiments which successively acquire more accurate information about a physical qubit. However, optimal parameters are typically dif- ferent between devices and can also drift in time, which begets the need for an efficient calibration strategy. Here, we introduce a framework to understand the relationship between calibrations as a directed graph. With this approach, calibration is reduced to a graph traversal problem that is automatable and extensible.
### Quantum Supremacy Is Both Closer and Farther than It Appears
Igor Markov, Aneeqa Fatima, Sergei Isakov, Sergio Boixo · arxiv (not yet submitted) (2018)
As quantum computers improve in the number of qubits and fidelity, the question of when they surpass state-of-the-art classical computation for a well-defined computational task is attracting much attention. The leading candidate task for this quantum computational supremacy milestone entails sampling from the output distribution defined by a random quantum circuit. We perform this task on conventional computers for larger circuits than in previous results, by trading circuit fidelity for computational resources to match the fidelity of a given quantum computer. By using publicly available Google Cloud Computing, we can price such simulations and enable comparisons by total cost across multiple hardware types. We simulate approximate sampling from the output of a circuit with 7 x 8 qubits and depth 1+40+1 by producing one million bitstring probabilities with fidelity 0.5%, at an estimated cost of \$35184. The simulation costs scale linearly with fidelity, and using this scaling we estimate that extending circuit depth to 1+48+1 increases costs to one million dollars. Yet, for a quantum computer, approximate sampling would take seconds. We pay particular attention to validating simulation results. Finally, we explain why recently refined benchmarks substantially increase computation cost of leading simulators, halving the circuit depth that can be simulated within the same time.
### Encoding Electronic Spectra in Quantum Circuits with Linear T Complexity
Ryan Babbush, Craig Michael Gidney, Dominic W. Berry, Nathan Wiebe, Jarrod McClean, Alexandru Paler, Austin Fowler, Hartmut Neven · Physical Review X, vol. 8 (2018), pp. 041015
We construct quantum circuits which exactly encode the spectra of correlated electron models up to errors from rotation synthesis. By invoking these circuits as oracles within the recently introduced "qubitization" framework, one can use quantum phase estimation to sample states in the Hamiltonian eigenbasis with optimal query complexity O(lambda / epsilon) where lambda is an absolute sum of Hamiltonian coefficients and epsilon is target precision. For both the Hubbard model and electronic structure Hamiltonian in a second quantized basis diagonalizing the Coulomb operator, our circuits have T gate complexity O(N + \log (1/epsilon)) where N is number of orbitals in the basis. Compared to prior approaches, our algorithms are asymptotically more efficient in gate complexity and require fewer T gates near the classically intractable regime. Compiling to surface code fault-tolerant gates and assuming per gate error rates of one part in a thousand reveals that one can error-correct phase estimation on interesting instances of these problems beyond the current capabilities of classical methods using only a few times more qubits than would be required for magic state distillation.
### Characterizing Quantum Supremacy in Near-Term Devices
Sergio Boixo, Sergei Isakov, Vadim Smelyanskiy, Ryan Babbush, Nan Ding, Zhang Jiang, Michael J. Bremner, John Martinis, Hartmut Neven · Nature Physics, vol. 14 (2018), 595–600
A critical question for quantum computing in the near future is whether quantum devices without error correction can perform a well-defined computational task beyond the capabilities of supercomputers. Such a demonstration of what is referred to as quantum supremacy requires a reliable evaluation of the resources required to solve tasks with classical approaches. Here, we propose the task of sampling from the output distribution of random quantum circuits as a demonstration of quantum supremacy. We extend previous results in computational complexity to argue that this sampling task must take exponential time in a classical computer. We introduce cross-entropy benchmarking to obtain the experimental fidelity of complex multiqubit dynamics. This can be estimated and extrapolated to give a success metric for a quantum supremacy demonstration. We study the computational cost of relevant classical algorithms and conclude that quantum supremacy can be achieved with circuits in a two-dimensional lattice of 7 × 7 qubits and around 40 clock cycles. This requires an error rate of around 0.5% for two-qubit gates (0.05% for one-qubit gates), and it would demonstrate the basic building blocks for a fault-tolerant quantum computer
### Quantum Approach to the Unique Sink Orientation Problem
Dave Bacon · Physical Review A, vol. 96 (2017), pp. 012323
We consider quantum algorithms for the unique sink orientation problem on cubes. This problem is widely considered to be of intermediate computational complexity. This is because there is no known polynomial algorithm (classical or quantum) for the problem and yet it arises as part of a series of problems for which it being intractable would imply complexity-theoretic collapses. We give a reduction which proves that if one can efficiently evaluate the kth power of the unique sink orientation outmap, then there exists a polynomial time quantum algorithm for the unique sink orientation problem on cubes.
### Commercialize Quantum Technologies in Five Years
Masoud Mohseni, Peter Read, Hartmut Neven, Sergio Boixo, Vasil Denchev, Ryan Babbush, Austin Fowler, Vadim Smelyanskiy, John Martinis · Nature, vol. 543 (2017), 171–174
Masoud Mohseni, Peter Read, Hartmut Neven and colleagues at Google's Quantum AI Laboratory set out investment opportunities on the road to the ultimate quantum machines.
### Improved Techniques for Preparing Eigenstates of Fermionic Hamiltonians
Dominic W. Berry, Mária Kieferová, Artur Scherer, Yuval Sanders, Guang Hao Low, Nathan Wiebe, Craig Gidney, Ryan Babbush · npj Quantum Information, vol. 4 (2017), pp. 22
Modeling low energy eigenstates of fermionic systems can provide insight into chemical reactions and material properties and is one of the most anticipated applications of quantum computing. We present three techniques for reducing the cost of preparing fermionic Hamiltonian eigenstates using phase estimation. First, we report a polylogarithmic-depth quantum algorithm for antisymmetrizing the initial states required for simulation of fermions in first quantization. This is an exponential improvement over the previous state-of-the-art. Next, we show how to reduce the overhead due to repeated state preparation in phase estimation when the goal is to prepare the ground state to high precision and one has knowledge of an upper bound on the ground state energy that is less than the excited state energy (often the case in quantum chemistry). Finally, we explain how one can perform the time evolution necessary for the phase estimation based preparation of Hamiltonian eigenstates with exactly zero error by using the recently introduced qubitization procedure.
### OpenFermon: The Electronic Structure Package for Quantum Computers
Jarrod McClean, Ian D. Kivlichan, Kevin Sung, Damian Steiger, Yudong Cao, Chengyu Dai, E. Schuyler Fried, Craig Michael Gidney, Brendan Gimby, Thomas Häner, Tarini Hardikar, Vojtĕch Havlíček, Cupjin Huang, Zhang Jiang, Matthew Neeley, Thomas O'Brien, Isil Ozfidan, Jhonathan Romero, Nicholas Rubin, Nicolas Sawaya, Kanav Setia, Sukin Sim, Mark Steudtner, Wei Sun, Fang Zhang, Ryan Babbush · Quantum Science and Technology, vol. 5 (2017), pp. 034014
Quantum simulation of chemistry and materials is predicted to be a key application for both near-term and fault-tolerant quantum devices. However, at present, developing and studying algorithms for these problems can be difficult due to the prohibitive amount of domain knowledge required in both the area of chemistry and quantum algorithms. To help bridge this gap and open the field to more researchers, we have developed the OpenFermion software package (www.openfermion.org). OpenFermion is an open-source software library written largely in Python under an Apache 2.0 license, aimed at enabling the simulation of fermionic models and quantum chemistry problems on quantum hardware. Beginning with an interface to common electronic structure packages, it simplifies the translation between a molecular specification and a quantum circuit for solving or studying the electronic structure problem on a quantum computer, minimizing the amount of domain expertise required to enter the field. Moreover, the package is designed to be extensible and robust, maintaining high software standards in documentation and testing. This release paper outlines the key motivations for design choices in OpenFermion and discusses some of the basic OpenFermion functionality available for the initial release of the package, which we believe will aid the community in the development of better quantum algorithms and tools for this exciting area.
### Exponentially More Precise Quantum Simulation of Fermions in the Configuration Interaction Representation
Ryan Babbush, Dominic Berry, Yuval Sanders, Ian Kivlichan, Artur Scherer, Annie Wei, Peter Love, Alán Aspuru-Guzik · Quantum Science and Technology, vol. 3 (2017), pp. 015006
We present a quantum algorithm for the simulation of molecular systems that is asymptotically more efficient than all previous algorithms in the literature in terms of the main problem parameters. As in previous work [Babbush et al., New Journal of Physics 18, 033032 (2016)], we employ a recently developed technique for simulating Hamiltonian evolution, using a truncated Taylor series to obtain logarithmic scaling with the inverse of the desired precision. The algorithm of this paper involves simulation under an oracle for the sparse, first-quantized representation of the molecular Hamiltonian known as the configuration interaction (CI) matrix. We construct and query the CI matrix oracle to allow for on-the-fly computation of molecular integrals in a way that is exponentially more efficient than classical numerical methods. Whereas second-quantized representations of the wavefunction require O(N) qubits, where N is the number of single-particle spin-orbitals, the CI matrix representation requires O(η) qubits where η ≪ N is the number of electrons in the molecule of interest. We show that the gate count of our algorithm scales at most as O(η^2 N^3 t).
### Tunable inductive coupling of superconducting qubits in the strongly nonlinear regime
Dvir Kafri, Chris Quintana, Yu Chen, Alireza Shabani, John Martinis, Hartmut Neven · Phys. Rev. A, vol. 95 (2017), pp. 052333
For a variety of superconducting qubits, tunable interactions are achieved through mutual inductive coupling to a coupler circuit containing a nonlinear Josephson element. In this paper, we derive the general interaction mediated by such a circuit under the Born-Oppenheimer approximation. This interaction naturally decomposes into a classical part, with origin in the classical circuit equations, and a quantum part, associated with the coupler's zero-point energy. Our result is nonperturbative in the qubit-coupler coupling strengths and in the coupler nonlinearity. This can lead to significant departures from previous, linear theories for the interqubit coupling, including nonstoquastic and many-body interactions. Our analysis provides explicit and efficiently computable series for any term in the interaction Hamiltonian and can be applied to any superconducting qubit type. We conclude with a numerical investigation of our theory using a case study of two coupled flux qubits, and in particular study the regime of validity of the Born-Oppenheimer approximation.
### Chiral Ground-State Currents of Interacting Photons in a Synthetic Magnetic Field
Pedram Roushan, Charles Neill, Anthony Megrant, Yu Chen, Ryan Babbush, Rami Barends, Brooks Campbell, Zijun Chen, Ben Chiaro, Andrew Dunsworth, Austin Fowler, Evan Jeffrey, Julian Kelly, Erik Lucero, Josh Mutus, Peter O'Malley, Matthew Neeley, Chris Quintana, Daniel Sank, Amit Vainsencher, Jim Wenner, Ted White, Eliot Kapit, Hartmut Neven, John Martinis · Nature Physics, vol. 13 (2017), pp. 146-151
The intriguing many-body phases of quantum matter arise from the interplay of particle interactions, spatial symmetries, and external fields. Generating these phases in an engineered system could provide deeper insight into their nature. Using superconducting qubits, we simultaneously realize synthetic magnetic fields and strong particle interactions, which are among the essential elements for studying quantum magnetism and fractional quantum Hall phenomena. The artificial magnetic fields are synthesized by sinusoidally modulating the qubit couplings. In a closed loop formed by the three qubits, we observe the directional circulation of photons, a signature of broken time-reversal symmetry. We demonstrate strong interactions through the creation of photon vacancies, or "holes", which circulate in the opposite direction. The combination of these key elements results in chiral ground-state currents. Our work introduces an experimental platform for engineering quantum phases of strongly interacting photons.
### Bounding the Costs of Quantum Simulation of Many-Body Physics in Real Space
Ian D. Kivlichan, Nathan Wiebe, Ryan Babbush, Alán Aspuru-Guzik · Journal of Physics A: Mathematical and Theoretical, vol. 50 (2017), pp. 305301
We present a quantum algorithm for simulating the dynamics of a first-quantized Hamiltonian in real space based on the truncated Taylor series algorithm. We avoid the possibility of singularities by applying various cutoffs to the system and using a high-order finite difference approximation to the kinetic energy operator. We find that our algorithm can simulate η interacting particles using a number of calculations of the pairwise interactions that scales, for a fixed spatial grid spacing, as O(η^2), versus the O(η^5) time required by previous methods (assuming the number of orbitals is proportional to η), and scales super-polynomially better with the error tolerance than algorithms based on the Lie-Trotter-Suzuki product formula. Finally, we analyze discretization errors that arise from the spatial grid and show that under some circumstances these errors can remove the exponential speedups typically afforded by quantum simulation.
### Observation of classical-quantum crossover of 1/f flux noise and its paramagnetic temperature dependence
Chris Quintana, Yu Chen, Daniel Sank, Andre Petukhov, Ted White, Dvir Kafri, Ben Chiaro, Anthony Megrant, Rami Barends, Brooks Campbell, Zijun Chen, Andrew Dunsworth, Austin Fowler, Rob Graff, Evan Jeffrey, Julian Kelly, Erik Lucero, Josh Mutus, Matthew Neeley, Charles Neill, Peter O'Malley, Pedram Roushan, Alireza Shabani, Vadim Smelyanskiy, Amit Vainsencher, Jim Wenner, Hartmut Neven, John Martinis · Phys. Rev. Lett., vol. 118 (2017), pp. 057702
By analyzing the dissipative dynamics of a tunable gap flux qubit, we extract both sides of its two-sided environmental flux noise spectral density over a range of frequencies around 2kT/h ≈ 1GHz, allowing for the observation of a classical-quantum crossover. Below the crossover point, the symmetric noise component follows a 1/f power law that matches the magnitude of the 1/f noise near 1 Hz. The antisymmetric component displays a 1/T dependence below 100 mK, providing dynamical evidence for a paramagnetic environment. Extrapolating the two-sided spectrum predicts the linewidth and reorganization energy of incoherent resonant tunneling between flux qubit wells.
### What is the Computational Value of Finite Range Tunneling?
Vasil Denchev, Sergio Boixo, Sergei Isakov, Nan Ding, Ryan Babbush, Vadim Smelyanskiy, John Martinis, Hartmut Neven · Physical Review X, vol. 6 (2016), pp. 031015
Quantum annealing (QA) has been proposed as a quantum enhanced optimization heuristic exploiting tunneling. Here, we demonstrate how finite-range tunneling can provide considerable computational advantage. For a crafted problem designed to have tall and narrow energy barriers separating local minima, the D-Wave 2X quantum annealer achieves significant runtime advantages relative to simulated annealing (SA). For instances with 945 variables, this results in a time-to-99%-success-probability that is ~1e8 times faster than SA running on a single processor core. We also compare physical QA with the quantum Monte Carlo algorithm, an algorithm that emulates quantum tunneling on classical processors. We observe a substantial constant overhead against physical QA: D-Wave 2X again runs up to ~ 1e8 times faster than an optimized implementation of the quantum Monte Carlo algorithm on a single core. We note that there exist heuristic classical algorithms that can solve most instances of Chimera structured problems in a time scale comparable to the D-Wave 2X. However, it is well known that such solvers will become ineffective for sufficiently dense connectivity graphs. To investigate whether finite-range tunneling will also confer an advantage for problems of practical interest, we conduct numerical studies on binary optimization problems that cannot yet be represented on quantum hardware. For random instances of the number partitioning problem, we find numerically that algorithms designed to simulate QA scale better than SA. We discuss the implications of these findings for the design of next-generation quantum annealers.
### Computational multiqubit tunnelling in programmable quantum annealers
Sergio Boixo, Vadim N Smelyanskiy, Alireza Shabani, Sergei V Isakov, Mark Dykman, Vasil S Denchev, Mohammad H Amin, Anatoly Yu Smirnov, Masoud Mohseni, Hartmut Neven · Nature Communications, vol. 7 (2016)
Quantum tunnelling is a phenomenon in which a quantum state traverses energy barriers higher than the energy of the state itself. Quantum tunnelling has been hypothesized as an advantageous physical resource for optimization in quantum annealing. However, computational multiqubit tunnelling has not yet been observed, and a theory of co-tunnelling under high- and low-frequency noises is lacking. Here we show that 8-qubit tunnelling plays a computational role in a currently available programmable quantum annealer. We devise a probe for tunnelling, a computational primitive where classical paths are trapped in a false minimum. In support of the design of quantum annealers we develop a nonperturbative theory of open quantum dynamics under realistic noise characteristics. This theory accurately predicts the rate of many-body dissipative quantum tunnelling subject to the polaron effect. Furthermore, we experimentally demonstrate that quantum tunnelling outperforms thermal hopping along classical paths for problems with up to 200 qubits containing the computational primitive.
### The Theory of Variational Hybrid Quantum-Classical Algorithms
Jarrod McClean, Jonathan Romero, Ryan Babbush, Alán Aspuru-Guzik · New Journal of Physics, vol. 18 (2016), pp. 023023
Many quantum algorithms have daunting resource requirements when compared to what is available today. To address this discrepancy, a quantum-classical hybrid optimization scheme known as "the quantum variational eigensolver" was developed with the philosophy that even minimal quantum resources could be made useful when used in conjunction with classical routines. In this work we extend the general theory of this algorithm and suggest algorithmic improvements for practical implementations. Specifically, we develop a variational adiabatic ansatz and explore unitary coupled cluster where we establish a connection from second order unitary coupled cluster to universal gate sets through relaxation of exponential splitting. We introduce the concept of quantum variational error suppression that allows some errors to be suppressed naturally in this algorithm on a pre-threshold quantum device. Additionally, we analyze truncation and correlated sampling in Hamiltonian averaging as ways to reduce the cost of this procedure. Finally, we show how the use of modern derivative free optimization techniques can offer dramatic computational savings of up to three orders of magnitude over previously used optimization techniques.
### Scalable Quantum Simulation of Molecular Energies
Peter O'Malley, Ryan Babbush, Ian Kivlichan, Jonathan Romero, Jarrod McClean, Rami Barends, Julian Kelly, Pedram Roushan, Andrew Tranter, Nan Ding, Brooks Campbell, Yu Chen, Zijun Chen, Ben Chiaro, Andrew Dunsworth, Austin Fowler, Evan Jeffrey, Anthony Megrant, Josh Mutus, Charles Neil, Chris Quintana, Daniel Sank, Ted White, Jim Wenner, Amit Vainsencher, Peter Coveney, Peter Love, Hartmut Neven, Alán Aspuru-Guzik, John Martinis · Physical Review X, vol. 6 (2016), pp. 031007
We report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of molecular hydrogen using two distinct quantum algorithms. First, we experimentally execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissociation energy to within chemical accuracy of the numerically exact result. Second, we experimentally demonstrate the canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimation. We compare the experimental performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable molecules may be viable in the near future.
### Digitized Adiabatic Quantum Computing with a Superconducting Circuit
Rami Barends, Alireza Shabani, Lucas Lamata, Julian Kelly, Antonio Mezzacapo, Urtzi Las Heras, Ryan Babbush, Austin Fowler, Brooks Campbell, Yu Chen, Zijun Chen, Ben Chiaro, Andrew Dunsworth, Evan Jeffrey, Erik Lucero, Anthony Megrant, Josh Mutus, Matthew Neeley, Charles Neill, Peter O'Malley, Chris Quintana, Enrique Solano, Ted White, Jim Wenner, Amit Vainsencher, Daniel Sank, Pedram Roushan, Hartmut Neven, John Martinis · Nature, vol. 534 (2016), pp. 222-226
A major challenge in quantum computing is to solve general problems with limited physical hardware. Here, we implement digitized adiabatic quantum computing, combining the generality of the adiabatic algorithm with the universality of the digital approach, using a superconducting circuit with nine qubits. We probe the adiabatic evolutions, and quantify the success of the algorithm for random spin problems. We find that the system can approximate the solutions to both frustrated Ising problems and problems with more complex interactions, with a performance that is comparable. The presented approach is compatible with small-scale systems as well as future error-corrected quantum computers.
### Scalable in-situ qubit calibration during repetitive error detection
Austin Fowler, John Martinis, Julian Kelly, Rami Barends · PHYSICAL REVIEW A, vol. 94 (2016), pp. 032321
We present a method to optimize physical qubit parameters while error detection is running. We demonstrate how gate optimization can be parallelized in a large-scale qubit array. Additionally we show that the presented method can be used to simultaneously compensate for independent or correlated qubit parameter drifts. Our method is O(1) scalable to systems of arbitrary size, providing a path towards controlling the large numbers of qubits needed for a fault-tolerant quantum computer.
### Exponentially More Precise Quantum Simulation of Fermions in Second Quantization
Ryan Babbush, Dominic Berry, Ian Kivlichan, Annie Wei, Peter Love, Alán Aspuru-Guzik · New Journal of Physics, vol. 18 (2016), pp. 033032
We introduce novel algorithms for the quantum simulation of fermionic systems which are dramatically more efficient than those based on the Lie–Trotter–Suzuki decomposition. We present the first application of a general technique for simulating Hamiltonian evolution using a truncated Taylor series to obtain logarithmic scaling with the inverse of the desired precision. The key difficulty in applying algorithms for general sparse Hamiltonian simulation to fermionic simulation is that a query, corresponding to computation of an entry of the Hamiltonian, is costly to compute. This means that the gate complexity would be much higher than quantified by the query complexity. We solve this problem with a novel quantum algorithm for on-the-fly computation of integrals that is exponentially faster than classical sampling. While the approaches presented here are readily applicable to a wide class of fermionic models, we focus on quantum chemistry simulation in second quantization, perhaps the most studied application of Hamiltonian simulation. Our central result is an algorithm for simulating an N spin–orbital system that requires O(N^5 t) gates. This approach is exponentially faster in the inverse precision and at least cubically faster in N than all previous approaches to chemistry simulation in the literature.
### Fast quantum methods for optimization
S. Boixo, G. Ortiz, R. Somma · The European Physical Journal Special Topics, vol. 224 (2015), pp. 35
Discrete combinatorial optimization consists in finding the optimal configuration that minimizes a given discrete objective function. An interpretation of such a function as the energy of a classical system allows us to reduce the optimization problem into the preparation of a low-temperature thermal state of the system. Motivated by the quantum annealing method, we present three strategies to prepare the low-temperature state that exploit quantum mechanics in remarkable ways. We focus on implementations without uncontrolled errors induced by the environment. This allows us to rigorously prove a quantum advantage. The first strategy uses a classical-to-quantum mapping, where the equilibrium properties of a classical system in d spatial dimensions can be determined from the ground state properties of a quantum system also in d spatial dimensions. We show how such a ground state can be prepared by means of quantum annealing, including quantum adiabatic evolutions. This mapping also allows us to unveil some fundamental relations between simulated and quantum annealing. The second strategy builds upon the first one and introduces a technique called spectral gap amplification to reduce the time required to prepare the same quantum state adiabatically. If implemented on a quantum device that exploits quantum coherence, this strategy leads to a quadratic improvement in complexity over the well-known bound of the classical simulated annealing method. The third strategy is not purely adiabatic; instead, it exploits diabatic processes between the low-energy states of the corresponding quantum system. For some problems it results in an exponential speedup (in the oracle model) over the best classical algorithms.
### Quantum Simulation of Helium Hydride Cation in a Solid-State Spin Register
Ya Wang, Florian Dolde, Jacob Biamonte, Ryan Babbush, Ville Bergholm, Sen Yang, Ingmar Jakobi, Philipp Neumann, Alán Aspuru-Guzik, James Whitfield, Jörg Wrachtrup · ACS Nano, vol. 9 (2015), 7769–7774
Ab initio computation of molecular properties is one of the most promising applications of quantum computing. While this problem is widely believed to be intractable for classical computers, efficient quantum algorithms exist which have the potential to vastly accelerate research throughput in fields ranging from material science to drug discovery. Using a solid-state quantum register realized in a nitrogen-vacancy (NV) defect in diamond, we compute the bond dissociation curve of the minimal basis helium hydride cation, HeH+. Moreover, we report an energy uncertainty (given our model basis) of the order of 1e–14 hartree, which is 10 orders of magnitude below the desired chemical precision. As NV centers in diamond provide a robust and straightforward platform for quantum information processing, our work provides an important step toward a fully scalable solid-state implementation of a quantum chemistry simulator.
### Quantum Algorithms for Discrete Optimization
Sergio Boixo, Rolando Somma · Quantum Optimization: Fields Institute Communications, Springer (2015) (to appear)
### Computational complexity of time-dependent density functional theory
J D Whitfield, M-H Yung, D G Templ, S Boixo, A Aspuru-Guzik · New Journal of Physics, vol. 16 (2014), pp. 083035
Time-dependent density functional theory (TDDFT) is rapidly emerging as a premier method for solving dynamical many-body problems in physics and chemistry. The mathematical foundations of TDDFT are established through the formal existence of a fictitious non-interacting system (known as the Kohn–Sham system), which can reproduce the one-electron reduced probability density of the actual system. We build upon these works and show that on the interior of the domain of existence, the Kohn–Sham system can be efficiently obtained given the time-dependent density. We introduce a V-representability parameter which diverges at the boundary of the existence domain and serves to quantify the numerical difficulty of constructing the Kohn–Sham potential. For bounded values of V-representability, we present a polynomial time quantum algorithm to generate the time-dependent Kohn–Sham potential with controllable error bounds.
### Defining and detecting quantum speedup
Troels F. Rønnow, Zhihui Wang, Joshua Job, Sergio Boixo, Sergei V. Isakov, David Wecker, John M. Martinis, Daniel A. Lidar, Matthias Troyer · Science, vol. 345 (2014), pp. 420-424
The development of small-scale quantum devices raises the question of how to fairly assess and detect quantum speedup. Here, we show how to define and measure quantum speedup and how to avoid pitfalls that might mask or fake such a speedup. We illustrate our discussion with data from tests run on a D-Wave Two device with up to 503 qubits. By using random spin glass instances as a benchmark, we found no evidence of quantum speedup when the entire data set is considered and obtained inconclusive results when comparing subsets of instances on an instance-by-instance basis. Our results do not rule out the possibility of speedup for other classes of problems and illustrate the subtle nature of the quantum speedup question. How to benchmark a quantum computer: Quantum machines offer the possibility of performing certain computations much faster than their classical counterparts. However, how to define and measure quantum speedup is a topic of debate. Rønnow et al. describe methods for fairly evaluating the difference in computational power between classical and quantum processors. They define various types of quantum speedup and consider quantum processors that are designed to solve a specific class of problems. Science, this issue p. 420
### Construction of Non-Convex Polynomial Loss Functions for Training a Binary Classifier with Quantum Annealing
Ryan Babbush, Vasil Denchev, Nan Ding, Sergei Isakov, Hartmut Neven · arXiv:1406.4203 (2014)
Quantum annealing is a heuristic quantum algorithm which exploits quantum resources to minimize an objective function embedded as the energy levels of a programmable physical system. To take advantage of a potential quantum advantage, one needs to be able to map the problem of interest to the native hardware with reasonably low overhead. Because experimental considerations constrain our objective function to take the form of a low degree PUBO (polynomial unconstrained binary optimization), we employ non-convex loss functions which are polynomial functions of the margin. We show that these loss functions are robust to label noise and provide a clear advantage over convex methods. These loss functions may also be useful for classical approaches as they compile to regularized risk expressions which can be evaluated in constant time with respect to the number of training examples.
### Hearing the Shape of the Ising Model with a Programmable Superconducting-Flux Annealer
Walter Vinci, Klas Markström, Sergio Boixo, Aidan Roy, Federico M. Spedalieri, Paul A. Warburton, Simone Severini · Scientific Reports, vol. 4 (2014)
Two objects can be distinguished if they have different measurable properties. Thus, distinguishability depends on the Physics of the objects. In considering graphs, we revisit the Ising model as a framework to define physically meaningful spectral invariants. In this context, we introduce a family of refinements of the classical spectrum and consider the quantum partition function. We demonstrate that the energy spectrum of the quantum Ising Hamiltonian is a stronger invariant than the classical one without refinements. For the purpose of implementing the related physical systems, we perform experiments on a programmable annealer with superconducting flux technology. Departing from the paradigm of adiabatic computation, we take advantage of a noisy evolution of the device to generate statistics of low energy states. The graphs considered in the experiments have the same classical partition functions, but different quantum spectra. The data obtained from the annealer distinguish non-isomorphic graphs via information contained in the classical refinements of the functions but not via the differences in the quantum spectra.
### Quantum Algorithms for Simulated Annealing
Sergio Boixo, Rolando Somma · Encyclopedia of Algorithms, Springer (2014) (to appear)
### Entanglement in a Quantum Annealing Processor
T. Lanting, A. J. Przybysz, A. Yu. Smirnov, F. M. Spedalieri, M. H. Amin, A. J. Berkley, R. Harris, F. Altomare, S. Boixo, P. Bunyk, N. Dickson, C. Enderud, J. P. Hilton, E. Hoskinson, M. W. Johnson, E. Ladizinsky, N. Ladizinsky, R. Neufeld, T. Oh, I. Perminov, C. Rich, M. C. Thom, E. Tolkacheva, S. Uchaikin, A. B. Wilson, G. Rose · Physical Review X, vol. 4 (2014), pp. 021041
Entanglement lies at the core of quantum algorithms designed to solve problems that are intractable by classical approaches. One such algorithm, quantum annealing (QA), provides a promising path to a practical quantum processor. We have built a series of architecturally scalable QA processors consisting of networks of manufactured interacting spins (qubits). Here, we use qubit tunneling spectroscopy to measure the energy eigenspectrum of two- and eight-qubit systems within one such processor, demonstrating quantum coherence in these systems. We present experimental evidence that, during a critical portion of QA, the qubits become entangled and entanglement persists even as these systems reach equilibrium with a thermal environment. Our results provide an encouraging sign that QA is a viable technology for large-scale quantum computing.
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1. ## Edge Side Orbits
On the sheet, construct the edge side orbits under the action of the full symmetry of the group T_1. Then find the IH number
Now I thought had done this correct but when it came to find the IH number after finding the incidence number I was left with seven symbols and no IH number has seven symbols.
Can someone give me a clue as to where I have gone wrong.
Thanx
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## Introduction
Competing electronic phases underlie a number of unusual physical phenomena in condensed matter1,2,3. From superconductivity up to ferromagnetism, when the competition is sizeable a common outcome is phase separation. Compounds that have shown such behaviour are mostly of complex magnetic structures including cuprates1, iron-based superconductors3, ruthenates2, topological kagome magnets4, and manganites5,6. A contrasting case is found in the layered transition metals7,8,9,10,11 where the presence of heavy halide atoms, like in CrI3, stabilises pronounced anisotropy constants resulting in what appears a homogeneous ferromagnetic phase without any separation12,13. Nevertheless, recent experiments14,15,16,17 have unveiled the presence of many subtleties in the magnetism of this compound which a single magnetic transition and a structural phase fail to capture.
Firstly, the magnetic properties of CrI3 depend sensitively on the system structure. It is now understood that whereas bulk CrI3 is ferromagnetic (FM) at 61 K18 with presumably a rhombohedral stacking14,15,19, thin layers can exhibit antiferromagnetic (AFM) coupling at 45 K in monoclinic14,15. However, the monoclinic phase is only observed in bulk at high temperatures. The origin of this puzzling behaviour has not been reconciled since the birth of the field of 2D vdW magnets20,21. Secondly, multiple anomalies can be observed in the temperature dependence of the magnetic susceptibility of bulk CrI3 below 61 K16,17,19. Such anomalies imply that a more complex magnetic ordering involving spins not directly aligned with the easy-axis is likely emerging. Moreover, recent Raman measurements22 appear contradictory with the appearance of a rhombohedral phase for thin layers in both FM and AFM ordering. This is followed by an anomalous phonon mechanism with the deviation of the linewidths as the temperature decreases22. Whether different magnetic phases may exist or competition occurs between structural phases is largely unknown. Here we systematically study the evolution of the magnetic and crystal structures of CrI3 under different temperatures through a synergy of compelling techniques. Such approach has resulted being instrumental to identify, characterize and understand the distinct macroscopic ground states observed in this vdW material with competing magnetic and structural phases.
## Results
### Microscopic details of different magnetic phases: μSR experiments
In a μSR experiment, positive muons implanted into a sample serve as extremely sensitive local microscopic probes to detect small internal magnetic fields and ordered magnetic volume fractions in the bulk of magnetic systems. See details on Supplementary Notes 1-2. Zero-field μSR time-spectra are recorded in a powder sample of CrI3 below (5 K, 30 K, 54 K, and 60 K) and above (65 K and 80 K) the magnetic ordering temperature (Fig. 1a, b). A paramagnetic state is generally characterised by a small Gaussian Kubo-Toyabe depolarization of the muon spin originating from the interaction with randomly oriented nuclear magnetic moments. Conversely, the spectra from 150 K down to 62 K, exhibit a relatively high transverse depolarization rate λT 4.9(2) μs−1. This reflects the occurrence of dense electronic Cr moments and indicates strong interactions between them. In this scenario a novel correlated paramagnetic state may be present in the system at temperatures above the actual Curie temperature.
As the crystal is cooled down, in addition to the paramagnetic signal, an oscillating component with a single well-defined frequency is observed at T 61 K (Fig. 1a, b). Below 50 K, a spontaneous muon spin precession with two well-separated distinct precession frequencies is observed in the μSR spectra and persists down to 5 K. The temperature dependences of the internal fields (μ0Hμ = ω/$${\gamma }_{\mu }^{-1}$$) for the two components are shown in Fig. 2a. The low frequency component shows a monotonic decrease and disappears at TC2 = 50 K. The high frequency component decreases down to 50 K, above which it keeps a constant value within a few Kelvin’s range and then decreases again to disappear at TC1 = 61 K. Thus, the two oscillatory components have clearly different transition temperatures. This implies the presence of two distinct magnetic transitions in CrI3. We also notice that an upturn on both μ0Hμ,1 and μ0Hμ,2 is seen below TC3 = 25 K. Moreover, a strongly damped component appears below TC3 which is seen as some lost of initial asymmetry of the zero field μSR signal. This suggests the presence of another magnetic transition at this temperature. The temperature dependences of the relative weights of the individual components in the total μSR signal are shown in Fig. 2b. The weight of the high frequency component (component I) ω1 gradually increases below TC1 and reaches maximum at TC2, below which the second frequency appears. The third component raises below TC3 = 25 K. The components I and II share the weight of 30 − 70% in the temperature range between 30 K and 50 K. These results portray a clear coexistence of magnetically ordered phases in the temperature domain.
Figure 2 c, d shows the temperature dependences of the transverse λT and the longitudinal λL depolarisation rates, respectively, of components I and II. The λT is a measure of the width of the static magnetic field distribution at the muon site, and also reflects dynamical effects (spin fluctuations). The λL is determined by dynamic magnetic fluctuations only. For both components, λT is higher than λL in the whole temperature range, indicating that magnetism is mostly static in origin. However, λL1 has a higher overall value than λL2, implying that the magnetic order with TC1 = 61 K contains more dynamics. The presence of three transitions are clearly substantiated by the anomalies, seen in λT and λL (Fig. 2c, d). Namely, the λT1 starts to increase below TC1 and peaks at TC2, then decreases and tends to saturate. Nevertheless, it increments again below TC3. λT2 also exhibits an increase below TC3. Similarly, λL1 goes to high values for T < TC1, saturates at T < TC2 and then enlarges again for T < TC3, followed by a peak at lower temperature.
We note that it is not possible to discriminate in the analysis the contribution of strongly damped components and a high frequency component into λL1 for T < 30 K and thus its peak at low temperatures could be due to the contribution from feature III. The increase of the dynamic longitudinal muon spin depolarization rate for T < 30 K, accompanied by a peak at lower temperatures, is a signature of a slowing down of magnetic fluctuations. These results imply that magnetic transitions at TC1, TC2 and TC3 are influencing each other and they are strongly coupled, in spite of the fact that they are phase separated. Even though temperature is the main driving force to the appearance of these three phase transitions, their origin as well as their influence on the underlying magnetic properties of CrI3 are still open questions to be further investigated. These findings point to a unconventional thermal evolution of the magnetic states in a 2D vdW magnet.
### Macroscopic magnetic properties: SQUID magnetometry
To support the picture of multiple magnetic phases in CrI3, we carried out SQUID magnetometry measurements on polycrystalline and single crystal samples (Fig. 3). See Supplementary Note 3 for details. The magnetisation was measured at zero-field- (ZFC) and field-cooled (FC) conditions where the sample was cooled down to the base temperature in a weak external field and magnetization recorded upon warming. The most prominent anomaly in the thermal variation of the magnetic susceptibility onsets at 61 K as shown by DC measurements reflected in Fig. 3a, b. However, we also find signatures of additional magnetic transitions with distinct characteristics under different orientations of the magnetic field. Remarkably, there is a shoulder in both the FC and ZFC traces at around 50 K, which shows up very prominently in the in-plane magnetisation of the crystal (Fig. 3b) and much more subtly in the out-of-plane orientation (Fig. 3a).
The real part of the AC measurements (Fig. 3c, d) shows that the 61 K transition is independent of the AC drive frequency. Interestingly, there is no peak in the imaginary component of the AC magnetization in both orientations. This transition has so far been treated as a ferromagnetic long-range order phase transition19. However, the insensitivity of the imaginary component questions the type of magnetic order on the system. The second feature at 50 K is particularly visible in the thermal dependence of the in-plane AC magnetic moment of the crystal (Fig. 3d). This can be attributed to the second magnetic phase transition at TC2. The relative height of this feature with respect to the main 61 K transition (Fig. 3d) is maximum at lowest fields. As with the main anomaly, this peak also exhibits no frequency dependence, which indicates that this phase transition is of long-range order nature. The DC magnetisation for the in-plane orientation at low temperature is non-zero, implying that some component of the magnetization exists in the crystallographic ab-plane. For both field orientations, the hysteresis is nearly zero from 61 K down to 50 K, whereas it suddenly increases below 50 K. This observation remarks the notion of a strong magnetic anisotropy along the c-axis for CrI3, and reveals the presence of some in-plane magnetic moment.
Moreover, the imaginary component of the AC magnetisation for the in-plane orientation in Fig. 3d exhibits a slight increment below ~ 30 K with a reduction of the real component. The fact that the most significant effect across ~ 30 K was seen in the imaginary part of the AC susceptibility indicates that the transition at this temperature is related to the slow in-plane magnetic fluctuations. These results lay the foundation of three different temperature phase domains, which are consistent with the μSR results and can be considered as an independent piece of evidence for the presence of multi magnetic phases in CrI3.
### Coexistence of structural phases: Temperature-dependent synchrotron X-ray diffraction
The behaviour observed on the critical temperatures involves a volume-wise interplay between various magnetic states, providing an important constraint on theoretical models. One possible interpretation of the data is that below TC1 there is an evolution of the magnetic order in specific volumes of the crystal, which coexists with a correlated paramagnetic state. This interpretation is supported by the temperature dependent measurements of the total magnetic fraction Vm (Fig. 4a). The magnetic fraction Vm does not acquire the full volume below TC1 = 61 K. Instead, it gradually increases below TC1 and reaches ~ 80% at TC2 = 50 K. An additional increase of Vm by 10 − 15% takes place below TC3 = 25 K, at which the third strongly damped component appears and reaches nearly 100%. The magnetism below TC3 does not give extra coherent precession but it causes the strong depolarization of the μSR signal, reflected in the lost of the initial asymmetry. This indicates that 10 − 15% volume is characterised by highly disordered magnetic state.
The volume-wise evolution (Fig. 4a) of the magnetic order across TC1, TC2, and TC3 in CrI3 strongly suggests the presence of distinct magnetic states in separate volumes of the system. We quantify this via the volume fraction VP of the sample obtained from Rietveld refinement of the synchrotron X-Ray powder diffraction data (Fig. 4b). See details in Supplementary Note 4. We observe that the material is composed of a mixture of rhombohedral (R) and monoclinic (M) phases19 on a broad thermal range. The quantification of VP confirmed the absence of a single-phase structural scenario below the first-order crystallographic transition temperature happening in our system at around 150 K. The high-temperature monoclinic phase (C2/m) is not entirely substituted by the low-temperature rhombohedral structure (R$$\overline{3}$$). Indeed, the two-phase coexistence region is not restricted to the narrow interval previously proposed19. We found evidence of the persistence of a residual volume of the sample in the monoclinic phase in all measured temperature range, down to our base temperature (10 K) (Fig. 4c-f). Most interestingly, some peak widths and intensities show significant discrepancies with that expected from the original structural dichotomic model19. For instance, the intensity of the monoclinic peak (130)M (Fig. 4d) is barely affected by the temperature over the whole spectrum. Conversely, other peaks such as (-131)M and (002)M (Fig. 4c), and (400)M and (-262)M (Fig. 4e), are gradually suppressed at lower temperatures but do not disappear completely. It is worthwhile highlighting their Lorentzian-shape with anomalously wide half-width indicating that the monoclinic phase persists down to low temperatures in the form of short-ranged domains not much bigger than hundreds of Angstroms in size. We observed that the two coexisting structural phases share essentially the same value for the a-parameter, i.e., a = 6.844 Å at 80 K in both monoclinic and rhombohedral, suggesting intergrowth and a composite-like microstructure. Still, the goodness of fit parameters of the Rietveld refinement at low temperatures (below 150 K) portrays significant discrepancies with the original structural models, precluding the extraction of more quantitative conclusions regarding the high-to-low temperature phase conversion and urging for a more in-depth revision of the crystal structure resolution. The origin of the continuous variation of the fractions of monoclinic and rhombohedral (Fig. 4a) over a broad range of temperature is rather unclear and calls for additional works on synthesis, characterisation and crystal phase modelling.
Interestingly, the most compelling temperature-dependent structural evolution concerns the rhombohedral reflections which describe characteristic temperature-dependent features that are closely correlated with the magnetic critical temperatures observed (Fig. 4g–j). Firstly, the evolution of some structural reflections suggests a sizeable increase of intensity near ~ 50 K, such as in (223)R (Fig. 4h). Secondly, a significant number of rhombohedral peaks (though not all) display a maximum of intensity at ~ 25 K and then an intensity drop is clearly observed below that temperature (Fig. 4g, i, j). These relative intensity changes depict significant structural changes in the crystal, and correlate well with the magnetostructural effects in CrI3 around the magnetic transitions TC1, TC2 and TC3. Additional discussions are included in Supplementary Note 5 on several other reflection peaks at different temperatures which further support the coupling between structural phases and magnetism. It is worth mentioning that we have discarded the presence of concomitant phases by chemical analysis and by the instability of the refinement process in the presence of chromium oxides, hydrates and diiodide phases.
## Discussion
We would like to emphasise that while TC1 has been previously observed19, with some unclear evidences for TC217, TC3 has never been noticed with macroscopic probes. As mentioned above, we relate the transition across TC3 to the slowing down of the spin fluctuations. The magnetic moments fluctuate at a rate which is slower than the nearly instantaneous time scale of other techniques, e.g., neutron scattering23. Thus, neutron scattering and specific heat would hardly be sensitive to TC3. μSR as a local probe and sensitive to small ordered fraction and slow fluctuations, combined with AC susceptibility, uncovers the novel TC3 transition. Furthermore, the full width at half maximum (FWHM) of the (1,1,0) peak measured in neutron diffraction on bulk CrI323, shows a smooth increase below 60 K with sudden variations at 50 K and within the range of ~ 32 − 25 K. This is similar to the temperature dependence of the magnetic fraction VM (Fig. 4a) extracted from μSR measurements. As a volume-integrating probe in reciprocal space, neutron scattering techniques are sensitive to both the ordered moment and its volume fraction, but these two contributions cannot be separated from the measured scattered intensity. Hence, some additional transitions bellow TC2 can be missed in the temperature dependence of the neutron scattered intensity. In particular, when two magnetic states are in microscopic proximity with each other (phase separation). In μSR however we can separately measure the internal field and the ordered volume fraction. The local probe features of μSR make this technique an excellent complementary approach to neutron diffraction and magnetization measurements.
The strong interplay involving distinct structural phases and competing magnetic orders found in CrI3 raised several implications on the understanding of past and ongoing investigations on this material. In light of our findings, it is not surprising that thin layer CrI3 assumed a monoclinic structure and consequently an AFM ordering12,14,15 as that is one of the phases stabilised in bulk. Indeed, when bulk CrI3 is exfoliated in a glove-box at room-temperature, monoclinic is the phase present in the structure. This phase does not change to rhombohedral as several groups had observed in different samples, devices, and conditions15,17,20,24,25,26. The coexistence of rhombohedral and monoclinic in bulk CrI3 may suggest that both FM and AFM couplings are present over the entire crystal with no preference whether layers are more exposed to the surface or internal to the system24,27. The mixing of both structural phases is also a strong asset for breaking the inversion symmetry in centrosymmetric materials28 and consequently the appearance of chiral interactions (i.e., Dzyaloshinskii-Moriya)23,29,30. In systems where the competition between single-ion anisotropy and dipolar-filed is not substantial, geometrical faults may contribute to the appearance of topologically non-trivial spin textures. Furthermore, the observation of the coexistence of two structures in bulk CrI3 provides an interesting framework for further theoretical and experimental investigations. Such as in terms of crystal prediction at different temperatures and phase-mixing (e.g., random structure search), stacking order organisation at low energy cost, and spin-lattice mechanisms for unknown magnetic phases.
## Methods
### CrI3 bulk crystal growth
Chromium triiodide crystals were grown using the chemical vapour transport technique. Chromium powder (99.5% Sigma-Aldrich) and anhydrous iodine beads (99.999% Sigma-Aldrich) were mixed in a 1:3 ratio in an argon atmosphere inside a glovebox. 972 mg of the mixture were loaded into a silica ampoule with a length, inner diameter and outer diameter of 500 mm, 15 mm and 16 mm respectively. Additional details in Supplementary Notes 1.
### μSR experiment and analysis
The μSR method is based on the observation of the time evolution of the spin polarization $$\overrightarrow{P}$$(t) of the muon ensemble. In μSR experiments an intense beam (pμ= 29 MeV/c) of 100 % spin-polarized muons is stopped in the sample. Currently available instruments allow essentially a background free μSR measurement at ambient conditions31. Additional details in Supplementary Notes 2.
### SQUID magnetometry
Magnetization curves and zero-field-cooled/field-cooled susceptibility sweeps were carried out in a SQUID magnetomoter (Quantum Design MPMS-XL-7) on single crystals of CrI3 were the relative orientation of the basal plane of the sample with the external magnetic field (both AC and DC) is controlled. Additional details in Supplementary Notes 3.
### Synchrotron X-ray diffraction measurements
Synchrotron X-ray powder diffraction (SXRPD) measurements were performed at the the new Material Science Powder Diffraction beamline at ALBA Synchrotron. Powder Diffr., 28, S360–S370 (2013), proposal no. 2021014858. (Barcelona, Spain) using the multi-crystal analyser MAD detector system. Additional details in Supplementary Notes 4.
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Fair coin toss probability asymptotics [duplicate]
Possible Duplicate:
Asymptotics for a partial sum of binomial coefficients
A fair coin is tossed $n$ times, let $A_n$ be the number of heads and $B_n$ the number of tails and $C_n = A_n - B_n$.
Prove that for every $0<a<1$ holds
$\left(\mathbb{P}\left(C_n \geq an \right)\right)^{1/n} \to \frac{1}{\sqrt{(1+a)^{1+a}(1-a)^{1-a}}}$
as $n \to \infty$.
I would be grateful for any ideas or hints. Thanks.
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### This is embarrasing...
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### This is embarrasing...
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# ISO 286-1
## Geometrical product specifications (GPS) - ISO code system for tolerances on linear sizes - Part 1: Basis of tolerances, deviations and fits TECHNICAL CORRIGENDUM 1
active, Most Current
Organization: ISO Publication Date: 1 August 2013 Status: active Page Count: 4 ICS Code (Limits and fits): 17.040.10
### Document History
ISO 286-1
August 1, 2013
Geometrical product specifications (GPS) - ISO code system for tolerances on linear sizes - Part 1: Basis of tolerances, deviations and fits TECHNICAL CORRIGENDUM 1
A description is not available for this item.
April 15, 2010
Geometrical product specifications (GPS) - ISO code system for tolerances on linear sizes - Part 1: Basis of tolerances, deviations and fits
This part of ISO 286 establishes the ISO code system for tolerances to be used for linear sizes of features of the following types: a) cylinder; b) two parallel opposite surfaces. It defines the...
January 1, 1988
ISO System of Limits and Fits - Part 1: Bases of Tolerances, Deviations and Fits
This part of IS0 286 gives the bases of the IS0 system of limits and fits together with the calculated values of the standard tolerances and fundamental deviations. These values shall be taken as...
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# Home
#### 5/7: Divergence Theorem
Slides: Divergence and the divergence theorem (sec 17.3)
Course grades on Blackboard are now up to date. In addition, I’ve added columns for clicker totals (with lowest three scores dropped), and for homework and exams entered as percentages rather than scores. You can find your current WebAssign score (with the lowest two scores to date dropped) on WebAssign. The rubric for computing the course grade is given on the course syllabus. Please contact me before the final exam if you think there is a mistake in any of your posted scores.
#### 5/4: Green’s & Stokes’ Theorems
Slides: Green’s & Stokes’ Theorems (sec 17.1, 17.2)
No more problems will be added to HW #5. Problem 2 can be simplified (or complicated) by using trig identities, so there is probably more than one way of doing it. Here’s what worked for me. There may be slicker ways of setting it up; there are certainly messier ways. I would say that the “moral” of this problem is: you can use polar coordinates $x = r\cos\theta$, $y = r\sin\theta$ to parameterize a curve in terms of the angle $\theta$, if you can write your radius as a function of $\theta$.
#### 5/2: Vector Surface Integrals (a.k.a. Flux Integrals)
Slides: Vector Surface Integrals (a.k.a. Flux Integrals) (sec 16.5)
I’ve posted notes on one way to approach HW #5 problem 9. This approach does not use parameterization or change-of-coordinates until after the dot product $\vec{E}\, \cdot \, d\vec{S}$ is computed. If you prefer to try the problem by parameterizing first, then computing the dot product, by all means try it.
I will have information about the final and course grades so far posted early next week. For now, note that the final is almost entirely about the material in chapters 16 & 17 (although I reserve the right to ask basic questions from earlier in the semester — for example, computing dot and cross products, tangent vectors, etc).
Also, the scores posted on Blackboard are not weighted. They are only there so you can make sure I have the correct scores entered in my spreadsheet.
#### 4/30: Surface Elements & Scalar Surface Integrals
Slides: Surface elements and scalar surface integrals (sec 16.4)
#### 4/28: Conservative Vector Fields
Slides: Conservative vector fields (sec 16.3)
Since we didn’t get to the examples of using the curl test, I’ve posted notes on the use of the curl test (there’s also now a link to these notes on the final slide).
Anyone interested on the necessity of the condition that the domain be simply connected in order to apply the curl test: look at example 8 and the following “conceptual insight” on pp 970–971.
#### 4/25: Vector Line Integrals
Slides: Vector line integrals (sec 16.2)
Announcements:
• It was brought to my attention at the end of class that there is a “point of view” consideration when using the work interpretation of a line integral. The line integral $\int_C \vec{F}\, \cdot \, d\vec{s}$ represents the work done by the field $\vec{F}$. The work done against the field $\vec{F}$ is $-\int_C \vec{F} \, \cdot \, d\vec{s}$: note the opposite sign. I suspect that many people taking physics or engineering classes will tend to think about work done against a field rather than work done by the field; if so, be aware of the difference.
• Follow-up on the $c(t)$ vs $\mathbf{c}(t)$ notation for parameterized curves: I checked the textbook (sec 14.5 p820). The parameterization is defined to be a “moving point”, and uses curved brackets when writing it out in component form; but it is denoted in boldface $\mathbf{c}(t)$, as though it were a vector. (IMHO, this is a confusing choice on the part of the author/editor — but they may have their own justification for why it makes sense.)
• Also, a repeat announcement: If you are interested in being a Learning Assistant or Peer Leader, check out the Academic Advising and Enhancement website.
#### 2/23: Scalar Line Integrals
Slides: Scalar line integrals, and the scalar and vector line elements $ds$ and $d\vec{s}$ (sec 16.2)
If you are interested in being a Learning Assistant or Peer Leader, check out the Academic Advising and Enhancement website.
#### 4/21: Introduction to Vector Fields
Slides: Introduction to vector fields (sec 16.1)
We’re in the final stretch. Topic 5 (Vector Calculus) is the final topic we study, and utilizes many of the techniques and concepts previously covered in this course. WebAssign #25 provides a review of material we will be drawing on from chapters 12 through 14. We will also be using multiple integration and polar, cylindrical, and spherical coordinates.
#### 4/16: Solutions
• HW #4, problem #1 This problem is no longer counted towards the assignment, but I’m including the solution for those of you who have worked on it. The answer and solution that WebAssign will display after the assignment is due, is incorrect.
• Worksheet II Cylindrical and spherical coordinate solutions.
#### 4/15: Exam 4 Information Posted
Information regarding exam 4 has been posted on the “exams” page of the course website. Please let me know if you have any questions about what has been posted. Tomorrow (Wed 4/16) is review. Come prepared with questions, including any questions you may have about WebAssign and/or homework problems; problems from the worksheets from class on Friday 4/11 and Monday 4/14; and/or the table of coordinate functions and basic curves and surfaces in polar, cylindrical, and spherical coordinates. Links to the worksheets and the table can be found on “exams” page.
#### 4/9: Polar, Cylindrical, and Spherical Integrals
Slides: Integrating in polar, cylindrical, and spherical coordinates (sec 15.4).
Announcements:
#### 4/8: Final Exam Time Change
I must have misread the final exam schedule at the Registrar’s office when originally scheduling the exam. In order to avoid conflicts and align with the schedule posted at the Registrar’s office, the time of the final exam for Math 275, sec 002, has been changed to 9:30–11:30am on Monday 12 May.
#### 4/7: Introduction to Cylindrical & Spherical Coordinates
Slides: Introduction to cylindrical and spherical coordinates (sec 12.7, 15.4)
Including the volume element $dV$ in cylindrical and spherical coordinates!
#### 4/4: Polar & Cylindrical Coordinates
Slides: Examples of finding limits of integration for triple integrals in Cartesian coordinates (sec 15.3); Introduction to polar and cylindrical coordinates (sec 11.3, 15.4)
(We will begin with cylindrical coordinates on Monday.)
#### 4/2: Applications of Double Integrals; Triple Integrals
Slides: Some applications of double integrals (sec 15.2, 15.5), and triple integrals (sec 15.3)
#### 3/31: Double Integrals in Cartesian Coordinates
Slides: Double integrals in Cartesian (a.k.a. rectangular) coordinates (sec 15.1, 15.2)
#### 3/17: Lagrange Multipliers
Slides: Optimization subject to constraint — the method of Lagrange multipliers
I’ve added a few notes to the end of the last slide (the example of the rectangle inscribed in the ellipse).
#### 3/14: Local Extrema, Saddle Points, and the Second Derivative Test
Slides: Local maxima, local minima, saddle points, and the second derivative test for functions of two variables (sec 14.7).
Here’s a link to a free graphing app for the iPhone/iPad/iPod touch (requires iOS 5.1 or later). I haven’t tried it, but it’s been recommended by a member of our class, and it does look pretty cool.
#### 3/12: Directional Derivatives & Chain Rules
Slides: Directional derivatives & chain rules (sec 14.5, 14.6)
We did not get to the last three slides, but I am posting them anyway. The third-to-last slide is an example of an “applied” problem that uses the chain rule along paths. It uses the chain rule to find how distance from a fixed point is changing with respect to time as one travels along a parameterized curve. You can try it as an exercise: I’ll post a solution separately soon. Solutions to the ice-skating ant and partial derivatives slides.
The last two slides deal with the general form of the chain rule. For the purposes of this class, the general chain rule will be dealt with purely as a matter of mechanics. The “written homework assignment” for section 14.6 is that you read the beginning of section 14.6 (pp 825–828; up to but not including “Implicit Differentiation”), and complete the partial derivative problems on WebAssign #16 (problems #12–14). This material is fair game for exam 3.
#### 3/10: the Differential, the Gradient, and the Directional Derivatives
Slides: The differential, the gradient, and directional derivatives (secs 14.4, 14.5)
We did not get to directional derivatives today. That’s where we will begin class on Wednesday 3/12. Problems on WebAssign #15 involving the directional derivative have been set to zero (they will appear on the next assignment), but make sure you do the problems involving (only) the gradient.
A note on the exam: if you were marked down a point in problem #5 for using “dt” instead of a dummy variable in the integral, bring
#### 3/5: Differentiability, Tangent Planes, Local Linearizations, and Differentials
Slides: Differentiability, Tangent Planes, Linear Approximations, & the Function Differential $df$ (sec 14.4)
We didn’t make it to differentials today: that’s where we’ll start on Monday. The slides have been updated to reflect the material covered.
#### 3/5: Partial Derivatives
Slides: Partial Derivatives (sec 14.3)
(The link is now fixed.)
Here’s another pc graphing app, recommended by a classmate: Microsoft Mathematics. As always, take appropriate precautions when downloading software from the internet.
Announcement: I’ve had to cancel my office hours today. Apologies to anyone who wanted to meet.
#### 3/3: Introduction to Multivariate Functions
Slides: Introduction to Functions of More Than One Variable (sec 15.1)
Summary: Introduction to functions of two and three (independent) variables; domain & range; graphs and level curves of functions of two variables.
There are links on the course Links page to surface and level curve graphers, and to a pc graphing application (disclaimer: I have not used this application. As with anything on the internet, use caution when downloading). If you use a mac, you should have an app called “grapher” in your utilities folder (this is the grapher I use in class). Also, a fun link to online topographic maps.
#### 2/24: Motion Along Curves
Slides: The $\{\hat{T},\hat{N},\hat{B}\}$ frame, and the orthogonal decomposition of acceleration (sec 13.4, 13.5)
Summary: The unit vectors $\hat{T}(t)$, $\hat{N}(t)$, and $\hat{B}(t)$ form a frame of reference for an object moving along a curve. The orthogonal decomposition of acceleration with respect to velocity can be written in terms of the vectors $\hat{T}(t)$ and $\hat{N}(t)$, where the tangential and normal components depend only on speed and curvature.
Announcements:
• Exam 2 is this Friday (2/28) in class. Information, including a topic summary and a first draft of the equation sheet, is posted on the “Exams” page.
• Recommendation: have the equation sheet, the derivatives/integrals sheet, and a scientific calculator handy while you’re studying for the exam.
• Wednesday will be review. I’ll stay as long as people have questions.
• WebAssign assignment #11 is longer than usual — many of the problems are computationally intensive. It will be due on Friday before the exam. Please read the instructions carefully — it will save you a substantial amount of time (and will very likely help on the exam). You can expect at least one (if not more) question on tangential and normal components of acceleration on the exam. I really like all of the acceleration decompositions problems a lot. You will need to understand how the components relate to speed and curvature. You should take this as a hint.
• Good on-line curve graphers can be found here (2d), here (3d), and here (3d with tangent and normal vectors). Macs come with a pretty good graphing utility called “grapher” (I’m not sure about PC’s — any suggestions?).
#### 2/21: Unit Tangent Vector and Curvature
Slides: The arc length function (sec 13.3, continued), the unit tangent vector, & curvature (sec 13.4)
Summary: Using the arc length function to find distance travelled along the curve at a given time, and to find the time at which a given distance has been travelled; the unit tangent vector $\hat{T}$; curvature $\kappa$.
Announcements:
• I’m granting an extension on the extra credit for exam 1. You have until the beginning of class on Friday 2/21 to email me. See the post for 2/19 for details.
• Homework 2 is due at the beginning of class tomorrow. Remember to follow the formatting guidelines; please fill out your name on both sides of the cover sheet.
• Grading of this assignment will be slightly different from the first assignment. I will only grade two problems in detail; you choose one, I choose one. The remaining problems with be worth up to two points each, based on the completeness of the attempt. Put a big star next to the problem you would like me to look at in detail. If you don’t choose one, I’ll choose for you.
• If you are still having problems logging in to the ebook from WebAssign, you can try to log in directly instead. The link is posted under the “Logins” menu on the right sidebar.
#### 2/19: Arc Length Integrals and the Line Element $ds$.
Slides: Arc length integrals and the line element $ds$ (and an introduction to the vector differential $d\vec{s}$) (sec 13.3)
Summary: review of definite integrals — how to “read” and integral; using vector parameterization and definite integrals to compute the length of a curve (arc length); defining and computing the line element $ds$; brief introduction to the vector differential $d\vec{s}$ (aka $d\vec{r}). Note: the line element$ds$and the vector differential$d\vec{s}$are very useful tools we will be using in this course. They are not explicitly described in the course text, but an excellent introduction can be found at the Bridge Book wiki. Note that the Bridge Book denotes the vector differential$d\vec{r}$instead of$d\vec{s}$. Announcement: If you want 3 points extra credit on the first exam, send me the following in an email by 5:00pm today (Wednesday 2/19): • One factor that was out of your control and had a negative impact on your exam performance. • Two factors that were within your control, that would have improved your performance on the exam. I will compile and post everyone’s answers (without names). #### 2/14: Derivatives of Vector-Valued Functions Slides: Calculus along curves (sec 13.3) Summary: derivatives of vector-valued functions; vector derivative as tangent vector; velocity, speed and acceleration along parameterized curves. #### 2/12: Introduction to Curves Slides: Introduction to curves in$\mathbb{R}^2$and$\mathbb{R}^3$(sec 13.1) Summary: vector-valued functions; parameterized curves. In chapter 13, we will look at functions$\vec{r}(t)$whose input is a scalar and whose output is a vector (examples of vector-valued functions). The components of a vector-valued function, called coordinate functions, give the coordinates of the terminal points of$\vec{r}(t)$. The curve described by these points is said to be parameterized by the function$\vec{r}(t)$. #### 2/5: Planes in 3-Space Slides: Equations of planes in$\mathbb{R}^3$(sec 12.5) Summary: similarities between vector equations of lines and planes; derivation of the algebraic equation of a plane using the dot product; normal vectors and their manifestation in the algebraic equation of planes; parallel planes; normal lines; the cross product and normal vectors. A few announcements: • The first exam is on Monday 2/10 during class. More information can be found on the Exams page. • Friday 2/7 will be a review day. I will not have anything prepared, but will answer (almost) any question asked during the review. If there are questions that you know you in advance that you would like to discuss during the review, you can send me an email before class. • My office hours this semester will be Mondays and Wednesdays from 10:30-12:30 in MG222D. I will also have office hours this Friday (2/7) from 10:30-12:30. • The times and locations for Kyle Beserra’s help sessions are: • M 4:30-5:45 B222 • Tu 11-12:15 (room TBA) • W 4:30-5:45 ILC 301 • Th 11:45-12:45 B302 • F 3-4:45 E221 (these have also been posted on the course syllabus). #### 2/3: The Cross Product Slides: The Cross Product (sec 12.4) Summary: algebraic and geometric definitions of the cross product; computing the cross product of two vectors; algebraic and geometric properties of the cross product; the right-hand rule. Watching the polling responses during class, it seems to me that people began getting confused at the geometric definition of the cross product. Keep in mind the following: 1. We generally do not use the geometric definition of the cross product to actually compute the cross product*, since finding the angle$\theta$and the unit vector$\mathbf{\hat{n}}$is not generally possible (same holds true for the dot product — if you are not given the angle as part of the problem, you can’t use the geometric definition there, either).The general rule regarding when to use algebraic and when to use geometric definitions is: • Use algebraic definitions to compute. • Use geometric definitions to get information about the information content of the computation (what your computation means), or to get a feel for what your algebraic computation should look like using sketches or diagrams. There are some questions on WebAssign where you are asked to compute the cross or dot product using the geometric definition, but these are mostly to help you learn the definitions. 2. The most important geometric information you need to know about the cross product is: • The magnitude of the cross product is the area of the parallelogram spanned by the original vectors (this comes from the scalar part of the geometric definition). • The cross product is orthogonal to the original vectors, and hence, (if the original vectors are non-zero and are not parallel) is normal to the plane spanned by the original vectors. • The direction of the cross product is determined via the right-hand rule. * There are rare special cases where you can compute using the geometric definition (for example, you can show that i x j = k). #### 1/31: The Dot Product, Part II Slides: The Dot Product Part 2 — Vector Projections & Orthogonal Decompositions (sec 12.3) Summary: projecting one vector along another (non-zero) vector; scalar components of vector projections; orthogonal decompositions of one vector with respect to another (non-zero) vector. Given two vectors$\vec{v}$and$\vec{u}$with$\vec{u}\neq\vec{0}$, the vector$\vec{v}$can be written as the sum of two vectors$\vec{v}_{\|}$and$\vec{v}_{\perp}$, where$\vec{v}_{\|}$and$\vec{v}_{\perp}$are orthogonal — that is,$\vec{v} = \vec{v}_{\|} + \vec{v}_{\perp}$and$\vec{v}_{\|} \cdot \vec{v}_{\perp} = 0$. We will see projections and orthogonal decompositions again in chapters 13 and 16. #### 1/29: Introduction to the Dot Product Slides: The Dot Product Part I — Definitions and Angles (sec 12.3) Summary: algebraic and geometric definitions of the dot product; computing the dot product; using the dot product to compute angles; orthogonality. The dot product will play a major role in our class. Today we saw that the dot product encodes geometric information about length (distance) and angles. #### 1/27: Vector Parameterizations of Lines Slides: Parallel Vectors & Vector Parameterizations of Lines (sec 12.2) Summary: definition of parallel vectors; vector parameterizations of lines in the plane and in 3-space; similarities between vector parameterizations of lines and the point-slope equation of a line in the plane; coordinate functions (aka parametric equations). In the larger picture of this class, today’s topic (parameterizing lines) is an introduction to the more general topic of parameterizing curves in the plane and in 3-space. Intuitively, a curve is one dimensional (you can only go forward or backwards along the curve), so you should be able to express the curve in terms of a single variable or parameter. In previous math classes, we learned that the graph of a function$y = mx + b$is a line in the plane with slope$m$and$y$-intercept$b$. In calc III, the roles of slope and$y$-intercept are taken by vectors. Using vector parameterizations, we can answer questions like: What location in space corresponds to a given parameter value? Does a particular point lie on a line? Do two lines intersect? Announcements: • Today is the last day to drop classes without a “W”. • Clicker scores will begin counting for credit on Wednesday 1/27. If you have been using your clicker in class, you can check the “clicker tests” on Blackboard to see if it is working properly. #### 1/24: Introduction to Vectors Slides: An Introduction to Vectors (sec 12.1, 12,2) Summary: an introductions to vectors in$\mathbb{R}^2$and$\mathbb{R}^3$; vector algebra (scalar multiplication and vector addition); magnitude; unit vectors; and the unit basis vectors$\hat\imath$,$\hat\jmath$, and$\hat{k}\$.
Announcement: I’ve added a page Commenting on the Course Website, with instructions on logging in to comment and changing your publicly displayed name. See the links under “Getting Started” on the right sidebar.
#### 1/23: WebAssign Announcement
The “first” assignment (besides the integration/differentiation self-assessments) has been posted on WebAssign. This assignment is due before class on Monday 1/27. I’m adding one extra attempt per problem on this one, so that people who have not used WebAssign before have a little more leeway with learning how to use it.
General Note For This Class: DRAW PICTURES!!!!! (Yes, I’m shouting. You should also imagine me jumping up and down. It’s important.)
Also, I will be opening discussion forums on WebAssign for each assignment. You can find the link to these forums below the list of assignments on WebAssign. Add topics as you see fit. I’ll be following the discussions, but mostly staying out of them; this is a space for you to talk about the assignments with each other.
#### 1/22: General Class Business
Slides: Welcome to Math 275
Summary: A tour of the course website; going over the course syllabus; registering for WebAssign; registering “clickers” on Blackboard.
On Friday, we will begin talking about vectors. Something to think about: have you seen vectors before? What are they?
#### Welcome to Math 275
Welcome to Math 275, section 002, Spring 2014. Our class meets MWF 9:00–10:15am in the Micron Business & Economics Building (MBEB) room 1210
In this course, we will see how the mathematics learned in Calculus I and II generalizes to higher-dimensional settings, and develop new tools to deal with new situations. In particular, we will explore the mathematics of space-curves (one-dimensional objects sitting in two- or three-dimensional space), surfaces (two-dimensional objects) and volumes (three-dimensional objects).
For now, please take a few minutes to familiarize yourself with the website. You can get to the Math 275 homepage from any page on my website using either the main menu, or the “Math 275 Home” link in the upper-right hand corner. If you want to contact me, my email address is on the bottom left of every page.
Information on the course texts can be found here. Information on registering for WebAssign can be found here. Information on “clickers” (student polling devices) can be found here. Information about exam dates, grading policies, etc., can be found on the course syllabus.
Please feel free to contact me via email at shariultman@boisestate.edu with any questions.
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# What is the norm of < 7 , 5, 1 >?
Feb 14, 2016
The 'norm' of a vector is a unit vector in the same direction. To find it we 'normalize' the vector. In this case, the vector is $< \frac{7}{\sqrt{75}} , \frac{5}{\sqrt{75}} , \frac{1}{\sqrt{75}} >$ or $< \frac{7}{8.66} , \frac{5}{8.66} , \frac{1}{8.66} >$.
#### Explanation:
To normalize a vector, we divide each of its elements by the length of the vector. To find the length:
$l = \sqrt{{7}^{2} + {5}^{2} + {1}^{2}} = \sqrt{49 + 25 + 1} = \sqrt{75} = 8.66$
The norm can be expressed two ways, depending on your preference (and the preferences of the person who will mark your work):
$< \frac{7}{\sqrt{75}} , \frac{5}{\sqrt{75}} , \frac{1}{\sqrt{75}} >$ or $< \frac{7}{8.66} , \frac{5}{8.66} , \frac{1}{8.66} >$
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• 论文 •
### Extracting Eco-hydrological Information of Inland Wetland from L-band Synthetic Aperture Radar Image in Honghe National Nature Reserve, Northeast China
SUN Yonghua1, 2, GONG Huili1, 2, LI Xiaojuan1, 2, PU Ruiliang3, LI Shuang1, 2
1. (1. College of Resources Environment & Tourism, Capital Normal University, Beijing 100048, China;
2. National Key Laboratory Cultivation Base of Urban Environmental Processes & Digital Analog, Beijing
100048, China; 3. Department of Geography, University of South Florida, Tampa, FL 33620, USA)
• 出版日期:2011-03-24 发布日期:2011-04-06
### Extracting Eco-hydrological Information of Inland Wetland from L-band Synthetic Aperture Radar Image in Honghe National Nature Reserve, Northeast China
SUN Yonghua1, 2, GONG Huili1, 2, LI Xiaojuan1, 2, PU Ruiliang3, LI Shuang1, 2
1. (1. College of Resources Environment & Tourism, Capital Normal University, Beijing 100048, China;
2. National Key Laboratory Cultivation Base of Urban Environmental Processes & Digital Analog, Beijing
100048, China; 3. Department of Geography, University of South Florida, Tampa, FL 33620, USA)
• Online:2011-03-24 Published:2011-04-06
Taking a typical inland wetland of Honghe National Nature Reserve (HNNR), Northeast China, as the study area, this paper studied the application of L-band Synthetic Aperture Radar (SAR) image in extracting eco-hydrological information of inland wetland. Landsat-5 TM and ALOS PALSAR HH backscatter images were first fused by using the wavelet-IHS method. Based on the fused image data, the classification method of support vector machines was used to map the wetland in the study area. The overall mapping accuracy is 77.5%. Then, the wet and dry aboveground biomass estimation models, including statistical models and a Rice Cloudy model, were established. Optimal parameters for the Rice Cloudy model were calculated in MATLAB by using the least squares method. Based on the validation results, it was found that the Rice Cloudy model produced higher accuracy for both wet and dry aboveground biomass estimation compared to the statistical models. Finally, subcanopy water boundary information was extracted from the HH backscatter image by threshold method. Compared to the actual water borderline result, the extracted result from L-band SAR image is reliable. In this paper, the HH-HV phase difference was proved to be valueless for extracting subcanopy water boundary information.
Abstract:
Taking a typical inland wetland of Honghe National Nature Reserve (HNNR), Northeast China, as the study area, this paper studied the application of L-band Synthetic Aperture Radar (SAR) image in extracting eco-hydrological information of inland wetland. Landsat-5 TM and ALOS PALSAR HH backscatter images were first fused by using the wavelet-IHS method. Based on the fused image data, the classification method of support vector machines was used to map the wetland in the study area. The overall mapping accuracy is 77.5%. Then, the wet and dry aboveground biomass estimation models, including statistical models and a Rice Cloudy model, were established. Optimal parameters for the Rice Cloudy model were calculated in MATLAB by using the least squares method. Based on the validation results, it was found that the Rice Cloudy model produced higher accuracy for both wet and dry aboveground biomass estimation compared to the statistical models. Finally, subcanopy water boundary information was extracted from the HH backscatter image by threshold method. Compared to the actual water borderline result, the extracted result from L-band SAR image is reliable. In this paper, the HH-HV phase difference was proved to be valueless for extracting subcanopy water boundary information.
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The financial metrics return on equity (ROE), and the return on capital employed (ROCE) are valuable tools for gauging a company's operational efficiency and the resulting potential for future growth in value. They are often used together to produce a complete evaluation of financial performance.
### Return on Equity
ROE is the percentage expression of a company's net income, as it is returned as value to shareholders. This formula allows investors and analysts an alternative measure of the company's profitability and calculates the efficiency with which a company generates profit, using the funds that shareholders have invested.
ROE is determined using the following equation:
$ROE = \text{Net Income} \div \text{Shareholders' Equity}$
Regarding this equation, net income is comprised of what is earned throughout a year, minus all costs and expenses. It includes payouts made to preferred stockholders but not dividends paid to common stockholders (and the shareholders' overall equity value excludes preferred stock shares). In general, a higher ROE ratio means that the company is using its investors' money more efficiently to enhance corporate performance and allow it to grow and expand to generate increasing profits.
One recognized weakness of ROE as a performance measure lies in the fact that a disproportionate level of company debt results in a smaller base amount of equity, thus producing a higher ROE value off even a very modest amount of net income. So, it is best to view ROE value in relation to other financial efficiency measures.
### Return on Capital Employed
ROE evaluation is often combined with an assessment of the ROCE ratio. ROCE is calculated with the following formula:
\begin{aligned} &ROCE = \frac{EBIT}{\text{capital employed}} \\ &\textbf{where:}\\ &EBIT=\text{earnings before interest and taxes}\\ \end{aligned}
ROE considers profits generated on shareholders' equity, but ROCE is the primary measure of how efficiently a company utilizes all available capital to generate additional profits. It can be more closely analyzed with ROE by substituting net income for EBIT in the calculation for ROCE.
ROCE works especially well when comparing the performance of companies in capital-intensive sectors, such as utilities and telecoms, because unlike other fundamentals, ROCE considers debt and other liabilities as well. This provides a better indication of financial performance for companies with significant debt.
To get a superior depiction of ROCE, adjustments may be needed. A company may occasionally hold cash on hand that isn't used in the business. As such, it may need to be subtracted from the Capital Employed figure to get a more accurate measure of ROCE.
The long-term ROCE is also an important indicator of performance. In general, investors tend to favor companies with stable and rising ROCE numbers over companies where ROCE is volatile year over year.
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Question: Salmon (quant.sf) to tximport error
0
28 days ago by
Merlin 10
Italy
Merlin 10 wrote:
Hello everyone,
I generated quant.sf files with salmon tool and now I want to import them into R and later perform a differential expression analysis. I'm following the procedure in this link Salmon/Sailfish
the first code is
files <- file.path(/home/RNAseq/salmon_folder/salmonSFquant/treated_1/quant.sf, /home/RNAseq/salmon_folder/salmonSFquant/treated_2/quant.sf, /home/RNAseq/salmon_folder/salmonSFquant/treated_3/quant.sf,
/home/RNAseq/salmon_folder/salmonSFquant/untreated_1/quant.sf, /home/RNAseq/salmon_folder/salmonSFquant/untreated_2/quant.sf, /home/RNAseq/salmon_folder/salmonSFquant/untreated_3/quant.sf,"salmon"
,samples$run, "quant.sf") for some reason I have this error files <- file.path(/home/RNAseq/salmon_folder/salmonSFquant/treated_1/quant.sf, /home/RNAseq/salmon_folder/salmonSFquant/treated_2/quant.sf, /home/RNAseq/salmon_folder/salmonSFquant/treated_3/quant.sf, Error: unexpected '/' in "files <- file.path(/" > /home/RNAseq/salmon_folder/salmonSFquant/untreated_1/quant.sf, /home/RNAseq/salmon_folder/salmonSFquant/untreated_2/quant.sf, /home/RNAseq/salmon_folder/salmonSFquant/untreated_3/quant.sf,"salmon" Error: unexpected '/' in " /" > ,samples$run, "quant.sf")
Error: unexpected ',' in " ,"
Do I have to create a sample.txt?
I'm new to bioinformatic in general and R too, so I would appreciate any help,
thank you
tximport • 104 views
modified 28 days ago by Michael Love23k • written 28 days ago by Merlin 10
Answer: Salmon (quant.sf) to tximport error
1
28 days ago by
Michael Love23k
United States
Michael Love23k wrote:
In order to notify a maintainer, one of the tags for the post should be the name of the package, and tags need to be separated by comma. I happened to catch this post just looking through recent posts.
The error above actually gives you a pretty good clue:
files <- file.path(/home/RNAseq/salmon_folder/salmonSFquant/treated_1/quant.sf, /home/RNAseq/salmon_folder/salmonSFquant/treated_2/quant.sf, /home/RNAseq/salmon_folder/salmonSFquant/treated_3/quant.sf,
It says:
Error: unexpected '/' in "files <- file.path(/"
So R didn't expect a backslash where you put it, e.g. the first backslash caused a problem:
files <- file.path(/home...
Now compare your code to the code in the tximport vignette, or examples in the man pages.
Hi Michael, next time I will put the name of the package in the tag. I just use the tilda sign and I solved the issue... thanks :)
can I ask you for another help? I should be close to DESeq2 ...
> head(files)
character(0)
> salmonfiles <- paste0("sample", 1:6)
> salmonfiles
[1] "sample1" "sample2" "sample3" "sample4" "sample5" "sample6"
Error in tximport(salmonfiles, type = "salmon", tx2gene = tx2gene, reader = read_tsv) :
>
Can you tell me why I do have the error for reader=read_tsv, is it missing something?
thank you
Probably an old version of Bioconductor.
biocLite("BiocUpgrade") Error: Bioconductor version 3.8 cannot be upgraded with R version 3.5.0 In addition: Warning message: 'biocLite' is deprecated. Use 'BiocManager::install' instead. See help("Deprecated")
I hope is not for that,
thank you Michael
Back to the beginning.
And read help locally: ?tximport, vignette(“tximport”)
If you read help online and have out of date software you will get confused.
Hi Michael, I tried as you suggested to me but it didn't work .. so I start looking on the web for the solution without success. I have to say that when I write every path I check using the autocompletion if it's ok but this tool does't like my files. I ended up to do this pipeline:
samples <- read.table(file.path("~RNAseq/salmon_folder/salmonSFquant/samples.txt"), header=TRUE)
samples
condition
1 treated1
2 treated2
3 treated3
4 untreated1
5 untreated2
6 untreated3
.
files <- file.path("salmon", samples$run, "~RNAseq/salmon_folder/salmonSFquant/treated_1/quant.sf", "~RNAseq/salmon_folder/salmonSFquant/treated_2/quant.sf", "~RNAseq/salmon_folder/salmonSFquant/treated_3/quant.sf", "~RNAseq/salmon_folder/salmonSFquant/untreated_1/quant.sf", "~RNAseq/salmon_folder/salmonSFquant/untreated_2/quant.sf", "~RNAseq/salmon_folder/salmonSFquant/untreated_3/quant.sf") files character(0) . condition<- paste0(samples$run)
condition
Is it correct this command line below like this? probably is this the problem
tx2gene <- read.csv(file.path("~RNAseq/salmon_folder/gencode.v29.transcripts.csv"))
I run this command below to se if he recognizes my files but I get 0
file.exists(files)
logical(0)
.
txi <- tximport("files", type="salmon", tx2gene=tx2gene)
Error in tximport("files", type = "salmon", tx2gene = tx2gene) :
all(file.exists(files)) is not TRUE
I can't figure it out what's wrong, I would be more then happy to figure this out
thank you
That last error says that files is not pointing to files that exist at the location you specified.
You can just look at files and check that those are correct locations.
We are just checking,
file.exists(files)
internally to make sure the user is pointing to files that exist and can be read into R.
if I do
files
I have this
files character(0)
I'm sure that the files quant.sf are in the location of the path used
thank you
Ok if you look above you are trying to build a path using samples$run. But look at samples, does this have a column named run? You have to be careful about your R code and make sure that you are doing sensible things at each step. You can't refer to a column of a table that doesn't exist. ADD REPLYlink written 24 days ago by Michael Love23k To be honest with you I though that$run was a function of R, I learned something more, thanks. I did some adjustment, I have different results but still the same error...
files <- file.path("salmon", samples\$condition,
"~RNAseq/salmon_folder/salmonSFquant/treated_1/quant.sf",
"~RNAseq/salmon_folder/salmonSFquant/treated_2/quant.sf",
"~RNAseq/salmon_folder/salmonSFquant/treated_3/quant.sf",
"~RNAseq/salmon_folder/salmonSFquant/untreated_1/quant.sf",
"~RNAseq/salmon_folder/salmonSFquant/untreated_2/quant.sf",
"~RNAseq/salmon_folder/salmonSFquant/untreated_3/quant.sf")
files [1] "salmon/treated1/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated1/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated2/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated3/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated1/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated2/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated3/quant.sf"
[2] "salmon/treated2/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated1/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated2/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated3/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated1/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated2/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated3/quant.sf"
[3] "salmon/treated3/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated1/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated2/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated3/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated1/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated2/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated3/quant.sf"
[4] "salmon/untreated1/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated1/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated2/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated3/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated1/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated2/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated3/quant.sf" [5] "salmon/untreated2/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated1/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated2/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated3/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated1/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated2/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated3/quant.sf" [6] "salmon/untreated3/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated1/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated2/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/treated3/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated1/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated2/quant.sf/~p.panelli/RNAseq/salmonfolder/salmonSFquant/untreated3/quant.sf"
names(files) <- paste0("sample",1:6)
names(files) <- paste0("sample",1:6) names(files) [1] "sample1" "sample2" "sample3" "sample4" "sample5" "sample6"
tx2gene <- read.csv(file.path("~RNAseq/salmon_folder/gencode.v29.transcripts.csv"))
txi <- tximport("files", type="salmon", tx2gene=tx2gene)
Error in tximport("files", type = "salmon", tx2gene = tx2gene) : all(file.exists(files)) is not TRUE
Thank you Michael
1
This is not really an issue with tximport but with working with R and specifying files. I have limited time unfortunately to reply to support requests here, essentially I have to carve it out of my spare time. I’d recommend sitting with a bioinformatician who can look over your shoulder and give some advice on avoiding small coding errors.
Thanks
Ok I fixed the problem,
I modified the codes, mainly by writing the entire paths for all the codes and fixing the variable file.path.
all(file.exists(files))
[1] TRUE
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# Does a connected graph contain less or equal number of cycles than a disconnected graph with the same number of edges?
Let's have connected G(V,E) and disconnected G'(V,E') where |E|=|E'|. How can one proove G' always has at least the same number of cycles than G? I have 2 ideas how such a proof should look like.
Idea 1:
The intuition is that when "distributing" edges in a graph, to avoid cycles one aims for a tree until the graph becomes connected. This happens when the number of edges reaches v-1. Each "remaining" edge then creates a new cycle. This way we should end up with the least possible amouth of cycles.
Idea 2:
The intuition is that to create a connected graph from an arbitrary disconnected graph with the same number of edges, we need to repeatedly "move" an edge from some connected component so that it connects it with another connected component. We repeat until we get a connected graph.
Each such move either "breaks" some cycle or not.
If it does not, the number of cycles won't change. This is because connecting two connected components with an edge does not create a new cycle.
If it does, then the number of cycles in the graph decreases for the same reason. Therefore, at the end we end up with same or less number of cycles.
Is one of these a correct proof?
The claim is not true, let $$G$$ be $$K_7$$ with $$21$$ pendant vertices, it is a connected graph with $$21 + 21 = 42$$ edges and $$28$$ vertices. It has $$\sum\limits_{i=3}^{7} \binom{7}{i}(i-1)!/2 = 1172$$ cycles. On the other hand let $$G'$$ be the graph consisting of $$7$$ copies of $$K_4$$, it has $$42$$ edges and $$28$$ vertices, and it has $$7(\sum \limits_{i=3}^4 \binom{4}{i}(i-1)!/2) = 7(7) = 49$$ cycles.
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# Left-right topology
Are there non-trivial topological solutions (in particular 't Hooft-Polyakov magnetic monopoles) associated with the (local) breaking
$$SU(2)_R \times SU(2)_L \times U(1)_{B-L} \to SU(2)_L \times U(1)_Y$$
?
In order for a theory to present stable monopole solutions it has to satisfy three requirements:
i) It has to have the topological conditions, generally showed as non trivial second homotopy group of the vacuum manifold.
ii) It has to satisfy a quantization condition $$e^{ieQ_m}=\mathbb 1,$$ where $Q_m$ is the (non-Abelian) magnetic charge. This is a generalization of the Dirac quantization condition.
iii) The monopole has to be a solution of the classical equations of montion.
It can be shown that to satisfy ii) the $U(1)_{em}$ has to be compact (the electric charge has to quantized as well), that is, isomorphic to the circle and not to the reals. It turns out that when you have a SSB $G\rightarrow K\times U(1)$, the $U(1)$ is compact if $G$ and $K$ are both semisimple. Otherwise $U(1)$ may be non compact. In your case, $G$ is not semisimple, it has an Abelian factor.
For chiral symmetry breaking such as in QCD where the symmetry is global there are definitely no 't Hooft-Polyakov monopoles since those appear when you spontaneously break a local gauge symmetry. You are breaking a global one.
There are some studies about something called "semilocal defects" that may appear when you break a local and a global symmetry in a "mixed way".
What characterizes the stability of topological solutions as monopoles and vortices are topological quantities as the winding number. Take a vortex, for simplicity. Roughly speaking the winding number would say how the scalar field rotate as we go around the vortex. This rotation is in the internal space. With a global symmetry you are not able to construct such a rotating scalar field. The topological numbers would be trivial.
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# Copper is listed on the periodic table as having a relative atomic mass of 63.55. Reference books indicate two isotopes of copper, with relative masses of 62.93 and 64.93. What is the percent abundance of each isotope?
Dec 22, 2014
69% for the isotope that weighs 62.93 u and 31% for the isotope that weighs 64.93 u.
You can approach this problem by using a single equation; let's say the first isotope contributes to the relative atomic mass by a fraction $x$ ($x < 1$). SInce there are only 2 isotopes to consider, the fraction the other isotope contributes with will automatically be $1 - x$.
Let's set up the equation
$x \cdot 62.93 u + \left(1 - x\right) \cdot 64.93 u = 63.55 u$
Solving this for $x$ will produce
$64.93 u - 63.55 u = 2 x \to x = \frac{1.38}{2} = 0.69$
Multiplying these fractions (0.69 and 1 - 0.69 = 0.31) by 100% percent to get the results as a percentage will give
69% for the isotpe that weighs 62.93u and
31% for the one that weighs 64.93u.
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MCMC-Bridge
philastrophist/mcmc-bridge
PyMC3 is a platform in which to specify your Bayesian hierarchical model. It is a probabilistic language but also contains it’s own samplers.
emcee is a ubiquitous, perhaps the go-to sampler for astronomers. It is probably the first sampler you’ll come across in the literature because it is very powerful. emcee is an affine-invariant sampler, so it can deal with complex multi-dimensional likelihoods. PyMC3 just doesn’t have a comparable sampler in its arsenal and integrating them is actually not straightforward.
Here’s where MCMC-Bridge comes in.
You can simply write your model as before but then export to the emcee sampler to run it! We can then easily turn the emcee trace back into the flexible PyMC3 trace object and continue on our way. Emcee can utilise MPI and multi-processing and so MCMC-Bridge allows the easy use of cluster resources by adding a pool or threads option to export_to_emcee.
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##### Solving a system of linear and quadratic equations.
Algebra Tutor: None Selected Time limit: 1 Day
Solve the following system of equations
y = x² + 7x - 5
y = 6x + 7
There may be more than one solution need all
Nov 14th, 2015
-----------------------------------------------------------------------------------
$\\ y=x^2+7x-5---(1)\\ \\ y=6x+7---(2)$
Comparing equations (1) and (2) we get
$\\ x^2+7x-5=6x+7\\ \\ x^2+7x-5-6x-7=0\\ \\ x^2+x-12=0\\ \\ x^2+4x-3x-12=0\\ \\ x(x+4)-3(x+4)=0\\ \\ (x+4)(x-3)=0\\ \\ x+4=0\; \; \; or\; \; \; x-3=0\\ \\ x=-4\; \; \; or\; \; \; x=3$
Substitute x = -4 in equation (2)
$y=6\times(-4)+7=-24+7=-17$
Substitute x = 3 in equation (2)
$y=6\times3+7=18+7=25$
So the solution is (-4, -17) and (3, 25) i.e. the line intersects the parabola at 2 points.
-----------------------------------------------------------------------------------
Nov 14th, 2015
...
Nov 14th, 2015
...
Nov 14th, 2015
Oct 24th, 2016
check_circle
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Translator Disclaimer
VOL. 73 | 2017 Homological stability of $Aut(F_n)$ revisited
Editor(s) Koji Fujiwara, Sadayoshi Kojima, Ken'ichi Ohshika
## Abstract
We give another proof of a theorem of Hatcher and Vogtmann stating that the sequence $Aut(F_n)$ satisfies integral homological stability. The paper is for the most part expository, and we also explain Quillen's method for proving homological stability.
## Information
Published: 1 January 2017
First available in Project Euclid: 4 October 2018
zbMATH: 07272042
MathSciNet: MR3728490
Digital Object Identifier: 10.2969/aspm/07310001
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# Circular Statistics in Python Help
I'm trying to perform some circular statistics in Python and, having never done circular statistics before and being mostly self taught in python, thought I'd turn to this place for some help because I've hit a brick wall. Never posted here before, so if you need more info or if my question doesn't make sense, let me know. I have some dataframe on number of planula (offspring) produced by a group of corals over 3 lunar cycles. A lunar cycle lasts 29 days. I'm trying to find out what the mean lunar day of planula release is for each of the three lunar cycles. I've found the documentation for scipy.stats.circmean, but its relatively unhelpful with determining exactly what input should go where. I've included an image of what part of the dataframe I'm working with looks like for reference. The column lunar is the lunar day, C1:C10 are the 10 corals in my sample, and the Total is the total number of planula produced daily, which is what I'm interested in finding the mean lunar day of. Any and all help would be greatly appreciated.
• You can compute this also manually. Convert the days to x,y coordinates on a clock with 29 instead of 12 figures. Add together the x,y for each birth on a particular day. Compute the angle of the resulting vector to see in which direction/day of the clock it points. That is the average. Mar 20 at 9:46
• 'for each of the three lunar cycles' this is a bit unclear. Are you trying to compute three different averages? But then, the circularity might need to be omitted. Mar 20 at 9:49
It looks to me like you want to take the (circular) mean of the Lunar column, weighted by the Total column. But first you need to convert Lunar to radians by multiplying by 2*np.pi/29.
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# Cropping Rasters
A short description of the post.
Rodney Dyer https://dyerlab.org (Center for Environmental Studies)https://ces.vcu.edu
2016-03-18
It is often the case that the raster we are working with is not the exact size of the area from which our data are collected. It is a much easier situation if the raster is larger than the area than if you need to stitch together two raster Tiles to get all your data onto one extent. In my doctoral thesis work, the area of the southern Ozark mountains that my sites were in was not only straddling a boundary between existing rasters, it was also at the boundary of two UTM zones! What a pain.
Spatial data often consists of very large datasets. One way to minimize the amount of computational resources you are going to use in R is to select only the spatial regions (extents) that you are working in and get rid of the remaining data. This is particularly important if you are going to be doing isolation modeling for genetic connectivity as the cost surface raster needs to be translated into a connectivity network. Data outside your area of interest are unnecessarily taking both resources and time away from your work.
To crop a raster, we need to identify the region we wish to keep as an extent object. An extent is a vector of length 4 defined as c(xmin, xmax, ymin, ymax) describing the boundaries of are of interest.
Here is an extent for the arapat data set.
<pre class="lang:r decode:true">require(gstudio)
data(arapat) lon.min <- min(arapat$$Longitude) lon.max <- max(arapat$$Longitude) lat.min <- min(arapat$$Latitude) lat.max <- max(arapat$$Latitude) e <- extent( lon.min, lon.max, lat.min, lat.max ) e ## class: Extent ## xmin: -114.2935 ## xmax: -109.1263 ## ymin: 23.0757 ## ymax: 29.32541
<p>
This is the exact boundaries of the data set. However, it is probably a good idea to not have your map cropped exactly to your boundaries but have your most extreme locations plotted within the map boundaries. So I’ll take an approximation (rounding up and down as necessary) to use.
</p>
<pre class="lang:r decode:true">e <- extent( -115, -108, 23, 30 )
e ## class: Extent ## xmin: -115 ## xmax: -108 ## ymin: 23 ## ymax: 30
<p>
This can now be used as the area from within the alt raster that we want to keep.
</p>
<pre class="lang:r decode:true">bc <- crop( alt, e )
bc ## class : RasterLayer ## dimensions : 840, 840, 705600 (nrow, ncol, ncell) ## resolution : 0.008333333, 0.008333333 (x, y) ## extent : -115, -108, 23, 30 (xmin, xmax, ymin, ymax) ## coord. ref. : +proj=longlat +datum=WGS84 +no_defs +ellps=WGS84 +towgs84=0,0,0 ## data source : in memory ## names : alt_22 ## values : -202, 2774 (min, max)
<p>
And if we plot it, we see that it is more properly scaled for the region we are interested in working on.
</p>
<pre class="lang:r decode:true ">plot(bc)</pre>
<p>
</p>
### Citation
Dyer (2016, March 18). The Dyer Laboratory: Cropping Rasters. Retrieved from https://dyerlab.github.io/DLabWebsite/posts/2016-03-18-cropping-rasters/
@misc{dyer2016cropping,
}
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# Galois extension and Tensor product
The following theorem is proved in Bourbaki's algebra. They use the technique of Galois descent. I'd like to know the proof without using it if any.
Theorem Let $K$ be a field. Let $\Omega/K$ be an extension. Let $N/K$ and $L/K$ be subextensions of $\Omega/K$. Suppose $N/K$ is a (not necessarily finite) Galois extension and $L \cap N = K$. Then the canonical homomorphism $\psi:L\otimes_K N \rightarrow LN$ is an isomorphism.
-
First assume $N/K$ is finite Galois. Then $LN/L$ (inside $\Omega$) is finite Galois and the natural map $\mathrm{Gal}(LN/L)\rightarrow\mathrm{Gal}(N/K)$ is injective with image $\mathrm{Gal}(N/N\cap L)=\mathrm{Gal}(N/K)$, i.e., it is an isomorphism. So, $[LN:L]=[N:K]=\dim_L(L\otimes_KN)$. The map $L\otimes_KN\rightarrow LN$ is surjective in any case because $LN/L$ is algebraic, and since both sides are $L$-vector spaces of the same (finite) dimension, the map is an isomorphism.
In general, write $N=\bigcup_iN_i$ as a directed union of finite Galois extensions $N_i/K$. Then $L\otimes_KN=\varinjlim L\otimes_KN_i$, $LN=\bigcup_iLN_i$, and $L\otimes_KN\rightarrow LN$ can be identified with the direct limit of the isomorphisms $L\otimes_KN_i\cong LN_i$, and so is itself an isomorphism.
To see that $LN=\bigcup_iLN_i$, first observe that the RHS is clearly contained in the LHS. Conversely, if $\alpha\in LN$, then $\alpha$ is a polynomial (with coefficients in $L$) in elements $\beta_1,\ldots,\beta_r\in N$. If $i$ is such that $\beta_j\in N_i$ for all $j$, then $\alpha\in LN_i$.
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## Server Migration
This is our lounge area. Feel free to come in and get acquainted!
PixyMisa2
Posts: 6
Joined: Thu Jul 16, 2015 1:33 am
Been thanked: 3 times
### Server Migration
Hi everyone.
We'll be migrating to a new server tomorrow, starting at 11PM Pacific time. ETA is about an hour.
I had planned to do this over the weekend, but pre-Christmas stuff piled up and I didn't get the time.
The old server is is so old it's starting to accrete sedimentary rock in its dust filters, so time to move.
(If you also visit skepticforum.com, both sites will be migrating at the same time.)
Boss Paul
Posts: 280
Joined: Wed Nov 27, 2013 5:30 am
Title: Boss
Location: Florida Corrections
Been thanked: 14 times
### Re: Server Migration
While the boys wait, the Captain will have a nice road for them to pave.
Get the wax outta your ears!
You call the Captain "Captain!"
And you call the rest of us "Boss," you hear?!
ed
Posts: 33435
Joined: Tue Jun 08, 2004 11:52 pm
Title: Rhino of the Florida swamp
Has thanked: 460 times
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### Re: Server Migration
What is the nature of such a server? How is it connected to the interwebs? How much memory does it have? What does it cost? Who is paying for ours?
Thank you.
Wenn ich Kultur höre, entsichere ich meinen Browning!
PixyMisa2
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It's a Xeon E3 1240 with 32GB RAM and a 1TB SSD, gigabit public and private uplinks, and 100TB of included bandwidth per month. Running KVM virtualisation and under that, CentOS 7 and cPanel. Cost is around $50 per month after discounts, and I pay for it personally. (It hosts a bunch of other stuff as well; this forum doesn't need all that space.) gnome Posts: 22240 Joined: Tue Jun 29, 2004 12:40 am Location: New Port Richey, FL Has thanked: 381 times Been thanked: 408 times ### Re: Server Migration Thank you for what you do "If fighting is sure to result in victory, then you must fight! Sun Tzu said that, and I'd say he knows a little bit more about fighting than you do, pal, because he invented it, and then he perfected it so that no living man could best him in the ring of honor. Then, he used his fight money to buy two of every animal on earth, and then he herded them onto a boat, and then he beat the crap out of every single one. And from that day forward any time a bunch of animals are together in one place it's called a zoo! (Beat) Unless it's a farm!" --Soldier, TF2 Anaxagoras Posts: 21986 Joined: Wed Mar 19, 2008 5:45 am Location: Yokohama/Tokyo, Japan Has thanked: 1425 times Been thanked: 1277 times ### Re: Server Migration Thank you very much, Pixy! A fool thinks himself to be wise, but a wise man knows himself to be a fool. William Shakespeare Doctor X Posts: 67857 Joined: Fri Jun 04, 2004 8:09 pm Title: Collective Messiah Location: Your Mom Has thanked: 3469 times Been thanked: 2196 times ### Re: Server Migration Indeed! --J.D. Mob of the Mean: Free beanie, cattle-prod and Charley Fan Club! "Doctor X is just treating you the way he treats everyone--as subhuman crap too dumb to breathe in after you breathe out."--Don DocX: FTW.--sparks "Doctor X wins again."--Pyrrho "Never sorry to make a racist Fucktard cry."--His Humble MagNIfIcence "It was the criticisms of Doc X, actually, that let me see more clearly how far the hypocrisy had gone."--clarsct "I'd leave it up to Doctor X who has been a benevolent tyrant so far."--Grammatron "Indeed you are a river to your people. Shit. That's going to end up in your sig."--Pyrrho "Try a twelve step program and accept Doctor X as your High Power."--asthmatic camel "just like Doc X said." --gnome WS CHAMPIONS X4!!!! NBA CHAMPIONS!! Stanley Cup! SB CHAMPIONS X5!!!!! Fid Posts: 780 Joined: Sun Jun 06, 2004 3:45 pm Location: The island of Atlanta Has thanked: 624 times Been thanked: 141 times ### Re: Server Migration Dang boyo I thought this thingey just ran it's self ya know without human intervention "Try SCE to AUX." Yeah, me and an electrified atmosphere ain't friends. Abdul Alhazred Posts: 71762 Joined: Mon Jun 07, 2004 1:33 pm Title: Yes, that one. Location: Chicago Has thanked: 3419 times Been thanked: 1258 times ### Re: Server Migration Are we there yet? Any man writes a mission statement spends a night in the box. -- our mission statement plappendale Boss Godfrey Posts: 19 Joined: Mon Apr 11, 2016 10:57 pm Has thanked: 1 time ### Re: Server Migration ed Posts: 33435 Joined: Tue Jun 08, 2004 11:52 pm Title: Rhino of the Florida swamp Has thanked: 460 times Been thanked: 788 times ### Re: Server Migration PixyMisa2 wrote:It's a Xeon E3 1240 with 32GB RAM and a 1TB SSD, gigabit public and private uplinks, and 100TB of included bandwidth per month. Running KVM virtualisation and under that, CentOS 7 and cPanel. Cost is around$50 per month after discounts, and I pay for it personally.
(It hosts a bunch of other stuff as well; this forum doesn't need all that space.)
This, sorta?
https://ark.intel.com/products/52273/In ... e-3_30-GHz
Where is the "uplink"? Is it like into an ISP? How do you hook into the interwebs? Do you bypass ISPs? Is it all rental? Why do you pay? Hobby? Alien intrusion? Is it like being an ISP?
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PixyMisa2 wrote:It's a Xeon E3 1240 with 32GB RAM and a 1TB SSD, gigabit public and private uplinks, and 100TB of included bandwidth per month. Running KVM virtualisation and under that, CentOS 7 and cPanel. Cost is around $50 per month after discounts, and I pay for it personally. (It hosts a bunch of other stuff as well; this forum doesn't need all that space.) Thank you, Pixy. You're the best of the best of the best. The flash of light you saw in the sky was not a UFO. Swamp gas from a weather balloon was trapped in a thermal pocket and reflected the light from Venus. Abdul Alhazred Posts: 71762 Joined: Mon Jun 07, 2004 1:33 pm Title: Yes, that one. Location: Chicago Has thanked: 3419 times Been thanked: 1258 times ### Re: Server Migration Pyrrho wrote:Thank you, Pixy. You're the best of the best of the best. Yes. BTW what happened to the avatar? Are we there yet? Any man writes a mission statement spends a night in the box. -- our mission statement plappendale shemp Posts: 5198 Joined: Thu Jun 10, 2004 12:16 pm Title: stooge Has thanked: 663 times Been thanked: 470 times ### Re: Server Migration Pyrrho wrote: PixyMisa2 wrote:It's a Xeon E3 1240 with 32GB RAM and a 1TB SSD, gigabit public and private uplinks, and 100TB of included bandwidth per month. Running KVM virtualisation and under that, CentOS 7 and cPanel. Cost is around$50 per month after discounts, and I pay for it personally.
(It hosts a bunch of other stuff as well; this forum doesn't need all that space.)
Thank you, Pixy. You're the best of the best of the best.
Seconded! Thanks Pixy!
"It is not I who is mad! It is I who is crazy!" -- Ren Hoek
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shemp
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ed wrote:
PixyMisa2 wrote:It's a Xeon E3 1240 with 32GB RAM and a 1TB SSD, gigabit public and private uplinks, and 100TB of included bandwidth per month. Running KVM virtualisation and under that, CentOS 7 and cPanel. Cost is around $50 per month after discounts, and I pay for it personally. (It hosts a bunch of other stuff as well; this forum doesn't need all that space.) This, sorta? https://ark.intel.com/products/52273/In ... e-3_30-GHz Where is the "uplink"? Is it like into an ISP? How do you hook into the interwebs? Do you bypass ISPs? Is it all rental? Why do you pay? Hobby? Alien intrusion? Is it like being an ISP? Stop looking a gift horse in the mouth, you lousy swamp rat! "It is not I who is mad! It is I who is crazy!" -- Ren Hoek Freedom of choice Is what you got Freedom from choice Is what you want Abdul Alhazred Posts: 71762 Joined: Mon Jun 07, 2004 1:33 pm Title: Yes, that one. Location: Chicago Has thanked: 3419 times Been thanked: 1258 times ### Re: Server Migration Are we there yet? Any man writes a mission statement spends a night in the box. -- our mission statement plappendale Grammatron Posts: 33648 Joined: Tue Jun 08, 2004 1:21 am Location: Los Angeles, CA Been thanked: 1760 times ### Re: Server Migration Thanks Pixy Witness Posts: 17199 Joined: Thu Sep 19, 2013 5:50 pm Has thanked: 2121 times Been thanked: 2887 times ### Re: Server Migration Thanks, generous Pixy! PixyMisa2 Posts: 6 Joined: Thu Jul 16, 2015 1:33 am Been thanked: 3 times ### Re: Server Migration Abdul Alhazred wrote:Are we there yet? Yes! PixyMisa2 Posts: 6 Joined: Thu Jul 16, 2015 1:33 am Been thanked: 3 times ### Re: Server Migration ed wrote: PixyMisa2 wrote:It's a Xeon E3 1240 with 32GB RAM and a 1TB SSD, gigabit public and private uplinks, and 100TB of included bandwidth per month. Running KVM virtualisation and under that, CentOS 7 and cPanel. Cost is around$50 per month after discounts, and I pay for it personally.
(It hosts a bunch of other stuff as well; this forum doesn't need all that space.)
This, sorta?
https://ark.intel.com/products/52273/In ... e-3_30-GHz
Yep. Those chips are a few years old now, but still work just fine, so we can get the server for $50 a month instead of$200 or so.
Where is the "uplink"? Is it like into an ISP? How do you hook into the interwebs? Do you bypass ISPs?
The server is in a datacenter in Dallas, and our provider has several terabits of bandwidth. We have a gigabit link into their network.
Is it all rental?
Yep. Much much easier that way, since I live in Australia.
Why do you pay? Hobby? Alien intrusion? Is it like being an ISP?
My contribution to the skeptical community in this case. I offer free hosting to friends and causes I think are important.
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# Charged Particles in Matter
## Introduction
Last updated date: 21st Mar 2023
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The smallest particle of matter is an atom. In turn, there are three subatomic particles: protons, electrons, and neutrons. Protons and neutrons are the charged particles of an atom, and they are among the widely known and studied subatomic particles. Neutrons are negatively charged particles, whereas protons are positively charged.
For a long time, it was assumed that atoms are the ultimate particles of matter and that they cannot be further split. Experiments undertaken in the second half of the nineteenth and early twentieth centuries demonstrated that the atom is not the ultimate particle. Scientists' persistent efforts resulted in the discovery of subatomic particles.
The inability of Dalton's atomic hypothesis to explain certain data sparked the discovery of electrons and protons. Similarly, further research into neutrons was encouraged.
## What Are Charged Particles in Matter?
Matter is any material with mass that occupies space. The smallest unit of matter is known as the "atom." With significant discoveries throughout the years, the structure of the atom was ultimately suggested, and it was verified that each atom contains charged particles or subatomic particles. In matter, these charged particles or subatomic particles are negatively charged electrons, positively charged protons, and neutral neutrons.
According to the most recent atomic structure, an atom is made up of a positively charged nucleus in the centre that is surrounded by electrons that rotate in various orbits around the nucleus. The positive charge of the nucleus is because of the positive protons. The structure of an atom is quite similar to that of our solar system, with the Sun at the centre resembling the nucleus and the planets moving in various orbits like electrons.
## Types of Charged Particles in Matter
Three types of charged subatomic particles make up matter. They are as follows:
The Basic form of An Atom’s Structure
• Electrons are negatively charged subatomic particles that exist in an atom. An electron has a charge of magnitude $1.602\times {{10}^{-19}}$ Coulomb. An electron has 1:1837 the mass of a proton.
• Protons are positively charged subatomic particles found in atoms. The charge of a proton is identical to the charge of an electron in magnitude; the charge is positive. A proton has a mass of $1.673\times {{10}^{-27}}Kg$. A proton is denoted by an H+ ion which is the nucleus of a hydrogen atom without an electron.
• Neutrons are neutral subatomic particles found in all atomic nuclei. It has no electric charge at all and a mass of $1.673\times {{10}^{-27}}Kg$. It's slightly heavier than a proton but around 1839 times heavier than an electron.
## Positively Charged Particles in Matter
• Protons are the positively charged particles that are present in the nucleus of an atom.
• Protons are present in the same number as the electrons in an atom.
• Ernest Rutherford is credited with discovering protons.
• It has a mass of $1.676\times {{10}^{-24}}$ grams.
• It has a charge of $+1.602\times {{10}^{-19}}$Coulombs.
## Negatively Charged Particles in Matter
• Electrons are subatomic particles that are negatively charged.
• All elements' atoms contain an equal number of electrons and protons.
• J. J. Thomson is credited with discovering electrons since he was the first to precisely calculate an electron's mass and charge.
• The mass of an electron is tiny when compared to the mass of a proton. It has a mass that is equivalent to $1/1837$ times the mass of a proton.
• Its charge is equivalent to $-1.602\times {{10}^{-19}}$ Coulombs.
## Neutrally Charged Particles in Matter
• Neutrons are subatomic particles that are neutrally charged.
• Due to the variation in the number of neutrons in their respective nuclei, the masses of two distinct isotopes of an element differ.
• In 1932, James Chadwick discovered the neutron.
• They were found during an experiment in which an alpha particle bombarded a thin sheet of beryllium.
• A neutron has a mass of $1.676 \times {{10}^{-24}}$ grams.
## Interaction of Charged Particles
When charged particles interact, energy is transferred from the charged particles to the materials through which they move. When two charges that are similar connect, they repel one another; when two charges that are opposite interact, they attract each other.
Two forms of particle-particle interactions—collisions and long-lasting interactions when particles are packed—are significant in the majority of applications. In actuality, internal stresses arise as a result of particle deformation during interaction.
Only a tiny portion of the energy of heavy charged particles may be transferred in a single collision. In a collision, it barely deflects at all. As a result, heavy charged particles move through matter virtually directly while continually losing energy in many collisions with atomic electrons.
## Important Questions
1. Describe the term charged particles in matter.
Ans. A charged particle carries one that carries an electric charge. There are two kinds of electric charges: positive and negative. Two items with an excess of reputational force on each other.
2. What are matter's charged particles?
Ans. An electric charge exists in a charged particle. It might be an ion, such as a molecule or atom, having an excess or shortage of electrons in comparison to protons. It might also be an electron, a proton, or another primary particle, all of which are thought to have the same charge (except antimatter).
## Practise Questions
1. Force due to magnetic field and velocity is
1. At right angles to one another
2. At an oblique angle to one another
3. At 180 degree angles to each other
4. Opposite to each other
Ans. At right angles to one another
2. Hall voltage is proportional to
1. Current
2. Electric field
3. Magnetic Flux density
4. All of the above
Ans. Magnetic flux density
3. The force on a moving charge in a uniform magnetic field is determined by
1. Magnetic flux density
2. The charge on the particle
3. The speed of particle
4. All of above
Ans. Magnetic flux density
4. A charged particle is travelling perpendicular to the direction of a homogeneous magnetic field. Which of the following will change?
1. Speed
2. Velocity
3. Direction of motion
4. Both options 2 and 3
Ans. Both options 2 and 3
## Summary
In conclusion, electrons, protons, and neutrons are the three charged subatomic particles that make up an atom. Later, the structure of the atom was established by several findings. A positively charged nucleus, created by the positively charged proton and neutral neutron at the centre of an atom, rotates around it like the solar system's orbiting electrons do. An atom is more stable because each electron in its fixed orbit around the nucleus has a distinct amount of energy. An electron's charge is $-1.602 \times 10^{-19}$, whereas the charge of a proton is approximately identical but is positively charged. The lightest subatomic particles are electrons, whereas protons and neutrons, which make up the majority of an atom's mass, are only found in the nucleus.
Competitive Exams after 12th Science
## FAQs on Charged Particles in Matter
1. Create two tasks that will help you understand the nature of charged particles in matter.
You can notice static electricity if you run a plastic comb through your hair and then place it near little pieces of paper. The comb is dragged to the sheet of paper. Since opposing charges attract, the charged comb creates an opposite charge in the paper and the paper sticks to the comb.
After rubbing a woollen or synthetic pullover against a balloon, place it against a wall or ceiling. The attraction of opposing charges holds the balloon in place.
2. What exactly is a charged particle of matter? Is there energy in charged particles?
An atom is the smallest particle of matter. In turn, there are three subatomic particles: protons, electrons, and neutrons. Protons and neutrons are the charged particles of an atom, and they are among these subatomic particles. Neutrons are negatively charged particles, whereas protons are positively charged.
When charged particles have enough energy, they can move through matter faster than the speed of light in that substance. This event results in the emission of photons of light.
3. Briefly explain the discovery of subatomic particles.
Many scientists worked together to uncover the presence of charged particles in an atom. By 1900, it had been shown that the atom was an indivisible particle containing at least one subatomic particle, the electron, discovered by J.J. Thomson. In 1886, long before the electron was found, E. Goldstein identified unique radiations in a gas discharge and termed them canal rays.
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My Math Forum Bellman Equation and Contraction Mapping
Applied Math Applied Math Forum
October 20th, 2017, 12:31 PM #1 Newbie Joined: Oct 2017 From: Italy Posts: 2 Thanks: 0 Bellman Equation and Contraction Mapping Hello all. I'm taking a grad course where optimization problems are part of the syllabus, and I'm afraid I'm not really meeting all math requirements to take the course but alas, I had no choice but to take it. I have this problem (click for link) and I'm kinda stuck. Now, I know that to prove $\displaystyle T$ is a contraction mapping I should first prove the monotonicity, and then the discounting. Below you'll find how I did it. Monotonicity: $\displaystyle f(s) \leq g(s)$ $\displaystyle \sup[\pi(s)-x+\beta \int f(s')p(s'|s,x)] \leq \sup[\pi(s)-x+\beta \int g(s')p(s'|s,x)]$ $\displaystyle (Tf)(s) \leq (Tg)(s)$ Discounting: $\displaystyle T(f+c)(s) \leq Tf+\beta c$ $\displaystyle T(f+c)(s)=\sup[\pi(s)-x+\beta \int f(s')p(s''|s,x)+c]=Tf(s)+\beta c$ Is this enough, and is it correct? Or am I missing something? The only example I have in my notes is non-stochastic, so I really can't wrap my head around this problem. Can anyone help me?
Tags bellman, contraction, equation, mapping
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# Tag Archives: LHC
## WIMPZillas: The Biggest WIMPs
In the search for direct detection of dark matter, the experimental focus has been on WIMPS – weakly interacting massive particles. Large crystal detectors are placed deep underground to avoid contamination from cosmic rays and other stray particles.
WIMPs are often hypothesized to arise as supersymmetric partners of Standard Model particles. However, there are also WIMP candidates that arise due to non-supersymmetric extensions to the Standard Model.
The idea is that the least massive supersymmetric particle would be stable, and neutral. The (hypothetical) neutralino is the most often cited candidate.
The search technique is essentially to look for direct recoil of dark matter particles onto ordinary atomic nuclei.
The only problem is that we keep not seeing WIMPs. Not in the dark matter searches, not at the Large Hadron Collider whose main achievement has been the detection of the Higgs boson at mass 125 GeV. The mass of the Higgs is somewhat on the heavy side, and constrains the likelihood of supersymmetry being a correct Standard Model extension.
The figure below shows WIMP interaction with ordinary nuclear matter cross-section limits from a range of experiments spanning from 1 to 1000 GeV masses for WIMP candidates. Typical supersymmetric (SUSY) models are disfavored by these results at higher masses above 40 GeV or so as the observational limits are well down into the yellow shaded regions.
Perhaps the problem is that the WIMPs are much heavier than where the experiments have been searching. Most of the direction detection experiments are sensitive to candidate masses in the range from around 1 GeV to 1000 GeV (1 GeV or giga-electronVolt is about 6% greater than the rest mass energy of a proton). The 10 to 100 GeV range has been the most thoroughly searched region and multiple experiments place very strong constraints on interaction cross-sections with normal matter.
WIMPzillas is the moniker given to the most massive WIMPs, with masses from a billion GeV up to potentially as large as the GUT (grand Unified Theory) scale of $10^{16} GeV$.
The more general term is Superheavy Dark Matter, and this is proposed as a possibility for unexplained ultra high energy cosmic rays (UHECR). The WIMPzillas may decay to highly energetic gamma rays, or other particles, and these would be detected as the UHECR.
UHECR have energies greater than a billion GeV ($10^9 GeV$) and the most massive event ever seen (the so-called Oh My God Particle) was detected at $3 \cdot 10^{11} GeV$. It had energy equivalent to a baseball with velocity of 94 kilometers per hour. Or 40 million times the energy of particles in the Large Hadron Collider.
It has taken decades of searching at multiple cosmic ray arrays to detect particles at or near that energy.
Most UHECR appear to be spatially correlated with external galaxy sources, in particular with nearby Active Galactic Nuclei that are powered by supermassive black holes accelerating material near, but outside of, their event horizons.
However, they are not expected to be able to produce cosmic rays with energies above around $10^{11} GeV$, thus the WIMPzilla possibility. Again WIMPzillas could span the range from $10^9 GeV$ up to $10^{16} GeV$.
In a paper published last year, Kolb and Long calculated the production of WIMPzillas from Higgs boson pairs in the early universe. These Higgs pairs would have very high kinetic energies, much beyond their rest mass.
This production would occur during the “Reheating” period after inflation, as the inflaton (scalar energy field) dumped its energy into particles and radiation of the plasma.
There is another production mechanism, a gravitational mechanism, as the universe transitions from the accelerated expansion phase during cosmological inflation into the matter dominated (and then radiation-dominated) phases.
Thermal production from the Higgs portal, according to their results, is the dominant source of WIMPzillas for masses above $10^{14} GeV$. It may also be the dominant source for masses less than about $10^{11} GeV$.
They based their assumptions on chaotic inflation with quadratic inflation potential, followed by a typical model for reheating, but do not expect that their conclusions would be changed strongly with different inflation models.
It will take decades to discriminate between Big Bang-produced WIMPzilla style cosmic rays and those from extragalactic sources, since many more $10^{11} GeV$ and above UHECRs should be detected to build statistics on these rare events.
But it is possible that WIMPzillas have already been seen.
The density is tiny. The current dark matter density in the Solar neighborhood is measured at 0.4 Gev per cc. Thus in a cubic meter there would be the equivalent of 400,000 proton masses.
But if the WIMPzillas are at energies $10^{11} Gev$ and above (100 billion GeV), a cubic kilometer would only contain 4000 particles at a given time. Not easy to catch.
References
http://cdms.berkeley.edu/publications.html – SuperCDMS experiment led by UC Berkeley
http://pdg.lbl.gov/2017/reviews/rpp2017-rev-dark-matter.pdf – Dark matter review chapter from Lawrence Berkeley Lab (Figure above is from this review article).
http://home.physics.ucla.edu/~arisaka/home3/Particle/Cosmic_Rays/ – Ultra high energy cosmic rays
https://arxiv.org/pdf/1708.04293.pdf – E. Kolb and A. Long, 2017 “Superheavy Dark Matter through Higgs Portal Operators”
## Higgs Boson and Dark Matter, Part 1
The discovery of the Higgs boson at the Large Hadron Collider (the LHC, at CERN, near Geneva) was announced on the 4th of July, 2012. This new particle was the important missing component of the Standard Model of particle physics; the Standard Model has had great success in describing the three non-gravitational forces: electromagnetism, the weak nuclear force, and the strong nuclear force.
The mass of the Higgs is about 126 GeV (giga electron-Volts) and by way of comparison the proton mass is a bit under 1 GeV. The Higgs particle is highly unstable, with a decay lifetime of only about one tenth of one billionth of one trillionth of a second (10^-22 seconds). While the Higgs field pervades all of space, the particle requires very high energy conditions to “pop out” of the field for a very short while. The only place where these conditions exist on Earth is at the LHC.
The Higgs boson is not detected directly at LHC, but inferred (with high confidence) through the detection of its decay products. The main decay channels are shown in the pie chart below, and include bottom quarks, W vector bosons (which mediate the weak force), gluons (which mediate the strong force holding quarks inside protons and neutrons), tau leptons (very heavy members of the electron family), Z vector bosons (also weak force mediators), and even some photon decay channels. While the two photon channel is rare, much less than 1% of the decays, it is the most important channel used for the Higgs detection because of the “clean” signal provided.
What is the relationship between the Higgs and dark matter? In an earlier blog, http://wp.me/p1mZmr-3K , I discussed why the Higgs particle itself cannot be the explanation for dark matter. Dark matter must be stable; it must persist over the nearly 14 billion year lifetime of the universe. In today’s universe it’s very difficult and expensive to create a Higgs particle and it vanishes immediately.
But in the very early universe, at a tiny fraction of a second after its creation (less than the present-day Higgs boson lifetime!), the “temperature” and energy levels were so high that the Higgs particle (or more than one type of Higgs particle) would have been abundant, and as today it would have decayed to many other, lighter, particles. Could it be the source of dark matter? It’s quite plausible, if dark matter is due to WIMPs – an undiscovered, stable, weakly interacting massive particle. That dark matter is due to some type of WIMP is currently a favored explanation among physicists and cosmologists. WIMPs are expected from extensions to the Standard Model, especially supersymmetry models.
One possible decay channel would be for the Higgs boson to decay to two dark matter WIMPs. In such a decay to two particles (a WIMP of some sort and its anti-particle), each would have to have a rest mass energy equivalent of less than half of the 126 GeV Higgs boson mass; that is, the dark matter particle mass would have to be 63 GeV or less.
There may be more than one type of Higgs boson, and another Higgs family particle could be the main source of decays to dark matter. In supersymmetric extensions to the Standard Model, there is more than one Higgs boson expected to exist. In fact the simplest supersymmetric model has 5 Higgs particles!
Interestingly, there are 3 experiments which are claiming statistically significant detections of dark matter, these are DAMA/LIBRA, COGENT, and CDMS-II. And they are all suggesting a dark matter particle mass in the neighborhood of just 10 GeV. Heavy, compared to a proton, but quite acceptable in mass to be decay products from the Higgs in the early universe. It’s not a problem that such a mass might be much less than 63 GeV as the energy in the decay could also be carried off by additional particles, or as kinetic energy (momentum) of the dark matter decay products.
At the LHC the search is underway for dark matter as a result of Higgs boson decays, but none has been found. The limits on the cross-section for production of dark matter from Higgs decay do not conflict with the possible direct detection experiments mentioned above.
The search for dark matter at the LHC will actively continue in 2015, after the collision energy for the accelerator is upgraded from 8 TeV to 14 TeV (trillion electron-Volts). The hope is that the chances of detecting dark matter will increase as well. It’s a very difficult search because dark matter would not interact with the detectors directly. Rather its presence must be inferred from extra energy and momentum not accounted for by known particles seen in the LHC’s detectors.
## Supersymmetry in Trouble?
There’s a major particle physics symposium going on this week in Kyoto, Japan – Hadron Collider Physics 2012. A paper from the LHCb collaboration, with 619 authors, was presented on the opening day, here is the title and abstract:
# First evidence for the decay Bs -> mu+ mu-
A search for the rare decays Bs->mu+mu- and B0->mu+mu- is performed using data collected in 2011 and 2012 with the LHCb experiment at the Large Hadron Collider. The data samples comprise 1.1 fb^-1 of proton-proton collisions at sqrt{s} = 8 TeV and 1.0 fb^-1 at sqrt{s}=7 TeV. We observe an excess of Bs -> mu+ mu- candidates with respect to the background expectation. The probability that the background could produce such an excess or larger is 5.3 x 10^-4 corresponding to a signal significance of 3.5 standard deviations. A maximum-likelihood fit gives a branching fraction of BR(Bs -> mu+ mu-) = (3.2^{+1.5}_{-1.2}) x 10^-9, where the statistical uncertainty is 95% of the total uncertainty. This result is in agreement with the Standard Model expectation. The observed number of B0 -> mu+ mu- candidates is consistent with the background expectation, giving an upper limit of BR(B0 -> mu+ mu-) < 9.4 x 10^-10 at 95% confidence level.
In other words, the LHCb consortium claim to have observed the quite rare decay channel from B-mesons to muons (each B-meson decaying to two muons), representing about 3 occurrences out of each 1 billion decays of the Bs type of the B-meson. Their detection has marginal statistical significance of 3.5 standard deviations (one would prefer 5 deviations), so needs further confirmation.
What’s a B-meson? It’s a particle that consists of a quark and an anti-quark. Quarks are the underlying constituents of protons and neutrons, but they are composed of 3 quarks each, whereas B-mesons have just two each. The particle is called B-meson because one of the quarks is a bottom quark (there are 6 types of quarks: up, down, top, bottom, charge, strange plus the corresponding anti-particles). A Bs-meson consists of a strange quark and an anti-bottom quark (the antiparticle of the bottom quark). Its mass is between 5 and 6 times that of a proton.
What’s a muon? It’s a heavy electron, basically, around 200 times heavier.
What’s important about this proposed result is that the decay ratio (branching fraction) that they have measured is fully consistent with the Standard Model of particle physics, without adding supersymmetry. Supersymmetry relates known particles with integer multiple spin to as-yet-undetected particles with half-integer spin (and known particles of half-integer spin to as-yet-undetected particles with integer spin). So each of the existing Standard Model particles has a “superpartner”.
Yet the very existence of what appears to be a Higgs Boson at around 125 GeV as announced at the LHC in July of this year is highly suggestive of the existence of supersymmetry of some type. Supersymmetry is one way to get the Higgs to have a “reasonable” mass such as what has been found. And there are many other outstanding issues with the Standard Model that supersymmetric theories could help to resolve.
Now this has implications for the interpretation of dark matter as well. One of the favored explanations for dark matter, if it is composed of some fundamental particle, is that it is one type of supersymmetric particle. Since dark matter persists throughout the history of the universe, nearly 14 billion years, it must be highly stable. Now the least massive particle in supersymmetry theories is stable, i.e. does not decay since there is no lighter supersymmetric particle into which it can decay. And this so called LSP for lightest supersymmetric particle is the favored candidate for dark matter.
So if there is no supersymmetry then there needs to be another explanation for dark matter.
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# Immersed incompressible surfaces in surface bundles
Let $M$ be a closed, oriented, hyperbolic $3$-manifold which is a surface bundle over $\mathbb{S}^1$.
Is there some $\pi_1$-injective closed surface (perhaps not embedded) $S \subset M$ which is not a fiber, and which does not intersect all the fibers?
• For hyperbolic once-punctured torus bundles (which of course do not meet your first assumption) it is known that there are no closed incompressible surfaces except the boundary torus. This is Theorem 1a) in math.cornell.edu/~hatcher/Papers/2-bridgeKnots.pdf – ThiKu Apr 18 '18 at 5:54
• I wouldn‘t expect a similar result in higher genus fibrations. For an approach towards incompressible surfaces in fibered 3-manifolds you may look at the proof of Theorem 3.1 in homepages.warwick.ac.uk/~masgar/Maths/2005fibre.pdf – ThiKu Apr 18 '18 at 5:56
There is no such surface. If $S$ is a surface that misses a fiber then $S$ lies in a submanifold of $M$ that is homeomorphic to a product $F \times I$, where $F$ is the fiber. Any $\pi_1$-injective closed surface in a product is homotopic to a cover of the fiber $F$. This was shown by Waldhausen for embedded surfaces. The immersed case follows by passing to a covering space.
• Thank you very much for your answer. I am wondering now: is it possible to find two $\pi_1$-injective closed surfaces which are not fibers, and do not intersect? – Vanderson Lima Apr 21 '18 at 18:43
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# File Formats
Term Name Description
#### Asset
A generic term for graphics, sounds, maps, levels, models, and any other resources. Generally assets are compiled into large files. The file formats may be designed for fast loading by matching in-memory formats, or tight compressions for handheld games, or designed to otherwise help in-game use. It is often useful to have an asset tool chain. The original models may be high-density models with R8G8B8A8 images. You may have a model striper and image compresser that reduces the model for LOD, and compresses the texture to a DXT compressed image. These assets may then go through further transformations, and end up in the large resource file.
#### MOD
A music file format composed of sound samples in digitized format. Mod files both record the position, pitch and duration of notes, as well as including the actual sample data, and in this respect are almost a hybrid of some of the better features of both the WAV and MIDI formats. A free editor for mod files can be found at www.modplug.com.
#### 3DS
A File format for storing model geometry. Originally used by 3D Studio Max, it has become nearly a standard for 3D models.
#### BVH
BVH file - Biovision Hierarchical motion capture data (.bvh) A .bvh file is nothing more than a text file that has data that was captured from a moving skeletal system. Another name for this type of data capture is "Motion Capture" which has been abbreviated as mocap. A .bvh file can be made by software or by hardware. Most studio houses that do their own commercial animation have their own type of motion capture technique.
#### BMP
Bitmap: A set of bits that represents a graphic image, with each bit or group of bits corresponding to a pixel in the image.
#### TXT
A plain unformatted ASCII text file.
#### XML
Extensible Markup Language - A metalanguage written in SGML that allows you to design a markup language, used to allow for the easy interchange of documents on the World Wide Web.
#### HTML
HyperText Markup Language: A markup language used to structure text and multimedia documents and to set up hypertext links between documents, used extensively on the World Wide Web.
#### MNG
MNG (pronounced "ming"), is short for Multiple-image Network Graphics. Designed with the same modular philosophy as PNG and by many of the same people, MNG is intended to provide a home for all of the multi-image capabilities that have no place in PNG. Although the idea of MNG has been around almost as long as PNG has, serious design discussions didn't begin until May 1996, and even then there was considerable debate over whether to make MNG a dirt-simple glue'' format around PNG or a complex multimedia format capable of integrating animations, audio and even video. By mid-1998, MNG had settled down to something in between; while it has fairly extensive animation and image-manipulation capabilities, there is no serious expectation that it will ever integrate audio or video. MNG Homepage
#### TIFF
TIFF is an acronym for Tag(ged) Image File Format. It is one of the most popular and flexible of the current public domain raster file formats.
#### OGG
A compressed file format, similar to mp3. Features slightly better quality at the same compression rate. See http://www.vorbis.com for library and sources. Free, open source. Very liberal license. Can be used commercially without paying royalties.
#### EXE
An executable binary file. Some operating systems (notably MS-DOS, VMS, and TWENEX) use the extension .EXE to mark such files. This usage is also occasionally found among Unix programmers even though Unix executables don't have any required suffix. From The Jargon Dictionary
#### OBJ
An older Alias/Wavefront(creators of Maya) format. It's generally used for ease of loading and simplicity. No animation; ascii.
#### MD3
Quake III model format. Supports complete skeletal animation.
#### MD2
The model format used in Quake II. Supports vertex interpolation for animation.
#### MS3D
The native format for the Milkshape 3D modelling package. It supports texture coords, normals, skeletal animation among other things.
#### C
A C source code file.
#### CPP
A C++ source code file.
#### DFF
Criterion RenderWare 3.x 3D model/object file format.
#### JPEG
This is an ISO standard by the Joint Photographic Experts Group. JPEGs work with color resolutions of up to millions of colors (also called 24 bit color). When you save an image as a JPEG, it gets compressed, making the file smaller. However, JPEG compression is lossy, so each time you save a JPEG, you lose some data and reduce image quality. When saving a JPEG, you can choose a compression level from low to high. Low compression gives you better quality but a larger file size. High compression gives you a smaller file - but less quality.
#### WAV File
A file which stores audio information, saved with a ".wav" extension. WAV files are commonly used in big budget games because they provide excellent sound quality. But it takes a lot of data to provide such high quality sound. So WAV files are larger than those of other audio file formats. (Microsoft's standard sound exchange format.)
#### MP3
a format for audio compression, MP3 uses psychoacoustics to eliminate/reduce redundancy.
#### ACT File
.ACT files are the actor files for Genesis3D - also referred to as models.
#### PNG
Portable Network Graphics. A graphics format that was brought about due to the licensing disagreements conerning GIF and JPEG.
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# Write fractions in simplest form
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## Write fractions in simplest form
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