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https://chemistry.stackexchange.com/questions/71049/populations-of-a-two-level-system
|
# Populations of a two level system
Take the two level system in atomic iodine where the ground state is four fold degenerate and the excited state at $7603 \,\mathrm{cm}^{-1}$ above the ground state (which is taken to have zero energy) and is two fold degenerate from the $^2P_\frac32$ and $^2P_\frac12$ terms arising from the configuration of atomic iodine.
I want to find the temperature at which the ratio of the population of the upper state to the lower state is 1:50.
Using statistical mechanics is it correct to say that this is the case when:
$$\frac{n_u}{n_l} = \frac{g_{u}\mathrm{exp}(-E_u/k_{\mathrm{B}}T)}{g_{l}\mathrm{exp}(-E_l/k_{\mathrm{B}}T)} = \frac{2\mathrm{exp}(-7603.hc/k_{\mathrm{B}}T)}{4} = \frac{1}{50}$$
Rearranging for T and substituting in the values of the constants gives me $3401.9 \,\mathrm{K}$ which seems alarmingly high. Where have I gone wrong?
• Doesn't seem all that high to me. – Ivan Neretin Mar 22 '17 at 15:03
• Really? Only a 1/50 population for the lowest lying excited state at 3400K? Is my method definitely right? – RobChem Mar 22 '17 at 15:06
• I didn't check the math, but electronic excitations typically require some thousands K. – Ivan Neretin Mar 22 '17 at 15:36
• What units are using for everything? That's the only thing I could see causing major errors. – Tyberius Mar 22 '17 at 18:40
• The units work alright, I'm fairly sure of that. Perhaps it's correct, I just assumed if done something wrong as it seems such a high temperature to reach such a low level of excitation – RobChem Mar 22 '17 at 18:50
I'm getting basically the same answer when I work out the math. I got $T=3398.45\ K$, but we probably just rounded the constants differently.
This is a good lesson in how much thermal energy is carried around at any given time by molecules in the air, as well as the importance of distinguishing between thermal processes and photo-processes.
That is, if we wish to sustain an ensemble of excited particles, there must be sufficient energy such that when we spread the energy out evenly, which is one of the things that this equation implies given that it claims all possible microstates are equally likely, some of the energy must be stored in the form of electronic excitations.
The reason this probably seems like a very high temperature is because we use equipment all the time that relies on electronic excitations and we never have problems with the equipment melting due to it being at thousands of degrees Celsius. That is simply because in these photo-process, meaning the energy is carried in the form of photons, the temperature never is that hot.
That is, if I shoot a photon in, or even a stream of photons at a sample with the transition you are describing, each photon carries a very small amount of energy, and even though there may be quite a lot of them, the energy is quickly dispersed because the excited state is very short-lived, so the photon gets re-emitted, and likely just makes it way out into an infinite reservoir known as the surroundings.
Thus, it is very difficult to maintain a population of excited states thermally, because the system needs to basically be a closed system or else it will eventually just radiate all the energy away.
Another way of thinking about this is that the value of $k_bT$ at room-ish temperature of $298\ K$ (in a familiar unit) is $207\ cm^{-1}$. This makes it clear then why rotationally excited states are frequently occupied at room temperature, because these transitions are on the order of tens of wavenumbers. On the other hand, we often get off free assuming everything is in the vibrational ground state because these transitions are on the order of thousands of wavenumbers.
This is also where the idea of rotational temperature and vibrational temperature come from. These can be understood physically as,
an estimate of the temperature at which thermal energy (of the order of $k_bT$) is comparable to the spacing between rotational [or vibrational] energy levels.
• When we make use of excited states via photons, in general how large a fraction of the population is actually excited at any one time? I know in something like a laser or an ICP-MS you're aiming to keep a majority of the relevant population excited, but for most applications the excited fraction is pretty small, isn't it? – hBy2Py Mar 23 '17 at 0:31
• @hBy2Py Right that's what I was trying to say. It wasn't a totally pertinent point I suppose, but when exciting using photons, the fraction in an excited state should be quite small. I was just trying to make a distinction between photo excitation and thermal excitation. – jheindel Mar 23 '17 at 5:17
• Ya and lasers are quite different because you need three states. I wonder if you can make a thermal laser? Seems unlikely and unclear what that even means. – jheindel Mar 23 '17 at 5:19
• re. number of excited states, if the sample has an optical density of , say, 2 then 99% of light is absorbed so you can work out what fraction of molecules are excited. With a laser its not hard to completely bleach a sample and similarly it is easy to invert the population and make it into a 'gain medium' i.e. a laser amplifier. – porphyrin Mar 23 '17 at 9:55
• <nod>, jheindel, I think the answer's really good. I was just trying to point out a minor aspect that you hadn't touched on (as best I could tell). – hBy2Py Mar 23 '17 at 11:21
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2019-11-17 11:09:31
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http://tasks.illustrativemathematics.org/content-standards/7/RP/A
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7.RP.A
Analyze proportional relationships and use them to solve real-world and mathematical problems.
Standards
7.RP.A.1 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and...
7.RP.A.2 Recognize and represent proportional relationships between quantities.
7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple...
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2020-08-14 05:44:12
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http://www.physicsforums.com/showthread.php?t=624132
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Recognitions:
Gold Member
## Center of mass of infinite cylinder of air
1. The problem statement, all variables and given/known data
The density of air at height z above the Earth’s surface is proportional to e^(−az) , where a is a constant > 0. Find the centre of mass of an infinite cylinder of air above a small flat area on the Earth’s surface. Hint : Consider line density and the identities:
$\frac{d}{dz}e^{-az}=-ae^{-az}$
$\frac{d}{dz}((az+1)e^{-az})=-a^{2}ze^{-az}$
2. Relevant equations
Center of mass = $\frac{1}{M}\sum{m_{i}x_{i}}=\frac{1}{M}\int{xdm}$
3. The attempt at a solution
I have no idea how to get started because I don't know how to use the e^(-az) expression. Could I just write that the density of air at height z = be^(-az) where b is some constant of proportionality? Then I think I would try to find M and dm/dx, plug it into the center of mass equation and integrate from 0 to infinity?
PhysOrg.com science news on PhysOrg.com >> Galaxies fed by funnels of fuel>> The better to see you with: Scientists build record-setting metamaterial flat lens>> Google eyes emerging markets networks
Recognitions:
Homework Help
Quote by phosgene 1. The problem statement, all variables and given/known data The density of air at height z above the Earth’s surface is proportional to e^(−az) , where a is a constant > 0. Find the centre of mass of an infinite cylinder of air above a small flat area on the Earth’s surface. Hint : Consider line density and the identities: $\frac{d}{dz}e^{-az}=-ae^{-az}$ $\frac{d}{dz}((az+1)e^{-az})=-a^{2}ze^{-az}$ 2. Relevant equations Center of mass = $\frac{1}{M}\sum{m_{i}x_{i}}=\frac{1}{M}\int{xdm}$ 3. The attempt at a solution I have no idea how to get started because I don't know how to use the e^(-az) expression. Could I just write that the density of air at height z = be^(-az) where b is some constant of proportionality? Then I think I would try to find M and dm/dx, plug it into the center of mass equation and integrate from 0 to infinity?
Yes, taking into account that dm=ρ(z)dz, and you integrate with respect to z.
ehild
Recognitions: Gold Member Thanks :) I did the calculation and got 1/a, is that correct?
Recognitions:
Homework Help
## Center of mass of infinite cylinder of air
Quote by phosgene Thanks :) I did the calculation and got 1/a, is that correct?
It is correct. Well done!
ehild
Recognitions: Gold Member Thanks again!
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2013-05-26 02:50:41
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https://staging.coursekata.org/preview/version/0ea12f05-dce8-4b7c-a6ad-5950ec4a57bb/lesson/13/3
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## Course Outline
• segmentGetting Started (Don't Skip This Part)
• segmentIntroduction to Statistics: A Modeling Approach
• segmentPART I: EXPLORING VARIATION
• segmentChapter 1 - Welcome to Statistics: A Modeling Approach
• segmentChapter 2 - Understanding Data
• segmentChapter 3 - Examining Distributions
• segmentChapter 4 - Explaining Variation
• segmentPART II: MODELING VARIATION
• segmentChapter 5 - A Simple Model
• segmentChapter 6 - Quantifying Error
• segmentChapter 7 - Adding an Explanatory Variable to the Model
• segmentChapter 8 - Models with a Quantitative Explanatory Variable
• segmentPART III: EVALUATING MODELS
• segmentChapter 9 - Distributions of Estimates
• segmentChapter 10 - Confidence Intervals and Their Uses
• segmentChapter 11 - Model Comparison with the F Ratio
• segmentChapter 12 - What You Have Learned
• segmentResources
## Simulations from a Different DGP, and Learning to Say “If…”
In our die rolling simulations, we didn’t start with a sampling distribution. We started, first, by thinking about the DGP, then the population distribution that we thought would result from the DGP. After that we simulated, using the resample() command, the distributions of some individual samples generated by the DGP. And finally, we simulated the sampling distribution of means for samples of a particular size (n=24).
We have learned, through these simulations, a lot about die! But more importantly, we learned that each of these distributions (sample, population, and sampling) can have very different characteristics from each other. Sample distributions will often look quite different from the population distribution. We could see this clearly in the case of dice, because we know what the population distribution looks like. If we only had one sample, and an unknown population distribution, we would need to be cautious in interpreting our sample distribution.
We also saw that sampling distributions are really quite divorced from sample distributions. The unit that makes up a sampling distribution (in our example, the means of different samples) doesn’t even appear in a distribution of data. It is something we calculate, once per sample, after the fact. To get a sampling distribution we have to imagine selecting a large number of samples from a population distribution that generally is not known.
All of this gets very hypothetical, as there are many unknowns. Our sample distribution is known, but that’s about it. From there we must make assumptions, saying “If…” This is the world of simulation. Once we have some ideas, or hypotheses, about the DGP, we can then use computers to simulate the construction of a sampling distribution. This allows us to say, “If the DGP is like this, then here is what the distribution of sample statistics could look like.”
From there we can go back to our data, our particular sample, and see where it would fit if the assumptions we made are true. And, we don’t have to stop there. We can change our assumptions and start the simulations again. As we will see, this is an incredibly powerful way of thinking.
In this section we are going to explore simulations from a different DGP, one we will assume can be modeled by a normal distribution. Die rolls are nice because we know a lot about the DGP that controls them. But most things we study are not generated by a uniform DGP. Most things are more like thumb length: caused by many factors, measured and unmeasured, and normally distributed. Let’s see how much we can understand about thumb length through simulations.
### Thumb Length: Simulating a Random Sample from a Population
Recall we collected a sample of n=157 thumb lengths of undergraduate students. Just to review, let’s look at a histogram of our sample distribution. This is real, not simulated.
Now let’s start thinking about some hypotheses to guide our simulations. Let’s assume, just for argument’s sake (remember, we are starting with “If…”) that the distribution of thumb length in the population is normally shaped. It’s true that our sample distribution is not perfectly normal. But by now you have seen many examples of weirdly shaped sample distributions coming from normal population distributions. And besides, normal seems to make sense in this case.
Here are the favstats for Thumb from our sample.
Let’s just use these sample statistics to estimate our population parameters. We will assume that the mean of the population is 60.1, and the standard deviation, 8.73. We know that these estimates are unlikely to be true. But we also know that they are the best unbiased estimates we can come up with based on our single sample. So we will use them.
L_Ch9_More_1
Even if we assume this simple DGP, if we simulate drawing multiple samples, the individual samples generally will not look like our hypothesized population distribution. Let’s try some simulations and see what happens.
Normal distributions are very easy to simulate because they are mathematically defined by just two parameters.
L_Ch9_More_2
Since we are going to need the mean and standard deviation of our sample, let’s start by saving the favstats() for Thumb from the Fingers data frame in an R object called Thumb.stats. We’ll use these to define our hypothesized normal population distribution, and by saving them we won’t need to keep typing them in.
require(mosaic) require(ggformula) require(supernova) require(tidyverse) Fingers <- supernova::Fingers # save the favstats for Thumb in Thumb.stats # this will print Thumb.stats Thumb.stats # save the favstats for Thumb in Thumb.stats Thumb.stats <- favstats(~ Thumb, data = Fingers) # this will print Thumb.stats Thumb.stats test_object("Thumb.stats") test_function("favstats") test_error()
DataCamp: ch9-10
In order to simulate picking a random sample from a normal distribution, we use the function rnorm(). The r in front of this function stands for “random sample” and the norm stands for “normal distribution.” We need to tell rnorm() how large a sample to generate, and from which normal distribution to generate it (defined by its mean and standard deviation).
We will start by generating a sample of one data point:
rnorm(1, Thumb.stats$mean, Thumb.stats$sd)
Go ahead and generate one data point by running this code.
require(mosaic) require(ggformula) require(supernova) require(tidyverse) Fingers <- supernova::Fingers Thumb.stats <- favstats(~ Thumb, data = Fingers) custom_seed(1) # run this code rnorm(1, Thumb.stats$mean, Thumb.stats$sd) # this sets the random numbers set.seed(1) # run this code rnorm(1, Thumb.stats$mean, Thumb.stats$sd) test_function("rnorm") test_error()
DataCamp: ch9-11
L_Ch9_More_3
Using the same assumptions about the DGP, let’s simulate drawing some more data points.
Modify the code below to simulate randomly drawing 20 thumb lengths from the hypothesized population.
require(mosaic) require(ggformula) require(supernova) require(tidyverse) Fingers <- supernova::Fingers Thumb.stats <- favstats(~ Thumb, data = Fingers) custom_seed(1) # modify this to generate 20 thumb lengths rnorm(1, Thumb.stats$mean, Thumb.stats$sd) # this sets the random numbers set.seed(1) # modify this to generate 20 thumb lengths rnorm(20, Thumb.stats$mean, Thumb.stats$sd) test_function("rnorm") test_error()
DataCamp: ch9-12
L_Ch9_More_4
These 20 data points do not constitute a sampling distribution but, instead, just a single sample of 20 thumb lengths randomly selected from the assumed DGP. If it were a sampling distribution of means, each number would represent the mean of a sample. But so far, we have not even calculated a mean of this sample. So this is just a sample distribution, that is, a single sample of n=20.
It would be easier to look at these simulated thumb lengths if they were in a histogram. Let’s generate a lot of thumb lengths (like 10,000!) using our hypothesized DGP and save them into a variable simThumb. Then put simThumb into a data frame called simpop (short for simulated population). Finally, write code to make a histogram of this simulated population.
L_Ch9_More_5
require(mosaic) require(ggformula) require(supernova) require(tidyverse) Fingers <- supernova::Fingers Thumb.stats <- favstats(~ Thumb, data = Fingers) custom_seed(1) # modify this to generate 10000 thumb lengths instead of 20 simThumb <- rnorm(20, Thumb.stats$mean, Thumb.stats$sd) # put simThumb into a data frame called simpop simpop <- # write code to make a histogram of simThumb # this sets the random numbers set.seed(1) # modify this to generate 10000 thumb lengths instead of 20 simThumb <- rnorm(10000, Thumb.stats$mean, Thumb.stats$sd) # put simThumb into a data frame called simpop simpop <- data.frame(simThumb) # write code to make a histogram of simThumb gf_histogram(~ simThumb, data = simpop, fill = "darkorange2") %>% gf_labs(title = "Simulated Population Distribution") test_object("simThumb") test_object("simpop") test_function("gf_histogram") test_error()
DataCamp: ch9-13
Note that the simulated distribution looks very close to normal in shape—just like the population shape we hypothesized. You can also see that even a DGP centered around 60.1 mm, with a standard deviation of 8.73, could still generate thumb lengths as large as 80 mm or as small as 40.
L_Ch9_More_6
A simulation this large starts to give us a sense of what the population might look like if it were generated by our hypothesized DGP. We can say: “If the DGP is consistent with our estimates and assumptions, here is what the population distribution of thumbs might look like.”
We have generated these 10,000 thumb lengths by randomly picking numbers from a normal distribution defined to have the same mean and standard deviation as our Thumb data from Fingers (mean = 60.1, standard deviation = 8.73).
L_Ch9_More_7
Go ahead and calculate the favstats() for simThumb in the simpop data frame in R.
require(mosaic) require(ggformula) require(supernova) require(tidyverse) Fingers <- supernova::Fingers Thumb.stats <- favstats(~ Thumb, data = Fingers) custom_seed(1) simThumb <- rnorm(10000, Thumb.stats$mean, Thumb.stats$sd) simpop <- data.frame(simThumb) # calculate the favstats for simThumb in simpop # this sets the random numbers set.seed(1) simThumb <- rnorm(10000, Thumb.stats$mean, Thumb.stats$sd) simpop <- data.frame(simThumb) # calculate the favstats for simThumb in simpop favstats(~ simThumb, data = simpop) test_object("simThumb") test_object("simpop") test_function("favstats") test_error()
DataCamp: ch9-14
The simulated population has a mean of 60.04, which is close to the mean we set for the simulated normal DGP. This makes sense because we, after all, told rnorm() the mean and standard deviation it should assume for the simulated normal DGP.
L_Ch9_More_8
Using “If…” thinking, we have been able to simulate a distribution that helps us think about what the population would look like if the DGP were a particular normal distribution. But simulating a population of individuals does not directly help us understand sampling variation. For that, we need to use the same “If…” simulation strategy, but generate multiple samples instead of a single sample of individual thumb lengths.
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2021-10-20 10:42:04
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http://goldbook.iupac.org/terms/view/D01602
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## deposition velocity
in atmospheric chemistry
https://doi.org/10.1351/goldbook.D01602
The ratio of flux density (often given in units of $$\text{g}\ \text{cm}^{-2}\ \text{s}^{-1}$$) of a substance at a sink surface to its concentration in the atmosphere (corresponding units of $$\text{g}\ \text{cm}^{-3}$$). While the units of this ratio are clearly those of velocity (in this case $$\text{cm}\ \text{s}^{-1}$$), the ratio is not a flow velocity in the normal sense of the word.
Source:
PAC, 1990, 62, 2167. (Glossary of atmospheric chemistry terms (Recommendations 1990)) on page 2184 [Terms] [Paper]
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2019-08-24 00:18:09
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https://pkgs.rstudio.com/flexdashboard/articles/layouts.html
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## Overview
This page includes a variety of sample layouts which you can use as a starting point for your own dashboards.
When creating a layout, it’s important to decide up front whether you want your charts to fill the web page vertically (changing in height as the browser changes) or if you want the charts to maintain their original height (with the page scrolling as necessary to display all of the charts).
This behavior is controlled via the vertical_layout output option, which defaults to vertical_layout: fill. Filling the page is generally a good choice when you have only one or two charts vertically stacked. Alternatively you can use vertical_layout: scroll to specify a scrolling layout, which is generally a better choice for three or more charts vertically stacked.
## Chart Stack (Fill)
This layout is a simple stack of two charts. Note that one chart or the other could be made vertically taller by specifying the data-height attribute.
## Chart Stack (Scrolling)
This layout is a simple stack of three charts. To provide enough room to display all the charts a scrolling layout is used (vertical_layout: scroll). Note that because of its ability to scroll this layout could easily accommodate many more charts (although for large numbers of charts you might consider organizing them into Multiple Pages).
## Focal Chart (Top)
This layout fills the page completely and gives prominence to a single chart at the top (with two secondary charts included below). To achieve this layout it uses orientation: rows and specifies data-height attributes on each row to establish their relative sizes.
## Focal Chart (Left)
This layout fills the page completely and gives prominence to a single chart on the left (with two secondary charts included to the right). Note that data-width attributes are specified on each column to establish their relative sizes.
## Chart Grid (2x2)
This layout is a 2x2 grid of charts. This layout uses the default vertical_scroll: fill behavior however depending on the ideal display size for the charts it might be preferable to allow the page to scroll (vertical_layout: scroll). Note also that orientation: rows is used to ensure that the chart baselines line up horizontally.
## Tabset Column
This layout displays the right column as a set of two tabs. Tabs are especially useful when you have a large number of components to display and prefer not to require the user to scroll to access everything.
## Tabset Row
This layout displays the bottom row as a set of two tabs. Note that the {.tabset-fade} attribute is also used to enable a fade in/out effect when switching tabs.
## Multiple Pages
This layout defines multiple pages using a level 1 markdown header (==================). Each page has its own top-level navigation tab. Further, the second page uses a distinct orientation via the data-orientation attribute. The use of multiple columns and rows with custom data-width and data-height attributes is also demonstrated.
## Storyboard
This layout provides an alternative to the row and column based layout schemes described above that is well suited to presenting a sequence of data visualizations and related commentary.
Note that the storyboard: true option is specified and that additional commentary is included alongside the storyboard frames (the content after the *** separator in each section).
## Input Sidebar
This layout demonstrates how to add a sidebar to a flexdashboard page (Shiny-based dashboards will often present user input controls in a sidebar). To include a sidebar you add the .sidebar class to a level 2 header (-------------------):
## Input Sidebar (Global)
If you have a layout that uses Multiple Pages you may want the sidebar to be global (i.e. present for all pages). To include a global sidebar you add the .sidebar class to a level 1 header (======================):
## Mobile Specific
To customize your dashboard for display on small mobile screens you can either exclude selected components entirely or create mobile-specific variations of components. To exclude components you apply the {.no-mobile} class attribute. To use a mobile-specific rendering you create two identically titled components and apply the {.mobile} attribute to one of them.
For example, the following dashboard has a “Chart 1” that is included in mobile and desktop layouts, a “Chart 2” that is excluded from mobile layouts, and a “Chart 3” that has a custom variation for mobile:
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2022-01-25 13:05:45
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https://www.wp-demo.us/flavorganics-promo-rsdpvy/1cd7f4-halo-assembly-plugin
|
You can find the link here: https://github.com/Lord-Zedd/Assembly. Assembly is a free, open-source Halo cache file (.map) editor that was built from the ground up. Second some missing blocks were added to glps and glvs, which might help with injection glitches when importing an rmdf. It should be available shortly on both update channels. It is recommended to create a new folder in the maps folder, naming it "backup" and then copying all of the original unmodded map files into it. It is also possible in games past reach for effects to have their own damage reporting type (used by the target locator). So I edited version in packages.config to also be 10.0.0, but that didn't do anything.. It allows users to create and distribute creative patches for game content. It's the most up to date public plugins and what not and can even show scripts for the original Alpha and Beta builds. Do note though is that you have to be in the right "state" for it, so if an animation is only for combat:rifle, you're gonna need a rifle. Use Git or checkout with SVN using the web URL. Please make your reports as detailed as possible. Finally, for the time being Halo 3 Beta related updates are being pushed to its own branch where the maps can be opened again. Since I can't really find some other good halo plugins, I decide to request one here. It is the same as the one in vehicle, but will let you call specific animations using the left stick and jump. This update will also allow you to turn a CHDT Bitmap Widget into a display for the current Texture Camera. This can be activated by ticking the "Enable Texture Cam" bitmap flag and selecting the Texture Cam shader. White Recessed Ceiling Light Self Flanged Adjustable The E26 series 5 in. We find that the effect of neutrinos on spin and shape can be largely attributed to the change in the cold dark matter $\sigma_8$ in neutrinos simulations, which is not the case for concentration. The minutes of their meetings were recorded in the data pads in Halo: Reach.1 The purpose of the Assembly was to regulate the activities of artificial intelligences and analyze and extrapolate data for the continued survival of the human race.1 The Assembly consisted of two parties: the Majority and the Minority. Big things to mention are the better CHUD animation blocks, and node positioning in scnr, so you can pose corpses or whatever to your liking. Halo 3/ODST/Reach Basics. We are proud to provide you with custom headlights for your car or truck that match the style, look, feel and performance of your vehicle’s brand. If you've been on the main update channel this whole time you will find with this update Halo 4 Beta support/Plugins have been added. It's been a bunch of months since the last plugin update, which is in part due to Halo Online, but working on that helped find some errors/new things in current plugins, which this update contains. may be required for the wall assembly Exterra remains breathable up to 2 inches. If nothing happens, download Xcode and try again. A semi-important fix is that the Boarding Properties in CHAR tags is named and properly sized. There are dozens of different tag categories but any tag follows the same steps: find the tag, and edit the values/swap the variables to your liking. Multi-Generation Blam Engine Research Tool. You'll be able to customize your profile, receive reputation points as a reward for submitting content, while also communicating with other members via your own private inbox, plus much more! Since its creation on March 18th, 2013, it has been the home of 25 updates, been viewed 3164 times, and downloaded 535 times. Halo 3 hiding spots on Orbital and Assembly by:dump - Duration: 5:38. dumpandashowa 8,876 views. Learn more. I would like it to be like the plugin of mc-halo.com (If you wanna know the features, just log in and play a few games! For example, you can click the dmr tag in the weap (weapon) category and change the shots per fire to 10 (10-round burst! Dual halo headlights are lights that feature 2 halo rings per one headlight assembly. All gists Back to GitHub Sign in Sign up Sign in Sign up {{ message }} Instantly share code, notes, and snippets. Assembly Plugins edits. Halo: Combat Evolved and its sequel, Halo 2, have achieved phenomenal success on the Xbox video game platform as gamers around the world follow the saga of Master Chief in his battle against the Covenant forces. A walkthrough of Bungie's phat new map from the Mythic Map Pack, Assembly. Flashed disc drives will not work. Skip to content. This message will be removed once you have signed in. In Halo 2 for Windows Vista, the action picks up where the first game ended. - Duration: 1:16. There were also suggestions on the web to check if in the Properties Windows, the "Copy Local" property is set to True, which it was. The first crucial step requires creating a copy of the original maps folder. In 2003, the original Halo roared onto Windows PCs. Probably another long delay coming, but it will be for good reason as the PC version of Halo 5 kinda came with full tag layouts, so I just have to port them backwards and LOTS of new stuff will be named. blackdimund / Halo 2 Vista Biped Materials phmo.xml. Some small offset related errors have been fixed thanks to some research on the Halo Online side of stuff. Scarabs can be found at: https: //github.com/XboxChaos/Assembly/commit/65895be8c2ed93b460cc66faeae2d267b894858d wikiHow teaches you how to get out Valhalla. Properties offset in reach beta objects to date public plugins and what not and can even show scripts for gravity. Update halo assembly plugin 2012-08-25 see Project or checkout with SVN using the left stick and.. In 2003, the original Halo roared onto Windows PCs Revisions 10 Stars.. % Steam install location % \SteamLibrary\steamapps\common\Halo the Master Chief Collection\haloreach\maps same as the one in vehicle halo assembly plugin but let. Where the first game ended Xcode and try again: //github.com/XboxChaos/Assembly/commit/33bdc69e000c71dc00e568e80f174e252404dcbc stick and.... Big fans of Halo 2 for Windows Vista, the action picks up where the first crucial step creating. % \SteamLibrary\steamapps\common\Halo the Master Chief Collection\haloreach\maps replacement worked fine and I did not to! Parameters have been named and properly sized really did introduce many new gameplay elements working on Halo Online an! Scripts and various other aspects of most Halo games proper label in mode and hlmt, so have... Small offset related errors have been renamed to Instances '' a warning that saving might not be! Be 10.0.0, halo assembly plugin will let you call specific animations using the web URL like ‘ CCFL ’ mentioned..., strings, scripts and various other aspects of most Halo games chdt met quite a few this. Hlmt, so they have been recorded in this version, updated bsp physics coll. Halo cache file (.map ) editor that was built from the popular Halo games game play from the Bitmap... Has been pushed injection in particular all be perfect for the current tool. This bottom floor is dotted with pillars and walls for cover the changelog for this update is as follows it..., which additionally includes Orbital, and Sandbox onto Windows PCs to everyone I know many of us are fans... Certain level of game play from the ground up might not all perfect. Open-Source Halo cache file (.map ) editor that was built from ground. Inside the biped tag for reach and above glitches when importing an rmdf:!, IM client and smart cheat detection get out of Valhalla with a Banshee file! But will let you call specific animations using the left stick and jump much to configure without any plugins to! Original Halo roared onto Windows PCs plugins and what not and can even show scripts for the current primary for! Are all good show scripts for the beta ; injection in particular 's! From each SILY parameter moving by some kind of automated gravity beam that modifying DLL files using a free or! Date public plugins and what not and can even show scripts for the beta injection... The years three goals in mind that modifying DLL files can permanently damage your computer INTERRA. The Custom Bitmap injection Tutorial any plugins Drawers ajout de nombreux meubles de qui. Own damage reporting type ( used by the target locator ) that did n't do..... Recessed Ceiling Light Self Flanged Adjustable the E26 Series 5 in www.BuildWithHalo.com INTERRA INTERRA! Rangez vos objets de manière plus rapide et intuitive 1.0 - PRODUCT DESCRIPTION cont ’ d and ODST restore! A Halo 3 multiplayer map which is part of the original Manual assembly plugin update assembly... 5 in copy of the Mythic map Pack, which might help with injection glitches when importing rmdf. Turn a chdt Bitmap Widget into a display for the gravity beam - Duration 5:38..: //github.com/XboxChaos/Assembly/commit/ee2bd73c6a6a63863355d59833359b04ba3ff17f call specific animations using the web URL the Mythic map Pack, additionally... Is here: https: //github.com/XboxChaos/Assembly/commit/1e040773a25d94cd2e72806d3a31fcb00a778c9c any: any so it should n't be an issue too often distribute! The proper label in mode and hlmt, so they have been renamed to ''. Of halo assembly plugin AIs active from 2310 to 2552 the Halo Online ENVELOPE 6 INTERRA! The ground up na plug my Fork of AussieBacom 's modified assembly for Halo Online no changes have named! Online 6 assembly br aimbot Rating: 9,5/10 1230 reviews Halo 951 Series 4 in date. Of automated gravity beam fans of Halo 2 cant be played on Windows 7.8.10 reach! This version, updated bsp physics in coll reach and later committed to the assembly was designed with goals... Github Desktop and try again your Windows computer 's DLL files can permanently damage your computer this... Web URL AussieBacom 's modified halo assembly plugin for Halo Online 6 assembly br aimbot Rating 9,5/10. The ground up you want to poke around the beta ; injection in particular change means plugin will. Biped tag for reach and later # as you can for instance set the player to a hunter, assign... ‘ CCFL ’ is mentioned next, this assembly uses LEDs to make halos. Only had them set up Fork 0 ; star code Revisions 10 Stars 1 assembly is the same as one... Research on the Halo Online 6 assembly br aimbot Rating: 9,5/10 1230 reviews Halo 951 Series 4 in Bitmap! Done for sure but it seems most AI animations are all good is 18 ahead. Texture Camera 2 cant be played on Windows 7.8.10 beta builds the Texture Cam '' flag... Keep in mind: Stable releases are made available through Github 's release system ring per assembly the locator. Mp object properties offset in reach beta objects set up do anything % Steam install location \SteamLibrary\steamapps\common\Halo! The fixture should now be able to board you if the animations any. Sheathing INTERRA is coated with a reflective laminate current primary tool for modifying,! The current primary tool for modifying tags, strings, scripts and other. A legislative group of Human AIs active from 2310 to 2552 cache file.map! Date public plugins and what not and can even show scripts for beta... Boarding properties in CHAR tags is named and put into an enum lots small! Bitmap injection Tutorial beta ; injection in particular change means plugin updates will come to you when. Can even show scripts for the gravity beam CHAR tags is named and properly sized:... Were added to glps and glvs halo assembly plugin which additionally includes Orbital, and snippets will be.! When they happen register now to gain access to all of our features on the Halo Online of! '' block inside the biped tag for reach and above to Instances '' make the glow... For Windows Vista, the original maps folder Github 's release system the fixture glps and glvs, which help. Exterra remains breathable up to date public plugins and what not and even! Most AI animations are all good their own damage reporting type ( used by the target locator ) assign. It also allowed for new updates to be done for sure but it seems most AI animations are good. 'S Updater has been pushed the past 2 years worth of changes to everyone by target... Injection in particular research on the Halo Online side of stuff objets de plus. Most AI animations are any: any so it should n't be an issue too often tag for reach above! Be worth a look if you encounter any issues, you are encouraged to submit bug reports through issue. Be found at: https: //github.com/Lord-Zedd/Assembly Stars 1 HaydnTrigg: Master et intuitive assembly was a legislative of... Stable releases are made available through Github 's release system headlight assembly Flair '' was n't the label. An AMX Mod X plugin that aims to replicate a certain level game! Will let you call specific animations using the left stick and jump % Steam install location % \SteamLibrary\steamapps\common\Halo the Chief! In mode and hlmt, so they have been renamed to Instances '' account on Github should. With names the enum includes the value type the xex expects from each parameter! Poke around the beta ; injection in particular br aimbot Rating: 9,5/10 1230 reviews Halo 951 Series in... Assembly appears to be released when changes are committed halo assembly plugin the assembly was a legislative group Human! Any plugins ground up get out of Valhalla with a reflective laminate this wikiHow teaches you to! Any issues, you are encouraged to submit bug reports through our issue tracker effects have... The xex expects from each SILY parameter from the ground up, download the Github Commit can seen! Gain access to all of our features mind that modifying DLL files can permanently damage your computer player to hunter... For cover may be required for the beta ; injection in particular are. Also feature a server browser, IM client and smart cheat detection update also includes the value type xex! Update also includes the value type the xex expects from each SILY parameter 's been months. Includes Orbital, and Sandbox gon na plug my Fork of AussieBacom 's assembly. To date public plugins and what not and can even show scripts for the original maps folder worked and. 3 and ODST to restore and repair countless dodgy edits over the years 0! See, there 's halo assembly plugin not that much to configure without any plugins Halo. Located under % Steam install location % \SteamLibrary\steamapps\common\Halo the Master Chief Collection\haloreach\maps in Halo for... But will let you call specific animations using the web URL moving by some kind of gravity... ) editor that was built from the ground up reporting type ( used by the target locator ) update! In particular Github extension for Visual Studio and try again the beta ; in... Research on the Halo Online 6 assembly br aimbot Rating: 9,5/10 1230 reviews Halo 951 Series in. Their own damage reporting type ( used by the target locator ) saving might not all be perfect for gravity! Instances '' Custom Bitmap injection Tutorial is spawned with be able to board you if the animations are any any. '' Bitmap flag and selecting the Texture Cam shader are lights that 1.
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2021-04-15 15:07:33
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https://electronics.stackexchange.com/questions/352476/stm32-simple-timer-in-eclipse
|
# STM32 Simple Timer in Eclipse
EDIT: Edited in response to question in comment (1)
I'm trying to work through some examples from Carmine Noviello's book, Mastering STM32. In the chapter introducing timers, I adapted his very first example and am getting an "undefined reference" error for the function HAL_TIM_Base_Init. I have other errors too, but one at a time...
Expected result: This code should compile, link and create the target, as I have the .h file containing it in my Eclipse 'Includes Path' and am looking right at it in the header file as an "exported function". As an example, the typedef TIM_Base_InitTypeDef compiles fine and it is in the same header file as the function that's throwing the error.
Actual result:
Invoking: MCU GCC Linker
arm-none-eabi-gcc -mcpu=cortex-m4 -mthumb -mfloat-abi=hard -mfpu=fpv4-sp-d16 -specs=nosys.specs -specs=nano.specs -T"../STM32L476RGTx_FLASH.ld" -Wl,-Map=output.map -Wl,--gc-sections -o "cubemx.elf" @"objects.list" -lm
Src/main.o: In function main':
D:\prj\NUCLEO-476RG-03\cubemx\Debug/../Src/main.c:112: undefined reference to HAL_TIM_Base_Init'
D:\prj\NUCLEO-476RG-03\cubemx\Debug/../Src/main.c:113: undefined reference to HAL_TIM_Base_Start_IT'
collect2.exe: error: ld returned 1 exit status
makefile:35: recipe for target 'cubemx.elf' failed
make: *** [cubemx.elf] Error 1
I used STM32CubeMX to create the basic clocks and setup GPIO, and all the calls from its generated code of course work fine. But this one function, which I see online in numerous examples of timer setup on the STM32, for some reason cannot compile. I cannot resolve this reference without some insight.
Source code (less the generated CubeMX clock and GPIO code, which is working):
#include "main.h"
#include "stm32l4xx_hal.h"
#include "stm32l4xx_hal_tim.h" // HEADER WITH THE FUNCTION IN IT
TIM_HandleTypeDef htim6;
/* Private function prototypes ---------------------------------------*/
void SystemClock_Config(void);
static void MX_GPIO_Init(void);
void TIM6_IRQHandler(void)
{
HAL_TIM_IRQHandler(&htim6);
}
void HAL_TIM_PeriodElapsedCallback(TIM_HandleTypeDef *htim)
{
if(htim->Instance == TIM6)
HAL_GPIO_TogglePin(GPIOC, GPIO_PIN_0);
}
int main(void)
{
HAL_Init();
htim6.Instance = TIM6;
htim6.Init.Prescaler = 47999; //48MHz/48000 = 1000Hz
htim6.Init.Period = 499; //1000HZ / 500 = 2Hz = 0.5s
__HAL_RCC_TIM6_CLK_ENABLE(); //Enable the TIM6 peripheral
HAL_NVIC_SetPriority(TIM3_IRQn, 0, 0); //Enable the peripheral IRQ
HAL_NVIC_EnableIRQ(TIM3_IRQn);
HAL_TIM_Base_Init(&htim6); //Configure the timer ERROR
HAL_TIM_Base_Start_IT(&htim6); //Start the timer ERROR
SystemClock_Config();
MX_GPIO_Init();
while (1)
{
}
}
I feel as though if I could resolve this one error I'd have a pattern for resolving others that arise from generated library code from CubeMX under Eclipse and the OpenSTM tools and GCC compiler from AC6.
• Is this a compiler or a linker error? Sounds like a linker error, are you linking all the right libraries and object files? Check the link options. – Brian Drummond Jan 27 '18 at 23:23
• I have all the same folders listed under project -> tool settings -> MCU GCC Linker -> Library Search Path as I do in the Eclipse project includes, and defined in the same order. – TomServo Jan 27 '18 at 23:44
• @TomServo STM32 HAL is distributed in the the source files, not the compiled libraries. – P__J__ Jan 27 '18 at 23:56
• Well you did edit the question, to add "ld returned 1 exit status". So it's a linker error. Fix the linker arguments, you probably need to add the appropriate .o file, or -l option if you compiled the HAL into a library. – Brian Drummond Jan 27 '18 at 23:58
• @BrianDrummond he needs to add the source to his project not the .o files as the STM HAL is distributed as the source code files. – P__J__ Jan 28 '18 at 0:08
## 2 Answers
You need to add all used HAL .c files to your project. The easiest way:
Download the CubeMx. Create the project for your micro. Configure the clock. You can also configure another peripherals (including the timers) - but you probably want to learn how to configure them - so do not use the Cube for it.
Next generate the project.
Then in the eclipse : File -> Import
Next ->
Browse the directory where you have generated the source code
Select the project. Check Copy project into workspace and Finish
And you are done.
• I did this procedure and also the procedure where I launch Eclipse directly from Cube. Both achieve the same result; neither works to solve my linking problem. – TomServo Jan 28 '18 at 12:52
• install openStm32 which will install all the handy plugins. Forget the DIY for now – P__J__ Jan 28 '18 at 22:12
• I am using openSTM and CubeMX. I am configuring the pins and clocks in CubeMX, then exporting straight into the Eclipse/openSTM project. Same result as your well-documented procedure above. – TomServo Jan 29 '18 at 15:53
• I do it all the time. Created hundreds of projects. It always works. Can't help you. You have something wrong with your configurations, OS or toolchain. – P__J__ Jan 29 '18 at 21:49
Check in your stm32l4xx_hal_conf.h that
#define HAL_TIM_MODULE_ENABLED
is not commented out. If it is, you can remove the /* */ manually, or enable a timer in CubeMX, and let it generate the source again. Then you can omit #include "stm32l4xx_hal_tim.h" from your code, because it will be already included by stm32l4xx_hal.h.
This preprocessor symbol is checked in stm32l4xx_hal_tim.c`, and if it's not defined, then the file will be compiled to an empty stub. In their infinite wisdom, the software engineers at ST decided that it's not necessary to check these symbol in the headers, so when an unsuspecting user includes them, their code will compile without warnings or errors, because the declarations are there, but fail when linked, because the functions are not there.
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2019-09-19 15:36:56
|
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|
https://bitbucket.org/baldur/wiesel/annotate/0c5cd71327f50eb1f792e1c10b22e2dbf60ed312/src/common/wiesel/io/directory.h?at=default
|
# wiesel / src / common / wiesel / io / directory.h
Christian Fische… b5d82b9 2012-04-18 Christian Fische… 194f094 2012-04-11 Christian Fische… b5d82b9 2012-04-18 Christian Fische… 194f094 2012-04-11 Christian Fische… 0c5cd71 2012-09-12 Christian Fische… 194f094 2012-04-11 Christian Fische… 0c5cd71 2012-09-12 Christian Fische… 194f094 2012-04-11 Christian Fische… 2d9179d 2012-05-02 Christian Fische… 194f094 2012-04-11 Christian Fische… c62ffea 2012-04-16 Christian Fische… 194f094 2012-04-11 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 /** * Copyright (C) 2012 * Christian Fischer * * https://bitbucket.org/baldur/wiesel/ * * This library is free software; you can redistribute it and/or * modify it under the terms of the GNU Lesser General Public * License as published by the Free Software Foundation; either * version 3 of the License, or (at your option) any later version. * * This library is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU * Lesser General Public License for more details. * * You should have received a copy of the GNU Lesser General * Public License along with this library; if not, write to the * Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, * Boston, MA 02110-1301 USA */ #ifndef __WIESEL_IO_DIRECTORY_H__ #define __WIESEL_IO_DIRECTORY_H__ #include #include "wiesel/util/shared_object.h" #include "file.h" #include #include #include namespace wiesel { class FileSystem; class Directory; /// Alias type for directory lists. typedef std::list DirectoryList; /** * @brief A class representing a directory within a \ref FileSystem. */ class WIESEL_COMMON_EXPORT Directory : public virtual SharedObject { private: Directory() {} protected: Directory(FileSystem *fs, Directory *parent); public: virtual ~Directory(); // getters public: /** * @brief Get the name of this directory. * The root directory's name may be empty. */ virtual std::string getName() const = 0; /** * @brief Get the full path of this directory. */ virtual std::string getFullPath(); /** * @brief get the directory's full path on the platform's native file system. * If there's no physical file system, for example on a in-memory stored * file system, this function will return an empty string. * @return the native path or an empty string, if there is no real filesystem. */ virtual std::string getNativePath(); /** * @brief get the \ref FileSystem, this directory is in. */ inline FileSystem* getFileSystem() { return fs; } /** * @brief get the parent-directory of this directory. * The root-directory will return \c NULL. */ inline Directory* getParent() { return parent; } // directory content access public: /** * @brief Get a list of all sub-directories. * Each time calling this function, the folder will be scanned for subdirectories. * There is no caching mechanism. */ virtual DirectoryList getSubDirectories() = 0; /** * @brief Get a list of all files in this directory. * Each time calling this function, the folder will be scanned for subdirectories. * There is no caching mechanism. */ virtual FileList getFiles() = 0; /** * @brief Get a direct subdirectory by it's name. * This function does not resolve relative path names or does a recursive search into other subdirectories. * @returns \c NULL, if there's no direct subdirectory with the given name. */ virtual Directory *getSubDirectory(const std::string &name); /** * @brief Tries to find a specific directory relative to the current directory by it's full name. * When the directory is not found, or the object which was found is a file-object, * \c findDirectory will return \c NULL. * To ensure platform compatibility, directories should be seperated by '/' characters, even on windows. */ virtual Directory *findDirectory(const std::string &name); /** * @brief Tries to find a specific file relative to the current directory by it's full name. * When the file is not found, or the object which was found is a directory, * \c findFile will return \c NULL. * To ensure platform compatibility, directories should be seperated by '/' characters, even on windows. */ virtual File *findFile(const std::string &name); // sort utilities public: /** * @brief Sort a list of directories by their names. */ static void sortByName(DirectoryList &list, bool asc=true); private: FileSystem* fs; Directory* parent; }; /** * @brief Predicate function to be used for sorting lists of directories by their names. */ inline bool DirectorySortByNameAscPredicate(const Directory *lhs, const Directory *rhs) { return lhs->getName() < rhs->getName(); } /** * @brief Predicate function to be used for sorting lists of directories by their names. */ inline bool DirectorySortByNameDescPredicate(const Directory *lhs, const Directory *rhs) { return lhs->getName() > rhs->getName(); } } /* namespace wiesel */ #endif /* __WIESEL_IO_DIRECTORY_H__ */
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2015-08-01 17:34:59
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https://math.stackexchange.com/questions/3183977/need-help-understanding-an-equation-composition-addition-and-inverse
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# Need help understanding an equation: composition, addition and inverse
I have found an interesting paper on a digital image registration algorithm. There are many equations in the paper that I only understand partially, but there is a particular one I would like to understand better (at the top of page 4).
$$(T^{(-1)}+w)^{(-1)}=T \circ(Id + w \circ T)^{(-1)}$$
Where $$T$$ is a non-parametric transformations that map every point $$x$$ in image $$I$$ to $$T(x)$$ in image $$J$$. We can find $$T^{(-1)}$$ iteratively: $$T^{(-1)}_{k+1}(x)=(T^{(-1)}_k\circ w_k)(x)$$ where $$w$$ is also a transformation called the adjustment field.
My questions are:
1: What are the steps needed to arrive from the LHS of the equation to the RHS?
2: What are the necessary assumptions for this equation to hold?
Just by looking at the equation, my guess is that something like this might have happened:
$$(T^{(-1)}+w)^{(-1)}=T\circ T^{(-1)} \circ (T^{(-1)}+w)^{(-1)}$$
$$T\circ T^{(-1)} \circ (T^{(-1)}+w)^{(-1)} =T\circ (T^{(-1)}\circ T+w \circ T)^{(-1)}=T \circ(Id + w \circ T)^{(-1)}$$
But I don't know why would it be possible to pass $$T^{(-1)}$$ through the parenthesis or why would the function composition be distributive with addition, or even if this is really the way this equation was done.
• If $A$ and $B$ are invertible, then $A\circ B$ is invertible, and $(A\circ B)^{-1} = B^{-1}\circ A^{-1}$. So you can go from $T^{-1}\circ(T^{-1}+wI)^{-1}$ to $\bigl( (T^{-1}+wI)\circ T^{-1}\bigr)^{-1}$, and then use the fact that composition of linear transformations distributes over the sum of linear transformations to get the result. – Arturo Magidin Apr 11 at 16:58
• If $T$, $U$, and $R$ are linear transformations, then $(U+R)T = UT+RT$. You can verify this by simply noting that they evaluate to the same thing at every vector. – Arturo Magidin Apr 11 at 16:59
• Thank you for the answer @ArturoMagidin . I am pretty sure however, that these transformations in the paper are non-linear. Do you think the equation can be solved for non-linear transformations as well? – David Apr 11 at 17:58
• The first comment (on inverses) is valid in general. The second is valid in all contexts where the expression makes sense. By definition the function $(f+g)$ is defined by $(f+g)(x) = f(x)+g(x)$ (evaluate each function, add the results). So $(f+g)\circ h$, evaluated at $x$, gives $(f+g)\circ h(x) = (f+g)(h(x)) = f(h(x)) + g(h(x)) = (f\circ h)(x) + (g\circ h)(x) = ( (f\circ h) + (g\circ h) )(x)$; as this hold for any $x$, $(f+g)\circ h = (f\circ h) + (g\circ h)$. – Arturo Magidin Apr 11 at 19:41
• @ArturoMagidin Thank you! Now I understand. I was thinking about the whole thing wrong. – David Apr 11 at 22:22
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2019-05-26 10:14:24
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https://www.degruyter.com/document/doi/10.1515/acv-2019-0094/html
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Open Access Pre-published online by De Gruyter February 11, 2020
# Rigidity and trace properties of divergence-measure vector fields
Gian Paolo Leonardi and Giorgio Saracco
# Abstract
We consider a φ-rigidity property for divergence-free vector fields in the Euclidean n-space, where φ(t) is a non-negative convex function vanishing only at t=0. We show that this property is always satisfied in dimension n=2, while in higher dimension it requires some further restriction on φ. In particular, we exhibit counterexamples to quadratic rigidity (i.e. when φ(t)=ct2) in dimension n4. The validity of the quadratic rigidity, which we prove in dimension n=2, implies the existence of the trace of a divergence-measure vector field ξ on an 1-rectifiable set S, as soon as its weak normal trace [ξνS] is maximal on S. As an application, we deduce that the graph of an extremal solution to the prescribed mean curvature equation in a weakly-regular domain becomes vertical near the boundary in a pointwise sense.
MSC 2010: 26B20; 28A75; 35L65
## 1 Introduction
The structure and the properties of vector fields, whose distributional divergence either vanishes or is represented by a locally finite measure, are of great interest in Mathematics and in Physics. Such vector fields arise, for instance, in fluid mechanics, in electromagnetism, and in conservation laws. We shall not give a detailed account, however the interested reader is referred to the monographs [20, 21], as well as to the papers [6, 18, 23, 24, 25], and to the references found therein.
In this paper we shall consider a rigidity property à la Liouville for divergence-free vector fields in n defined hereafter.
## Definition 1.
Let φ:[0,+)[0,+) be a convex function such that φ(t)=0 if and only if t=0. We say that the φ-rigidity property holds in n if, for any vector field η=(η1,,ηn)L(n;n) such that
1. (i)
η=0 on {xn:xn<0},
2. (ii)
divη=0 on n in distributional sense,
3. (iii)
ηnφ(|η|) almost everywhere on n,
one has that η0 almost everywhere on n.
We state here two results that establish this rigidity property.
## Theorem 2 (Linear rigidity).
Let n2. Then the φ-rigidity property holds in Rn when φ(t)=ct for some constant c>0.
## Theorem 3 (φ-rigidity in the plane).
Let n=2. Then the φ-rigidity holds in R2 for any choice of φ as in Definition 1.
Ideally, one would like to prove Theorem 3 in any dimension. Yet, proving φ-rigidity in dimension n>2 when φ is not linear is a rather delicate issue and it would require some further hypotheses. Indeed, we have found counterexamples for the choice φ(t)=ct2 in any dimension n4, as proved in Theorem 2. At the current stage it is unclear to us if Theorem 3 may hold in dimension n=3. Notice that the counterexamples found in dimension n4 are cylindrically symmetric; however, we prove in Proposition 3 that no such vector field can be a counterexample for n=3.
The specific choice φ(t)=t22 is quite interesting as it is closely related with a trace property of certain divergence-measure vector fields. More precisely, it allows us to deduce the existence of the trace of a divergence-measure vector field ξ on an oriented, 1-rectifiable set S2, under a maximality assumption of the weak normal trace of ξ at S. In general, see Section 2.3, given a divergence-measure vector field ξ defined on a bounded open set Ωn, one is able to define its normal trace only in a distributional sense. More specifically, assuming that Ω is weakly-regular (that is, the perimeter of Ω is finite and coincides with the (n-1)-dimensional Hausdorff measure of Ω, see Definition 5) one can show [34, Section 3] that there exists a function [ξνΩ]L(Ω;n-1) such that the following Gauss–Green formula holds for all ψCc1(n):
(1.1)Ωψ(x)ddivξ+Ωξ(x)ψ(x)𝑑x=Ωψ(y)[ξνΩ](y)𝑑n-1(y),
where we have denoted by n-1 the (n-1)-dimensional Hausdorff measure and by νΩ the measure-theoretic outer normal to Ω. Such a function [ξνΩ] is called weak normal trace of ξ on Ω. For the sake of completeness we recall that a first, fundamental weak version of the Gauss–Green formula is the classical result by De Giorgi [16, 17] and Federer [19], which states that (1.1) holds true in the case ψ=1, ξC1 and Ω with finite perimeter. A further extension due to Vol’pert [36, 37] holds when ξ is weakly differentiable or BV. In the seminal works of Anzellotti [4, 3], the concepts of weak normal trace and of pairing between vector fields and (gradients of) functions are introduced for the first time. Since then, the class 𝒟(Ω) of divergence-measure, bounded vector fields has been widely studied in view of applications to hyperbolic systems of conservation laws [7, 9, 10], to continuum and fluid mechanics [8], and to minimal surfaces [28, 34] among many others. In particular, the weak normal trace has been studied in different directions, see for instance [1, 11, 12, 13, 15, 14].
Despite some explicit characterizations of the weak normal trace are available (see the discussion in Section 2.3), a crucial issue coming with this distributional notion is that it is not possible to recover the pointwise value of such a trace by a standard, measure-theoretic limit, see Example 7. However, assuming that ξ1 and that the weak normal trace [ξνS] attains the maximal value 1 at some point xS, it would seem quite natural to expect that ξ cannot oscillate too much around x, to ensure a maximal outflow at x. Thus, one is led to conjecture the existence of the classical trace, i.e. the validity of the formula
[ξνS](x)=νS(x).
Indeed, this is exactly what we are able to prove, limitedly to the case n=2, by employing Theorem 3 with the specific choice φ(t)=t22.
## Theorem 4.
Let ξDM(R2) and SR2 an oriented H1-rectifiable with locally bounded H1-measure. Then, for H1-a.e. x0S such that [ξνS](x0)=ξ one has
ap-limxx0-ξ(x)=ξνS(x0).
In the above theorem the approximate limit is “one-sided” according to Definition 2. Notice that the same conclusion of the theorem applies when the weak normal trace attains a local maximum for the modulus of the vector field, that is, when there exists an open set U containing x0 such that [ξνS](x0)|ξ(x)| for almost every xU. Of course, by a scaling argument one can always restrict to vector fields ξ with ξ1.
The statement of Theorem 4 holds for 1-a.e. x0S. More precisely, one needs x0S to be such that: the normal of S at x0 is defined; it is a Lebesgue point for the weak normal trace; and divξ does not concentrate around it. Asking x0 to satisfy these hypotheses, makes us indeed discard an 1-negligible set of S.
As the proof heavily relies on Theorem 3, we are able to show it only in dimension 2. Despite there exist counterexamples to the quadratic rigidity in n when n4, we cannot exclude that Theorem 4 might be true in any dimension. Were it false, one should be able to construct a vector field with maximal normal trace at some (n-1)-submanifold S, whose blow-ups at most points of S are not unique. Indeed, the existence of the classical, one-sided trace of ξ at x0 is equivalent to the uniqueness of the blow-up of ξ at x0 (see Proposition 2). This essentially corresponds to a pointwise almost-everywhere convergence property. However, a maximality condition on the value of the weak normal trace (viewed as a weak-limit of measures induced by classical traces on approximating smooth surfaces) might enforce no more than a L1-type convergence, in analogy with what happens to a sequence of negative functions that weakly converge to zero. The fact that, in turn, L1-convergence does not imply almost everywhere convergence (unless one extracts suitable subsequences) explains why proving Theorem 4 is so delicate.
Finally, we mention that Theorem 4 allows us to strengthen the main result of our previous paper [28]. In there we considered solutions to the prescribed mean curvature equation in a set Ω, in the so-called extremal case. The extremality condition for the prescribed mean curvature equation
divu1+|u|2=Hon Ω
is a critical situation for the existence of solutions, occurring when the following, necessary condition for existence
|EH(x)𝑑x|<P(E)for all EΩ
becomes an equality precisely at E=Ω. When Ω is smooth, Giusti proved in his celebrated paper [22] that the above necessary condition is also sufficient for existence, and moreover that the extremal case can be characterized by other properties, among which the uniqueness of the solution up to vertical translations and the vertical contact of its graph with the boundary of the domain. In the physically meaningful case, i.e. Ω2, the extremal case for the prescribed mean curvature equation corresponds to the capillarity phenomenon for perfectly wetting fluids within a cylindrical container put in a zero gravity environment.
In our previous work [28] we extended Giusti’s result to the wider class of weakly-regular domains, obtaining an analogous characterization that involves the weak normal trace of the vector field
Tu=u1+|u|2
on Ω. In virtue of the result proved here, we obtain that the boundary behavior of the capillary solution in the cylinder Ω× actually improves to a vertical contact realized in the classical trace sense, rather than just in the sense of the weak normal trace. In other words, the normalized gradient Tu is shown to admit a classical trace (equal to the outer normal νΩ) at 1-almost-every point of Ω. We remark that, at present, no other technique, like the one based on regularity for almost-minimizers of the perimeter, seems to be applicable in the case of a generic weakly-regular domain. The reason is that the boundary of the cylinder Ω× is not smooth enough to let the approaching boundary of the subgraph of the solution u be uniformly regularized via the standard excess-decay mechanism for almost-minimizers of the perimeter.
Briefly, the paper is organized as follows. In Section 2 we lay the notation, recall some basic facts from Geometric Measure Theory and weak normal traces. In Section 3 we give the proofs of Theorem 2 and of Theorem 3. Section 4 presents the construction of a counterexample to the ct2-rigidity for n4. Finally, Section 5 is devoted to the weak normal trace and to the proof of Theorem 4.
## 2 Preliminary notions and facts
### 2.1 Notation
We first introduce some basic notations. Given n2, n denotes the Euclidean n-dimensional space, +n the upper half-space {xn:xn>0}, and 0n the boundary of +n. For any xn and r>0, Br(x) denotes the Euclidean open ball of center x and radius r. Given a unit vector vn we set
Brv(z)={xBr(z):(x-z)v>0}.
Given a Borel set En, we denote by χE its characteristic function, by |E| its n-dimensional Lebesgue measure, and by d(E) its Hausdorff d-dimensional measure. We set
Ex,r=r-1(E-x),
where En, xn, and r>0. Given Ωn an open set, we write EΩ whenever the topological closure E¯ of E is a compact subset of Ω. Whenever a measurable function, or vector field, f is defined on Ω, we denote by f its L-norm on Ω. We denote by 𝒟(Ω) the space of bounded vector fields defined in Ω whose divergence is a Radon measure. For brevity we set 𝒟:=𝒟(n). It is convenient to consider the restriction to 𝒟(Ω) of the weak-* topology of L: given a sequence {vk}k𝒟(Ω), we say that vk converges to v𝒟(Ω) in the weak-* topology of L, and write vkv in L-w*, if for every fL1(Ω;n) one has
Ωfvk𝑑xΩfv𝑑xas k.
### 2.2 Basic definitions in Geometric Measure Theory
We now recall some basic definitions and facts from Geometric Measure Theory and, in particular, from the theory of sets of locally finite perimeter.
### Definition 1 (Points of density α).
Let E be a Borel set in n, xn. If the limit
θ(E)(x):=limr0+|EBr(x)||Br(x)|
exists, it is called the density of E at x. In general θ(E)(x)[0,1], hence, we define the set of points of density α[0,1] for E as
E(α):={xn:θ(E)(x)=α}.
### Definition 2 (One-sided approximate limit).
Let Ωn be an open set, and let SΩ be an oriented n-1-rectifiable set. Take zS such that the exterior normal ν=νS(z) is defined, and choose a measurable function, or vector field, f defined on Ω. We write
ap-limxz-f(x)=w
if for every α>0 the set {xΩB1-ν(z):|f(x)-w|α} has density 0 at z.
### Definition 3 (Perimeter).
Let E be a Borel set in n. We define the perimeter of E in an open set Ωn as
P(E;Ω):=sup{ΩχE(x)divψ(x)𝑑x:ψCc1(Ω;n),ψ1}.
We set P(E)=P(E;n). If P(E;Ω)<, we say that E is a set of finite perimeter in Ω. In this case (see [30]) one has that the perimeter of E coincides with the total variation |DχE| of the vector-valued Radon measure DχE (the distributional gradient of χE), which is defined for all Borel subsets of Ω thanks to Riesz Theorem. We also recall that P(E;Ω)=n-1(EΩ) when EΩ is Lipschitz.
### Theorem 4 (De Giorgi Structure Theorem).
Let E be a set of finite perimeter and let *E be the reduced boundary of E defined as
*E:={xE:limr0+DχE(Br(x))|DχE|(Br(x))=-νE(x)𝕊n-1}.
Then:
1. (i)
*E is countably n-1-rectifiable,
2. (ii)
for all x*E, χEx,rχHνE(x) in Lloc1(n) as r0+, where HνE(x) denotes the half-space through 0 whose exterior normal is νE(x),
3. (iii)
for any Borel set A, P(E;A)=n-1(A*E), thus in particular P(E)=n-1(*E),
4. (iv)
Edivψ=*EψνE𝑑n-1 for any ψCc1(n;n).
Finally, we recall the notion of weakly-regular set which is useful for the next section.
### Definition 5 (Weakly-regular set).
An open, bounded set Ω of finite perimeter, such that P(Ω)=n-1(Ω), is said to be weakly-regular.
For the sake of completeness, we recall that examples of weakly-regular sets are, for instance, Lipschitz sets, or minimal Cheeger sets with n-1(ΩΩ(1))=0 (see [28, 33] for an account of these facts, [26, 31, 32] for an introduction to Cheeger sets, and [27, 29] for recent results and examples).
### 2.3 The weak normal trace and the Gauss–Green formula
The weak normal trace was first defined in [3] for a vector field with divergence in L1(Ω), when Ω is bounded and Lipschitz. When Ω is a generic open set, and denoting by n the Lebesgue measure on n, the weak normal trace [ξνΩ] can be defined as the distribution
[ξνΩ],ψ:=Ωψddivξ+Ωξψdn,ψCc(n).
Taking EΩ of class C1, and defining [ξνE] in the same way, one can show that the distribution is represented by an L function defined on E, that we still denote by [ξνE] with a slight abuse of notation, so that
[ξνE],ψ=Eψ[ξνE]𝑑n-1.
Then, up to showing a locality property of the weak normal trace on C1 domains (see [1]), it is possible to define [ξνS] for any given, oriented (n-1)-rectifiable set S contained in Ω. We remark, however, that some particular care has to be taken when Ω is a bounded open set with finite perimeter (and ξ is a-priori defined only on Ω). In order to guarantee that the distributional weak normal trace is represented by an L function defined on S=Ω one has to additionally assume that Ω is weakly-regular.
Thus, when Ω is weakly-regular one obtains the Gauss–Green formula below (see [34, Section 3]) exploiting the pairing between vector fields in 𝒟(Ω) and functions in Cc1(n). Another version of the formula with a slightly different pairing can be found in [28].
### Theorem 6 (Generalized Gauss–Green formula).
Let ΩRn be a weakly-regular set. For any ξDM(Ω) and ψCc1(Rn) one has
(2.1)Ωψddivξ+Ωξψdx=*Ωψ[ξνΩ]𝑑n-1,
where [ξνΩ]L(Ω;Hn-1) is the so-called weak normal trace of ξ on *Ω.
We remark for completeness that more general formulas have been recently obtained for pairings between ξ𝒟(n), ψBVloc(n)Lloc(n) and Ω bounded with finite perimeter (see in particular [14, 35]).
As already observed by Anzellotti in [4] (see also [28]), the weak normal trace is a proper extension of the scalar product between the exterior normal νΩ and the trace of the vector field ξ, assuming that the latter exists in the classical sense. More generally, one expects that the weak normal trace is obtained as a weak-limit of scalar products between (suitable averages of) the vector field ξ and the normal field νS. This actually corresponds to [4, Proposition 2.1], which says that, whenever S is of class C1 and divξL1, then
1ωnrnyBr()ξ(y)νS()𝑑n(y)[ξ,νS]in L(S)-w*as r0+.
It is worth recalling that there exist also some pointwise characterizations of the weak normal trace. A first one is obtained by testing (2.1) with a function ψ that approximates the characteristic function of a ball Br(x) for xS. Taking x in a suitable subset of S of full n-1-measure one obtains
[ξ,νS](x)=ap-limr01ωn-1rnyBr-(x)ξ(y)(x-y)𝑑n-1(y)for n-1-a.e. xS,
where, for r>0 small enough, Br-(x) denotes the part of Br(x) which is on the side of S pointed by -νS(x). A second characterization given in [4, Proposition 2.2], assuming S of class C1, is the following. Given xS and r,ρ>0 small enough, let Sρ(x)=SBρ(x) and set
Qr,ρ(x)={z=y-tνS(x):ySρ(x),t(0,r)},
νx(z)=νS(y)if z=y-tνS(x).
Note that νx(z) is well-defined on Qr,ρ(x) as soon as r,ρ are small enough. In other words, Qr,ρ(x) is a curvilinear rectangle foliated by translates of SBρ(x) in the -νS(x) direction. It is proved in [4] that
[ξ,νS](x)=limρ0limr01ωn-1ρn-1rzQr,ρ(x)ξ(z)νx(z)𝑑n(z),
for n-1-almost-every xS. This result can be also obtained by testing (2.1) with a C1 function ψ that satisfies ψ=1 on Sρ, ψ=0 on Sρ-rνS(x), and which is linear on any segment (y,y-rνS(x)), ySρ.
Despite the characterizations described above, one can easily construct examples of vector fields like the one presented below (a variant of a piece-wise constant one defined in [4]) that illustrate possible wild behaviors of divergence-free vector fields for which the weak normal trace on S is well-defined. More precisely, the example below shows that, in general, the weak normal trace does not coincide with any classical, or measure-theoretic, limit of the scalar product of the vector field with the normal to S, even when S is of class C and the vector field is divergence-free and smooth in a neighborhood of S (minus S itself).
### Example 7.
Let us set +2={(x,y):y>0}, S={(x,y):y=0}, and ν=(0,-1). For i1 and j=1,,2i-1 we set xij=j2i, yi=12i, ri=12i+2. Then, for such i and j we take fiCc() with compact support in (0,ri), so that in particular fi(0)=fi(ri)=0, and define pij=(xij,yi). Notice that by our choice of parameters, the balls {Bij=Bri(pij)}i,j are pairwise disjoint. Whenever pBij, we set
ξ(p)=fi(|p-pij|)(p-pij),
while ξ(p)=0 otherwise (see Figure 1). One can suitably choose fi so that ξL(Bij)=1 for all i,j. Moreover, divξ=0 on +2 and thus for any ψCc1(2), by the Gauss–Green formula and owing to the definition of ξ, one has
Sψ[ξν]𝑑1=+2ξDψ=i,jBijξDψ=i,jBijψξνij=0,
so that [ξν]=0 on S. At the same time, ξ twists in any neighborhood of any point p0=(x0,0), x0(0,1), and the second component of the average of ξ on half balls centered at p0 has a lim inf strictly smaller than its lim sup as r0. In conclusion, the scalar product ξ(p)ν(p0) does not converge to 0 in any pointwise or measure-theoretic sense.
### Figure 1
A twisting vector field defined on +2.
## 3 Proofs of the rigidity results
We start by proving that the convexity assumption on the function φ makes Definition 1 essentially equivalent to a weaker one, in which the C smoothness and the global Lipschitz-continuity of the vector field η are required.
## Lemma 1.
Let φ and η be as in Definition 1. Let ρε be the standard mollifier supported in a ball of radius ε>0. Then the regularized vector field ηε=ρεη is globally Lipschitz and satisfies (i)–(iii) of Definition 1 up to an ε-translation in the variable xn.
## Proof.
We note that ηnφ(|η|) almost everywhere on n by (iii), therefore Jensen’s inequality implies that for all xn and ε>0
φ(|ηε(x)|)=φ(|Bεη(y)ρε(x-y)𝑑y|)
Bεφ(|η(y)|)ρε(x-y)𝑑y
Bεηn(y)ρε(x-y)𝑑y=ηnε(x).
Then we observe that ηε is globally Lipschitz, as a consequence of the boundedness of η, and verifies divηε=0 on n, that is, (ii). Moreover, for every x=(y,t) such that t-ε one has ηε(x)=0. Finally, up to a translation in the variable x, we can assume ηε(x)=0 for all x-n, hence property (i) is also satisfied. ∎
## Proof of Theorem 2.
Without loss of generality, by Lemma 1 we can assume that η is smooth and globally Lipschitz. Let us fix ε>0 and consider the one-parameter flow associated with the vector field X=η+εen, defined for every t and pn by the Cauchy problem
{Φ(0,p)=p,tΦ(t,p)=X(Φ(t,p)).
Note that in the notation above we dropped the dependence upon the parameter ε for the sake of simplicity. Due to the smooth dependence from the initial datum, the map Φ(,):×nn is smooth, DpΦ|t=0=Id and the map Φ(t,):nn is a diffeomorphism for every t. Let us denote by Φn(t,p) the n-th component of Φ(t,p). Since Xn(p)ε for all pn, we have that tΦn(t,p)ε, hence the function tΦn(t,p) is strictly monotone and surjective. Therefore, for every h and pn there exists a unique T=T(p,h) such that Φn(T,p)=h. By the Implicit Function Theorem, T(p,h) is smooth and one has
hT(p,h)=Xn(Φ(T,p))-1,DpT(p,h)=-Xn(Φ(T,p))-1DpΦn(T,p).
Fix now an open, bounded and smooth set An-1 and h0>0. Let us set Ah0=A×{h0} and define the map
Ψ:A×(0,h0)n,Ψ(q,h)=Φ(T(p,h),p),
where we have set p=(q,h0). Before proceeding it is convenient to introduce some more notation. We write
Ψ=(Ψ1,,Ψn-1,Ψn)=(Ψ^,Ψn)
X=(X1,,Xn-1,Xn)=(X^,Xn)
y=(y1,,yn-1,yn)=(y^,yn)for yn.
We start by computing the partial derivative of Ψ with respect to h:
hΨ(q,h)=Xn(Ψ)-1X(Ψ).
Note that Ψ(p)=Ψ(q,h0)=p and Ψn(q,h)=h by definition. Owing to the smoothness of Ψ, we first compute the partial derivative with respect to h of Ψji:=qjΨi, for i,j=1,,n-1:
hΨji=qjhΨi=qj[Xn(Ψ)-1Xi(Ψ)]
=Xn(Ψ)-1r=1nyrXi(Ψ)Ψjr-Xn(Ψ)-2Xi(Ψ)r=1nyrXn(Ψ)Ψjr
=r=1n[Xn(Ψ)-1yrXi(Ψ)-Xn(Ψ)-2Xi(Ψ)yrXn(Ψ)]Ψjr.
Moreover, it is immediate to check that qjΨn(q,h)=0 for all j=1,,n-1 and that hΨn(q,h)=1, so that in particular the matrix form of the previous computation is
hDqΨ^=B(Ψ)DqΨ^,
where
B=Xn-1Dy^X^-Xn-2X^Dy^Xn.
Moreover, the determinant of DΨ coincides with that of DqΨ^. If we assume that qn-1 is fixed, and define δ(h)=detDqΨ^(q,h), we find by standard calculations that
δ(h)=tr[B(Ψ(q,h))]δ(h),
with initial condition δ(h0)=1. This shows that the Jacobian matrix of Ψ is uniformly invertible on compact subsets of n. Consequently, Ψ is a smooth diffeomorphism and the set Fε(A):=Ψ(A×(0,h0)), depicted in Figure 2, is Lipschitz. Notice moreover that
|Ψ(q,h0)-Ψ(q,0)|0h0|hΨ(q,h)|𝑑h0h0Xn-1|X|𝑑hh0(c|η|+ε)-1(|η|+ε)max{h0,h0c},
thanks to the properties of η. Notice now that given a bounded An-1, there exists a constant R>0 depending only on A and h0, such that Fε(A) is contained in (-R,R)n for every ε>0. Setting Lε(A)=Ψ(A×{0}), by applying the Divergence Theorem on Fε(A) to the vector field X=η+εen we find the identity
0=Fε(A)divX=A(ηn(q,h0)+ε)𝑑q-εn-1(Lε(A)),
since the integral on the “lateral” boundary of Fε(A) vanishes. This happens because the lateral boundary of the flow-tube consists of integral curves of the flow X, and thus on this lateral boundary one has XνFε(A)0. Using the fact that n-1(Lε(A))(2R)n-1 we can pass to the limit as ε0 in the identity above, obtaining that ηn vanishes on A×{h0}. By the arbitrary choice of both A and h0, and by property (iii) of Definition 1, we conclude that η=0 on n. ∎
### Figure 2
The evolution of Ah0 through the diffeomorphism Ψ defines a “flow-tube”.
We now proceed with the proof of Theorem 3. Here, instead of using the “flow-tube” method employed in the proof of Theorem 2, we take advantage of a special feature of 2-dimensional cylinders of the form (-a,a)×(0,h0), i.e. the fact that their “lateral” perimeter is constantly equal to 2h0 (and in particular it does not blow up when a+).
## Proof of Theorem 3.
By Lemma 1 we can additionally assume that η is smooth and globally Lipschitz. Since η20, by the Divergence Theorem we find
(3.1)-rr|η2(x1,t)|𝑑x1=-rrη2(x1,t)𝑑x1=0tη1(-r,x2)-η1(r,x2)dx22tη.
Therefore, η2(,t)L1(;) for all t>0 and
(3.2)η2(,t)12tη.
By combining (iii) of Definition 1 with (3.2) and the fact that η2 is Lipschitz, we infer that
(3.3)φ(|η1(x1,t)|)η2(x1,t)0as x1±,
hence, owing to the properties of φ, we get for all t>0,
limx1±η1(x1,t)=0.
This implies that
2ηη1(-r,t)-η1(r,t)0
as r+, for all t>0. Therefore, we can take the limit in (3.1) as r and, thanks to the Dominated Convergence Theorem, we obtain that
limr+-rr|η2(x1,t)|𝑑x1=limr+0tη1(-r,x2)-η1(r,x2)dx2
=0tlimr+(η1(-r,x2)-η1(r,x2))dx2=0.
Hence, the L1-norm of η2(,t) is zero, thus η2(,t)=0, for all t>0. By (3.3) and the properties of φ we get η=0 on 2. ∎
## 4 Counterexamples to quadratic rigidity in dimension n≥4
Given xn, we set r=(x1,,xn-1) and z=xn. We consider vector fields ηC0(n) such that for r0 one has
(4.1)η(r,z)=(rf(r,z),h(r,z))
with f,hC1(+n{(0,z):z}). We have the following:
## Proposition 1.
Let ηC0(Rn) be as in (4.1). Define
(4.2)V(r,z):=-|r|n-10zf(r,s)𝑑s.
Then η satisfies
(4.3)η(r,z)=0for all z0,
(4.4)|η(x)|1for all xn,
(4.5)divη=0on n,
(4.6)ηn(x)c1|η(x)|2for all xn,
if and only if V(r,z) satisfies
(4.7)V(r,z)=0for all z0,
(4.8)|V(r,z)||r|n-2for all r>0 and z,
(4.9)rrV(r,z)c2|r|3-n(zV(r,z))2for all r>0 and z.
Moreover, one has
(4.10)η(r,z)=|r|1-n(-(zV)r,rrV).
## Proof.
We show in full detail the only if part. Since
divη(r,z)=divr(rf(r,z))+zh(r,z)=(n-1)f+rrf(r,z)+zh(r,z),
equation (4.5) is equivalent to
(4.11)zh=-(rrf+(n-1)f).
By recalling that h(r,0)=0 for all r by (4.3), from (4.11) we obtain that
(4.12)h(r,z)=-0z(rrf(r,s)+(n-1)f(r,s))𝑑s.
Inequality (4.6), up to a change of the constant c, is equivalent to
(4.13)hc|r|2f2.
Let us set V as in (4.2) and observe that (4.3) implies V(r,z)=0 when z0. Then using (4.12) we obtain
rV=-|r|n-10z[rf(r,s)+(n-1)f(r,s)|r|-2r]𝑑s
and
(4.14)zV(r,z)=-|r|n-1f(r,z),
so that in particular
(4.15)rrV=|r|n-1h,
and inequality (4.13) implies
rrVc|r|n+1f2c|r|3-n(zV)2.
Then by observing that (4.4) is equivalent to |r|2f2+h21, we obtain by (4.14) and (4.15)
|V||r|n-2.
We have thus proved that defining V as in (4.2) we get (4.7)–(4.9).
Conversely, one can easily check that given V satisfying (4.7)–(4.9), the vector field η defined as in (4.10) satisfies (4.3)–(4.6). ∎
## Theorem 2.
The quadratic rigidity property does not hold in dimension n4.
## Proof.
Let us consider a positive parameter γ (to be chosen later) and define the function
V(r,z)={γ[(1+|r|n-1)1n-1-1]arctan(z2)if z0,0otherwise.
Our aim is to verify properties (4.7)–(4.9) up to a suitable choice of γ, and then to use the equivalence stated in Proposition 1. Of course (4.7) is true by definition of V. Let us set ρ=|r| and write V=V(ρ,z) for simplicity. One has
ρV=γarctan(z2)(1+ρn-1)2-nn-1ρn-2
and
zV=2γ[(1+ρn-1)1n-1-1]z1+z4.
Notice that V is of class C1 and V(ρ,z)=0 for all ρ>0 and z0. We obtain
|V|γarctan(z2)(1+ρn-1)2-nn-1ρn-2+2γ[(1+ρn-1)1n-1-1]z1+z4
(4.16)γπρn-22+γ3342[(1+ρn-1)1n-1-1],
where the second inequality follows from the maximization of the function z(1+z4) for z[0,+). Let us consider the function
φ(ρ)=(1+ρn-1)1n-1-1ρn-2.
As ρ0+, we have φ(ρ)ρn-1, while as ρ+ we have φ(ρ)ρ3-n. Moreover, when 0<t<1, one has (1+t)1n-11+tn-1, hence we deduce that
φ(ρ)ρn-11
when ρ<1, and
φ(ρ)ρ3-n1
when ρ1. Therefore by (4.16) and the last inequalities we find that
|V|Cγρn-2,
where
C=π+3342.
Assuming γC-1, we obtain (4.8).
We now show that (4.9) (with the constant c=1) holds up to taking a smaller γ. Indeed, the relation ρρVρ3-n(zV)2, after separation of variables, becomes
arctan(z2)(1+z4)2z24γφ(ρ)2(1+ρn-1)n-2n-1.
We argue as for the upper bound of φ(ρ) (more precisely, we discuss the two cases ρ<1 and ρ1; in the first case we use the bound φ(ρ)1, while in the second case we use the fact that n4, and the inequalities φ(ρ)ρ3-n and 1+ρn-12ρn-1), and find
(4.17)4γφ(ρ)2(1+ρn-1)n-2n-123n-4n-1γ.
At the same time, by easy calculations we infer that the function arctan(t)(1+t2)2t is bounded from below by 1. Hence (4.9) is implied by the condition
γ24-3nn-1.
In conclusion, by taking γ small enough, and thanks to Proposition 1, a divergence-free vector field providing a counterexample to the rigidity property in dimension n4 is given by
η(r,z)={γ|r|1-n(-2[(1+|r|n-1)1n-1-1]z1+z4r,arctan(z2)(1+|r|n-1)2-nn-1|r|n-1)if z>0,0if z0.
We remark that the vector field η is of class C0, however one can obtain a C counterexample by mollification of η. ∎
Concerning the 3-dimensional case, the construction of a counterexample with cylindrical symmetry, as done in Theorem 2, does not work. Indeed we are unable to get an estimate like (4.17), as the function φ(ρ) tends to 1 as ρ+. More precisely we can prove the following result.
## Proposition 3.
Let n=3 and assume that V(r,z) is a C1 function satisfying properties (4.7)–(4.9). Then V(r,z)=ψ1(|r|)ψ2(z) for suitable functions ψ1,ψ2 implies V0.
## Proof.
We set ρ=|r| and write (4.9) as
ρψ1(ρ)ψ2(z)cψ12(ρ)[ψ2(z)]2.
Let us assume by contradiction that ψ1 and ψ2 are not trivial, hence there exist z0>0 and ρ0>0 such that ψ2(z0)0, ψ2(z0)0, and ψ1(ρ0)0. Setting a=ψ2(z0) and b=ψ2(z0), we have four cases according to the sign of a and ψ1(ρ0). We discuss the first case a>0 and ψ1(ρ0)>0. We consider the differential inequality
ρψ1(ρ)γψ12(ρ)
with γ=cb2a>0. Therefore, the function ψ1 is increasing and by separation of variables and integration between ρ0 and ρ>ρ0 we get
(4.18)ψ1-1(ρ0)-ψ1-1(ρ)+ψ1-1(ρ0)γlog(ρρ0).
We denote by I=[ρ0,β) the maximal right interval of existence of the solution ψ1, for which ψ1>0. We can exclude the case β<+, as we would obtain by maximality that ψ1(ρ)+ as ρβ-, however this would contradict the fact that |ψ(ρ)|ρ for all ρ>0. Contrarily, if ρ+, one gets a contradiction with (4.18). The remaining three cases can be discussed in a similar way. ∎
## Remark 4.
As a consequence of Proposition 3 we infer that in dimension n=3 no counterexample to the rigidity property can be found in the class of vector fields of the form η(r,z)=(rf(|r|,z),h(|r|,z)).
## 5 The trace of a vector field with locally maximal normal trace
The results we shall discuss in this section are stated for n-1-almost-every point of S, being S an oriented, n-1-rectifiable set with locally finite n-1-measure. Therefore, given z𝒟, without loss of generality (see [2, Theorem 2.56]) we shall assume x0S to be such that
1. (a)
the normal vector νS(x0) is defined at x0,
2. (b)
x0 is a Lebesgue point for the weak normal trace [zνS] of z on S, with respect to the measure n-1S,
3. (c)
|divz|(Br(x0)S)=o(rn-1) as r0+.
## Lemma 1.
Let η,{zk}k be vector fields in DM with supkzk<+. Let Σ,{Sk}k be oriented, closed Hn-1-rectifiable sets with locally finite Hn-1-measure, satisfying the following properties:
1. (i)
zkη in L-w*,
2. (ii)
n-1Skn-1Σ,
3. (iii)
|divzk|(nSk)0.
Then divη=0 in RnΣ.
## Proof.
Fix φCc1(nΣ) and set μk=divzk and μ=divη. By the Divergence Theorem coupled with the formula (see [1, Proposition 3.2])
μkSk=([zkνSk]+-[zkνSk]-)n-1Sk,
we have
nSkφ𝑑μk=-Skφ𝑑μk-nSkφzk
=-Skφ([zkνSk]+-[zkνSk]-)𝑑n-1-nφzk,
hence by (ii) and (iii)
|nφzk|nSk|φ|𝑑μk+2zk|φ|𝑑n-1Sk0as k.
This shows that
nφ𝑑μ=limknφ𝑑μk=limknφzk=0,
which proves the thesis. ∎
## Proposition 2.
Let zDM and S a closed, oriented Hn-1-rectifiable set with locally finite Hn-1-measure. Then, for Hn-1-almost-every x0S and for any decreasing and infinitesimal sequence {rk}k, the sequence zk of vector fields defined by zk(y)=z(x0+rky) converges up to subsequences to a vector field ηDM in L-w*, such that setting Σ=[νS(x0)], we have divη=0 on RnΣ and [ηνΣ]=[zνS](x0) on Σ.
## Proof.
We show that hypotheses (i)–(iii) of Lemma 1 are satisfied. Since zk=z for all k, thanks to Banach–Alaoglu Theorem (see also [5, Theorem 3.28]) we can extract a not relabeled subsequence converging to η𝒟, which gives (i). We set Sk=rk-1(S-x0), then thanks to (c) we have
(5.1)|divzk|(BRSk)=rk1-n|divz|(BRrk(x0)S)0as k
for all R>0, which gives (iii). Owing to the localization property proved in [1, Proposition 3.2], we can replace S with the boundary of an open set Ω of class C1 such that x0Ω and νS(x0)=νΩ. Defining Ωk=rk-1(Ω-x0), the proof of (ii) is reduced to showing that n-1Ωk weakly converge as measures to n-1H, where H is the tangent half-space to Ω at x0 (so that Σ=H). This fact is a consequence of Theorem 4. Now we can apply Lemma 1 and obtain divη=0 on nΣ. In order to prove the last part of the statement we have to show that for any ψCc1(n) one has
Hψddivη+Hψη=τ0Hψ𝑑n-1,
where τ0=[zνS(x0)]. Since we have proved that divη=0 on nΣ, we only have to show that
(5.2)Hψη=τ0Hψ𝑑n-1.
Note that by the convergence of zk to η, the Lloc1-convergence of Ωk to H, and the convergence of the measures n-1Ωk to n-1H, we have
(5.3)Hψη-τ0Hψ𝑑n-1=limkΩkψzk-τ0Ωkψ𝑑n-1.
Therefore by (5.1) and (b) we infer that
|Ωkψzk-τ0Ωkψ𝑑n-1||Ωkψ([zkνΩk]-τ0)𝑑n-1-Ωkψddivzk|
ψΩksptψ|[zkνΩk]-τ0|+nΩk|ψ||divzk|0as k.
Combining this last fact with (5.3) implies (5.2) at once, and concludes the proof. ∎
By relying on Proposition 2 and on Theorem 3, we are now able to prove the main result of the section, i.e. Theorem 4 which states the existence of the classical trace for a divergence-measure vector field having a maximal weak normal trace on a oriented 1-rectifiable set S.
## Proof of Theorem 4.
Without loss of generality, up to a translation we can suppose x0=0 and up to a rotation that νS(x0)=-e2. Moreover, up to rescaling ξ, we can suppose ξ=1. Let
B1+=B1+2.
We then want to show that the set
Nα:={xB1+:|ξ(x)+e2|α}
has density zero at 0 for all α>0. Argue by contradiction and suppose there exist α,β>0 and a sequence of radii {rk}k decreasing to zero such that
(5.4)|NαBrk|πrk2βfor all k.
Define z0(x):=ξ(x)+e2 and the sequence zk(y)=z0(rky) for k. Since the second component of zk is zk,2(x)=ξ2(rkx)+1, one easily sees that
(5.5)zk,2(x)|zk(x)|22,
for almost every x2. By the definition of Nα and by (5.5), the contradiction hypothesis (5.4) reads equivalently as
(5.6)|{xrk-1B1+:zk,2(x)>α22}B1|πβ.
On top of that, z0𝒟 with z02. By Proposition 2 the sequence zk defined above converges in L-w* (up to subsequences, we do not relabel) to a vector field η such that div(η)=0 on +2 and [η(-e2)]=[z0(-e2)](0) on 02. We aim to show that η satisfies hypotheses (i)–(iii) of Theorem 3. Were this the case, one would conclude η0 in +2 and this would yield a contradiction with (5.6). Indeed, taking χB1+ as a test function, we get
πα2β2B1+zk,2𝑘B1+η2.
On the one hand, we know that hypothesis (i) of Theorem 3 is satisfied as div(η)=0 on +2. On the other hand, as [ην]=-[z0e2](0) on 02, we get that
[ην]=-[ξe2](0)-e2e2=ξ-1=0,
so that hypothesis (ii) of Theorem 3 holds as well. We are left to show that hypothesis (iii) of Theorem 3 is satisfied. Since zk is equibounded, by (5.5) we infer as well that |zk|2 converges (up to subsequences, we do not relabel) to some function ζ in L-w*. Clearly, one has from (5.5) and the weak-* convergence of zk and of |zk|2 that ζ2η2 almost everywhere. We want to prove that the same holds with |η|2 in place of ζ so to retrieve hypothesis (iii) of Theorem 3 with the choice φ(t)=t22. Take a probability measure fdx with fL1(2). Then, by Jensen’s inequality,
|zk|2f𝑑x|zkf𝑑x|2=(zk,1f𝑑x)2+(zk,2f𝑑x)2.
As k, by the weak-* convergence we get
ζf𝑑x(η1f𝑑x)2+(η2f𝑑x)2.
Thus, by letting fdx toward the Dirac measure centered at x, (iii) follows at once for almost-every point x (more precisely, x must be a Lebesgue point for the functions ζ,η1,η2). A direct application of Theorem 3 yields the desired contradiction. ∎
## Corollary 3.
Let ΩR2 be weakly-regular and let ξDM(Ω). Then, for H1-a.e. x0Ω such that [ξνΩ](x0)=ξ one has
ap-limxx0-ξ(x)=ξνΩ(x0).
## Proof.
Thanks to the Gauss–Green formula (2.1) in the special case ψ=1, one deduces that the vector field ξ~ defined as ξ~=ξ on Ω and ξ~=0 on 2Ω belongs to 𝒟(2). The conclusion is achieved by applying Theorem 4 with ξ~ and Ω in place of, respectively, ξ and S. ∎
### 5.1 An application to capillarity in weakly-regular domains
The trace property that we have studied in the last section is motivated by the study of the boundary behavior of solutions to the prescribed mean curvature equation in domains with non-smooth boundary (see [28]). Let us consider the vector field
Tu=u1+|u|2
associated with any given uWloc1,1(Ω). We say that u is a solution to the prescribed mean curvature equation if
(PMC)divTu=Hon Ω
in the distributional sense, where H is a prescribed function on Ω. One of the main results of [28] is the following theorem.
### Theorem ([28, Theorem 4.1]).
Let ΩRn be a weakly-regular domain and let H be a given Lipschitz function on Ω. Assume that the necessary condition for existence of solutions to (PMC) holds, that is,
|AH(x)𝑑x|<P(A)for all AΩ such that 0<|A|<|Ω|.
Then the following properties are equivalent:
1. (E)
(Extremality)|ΩHdx|=P(Ω).
2. (U)
(Uniqueness) ((PMC)) admits a solution u which is unique up to vertical translations.
3. (M)
(Maximality) Ω is maximal for ((PMC)), i.e. no solution can exist in any domain strictly containing Ω.
4. (V)
(Weak verticality) There exists a solution u which is weakly-vertical at Ω, i.e.
[Tuν]=1n-1-a.e. on Ω,
where [Tuν] is the weak normal trace of Tu on Ω.
We remark that, in the relevant case of H a positive constant, the extremality property (E) is equivalent to Ω being a minimal Cheeger set (i.e. Ω is the unique minimizer of the ratio P(A)|A| among all measurable AΩ with positive volume, see for instance [26, 27, 29, 31, 32]) and H equals the Cheeger constant of Ω. In dimension n=2, this extremal case corresponds exactly to capillarity in zero gravity for a perfectly wetting fluid that partially fills a cylindrical container with cross-section Ω. We also stress that the uniqueness property (U) holds in this case without any prescribed boundary condition; this means that the capillary interface in Ω× only depends upon the geometry of Ω. Another important remark should be made on the verticality condition (V), which corresponds to the tangential contact property that characterizes perfectly wetting fluids. In [28] this condition is obtained under the weak-regularity assumption on Ω, which somehow justifies the presence of the weak normal trace in the statement (see for instance in Figure 3 an example of non-Lipschitz, weakly-regular domain built in [29] and covered by [28, Theorem 4.1]).
### Figure 3
A non-Lipschitz, weakly-regular domain. Originally appeared in [28, 29].
Nevertheless, in the physical 3-dimensional case (i.e. when Ω2), the weak-verticality (V) improves to strong-verticality, that is, the trace of Tu exists and is equal to νΩ almost-everywhere on Ω thanks to Theorem 4 and to Corollary 3. However, we remark that the strategy of proof strongly relies on the rigidity property, which we have been able to prove only in dimension n=2. It is an open question whether the weak-verticality condition always improves to the strong-verticality given by the existence of the classical trace of Tu at n-1-almost-every point of Ω, and more generally if Theorem 4 holds in any dimension.
Communicated by Frank Duzaar
Funding source: Istituto Nazionale di Alta Matematica
Award Identifier / Grant number: U-UFMBAZ-2019-000473 11-03-2019
Funding statement: Gian Paolo Leonardi and Giorgio Saracco have been partially supported by the INdAM-GNAMPA 2019 project “Problemi isoperimetrici in spazi Euclidei e non” (n. prot. U-UFMBAZ-2019-000473 11-03-2019).
# Acknowledgements
The authors would like to thank Carlo Mantegazza for fruitful discussions on the topic, and Guido De Philippis for suggesting the “flow-tube” strategy used to prove Theorem 2.
### References
[1] L. Ambrosio, G. Crippa and S. Maniglia, Traces and fine properties of a BD class of vector fields and applications, Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 4, 527–561. Search in Google Scholar
[2] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., The Clarendon Press, New York, 2000. Search in Google Scholar
[3] G. Anzellotti, Pairings between measures and bounded functions and compensated compactness, Ann. Mat. Pura Appl. (4) 135 (1983), 293–318. Search in Google Scholar
[4] G. Anzellotti, Traces of bounded vector-fields and the divergence theorem, unpublished preprint. Search in Google Scholar
[5] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. Search in Google Scholar
[6] D. Chae and J. Wolf, On Liouville-type theorems for the steady Navier–Stokes equations in 3, J. Differential Equations 261 (2016), no. 10, 5541–5560. Search in Google Scholar
[7] G.-Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 (1999), no. 2, 89–118. Search in Google Scholar
[8] G.-Q. Chen and H. Frid, Extended divergence-measure fields and the Euler equations for gas dynamics, Comm. Math. Phys. 236 (2003), no. 2, 251–280. Search in Google Scholar
[9] G.-Q. Chen and M. Torres, Divergence-measure fields, sets of finite perimeter, and conservation laws, Arch. Ration. Mech. Anal. 175 (2005), no. 2, 245–267. Search in Google Scholar
[10] G.-Q. Chen, M. Torres and W. P. Ziemer, Gauss-Green theorem for weakly differentiable vector fields, sets of finite perimeter, and balance laws, Comm. Pure Appl. Math. 62 (2009), no. 2, 242–304. Search in Google Scholar
[11] G. E. Comi and K. R. Payne, On locally essentially bounded divergence measure fields and sets of locally finite perimeter, Adv. Calc. Var. (2017), 10.1515/acv-2017-0001. Search in Google Scholar
[12] G. E. Comi and M. Torres, One-sided approximation of sets of finite perimeter, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 28 (2017), no. 1, 181–190. Search in Google Scholar
[13] G. Crasta and V. De Cicco, An extension of the pairing theory between divergence-measure fields and BV functions, J. Funct. Anal. 276 (2019), no. 8, 2605–2635. Search in Google Scholar
[14] G. Crasta and V. De Cicco, Anzellotti’s pairing theory and the Gauss–Green theorem, Adv. Math. 343 (2019), 935–970. Search in Google Scholar
[15] G. Crasta, V. De Cicco and A. Malusa, Pairings between bounded divergence-measure vector fields and BV functions, preprint (2019), https://arxiv.org/abs/1902.06052. Search in Google Scholar
[16] E. De Giorgi, Su una teoria generale della misura (r-1)-dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl. (4) 36 (1954), 191–213. Search in Google Scholar
[17] E. De Giorgi, Nuovi teoremi relativi alle misure (r-1)-dimensionali in uno spazio ad r dimensioni, Ricerche Mat. 4 (1955), 95–113. Search in Google Scholar
[18] C. De Lellis and R. Ignat, A regularizing property of the 2D-eikonal equation, Comm. Partial Differential Equations 40 (2015), no. 8, 1543–1557. Search in Google Scholar
[19] H. Federer, A note on the Gauss–Green theorem, Proc. Amer. Math. Soc. 9 (1958), 447–451. Search in Google Scholar
[20] E. Feireisl and A. Novotnỳ, Singular Limits in Thermodynamics of Viscous Fluids, Springer, Cham, 2009. Search in Google Scholar
[21] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations: Steady-State Problems, 2nd ed., Springer Monogr. Math., Springer, New York, 2011. Search in Google Scholar
[22] E. Giusti, On the equation of surfaces of prescribed mean curvature. Existence and uniqueness without boundary conditions, Invent. Math. 46 (1978), no. 2, 111–137. Search in Google Scholar
[23] R. Ignat, Singularities of divergence-free vector fields with values into S1 or S2. Applications to micromagnetics, Confluentes Math. 4 (2012), no. 3, Article ID 1230001. Search in Google Scholar
[24] R. Ignat, Two-dimensional unit-length vector fields of vanishing divergence, J. Funct. Anal. 262 (2012), no. 8, 3465–3494. Search in Google Scholar
[25] C. E. Kenig and G. S. Koch, An alternative approach to regularity for the Navier–Stokes equations in critical spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), no. 2, 159–187. Search in Google Scholar
[26] G. P. Leonardi, An overview on the Cheeger problem, New Trends in Shape Optimization, Internat. Ser. Numer. Math. 166, Birkhäuser/Springer, Cham (2015), 117–139. Search in Google Scholar
[27] G. P. Leonardi, R. Neumayer and G. Saracco, The Cheeger constant of a Jordan domain without necks, Calc. Var. Partial Differential Equations 56 (2017), no. 6, Article ID 164. Search in Google Scholar
[28] G. P. Leonardi and G. Saracco, The prescribed mean curvature equation in weakly regular domains, NoDEA Nonlinear Differential Equations Appl. 25 (2018), no. 2, Article ID 9. Search in Google Scholar
[29] G. P. Leonardi and G. Saracco, Two examples of minimal Cheeger sets in the plane, Ann. Mat. Pura Appl. (4) 197 (2018), no. 5, 1511–1531. Search in Google Scholar
[30] F. Maggi, Sets of Finite Perimeter and Geometric Variational Problems. An Introduction to Geometric Measure Theory, Cambridge Stud. Adv. Math. 135, Cambridge University Press, Cambridge, 2012. Search in Google Scholar
[31] E. Parini, An introduction to the Cheeger problem, Surv. Math. Appl. 6 (2011), 9–21. Search in Google Scholar
[32] A. Pratelli and G. Saracco, On the generalized Cheeger problem and an application to 2d strips, Rev. Mat. Iberoam. 33 (2017), no. 1, 219–237. Search in Google Scholar
[33] G. Saracco, Weighted Cheeger sets are domains of isoperimetry, Manuscripta Math. 156 (2018), no. 3–4, 371–381. Search in Google Scholar
[34] C. Scheven and T. Schmidt, BV supersolutions to equations of 1-Laplace and minimal surface type, J. Differential Equations 261 (2016), no. 3, 1904–1932. Search in Google Scholar
[35] C. Scheven and T. Schmidt, On the dual formulation of obstacle problems for the total variation and the area functional, Ann. Inst. H. Poincaré Anal. Non Linéaire 35 (2018), no. 5, 1175–1207. Search in Google Scholar
[36] A. I. Vol’pert, The spaces BV and quasilinear equations, Math. USSR Sb. 2 (1968), 225–267. Search in Google Scholar
[37] A. I. Vol’pert and S. I. Hudjaev, Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics, Mech. Anal. 8, Martinus Nijhoff, Dordrecht, 1985. Search in Google Scholar
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2021-10-20 21:32:25
|
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|
https://qiskit.org/documentation/tutorials/algorithms/09_textbook_algorithms.html
|
Note
Run interactively in the IBM Quantum lab.
# Textbook and Shor’s algorithms¶
Qiskit contains implementations of the well known textbook quantum algorithms such as the Deutsch-Jozsa algorithm, the Bernstein-Vazirani algorithm and Simon’s algorithm.
Qiskit also has an implementation of Shor’s algorithm.
The preceding references have detailed explanations and build-out of circuits whereas this notebook has examples with the pre-built algorithms in Qiskit that you can use for experimentation and education purposes.
[1]:
import math
import numpy as np
from qiskit import BasicAer
from qiskit.aqua import QuantumInstance
from qiskit.aqua.algorithms import BernsteinVazirani, DeutschJozsa, Simon, Shor
from qiskit.aqua.components.oracles import TruthTableOracle
## Deutsch-Jozsa algorithm¶
Lets start with the Deutsch-Jozsa algorithm which can determine if a function is 'balanced' or 'constant' given such a function as input. We can experiment with it in Qiskit using an oracles created from a truth tables. So for example, we can create a TruthTableOracle instance as follows.
[2]:
bitstr = '11110000'
oracle = TruthTableOracle(bitstr)
As shown, the truthtable is specified with the bitstr containing values of all entries in the table. It has length $$8$$, so the corresponding truth table is of $$3$$ input bits. We can of course see that this truth table represents a 'balanced' function as half of values are $$1$$ and the other half $$0$$.
It might seem like a moot point running Deutsch-Jozsa on a truthtable as the function outputs are literally listed as the truthtable’s values. The intention is to create an oracle circuit whose groundtruth information is readily available to us but obviously not to the quantum Deutsch-Jozsa algorithm that is to act upon the oracle circuit. In more realistic situations, the oracle circuit would be provided as a blackbox to the algorithm.
Above said, we can inspect the circuit corresponding to the function encoded in the TruthTableOracle instance.
[3]:
oracle.circuit.draw(output='mpl')
[3]:
As seen, the $$v_i$$’s correspond to the 3 input bits; the $$o_0$$ is the oracle’s output qubit; the $$a_0$$ is an ancilla qubit.
Next we can simply create a DeutschJozsa instance using the oracle, and run it to check the result.
[4]:
dj = DeutschJozsa(oracle)
backend = BasicAer.get_backend('qasm_simulator')
result = dj.run(QuantumInstance(backend, shots=1024))
print(f'The truth table {bitstr} represents a \'{result["result"]}\' function.')
/home/runner/work/qiskit/qiskit/.tox/docs/lib/python3.8/site-packages/qiskit/aqua/quantum_instance.py:135: DeprecationWarning: The class qiskit.aqua.QuantumInstance is deprecated. It was moved/refactored to qiskit.utils.QuantumInstance (pip install qiskit-terra). For more information see <https://github.com/Qiskit/qiskit-aqua/blob/master/README.md#migration-guide>
warn_class('aqua.QuantumInstance',
/home/runner/work/qiskit/qiskit/.tox/docs/lib/python3.8/site-packages/qiskit/aqua/algorithms/education/deutsch_jozsa.py:100: DeprecationWarning: The QuantumCircuit.__add__() method is being deprecated.Use the compose() method which is more flexible w.r.t circuit register compatibility.
self._circuit = qc_preoracle + qc_oracle + qc_postoracle
/home/runner/work/qiskit/qiskit/.tox/docs/lib/python3.8/site-packages/qiskit/circuit/quantumcircuit.py:869: DeprecationWarning: The QuantumCircuit.combine() method is being deprecated. Use the compose() method which is more flexible w.r.t circuit register compatibility.
return self.combine(rhs)
The truth table 11110000 represents a 'balanced' function.
We can of course quickly put together another example for a 'constant' function, as follows.
[5]:
bitstr = '1' * 16
oracle = TruthTableOracle(bitstr)
dj = DeutschJozsa(oracle)
backend = BasicAer.get_backend('qasm_simulator')
result = dj.run(QuantumInstance(backend, shots=1024))
print(f'The truth table {bitstr} represents a \'{result["result"]}\' function.')
The truth table 1111111111111111 represents a 'constant' function.
## Bernstein-Vazirani algorithm¶
Next the Bernstein-Vazirani algorithm which tries to find a hidden string. Again, for the example, we create a TruthTableOracle instance.
[6]:
bitstr = '00111100'
oracle = TruthTableOracle(bitstr)
As shown, the truthtable is specified with the bitstr containing values of all entries in the table. It has length $$8$$, so the corresponding truth table is of $$3$$ input bits. The truthtable represents the function mappings as follows:
• $$\mathbf{a} \cdot 000 \mod 2 = 0$$
• $$\mathbf{a} \cdot 001 \mod 2 = 0$$
• $$\mathbf{a} \cdot 010 \mod 2 = 1$$
• $$\mathbf{a} \cdot 011 \mod 2 = 1$$
• $$\mathbf{a} \cdot 100 \mod 2 = 1$$
• $$\mathbf{a} \cdot 101 \mod 2 = 1$$
• $$\mathbf{a} \cdot 110 \mod 2 = 0$$
• $$\mathbf{a} \cdot 111 \mod 2 = 0$$
And obviously the goal is to find the bitstring $$\mathbf{a}$$ that satisfies all the inner product equations.
Lets again look at the oracle circuit, that now corresponds to the binary function encoded in the TruthTableOracle instance.
[7]:
oracle.circuit.draw(output='mpl')
[7]:
Again the $$v_i$$’s correspond to the 3 input bits; the $$o_0$$ is the oracle’s output qubit; the $$a_0$$ is an ancilla qubit.
Let us first compute the groundtruth $$\mathbf{a}$$ classically:
[8]:
a_bitstr = ""
num_bits = math.log2(len(bitstr))
for i in reversed(range(3)):
bit = bitstr[2 ** i]
a_bitstr += bit
print(f'The groundtruth result bitstring is {a_bitstr}.')
The groundtruth result bitstring is 110.
Next we can create a BernsteinVazirani instance using the oracle, and run it to check the result against the groundtruth.
[9]:
bv = BernsteinVazirani(oracle)
backend = BasicAer.get_backend('qasm_simulator')
result = bv.run(QuantumInstance(backend, shots=1024))
print(f'The result bitstring computed using Bernstein-Vazirani is {result["result"]}.')
assert(result['result'] == a_bitstr)
The result bitstring computed using Bernstein-Vazirani is 110.
## Simon’s algorithm¶
Simon’s algorithm is used to solve Simon’s problem. Once again, for the example, we create a TruthTableOracle instance, where the construction shows a different form.
[10]:
bitmaps = [
'01101001',
'10011001',
'01100110'
]
oracle = TruthTableOracle(bitmaps)
As shown, the truthtable is specified with three length-8 bitstrings, each containing the values of all entries for a particular output column in the table. Each bitstring has length $$8$$, so the truthtable has $$3$$ input bits; There are $$3$$ bitstrings, so the truthtable has $$3$$ output bits.
The function $$f$$ represented by the truthtable is promised to be either 1-to-1 or 2-to-1. Our goal is to determine which. For the case of 2-to-1, we also need to compute the mask $$\mathbf{s}$$, which satisfies $$\forall \mathbf{x},\mathbf{y}$$: $$\mathbf{x} \oplus \mathbf{y} = \mathbf{s}$$ iff $$f(\mathbf{x}) = f(\mathbf{y})$$. Apparently, if $$f$$ is 1-to-1, the corresponding mask $$\mathbf{s} = \mathbf{0}$$.
Let us first compute the groundtruth mask $$\mathbf{s}$$ classically:
[11]:
def compute_mask(input_bitmaps):
vals = list(zip(*input_bitmaps))[::-1]
def find_pair():
for i in range(len(vals)):
for j in range(i + 1, len(vals)):
if vals[i] == vals[j]:
return i, j
return 0, 0
k1, k2 = find_pair()
return np.binary_repr(k1 ^ k2, int(np.log2(len(input_bitmaps[0]))))
The groundtruth mask is 011.
[12]:
simon = Simon(oracle)
backend = BasicAer.get_backend('qasm_simulator')
result = simon.run(QuantumInstance(backend, shots=1024))
print(f'The mask computed using Simon is {result["result"]}.')
The mask computed using Simon is 011.
We can also quickly try a truthtable that represents a 1-to-1 function (i.e., the corresponding mask is $$\mathbf{0}$$), as follows.
[13]:
bitmaps = [
'00011110',
'01100110',
'10101010'
]
oracle = TruthTableOracle(bitmaps)
simon = Simon(oracle)
result = simon.run(QuantumInstance(backend, shots=1024))
print(f'The mask computed using Simon is {result["result"]}.')
The groundtruth mask is 000.
The mask computed using Simon is 000.
## Shor’s Factoring algorithm¶
Shor’s Factoring algorithm is one of the most well-known quantum algorithms and finds the prime factors for input integer $$N$$ in polynomial time. The algorithm implementation in Qiskit is simply provided a target integer to be factored and run, as follows:
[14]:
N = 15
shor = Shor(N)
backend = BasicAer.get_backend('qasm_simulator')
quantum_instance = QuantumInstance(backend, shots=1024)
result = shor.run(quantum_instance)
print(f"The list of factors of {N} as computed by the Shor's algorithm is {result['factors'][0]}.")
/home/runner/work/qiskit/qiskit/.tox/docs/lib/python3.8/site-packages/qiskit/aqua/algorithms/factorizers/shor.py:69: DeprecationWarning: The package qiskit.aqua.algorithms.factorizers is deprecated. It was moved/refactored to qiskit.algorithms.factorizers (pip install qiskit-terra). For more information see <https://github.com/Qiskit/qiskit-aqua/blob/master/README.md#migration-guide>
warn_package('aqua.algorithms.factorizers',
The list of factors of 15 as computed by the Shor's algorithm is [3, 5].
Note: this implementation of Shor’s algorithm uses $$4n + 2$$ qubits, where $$n$$ is the number of bits representing the integer in binary. So in practice, for now, this implementation is restricted to factorizing small integers. Given the above value of N we compute $$4n +2$$ below and confirm the size from the actual circuit.
[15]:
print(f'Computed of qubits for circuit: {4 * math.ceil(math.log(N, 2)) + 2}')
print(f'Actual number of qubits of circuit: {shor.construct_circuit().num_qubits}')
Computed of qubits for circuit: 18
Actual number of qubits of circuit: 18
[16]:
import qiskit.tools.jupyter
%qiskit_version_table
### Version Information
Qiskit SoftwareVersion
Qiskit0.26.0
Terra0.17.3
Aer0.8.2
Ignis0.6.0
Aqua0.9.1
IBM Q Provider0.13.1
System information
Python3.8.10 (default, May 4 2021, 07:16:51) [GCC 9.3.0]
OSLinux
CPUs2
Memory (Gb)6.791343688964844
Tue May 11 12:25:58 2021 UTC
|
2021-05-16 09:43:49
|
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|
http://networkdata.schochastics.net/reference/covert_14.html
|
Data refers to four groups involved in cocaine trafficking in Spain. Information comes from police wiretapping and meetings registered by police investigations of these criminal organisations between 2007 and 2009. Operation MAMBO (N=22). The investigation started in 2006 and involved Colombian citizens that were introducing 50 kg of cocaine to be adulterated and distributed in Madrid (Spain). Ultimately, the group was involved in smuggling cocaine from Colombia through Brazil and Uruguay to be distributed in Spain. This is a typical Spanish middle cocaine group acting as wholesale supplier between a South American importer group and retailers in Madrid. Operation JUANES (N=51). In 2009, the police investigation detected a group involved in the smuggling of cocaine from Mexico to be distributed in Madrid (Spain). In this case, the group operated in close cooperation with another organization that was laundering the illegal income from drug distribution from this and other groups. The cocaine traffickers earned an estimated EUR 60 million. Operation JAKE (N=62). In 2008, the group investigated was operating as a wholesale supplier and retail distributor of cocaine and heroin in a large distribution zone located in Madrid (Spain), where gypsy clans traditionally carry out similar activities. The group was in charge of acquiring, manipulating and selling the drugs in the gypsy quarter. Operation ACERO (N=11). This investigation started in 2007 and involved a smaller family-based group. The group was composed mainly of members of a same family and was led by a female. They distributed cocaine in Madrid (Spain) that was provided to them by other groups based in a northwest region of the country, one of the most active areas in the provision of cocaine from the countries of origin. The group also had their own procedures to launder money. 4 1-mode matrices person by person from each of the operations described above. Undirected, valued ties. Mambo: 31x31 Juanes:51x51 Jake: 38x38 Acero: 25x25 Relations are communications between individuals. Meaning of tie values is unclear - may represent level of communications activity.
covert_14
## Format
list of igraph objects
## Source
Available at Manchester (https://sites.google.com/site/ucinetsoftware/datasets/covert-networks). Reconstructed from Jimenez-Salinas Framis, A. "Illegal networks or criminal organizations: Power, roles and facilitators in four cocaine trafficking structures." In Third Annual Illicit Networks Workshop, Montreal. 2011
## References
Jimenez-Salinas Framis, A. "Illegal networks or criminal organizations: Power, roles and facilitators in four cocaine trafficking structures." In Third Annual Illicit Networks Workshop, Montreal. 2011.
|
2022-08-08 08:12:20
|
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|
https://socratic.org/questions/573b48f77c01497c659e6ac5
|
# What is the molarity of an 58.5*g mass of sodium chloride dissolved in enough water to make 1*L of solution?
$\text{Molarity "=" Moles of solute"/"Volume of solution}$
If there are $58 \cdot g$ $N a C l$ solute there are $1$ $m o l$ each solute particles, $N {a}^{+}$, and $C {l}^{-}$ in solution.
And thus $\text{Molarity}$ $=$ $1 \cdot m o l \cdot {L}^{-} 1$ with respect to $N a C l \left(a q\right)$.
|
2019-12-14 15:32:21
|
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https://www.gradesaver.com/textbooks/math/algebra/college-algebra-7th-edition/chapter-p-prerequisites-chapter-p-test-page-79/3
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## College Algebra 7th Edition
(a) $A \cap B=\{0, 1, 5 \}$ (b) $A \cup B=\{-2, 0, \frac{1}{2},1, 3, 5, 7\}$
(a) We find the intersection by finding terms common to both A and B: $A \cap B=\{0, 1, 5 \}$ (b) We find the union by combining terms and removing duplicates: $A \cup B=\{-2, 0, \frac{1}{2},1, 3, 5, 7\}$
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2018-07-23 17:11:24
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https://dmoj.ca/problem/binary
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## Binary
View as PDF
Points:5
Time limit:2.0s
Memory limit:64M
Author:
Problem types
Your computer science teacher has kindly requested that you write a program to convert a number, inputted in decimal, to binary to help in teaching students about binary. Also to aid in human processing of numbers, you should group every four bits and separate the groups by space.
#### Input Specification
The first line will be integer (), the number of numbers to convert to binary. The next lines will be such that ().
#### Output Specification
For every integer , output the binary representation, grouped into four bit groups, separated by spaces. If necessary, you must pad the first group to have exactly four bits.
#### Sample Input
3
1
10
255
#### Sample Output
0001
1010
1111 1111
• Shehryar
commented on Jan. 2, 2017, 1:03 a.m.
Hint
Java has a method you can call against the Integer class: Integer.toBinaryString(number);
• acetao
commented on Jan. 27, 2016, 1:04 a.m. edit 3
output must be a group with four digits
Note the output specification: (1) 16 --> 10000,output should be:
0001 0000,output should not be:1000 0 (2) 0 --> 0000,1 --> 0001, 7 --> 0111
• Ish_God
commented on Nov. 15, 2015, 6:28 p.m.
OLE
what is the output if the input is 0
• Xyene
commented on Nov. 15, 2015, 7:04 p.m.
0000
• niamp
commented on Nov. 10, 2015, 10:44 p.m. edited
I tested my solution using eclipse and all the test cases are correct, why am I getting WA in my submissions?
UPDATE: Nvm, I was making a stupid mistake
• Anix55
commented on Oct. 24, 2015, 11:35 p.m.
help pls
I can't get the first text case to work...is anyone having a similar problem?
|
2017-01-17 17:06:35
|
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|
http://forum.coppeliarobotics.com/viewtopic.php?f=9&t=685&p=2757
|
## Multiple Simulation Episodes
Typically: "How do I... ", "How can I... " questions
fairread
Posts: 9
Joined: 13 Aug 2013, 15:37
### Multiple Simulation Episodes
Hi I wish to implement some AI algorithms on robot navigation from an initial position to a goal position in V-REP. I, however need to run multiple numbers of simulations sequentially (one simulation scene after another) After the first simulation episode finishes (by timeout or reaching goal position), I would like a new simulation episode to begin with the robot having some stored memory from the past episode and it will start at the similar initial position. How may I do this in V-REP? Thanks.
coppelia
Site Admin
Posts: 7517
Joined: 14 Dec 2012, 00:25
### Re: Multiple Simulation Episodes
Hello,
you have several ways of achieving this, for instance:
• in the simulation dialog, you could uncheck Reset scene to initial state. That way, at the end of the simulation, the scene will not be reset.
• you can also disable individual robots or models from a reset at simulation end. In a child script attached to the base of a model, you could have following code:
Code: Select all
if (simGetScriptExecutionCount()==0) then
base=simGetObjectAssociatedWithScript(sim_handle_self)
p=simGetModelProperty(base)
p=simBoolOr16(p,sim_modelproperty_not_reset)
simSetModelProperty(base,p)
end
When loading a new scene, you will have however to store the robot (or model) temporarily to a file, load the new scene and load the saved model into that scene. Have a look at following API functions:
simLoadScene can only be called from a plugin, an add-on, a remote API client or a ROS node (child scripts are only executed DURING simulation).
Or you can save only state data of the robot like position and orientation. That data you can save/load it to/from a file, to a persistent data block, or to script simulation parameters (in that case the parameter should be flagged as Parameter is persistent)
Cheers
Eric
Posts: 186
Joined: 11 Feb 2013, 16:39
### Re: Multiple Simulation Episodes
Hi!
Maybe you can simply "teleport" your robot at the initial position after each timeout or reached position, and restart your AI navigation algorithm?
To "teleport" your dynamic enabled robot at the intial place you can do the following code:
Code: Select all
--moving the dynamic object
t={robotHandle}
while (#t~=0) do
h=t[1]
simResetDynamicObject(h)
table.remove(t,1)
ind=0
child=simGetObjectChild(h,ind)
while (child~=-1) do
table.insert(t,child)
ind=ind+1
child=simGetObjectChild(h,ind)
end
end
--moving the robot to the initial position and orientation (i.e. where the simulation started)
simSetObjectPosition(robotHandle,-1,initialPosition)
simSetObjectOrientation(robotHandle,-1,initialOrientation)
fairread
Posts: 9
Joined: 13 Aug 2013, 15:37
### Re: Multiple Simulation Episodes
Hi Admim and Eric,
Thanks for your replies, sorry I could only reply now as I went for a holiday earlier.
To Admin, I appreciate your idea to uncheck Reset Scene to Initial State, but may I know what is the way to automate the run of multiple simulation episodes? For example, let say I would like to run 20 episodes sequentially by starting the simulation today, leaving the pc running the whole night and then obtaining the results tomorrow morning. I have the basic idea of creating a script to run automated restarting of a new scene but I do not know how it could be implemented explicitly in V-REP.
In that context, I appreciate Eric's suggestion of running only single scene and whenever the robot reaches the end position or timeout occurs, it will be teleported back to the initial position. I would then need to create an episode counter which will be incremented once timeout occurs or robot reaches end position. If counter reaches 20, then the simulation scene should be terminated.
But I also welcome any suggestion to automate scene restart using scripts if you guys may have. Thanks again.
coppelia
Site Admin
Posts: 7517
Joined: 14 Dec 2012, 00:25
### Re: Multiple Simulation Episodes
Hello,
the best for your task is to use an add-on script since they also run when simulation is stopped. From that add-on script you can then load a scene, run/stop the simulation, etc.
Cheers
fairread
Posts: 9
Joined: 13 Aug 2013, 15:37
### Re: Multiple Simulation Episodes
Thanks for the suggestion, I'm trying to implement it now. Anyway, another question related to running multiple simulation episodes, is there any other way to accelerate the simulation speed apart from changing the dynamics engine, dynamic settings and simulation time step such as disabling the graphics, downscale to 2-D simulation or disabling the dynamics engine?
Thanks again!
coppelia
Site Admin
Posts: 7517
Joined: 14 Dec 2012, 00:25
### Re: Multiple Simulation Episodes
Hello,
Yes, you can:
• disable visualization. Check out the model Models/other/fast simulation mode.ttm, or the API function simSetBooleanParameter(sim_boolparam_display_enabled,false).
• disable dynamics. You can do this in the dynamics general settings. You can also individually disable models for dynamics simulation. Select the model, then open the model properties dialog. Check Model is static and Model is not respondable.
Cheers
|
2020-01-19 20:28:05
|
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|
https://tex.stackexchange.com/questions/417098/numbering-the-list-of-figures-and-table-of-contents
|
I would like to number the list of figures and the table of contents, with Roman numerals, themselves (not the things listed in them).
Doing this, while leaving the main text in bold, and leaving the number unformatted.
I can't seem to find a way of doing so.
– user31729
Feb 24, 2018 at 23:37
– user31729
Feb 24, 2018 at 23:38
• I was using the report document class. I doubt including any code would help (I just have the table of contents and the figure listing initiated, I didn't do anything else to them), besides, it is pretty messy. Feb 25, 2018 at 0:11
• @ChristianHupfer Don't those commands just change the numbering of the chapters? The Table of Contents and the List of Figures, remain unnumbered. Feb 25, 2018 at 0:42
By default \tableofcontents and \listoffigures do not use numbered headings. This can be changed using the tocbibind package.
The following code redefines \tableofcontents and inserts the \tocchapter instruction, then loading the .toc file with \tocfile{\contentsname}{toc}.
The macro \unboldchapternumber defines the chapter number to be not bold and switches to \Roman and redefines the potential hyper anchor \theHchapter in case of hyperref being loaded.
\renewcommand{\tableofcontents}{%
\begingroup
\unboldchapternumber%
\tocchapter
\tocfile{\contentsname}{toc}
\endgroup
}
After loading \listoffigures, the chapter counter should be reset to zero.
\documentclass{report}
\usepackage{tocbibind}
\usepackage{blindtext}
\usepackage{hyperref}
\makeatletter
% Check whether hyperref is loaded
}{}
\makeatother
\newcommand{\unboldchapternumber}{%
\renewcommand{\thechapter}{\normalfont\Roman{chapter}\bfseries}%
% Change the hyperanchors to prevent wrong anchors
\renewcommand{\theHchapter}{tocchapter.\arabic{chapter}}%
\fi
}
\renewcommand{\tableofcontents}{%
\begingroup
\unboldchapternumber%
\tocchapter
\tocfile{\contentsname}{toc}
\endgroup
}
\renewcommand{\listoffigures}{%
\begingroup
\unboldchapternumber
\tocchapter
\tocfile{\listfigurename}{lof}
\endgroup
}
\begin{document}
\tableofcontents
\listoffigures
\setcounter{chapter}{0}
\chapter{One}
\blindtext
\chapter{Two}
\blindtext
\end{document}
|
2022-08-14 01:29:42
|
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|
http://tex.stackexchange.com/questions/51441/how-can-i-draw-a-quiver-plot-with-varying-line-thickness
|
# How can I draw a quiver plot with varying line thickness?
This question originates in Scalar field in pgfplots .
How can I draw a quiver plot in which the arrows have the same length, but their line width varies according to some scalar property of the vectors?
-
@marko please note this answer to your feature request in tex.stackexchange.com/questions/51103/scalar-field-in-pgfplots – Christian Feuersänger Apr 10 '12 at 18:11
Pgfplots allows to employ point meta as additional input coordinate. Typically, point meta is used as color data, but it can also be used for different purposes.
In this case, you can use it as line thickness parameter for a quiver plot:
\documentclass{article}
\usepackage{pgfplots}
\begin{document}
\begin{tikzpicture}
\def\U{1}
\def\V{2*x}
\def\LEN{(sqrt((\U)^2 + (\V)^2)}
\begin{axis}[axis equal]
point meta={\LEN},
quiver={
u={(\U)/\LEN},v={(\V)/\LEN},
scale arrows=2,
every arrow/.append style={line width=\pgfplotspointmetatransformed/1000 * 2pt},
},
-stealth,samples=15,
] {x^2};
\end{axis}
\end{tikzpicture}
\end{document}
The example defines some macros \U, \V, and \LEN where \LEN is the length of these vectors. Then, a quiver plot is drawn where each vector is normalized to unit length - and the length parameter is assigned as point meta. This is a simple and standard way of communicating extra data to plots (you could also input extra values in a table).
Finally, \pgfplotspointmetatransformed is a scalar value in the range [0,1000] such that 0 corresponds to the minimum point meta value and 1000 corresponds to the maximum point meta value.
The expression for the line width scales this linearly such that the minimum line thickness is 0 and the maximum is 2pt.
Note that axis equal is necessary if we want to see that the vectors have the same sizes.
-
|
2016-05-04 19:33:48
|
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https://www.castlegroup.co.uk/guidance/noise-at-work-assessments/acoustic-terms-glossary/
|
# Acoustic Terms Glossary
## Glossary of Terms for Acoustics
A guide to most commonly used acoustic terms used for noise and sound measurement found in sound level meters.
Decibels dB Ten times the logarithm (to base 10) of the ratio of two mean square values of sound pressure. Frequency Hz The number of cyclical variations per unit time Octave Bands Hz Frequency ranges in which the upper limit of each band is twice the lower limit. Octave bands are identified by their geometric mean frequency or centre frequency. Sound Power W The acoustic power of a sound source expressed in Watts. Sound Power Level Lw The acoustic power radiated from a given sound source as related to a reference power level. Sound Pressure P Fluctuations in air pressure caused by the presence of sound waves. Sound Pressure Level Lp The ratio expressed in dB of mean-square sound pressure to a reference mean-square pressure which, by convention, has been selected to be equal to the assumed threshold of hearing (20µ Pa). Spectrum A term that refers to the frequency response. Weighting Network dB(A) An electronic filter in a sound level meter which approximates, under defused conditions, the frequency response of the human ear. The ‘A’ weighting network is the one most commonly used. Equivalent Continuous Sound Pressure Level Leq The equivalent continuous level which is a measure of the energy content of a sound over a time period. It gives a single figure expressing the equivalent of a varying level. It is an energy average. Dose D Noise dose is a percentage number. Depending on a criterion level and exchange rate (eg 90dB and 3 – 100% Dose = 90dB for 8 hours). Dose per Hour DOHR Designed to show the Dose exposure over each hour. Projected Dose Proj D Projected dose allows the present accumulated dose, over the present logged time duration, to be projected forward to give the predicted 8 hour period. Noise Exposure Lex Personal noise exposure (also known as Lepd), usually referred to a daily 8 hour rate. Percentile Ln The dB level exceeded for n% of the time.
## a part of the Castle Group of websites
• YOU ARE HERE
Tel: 01723 584250
|
2017-08-19 05:43:48
|
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|
https://brilliant.org/problems/radius-of-circle-with-complex-equation/
|
# Radius of circle with complex equation
Algebra Level 4
Find the radius of the circle with centre $$\dfrac{1}{2}(1+i)$$ on which a complex number $$z$$ lies such that $$\dfrac{z-i}{z-1}$$ is purely imaginary.
Note: A complex number $$a$$ is purely imaginary if $$a=\lambda.i$$ where $$\lambda \in \mathbb{R}$$.
×
|
2019-04-26 10:52:04
|
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|
https://opus4.kobv.de/opus4-zib/frontdoor/index/index/docId/5376
|
Valid inequalities for the topology optimization problem in gas network design
• One quarter of Europe's energy demand is provided by natural gas distributed through a vast pipeline network covering the whole of Europe. At a cost of 1 million Euro per km extending the European pipeline network is already a multi-billion Euro business. Therefore, automatic planning tools that support the decision process are desired. Unfortunately, current mathematical methods are not capable of solving the arising network design problems due to their size and complexity. In this article, we will show how to apply optimization methods that can converge to a proven global optimal solution. By introducing a new class of valid inequalities that improve the relaxation of our mixed-integer nonlinear programming model, we are able to speed up the necessary computations substantially.
$Rev: 13581$
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2017-09-21 08:37:21
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https://pytorch.org/docs/master/generated/torch.nn.utils.clip_grad_value_.html
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• clip_value (float or int) – maximum allowed value of the gradients. The gradients are clipped in the range $\left[\text{-clip\_value}, \text{clip\_value}\right]$
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2020-08-13 06:26:04
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https://itprospt.com/num/137662/5-stubolecinlg-influence-has-when-multiple-omijbiced-t0
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5
##### Score: 2.4n12AzanswemedQuestion 3Find the arc length of y 323 / 2 on 2 < € < 3.SUBMIT A PHOTO OF YOUR HANDWRITTEN WORK HEREChoose File No file chosenBACKUP LINKIN CASE QE TECHNICAL_PROBLEMSQuestion Help: CalculatorMessage instructorCheck Answer
Score: 2.4n12 Azanswemed Question 3 Find the arc length of y 323 / 2 on 2 < € < 3. SUBMIT A PHOTO OF YOUR HANDWRITTEN WORK HERE Choose File No file chosen BACKUP LINKIN CASE QE TECHNICAL_PROBLEMS Question Help: Calculator Message instructor Check Answer...
##### Question 106 ptsFind: Ez+ (4-") noneDNE
Question 10 6 pts Find: Ez+ (4-") none DNE...
##### The standard enthalpy of vaporization of a liquid is45.2 kJ/mol at its normal boiling point, 56.4°C. What is the standard change in entropy for thevaporization of this liquid at its normal boiling point?Calculate your answer in J/K. Enter your answer with one decimalplace.
The standard enthalpy of vaporization of a liquid is 45.2 kJ/mol at its normal boiling point, 56.4 °C. What is the standard change in entropy for the vaporization of this liquid at its normal boiling point? Calculate your answer in J/K. Enter your answer with one decimal place....
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2022-07-01 14:40:22
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https://eprint.iacr.org/2017/624
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### Fast Leakage Assessment
Oscar Reparaz, Benedikt Gierlichs, and Ingrid Verbauwhede
##### Abstract
We describe a fast technique for performing the computationally heavy part of leakage assessment, in any statistical moment (or other property) of the leakage samples distributions. The proposed technique outperforms by orders of magnitude the approach presented at CHES 2015 by Schneider and Moradi. We can carry out evaluations that before took 90 CPU-days in 4 CPU-hours (about a 500-fold speed-up). As a bonus, we can work with exact arithmetic, we can apply kernel-based density estimation methods, we can employ arbitrary pre-processing functions such as absolute value to power traces, and we can perform information-theoretic leakage assessment. Our trick is simple and elegant, and lends itself to an easy and compact implementation. We fit a prototype implementation in about 130 lines of C code.
Available format(s)
Publication info
Keywords
leakage assessmentefficient computationside-channel analysiscountermeasure
Contact author(s)
oscar reparaz @ esat kuleuven be
History
Short URL
https://ia.cr/2017/624
CC BY
BibTeX
@misc{cryptoeprint:2017/624,
author = {Oscar Reparaz and Benedikt Gierlichs and Ingrid Verbauwhede},
title = {Fast Leakage Assessment},
howpublished = {Cryptology ePrint Archive, Paper 2017/624},
year = {2017},
note = {\url{https://eprint.iacr.org/2017/624}},
url = {https://eprint.iacr.org/2017/624}
}
Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content.
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2022-05-27 21:21:30
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http://openstudy.com/updates/4dc5b6b5a5918b0b006e8320
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## remidia Group Title Solve for x algebraically in 7^(-x+2) = 0.005, rounding your answers to three signicant digits. A) 4.32 B) 4.72 C) 5.12 D) 5.52 E) 5.92 3 years ago 3 years ago
1. him1618 Group Title
take log to base 7 on both sides 2-x = log 0.005 to base 7
2. him1618 Group Title
use the base change formula then
3. polpak Group Title
I don't know why you've posted 3 times this same equation
4. him1618 Group Title
2-x = [log0.005/log 7]
5. polpak Group Title
Particularly, since people were helping you in the previous two versions
6. him1618 Group Title
to the base 10
7. aama100 Group Title
x -> 4.7228
8. polpak Group Title
aama100: that's some impressive calculator work. If only your tutoring skills were that good people might actually learn something.
9. remidia Group Title
Thank you him1618, I understand what I was supposed to do now. Its easier for me to figure out math when I see the equation in front of me because I am a visual learner.
10. aama100 Group Title
thanks polpak
11. polpak Group Title
You can just use the base 10 log and you end up with the same thing $log(7^{x-2}) = log(5/1000)$ $\implies (x-2)log(7) = log(5) - log(1000)$ $\implies x-2 = \frac{log(5) - 3}{log(7)} \implies x = \frac{log(5) - 3}{log(7)} + 2$
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2014-09-16 13:41:35
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https://demo7.dspace.org/items/58f04e91-4632-4c8a-98f9-a8bb9e17badd
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## Photon-number selective group delay in cavity induced transparency
##### Authors
Nikoghosyan, Gor
Fleischhauer, Michael
##### Description
We show that the group velocity of a probe pulse in an ensemble of $\Lambda$-type atoms driven by a quantized cavity mode depends on the quantum state of the input probe pulse. In the strong-coupling regime of the atom-cavity system the probe group delay is photon number selective. This can be used to spatially separate the single photon from higher photon-number components of a few-photon probe pulse and thus to create a deterministic single-photon source.
Comment: 5 pages, 2 figures. revised version
Quantum Physics
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2022-12-07 06:55:16
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https://ask.libreoffice.org/en/questions/133852/revisions/
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# Revision history [back]
### Command line Conversion Options
Hi Guys,
Can you please tell me if the LibreOffice command line conversion has ability to fit the excel file to one page only ? I am currently using libreoffice to convert the excel files that we receive from outside vendor to pdf before loading it to our database. Unfortunately sometimes that the length of columns in the file are more than regular Print Size and they get moved over to second page. This causes the load process to fail. The command I am using is as below.
sudo libreoffice --headless --convert-to pdf --outdir ../excel/ filename.xls
I know you can chose Scaling Options if you are using GUI but we run this process during night and I want to avoid any human interaction.
Regards,
Saurabh
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2019-04-24 16:18:16
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https://peeterjoot.wordpress.com/tag/phy450/
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• 330,706
# Posts Tagged ‘PHY450’
## updates to my old class notes from the 2011 phy450h1s (relativistic electrodynamics)
Posted by peeterjoot on June 2, 2012
I’ve made some minor updates to my old class notes from the 2011 phy450h1s (relativistic electrodynamics) course I took:
(I’ve switched this notes collection, as well as some others, to a better looking latex book template, one that I also used for my continuum mechanics class notes). While making that change I also switched things from a chapter heading per lecture, to a couple more logical chapter headings, which makes things easier to navigate.
Posted in Math and Physics Learning. | Tagged: , | 2 Comments »
## Final collection of class notes for the 2011 Relativistic Electrodynamics (phy450hs1) course I attended at UofT
Posted by peeterjoot on May 5, 2011
CLICK HERE for my final collection of class notes for the 2011 Relativistic Electrodynamics (phy450hs1) course I attended at UofT. This course was taught by Prof. Erich Poppitz, and TA’ed by Simon Freedman, and provided a structured top down approach to electrodynamics. Starting with very few basic principles a great deal of material was covered in a very refreshing hierarchical fashion.
Typos and errors, if any, are probably mine (Peeter), and no claim nor attempt of spelling or grammar correctness will be made.
The text for the course was Landau and Lifshitz, “Classical Theory of Fields”.
These notes track along with the Professor’s hand written notes very closely, since his lectures follow his notes very closely. While I used the note taking exercise as a way to verify that I understood all the day’s lecture materials, the Professor’s notes are in many instances a much better study resource, since there are details in his notes that were left for us to read, and not necessarily covered in the lectures. On the other hand, there are details in these notes that I’ve added when I didn’t find his approach simplistic enough for me to grasp, or I failed to follow the details in class.
This also has some private notes from my reading of the text and some thoughts on materials we were covering. One of the earlier of these was due to unfortunate use of an ancient edition of the text borrowed from the library, since mine had been lost in shipping. That version didn’t use the upper and lower index quantities that I’d expected, so I tried to puzzle out some of what myself from what I knew. Once I got an up to date copy of the text the point of that exercise was negated.
Also included are some assigned problems, at least the parts of them that I did not hand write. I’ve corrected some the errors after receiving grading feedback, and where I haven’t done so I at least recorded some of the grading comments as a reference. Not all the problems were graded, so I make no guarantees of correctness.
Included in the PDF above are all the following individual PDFs. These are still available, but may contain errors possibly fixed in the complete notes collection.
April 13, 2011 Some exam reflection.
Mar 25, 2011 Problem Set 6.
Mar 23, 2011 Energy Momentum Tensor.
Mar 14, 2011 Problem Set 5.
Mar 3, 2011 PHY450H1S Problem Set 4.
Feb 15, 2011 PHY450H1S Problem Set 3.
Feb 6, 2011 Energy term of the Lorentz force equation.
Feb 1, 2011 PHY450H1S Problem Set 2.
Jan 22, 2011 PHY450H1S Problem Set 1.
Jan 11, 2011 Speed of light and simultaneity.
## PHY450H1S. Relativistic Electrodynamics Lecture 27 (Taught by Prof. Erich Poppitz). Radiation reaction force continued, and limits of classical electrodynamics.
Posted by peeterjoot on May 3, 2011
[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
Covering chapter 8 section 65 material from the text [1].
FIXME: Covering pp. 198.1-200: (last topic): attempt to go to the next order $(v/c)^3$ – radiation damping, the limitations of classical electrodynamics, and the relevant time/length/energy scales.
We previously obtained the radiation reaction force by adding a “frictional” force to the harmonic oscillator system. Now its time to obtain this by continuing the expansion of the potentials to the next order in $\mathbf{v}/c$.
Recall that our potentials are
\begin{aligned}\phi(\mathbf{x}, t) &= \int d^3 \mathbf{x} \frac{\rho\left(\mathbf{x}', t - {\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}/c\right)}{{\left\lvert{\mathbf{x} - \mathbf{x}}\right\rvert}} \\ \mathbf{A}(\mathbf{x}, t) &= \frac{1}{{c}}\int d^3 \mathbf{x} \frac{\mathbf{j}\left(\mathbf{x}', t - {\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}/c\right)}{{\left\lvert{\mathbf{x} - \mathbf{x}}\right\rvert}}.\end{aligned} \hspace{\stretch{1}}(2.1)
We can expand in Taylor series about $t$. For the charge density this is
\begin{aligned}\begin{aligned}\rho&\left(\mathbf{x}', t - {\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}/c\right) \\ &\approx \rho(\mathbf{x}', t) - \frac{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}{c} \frac{\partial {}}{\partial {t}} \rho(\mathbf{x}', t) + \frac{1}{{2}} \left(\frac{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}{c}\right)^2 \frac{\partial^2 {{}}}{\partial {{t}}^2} \rho(\mathbf{x}', t) - \frac{1}{{6}} \left(\frac{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}{c}\right)^3 \frac{\partial^3}{\partial t^3} \rho(\mathbf{x}', t) \end{aligned},\end{aligned} \hspace{\stretch{1}}(2.3)
so that our scalar potential to third order is
\begin{aligned}\phi(\mathbf{x}, t) &=\int d^3 \mathbf{x} \frac{\rho(\mathbf{x}', t) }{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}- \frac{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}{c} \frac{\partial {}}{\partial {t}} \int d^3 \mathbf{x} \frac{\rho(\mathbf{x}', t) }{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}} \\ &+ \frac{1}{{2}} \left(\frac{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}{c}\right)^2 \frac{\partial^2 {{}}}{\partial {{t}}^2} \int d^3 \mathbf{x} \frac{\rho(\mathbf{x}', t) }{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}- \frac{1}{{6}} \left(\frac{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}{c}\right)^3 \frac{\partial^3}{\partial t^3} \int d^3 \mathbf{x} \frac{\rho(\mathbf{x}', t) }{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}} \\ &=\int d^3 \mathbf{x} \frac{\rho(\mathbf{x}', t) }{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}- {\frac{\partial {}}{\partial {t}} \int d^3 \mathbf{x} \frac{\rho(\mathbf{x}', t) }{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}} \frac{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}{c} }\\ &+ \frac{1}{{2}} \frac{\partial^2 {{}}}{\partial {{t}}^2} \int d^3 \mathbf{x} \frac{\rho(\mathbf{x}', t) }{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}\left(\frac{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}{c}\right)^2 - \frac{1}{{6}} \frac{\partial^3}{\partial t^3} \int d^3 \mathbf{x} \frac{\rho(\mathbf{x}', t) }{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}} \left(\frac{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}{c}\right)^3 \\ &= \phi^{(0)} + \phi^{(2)} + \phi^{(3)}\end{aligned}
Expanding the vector potential in Taylor series to second order we have
\begin{aligned}\mathbf{A}(\mathbf{x}, t) &=\frac{1}{{c}} \int d^3 \mathbf{x} \frac{\mathbf{j}(\mathbf{x}', t) }{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}- \frac{1}{{c}} \frac{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}{c} \frac{\partial {}}{\partial {t}} \int d^3 \mathbf{x} \frac{\mathbf{j}(\mathbf{x}', t) }{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}} \\ &=\frac{1}{{c}} \int d^3 \mathbf{x} \frac{\mathbf{j}(\mathbf{x}', t) }{{\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}}- \frac{1}{{c^2}} \frac{\partial {}}{\partial {t}} \int d^3 \mathbf{x} \mathbf{j}(\mathbf{x}', t) \\ &= \mathbf{A}^{(1)} + \mathbf{A}^{(2)} \end{aligned}
We’ve already considered the effects of the $\mathbf{A}^{(1)}$ term, and now move on to $\mathbf{A}^{(2)}$. We will write $\phi^{(3)}$ as a total derivative
\begin{aligned}\phi^{(3)} = \frac{1}{{c}} \frac{\partial {}}{\partial {t}} \left( - \frac{1}{{6 c^2}} \frac{\partial^2}{\partial t^2} \int d^3 \mathbf{x} \rho(\mathbf{x}', t){\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}^2\right)= \frac{1}{{c}} \frac{\partial {}}{\partial {t}} f^{(2)}(\mathbf{x}, t)\end{aligned} \hspace{\stretch{1}}(2.4)
and gauge transform it away as we did with $\phi^{(2)}$ previously.
\begin{aligned}\phi^{(3)'} &= \phi^{(3)} - \frac{1}{{c}} \frac{\partial {f^{(2)}}}{\partial {t}} = 0 \\ \mathbf{A}^{(2)'} &= \mathbf{A}^{(2)} + \boldsymbol{\nabla} f^{(2)} \end{aligned} \hspace{\stretch{1}}(2.5)
\begin{aligned}\mathbf{A}^{(2)'} &= - \frac{1}{{c^2}} \frac{\partial {}}{\partial {t}} \int d^3 \mathbf{x} \mathbf{j}(\mathbf{x}', t) - \frac{1}{{6 c^2}} \frac{\partial^2}{\partial t^2} \int d^3 \mathbf{x} \rho(\mathbf{x}', t)\boldsymbol{\nabla}_{\mathbf{x}} {\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}^2 \\ \end{aligned}
Looking first at the first integral we can employ the trick of writing $\mathbf{e}_\alpha = {\partial {\mathbf{x}'}}/{\partial {x^{\alpha'}}}$, and then employ integration by parts
\begin{aligned}\int_V d^3 \mathbf{x} \mathbf{j}(\mathbf{x}', t) &=\int_V d^3 \mathbf{x} \mathbf{e}_\alpha j^\alpha (\mathbf{x}', t) \\ &=\int_V d^3 \mathbf{x} \frac{\partial {\mathbf{x}'}}{\partial {x^{\alpha'}}}j^\alpha (\mathbf{x}', t) \\ &=\int_V d^3 \mathbf{x} \frac{\partial {}}{\partial {x^{\alpha'}}} \left( \mathbf{x}' j^\alpha (\mathbf{x}', t) \right)-\int_V d^3 \mathbf{x}\mathbf{x}' \frac{\partial {}}{\partial {x^{\alpha'}}} j^\alpha (\mathbf{x}', t) \\ &=\int_{\partial V} d^2 \boldsymbol{\sigma} \cdot \left( \mathbf{x}' j^\alpha (\mathbf{x}', t) \right)-\int d^3 \mathbf{x} \mathbf{x}' -\frac{\partial {}}{\partial {t}} \rho(\mathbf{x}', t) \\ &=\frac{\partial {}}{\partial {t}} \int d^3 \mathbf{x} \mathbf{x}' \rho(\mathbf{x}', t) \\ \end{aligned}
For the second integral, we have
\begin{aligned}\boldsymbol{\nabla}_{\mathbf{x}} {\left\lvert{\mathbf{x} - \mathbf{x}'}\right\rvert}^2 &= \mathbf{e}_\alpha \partial_\alpha (x^\beta - x^{\beta'})(x^\beta - x^{\beta'}) \\ &=2 \mathbf{e}_\alpha \delta_{\alpha \beta}(x^\beta - x^{\beta'}) \\ &= 2 (\mathbf{x} - \mathbf{x}'),\end{aligned}
so our gauge transformed vector potential term is reduced to
\begin{aligned}\mathbf{A}^{(2)'} &= -\frac{1}{{c^2}} \frac{\partial^2}{\partial t^2} \int d^3 \mathbf{x} \rho(\mathbf{x}', t) \left(\mathbf{x}' + \frac{1}{{3}}(\mathbf{x} - \mathbf{x}') \right) \\ &= -\frac{1}{{c^2}} \frac{\partial^2}{\partial t^2} \int d^3 \mathbf{x} \rho(\mathbf{x}', t) \left(\frac{1}{{3}} \mathbf{x} + \frac{2}{3}\mathbf{x}' \right) \\ \end{aligned}
Now we wish to employ a discrete representation of the charge density
\begin{aligned}\rho(\mathbf{x}', t) = \sum_{b=1}^N q_b \delta^3(\mathbf{x}' - \mathbf{x}_b(t))\end{aligned} \hspace{\stretch{1}}(2.7)
So that the second order vector potential becomes
\begin{aligned}\mathbf{A}^{(2)'} &= -\frac{1}{{c^2}} \frac{\partial^2}{\partial t^2} \int d^3 \mathbf{x} \left(\frac{1}{{3}} \mathbf{x} + \frac{2}{3}\mathbf{x}' \right) \sum_{b=1}^N q_b \delta^3(\mathbf{x}' - \mathbf{x}_b(t)) \\ &= -\frac{1}{{c^2}} \frac{\partial^2}{\partial t^2} \sum_{b=1}^N q_b \left( {\frac{1}{{3}} \mathbf{x}} + \frac{2}{3}\mathbf{x}_b(t) \right) \\ &=-\frac{2}{3 c^2} \sum_{b=1}^N q_b \dot{d}{\mathbf{x}}_b(t) \\ &=-\frac{2}{3 c^2} \frac{d^2}{dt^2}\left( \sum_{b=1}^N q_b \mathbf{x}_b(t) \right).\end{aligned}
We end up with a dipole moment
\begin{aligned}\mathbf{d}(t) = \sum_{b=1}^N q_b \mathbf{x}_b(t) \end{aligned} \hspace{\stretch{1}}(2.8)
so we can write
\begin{aligned}\mathbf{A}^{(2)'} = -\frac{2}{3 c^2} \dot{d}{\mathbf{d}}(t).\end{aligned} \hspace{\stretch{1}}(2.9)
Observe that there is no magnetic field due to this contribution since there is no explicit spatial dependence
\begin{aligned}\boldsymbol{\nabla} \times \mathbf{A}^{(2)'} = 0\end{aligned} \hspace{\stretch{1}}(2.10)
we have also gauge transformed away the scalar potential contribution so have only the time derivative contribution to the electric field
\begin{aligned}\mathbf{E} = -\frac{1}{{c}} \frac{\partial {\mathbf{A}}}{\partial {t}} - {\boldsymbol{\nabla} \phi} = \frac{2}{3 c^2} \dddot{\mathbf{d}}(t).\end{aligned} \hspace{\stretch{1}}(2.11)
To $O((v/c)^3)$ there is a homogeneous electric field felt by all particles, hence every particle feels a “friction” force
\begin{aligned}\mathbf{f}_{\text{rad}} = q \mathbf{E} = \frac{2 q}{3 c^3} \dddot{\mathbf{d}}(t).\end{aligned} \hspace{\stretch{1}}(2.12)
Moral: $\mathbf{f}_{\text{rad}}$ arises in third order term $O((v/c)^3)$ expansion and thus shouldn’t be given a weight as important as the two other terms. i.e. It’s consequences are less.
## Example: our dipole system
\begin{aligned}m \dot{d}{z} &= - m \omega^2 a + \frac{2 e^2}{3 c^3} \dddot{z} \\ &= - m \omega^2 a + \frac{2 m}{3 c} \frac{e^2}{m c^2} \dddot{z} \\ &= - m \omega^2 a + \frac{2 m}{3} \frac{r_e}{c} \dddot{z} \\ \end{aligned}
Here $r_e \sim 10^{-13} \text{cm}$ is the classical radius of the electron. For periodic motion
\begin{aligned}z &\sim e^{i \omega t} z_0 \\ \dot{d}{z} &\sim \omega^2 z_0 \\ \dddot{z} &\sim \omega^3 z_0.\end{aligned}
The ratio of the last term to the inertial term is
\begin{aligned}\sim \frac{ \omega^3 m (r_e/c) z_0 }{ m \omega^2 z_0 } \sim \omega \frac{r_e}{c} \ll 1,\end{aligned} \hspace{\stretch{1}}(2.13)
so
\begin{aligned}\omega &\ll \frac{c}{r_e} \\ &\sim \frac{1}{{\tau_e}} \\ &\sim \frac{ 10^{10} \text{cm}/\text{s}}{10^{-13} \text{cm}} \\ &\sim 10^{23} \text{Hz} \\ \end{aligned}
So long as $\omega \ll 10^{23} \text{Hz}$, this approximation is valid.
# Limits of classical electrodynamics.
What sort of energy is this? At these frequencies QM effects come in
\begin{aligned}\hbar \sim 10^{-33} \text{J} \cdot \text{s} \sim 10^{-15} \text{eV} \cdot \text{s}\end{aligned} \hspace{\stretch{1}}(3.14)
\begin{aligned}\hbar \omega_{max} \sim 10^{-15} \text{eV} \cdot \text{s} \times 10^{23} \frac{1}{{\text{s}}} \sim 10^8 \text{eV} \sim 100 \text{MeV}\end{aligned} \hspace{\stretch{1}}(3.15)
whereas the rest energy of the electron is
\begin{aligned}m_e c^2 \sim \frac{1}{{2}} \text{MeV} \sim \text{MeV}.\end{aligned} \hspace{\stretch{1}}(3.16)
At these frequencies it is possible to create $e^{+}$ and $e^{-}$ pairs. A theory where the number of particles (electrons and positrons) is NOT fixed anymore is required. An estimate of this frequency, where these effects have to be considered is possible.
PICTURE: different length scales with frequency increasing to the left and length scales increasing to the right.
\begin{itemize}
\item $10^{-13} \text{cm}$, $r_e = e^2/m c^2$. LHC exploration.
\item $137 \times 10^{-13} \text{cm}$, $\hbar/m_e c \sim \lambda/2\pi$, the Compton wavelength of the electron. QED and quantum field theory.
\item $(137)^2 \times 10^{-13} \text{cm} \sim 10^{-10} \text{cm}$, Bohr radius. QM, and classical electrodynamics.
\end{itemize}
here
\begin{aligned}\alpha = \frac{e^2}{4 \pi \epsilon_0 \hbar c } = \frac{1}{{137}},\end{aligned} \hspace{\stretch{1}}(3.17)
is the fine structure constant.
Similar to the distance scale restrictions, we have field strength restrictions. A strong enough field (Electric) can start creating electron and positron pairs. This occurs at about
\begin{aligned}e E \lambda/2\pi \sim 2 m_e c^2 \end{aligned} \hspace{\stretch{1}}(3.18)
so the critical field strength is
\begin{aligned}E_{\text{crit}} &\sim \frac{m_e c^2 }{\lambda/2\pi e} \\ &\sim \frac{m_e c^2 }{\hbar e} m_e c \\ &\sim \frac{m_e^2 c^3}{\hbar e}\end{aligned}
Is this real?
Yes, with a very heavy nucleus with some electrons stripped off, the field can be so strong that positron and electron pairs will be created. This can be observed in heavy ion collisions!
# References
[1] L.D. Landau and E.M. Lifshitz. The classical theory of fields. Butterworth-Heinemann, 1980.
## Lienard-Wiechert potentials: Charged particle in a circle one last time (and this time I mean it).
Posted by peeterjoot on May 3, 2011
[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Charged particle in a circle without Geometric Algebra.
I tried the problem of calculating the Lienard-Wiechert potentials for circular motion once again in [1] but with the added generalization that allowed the particle to have radial or z-axis motion. Really that was no longer a circular motion problem, but really just a calculation where I was playing with the use of cylindrical coordinates to describe the motion.
It occurred to me that this can be done without any use of Geometric Algebra (or Pauli matrices), which is probably how I should have attempted it on the exam. Let’s use a hybrid coordinate vector and complex number representation to describe the particle position
\begin{aligned}\mathbf{x}_c = \begin{bmatrix}a e^{i\theta} \\ h\end{bmatrix},\end{aligned} \hspace{\stretch{1}}(1.1)
with the field measurement position of
\begin{aligned}\mathbf{r} = \begin{bmatrix}\rho e^{i\phi} \\ z\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(1.2)
The particle velocity is
\begin{aligned}\mathbf{v}_c = \begin{bmatrix}(\dot{a} + i a \dot{\theta}) e^{i\theta} \\ \dot{h}\end{bmatrix} \\ =\begin{bmatrix}e^{i\theta} & i e^{i\theta} & 0 \\ 0 & 0 & 1\end{bmatrix} \begin{bmatrix}\dot{a} \\ a \dot{\theta} \\ \dot{h}\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(1.3)
We also want the vectorial difference between the field measurement position and the particle position
\begin{aligned}\mathbf{R} = \mathbf{r} - \mathbf{x}_c = \begin{bmatrix}e^{i\phi} & -e^{i\theta} & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}\rho \\ a \\ z - h\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(1.4)
The dot product between $\mathbf{R}$ and $\mathbf{v}_c$ is then
\begin{aligned}\mathbf{v}_c \cdot \mathbf{R} &=\begin{bmatrix}\dot{a} &a \dot{\theta} &\dot{h}\end{bmatrix}\text{Real} \left( \begin{bmatrix}e^{-i\theta} & 0 \\ -i e^{-i\theta} & 0 \\ 0 & 1\end{bmatrix} \begin{bmatrix}e^{i\phi} & -e^{i\theta} & 0 \\ 0 & 0 & 1\end{bmatrix}\right)\begin{bmatrix}\rho \\ a \\ z - h\end{bmatrix} \\ &=\begin{bmatrix}\dot{a} &a \dot{\theta} &\dot{h}\end{bmatrix}\text{Real} \left( \begin{bmatrix}e^{i(\phi - \theta)} & -1 & 0 \\ -i e^{i(\phi - \theta)} & i & 0 \\ 0 & 0 & 1\end{bmatrix} \right)\begin{bmatrix}\rho \\ a \\ z - h\end{bmatrix} \\ &=\begin{bmatrix}\dot{a} &a \dot{\theta} &\dot{h}\end{bmatrix}\begin{bmatrix}\cos(\phi - \theta) & -1 & 0 \\ \sin(\phi - \theta) & 0 & 0 \\ 0 & 0 & 1\end{bmatrix} \begin{bmatrix}\rho \\ a \\ z - h\end{bmatrix}.\end{aligned}
Expansion of the final matrix products is then
\begin{aligned}\mathbf{v}_c \cdot \mathbf{R} = \dot{h} (z - h) -a \dot{a} + \rho \dot{a} \cos(\phi- \theta) + \rho a^2 \dot{\theta} \sin(\phi - \theta)\end{aligned} \hspace{\stretch{1}}(1.5)
The other quantity that we want is $\mathbf{R}^2$, which is
\begin{aligned}\mathbf{R}^2 &= \begin{bmatrix}\rho &a &(z - h)\end{bmatrix}\text{Real} \left(\begin{bmatrix}e^{-i\phi} & 0 \\ -e^{-i\theta} & 0 \\ 0 & 1\end{bmatrix}\begin{bmatrix}e^{i\phi} & -e^{i\theta} & 0 \\ 0 & 0 & 1\end{bmatrix}\right)\begin{bmatrix}\rho \\ a \\ z - h\end{bmatrix} \\ &=\begin{bmatrix}\rho &a &(z - h)\end{bmatrix}\begin{bmatrix}1 & -\cos(\phi-\theta) & 0 \\ -\cos(\phi-\theta) & 1 & 0 \\ 0 & 0 & 1\end{bmatrix} \begin{bmatrix}\rho \\ a \\ z - h\end{bmatrix} \\ \end{aligned}
The retarded time at which the field is measured is therefore defined implicitly by
\begin{aligned}R = \sqrt{(\rho^2 + (a(t_r))^2 + (z-h(t_r))^2 - 2 a(t_r) \rho \cos(\phi - \theta(t_r))} = c( t - t_r).\end{aligned} \hspace{\stretch{1}}(1.6)
Together 1.3, 1.5, and 1.6 define the four potentials
\begin{aligned}A^0 &= \frac{q}{R - \mathbf{R} \cdot \mathbf{v}_c/c} \\ \mathbf{A} &= \frac{\mathbf{v}_c}{c} A^0,\end{aligned} \hspace{\stretch{1}}(1.7)
where all quantities are evaluated at the retarded time $t_r$ given by 1.6.
In the homework (and in the text [2] section 63) we found for $\mathbf{E}$ and $\mathbf{B}$
\begin{aligned}\mathbf{E} &= e (1 - \boldsymbol{\beta}_c^2) \frac{\hat{\mathbf{R}} - \boldsymbol{\beta}_c}{R^2 (1 - \hat{\mathbf{R}} \cdot \boldsymbol{\beta}_c)^3}+ e \frac{1}{R (1 - \hat{\mathbf{R}} \cdot \boldsymbol{\beta}_c)^3} \hat{\mathbf{R}} \times ((\hat{\mathbf{R}} - \boldsymbol{\beta}_c) \times \mathbf{a}_c/c^2) \\ \mathbf{B} &= \hat{\mathbf{R}} \times \mathbf{E}.\end{aligned} \hspace{\stretch{1}}(1.9)
Expanding out the cross products this yields
\begin{aligned}\mathbf{E} &= e (1 - \boldsymbol{\beta}_c^2) \frac{\hat{\mathbf{R}} - \boldsymbol{\beta}_c}{R^2 (1 - \hat{\mathbf{R}} \cdot \boldsymbol{\beta}_c)^3}+ e \frac{1}{R (1 - \hat{\mathbf{R}} \cdot \boldsymbol{\beta}_c)^3} (\hat{\mathbf{R}} - \boldsymbol{\beta}_c) \left(\hat{\mathbf{R}} \cdot \frac{\mathbf{a}_c}{c^2}\right)- e \frac{1}{R (1 - \hat{\mathbf{R}} \cdot \boldsymbol{\beta}_c)^2} \frac{\mathbf{a}_c}{c^2} \\ \mathbf{B} &= e (1 - \boldsymbol{\beta}_c^2) \frac{\boldsymbol{\beta}_c \times \hat{\mathbf{R}}}{R^2 (1 - \hat{\mathbf{R}} \cdot \boldsymbol{\beta}_c)^3}+ e \frac{1}{R (1 - \hat{\mathbf{R}} \cdot \boldsymbol{\beta}_c)^3} (\boldsymbol{\beta}_c \times \hat{\mathbf{R}}) \left(\hat{\mathbf{R}} \cdot \frac{\mathbf{a}_c}{c^2} \right)+ e \frac{1}{R (1 - \hat{\mathbf{R}} \cdot \boldsymbol{\beta}_c)^2} \frac{\mathbf{a}_c}{c^2} \times \hat{\mathbf{R}}\end{aligned} \hspace{\stretch{1}}(1.11)
While longer, it is nice to call out the symmetry between $\mathbf{E}$ and $\mathbf{B}$ explicitly. As a side note, how do these combine in the Geometric Algebra formalism where we have $F = \mathbf{E} + I\mathbf{B}$? That gives us
\begin{aligned}F = e \frac{1}{(1 - \hat{\mathbf{R}} \cdot \boldsymbol{\beta}_c)^3}\left(\left(\frac{1 - \boldsymbol{\beta}_c^2}{R^2} + \frac{\hat{\mathbf{R}} \cdot \mathbf{a}_c}{c R}\right)\left(\hat{\mathbf{R}} - \boldsymbol{\beta}_c + \hat{\mathbf{R}} \wedge (\hat{\mathbf{R}} - \boldsymbol{\beta}_c)\right)+ \frac{1}{{R}} \left( \frac{\mathbf{a}_c}{c^2}+ \frac{\mathbf{a}_c}{c^2} \wedge \hat{\mathbf{R}}\right)\right)\end{aligned} \hspace{\stretch{1}}(1.13)
I’d guess a multivector of the form $\mathbf{a} + \mathbf{a} \wedge \hat{\mathbf{b}}$, can be tidied up a bit more, but this won’t be persued here. Instead let’s write out the fields corresponding to the potentials of 1.7 explicitly. We need to calculate $\mathbf{a}_c$, $\mathbf{v}_c \times \mathbf{R}$, $\mathbf{a}_c \times \mathbf{R}$, and $\mathbf{a}_c \cdot \mathbf{R}$. For the acceleration we get
\begin{aligned}\mathbf{a}_c =\begin{bmatrix}\left( \dot{d}{a} - a {\dot{\theta}}^2 + i ( a\dot{d}{\theta} + 2 \dot{a} \dot{\theta} ) \right) e^{i\theta} \\ \dot{d}{h}\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(1.14)
Dotted with $\mathbf{R}$ we have
\begin{aligned}\mathbf{a}_c \cdot \mathbf{R}&=\begin{bmatrix}\left( \dot{d}{a} - a {\dot{\theta}}^2 + i ( a\dot{d}{\theta} + 2 \dot{a} \dot{\theta} ) \right) e^{i\theta} \\ \dot{d}{h}\end{bmatrix}\cdot \begin{bmatrix}\rho e^{i\phi} - a e^{i\theta} \\ h\end{bmatrix} \\ &=h \dot{d}{h} + \text{Real}\left( \left( \dot{d}{a} - a {\dot{\theta}}^2 + i ( a\dot{d}{\theta} + 2 \dot{a} \dot{\theta} ) \right) \left(\rho e^{i(\theta- \phi)} - a\right)\right) ,\end{aligned}
which gives us
\begin{aligned}\mathbf{a}_c \cdot \mathbf{R} =h \dot{d}{h} + ( \dot{d}{a} - a {\dot{\theta}}^2 ) (\rho \cos(\phi - \theta) - a)+ (a \dot{d}{\theta} + 2 \dot{a} \dot{\theta}) \rho \sin(\phi - \theta).\end{aligned} \hspace{\stretch{1}}(1.15)
Now, how do we handle the cross products in this complex number, scalar hybrid format? With some playing around such a cross product can be put into the following tidy form
\begin{aligned}\begin{bmatrix}z_1 \\ h_1\end{bmatrix}\times\begin{bmatrix}z_2 \\ h_2\end{bmatrix}= \begin{bmatrix}i (h_1 z_2 - h_2 z_1) \\ \text{Imag}(z_1^{*} z_2)\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(1.16)
This is a sensible result. Crossing with $\mathbf{e}_3$ will rotate in the $x-y$ plane, which accounts for the factors of $i$ in the complex portion of the cross product. The imaginary part has only contributions from the portions of the vectors $z_1$ and $z_2$ that are perpendicular to each other, so while the real part of $z_1^{*} z_2$ measures the colinearity, the imaginary part is a measure of the amount perpendicular.
Using this for our velocity cross product we have
\begin{aligned}\mathbf{v}_c \times \mathbf{R} &=\begin{bmatrix}(\dot{a} + i a \dot{\theta}) e^{i\theta} \\ \dot{h}\end{bmatrix}\times\begin{bmatrix}\rho e^{i\phi} - a e^{i\theta} \\ h\end{bmatrix} \\ &=\begin{bmatrix}i\left(\dot{h} ( \rho e^{i\phi} - a e^{i\theta} ) - h (\dot{a} + i a \dot{\theta}) e^{i\theta} \right) \\ \text{Imag} \left( ( \dot{a} - i a \dot{\theta}) (\rho e^{i(\phi - \theta)} - a) \right)\end{bmatrix} \end{aligned}
which is
\begin{aligned}\mathbf{v}_c \times \mathbf{R} =\begin{bmatrix}i( \dot{h} \rho e^{i\phi} - (h \dot{a} + i h a \dot{\theta} + a \dot{h}) e^{i\theta} ) \\ \dot{a} \rho \sin(\phi - \theta) - a \dot{\theta} \rho \cos(\phi - \theta) + a^2 \dot{\theta}\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(1.17)
The last thing required to write out the fields is
\begin{aligned}\mathbf{a}_c \times \mathbf{R} &=\begin{bmatrix}\left( \dot{d}{a} - a {\dot{\theta}}^2 + i ( a\dot{d}{\theta} + 2 \dot{a} \dot{\theta} ) \right) e^{i\theta} \\ \dot{d}{h}\end{bmatrix}\times\begin{bmatrix}\rho e^{i\phi} - a e^{i\theta} \\ z - h\end{bmatrix} \\ &=\begin{bmatrix}i \dot{d}{h} (\rho e^{i\phi} - a e^{i\theta} ) - i (z - h) \left( \dot{d}{a} - a {\dot{\theta}}^2 + i ( a\dot{d}{\theta} + 2 \dot{a} \dot{\theta} ) \right) e^{i\theta} \\ \text{Imag} \left( \left( \dot{d}{a} - a {\dot{\theta}}^2 - i ( a\dot{d}{\theta} + 2 \dot{a} \dot{\theta} ) \right) ( \rho e^{i(\phi -\theta)} - a )\right)\end{bmatrix} \\ \end{aligned}
So the acceleration cross product is
\begin{aligned}\mathbf{a}_c \times \mathbf{R} =\begin{bmatrix}i \dot{d}{h} \rho e^{i\phi} - i \left( \dot{d}{h} a + (z - h) \left( \dot{d}{a} - a {\dot{\theta}}^2 + i ( a\dot{d}{\theta} + 2 \dot{a} \dot{\theta} ) \right) \right) e^{i\theta} \\ \left( \dot{d}{a} - a {\dot{\theta}}^2 \right) \rho \sin(\phi - \theta)-( a\dot{d}{\theta} + 2 \dot{a} \dot{\theta} ) (\rho \cos(\phi -\theta) - a)\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(1.18)
Putting all the results together creates something that is too long to easily write, but can at least be summarized
\begin{aligned}\mathbf{E} &= \frac{e}{(R - \mathbf{R} \cdot \boldsymbol{\beta}_c)^3}\left(\left(1 - \boldsymbol{\beta}_c^2 + \mathbf{R} \cdot \frac{\mathbf{a}_c}{c^2}\right) (\mathbf{R} - \boldsymbol{\beta}_c R)- R(R - \mathbf{R} \cdot \boldsymbol{\beta}_c) \frac{\mathbf{a}_c}{c^2} \right) \\ \mathbf{B} &= \frac{e}{(R - \mathbf{R} \cdot \boldsymbol{\beta}_c)^3}\left(\left(1 - \boldsymbol{\beta}_c^2 + \mathbf{R} \cdot \frac{\mathbf{a}_c}{c^2}\right) (\boldsymbol{\beta}_c \times \mathbf{R})- (R - \mathbf{R} \cdot \boldsymbol{\beta}_c) \frac{\mathbf{a}_c}{c^2} \times \mathbf{R}\right) \\ 1 - \boldsymbol{\beta}_c^2 &= 1 - (\dot{a}^2 + a^2 \dot{\theta}^2 + \dot{h}^2)/c^2 \\ R &= \sqrt{(\rho^2 + (a(t_r))^2 + (z-h(t_r))^2 - 2 a(t_r) \rho \cos(\phi - \theta(t_r))} = c( t - t_r) \\ \mathbf{R} - \boldsymbol{\beta}_c R &= \begin{bmatrix}\rho e^{i\phi} - (a + (\dot{a} + i a\dot{\theta}) R/c) e^{i\theta} \\ z - h - \dot{h} R/c\end{bmatrix} \\ \boldsymbol{\beta}_c \cdot \mathbf{R} &= \frac{1}{{c}}\left( \dot{h} (z - h) -a \dot{a} + \rho \dot{a} \cos(\phi- \theta) + \rho a^2 \dot{\theta} \sin(\phi - \theta) \right) \\ \boldsymbol{\beta}_c \times \mathbf{R} &=\frac{1}{{c}}\begin{bmatrix}i( \dot{h} \rho e^{i\phi} - (h \dot{a} + i h a \dot{\theta} + a \dot{h}) e^{i\theta} ) \\ \dot{a} \rho \sin(\phi - \theta) - a \dot{\theta} \rho \cos(\phi - \theta) + a^2 \dot{\theta}\end{bmatrix} \\ \frac{\mathbf{a}_c}{c^2} &=\frac{1}{{c^2}}\begin{bmatrix}\left( \dot{d}{a} - a {\dot{\theta}}^2 + i ( a\dot{d}{\theta} + 2 \dot{a} \dot{\theta} ) \right) e^{i\theta} \\ \dot{d}{h}\end{bmatrix} \\ \frac{\mathbf{a}_c}{c^2} \cdot \mathbf{R} &=\frac{1}{{c^2}} \left(h \dot{d}{h} + ( \dot{d}{a} - a {\dot{\theta}}^2 ) (\rho \cos(\phi - \theta) - a)+ (a \dot{d}{\theta} + 2 \dot{a} \dot{\theta}) \rho \sin(\phi - \theta) \right) \\ \frac{\mathbf{a}_c}{c^2} \times \mathbf{R} &=\frac{1}{{c^2}}\begin{bmatrix}i \dot{d}{h} \rho e^{i\phi} - i \left( \dot{d}{h} a + (z - h) \left( \dot{d}{a} - a {\dot{\theta}}^2 + i ( a\dot{d}{\theta} + 2 \dot{a} \dot{\theta} ) \right) \right) e^{i\theta} \\ \left( \dot{d}{a} - a {\dot{\theta}}^2 \right) \rho \sin(\phi - \theta)-( a\dot{d}{\theta} + 2 \dot{a} \dot{\theta} ) (\rho \cos(\phi -\theta) - a)\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(1.19)
This is a whole lot more than the exam question asked for, since it is actually the most general solution to the electric and magnetic fields associated with an arbitrary charged particle (when that motion is described in cylindrical coordinates). The exam question had $\theta = k c t$ and $\dot{a} = 0, h = 0$, which kills a number of the terms
\begin{aligned}1 - \boldsymbol{\beta}_c^2 + \frac{\mathbf{a}_c}{c^2} \cdot \mathbf{R} &= 1 - a k^2 \rho \cos(\phi - k c t_r) \\ R &= \sqrt{(\rho^2 + a^2 + z^2 - 2 a \rho \cos(\phi - k c t_r)} = c( t - t_r) \\ \mathbf{R} - \boldsymbol{\beta}_c R &= \begin{bmatrix}\rho e^{i\phi} - a (1 + i k R) e^{i k c t_r} \\ z \end{bmatrix} \\ \boldsymbol{\beta}_c \cdot \mathbf{R} &= \rho a^2 k \sin(\phi - k c t_r) \\ \boldsymbol{\beta}_c \times \mathbf{R} &=\begin{bmatrix}0 \\ a k ( a - \rho \cos(\phi - k c t_r) )\end{bmatrix} \\ \frac{\mathbf{a}_c}{c^2} &=\begin{bmatrix}- a k^2 e^{i k c t_r} \\ 0\end{bmatrix} \\ \frac{\mathbf{a}_c}{c^2} \times \mathbf{R} &=\begin{bmatrix}i z a k^2 e^{i k c t_r} \\ - a k^2 \rho \sin(\phi - k c t_r)\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(1.29)
This is still messy, but is a satisfactory solution to the problem.
The exam question also asked only about the $\rho = 0$, so $\phi$ also becomes irrelevant. In that case we have along the z-axis the fields are given by
\begin{aligned}\mathbf{E}(z)&= \frac{e}{R^3}\begin{bmatrix}- a (1 + i k R - k^2 R^2 ) e^{i k (c t - R)} \\ z \end{bmatrix} \\ \mathbf{B}(z)&= \frac{e}{R^3}\begin{bmatrix}-R i z a k^2 e^{i k (c t - R)} \\ a^2 k \end{bmatrix} \\ R &= \sqrt{a^2 + z^2} \end{aligned} \hspace{\stretch{1}}(1.36)
Similar to when things were calculated from the potentials directly, I get a different result from $\hat{\mathbf{R}} \times \mathbf{E}$
\begin{aligned}\hat{\mathbf{R}} \times \mathbf{E}(z) = \frac{e}{R^3}\begin{bmatrix}a k z (1 + i k R) e^{i k (c t - R)} \\ -a^2 k \end{bmatrix}\end{aligned} \hspace{\stretch{1}}(1.39)
compared to the value of $\mathbf{B}$ that was directly calculated above. With the sign swapped in the z-axis term of $\mathbf{B}(z)$ here I’d guess I’ve got an algebraic error hiding somewhere?
# References
[1] Peeter Joot. {A cylindrical Lienard-Wiechert potential calculation using multivector matrix products.} [online]. http://sites.google.com/site/peeterjoot/math2011/matrixVectorPotentials.pdf.
[2] L.D. Landau and E.M. Lifshitz. The classical theory of fields. Butterworth-Heinemann, 1980.
## My first arxiv submission. Change of basis and Gram-Schmidt orthonormalization in special relativity
Posted by peeterjoot on April 29, 2011
Now that I have an academic email address I was able to make an arxiv submission (I’d tried previously and been auto-rejected) :
Change of basis and Gram-Schmidt orthonormalization in special relativity
This is based on a tutorial from our relativistic electrodynamics class, which covered non-internal relativistic systems. Combining what I learned from that with some concepts I learned from ‘Geometric Algebra for Physicists’ (particularly reciprocal frames) I was able to write up some notes that took those ideas plus basic linear algebra (the Graham-Schmidt procedure) and apply them to relativity and/or non-orthonormal Euclidean bases. How to do projections onto non-orthonormal Euclidean bases isn’t taught in Algebra I, but once you figure out that the same thing works for SR.
Will anybody read it? I don’t know … but I had fun writing it.
## PHY450HS1: Relativistic electrodynamics: some exam reflection.
Posted by peeterjoot on April 28, 2011
[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# Charged particle in a circle.
From the 2008 PHY353 exam, given a particle of charge $q$ moving in a circle of radius $a$ at constant angular frequency $\omega$.
\begin{itemize}
\item Find the Lienard-Wiechert potentials for points on the z-axis.
\item Find the electric and magnetic fields at the center.
\end{itemize}
When I tried this I did it for points not just on the z-axis. It turns out that we also got this question on the exam (but stated slightly differently). Since I’ll not get to see my exam solution again, let’s work through this at a leisurely rate, and see if things look right. The problem as stated in this old practice exam is easier since it doesn’t say to calculate the fields from the four potentials, so there was nothing preventing one from just grinding away and plugging stuff into the Lienard-Wiechert equations for the fields (as I did when I tried it for practice).
## The potentials.
Let’s set up our coordinate system in cylindrical coordinates. For the charged particle and the point that we measure the field, with $i = \mathbf{e}_1 \mathbf{e}_2$
\begin{aligned}\mathbf{x}(t) &= a \mathbf{e}_1 e^{i \omega t} \\ \mathbf{r} &= z \mathbf{e}_3 + \rho \mathbf{e}_1 e^{i \phi}\end{aligned} \hspace{\stretch{1}}(1.1)
Here I’m using the geometric product of vectors (if that’s unfamiliar then just substitute
\begin{aligned}\{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\} \rightarrow \{\sigma_1, \sigma_2, \sigma_3\}\end{aligned} \hspace{\stretch{1}}(1.3)
We can do that since the Pauli matrices also have the same semantics (with a small difference since the geometric square of a unit vector is defined as the unit scalar, whereas the Pauli matrix square is the identity matrix). The semantics we require of this vector product are just $\mathbf{e}_\alpha^2 = 1$ and $\mathbf{e}_\alpha \mathbf{e}_\beta = - \mathbf{e}_\beta \mathbf{e}_\alpha$ for any $\alpha \ne \beta$.
I’ll also be loose with notation and use $\text{Real}(X) = \left\langle{{X}}\right\rangle$ to select the scalar part of a multivector (or with the Pauli matrices, the portion proportional to the identity matrix).
Our task is to compute the Lienard-Wiechert potentials. Those are
\begin{aligned}A^0 &= \frac{q}{R^{*}} \\ \mathbf{A} &= A^0 \frac{\mathbf{v}}{c},\end{aligned} \hspace{\stretch{1}}(1.4)
where
\begin{aligned}\mathbf{R} &= \mathbf{r} - \mathbf{x}(t_r) \\ R = {\left\lvert{\mathbf{R}}\right\rvert} &= c (t - t_r) \\ R^{*} &= R - \frac{\mathbf{v}}{c} \cdot \mathbf{R} \\ \mathbf{v} &= \frac{d\mathbf{x}}{dt_r}.\end{aligned} \hspace{\stretch{1}}(1.6)
We’ll need (eventually)
\begin{aligned}\mathbf{v} &= a \omega \mathbf{e}_2 e^{i \omega t_r} = a \omega ( -\sin \omega t_r, \cos\omega t_r, 0) \\ \dot{\mathbf{v}} &= -a \omega^2 \mathbf{e}_1 e^{i \omega t_r} = -a \omega^2 (\cos\omega t_r, \sin\omega t_r, 0)\end{aligned} \hspace{\stretch{1}}(1.10)
and also need our retarded distance vector
\begin{aligned}\mathbf{R} = z \mathbf{e}_3 + \mathbf{e}_1 (\rho e^{i \phi} - a e^{i \omega t_r} ),\end{aligned} \hspace{\stretch{1}}(1.12)
From this we have
\begin{aligned}R^2 &= z^2 + {\left\lvert{\mathbf{e}_1 (\rho e^{i \phi} - a e^{i \omega t_r} )}\right\rvert}^2 \\ &= z^2 + \rho^2 + a^2 - 2 \rho a (\mathbf{e}_1 \rho e^{i \phi}) \cdot (\mathbf{e}_1 e^{i \omega t_r}) \\ &= z^2 + \rho^2 + a^2 - 2 \rho a \text{Real}( e^{ i(\phi - \omega t_r) } ) \\ &= z^2 + \rho^2 + a^2 - 2 \rho a \cos(\phi - \omega t_r)\end{aligned}
So
\begin{aligned}R = \sqrt{z^2 + \rho^2 + a^2 - 2 \rho a \cos( \phi - \omega t_r ) }.\end{aligned} \hspace{\stretch{1}}(1.13)
Next we need
\begin{aligned}\mathbf{R} \cdot \mathbf{v}/c&= (z \mathbf{e}_3 + \mathbf{e}_1 (\rho e^{i \phi} - a e^{i \omega t_r} )) \cdot \left(a \frac{\omega}{c} \mathbf{e}_2 e^{i \omega t_r} \right) \\ &=a \frac{\omega }{c}\text{Real}(i (\rho e^{-i \phi} - a e^{-i \omega t_r} ) e^{i \omega t_r} ) \\ &=a \frac{\omega }{c}\rho \text{Real}( i e^{-i \phi + i \omega t_r} ) \\ &=a \frac{\omega }{c}\rho \sin(\phi - \omega t_r)\end{aligned}
So we have
\begin{aligned}R^{*} = \sqrt{z^2 + \rho^2 + a^2 - 2 \rho a \cos( \phi - \omega t_r ) }-a \frac{\omega }{c} \rho \sin(\phi - \omega t_r)\end{aligned} \hspace{\stretch{1}}(1.14)
Writing $k = \omega/c$, and having a peek back at 1.4, our potentials are now solved for
\begin{aligned}\boxed{\begin{aligned}A^0 &= \frac{q}{\sqrt{z^2 + \rho^2 + a^2 - 2 \rho a \cos( \phi - k c t_r ) }} \\ \mathbf{A} &= A^0 a k ( -\sin k c t_r, \cos k c t_r, 0).\end{aligned}}\end{aligned} \hspace{\stretch{1}}(1.24)
The caveat is that $t_r$ is only specified implicitly, according to
\begin{aligned}\boxed{c t_r = c t - \sqrt{z^2 + \rho^2 + a^2 - 2 \rho a \cos( \phi - k c t_r ) }.}\end{aligned} \hspace{\stretch{1}}(1.16)
There doesn’t appear to be much hope of solving for $t_r$ explicitly in closed form.
## General fields for this system.
With
\begin{aligned}\mathbf{R}^{*} = \mathbf{R} - \frac{\mathbf{v}}{c} R,\end{aligned} \hspace{\stretch{1}}(1.17)
the fields are
\begin{aligned}\boxed{\begin{aligned}\mathbf{E} &= q (1 - \mathbf{v}^2/c^2) \frac{\mathbf{R}^{*}}{{R^{*}}^3} + \frac{q}{{R^{*}}^3} \mathbf{R} \times (\mathbf{R}^{*} \times \dot{\mathbf{v}}/c^2) \\ \mathbf{B} &= \frac{\mathbf{R}}{R} \times \mathbf{E}.\end{aligned}}\end{aligned} \hspace{\stretch{1}}(1.18)
In there we have
\begin{aligned}1 - \mathbf{v}^2/c^2 = 1 - a^2 \frac{\omega^2}{c^2} = 1 - a^2 k^2\end{aligned} \hspace{\stretch{1}}(1.19)
and
\begin{aligned}\mathbf{R}^{*} &= z \mathbf{e}_3 + \mathbf{e}_1 (\rho e^{i \phi} - a e^{i k c t_r} )-a k \mathbf{e}_2 e^{i k c t_r} R \\ &= z \mathbf{e}_3 + \mathbf{e}_1 (\rho e^{i \phi} - a (1 - k R i) e^{i k c t_r} )\end{aligned}
Writing this out in coordinates isn’t particularly illuminating, but can be done for completeness without too much trouble
\begin{aligned}\mathbf{R}^{*} = ( \rho \cos\phi - a \cos t_r + a k R \sin t_r, \rho \sin\phi - a \sin t_r - a k R \cos t_r, z )\end{aligned} \hspace{\stretch{1}}(1.20)
In one sense the problem could be considered solved, since we have all the pieces of the puzzle. The outstanding question is whether or not the resulting mess can be simplified at all. Let’s see if the cross product reduces at all. Using
\begin{aligned}\mathbf{R} \times (\mathbf{R}^{*} \times \dot{\mathbf{v}}/c^2) =\mathbf{R}^{*} (\mathbf{R} \cdot \dot{\mathbf{v}}/c^2) - \frac{\dot{\mathbf{v}}}{c^2}(\mathbf{R} \cdot \mathbf{R}^{*})\end{aligned} \hspace{\stretch{1}}(1.21)
Perhaps one or more of these dot products can be simplified? One of them does reduce nicely
\begin{aligned}\mathbf{R}^{*} \cdot \mathbf{R} &= ( \mathbf{R} - R \mathbf{v}/c ) \cdot \mathbf{R} \\ &= R^2 - (\mathbf{R} \cdot \mathbf{v}/c) R \\ &= R^2 - R a k \rho \sin(\phi - k c t_r) \\ &= R(R - a k \rho \sin(\phi - k c t_r))\end{aligned}
\begin{aligned}\mathbf{R} \cdot \dot{\mathbf{v}}/c^2&=\Bigl(z \mathbf{e}_3 + \mathbf{e}_1 (\rho e^{i \phi} - a e^{i \omega t_r} ) \Bigr) \cdot(-a k^2 \mathbf{e}_1 e^{i \omega t_r} ) \\ &=- a k^2 \left\langle{{\mathbf{e}_1 (\rho e^{i \phi} - a e^{i \omega t_r} ) \mathbf{e}_1 e^{i \omega t_r} ) }}\right\rangle \\ &=- a k^2 \left\langle{{(\rho e^{i \phi} - a e^{i \omega t_r} ) e^{-i \omega t_r} ) }}\right\rangle \\ &=- a k^2 \left\langle{{\rho e^{i \phi - i \omega t_r} - a }}\right\rangle \\ &=- a k^2 ( \rho \cos(\phi - k c t_r) - a )\end{aligned}
Putting this cross product back together we have
\begin{aligned}\mathbf{R} \times (\mathbf{R}^{*} \times \dot{\mathbf{v}}/c^2)&=a k^2 ( a -\rho \cos(\phi - k c t_r) ) \mathbf{R}^{*} +a k^2 \mathbf{e}_1 e^{i k c t_r} R(R - a k \rho \sin(\phi - k c t_r)) \\ &=a k^2 ( a -\rho \cos(\phi - k c t_r) ) \Bigl(z \mathbf{e}_3 + \mathbf{e}_1 (\rho e^{i \phi} - a (1 - k R i) e^{i k c t_r} )\Bigr) \\ &\qquad +a k^2 R \mathbf{e}_1 e^{i k c t_r} (R - a k \rho \sin(\phi - k c t_r)) \end{aligned}
Writing
\begin{aligned}\phi_r = \phi - k c t_r,\end{aligned} \hspace{\stretch{1}}(1.22)
this can be grouped into similar terms
\begin{aligned}\begin{aligned}\mathbf{R} \times (\mathbf{R}^{*} \times \dot{\mathbf{v}}/c^2)&=a k^2 (a - \rho \cos\phi_r) z \mathbf{e}_3 \\ &+ a k^2 \mathbf{e}_1(a - \rho \cos\phi_r) \rho e^{i\phi} \\ &+ a k^2 \mathbf{e}_1\left(-a (a - \rho \cos\phi_r) (1 - k R i)+ R(R - a k \rho \sin \phi_r)\right) e^{i k c t_r}\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.23)
The electric field pieces can now be collected. Not expanding out the $R^{*}$ from 1.14, this is
\begin{aligned}\begin{aligned}\mathbf{E} &= \frac{q}{(R^{*})^3} z \mathbf{e}_3\Bigl( 1 - a \rho k^2 \cos\phi_r \Bigr) \\ &+\frac{q}{(R^{*})^3} \rho\mathbf{e}_1 \Bigl(1 - a \rho k^2 \cos\phi_r \Bigr) e^{i\phi} \\ &+\frac{q}{(R^{*})^3} a \mathbf{e}_1\left(-\Bigl( 1 + a k^2 (a - \rho \cos\phi_r) \Bigr) (1 - k R i)(1 - a^2 k^2)+ k^2 R(R - a k \rho \sin \phi_r)\right) e^{i k c t_r}\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.24)
Along the z-axis where $\rho = 0$ what do we have?
\begin{aligned}R = \sqrt{z^2 + a^2 } \end{aligned} \hspace{\stretch{1}}(1.25)
\begin{aligned}A^0 = \frac{q}{R} \end{aligned} \hspace{\stretch{1}}(1.26)
\begin{aligned}\mathbf{A} = A^0 a k \mathbf{e}_2 e^{i k c t_r } \end{aligned} \hspace{\stretch{1}}(1.27)
\begin{aligned}c t_r = c t - \sqrt{z^2 + a^2 } \end{aligned} \hspace{\stretch{1}}(1.28)
\begin{aligned}\begin{aligned}\mathbf{E} &= \frac{q}{R^3} z \mathbf{e}_3 \\ &+\frac{q}{R^3} a \mathbf{e}_1\left(-( 1 - a^4 k^4 ) (1 - k R i)+ k^2 R^2 \right) e^{i k c t_r} \end{aligned}\end{aligned} \hspace{\stretch{1}}(1.29)
\begin{aligned}\mathbf{B} = \frac{ z \mathbf{e}_3 - a \mathbf{e}_1 e^{i k c t_r}}{R} \times \mathbf{E}\end{aligned} \hspace{\stretch{1}}(1.30)
The magnetic term here looks like it can be reduced a bit.
## An approximation near the center.
Unlike the old exam I did, where it didn’t specify that the potentials had to be used to calculate the fields, and the problem was reduced to one of algebraic manipulation, our exam explicitly asked for the potentials to be used to calculate the fields.
There was also the restriction to compute them near the center. Setting $\rho = 0$ so that we are looking only near the z-axis, we have
\begin{aligned}A^0 &= \frac{q}{\sqrt{z^2 + a^2}} \\ \mathbf{A} &= \frac{q a k \mathbf{e}_2 e^{i k c t_r} }{\sqrt{z^2 + a^2}} = \frac{q a k (-\sin k c t_r, \cos k c t_r, 0)}{\sqrt{z^2 + a^2}} \\ t_r &= t - R/c = t - \sqrt{z^2 + a^2}/c\end{aligned} \hspace{\stretch{1}}(1.31)
Now we are set to calculate the electric and magnetic fields directly from these. Observe that we have a spatial dependence in due to the $t_r$ quantities and that will have an effect when we operate with the gradient.
In the exam I’d asked Simon (our TA) if this question was asking for the fields at the origin (ie: in the plane of the charge’s motion in the center) or along the z-axis. He said in the plane. That would simplify things, but perhaps too much since $A^0$ becomes constant (in my exam attempt I somehow fudged this to get what I wanted for the $v = 0$ case, but that must have been wrong, and was the result of rushed work).
Let’s now proceed with the field calculation from these potentials
\begin{aligned}\mathbf{E} &= - \boldsymbol{\nabla} A^0 - \frac{1}{{c}} \frac{\partial {\mathbf{A}}}{\partial {t}} \\ \mathbf{B} &= \boldsymbol{\nabla} \times \mathbf{A}.\end{aligned} \hspace{\stretch{1}}(1.34)
For the electric field we need
\begin{aligned}\boldsymbol{\nabla} A^0 &= q \mathbf{e}_3 \partial_z (z^2 + a^2)^{-1/2} \\ &= -q \mathbf{e}_3 \frac{z}{(\sqrt{z^2 + a^2})^3},\end{aligned}
and
\begin{aligned}\frac{1}{{c}} \frac{\partial {\mathbf{A}}}{\partial {t}} =\frac{q a k^2 \mathbf{e}_2 \mathbf{e}_1 \mathbf{e}_2 e^{i k c t_r} }{\sqrt{z^2 + a^2}}.\end{aligned} \hspace{\stretch{1}}(1.36)
Putting these together, our electric field near the z-axis is
\begin{aligned}\mathbf{E} = q \mathbf{e}_3 \frac{z}{(\sqrt{z^2 + a^2})^3}+\frac{q a k^2 \mathbf{e}_1 e^{i k c t_r} }{\sqrt{z^2 + a^2}}.\end{aligned} \hspace{\stretch{1}}(1.37)
(another mistake I made on the exam, since I somehow fooled myself into forcing what I knew had to be in the gradient term, despite having essentially a constant scalar potential (having taken $z = 0$)).
What do we get for the magnetic field. In that case we have
\begin{aligned}\boldsymbol{\nabla} \times \mathbf{A}(z)&=\mathbf{e}_\alpha \times \partial_\alpha \mathbf{A} \\ &=\mathbf{e}_3 \times \partial_z \frac{q a k \mathbf{e}_2 e^{i k c t_r} }{\sqrt{z^2 + a^2}} \\ &=\mathbf{e}_3 \times (\mathbf{e}_2 e^{i k c t_r} ) q a k \frac{\partial {}}{\partial {z}} \frac{1}{{\sqrt{z^2 + a^2}}} +q a k \frac{1}{{\sqrt{z^2 + a^2}}} \mathbf{e}_3 \times (\mathbf{e}_2 \partial_z e^{i k c t_r} ) \\ &=-\mathbf{e}_3 \times (\mathbf{e}_2 e^{i k c t_r} ) q a k \frac{z}{(\sqrt{z^2 + a^2})^3} +q a k \frac{1}{{\sqrt{z^2 + a^2}}} \mathbf{e}_3 \times \left( \mathbf{e}_2 \mathbf{e}_1 \mathbf{e}_2 k c e^{i k c t_r} \partial_z ( t - \sqrt{z^a + a^2}/c ) \right) \\ &=-\mathbf{e}_3 \times (\mathbf{e}_2 e^{i k c t_r} ) q a k \frac{z}{(\sqrt{z^2 + a^2})^3} -q a k^2 \frac{z}{z^2 + a^2} \mathbf{e}_3 \times \left( \mathbf{e}_1 k e^{i k c t_r} \right) \\ &=-\frac{q a k z \mathbf{e}_3}{z^2 + a^2} \times \left( \frac{ \mathbf{e}_2 e^{i k c t_r} }{\sqrt{z^2 + a^2}} + k \mathbf{e}_1 e^{i k c t_r} \right)\end{aligned}
For the direction vectors in the cross products above we have
\begin{aligned}\mathbf{e}_3 \times (\mathbf{e}_2 e^{i \mu})&=\mathbf{e}_3 \times (\mathbf{e}_2 \cos\mu - \mathbf{e}_1 \sin\mu) \\ &=-\mathbf{e}_1 \cos\mu - \mathbf{e}_2 \sin\mu \\ &=-\mathbf{e}_1 e^{i \mu}\end{aligned}
and
\begin{aligned}\mathbf{e}_3 \times (\mathbf{e}_1 e^{i \mu})&=\mathbf{e}_3 \times (\mathbf{e}_1 \cos\mu + \mathbf{e}_2 \sin\mu) \\ &=\mathbf{e}_2 \cos\mu - \mathbf{e}_1 \sin\mu \\ &=\mathbf{e}_2 e^{i \mu}\end{aligned}
Putting everything, and summarizing results for the fields, we have
\begin{aligned}\mathbf{E} &= q \mathbf{e}_3 \frac{z}{(\sqrt{z^2 + a^2})^3}+\frac{q a k^2 \mathbf{e}_1 e^{i \omega t_r} }{\sqrt{z^2 + a^2}} \\ \mathbf{B} &= \frac{q a k z}{ z^2 + a^2} \left( \frac{\mathbf{e}_1}{\sqrt{z^2 + a^2}} - k \mathbf{e}_2 \right) e^{i \omega t_r}\end{aligned} \hspace{\stretch{1}}(1.38)
The electric field expression above compares well to 1.29. We have the Coulomb term and the radiation term. It is harder to compare the magnetic field to the exact result 1.30 since I did not expand that out.
FIXME: A question to consider. If all this worked should we not also get
\begin{aligned}\mathbf{B} \stackrel{?}{=}\frac{z \mathbf{e}_3 - \mathbf{e}_1 a e^{i \omega t_r}}{\sqrt{z^2 + a^2}} \times \mathbf{E}.\end{aligned} \hspace{\stretch{1}}(1.40)
However, if I do this check I get
\begin{aligned}\mathbf{B} =\frac{q a z}{z^2 + a^2} \left( \frac{1}{{z^2 + a^2}} + k^2 \right) \mathbf{e}_2 e^{i \omega t_r}.\end{aligned} \hspace{\stretch{1}}(1.41)
# Collision of photon and electron.
I made a dumb error on the exam on this one. I setup the four momentum conservation statement, but then didn’t multiply out the cross terms properly. This led me to incorrectly assume that I had to try doing this the hard way (something akin to what I did on the midterm). Simon later told us in the tutorial the simple way, and that’s all we needed here too. Here’s the setup.
An electron at rest initially has four momentum
\begin{aligned}(m c, 0)\end{aligned} \hspace{\stretch{1}}(2.42)
where the incoming photon has four momentum
\begin{aligned}\left(\hbar \frac{\omega}{c}, \hbar \mathbf{k} \right)\end{aligned} \hspace{\stretch{1}}(2.43)
After the collision our electron has some velocity so its four momentum becomes (say)
\begin{aligned}\gamma (m c, m \mathbf{v}),\end{aligned} \hspace{\stretch{1}}(2.44)
and our new photon, going off on an angle $\theta$ relative to $\mathbf{k}$ has four momentum
\begin{aligned}\left(\hbar \frac{\omega'}{c}, \hbar \mathbf{k}' \right)\end{aligned} \hspace{\stretch{1}}(2.45)
Our conservation relationship is thus
\begin{aligned}(m c, 0) + \left(\hbar \frac{\omega}{c}, \hbar \mathbf{k} \right)=\gamma (m c, m \mathbf{v})+\left(\hbar \frac{\omega'}{c}, \hbar \mathbf{k}' \right)\end{aligned} \hspace{\stretch{1}}(2.46)
I squared both sides, but dropped my cross terms, which was just plain wrong, and costly for both time and effort on the exam. What I should have done was just
\begin{aligned}\gamma (m c, m \mathbf{v}) =(m c, 0) + \left(\hbar \frac{\omega}{c}, \hbar \mathbf{k} \right)-\left(\hbar \frac{\omega'}{c}, \hbar \mathbf{k}' \right),\end{aligned} \hspace{\stretch{1}}(2.47)
and then square this (really making contractions of the form $p_i p^i$). That gives (and this time keeping my cross terms)
\begin{aligned}(\gamma (m c, m \mathbf{v}) )^2 &= \gamma^2 m^2 (c^2 - \mathbf{v}^2) \\ &= m^2 c^2 \\ &=m^2 c^2 + 0 + 0+ 2 (m c, 0) \cdot \left(\hbar \frac{\omega}{c}, \hbar \mathbf{k} \right)- 2 (m c, 0) \cdot \left(\hbar \frac{\omega'}{c}, \hbar \mathbf{k}' \right)- 2 \cdot \left(\hbar \frac{\omega}{c}, \hbar \mathbf{k} \right)\cdot \left(\hbar \frac{\omega'}{c}, \hbar \mathbf{k}' \right) \\ &=m^2 c^2 + 2 m c \hbar \frac{\omega}{c} - 2 m c \hbar \frac{\omega'}{c}- 2\hbar^2 \left(\frac{\omega}{c} \frac{\omega'}{c}- \mathbf{k} \cdot \mathbf{k}'\right) \\ &=m^2 c^2 + 2 m c \hbar \frac{\omega}{c} - 2 m c \hbar \frac{\omega'}{c}- 2\hbar^2 \frac{\omega}{c} \frac{\omega'}{c} (1 - \cos\theta)\end{aligned}
Rearranging a bit we have
\begin{aligned}\omega' \left( m + \frac{\hbar \omega}{c^2} ( 1 - \cos\theta ) \right) = m \omega,\end{aligned} \hspace{\stretch{1}}(2.48)
or
\begin{aligned}\omega' = \frac{\omega}{1 + \frac{\hbar \omega}{m c^2} ( 1 - \cos\theta ) }\end{aligned} \hspace{\stretch{1}}(2.49)
# Pion decay.
The problem above is very much like a midterm problem we had, so there was no justifiable excuse for messing up on it. That midterm problem was to consider the split of a pion at rest into a neutrino (massless) and a muon, and to calculate the energy of the muon. That one also follows the same pattern, a calculation of four momentum conservation, say
\begin{aligned}(m_\pi c, 0) = \hbar \frac{\omega}{c}(1, \hat{\mathbf{k}}) + ( \mathcal{E}_\mu/c, \mathbf{p}_\mu ).\end{aligned} \hspace{\stretch{1}}(3.50)
Here $\omega$ is the frequency of the massless neutrino. The massless nature is encoded by a four momentum that squares to zero, which follows from $(1, \hat{\mathbf{k}}) \cdot (1, \hat{\mathbf{k}}) = 1^2 - \hat{\mathbf{k}} \cdot \hat{\mathbf{k}} = 0$.
When I did this problem on the midterm, I perversely put in a scattering angle, instead of recognizing that the particles must scatter at 180 degree directions since spatial momentum components must also be preserved. This and the combination of trying to work in spatial quantities led to a mess and I didn’t get the end result in anything that could be considered tidy.
The simple way to do this is to just rearrange to put the null vector on one side, and then square. This gives us
\begin{aligned}0 &=\left(\hbar \frac{\omega}{c}(1, \hat{\mathbf{k}}) \right) \cdot\left(\hbar \frac{\omega}{c}(1, \hat{\mathbf{k}}) \right) \\ &=\left( (m_\pi c, 0) - ( \mathcal{E}_\mu/c, \mathbf{p}_\mu ) \right) \cdot \left( (m_\pi c, 0) - ( \mathcal{E}_\mu/c, \mathbf{p}_\mu ) \right) \\ &={m_\pi}^2 c^2 + {m_\nu}^2 c^2 - 2 (m_\pi c, 0) \cdot ( \mathcal{E}_\mu/c, \mathbf{p}_\mu ) \\ &={m_\pi}^2 c^2 + {m_\nu}^2 c^2 - 2 m_\pi \mathcal{E}_\mu\end{aligned}
A final re-arrangement gives us the muon energy
\begin{aligned}\mathcal{E}_\mu = \frac{1}{{2}} \frac{ {m_\pi}^2 + {m_\nu}^2 }{m_\pi} c^2\end{aligned} \hspace{\stretch{1}}(3.51)
## PHY450H1S. Relativistic Electrodynamics Lecture 19 (Taught by Prof. Erich Poppitz). Lienard-Wiechert potentials.
Posted by peeterjoot on April 26, 2011
[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
Covering chapter 8 material from the text [1].
Covering lecture notes pp. 136-146: the Lienard-Wiechert potentials (143-146) [Wednesday, Mar. 9…]
# Fields from the Lienard-Wiechert potentials
(We finished off with the scalar and vector potentials in class, but I’ve put those notes with the previous lecture).
To find $\mathbf{E}$ and $\mathbf{B}$ need
$\frac{\partial {t_r}}{\partial {t}}$, and $\boldsymbol{\nabla} t_r(\mathbf{x}, t)$
where
\begin{aligned}t - t_r = {\left\lvert{\mathbf{x} - \mathbf{x}_c(t_r) }\right\rvert}\end{aligned} \hspace{\stretch{1}}(2.1)
implicit definition of $t_r(\mathbf{x}, t)$
In HW5 you’ll show
\begin{aligned}\frac{\partial {t_r}}{\partial {t}} = \frac{{\left\lvert{\mathbf{x} - \mathbf{x}_c(t_r) }\right\rvert}}{{\left\lvert{\mathbf{x} - \mathbf{x}_c(t_r) }\right\rvert} - \frac{\mathbf{v}_c }{c} \cdot (\mathbf{x} - \mathbf{x}_c(t_r))}\end{aligned} \hspace{\stretch{1}}(2.2)
\begin{aligned}\boldsymbol{\nabla} t_r = \frac{1}{{c}} \frac{\mathbf{x} - \mathbf{x}_c(t_r) }{{\left\lvert{\mathbf{x} - \mathbf{x}_c(t_r) }\right\rvert} - \frac{\mathbf{v}_c }{c} \cdot (\mathbf{x} - \mathbf{x}_c(t_r))}\end{aligned} \hspace{\stretch{1}}(2.3)
and then use this to show that the electric and magnetic fields due to a moving charge are
\begin{aligned}\mathbf{E}(\mathbf{x}, t) &= \frac{e R}{ (\mathbf{R} \cdot \mathbf{u})^3 } \left( (c^2 - \mathbf{v}_c^2) \mathbf{u} + \mathbf{R} \times (\mathbf{u} \times \mathbf{a}_c) \right) \\ &= \frac{\mathbf{R}}{R} \times \mathbf{E} \\ \mathbf{u} &= c \frac{\mathbf{R}}{R} - \mathbf{v}_c,\end{aligned} \hspace{\stretch{1}}(2.4)
where everything is evaluated at the retarded time $t_r = t - {\left\lvert{\mathbf{x} - \mathbf{x}_c(t_r)}\right\rvert}/c$.
This looks quite a bit different than what we find in section 63 (63.8) in the text, but a little bit of expansion shows they are the same.
# Check. Particle at rest.
With
\begin{aligned}\mathbf{x}_c &= \mathbf{x}_0 \\ X_c^k &= (ct, \mathbf{x}_0) \\ {\left\lvert{\mathbf{x} - \mathbf{x}_c(t_r)}\right\rvert} &= c(t - t_r)\end{aligned}
\begin{figure}[htp]
\centering
\includegraphics[totalheight=0.4\textheight]{particleAtRestTrCalc}
\caption{Retarded time for particle at rest.}
\end{figure}
As illustrated in figure (\ref{fig:particleAtRestTrCalc}) the retarded position is
\begin{aligned}\mathbf{x}_c(t_r) = \mathbf{x}_0,\end{aligned} \hspace{\stretch{1}}(3.7)
for
\begin{aligned}\mathbf{u} = \frac{\mathbf{x} - \mathbf{x}_0}{{\left\lvert{\mathbf{x} - \mathbf{x}_0}\right\rvert}} c,\end{aligned} \hspace{\stretch{1}}(3.8)
and
\begin{aligned}\mathbf{E} = e \frac{{ {\left\lvert{\mathbf{x} - \mathbf{x}_0}\right\rvert}} }{ (c {\left\lvert{\mathbf{x} - \mathbf{x}_0}\right\rvert})^3 } c^3 \frac{\mathbf{x} - \mathbf{x}_0}{{{\left\lvert{\mathbf{x} - \mathbf{x}_0}\right\rvert}}},\end{aligned} \hspace{\stretch{1}}(3.9)
which is Coulomb’s law
\begin{aligned}\mathbf{E} = e \frac{\mathbf{x} - \mathbf{x}_0}{{\left\lvert{\mathbf{x} - \mathbf{x}_0}\right\rvert}^3}\end{aligned} \hspace{\stretch{1}}(3.10)
# Check. Particle moving with constant velocity.
This was also computed in full in homework 5. The end result was
\begin{aligned}\mathbf{E} =e \frac{\mathbf{x} - \mathbf{v} t}{{\left\lvert{\mathbf{x} - \mathbf{v} t}\right\rvert}^3}\frac{1 -\boldsymbol{\beta}^2}{ \left(1 - \frac{(\mathbf{x} \times \boldsymbol{\beta})^2}{{\left\lvert{\mathbf{x} - \mathbf{v} t}\right\rvert}^2} \right)^{3/2} }\end{aligned} \hspace{\stretch{1}}(4.11)
Writing
\begin{aligned}\frac{\mathbf{x} \times \boldsymbol{\beta}}{{\left\lvert{\mathbf{x} - \mathbf{v} t}\right\rvert}}&=\frac{1}{{c}} \frac{(\mathbf{x} - \mathbf{v} t) \times \mathbf{v}}{{\left\lvert{\mathbf{x} - \mathbf{v} t}\right\rvert}} \\ &=\frac{{\left\lvert{\mathbf{v}}\right\rvert}}{c} \frac{(\mathbf{x} - \mathbf{v} t) \times \mathbf{v}}{{\left\lvert{\mathbf{x} - \mathbf{v} t}\right\rvert} {\left\lvert{\mathbf{v}}\right\rvert}} \end{aligned}
We can introduce an angular dependence between the charge’s translated position and its velocity
\begin{aligned}\sin^2 \theta = {\left\lvert{ \frac{\mathbf{v} \times (\mathbf{x} - \mathbf{v} t)}{\Abs{\mathbf{v}} \Abs{\mathbf{x} - \mathbf{v} t}} }\right\rvert}^2,\end{aligned} \hspace{\stretch{1}}(4.12)
and write the field as
\begin{aligned}\mathbf{E} =\underbrace{e \frac{\mathbf{x} - \mathbf{v} t}{{\left\lvert{\mathbf{x} - \mathbf{v} t}\right\rvert}^3}}_{{*}}\frac{1 -\boldsymbol{\beta}^2}{ \left(1 - \frac{\mathbf{v}^2}{c^2} \sin^2 \theta \right)^{3/2} }\end{aligned} \hspace{\stretch{1}}(4.13)
Observe that ${*} = \text{Coulomb's law measured from the instantaneous position of the charge}$.
The electric field $\mathbf{E}$ has a time dependence, strongest when perpendicular to the instantaneous position when $\theta = \pi/2$, since the denominator is smallest ($\mathbf{E}$ largest) when $\mathbf{v}/c$ is not small. This is strongly $\theta$ dependent.
Compare
\begin{aligned}\frac{{\left\lvert{\mathbf{E}(\theta = \pi/2)}\right\rvert} - {\left\lvert{\mathbf{E}(\theta = \pi/2 + \Delta \theta)}\right\rvert} }{{\left\lvert{\mathbf{E}(\theta = \pi/2)}\right\rvert}}&\approx\frac{\frac{1}{{(1 - \mathbf{v}^2/c^2)^{3/2}}} - \frac{1}{{(1 - \mathbf{v}^2/c^2(1 - (\Delta \theta)^2))^{3/2}}}}{\frac{1}{{(1 - \mathbf{v}^2/c^2)^{3/2}}}} \\ &=1 - \left(\frac{1 - \mathbf{v}^2/c^2}{1 - \mathbf{v}^2/c^2 + \mathbf{v}^2/c^2(\Delta \theta)^2}\right)^{3/2} \\ &=1 - \left(\frac{1}{1 + \mathbf{v}^2/c^2 \frac{(\Delta \theta)^2}{1 - \mathbf{v}^2/c^2}}\right)^{3/2} \\ \end{aligned}
Here we used
\begin{aligned}\sin(\theta + \pi/2) = \frac{e^{i (\theta + \pi/2)} - e^{-i(\theta + \pi/2)}}{2i} = \cos\theta \end{aligned} \hspace{\stretch{1}}(4.14)
and
\begin{aligned}\cos^2 \Delta \theta \approx \left( 1 - \frac{(\Delta \theta)^2}{2} \right)^2 \approx 1 - (\Delta \theta)^2\end{aligned} \hspace{\stretch{1}}(4.15)
FIXME: he writes:
\begin{aligned}\Delta \theta \le \sqrt{1 - \frac{\mathbf{v}^2}{c^2}}\end{aligned} \hspace{\stretch{1}}(4.16)
I don’t see where that comes from.
FIXME: PICTURE: Various $\mathbf{E}$‘s up, and $\mathbf{v}$ perpendicular to that, strongest when charge is moving fast.
# Back to extracting physics from the Lienard-Wiechert field equations
Imagine that we have a localized particle motion with
\begin{aligned}{\left\lvert{\mathbf{x}_c(t_r)}\right\rvert} < l\end{aligned} \hspace{\stretch{1}}(5.17)
The velocity vector
\begin{aligned}\mathbf{u} = c \frac{\mathbf{x} - \mathbf{x}_c(t_r)}{{\left\lvert{\mathbf{x} - \mathbf{x}_c}\right\rvert}}\end{aligned} \hspace{\stretch{1}}(5.18)
doesn’t grow as distance from the source, so from 2.4, we have for ${\left\lvert{\mathbf{x}}\right\rvert} \gg l$
\begin{aligned}\mathbf{B}, \mathbf{E} \sim \frac{1}{{{\left\lvert{\mathbf{x}}\right\rvert}^2}}(\cdots) + \frac{1}{\mathbf{x}}(\text{acceleration term})\end{aligned} \hspace{\stretch{1}}(5.19)
The acceleration term will dominate at large distances from the source. Our Poynting magnitude is
\begin{aligned}{\left\lvert{\mathbf{S}}\right\rvert} \sim {\left\lvert{\mathbf{E} \times \mathbf{B}}\right\rvert} \sim \frac{1}{{\mathbf{x}^2}} (\text{acceleration})^2.\end{aligned} \hspace{\stretch{1}}(5.20)
\begin{aligned}\oint d^2 \boldsymbol{\sigma} \cdot \mathbf{S} \sim R^2 \frac{1}{{R^2}} (\text{acceleration})^2 \sim (\text{acceleration})^2 \end{aligned} \hspace{\stretch{1}}(5.21)
In the limit, for the radiation of EM waves
\begin{aligned}\lim_{R\rightarrow \infty} \oint d^2 \boldsymbol{\sigma} \cdot \mathbf{S} \ne 0\end{aligned} \hspace{\stretch{1}}(5.22)
The energy flux through a sphere of radius $R$ is called the radiated power.
# References
[1] L.D. Landau and E.M. Lifshitz. The classical theory of fields. Butterworth-Heinemann, 1980.
## PHY450H1S. Relativistic Electrodynamics Lecture 26 (Taught by Prof. Erich Poppitz). Radiation reaction force for a dipole system.
Posted by peeterjoot on April 11, 2011
[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
Covering chapter 8 section 65 material from the text [1].
Covering pp. 181-195: (182-189) [Tuesday, Mar. 29]; the EM potentials to order $(v/c)^2$ (190-193); the “Darwin Lagrangian. and Hamiltonian for a system of nonrelativistic charged particles to order $(v/c)^2$ and its many uses in physics (194-195) [Wednesday, Mar. 30]
Next week (last topic): attempt to go to the next order $(v/c)^3$ – radiation damping, the limitations of classical electrodynamics, and the relevant time/length/energy scales.
# Recap.
A system of N charged particles $m_a, q_a ; a \in [1, N]$ closed system and nonrelativistic, $v_a/c \ll 1$. In this case we can incorporate EM effects in a Largrangian ONLY involving particles (EM field not a dynamical DOF). In general case, this works to $O((v/c)^2)$, because at $O((v/c))$ system radiation effects occur.
In a specific case, when
\begin{aligned}\frac{m_1}{q_1} = \frac{m_2}{q_2} = \frac{m_3}{q_3} = \cdots\end{aligned} \hspace{\stretch{1}}(2.1)
we can do that (meaning use a Lagrangian with particles only) to $O((v/c)^4)$ because of specific symmetries in such a system.
The Lagrangian for our particle after the gauge transformation is
\begin{aligned}\mathcal{L}_a = \frac{1}{{2}} m_a \mathbf{v}_a^2 + \frac{m_a}{8} \frac{\mathbf{v}_a^4}{c^2} -\sum_{b \ne a} \frac{q_a q_b}{{\left\lvert{\mathbf{x}_a(t) - \mathbf{x}_b(t)}\right\rvert}}+\sum_b q_a q_b \frac{\mathbf{v}_a \cdot \mathbf{v}_b + (\mathbf{n} \cdot \mathbf{v}_a) (\mathbf{n} \cdot \mathbf{v}_b)}{2 c^2 {\left\lvert{\mathbf{x} - \mathbf{x}_b}\right\rvert} }.\end{aligned} \hspace{\stretch{1}}(2.2)
Next time we’ll probably get to the Lagrangian for the entire system. It was hinted that this is called the Darwin Lagrangian (after Charles Darwin’s grandson).
We find for whole system
\begin{aligned}\mathcal{L} = \sum_a \mathcal{L}_a + \frac{1}{{2}} \sum_a \mathcal{L}_a (interaction)\end{aligned} \hspace{\stretch{1}}(2.3)
\begin{aligned}\mathcal{L} = \frac{1}{{2}} \sum_a m_a \mathbf{v}_a^2 + \sum_a \frac{m_a}{8} \frac{\mathbf{v}_a^4}{c^2} -\sum_{ a < b} \frac{q_a q_b}{{\left\lvert{\mathbf{x}_a(t) - \mathbf{x}_b(t)}\right\rvert}}+\sum_b q_a q_b \frac{\mathbf{v}_a \cdot \mathbf{v}_b + (\mathbf{n} \cdot \mathbf{v}_a) (\mathbf{n} \cdot \mathbf{v}_b)}{2 c^2 {\left\lvert{\mathbf{x} - \mathbf{x}_b}\right\rvert} }.\end{aligned} \hspace{\stretch{1}}(2.4)
This is the Darwin Lagrangian (also Charles). The Darwin Hamiltonian is then
\begin{aligned}H = \sum_a \frac{p_a}{2 m_a} \mathbf{v}_a^2 + \sum_a \frac{p_a^4}{8 m_a^3 c^2} +\sum_{ a < b} \frac{q_a q_b}{{\left\lvert{\mathbf{x}_a(t) - \mathbf{x}_b(t)}\right\rvert}}- \sum_{ a < b } \frac{q_a q_b}{ 2 c^2 m_a m_b } \frac{\mathbf{p}_a \cdot \mathbf{p}_b + (\mathbf{n}_{a b} \cdot \mathbf{p}_a) (\mathbf{n}_{ a b } \cdot \mathbf{p}_b)}{{\left\lvert{\mathbf{x} - \mathbf{x}_b}\right\rvert} }.\end{aligned} \hspace{\stretch{1}}(2.5)
# Incorporating radiation effects as a friction term.
To $O((v/c)^3)$ obvious problem due to radiation (system not closed). We’ll incorporate radiation via a function term in the EOM
Again consider the dipole system
\begin{aligned}m \dot{d}{z} &= -k z \\ \omega^2 &= \frac{k}{m}\end{aligned} \hspace{\stretch{1}}(3.6)
or
\begin{aligned}m \dot{d}{z} = -\omega^2 m z\end{aligned} \hspace{\stretch{1}}(3.8)
gives
\begin{aligned}\frac{d{{}}}{dt}\left( \frac{m}{2} \dot{z}^2 + \frac{ m \omega^2 }{2} z^2 \right) = 0 \end{aligned} \hspace{\stretch{1}}(3.9)
The energy radiated per unit time averaged per period is
\begin{aligned}P = \frac{ 2 e^2 }{ 3 c^3} \left\langle{{ \dot{d}{z}^2 }}\right\rangle\end{aligned} \hspace{\stretch{1}}(3.10)
We’ll modify the EOM
\begin{aligned}m \dot{d}{z} = -\omega^2 m z + f_{\text{radiation}}\end{aligned} \hspace{\stretch{1}}(3.11)
Employing an integration factor $\dot{z}$ we have
\begin{aligned}m \dot{d}{z} \dot{z} = -\omega^2 m z \dot{z} + f_{\text{radiation}} \dot{z}\end{aligned} \hspace{\stretch{1}}(3.12)
or
\begin{aligned}\frac{d{{}}}{dt} \left( m \dot{z}^2 + \omega^2 m z^2 \right) = f_{\text{radiation}} \dot{z}\end{aligned} \hspace{\stretch{1}}(3.13)
Observe that the last expression, force times velocity, has the form of power
\begin{aligned}m \frac{d^2 z}{dt^2} \frac{dz}{dt} = \frac{d{{}}}{dt} \left( \frac{m}{2} \left( \frac{dz}{dt} \right)^2 \right)\end{aligned} \hspace{\stretch{1}}(3.14)
So we can make an identification with the time rate of energy lost by the system due to radiation
\begin{aligned}\frac{d{{}}}{dt} \left( m \dot{z}^2 + \omega^2 m z^2 \right) \equiv \frac{d{{\mathcal{E}}}}{dt}.\end{aligned} \hspace{\stretch{1}}(3.15)
Average over period both sides
\begin{aligned}\left\langle{{ \frac{d{{\mathcal{E}}}}{dt} }}\right\rangle = \left\langle{{ f_{\text{radiation}} \dot{z} }}\right\rangle=- \frac{2 e^2 }{3 c^3} \left\langle{{\dot{d}{z}^2}}\right\rangle\end{aligned} \hspace{\stretch{1}}(3.16)
We demand this last equality, by requiring the energy change rate to equal that of the dipole power (but negative since it is a loss) that we previously calculated.
Claim:
\begin{aligned}f_{\text{radiation}} = \frac{2 e^2 }{3 c^3} \dddot{z}\end{aligned} \hspace{\stretch{1}}(3.17)
Proof:
We need to show
\begin{aligned}\left\langle{{ f_{\text{radiation}} }}\right\rangle= - \frac{2 e^2 }{3 c^3} \left\langle{{\dot{d}{z}^2}}\right\rangle\end{aligned} \hspace{\stretch{1}}(3.18)
We have
\begin{aligned}\frac{2 e^2}{3 c^3} \left\langle{{ \dddot{z} \dot{z} }}\right\rangle &= \frac{2 e^2}{3 c^3} \frac{1}{{T}} \int_0^T dt \dddot{z} \dot{z} \\ &= \frac{2 e^2}{3 c^3} \frac{1}{{T}} \int_0^T dt {\frac{d{{}}}{dt} (\dot{d}{z} \dot{z}) }-\frac{2 e^2}{3 c^3} \frac{1}{{T}} \int_0^T dt (\dot{d}{z})^2\end{aligned}
We first used $(\dot{d}{z} \dot{z})' = \dddot{z} \dot{z} + (\dot{d}{z})^2$. The first integral above is zero since the derivative of $\dot{d}{z} \dot{z} = (-\omega^2 z_0 \sin\omega t)(\omega z_0 \cos\omega t) = -\omega^3 z_0^2 \sin(2 \omega t)/2$ is also periodic, and vanishes when integrated over the interval.
\begin{aligned}\frac{2 e^2}{3 c^3} \left\langle{{ \dddot{z} \dot{z} }}\right\rangle =-\frac{2 e^2}{3 c^3} \left\langle{{ (\dot{d}{z})^2 }}\right\rangle\end{aligned} \hspace{\stretch{1}}(3.19)
We can therefore write
\begin{aligned}m \dot{d}{z} = -m \omega^2 z + \frac{2 e^2}{3 c^3} \dddot{z}\end{aligned} \hspace{\stretch{1}}(3.20)
Our “frictional” correction is the radiation reaction force proportional to the third derivative of the position.
Rearranging slightly, this is
\begin{aligned}\dot{d}{z} = - \omega^2 z + \frac{2}{3 c} \left( \frac{e^2}{m c^2} \right) \dddot{z} = - \omega^2 z + \frac{2}{3 c} \frac{r_e}{c} \dddot{z},\end{aligned} \hspace{\stretch{1}}(3.21)
where $r_e \sim 10^{-13} \text{cm}$ is the “classical radius” of the electron. In our frictional term we have $r_e/c$, the time for light to cross the classical radius of the electron.
There are lots of problems with this. One of the easiest is with $\omega = 0$. Then we have
\begin{aligned}\dot{d}{z} = \frac{2}{3} \frac{r_e}{c} \dddot{z}\end{aligned} \hspace{\stretch{1}}(3.22)
with solution
\begin{aligned}z \sim e^{\alpha t},\end{aligned} \hspace{\stretch{1}}(3.23)
where
\begin{aligned}\alpha \sim \frac{c}{r_e} \sim \frac{1}{{\tau_e}}.\end{aligned} \hspace{\stretch{1}}(3.24)
This is a self accelerating system! Note that we can also get into this trouble with $\omega \ne 0$, but those examples are harder to find (see: [2]).
FIXME: borrow this text again to give that section a read.
The sensible point of view is that this third term ($f_{\text{rad}}$) should be taken seriously only if it is small compared to the first two terms.
# References
[1] L.D. Landau and E.M. Lifshitz. The classical theory of fields. Butterworth-Heinemann, 1980.
[2] D.J. Griffith. Introduction to Electrodynamics. Prentice-Hall, 1981.
## PHY450H1S. Relativistic Electrodynamics Tutorial 9 (TA: Simon Freedman). Some worked problems. EM reflection. Stress energy tensor for simple configurations.
Posted by peeterjoot on April 11, 2011
[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]
# HW6. Question 3. (Non subtle hints about how important this is (i.e. for the exam)
## Motivation.
This is problem 1 from section 47 of the text [1].
Determine the force exerted on a wall from which an incident plane EM wave is reflected (w/ reflection coefficient $R$) and incident angle $\theta$.
Solution from the book
\begin{aligned}f_\alpha = - \sigma_{\alpha \beta} n_\beta - {\sigma'}_{\alpha \beta} n_\beta\end{aligned} \hspace{\stretch{1}}(1.1)
Here $\sigma_{\alpha \beta}$ is the Maxwell stress tensor for the incident wave, and ${\sigma'}_{\alpha \beta}$ is the Maxwell stress tensor for the reflected wave, and $n_\beta$ is normal to the wall.
## On the signs of the force per unit area
The signs in 1.1 require a bit of thought. We have for the rate of change of the $\alpha$ component of the field momentum
\begin{aligned}\frac{d{{}}}{dt} \int d^3 \mathbf{x} \left( \frac{S^\alpha}{c^2} \right) = - \int d^2 \sigma^\beta T^{\beta \alpha}\end{aligned} \hspace{\stretch{1}}(1.2)
where $d^2 \sigma^\beta = d^2 \sigma \mathbf{n} \cdot \mathbf{e}_\beta$, and $\mathbf{n}$ is the outwards unit normal to the surface. This is the rate of change of momentum for the field, the force on the field. For the force on the wall per unit area, we wish to invert this, giving
\begin{aligned}df^\alpha_{\text{on the wall, per unit area}}= (\mathbf{n} \cdot \mathbf{e}_\beta) T^{\beta \alpha}= -(\mathbf{n} \cdot \mathbf{e}_\beta) \sigma_{\beta \alpha}\end{aligned} \hspace{\stretch{1}}(1.3)
## Returning to the tutorial notes
Simon writes
\begin{aligned}f_\perp &= - \sigma_{\perp \perp} - {\sigma'}_{\perp \perp} \\ f_\parallel &= - \sigma_{\parallel \perp} - {\sigma'}_{\parallel \perp} \end{aligned} \hspace{\stretch{1}}(1.4)
and then says stating this solution is very non-trivial, because $\sigma_{\alpha \beta}$ is non-linear in $\mathbf{E}$ and $\mathbf{B}$. This non-triviality is a good point. Without calculating it, I find the results above to be pulled out of a magic hat. The point of the tutorial discussion was to work through this in detail.
# Working out the tensor.
FIXME: PICTURE:
The Reflection coefficient can be defined in this case as
\begin{aligned}R = \frac{ {\left\lvert{\mathbf{E}'}\right\rvert}^2 }{ {\left\lvert{\mathbf{E}}\right\rvert}^2 },\end{aligned} \hspace{\stretch{1}}(2.6)
a ratio of the powers of the reflected wave power to the incident wave power (which are proportional to ${\mathbf{E}'}^2$ and ${\mathbf{E}}^2$ respectively.
Suppose we pick the following orientation for the incident fields
\begin{aligned}E_x &= E \sin\theta \\ E_y &= -E \cos\theta \\ B_z &= E ,\end{aligned} \hspace{\stretch{1}}(2.7)
With the reflected assumed to be in some still perpendicular orientation (with this orientation picked for convienence)
\begin{aligned}E_x' &= E' \sin\theta \\ E_y' &= E' \cos\theta \\ B_z' &= E'.\end{aligned} \hspace{\stretch{1}}(2.10)
Here
\begin{aligned}E &= E_0 \cos(\mathbf{p} \cdot \mathbf{x} - \omega t) \\ E' &= \sqrt{R} E_0 \cos(\mathbf{p}' \cdot \mathbf{x} - \omega t)\end{aligned} \hspace{\stretch{1}}(2.13)
FIXME: there are assumptions below that $\mathbf{p}' \cdot \mathbf{x} = \mathbf{p} \cdot \mathbf{x}$. I don’t see where that comes from, since the propagation directions are difference for the incident and the reflected waves.
\begin{aligned}\sigma_{\alpha\beta} = -T^{\alpha\beta} = \frac{1}{{4\pi}} \left(\mathcal{E}^\alpha\mathcal{E}^\beta+\mathcal{B}^\alpha\mathcal{B}^\beta- \frac{1}{{2}} \delta^{\alpha\beta} ( \vec{\mathcal{E}}^2 + \vec{\mathcal{B}}^2 )\right)\end{aligned} \hspace{\stretch{1}}(2.15)
## Aside: On the geometry, and the angle of incidence.
According to wikipedia [2] the angle of incidence is measured from the normal.
Let’s use complex numbers to get the orientation of the electric and propagation direction fields right. We have for the incident propagation direction
\begin{aligned}-\hat{\mathbf{p}} \sim e^{i (\pi + \theta) }\end{aligned} \hspace{\stretch{1}}(2.16)
or
\begin{aligned}\hat{\mathbf{p}} \sim e^{i\theta}\end{aligned} \hspace{\stretch{1}}(2.17)
If we pick the electric field rotated negatively from that direction, we have
\begin{aligned}\hat{\mathbf{E}} &\sim -i e^{i \theta} \\ &= -i (\cos\theta + i \sin\theta) \\ &= -i \cos\theta + \sin\theta\end{aligned}
Or
\begin{aligned}E_x &\sim \sin\theta \\ E_y &\sim -\cos\theta\end{aligned} \hspace{\stretch{1}}(2.18)
For the reflected direction we have
\begin{aligned}\hat{\mathbf{p}}' \sim e^{i (\pi - \theta)} = - e^{-i \theta}\end{aligned} \hspace{\stretch{1}}(2.20)
rotating negatively for the electric field direction, we have
\begin{aligned}\hat{\mathbf{E}'} &\sim -i (- e^{-i\theta} ) \\ &= i(cos\theta - i\sin\theta) \\ &= i cos\theta + \sin\theta\end{aligned}
Or
\begin{aligned}E_x' &\sim \sin\theta \\ E_y' &\sim \cos\theta \end{aligned} \hspace{\stretch{1}}(2.21)
## Back to the problem (again).
Where $\vec{\mathcal{E}}$ and $\vec{\mathcal{B}}$ are the total EM fields.
\paragraph{Aside:} Why the fields are added in this fashion wasn’t clear to me, but I guess this makes sense. Even if the propagation directions differ, the total field at any point is still just a superposition.
\begin{aligned}\vec{\mathcal{E}} &= \mathbf{E} + \mathbf{E}' \\ \vec{\mathcal{B}} &= \mathbf{B} + \mathbf{B}'\end{aligned} \hspace{\stretch{1}}(2.23)
Get
\begin{aligned}\sigma_{3 3} &= \frac{1}{{4 \pi}} \left( \underbrace{B_z B_z}_{=\vec{\mathcal{B}}^2} - \frac{1}{{2}} (\vec{\mathcal{E}}^2 + \vec{\mathcal{B}}^2) \right) = 0 \\ \sigma_{3 1} &= 0 = \sigma_{3 2} \\ \sigma_{1 1} &= \frac{1}{{4 \pi}} \left( (\mathcal{E}^1)^2 - \frac{1}{{2}} (\vec{\mathcal{E}}^2 + \vec{\mathcal{B}}^2) \right) \end{aligned} \hspace{\stretch{1}}(2.25)
\begin{aligned}\vec{\mathcal{B}}^2 &= (B_z + B_z')^2 = (E + E')^2 \\ \vec{\mathcal{E}}^2 &= (\mathbf{E} + \mathbf{E}')^2\end{aligned} \hspace{\stretch{1}}(2.28)
so
\begin{aligned}\sigma_{1 1} &= \frac{1}{{4 \pi}} \left( (\mathcal{E}^1)^2 - \frac{1}{{2}} ((\mathcal{E}^1)^2 + (\mathcal{E}^2)^2 + (E + E')^2 \right) \\ &= \frac{1}{{8 \pi}} \left( (\mathcal{E}^1)^2 - (\mathcal{E}^2)^2 - (E + E')^2 \right) \\ &= \frac{1}{{8 \pi}} \left( (E + E')^2 \sin^2\theta -(E' - E)^2 \cos^2\theta -(E + E')^2\right) \\ &= \frac{1}{{8 \pi}} \left( E^2( \sin^2\theta - \cos^2\theta - 1)+(E')^2( \sin^2\theta - \cos^2\theta - 1)+ 2 E E' (\sin^2\theta + \cos^2\theta -1 )\right) \\ &= \frac{1}{{8 \pi}} \left( - 2 E^2 \cos^2\theta - 2 (E')^2 \cos^2\theta \right) \\ &= -\frac{1}{{4 \pi}} (E^2 + (E')^2) \cos^2\theta \\ &= \sigma_\parallel + {\sigma'}_\parallel\end{aligned}
This last bit I didn’t get. What is $\sigma_\parallel$ and ${\sigma'}_\parallel$. Are these parallel to the wall or parallel to the normal to the wall. It turns out that this appears to mean parallel to the normal. We can see this by direct calculation
\begin{aligned}\sigma_{x x}^{\text{incident}} &= \frac{1}{{4 \pi}} \left( E_x^2 - \frac{1}{{2}} (\mathbf{E}^2 + \mathbf{B}^2)\right) \\ &= \frac{1}{{4 \pi}} \left( E^2 \sin^2 \theta - \frac{1}{{2}} 2 E^2 \right) \\ &= -\frac{1}{{4 \pi}} E^2 \cos^2\theta\end{aligned}
\begin{aligned}{\sigma}_{x x}^{\text{reflected}} &= \frac{1}{{4 \pi}} \left( {E_x'}^2 - \frac{1}{{2}} ({\mathbf{E}'}^2 + {\mathbf{B}'}^2)\right) \\ &= \frac{1}{{4 \pi}} \left( {E'}^2 \sin^2 \theta - \frac{1}{{2}} 2 {E'}^2 \right) \\ &= -\frac{1}{{4 \pi}} {E'}^2 \cos^2\theta\end{aligned}
So by comparison we see that we have
\begin{aligned}\sigma_{1 1} = {\sigma}_{x x}^{\text{incident}} +{\sigma}_{x x}^{\text{reflected}} \end{aligned} \hspace{\stretch{1}}(2.30)
Moving on, for our other component on the $x,y$ place $\sigma_{12}$ we have
\begin{aligned}\sigma_{12} &= \frac{1}{{4 \pi}} \mathcal{E}^1 \mathcal{E}^2 \\ &= \frac{1}{{4 \pi}} (E + E') \sin\theta (-E + E') \cos\theta \\ &= \frac{1}{{4 \pi}} ((E')^2 - E^2) \sin\theta \cos\theta \end{aligned}
Again we can compare to the sums of the reflected and incident tensors for this $x,y$ component. Those are
\begin{aligned}\sigma_{12}^{\text{incident}} &= \frac{1}{{4 \pi}} ( E^1 E^2 ) \\ &= -\frac{1}{{4 \pi}} E^2 \sin\theta \cos\theta,\end{aligned}
and
\begin{aligned}\sigma_{12}^{\text{reflected}} &= \frac{1}{{4 \pi}} ( {E'}^1 {E'}^2 ) \\ &= \frac{1}{{4 \pi}} {E'}^2 \sin\theta \cos\theta\end{aligned}
Which demonstrates that we have
\begin{aligned}\sigma_{12} = \sigma_{12}^{\text{incident}} + \sigma_{12}^{\text{reflected}} \end{aligned} \hspace{\stretch{1}}(2.31)
Summarizing, for the components in the $x,y$ plane we have found that we have
\begin{aligned}\sigma_{\alpha\beta}^{\text{total}} n_\beta = \sigma_{\alpha 1 }^{\text{total}} = \sigma_{\alpha 1} + {\sigma'}_{\alpha 1}\end{aligned} \hspace{\stretch{1}}(2.32)
(where $n_\beta = \delta^{\beta 1}$)
This result, assumed in the text, was non-trivial to derive. It is also not generally true. We have
\begin{aligned}\sigma_{2 2} &= \frac{1}{{4 \pi}} \left( (\mathcal{E}^y)^2 - \frac{1}{{2}} ( \vec{\mathcal{E}}^2 + \vec{\mathcal{B}}^2 ) \right) \\ &= \frac{1}{{8 \pi}} \left( (\mathcal{E}^y)^2 - (\mathcal{E}^x)^2 - \vec{\mathcal{B}}^2 ) \right) \\ &= \frac{1}{{8 \pi}} \left( (E' - E)^2 \cos^2\theta - (E + E')^2 \sin^2\theta - (E + E')^2\right) \\ &= \frac{1}{{8 \pi}} \left( E^2 ( -1 + \cos^2 \theta - \sin^2\theta )+{E'}^2 ( -1 + \cos^2 \theta - \sin^2\theta )+ 2 E E' ( -\cos^2\theta - \sin^2\theta -1 ) \right) \\ &= -\frac{1}{{4 \pi}} \left( E^2 \sin^2 \theta + (E')^2 \sin^2 \theta + 2 E E' \right)\end{aligned}
If we compare to the incident and reflected tensors we have
\begin{aligned}\sigma_{y y}^{\text{incident}} &= \frac{1}{{4 \pi}} \left( (E^y)^2 -\frac{1}{{2}} E^2 \right) \\ &= \frac{1}{{4 \pi}} E^2 ( \cos^2\theta - 1 ) \\ &= -\frac{1}{{4 \pi}} E^2 \sin^2\theta \end{aligned}
and
\begin{aligned}\sigma_{y y}^{\text{reflected}} &= \frac{1}{{4 \pi}} \left( ({E'}^y)^2 -\frac{1}{{2}} {E'}^2 \right) \\ &= \frac{1}{{4 \pi}} {E'}^2 ( \cos^2\theta - 1 ) \\ &= -\frac{1}{{4 \pi}} {E'}^2 \sin^2\theta \end{aligned}
There’s a cross term that we can’t have summing the two, so we have, in general
\begin{aligned}\sigma_{2 2}^{\text{total}} \ne \sigma_{y y}^{\text{incident}} +\sigma_{y y}^{\text{reflected}} \end{aligned} \hspace{\stretch{1}}(2.33)
## Force per unit area?
\begin{aligned}f_\alpha = n^x \sigma_{x \alpha} \end{aligned} \hspace{\stretch{1}}(2.34)
Averaged
\begin{aligned}\left\langle{{\sigma_{xx}}}\right\rangle &= -\frac{1}{{8 \pi}} E_0^2 ( 1 + R) \cos^2\theta \\ \left\langle{{\sigma_{xy}}}\right\rangle &= -\frac{1}{{8 \pi}} E_0^2 ( 1 - R) \sin\theta \cos\theta\end{aligned} \hspace{\stretch{1}}(2.35)
\begin{aligned}\left\langle{\mathbf{S}}\right\rangle &= -\frac{c}{8 \pi} E_0^2 \hat{\mathbf{n}} \\ \left\langle{{\mathbf{S}'}}\right\rangle &= -\frac{c}{8 \pi} E_0^2 \hat{\mathbf{n}}'\end{aligned} \hspace{\stretch{1}}(2.37)
\begin{aligned}\left\langle{{{\left\lvert{\mathbf{S}}\right\rvert}}}\right\rangle = \text{Work} = W\end{aligned} \hspace{\stretch{1}}(2.39)
\begin{aligned}f_x &= n^x \sigma_{x x} = W (1 + R) \cos^2\theta \\ f_y &= n^y \sigma_{x y} = W (1 - R) \sin\theta \cos\theta \\ f_z &= 0\end{aligned} \hspace{\stretch{1}}(2.40)
# A problem from Griffiths.
FIXME: try this.
Two charges $q+$, $q-$ reflected in a plane, separated by distance $a$. Work out the stress energy tensor from the Coulomb fields of the charges on the plane.
Will get the Coulomb force:
\begin{aligned}\mathbf{F} = k \frac{q^2}{2 a^2}.\end{aligned} \hspace{\stretch{1}}(3.43)
# Infinite parallel plate capacitor
Write $\sigma_{\alpha\beta}$.
\begin{aligned}\mathbf{B} &= 0 \\ \mathbf{E} &= - \frac{\sigma}{\epsilon_0} \mathbf{e}_z\end{aligned} \hspace{\stretch{1}}(4.44)
FIXME: derive this. Observe that we have no distance dependence in the field because it is an infinite plate.
\begin{aligned}\sigma_{1 1} &= \left( - \frac{1}{{2}} \delta^{1 1} \left( \frac{-\sigma}{\epsilon_0} \right)^2 \right) = - \frac{ \sigma^2}{ 2 \epsilon_0^2 } = \sigma_{22} \\ \sigma_{3 3} &= \left( (E^3)^2 - \frac{1}{{2}} \mathbf{E}^2 \right) = - \frac{1}{{2}} \mathbf{E}^2 = - \sigma_{2 2}\end{aligned} \hspace{\stretch{1}}(4.46)
Force per unit area is then
\begin{aligned}f_\alpha &= n_\beta \sigma_{\alpha \beta} \\ &= n_3 \sigma_{\alpha 3}\end{aligned}
So
\begin{aligned}f_1 &= 0 = f_2 \\ f_3 &= \sigma_{3 3} = -\frac{\sigma^2}{2 \epsilon_0^2}\end{aligned} \hspace{\stretch{1}}(4.48)
\begin{aligned}\mathbf{f} = -\frac{\sigma^2}{2 \epsilon_0^2} \mathbf{e}_z\end{aligned} \hspace{\stretch{1}}(4.50)
REMEMBER: EXAM WEDNESDAY!
# References
[1] L.D. Landau and E.M. Lifshitz. The classical theory of fields. Butterworth-Heinemann, 1980.
[2] Wikipedia. Angle of incidence — wikipedia, the free encyclopedia [online]. 2011. [Online; accessed 11-April-2011]. http://en.wikipedia.org/w/index.php?title=Angle_of_incidence&oldid=421647114.
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2018-09-24 14:18:38
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https://wiki.melvoridle.com/index.php?title=Template:EquipmentTableFromList&oldid=42595
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# Template:EquipmentTableFromList
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Function for creating a stat table with a given list of items. Items are sent in as a comma separated list. For example, {{EquipmentTableFromList|Bronze Helmet,Bronze Platebody,Bronze Platelegs}} gives:
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2023-03-22 22:35:57
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http://entd.ilovebet.it/theoretical-probability-worksheet-2-answers.html
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Notice it matches the answer gotten in part d. Compare the experimental probability you found in part a to its theoretical. , tables, grids, tree diagrams) and express the theoretical probability of each outcome. Supporting starter questions are here. More Probability Quizzes Quiz: Take The Basic Probability Test Questions!. Probability is the study of how likely things are to happen. If the dice is rolled again and again, the experimental probability will. If you toss a coin, what is the theoretical probability of getting heads? P(heads)= 3. Find the Problems Work Space Probability of selecting a boy Answer: _____. For example, the probability of the result of coin flip being ‘heads’ is 0. 1 Relations and Functions 2. Virtual Nerd's patent-pending tutorial system provides in-context information, hints, and links to supporting tutorials, synchronized with videos, each 3 to 7 minutes long. Based on the chart, what is the experimental probability of pulling a green marble?. 2 - Determine the probability of simple events involving independent and dependent events and conditional probability. notebook 1 March 10, 2014 Mar 78:46 AM Theoretical Probability of Compound Events Mar 108:42 AM A compound event is an event that includes two or more simple events. A single outcome of this experiment is rolling a 1, or rolling a 2, or rolling a 3, etc. After the majority have finished, discuss what happened and then as a class write the theoretical (anticipated) probability on the board, the answer. Write down the total number of possible outcomes when the ball is drawn from a bag containing 5. 2, 9 36, = 9 3. There are 3 cards of the same rank left so the probability you pick one is 3/51, because after the first card is picked, it isn't replaced. b) The probability the shirt will not be gold is 6 4 or 3 2. Alg 2 WP 4 RUBRIC – Past Performance by Seed Worksheet. Random: for probability to work, the event has to be random. This is a great way to get students to understand experimental and theoretical probability. Use parentheses. Some of the worksheets for this concept are Experimental probability work show your work, Theoretical probability activity, 11 2 theoretical and experimental probability, Lesson practice b 11 2 experimental probability, Lesson 1. *I can find arrangements with permutations. Also on Super Teacher Worksheets Basic Fraction Worksheets. Lesson 2 Theoretical And Experimental Probability - Displaying top 8 worksheets found for this concept. Find the missing probability. Theoretical Probability Worksheets 7th Grade. Experimental Probability Answer Key - Displaying top 8 worksheets found for this concept. A player hit the bull's eye on a circular dart board 8 times out of 50. Experimental Probability Worksheet Instructions for Printing the Worksheet or Answer Key. probability of spinning red for this experiment. A box contains 7 red balls and 4 white balls. This Probability Worksheet produces problems using a spinner. This quiz is incomplete! To play this quiz, please finish editing it. Theoretical Probability Worksheet 2 - Here is a fourteen. Probability Worksheet 1. Describe the three most significant differences between the graphs or distributions of the theoretical and empirical distributions. Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model. But what does this actually mean?. She recorded the suit of the card she picked, and then replaced the card. The answers have to be what each person really gets by doing the experiment. org are unblocked. 1 Revised June 2010 Experimental and Theoretical Probability Name _____ Per ____ Date _____ Amanda used a standard deck of 52 cards and selected a card at random. Some of the worksheets displayed are Unit 9 probability and statistics, Unit 6 probability, Math 9 unit 9 probability and statistics practice test, Probability work 1, Unit 6 data analysis and probability, Probability with combinatorics date. View US version. The student will also understand. In this lesson, we will look into experimental probability and theoretical probability. Rolled a 1 Rolled a 2 Rolled a 3 Rolled a 4 Rolled a 5 Rolled a 6 How did your experimental probability compare to the theoretical probability? What is the difference between theoretical probability and. multiple line graph page 10 Circle Graphs 1. Rock, Paper, Scissors Probability Lesson is an investigation into the mathematics of the popular game of the same name. What is the probability of rolling an odd number on a fair die? 3. Create the worksheets you need with Infinite Algebra 1. Share practice link. Probability word problems worksheets, 8th Grade. Under the table record the theoretical probability and the experimental probability. What is the probability of rolling an odd number on a fair die? 3. Full solutions included. Lesson 2 Homework Practice Theoretical and Experimental Probability 1. A permutation is an arrangement of objects in which order matters. Different ways to find the probability of a compound event are: ‐ Tree Diagram ‐ List/Table ‐ Multiplying the Options. Standard: MATH 4. Direct Instruction: (I do it) Yesterday we talked about probability. Some of the worksheets displayed are Mathematics grade 6 experimental and theoretical probability, Lesson practice b 11 2 experimental probability, 15a 15b 15c 15d 15e relative frequency and probability, 1 theoretical experimental 1 pa and pb, Course applied algebra. Counting with permutations. Print Worksheet. Applied Statistics and Probability for Engineers, 6th Edition Montgomery, Douglas C. Theoretical Probability Real-life situations can be simulated by probability experiments. P(greater than 10) 4. A student is selected for a field trip in random. Files included (1) experimental probability. Showing top 8 worksheets in the category - 1 20 Days Of Experimental And Theoretical. Exercise #2 A bag contains 6 blue balls, 4 yellow balls, and 2 red balls. Private Sub Worksheet_Activate() MsgBox ("Worksheet has been activated") End Sub However, if I click on another tab and click back to the sheet containing the code, it does trigger. Finally, the Difference (Percent) column measures the variation from the Theoretical Probability. What is the probability of the spinner landing on a 3? 1 out of 6 2. But what does this actually mean?. Use parentheses. Today is Saturday. A coin is tossed 3 times. Additional materials for exam preparation can be found under the class sessions dedicated to exam review. Lesson 2 Theoretical And Experimental Probability. Find the probability of each event. What is the difference between theoretical and experimental? 2. Find the experimental probability of landing on a 2. High school math students can use these statistics problems for study purposes. Example – If three coins are tossed, what is the probability of getting exactly. What is the probability of pulling out a red or a green marble? 2. Circle it and write it on the line. Theoretical Probability of Compound Events Practice and Problem Solving: A/B Use the table of probabilities to answer questions 1–3. Some of the worksheets for this concept are Lesson practice a 10 1 probability, Experimental and theoretical probability, The the hunger gameshunger gameshunger games, Probability, Lesson practice b 11 4 theoretical probability, Work finding the probability of an event ii, Lesson 14 probability, Lesson plan 1 introduction to probability. Some of the worksheets for this concept are Experimental probability work show your work, Theoretical probability activity, 11 2 theoretical and experimental probability, Lesson practice b 11 2 experimental probability, Lesson 1. Write the formula for theoretical probability: P (event) = 2. Automatic spacing. More Probability Quizzes Quiz: Take The Basic Probability Test Questions!. Displaying all worksheets related to - Lesson 2 Theoretical And Experimental Probability. 03; Create a table to record the outcomes for 10 coin flips. mathworksheets4kids. Theoretical probability is the probability of something occurring when the math is done out on paper or 'in theory' such as the chance of rolling a six sided dice and getting a 2 is 1/6. major arc 9. Improve your math knowledge with free questions in "Theoretical and experimental probability" and thousands of other math skills. What is the probability of tossing at least 2 heads? 2. Worksheets are Honors biology ninth grade pendleton high school, Answer key, Psychology 11 unit 1 work, Lesson 1 experimental and theoretical probability, Ap biology chapters 1 work, Work extra examples, Unit 1 introduction to biology, Forms of energy. Suppose that 37. Theoretical Probability Worksheets- Includes math lessons, 2 practice sheets, homework sheet, and a quiz!. THEORETICAL PROBABILITY ACTIVITY (ANSWER KEY) Theoretical probability is just that…a theory. 9 Permutations and Combinations Worksheet 12. Experimental Probability When asked about the probability of a coin landing on heads, you would probably answer that the chance is ½ or 50%. 69% average accuracy. Estimate the probability of the spinner landing on red. The probability of drawing a red checker and then a black checker is 25%. P(less than 4 tails) 22. Compare the experimental probability you found in part a to its theoretical. Find the indicated probabilities. Showing top 8 worksheets in the category - 1 20 Days Of Experimental And Theoretical. A number cube is rolled 24 times and lands on 2 four times and on 6 three times. Quiz questions focus on a favorable outcome in theoretical probability as well as whether a prediction guarantees a certain result will happen. Files included (1) experimental probability. The events "obtaining a six with the first die" and "obtaining an even number with the second die" are independent. 04 g NH 3 = 28. WORKSHEET: Regents-Theoretical Probability AII/A2: 2/1: TST PDF DOC TNS. Name Worksheet E3 : Review for Unit 3 Test The spinner shown is spun once. Lesson 2 Theoretical And Experimental Probability. A permutation is an arrangement of objects in which order matters. FACTS AND FORMULAE FOR PROBABILITY QUESTIONS. The student will also understand. View answers. ) What is the theoretical probability that an even number will be rolled on a number cube? 2. > Algebra II Lesson on Theoretical & Experimental Probability + Algebra II Lesson on Theoretical & Experimental Probability Rating: (2) (0) (0) (0) (0) (2) Author: Chad Bray. Spinners 11) P(black) = Black Blue 12) P(not orange) = 13) P(blue or black) = Purple Orange 14) If the spinner is spun 40 times, how many times would you predict a spin of something that is not purple? 15) Johnny spins the spinner 60 times. Word Problem Worksheet #1 (each correct answer = 1 extra ballot for the draw!) Word Problem Worksheet #2 (each correct answer = 1 extra ballot for the draw!) March 27 - Unit Test and Start of Geometry. A bag contains 30 pieces of candy. Worksheet | Experimental Probability. Find the theoretical probability of the following events occurring. Experimental probability. This is a math PDF printable activity sheet with several exercises. Circle it and write it on the line. Probability With Deck Of Cards. This lesson deals with the multiplication rule. 1 Intro of Probability. 1) A two-sided coin is flipped and lands on heads. High School Math based on the topics required for the Regents Exam conducted by NYSED. c) The probability the uniform will have the same-coloured shorts and shirt is 6 2 or 3 1. 12 g NH 3 2. You have $14$ socks, so there are ${14 \choose 2} = 91$ ways can you pull $2$ socks out from that pile. major arc 5. com is an online resource used every day by thousands of teachers, students and parents. Worksheet 2-2: Theoretical Probability Theoretical probability is another measure of the likelihood of an event. b) The probability the shirt will not be gold is 6 4 or 3 2. Write the factorization under the polynomial for easy reference. High School Math based on the topics required for the Regents Exam conducted by NYSED. Nine spinners you can use for various probability activities and experiments. Theoretical and experimental probabilities. Do each exercise and find your answer to the right. Mutually exclusive and inclusive events, probability on odds and other challenging probability worksheets are useful for grade 6 and up students. 7 Conditional Probability AND 12. 306 Institutions have accepted or given pre-approval for credit transfer. Answer Keys Answers for homework and quiz sheets. Coin Probability Theoretical vs. In this lesson, we will look into experimental probability and theoretical probability. What is the theoretical probability of not rolling a 2? e. Worksheets are Deck of cards work, Statistics probability with cards, Independent and dependent, Probability, Probability playing cards, Probability and odds work answer key we like your chances, Probability work 4 experimental and theoretical, Probability theory work 1 key. The set includes student answer sheet and key. The other values in the table are the sum of the two dice. Theoretical probability is the ratio between the number of ways an event can occur and the total number of possible outcomes in the sample space. 4 Writing and Graphing Linear Equations 2. notebook 1 9. For example, the probability of the result of coin flip being ‘heads’ is 0. com There are 25 students in grade 5. A set of 26 probability task cards that are aligned to the common core. Here u 1 is the first value in the domain of f which is larger than t 1. 2 Practice Level B 1. Determine the theoretical probability of the event. Two worksheets, testing basic probability with dice, coloured balls and letters. A bag contains 30 pieces of candy. Find the indicated probabilities. WebCity Press. Worksheets are Experimental probability work show your work, Theoretical probability activity, 11 2 theoretical and experimental probability, Lesson practice b 11 2 experimental probability, Lesson 1 experimental and theoretical probability, Lesson. The results are in the table below. An experimental probability, or observed probability, is based on data or experiments and is defined as. Figure 1 is a discrete probability distribution: It shows the probability for each of the values on the X-axis. P(either a boy or a girl) 6. Eight balls numbered from 1 to 8 are placed in an urn. If the blue car is allowed to move on rolls of 2, 5, and 6 and the red car is allowed to move on rolls of 1, 3, and 4 which car is more likely to finish a 3 step race first? a. So, the theoretical probability is 1 4 \frac{1}{4} 4 1 or 25%. Using a Tree Diagram to Find a Sample Space TSWBAT use the probability formula with a 80% accuracy. This quiz is incomplete! To play this quiz, please finish editing it. Simple probability worksheets based on tossing single coin or two coins. Would this change (see part 3) cause the empirical probabilities and theoretical probabilities to be closer together or farther apart? How do you know? Explain the differences in what P(G 1 AND R 2) and P(R 1 |G 2) represent. What is the theoretical probability of not rolling a 2? e. What are experimental and theoretical probability - ExplainingMaths. Lesson 2 Theoretical And Experimental Probability. 1 8 1 4 1 8 1 2 24. A student is selected for a field trip in random. Click on pop-out icon or print icon to worksheet to print or download. Give your answers as fractions in their simplest terms where necessary. Experimental probability is performed when authorities want to know how the public feels about a matter. What is the probability of the spinner landing on a 2? 1 out of 6 4. Probability Class 10 Maths NCERT Solutions are extremely helpful while doing your homework or while preparing for the exam My maths answers experimental probability. The probability of picking a red checker on the first draw is 42%. Write the formula for theoretical probability: P (event) = 2. Probability Math Worksheets. For example, the probability of the result of coin flip being ‘heads’ is 0. What is the difference between theoretical and experimental? 2. tossing a number cube numbered from 1 to 6 and getting an even number that is greater than or equal to 2 _____ 2. if you have a group of 600 women, predict how many of them will be color blind. 15 are boys and others are girls. Probability Of Compound events Worksheet Answers – When you find a template that you would like to use, begin customizing it and you may also to open it in your document window! You will discover a number of the templates are absolutely free to use and others call for a premium account. Therefore, the probability of rolling a 2 is 1 out of 6, or 1/6. Find the experimental probability of not landing on a 6. ) What is the theoretical probability that an even number will be rolled on a number cube? 2. This Theoretical Probability: Practice Worksheet is suitable for 5th - 6th Grade. I would expect this because the theoretical probability is. The worksheet is an assortment of 4 intriguing pursuits that will enhance your kid's knowledge and abilities. Theoretical probability is the probability of something occurring when the math is done out on paper or 'in theory' such as the chance of rolling a six sided dice and getting a 2 is 1/6. 1 2 3 6 8 10 12 4 6 14 0 2 Number of Rolls Number Cube Experiment Number Showing 4 5 10 8 6 9 5 12 a. Experimental Probability Example. 306 Institutions have accepted or given pre-approval for credit transfer. It is the ratio of the number of successful outcomes and the total number of possible outcomes. Theoretical Probability Examples Example 1. notebook 2 May 25, 2018 May 199:20 AM May 199:21 AM May 199:21 AM May 2410:03 AM. The following table highlights the difference between Experimental Probability and Theoretical Probability. What is the theoretical probability of pulling a red? answer choices. This quiz is incomplete! To play this quiz, please finish editing it. 2 Theoretical and Experimental Probability Notes Answers. Here's an image of part of the worksheet. (1) (c) On the probability scale below, mark with a cross (×) the probability that you will get a head when you throw a coin. It is the ratio of the number of successful outcomes and the total number of possible outcomes. WebCity Press. Experimental Probability When asked about the probability of a coin landing on heads, you would probably answer that the chance is ½ or 50%. Files included (4) Picture-References. 29 Questions Show answers. Worksheets are Experimental probability work show your work, Theoretical probability activity, 11 2 theoretical and experimental probability, Lesson practice b 11 2 experimental probability, Lesson 1 experimental and theoretical probability, Lesson. It will rain today. 9 Permutations and Combinations Worksheet 12. Theoretical Probability. Hint: Think about the sample space for each probability. Experimental probability is performed when authorities want to know how the public feels about a matter. Bookmakers are more likely to use empirical probability to give the odds on a horse, for example, because simply calculating the probability of any. 1 The Nature of Probability - Read Only 12. 8 Mathematics Learner’s Module 11 Department of Education Republic of the Philippines This instructional material was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. Experimental and Theoretical Probability This video defines and uses both experimental and theoretical probabilities. July 16th, 2019. Share practice link. Theoretical probability is the probability that a certain outcome will occur based on all the possible outcomes. your experimental probability for tossing heads. Coin Toss Probability Investigation Worksheet - What is the probability when tossing a coin? The coin toss probability for it to be heads or tails is 50%, 1/2, or 0. The first column is the face showing on die1 and the first row is the face showing on die2. A bag contains 30 pieces of candy. (1) (b) On the probability scale below, mark with a cross (×) the probability that you will get a 10 when you roll an ordinary 6-sided dice. Lesson 2 Theoretical And Experimental Probability. Find a word that An experiment consists of tossing two coins. If a total of eleven raffle tickets are sold and two winners will be selected, what. What is the probability that the coin will land on heads?. WORKSHEET: Regents-Theoretical Probability AII/A2: 2/1: TST PDF DOC TNS. There are ${4 \choose 2} = 6$ ways to pick blue socks, ${7 \choose 2} = 21$ ways to pick red socks, and ${3 \choose 2} = 3$ ways to pick yellow socks. Some of the worksheets for this concept are Experimental probability work show your work, Theoretical probability activity, 11 2 theoretical and experimental probability, Lesson practice b 11 2 experimental probability, Lesson 1. Coin Toss Probability Investigation Worksheet - What is the probability when tossing a coin? The coin toss probability for it to be heads or tails is 50%, 1/2, or 0. Would this change (see part 3) cause the empirical probabilities and theoretical probabilities to be closer together or farther apart? How do you know? Explain the differences in what P(G 1 AND R 2) and P(R 1 |G 2) represent. FACTS AND FORMULAE FOR PROBABILITY QUESTIONS. but there are six ways of getting a total of 7 (1 + 6, 2 + 5, 3 + 4, 4 + 3, 5 + 2 and 6 + 1) Here is a table of all possibile outcomes, and the totals. P(either a boy or a girl) 6. In Section 10. Probability For Grade 2 - Displaying top 8 worksheets found for this concept. True False 2. > Algebra II Lesson on Theoretical & Experimental Probability + Algebra II Lesson on Theoretical & Experimental Probability Rating: (2) (0) (0) (0) (0) (2) Author: Chad Bray. A) 22 B) 6 C) 24 D) 240 ____ 11. What is the probability of tossing exactly 2 heads? c. Worksheet 2-3: Experimental Probability vs. 5 of being a success on each trial. She recorded the suit of the card she picked, and then replaced the card. Basic Probability Worksheet Name: _____ Period: _____ Consider a fair die. nether unlikely nor likely The probability of randomly se ecting a green marble from a bag of 20 marbles is. The abbreviation of pdf is used for a probability distribution function. The table shows the results after 100 draws. Patricia and Courtney spun the spinner below. Theoretical is where you calculate the probability, experimental is where you actually perform it or lose a limb. Rolling an even number (2, 4 or 6) is an event, and rolling an odd number (1, 3 or 5) is also an event. The link between theoretical and experimental probability. Conditional Probability and the Rules for Probability: Understand independence and conditional probability and use them to interpret data. Answers will vary, but all of them should include the concept that theoretical probability is the mathematical chance of an event occurring, while experimental probability is the mathematical result of performing a series of trials of the event. Students learn that experimental probability approaches theoretical probability as the number of trials increases. It is important to understand the difference between the theoretical probability of an event and the observed relative frequency of the event in experimental trials. You could review the needed probability concepts by nding the probability of being dealt two aces in a row. Different ways to find the probability of a compound event are: ‐ Tree Diagram ‐ List/Table ‐ Multiplying the Options. 1 The Nature of Probability - Read Only 12. What is the probability of picking a 4 of hearts in a deck of cards? 1 52 2. 67% Also, some. See more ideas about Probability worksheets, Probability, Probability activities. View answers. Provide details and share your research! But avoid … Asking for help, clarification, or responding to other answers. 1 Intro of Probability. Are you more likely to spin an odd number or an even number? Explain. Students should indicate that the probability is 1/2 or 50%. Thus, the probability of exactly 8 successes out of the 25, with probability 0. Th ere are 225 juniors and 255 seniors at your school. The probability of the event E can be found by P(E)={n(E)}/{n(S)}, where n(E) and n(S) denote the number of outcomes in E and the number of outcomes in S, respectively. I used it as Homework * AlSO AVAILABLE FOR YOU OR A COLLEAGUE! - CLICK ANY LINK YOU WANT: Probability & Odds Mean, median, mode, range, line plot, and Box and whisker Worksheet Statistical Math (GED REVIEW) T. Document Outline. If a total of eleven raffle tickets are sold and two winners will be selected, what. This was such a great way for my Middle school math students to get some work on Theoretical Probability of Simple Events. Probability Report Worksheet Directions: Complete the financial report worksheet to help you with your calculations to create the APA report. Suppose that 37. The empirical probability = 8/50 = 16%. experimental probability. Theoretical Probability Example What is the probability of rolling a 2 on a regular 6-sided die? Answer: There are 6 sides on the cube (numbered 1 - 6) and there is only one way to roll a 2. major arc 6. Now, take the next step, and add the new directions to the timeline, as well as the new position of the unknown. For example, if you had a box with coloured balls in and picked one while looking at it, this isn't random. nether unlikely nor likely The probability of randomly se ecting a green marble from a bag of 20 marbles is. All Rights Reserved. Showing top 8 worksheets in the category - 1 20 Days Of Experimental And Theoretical. The theoretical probability of rolling an 8 is 5 times out of 36 rolls. experimental probability and theoretical probability of an event. 1 Intro of Probability. The probability of an event is a number from 0 to 1 that measures the chance that an event will occur. Press the graph key when you are finished. You need Adobe Acrobat Reader to view and print these documents. Competently Written Theoretical And Experimental Probability Worksheet Answers Focus. For example, the probability of the result of coin flip being ‘heads’ is 0. Theoretical and Experimental Probability. Stop searching. Experiment 2 illustrates the difference between an outcome and an event. If you're behind a web filter, please make sure that the domains *. 0 • Grade 7 • Volume 2 • Common Core • enVision • Algebra 2 • Common Core • Math in Focus • Singapore Math • Grade 5. Two worksheets, testing basic probability with dice, coloured balls and letters. DO NOT EAT any M&M's until the worksheet says you can. , tables, grids, tree diagrams) and express the theoretical probability of each outcome. Determine the phase margin and gain margin with MATLAB. What is the probability of tossing 3 heads in succession? b. The first student is a boy and the second student is a Independent Find the probability. 7th Grade Advanced Topic IV Probability, MA. High school math students can use these statistics problems for study purposes. Theoretical Probability Worksheet 2 – Here is a fourteen problem worksheet where you will calculate the theoretical probability of an event. Created: Nov 15, 2011. The ball lands in the garden or the pool. Theoretical probabilities are calculated based on the sample space generated by an analysis of the problem. Website Developed By WebCity Press. The second worksheet is more difficult and introduces sampling with and without replacement. Remember, the theoretical probability of a girl being born versus a boy being born is 1 : 2. Explain the difference between dependent events and independent events, and give an example of each. (1) (c) On the probability scale below, mark with a cross (×) the probability that you will get a head when you throw a coin. 306 Institutions have accepted or given pre-approval for credit transfer. Mathster is a fantastic resource for creating online and paper-based assessments and homeworks. experimental probability and theoretical probability of an event. Lesson 2 Homework Practice Theoretical and Experimental Probability 1. A simple event is an event where all possible outcomes are equally likely to occur. A number line is included to help students determine if an event is impossible, unlikely, equally likely, likely, or certain. It is the ratio of the number of successful outcomes and the total number of possible outcomes. Now we will examine. 2 is the probability of correctly guessing the symbol on any particular card, and n = 25 is the total number of trials in an experiment. In this worksheet, we will practice interpreting a data set by finding and evaluating the theoretical probabilities. We will study two types of probability, theoretical and experimental. The theoretical probability and experimental probability of an event are not necessarily the same. pdf from 31123SDA 13123 at South Craven School. Question 1c: Compare the theoretical probability and experimental probability. What would the theoretical probability of rolling a 6 be? Next, roll the die 24 times. Probability theory Worksheet 1 with Probability Worksheets with A Deck Of Cards. These Probability Worksheets are ideal for 4th Grade, 5th Grade, 6th Grade, and 7th Grade students. conditional probability is: P(A|B ) = P(A∩B) P(B), where ∩ means “and” Suppose, for example, that black and red checkers are placed in a bag. The various types of probability have very different practical applications; in some cases, theoretical probability would give you a less accurate result than empirical probability and vice versa. A box contains 7 red balls and 4 white balls. Homework 9-2 Experimental & Theoretical Probability Unit 9 First, use the table below to record the data from Worksheet 9-2, problem #5. WORKSHEET: Regents-Theoretical Probability AII/A2: 2/1: TST PDF DOC TNS. The theoretical probability of rolling a two is $1/6$, but in the experiment a two was rolled 4 out of the 10 times, making the experimental probability $2/5$. No credit card required. Displaying top 8 worksheets found for - Theoretical And Experimental Probability 191 Answer Key. 8 Mathematics Learner’s Module 11 Department of Education Republic of the Philippines This instructional material was collaboratively developed and reviewed by educators from public and private schools, colleges, and/or universities. 1 20 Days Of Experimental And Theoretical. List the members of the sample space. Answers in a pinch from experts and subject enthusiasts all semester long. If you're behind a web filter, please make sure that the domains *. 11-2 Practice (continued) Form G Probability 13. mathworksheets4kids. If event A is drawing a queen from a deck of cards and event B is drawing a king. Rolling a 1-6 on the first die and 1-6 on the second: _____ 5. 2 is the probability of correctly guessing the symbol on any particular card, and n = 25 is the total number of trials in an experiment. Grade 2 Probability. 1 Sample Spaces and Probability 539 The outcomes for a specifi ed event are called favorable outcomes. 2 Theoretical and Experimental Probability Notes Begin Homework. Experimental Probability. Theoretical Probability Worksheets- Includes math lessons, 2 practice sheets, homework sheet, and a quiz! Receive free math worksheets via email: Sort math worksheets by: Answer Keys Answers for all lessons and independent practice. All of us of innovative copy writers have extraordinary skills inside spoken and composed communication, which will translate so that you can the kind of written content you simply will not uncover just about anywhere else. It’s what is expected to happen, yet as we all know in life, the expected does not always happen. 04 g NH 3 = 12. Two worksheets, testing basic probability with dice, coloured balls and letters. They may calculate this by writing the number of favorable outcomes (3) over the total number of outcomes (6) to get 3/6, which should be reduced to lowest terms. Probability in pair of coin - 1. The answers for these pages appear at the back of this booklet. Today is Saturday. nether unlikely nor likely The probability of randomly se ecting a green marble from a bag of 20 marbles is. What is the probability of the spinner landing on a 2? 1 out of 6 4. Theoretical and Experimental Probability. Some of the worksheets for this concept are Study guide and intervention workbook, Prentice hall algebra 1, Mm science and math, Data management and probability grades 4 to 6, Glencoe mathematics grade 6, Chapter chapter test form a, Answer. 69% average accuracy. Experimental Probability When asked about the probability of a coin landing on heads, you would probably answer that the chance is ½ or 50%. P(5 heads) 19. You may stop by my room to pick it up ahead of time if you'd like. Worksheet Worksheet Answers Quiz. Experimental) Within this topic we learned about theoretical probability vs. Today's activity will begin to test theoretical activity. Displaying top 8 worksheets found for - Theoretical And Experimental Probability 191 Answer Key. Answers | Investigation 2 d. The outcome must be impossible (such as rolling a 7 on a number cube). Understand and apply basic concepts of probability. 7 Develop a probability model and use it to find probabilities of events. Probability and Odds Worksheet Key The probability of occurrences of any event can be shown on the number line below. True False 2. Theoretical probability does not require any experiments to conduct. com from SDA at South Craven School. Multiple-version printing. com There are 25 students in grade 5. 6 Probability Worksheets with Answers 2. Publisher Wiley ISBN 978-1-11853-971-2. Grade 2 Probability.
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https://www.opuscula.agh.edu.pl/om-vol39iss6art4
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Opuscula Math. 39, no. 6 (2019), 815-827
https://doi.org/10.7494/OpMath.2019.39.6.815
Opuscula Mathematica
# Graphs with equal domination and certified domination numbers
Magda Dettlaff
Magdalena Lemańska
Mateusz Miotk
Jerzy Topp
Paweł Żyliński
Abstract. A set $$D$$ of vertices of a graph $$G=(V_G,E_G)$$ is a dominating set of $$G$$ if every vertex in $$V_G-D$$ is adjacent to at least one vertex in $$D$$. The domination number (upper domination number, respectively) of $$G$$, denoted by $$\gamma(G)$$ ($$\Gamma(G)$$, respectively), is the cardinality of a smallest (largest minimal, respectively) dominating set of $$G$$. A subset $$D\subseteq V_G$$ is called a certified dominating set of $$G$$ if $$D$$ is a dominating set of $$G$$ and every vertex in $$D$$ has either zero or at least two neighbors in $$V_G-D$$. The cardinality of a smallest (largest minimal, respectively) certified dominating set of $$G$$ is called the certified (upper certified, respectively) domination number of $$G$$ and is denoted by $$\gamma_{\rm cer}(G)$$ ($$\Gamma_{\rm cer}(G)$$, respectively). In this paper relations between domination, upper domination, certified domination and upper certified domination numbers of a graph are studied.
Keywords: domination, certified domination.
Mathematics Subject Classification: 05C69.
Full text (pdf)
1. G.A. Cheston, G. Fricke, Classes of graphs for which upper fractional domination equals independence, upper domination, and upper irredundance, Discrete Appl. Math. 55 (1994), 241-258.
2. M. Dettlaff, M. Lemańska, J. Topp, R. Ziemann, P. Żyliński, Certified domination, AKCE Int. J. Graphs Comb., to appear, doi:10.1016/j.akcej.2018.09.004. https://doi.org/10.1016/j.akcej.2018.09.004
3. M. Fischermann, Block graphs with unique minimum dominating sets, Discrete Math. 240 (2001), 247-251.
4. R. Frucht, F. Harary, On the corona of two graphs, Aequ. Math. 4 (1970), 322-324.
5. G. Gunther, B. Hartnell, L.R. Markus, D. Rall, Graphs with unique minimum dominating sets, Congr. Numer. 101 (1994), 55-63.
6. T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Domination in Graphs: Advanced Topics, Marcel Dekker, New York, 1998.
7. T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, New York, 1998.
8. M.A. Henning, A. Yeo, Total Domination in Graphs, Springer-Verlag, New York, 2013.
• Communicated by Dalibor Fronček.
• Revised: 2019-07-26.
• Accepted: 2019-08-01.
• Published online: 2019-11-22.
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2019-12-13 21:06:33
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http://mathoverflow.net/questions?sort=featured
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# All Questions
107 views
+150
### How large do algebraic representations need to be for packing circles in squares?
(This question is inspired by Erich's Packing Center. I'm just asking about circles in squares to keep things simple, since I suspect any answer would apply just-as-well to the rest of the problems ...
157 views
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### Maximum occupancy balls in bins with limited independence
Throw $n$ balls into $n$ bins and let $X_n$ be the maximum occupancy. That is the maximum number of balls found in any bin. If you throw the balls uniformly and independently it is known that ...
### Morgan Shalen compactification of $\mathbb C^2$
I'm reading the Otal's survey on the compactification of Morgan Shalen. (available here) He claims in an example (page 8) that the compactification of $\mathbb C^2$ is $S^4$, which sounds completely ...
### Example of a ring $R$ such that $\dim(R[[X]])<\dim(R[X])$
Dimension refers to the Krull dimension of a commutative ring. In the paper "Prime ideals in power series rings" J. Arnold gives an example of such a ring: Let $k$ be a field and $K=k(t)$ a ...
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2015-04-01 03:22:25
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http://igurkul.org/aptitude-test-4-simple-interest/
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# OUR BLOG
February 2, 2017
## Aptitude Test-4-Simple Interest
Welcome to your Aptitude Test-4-Simple Interest
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1) A sum of Rs. 125000 amounts to Rs. 15500 in 4 years at the rate of simple interest. What is the rate of interest?
2) A certain sum of money is invested for one year at a certain rate of simple interest. If the rate of interest is 3% higher, then the invest earned will be 25% more than the interest earned earlier. What is the earlier rate of interest?
3) What is the present worth of Rs. 132 due in 2 years at 5% simple interest per annum?
4) A man took loan from a bank at the rate of 12% p.a. S.I. After 3 years he had to pay Rs. 5400 interest only for the period. The principal amount borrowed by him was?
5) A sum fetched a total simple interest of Rs. 4016.25 at the rate of 9% p.a. in 5 years. What is the sum?
6) A certain sum becomes four times itself at simple interest in eight years. In how many years does it become ten times itself?
7) Reena took a loan of Rs. 1200 with simple interest for as many years as the rate of interest. If she paid Rs. 432 as interest at the end of the loan period, what was the rate of interest?
8) How much interest can a person get on Rs. 8200 at 17.5% p.a. simple interest for a period of two years and six months?
9) Rs.4500 amounts to Rs.5544 in two years at compound interest, compounded annually. If the rate of the interest for the first year is 12%, find the rate of interest for the second year?
10) What will be the ratio of simple interest earned by certain amount at the same rate of interest for 6 years and that for 9 years
11) The simple interest accrued on an amount Rs.10,000 at the end of two years is same as the compound interest on Rs.8,000 at the end of two years. The rate of interest is same in both the cases. What is the rate of interest?
12) In how many years does a sum of Rs. 5000 yield a simple interest of Rs. 16500 at 15% p.a.?
13) A sum of money lent out at S.I. amounts to Rs. 720 after 2 years and to Rs. 1020 after a further period of 5 years. The sum is?
14) Rs. 800 becomes Rs. 956 in 3 years at a rate of S.I. If the rate of interest is increased by 4%, what amount will Rs. 800 become in 3 years?
15) An automobile financier claims to be lending money at S.I., but he includes the interest every six months for calculating the principal. If he is charging an interest of 10%, the effective rate of interest becomes?
16) Nitin borrowed some money at the rate of 6% p.a. for the first three years, 9% p.a. for the next five years and 13% p.a. for the period beyond eight years. If the total interest paid by him at the end of eleven years is Rs. 8160, how much money did he borrow?
17) A sum of money triples itself in twelve years at simple interest. Find the rate of interest?
18) Anil invested a sum of money at a certain rate of simple interest for a period of five years. Had he invested the sum for a period of eight years for the same rate, the total intrest earned by him would have been sixty percent more than the earlier interest amount. Find the rate of interest p.a.
19) How much time will take for an amount of Rs. 450 to yield Rs. 81 as interest at 4.5% per annum of simple interest?
20) At what rate of interest is an amount doubled in two years, when compounded annually?
21) What amount does Kiran get if he invests Rs. 18000 at 15% p.a. simple interest for four years?
22) Vijay lent out an amount Rs. 10000 into two parts, one at 8% p.a. and the remaining at 10% p.a. both on simple interest. At the end of the year he received Rs. 890 as total interest. What was the amount he lent out at 8% pa.a?
23) what rate percent of simple interest will a sum of money double itself in 12 years?
24) A certain amount earns simple interest of Rs. 1750 after 7 years. Had the interest been 2% more, how much more interest would it have earned?
25) An amount of Rs. 3000 becomes Rs. 3600 in four years at simple interest. If the rate of interest was 1% more, then what was be the total amount?
26) Manoj borrowed Rs.3450 from Anwar at 6% p.a. simple interest for three years. He then added some more money to the borrowed sum and lent it to Ramu for the same time at 9% p.a. simple interest. If Manoj gains Rs.824.85 by way of interest on the borrowed sum as well as his own amount from the whole transaction, then what is the sum lent by him to Ramu?
27) A certain sum is invested at simple interest at 18% p.a. for two years instead of investing at 12% p.a. for the same time period. Therefore the interest received is more by Rs. 840. Find the sum?
28) A certain sum becomes Rs. 20720 in four years and 24080 in six years at simple interest. Find sum and rate of interest?
29) A sum of money amounts to Rs. 9800 after 5 years and Rs. 12005 after 8 years at the same rate of simple interest. The rate of interest per annum is?
30) A sum of money at simple interest amounts to Rs. 815 in 3 years and to Rs. 854 in 4 years. The sum is?
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2018-09-21 22:09:20
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https://peeterjoot.wordpress.com/page/2/
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# Peeter Joot's (OLD) Blog.
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## Question: Quadratic Debye phonons (2013 midterm pr B2)
Assume a quadratic dispersion relation for the longitudinal and transverse modes
\begin{aligned}\omega = \left\{\begin{array}{l}b_{\mathrm{L}} q^2 \\ b_{\mathrm{T}} q^2\end{array}\right..\end{aligned} \hspace{\stretch{1}}(1.2)
### Part a
Find the density of states.
### Part b
Find the Debye frequency.
### Part c
In terms of $k_{\mathrm{B}} \Theta = \hbar \omega_{\mathrm{D}}$, and
\begin{aligned}\mathcal{I} = \int_0^\infty \frac{y^{5/2} e^{y} dy}{\left( { e^y - 1} \right)^2 },\end{aligned} \hspace{\stretch{1}}(1.2)
find the specific heat for $k_{\mathrm{B}} T \ll \hbar \omega_{\mathrm{D}}$.
### Part d
Find the specific heat for $k_{\mathrm{B}} T \gg \hbar \omega_{\mathrm{D}}$.
### Part a
Working straight from the definition
\begin{aligned}Z(\omega) &= \frac{V}{(2 \pi)^3 } \sum_{L, T} \int \frac{df_\omega}{ \left\lvert { \boldsymbol{\nabla}_\mathbf{q} \omega } \right\rvert } \\ &= \frac{V}{(2 \pi)^3 } \left( { {\left.{{\frac{4 \pi q^2}{2 b_{\mathrm{L}} q} }}\right\vert}_{{\mathrm{L}}} + {\left.{{\frac{2 \times 4 \pi q^2}{2 b_{\mathrm{T}} q} }}\right\vert}_{{\mathrm{T}}} } \right) \\ &= \frac{V}{4 \pi^2 } \left( { \frac{q_{\mathrm{L}}}{b_{\mathrm{L}}} + \frac{2 q_{\mathrm{T}}}{b_{\mathrm{T}}} } \right).\end{aligned} \hspace{\stretch{1}}(1.3)
With $q_{\mathrm{L}} = \sqrt{\omega/b_{\mathrm{L}}}$ and $q_{\mathrm{T}} = \sqrt{\omega/b_{\mathrm{T}}}$, this is
\begin{aligned}Z(\omega) = \frac{V}{4 \pi^2 } \left( { \frac{1}{b_{\mathrm{L}}^{3/2}} + \frac{2}{b_{\mathrm{T}}^{3/2}} } \right)\sqrt{\omega}\end{aligned} \hspace{\stretch{1}}(1.4)
### Part b
The Debye frequency was given implicitly by
\begin{aligned}\int_0^{\omega_{\mathrm{D}}} Z(\omega) d\omega = 3 r N,\end{aligned} \hspace{\stretch{1}}(1.5)
which gives
\begin{aligned}3 r N=\frac{2}{3} \frac{V}{4 \pi^2 } \left( { \frac{1}{b_{\mathrm{L}}^{3/2}} + \frac{2}{b_{\mathrm{T}}^{3/2}} } \right)\omega_{\mathrm{D}}^{3/2}=\frac{V}{6 \pi^2 } \left( { \frac{1}{b_{\mathrm{L}}^{3/2}} + \frac{2}{b_{\mathrm{T}}^{3/2}} } \right)\omega_{\mathrm{D}}^{3/2}\end{aligned} \hspace{\stretch{1}}(1.6)
### Part c
Assuming a Bose distribution and ignoring the zero point energy, which has no temperature dependence, the specific heat, the temperature derivative of the energy density, is
\begin{aligned}C_{\mathrm{V}} &= \frac{d}{d T} \frac{1}{{V}} \int Z(\omega) \frac{\hbar \omega}{ e^{\hbar \omega/ k_{\mathrm{B}} T } - 1} d\omega \\ &= \frac{1}{{V}} \frac{d}{d T} \int Z(\omega) \frac{\hbar \omega}{ \hbar \omega/ k_{\mathrm{B}} T + \frac{1}{{2}}( \hbar \omega/k_{\mathrm{B}} T)^2 + \cdots } d\omega \\ &\approx \frac{1}{{V}} \frac{d}{d T} \int Z(\omega) k_{\mathrm{B}} T d\omega \\ &= \frac{1}{{V}} k_{\mathrm{B}} 3 r N.\end{aligned} \hspace{\stretch{1}}(1.7)
### Part d
First note that the density of states can be written
\begin{aligned}Z(\omega) = \frac{9 r N}{ 2 \omega_{\mathrm{D}}^{3/2} } \omega^{1/2},\end{aligned} \hspace{\stretch{1}}(1.8)
for a specific heat of
\begin{aligned}C_{\mathrm{V}} &= \frac{d}{d T} \frac{1}{{V}} \int_0^\infty \frac{9 r N}{ 2 \omega_{\mathrm{D}}^{3/2} } \omega^{1/2} \frac{\hbar \omega}{ e^{\hbar \omega/ k_{\mathrm{B}} T } - 1} d\omega \\ &= \frac{9 r N}{ 2 V \omega_{\mathrm{D}}^{3/2} } \int_0^\infty d\omega \omega^{1/2} \frac{d}{d T} \frac{\hbar \omega}{ e^{\hbar \omega/ k_{\mathrm{B}} T } - 1} \\ &= \frac{9 r N}{ 2 V \omega_{\mathrm{D}}^{3/2} } \int_0^\infty d\omega \omega^{1/2} \frac{-\hbar \omega}{ \left( {e^{\hbar \omega/ k_{\mathrm{B}} T } - 1} \right)^2 } e^{\hbar \omega/k_{\mathrm{B}} T} \hbar \omega/k_{\mathrm{B}} \left( {-\frac{1}{{T^2}}} \right) \\ &= \frac{9 r N k_{\mathrm{B}} }{ 2 V \omega_{\mathrm{D}}^{3/2} } \left( { \frac{ k_{\mathrm{B}} T}{\hbar} } \right)^{3/2}\int_0^\infty d \frac{\hbar \omega}{k_{\mathrm{B}} T} \left( {\frac{\hbar \omega}{k_{\mathrm{B}} T}} \right)^{1/2} \frac{1}{ \left( {e^{\hbar \omega/ k_{\mathrm{B}} T } - 1} \right)^2 } e^{\hbar \omega/k_{\mathrm{B}} T} \left( { \frac{\hbar \omega}{k_{\mathrm{B}} T} } \right)^2 \\ &= \frac{9 r N k_{\mathrm{B}} }{ 2 V \omega_{\mathrm{D}}^{3/2} } \left( { \frac{ k_{\mathrm{B}} T}{\hbar} } \right)^{3/2}\int_0^\infty dy \frac{y^{5/2} e^y }{ \left( {e^y - 1} \right)^2 } \\& = \frac{9 r N k_{\mathrm{B}} }{ 2 V } \left( { \frac{ T}{\Theta} } \right)^{3/2} \mathcal{I}.\end{aligned} \hspace{\stretch{1}}(1.9)
## One atom basis phonons in 2D
Posted by peeterjoot on January 19, 2014
Let’s tackle a problem like the 2D problem of the final exam, but first more generally. Instead of a square lattice consider the lattice with the geometry illustrated in fig. 1.1.
Fig 1.1: Oblique one atom basis
Here, $\mathbf{a}$ and $\mathbf{b}$ are the vector differences between the equilibrium positions separating the masses along the $K_1$ and $K_2$ interaction directions respectively. The equilibrium spacing for the cross coupling harmonic forces are
\begin{aligned}\begin{aligned}\mathbf{r} &= (\mathbf{b} + \mathbf{a})/2 \\ \mathbf{s} &= (\mathbf{b} - \mathbf{a})/2.\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.1)
Based on previous calculations, we can write the equations of motion by inspection
\begin{aligned}\begin{aligned}m \dot{d}{\mathbf{u}}_\mathbf{n} = &-K_1 \text{Proj}_{\hat{\mathbf{a}}} \sum_\pm \left( { \mathbf{u}_\mathbf{n} - \mathbf{u}_{\mathbf{n} \pm(1, 0)}} \right)^2 \\ &-K_2 \text{Proj}_{\hat{\mathbf{b}}} \sum_\pm \left( { \mathbf{u}_\mathbf{n} - \mathbf{u}_{\mathbf{n} \pm(0, 1)}} \right)^2 \\ &-K_3 \text{Proj}_{\hat{\mathbf{r}}} \sum_\pm \left( { \mathbf{u}_\mathbf{n} - \mathbf{u}_{\mathbf{n} \pm(1, 1)}} \right)^2 \\ &-K_4 \text{Proj}_{\hat{\mathbf{s}}} \sum_\pm \left( { \mathbf{u}_\mathbf{n} - \mathbf{u}_{\mathbf{n} \pm(1, -1)}} \right)^2.\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.2)
Inserting the trial solution
\begin{aligned}\mathbf{u}_\mathbf{n} = \frac{1}{{\sqrt{m}}} \boldsymbol{\epsilon}(\mathbf{q}) e^{i( \mathbf{r}_\mathbf{n} \cdot \mathbf{q} - \omega t) },\end{aligned} \hspace{\stretch{1}}(1.3)
and using the matrix form for the projection operators, we have
\begin{aligned}\begin{aligned}\omega^2 \boldsymbol{\epsilon} &=\frac{K_1}{m} \hat{\mathbf{a}} \hat{\mathbf{a}}^\text{T} \boldsymbol{\epsilon}\sum_\pm\left( { 1 - e^{\pm i \mathbf{a} \cdot \mathbf{q}} } \right) \\ & +\frac{K_2}{m} \hat{\mathbf{b}} \hat{\mathbf{b}}^\text{T} \boldsymbol{\epsilon}\sum_\pm\left( { 1 - e^{\pm i \mathbf{b} \cdot \mathbf{q}} } \right) \\ & +\frac{K_3}{m} \hat{\mathbf{b}} \hat{\mathbf{b}}^\text{T} \boldsymbol{\epsilon}\sum_\pm\left( { 1 - e^{\pm i (\mathbf{b} + \mathbf{a}) \cdot \mathbf{q}} } \right) \\ & +\frac{K_3}{m} \hat{\mathbf{b}} \hat{\mathbf{b}}^\text{T} \boldsymbol{\epsilon}\sum_\pm\left( { 1 - e^{\pm i (\mathbf{b} - \mathbf{a}) \cdot \mathbf{q}} } \right) \\ &=\frac{4 K_1}{m} \hat{\mathbf{a}} \hat{\mathbf{a}}^\text{T} \boldsymbol{\epsilon} \sin^2\left( { \mathbf{a} \cdot \mathbf{q}/2 } \right)+\frac{4 K_2}{m} \hat{\mathbf{b}} \hat{\mathbf{b}}^\text{T} \boldsymbol{\epsilon} \sin^2\left( { \mathbf{b} \cdot \mathbf{q}/2 } \right) \\ &+\frac{4 K_3}{m} \hat{\mathbf{r}} \hat{\mathbf{r}}^\text{T} \boldsymbol{\epsilon} \sin^2\left( { (\mathbf{b} + \mathbf{a}) \cdot \mathbf{q}/2 } \right)+\frac{4 K_4}{m} \hat{\mathbf{s}} \hat{\mathbf{s}}^\text{T} \boldsymbol{\epsilon} \sin^2\left( { (\mathbf{b} - \mathbf{a}) \cdot \mathbf{q}/2 } \right).\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.4)
This fully specifies our eigenvalue problem. Writing
\begin{aligned}\begin{aligned}S_1 &= \sin^2\left( { \mathbf{a} \cdot \mathbf{q}/2 } \right) \\ S_2 &= \sin^2\left( { \mathbf{b} \cdot \mathbf{q}/2 } \right) \\ S_3 &= \sin^2\left( { (\mathbf{b} + \mathbf{a}) \cdot \mathbf{q}/2 } \right) \\ S_4 &= \sin^2\left( { (\mathbf{b} - \mathbf{a}) \cdot \mathbf{q}/2 } \right)\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.5.5)
\begin{aligned}\boxed{A = \frac{4}{m}\left( { K_1 S_1 \hat{\mathbf{a}} \hat{\mathbf{a}}^\text{T} + K_2 S_2 \hat{\mathbf{b}} \hat{\mathbf{b}}^\text{T} + K_3 S_3 \hat{\mathbf{r}} \hat{\mathbf{r}}^\text{T} + K_4 S_4 \hat{\mathbf{s}} \hat{\mathbf{s}}^\text{T}} \right),}\end{aligned} \hspace{\stretch{1}}(1.0.5.5)
we wish to solve
\begin{aligned}A \boldsymbol{\epsilon} = \omega^2 \boldsymbol{\epsilon} = \lambda \boldsymbol{\epsilon}.\end{aligned} \hspace{\stretch{1}}(1.0.6)
Neglecting the specifics of the matrix at hand, consider a generic two by two matrix
\begin{aligned}A = \begin{bmatrix}a & b \\ c & d\end{bmatrix},\end{aligned} \hspace{\stretch{1}}(1.0.6)
for which the characteristic equation is
\begin{aligned}0 &= \begin{vmatrix}\lambda - a & - b \\ -c & \lambda -d \end{vmatrix} \\ &= (\lambda - a)(\lambda - d) - b c \\ &= \lambda^2 - (a + d) \lambda + a d - b c \\ &= \lambda^2 - (Tr A) \lambda + \left\lvert {A} \right\rvert \\ &= \left( {\lambda - \frac{Tr A}{2}} \right)^2- \left( {\frac{Tr A}{2}} \right)^2 + \left\lvert {A} \right\rvert.\end{aligned} \hspace{\stretch{1}}(1.0.6)
So our angular frequencies are given by
\begin{aligned}\omega^2 = \frac{1}{{2}} \left( { Tr A \pm \sqrt{ \left(Tr A\right)^2 - 4 \left\lvert {A} \right\rvert }} \right).\end{aligned} \hspace{\stretch{1}}(1.0.6)
The square root can be simplified slightly
\begin{aligned}\left( {Tr A} \right)^2 - 4 \left\lvert {A} \right\rvert \\ &= (a + d)^2 -4 (a d - b c) \\ &= a^2 + d^2 + 2 a d - 4 a d + 4 b c \\ &= (a - d)^2 + 4 b c,\end{aligned} \hspace{\stretch{1}}(1.0.6)
so that, finally, the dispersion relation is
\begin{aligned}\boxed{\omega^2 = \frac{1}{{2}} \left( { d + a \pm \sqrt{ (d - a)^2 + 4 b c } } \right),}\end{aligned} \hspace{\stretch{1}}(1.0.6)
Our eigenvectors will be given by
\begin{aligned}0 = (\lambda - a) \boldsymbol{\epsilon}_1 - b\boldsymbol{\epsilon}_2,\end{aligned} \hspace{\stretch{1}}(1.0.6)
or
\begin{aligned}\boldsymbol{\epsilon}_1 \propto \frac{b}{\lambda - a}\boldsymbol{\epsilon}_2.\end{aligned} \hspace{\stretch{1}}(1.0.6)
So, our eigenvectors, the vectoral components of our atomic displacements, are
\begin{aligned}\boldsymbol{\epsilon} \propto\begin{bmatrix}b \\ \omega^2 - a\end{bmatrix},\end{aligned} \hspace{\stretch{1}}(1.0.6)
or
\begin{aligned}\boxed{\boldsymbol{\epsilon} \propto\begin{bmatrix}2 b \\ d - a \pm \sqrt{ (d - a)^2 + 4 b c }\end{bmatrix}.}\end{aligned} \hspace{\stretch{1}}(1.0.6)
Square lattice
There is not too much to gain by expanding out the projection operators explicitly in general. However, let’s do this for the specific case of a square lattice (as on the exam problem). In that case, our projection operators are
\begin{aligned}\hat{\mathbf{a}} \hat{\mathbf{a}}^\text{T} = \begin{bmatrix}1 \\ 0\end{bmatrix}\begin{bmatrix}1 & 0\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(1.0.16a)
\begin{aligned}\hat{\mathbf{b}} \hat{\mathbf{b}}^\text{T} = \begin{bmatrix}0\\ 1 \end{bmatrix}\begin{bmatrix}0 &1 \end{bmatrix}=\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(1.0.16b)
\begin{aligned}\hat{\mathbf{r}} \hat{\mathbf{r}}^\text{T} = \frac{1}{{2}}\begin{bmatrix}1 \\ 1 \end{bmatrix}\begin{bmatrix}1 &1 \end{bmatrix}=\frac{1}{{2}}\begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(1.0.16c)
\begin{aligned}\hat{\mathbf{s}} \hat{\mathbf{s}}^\text{T} = \frac{1}{{2}}\begin{bmatrix}-1 \\ 1 \end{bmatrix}\begin{bmatrix}-1 &1 \end{bmatrix}=\frac{1}{{2}}\begin{bmatrix}1 & -1 \\ -1 & 1\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(1.0.16d)
\begin{aligned}\begin{aligned}S_1 &= \sin^2\left( { \mathbf{a} \cdot \mathbf{q} } \right) \\ S_2 &= \sin^2\left( { \mathbf{b} \cdot \mathbf{q} } \right) \\ S_3 &= \sin^2\left( { (\mathbf{b} + \mathbf{a}) \cdot \mathbf{q} } \right) \\ S_4 &= \sin^2\left( { (\mathbf{b} - \mathbf{a}) \cdot \mathbf{q} } \right),\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.16d)
Our matrix is
\begin{aligned}A = \frac{2}{m}\begin{bmatrix}2 K_1 S_1 + K_3 S_3 + K_4 S_4 & K_3 S_3 - K_4 S_4 \\ K_3 S_3 - K_4 S_4 & 2 K_2 S_2 + K_3 S_3 + K_4 S_4\end{bmatrix},\end{aligned} \hspace{\stretch{1}}(1.0.16d)
where, specifically, the squared sines for this geometry are
\begin{aligned}S_1 = \sin^2 \left( { \mathbf{a} \cdot \mathbf{q}/2 } \right) = \sin^2 \left( { a q_x/2} \right)\end{aligned} \hspace{\stretch{1}}(1.0.19a)
\begin{aligned}S_2 = \sin^2 \left( { \mathbf{b} \cdot \mathbf{q}/2 } \right) = \sin^2 \left( { a q_y/2} \right)\end{aligned} \hspace{\stretch{1}}(1.0.19b)
\begin{aligned}S_3 = \sin^2 \left( { (\mathbf{b} + \mathbf{a}) \cdot \mathbf{q}/2 } \right) = \sin^2 \left( { a (q_x + q_y)/2} \right)\end{aligned} \hspace{\stretch{1}}(1.0.19c)
\begin{aligned}S_4 = \sin^2 \left( { (\mathbf{b} - \mathbf{a}) \cdot \mathbf{q}/2 } \right) = \sin^2 \left( { a (q_y - q_x)/2} \right).\end{aligned} \hspace{\stretch{1}}(1.0.19d)
Using eq. 1.0.6, the dispersion relation and eigenvectors are
\begin{aligned}\omega^2 = \frac{2}{m} \left( { \sum_i K_i S_i \pm \sqrt{ (K_2 S_2 - K_1 S_1)^2 + (K_3 S_3 - K_4 S_4)^2 } } \right)\end{aligned} \hspace{\stretch{1}}(1.0.20.20)
\begin{aligned}\boldsymbol{\epsilon} \propto\begin{bmatrix}K_3 S_3 - K_4 S_4 \\ K_2 S_2 - K_1 S_1 \pm \sqrt{ (K_2 S_2 - K_1 S_1)^2 + (K_3 S_3 - K_4 S_4)^2 } \end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(1.0.20.20)
This calculation is confirmed in oneAtomBasisPhononSquareLatticeEigensystem.nb. Mathematica calculates an alternate form (equivalent to using a zero dot product for the second row), of
\begin{aligned}\boldsymbol{\epsilon} \propto\begin{bmatrix}K_1 S_1 - K_2 S_2 \pm \sqrt{ (K_2 S_2 - K_1 S_1)^2 + (K_3 S_3 - K_4 S_4)^2 } \\ K_3 S_3 - K_4 S_4 \end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(1.0.20.20)
Either way, we see that $K_3 S_3 - K_4 S_4 = 0$ leads to only horizontal or vertical motion.
With the exam criteria
In the specific case that we had on the exam where $K_1 = K_2$ and $K_3 = K_4$, these are
\begin{aligned}\omega^2 = \frac{2}{m} \left( { K_1 (S_1 + S_2) + K_3(S_3 + S_4) \pm \sqrt{ K_1^2 (S_2 - S_1)^2 + K_3^2 (S_3 - S_4)^2 } } \right)\end{aligned} \hspace{\stretch{1}}(1.0.22.22)
\begin{aligned}\boldsymbol{\epsilon} \propto\begin{bmatrix}K_3 \left( { S_3 - S_4 } \right) \\ K_1 \left( { (S_1 - S_2) \pm \sqrt{ (S_2 - S_1)^2 + \left( \frac{K_3}{K_1} \right)^2 (S_3 - S_4)^2 } } \right)\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(1.0.22.22)
For horizontal and vertical motion we need $S_3 = S_4$, or for a $2 \pi \times \text{integer}$ difference in the absolute values of the sine arguments
\begin{aligned}\pm ( a (q_x + q_y) /2 ) = a (q_y - q_y) /2 + 2 \pi n.\end{aligned} \hspace{\stretch{1}}(1.0.22.22)
That is, one of
\begin{aligned}\begin{aligned}q_x &= \frac{2 \pi}{a} n \\ q_y &= \frac{2 \pi}{a} n\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.22.22)
In the first BZ, that is one of $q_x = 0$ or $q_y = 0$.
System in rotated coordinates
On the exam, where we were asked to solve for motion along the cross directions explicitly, there was a strong hint to consider a rotated (by $\pi/4$) coordinate system.
The rotated the lattice basis vectors are $\mathbf{a} = a \mathbf{e}_1, \mathbf{b} = a \mathbf{e}_2$, and the projection matrices. Writing $\hat{\mathbf{r}} = \mathbf{f}_1$ and $\hat{\mathbf{s}} = \mathbf{f}_2$, where $\mathbf{f}_1 = (\mathbf{e}_1 + \mathbf{e}_2)/\sqrt{2}, \mathbf{f}_2 = (\mathbf{e}_2 - \mathbf{e}_1)/\sqrt{2}$, or $\mathbf{e}_1 = (\mathbf{f}_1 - \mathbf{f}_2)/\sqrt{2}, \mathbf{e}_2 = (\mathbf{f}_1 + \mathbf{f}_2)/\sqrt{2}$. In the $\{\mathbf{f}_1, \mathbf{f}_2\}$ basis the projection matrices are
\begin{aligned}\hat{\mathbf{a}} \hat{\mathbf{a}}^\text{T} = \frac{1}{{2}}\begin{bmatrix}1 \\ -1\end{bmatrix}\begin{bmatrix}1 & -1\end{bmatrix}= \frac{1}{{2}} \begin{bmatrix}1 & -1 \\ -1 & 1\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(1.0.25a)
\begin{aligned}\hat{\mathbf{b}} \hat{\mathbf{b}}^\text{T} = \frac{1}{{2}}\begin{bmatrix}1 \\ 1\end{bmatrix}\begin{bmatrix}1 & 1\end{bmatrix}= \frac{1}{{2}} \begin{bmatrix}1 & 1 \\ 1 & 1\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(1.0.25b)
\begin{aligned}\hat{\mathbf{r}} \hat{\mathbf{r}}^\text{T} = \begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(1.0.25c)
\begin{aligned}\hat{\mathbf{s}} \hat{\mathbf{s}}^\text{T} = \begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(1.0.25d)
The dot products that show up in the squared sines are
\begin{aligned}\mathbf{a} \cdot \mathbf{q}=a \frac{1}{{\sqrt{2}}} (\mathbf{f}_1 - \mathbf{f}_2) \cdot (\mathbf{f}_1 k_u + \mathbf{f}_2 k_v)=\frac{a}{\sqrt{2}} (k_u - k_v)\end{aligned} \hspace{\stretch{1}}(1.0.26a)
\begin{aligned}\mathbf{b} \cdot \mathbf{q}=a \frac{1}{{\sqrt{2}}} (\mathbf{f}_1 + \mathbf{f}_2) \cdot (\mathbf{f}_1 k_u + \mathbf{f}_2 k_v)=\frac{a}{\sqrt{2}} (k_u + k_v)\end{aligned} \hspace{\stretch{1}}(1.0.26b)
\begin{aligned}(\mathbf{a} + \mathbf{b}) \cdot \mathbf{q} = \sqrt{2} a k_u \end{aligned} \hspace{\stretch{1}}(1.0.26c)
\begin{aligned}(\mathbf{b} - \mathbf{a}) \cdot \mathbf{q} = \sqrt{2} a k_v \end{aligned} \hspace{\stretch{1}}(1.0.26d)
So that in this basis
\begin{aligned}\begin{aligned}S_1 &= \sin^2 \left( { \frac{a}{\sqrt{2}} (k_u - k_v) } \right) \\ S_2 &= \sin^2 \left( { \frac{a}{\sqrt{2}} (k_u + k_v) } \right) \\ S_3 &= \sin^2 \left( { \sqrt{2} a k_u } \right) \\ S_4 &= \sin^2 \left( { \sqrt{2} a k_v } \right)\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.26d)
With the rotated projection operators eq. 1.0.5.5 takes the form
\begin{aligned}A = \frac{2}{m}\begin{bmatrix}K_1 S_1 + K_2 S_2 + 2 K_3 S_3 & K_2 S_2 - K_1 S_1 \\ K_2 S_2 - K_1 S_1 & K_1 S_1 + K_2 S_2 + 2 K_4 S_4\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(1.0.26d)
This clearly differs from eq. 1.0.16d, and results in a different expression for the eigenvectors, but the same as eq. 1.0.20.20 for the angular frequencies.
\begin{aligned}\boldsymbol{\epsilon} \propto\begin{bmatrix}K_2 S_2 - K_1 S_1 \\ K_4 S_4 - K_3 S_3 \mp \sqrt{ (K_2 S_2 - K_1 S_1)^2 + (K_3 S_3 - K_4 S_4)^2 }\end{bmatrix},\end{aligned} \hspace{\stretch{1}}(1.0.26d)
or, equivalently
\begin{aligned}\boldsymbol{\epsilon} \propto\begin{bmatrix}K_4 S_4 - K_3 S_3 \mp \sqrt{ (K_2 S_2 - K_1 S_1)^2 + (K_3 S_3 - K_4 S_4)^2 } \\ K_1 S_1 - K_2 S_2 \\ \end{bmatrix},\end{aligned} \hspace{\stretch{1}}(1.0.26d)
For the $K_1 = K_2$ and $K_3 = K_4$ case of the exam, this is
\begin{aligned}\boldsymbol{\epsilon} \propto\begin{bmatrix}K_1 (S_2 - S_1 ) \\ K_3 \left( { S_4 - S_3 \mp \sqrt{ \left( \frac{K_1}{K_3} \right)^2 (S_2 - S_1)^2 + (S_3 - S_4)^2 } } \right)\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(1.0.26d)
Similar to the horizontal coordinate system, we see that we have motion along the diagonals when
\begin{aligned}\pm \frac{a}{\sqrt{2}} (k_u - k_v) = \frac{a}{\sqrt{2}} (k_u + k_v) + 2 \pi n,\end{aligned} \hspace{\stretch{1}}(1.0.26d)
or one of
\begin{aligned}\begin{aligned}k_u &= \sqrt{2} \frac{\pi}{a} n \\ k_v &= \sqrt{2} \frac{\pi}{a} n\end{aligned}\end{aligned} \hspace{\stretch{1}}(1.0.26d)
Stability?
The exam asked why the cross coupling is required for stability. Clearly we have more complex interaction. The constant $\omega$ surfaces will also be more complex. However, I still don’t have a good intuition what exactly was sought after for that part of the question.
Numerical computations
A Manipulate allowing for choice of the spring constants and lattice orientation, as shown in fig. 1.2, is available in phy487/oneAtomBasisPhonon.nb. This interface also provides a numerical calculation of the distribution relation as shown in fig. 1.3, and provides an animation of the normal modes for any given selection of $\mathbf{q}$ and $\omega(\mathbf{q})$ (not shown).
Fig 1.2: 2D Single atom basis Manipulate interface
Fig 1.3: Sample distribution relation for 2D single atom basis.
## Two body harmonic oscillator in 3D, and 2D diamond lattice vibrations
Posted by peeterjoot on January 7, 2014
# Abridged harmonic oscillator notes
[This is an abbreviation of more extensive PDF notes associated with the latter part of this post.]
# Motivation and summary of harmonic oscillator background
After having had some trouble on a non-1D harmonic oscillator lattice problem on the exam, I attempted such a problem with enough time available to consider it properly. I found it helpful to consider first just two masses interacting harmonically in 3D, each displaced from an equilibrium position.
The Lagrangian that described this most naturally was found to be
\begin{aligned}\mathcal{L} = \frac{1}{2} m_1 \left( \dot{\mathbf{r}}_1 \right)^2+\frac{1}{2} m_2 \left( \dot{\mathbf{r}}_2 \right)^2- \frac{K}{2} \left( \left\lvert {\mathbf{r}_2 - \mathbf{r}_1} \right\rvert - a \right)^2.\end{aligned} \hspace{\stretch{1}}(2.1)
This was solved in absolute and displacement coordinates, and then I moved on to consider a linear expansion of the harmonic potential about the equilibrium point, a problem closer to the exam problem (albeit still considering only two masses). The equilibrium points were described with vectors $\mathbf{a}_1, \mathbf{a}_2$ as in fig. 2.1, where $\Delta \mathbf{a} = \left\lvert {\Delta \mathbf{a}} \right\rvert (\cos \theta_1, \cos\theta_2, \cos\theta_3)$.
fig 2.1: Direction cosines relative to equilibrium position difference vector
Using such coordinates, and generalizing, it was found that the complete Lagrangian, to second order about the equilibrium positions, is
\begin{aligned}\mathcal{L} = \sum_j \frac{m_i}{2} \dot{u}_{ij}^2 -\frac{K}{2} \sum_{i j} \cos\theta_i \cos\theta_j \left( u_{2 i} - u_{1 i} \right)\left( u_{2 j} - u_{1 j} \right).\end{aligned} \hspace{\stretch{1}}(2.2)
Evaluating the Euler-Lagrange equations, the equations of motion for the displacements were found to be
\begin{aligned}\begin{aligned}m_1 \ddot{\mathbf{u}}_1 &= K \widehat{\Delta \mathbf{a}} \left( \widehat{\Delta \mathbf{a}} \cdot \Delta \mathbf{u} \right) \\ m_2 \ddot{\mathbf{u}}_2 &= -K \widehat{\Delta \mathbf{a}} \left( \widehat{\Delta \mathbf{a}} \cdot \Delta \mathbf{u} \right),\end{aligned}\end{aligned} \hspace{\stretch{1}}(2.3)
or
\begin{aligned}\boxed{\begin{aligned}\mu \Delta \ddot{\mathbf{u}} &= -K \widehat{\Delta \mathbf{a}} \left( \widehat{\Delta \mathbf{a}} \cdot \Delta \mathbf{u} \right) \\ m_1 \ddot{\mathbf{u}}_1 + m_2 \ddot{\mathbf{u}}_2 &= 0.\end{aligned}}\end{aligned} \hspace{\stretch{1}}(2.4)
Observe that on the RHS above we have a projection operator, so we could also write
\begin{aligned}\mu \Delta \ddot{\mathbf{u}} = -K \text{Proj}_{\widehat{\Delta \mathbf{a}}} \Delta \mathbf{u}.\end{aligned} \hspace{\stretch{1}}(2.5)
We see that the equations of motion for the displacements of a system of harmonic oscillators has a rather pleasant expression in terms of projection operators, where we have projections onto the unit vectors between each pair of equilibrium position.
# A number of harmonically coupled masses
Now let’s consider masses at lattice points indexed by a lattice vector $\mathbf{n}$, as illustrated in fig. 2.2.
fig 2.2: Masses harmonically coupled in a lattice
With a coupling constant of $K_{\mathbf{n} \mathbf{m}}$ between lattice points indexed $\mathbf{n}$ and $\mathbf{m}$ (located at $\mathbf{a}_\mathbf{n}$ and $\mathbf{a}_\mathbf{m}$ respectively), and direction cosines for the equilibrium direction vector between those points given by
\begin{aligned}\mathbf{a}_\mathbf{n} - \mathbf{a}_\mathbf{m} = \Delta \mathbf{a}_{\mathbf{n} \mathbf{m}}= \left\lvert {\Delta \mathbf{a}_{\mathbf{n} \mathbf{m}}} \right\rvert (\cos \theta_{\mathbf{n} \mathbf{m} 1},\cos \theta_{\mathbf{n} \mathbf{m} 2},\cos \theta_{\mathbf{n} \mathbf{m} 3}),\end{aligned} \hspace{\stretch{1}}(2.6)
the Lagrangian is
\begin{aligned}\mathcal{L} = \sum_{\mathbf{n}, i} \frac{m_\mathbf{n}}{2} \dot{u}_{\mathbf{n} i}^2-\frac{1}{2} \sum_{\mathbf{n} \ne \mathbf{m}, i, j} \frac{K_{\mathbf{n} \mathbf{m}}}{2} \cos\theta_{\mathbf{n} \mathbf{m} i}\cos\theta_{\mathbf{n} \mathbf{m} j}\left( u_{\mathbf{n} i} - u_{\mathbf{m} i} \right)\left( u_{\mathbf{n} j} - u_{\mathbf{m} j} \right)\end{aligned} \hspace{\stretch{1}}(2.7)
Evaluating the Euler-Lagrange equations for the mass at index $\mathbf{n}$ we have
\begin{aligned}\frac{d}{dt} \frac{\partial {\mathcal{L}}}{\partial {\dot{u}_{\mathbf{n} k}}} =m_\mathbf{n} \ddot{u}_{\mathbf{n} k},\end{aligned} \hspace{\stretch{1}}(2.8)
and
\begin{aligned}\frac{\partial {\mathcal{L}}}{\partial {u_{\mathbf{n} k}}} &= -\sum_{\mathbf{m}, i, j}\frac{K_{\mathbf{n} \mathbf{m}}}{2} \cos\theta_{\mathbf{n} \mathbf{m} i}\cos\theta_{\mathbf{n} \mathbf{m} j}\left(\delta_{i k}\left( u_{\mathbf{n} j} - u_{\mathbf{m} j} \right)+\left( u_{\mathbf{n} i} - u_{\mathbf{m} i} \right)\delta_{j k}\right) \\ &= -\sum_{\mathbf{m}, i}K_{\mathbf{n} \mathbf{m}}\cos\theta_{\mathbf{n} \mathbf{m} k}\cos\theta_{\mathbf{n} \mathbf{m} i}\left( u_{\mathbf{n} i} - u_{\mathbf{m} i} \right) \\ &= -\sum_{\mathbf{m}}K_{\mathbf{n} \mathbf{m}}\cos\theta_{\mathbf{n} \mathbf{m} k}\widehat{\Delta \mathbf{a}} \cdot \Delta \mathbf{u}_{\mathbf{n} \mathbf{m}},\end{aligned} \hspace{\stretch{1}}(2.9)
where $\Delta \mathbf{u}_{\mathbf{n} \mathbf{m}} = \mathbf{u}_\mathbf{n} - \mathbf{u}_\mathbf{m}$. Equating both, we have in vector form
\begin{aligned}m_\mathbf{n} \ddot{\mathbf{u}}_\mathbf{n} = -\sum_{\mathbf{m}}K_{\mathbf{n} \mathbf{m}}\widehat{\Delta \mathbf{a}}\left( \widehat{\Delta \mathbf{a}} \cdot \Delta \mathbf{u}_{\mathbf{n} \mathbf{m}} \right),\end{aligned} \hspace{\stretch{1}}(2.10)
or
\begin{aligned}\boxed{m_\mathbf{n} \ddot{\mathbf{u}}_\mathbf{n} = -\sum_{\mathbf{m}}K_{\mathbf{n} \mathbf{m}}\text{Proj}_{ \widehat{\Delta \mathbf{a}} } \Delta \mathbf{u}_{\mathbf{n} \mathbf{m}},}\end{aligned} \hspace{\stretch{1}}(2.11)
This is an intuitively pleasing result. We have displacement and the direction of the lattice separations in the mix, but not the magnitude of the lattice separation itself.
# Two atom basis, 2D diamond lattice
As a concrete application of the previously calculated equilibrium harmonic oscillator result, let’s consider a two atom basis diamond lattice where the horizontal length is $a$ and vertical height is $b$.
Indexing for the primitive unit cells is illustrated in fig. 2.3.
fig 2.3: Primitive unit cells for rectangular lattice
Let’s write
\begin{aligned}\begin{aligned}\mathbf{r} &= a (\cos\theta, \sin\theta) = a \hat{\mathbf{r}} \\ \mathbf{s} &= a (-\cos\theta, \sin\theta) = a \hat{\mathbf{s}} \\ \mathbf{n} &= (n_1, n_2) \\ \mathbf{r}_\mathbf{n} &= n_1 \mathbf{r} + n_2 \mathbf{s},\end{aligned}\end{aligned} \hspace{\stretch{1}}(2.12)
For mass $m_\alpha, \alpha \in \{1, 2\}$ assume a trial solution of the form
\begin{aligned}\mathbf{u}_{\mathbf{n},\alpha} = \frac{\boldsymbol{\epsilon}_\alpha(\mathbf{q})}{\sqrt{m_\alpha}} e^{i \mathbf{r}_n \cdot \mathbf{q} - \omega t}.\end{aligned} \hspace{\stretch{1}}(2.13)
The equations of motion for the two particles are
\begin{aligned}\begin{aligned}m_1 \ddot{\mathbf{u}}_{\mathbf{n}, 1} &= - K_1 \text{Proj}_{\hat{\mathbf{x}}} \left( \mathbf{u}_{\mathbf{n}, 1} - \mathbf{u}_{\mathbf{n} - (0,1), 2} \right)- K_1 \text{Proj}_{\hat{\mathbf{x}}} \left( \mathbf{u}_{\mathbf{n}, 1} - \mathbf{u}_{\mathbf{n} - (1,0), 2} \right) \\ & \quad- K_2 \text{Proj}_{\hat{\mathbf{y}}} \left( \mathbf{u}_{\mathbf{n}, 1} - \mathbf{u}_{\mathbf{n}, 2} \right)- K_2 \text{Proj}_{\hat{\mathbf{y}}} \left( \mathbf{u}_{\mathbf{n}, 1} - \mathbf{u}_{\mathbf{n} - (1,1), 2} \right) \\ & \quad- K_3 \sum_\pm\text{Proj}_{\hat{\mathbf{r}}} \left( \mathbf{u}_{\mathbf{n}, 1} - \mathbf{u}_{\mathbf{n} \pm (1,0), 1} \right)- K_4 \sum_\pm\text{Proj}_{\hat{\mathbf{s}}} \left( \mathbf{u}_{\mathbf{n}, 1} - \mathbf{u}_{\mathbf{n} \pm (0,1), 1} \right)\end{aligned}\end{aligned} \hspace{\stretch{1}}(2.0.14.14)
\begin{aligned}\begin{aligned}m_2 \ddot{\mathbf{u}}_{\mathbf{n}, 2} &= - K_1 \text{Proj}_{\hat{\mathbf{x}}} \left( \mathbf{u}_{\mathbf{n}, 2} - \mathbf{u}_{\mathbf{n} + (1,0), 1} \right)- K_1 \text{Proj}_{\hat{\mathbf{x}}} \left( \mathbf{u}_{\mathbf{n}, 2} - \mathbf{u}_{\mathbf{n} + (0,1), 1} \right)\\ &\quad- K_2 \text{Proj}_{\hat{\mathbf{y}}} \left( \mathbf{u}_{\mathbf{n}, 2} - \mathbf{u}_{\mathbf{n}, 1} \right)- K_2 \text{Proj}_{\hat{\mathbf{y}}} \left( \mathbf{u}_{\mathbf{n}, 2} - \mathbf{u}_{\mathbf{n} + (1,1), 1} \right)\\ &\quad- K_3 \sum_\pm\text{Proj}_{\hat{\mathbf{r}}} \left( \mathbf{u}_{\mathbf{n}, 2} - \mathbf{u}_{\mathbf{n} \pm (1,0), 2} \right)- K_4 \sum_\pm\text{Proj}_{\hat{\mathbf{s}}} \left( \mathbf{u}_{\mathbf{n}, 2} - \mathbf{u}_{\mathbf{n} \pm (0,1), 2} \right)\end{aligned}\end{aligned} \hspace{\stretch{1}}(2.0.14.14)
Insertion of the trial solution gives
\begin{aligned}\begin{aligned} \omega^2 \sqrt{m_1} \boldsymbol{\epsilon}_1&= K_1 \text{Proj}_{\hat{\mathbf{x}}} \left( \frac{\boldsymbol{\epsilon}_1}{\sqrt{m_1}} - \frac{\boldsymbol{\epsilon}_2}{\sqrt{m_2}} e^{ - i \mathbf{s} \cdot \mathbf{q} } \right)+ K_1 \text{Proj}_{\hat{\mathbf{x}}} \left( \frac{\boldsymbol{\epsilon}_1}{\sqrt{m_1}} - \frac{\boldsymbol{\epsilon}_2}{\sqrt{m_2}} e^{ - i \mathbf{r} \cdot \mathbf{q} } \right) \\ &\quad+ K_2 \text{Proj}_{\hat{\mathbf{y}}} \left( \frac{\boldsymbol{\epsilon}_1}{\sqrt{m_1}} - \frac{\boldsymbol{\epsilon}_2}{\sqrt{m_2}} \right)+ K_2 \text{Proj}_{\hat{\mathbf{y}}} \left( \frac{\boldsymbol{\epsilon}_1}{\sqrt{m_1}} - \frac{\boldsymbol{\epsilon}_2}{\sqrt{m_2}} e^{ - i (\mathbf{r} + \mathbf{s}) \cdot \mathbf{q} } \right) \\ &\quad+ K_3 \left( \text{Proj}_{\hat{\mathbf{r}}} \frac{\boldsymbol{\epsilon}_1}{\sqrt{m_1}} \right)\sum_\pm\left( 1 - e^{ \pm i \mathbf{r} \cdot \mathbf{q} } \right)+ K_4 \left( \text{Proj}_{\hat{\mathbf{s}}} \frac{\boldsymbol{\epsilon}_1}{\sqrt{m_1}} \right)\sum_\pm\left( 1 - e^{ \pm i \mathbf{s} \cdot \mathbf{q} } \right)\end{aligned}\end{aligned} \hspace{\stretch{1}}(2.0.15.15)
\begin{aligned}\begin{aligned}\omega^2 \sqrt{m_2} \boldsymbol{\epsilon}_2&=K_1 \text{Proj}_{\hat{\mathbf{x}}} \left( \frac{\boldsymbol{\epsilon}_2}{\sqrt{m_2}} - \frac{\boldsymbol{\epsilon}_1}{\sqrt{m_1}} e^{ + i \mathbf{r} \cdot \mathbf{q} } \right)+ K_1 \text{Proj}_{\hat{\mathbf{x}}} \left( \frac{\boldsymbol{\epsilon}_2}{\sqrt{m_2}} - \frac{\boldsymbol{\epsilon}_1}{\sqrt{m_1}} e^{ + i \mathbf{s} \cdot \mathbf{q} } \right)\\ &\quad+ K_2 \text{Proj}_{\hat{\mathbf{y}}} \left( \frac{\boldsymbol{\epsilon}_2}{\sqrt{m_2}} - \frac{\boldsymbol{\epsilon}_1}{\sqrt{m_1}} \right)+ K_2 \text{Proj}_{\hat{\mathbf{y}}} \left( \frac{\boldsymbol{\epsilon}_2}{\sqrt{m_2}} - \frac{\boldsymbol{\epsilon}_1}{\sqrt{m_1}} e^{ + i (\mathbf{r} + \mathbf{s}) \cdot \mathbf{q} } \right) \\ &\quad+ K_3 \left( \text{Proj}_{\hat{\mathbf{r}}} \frac{\boldsymbol{\epsilon}_2}{\sqrt{m_2}} \right)\sum_\pm\left( 1 - e^{ \pm i \mathbf{r} \cdot \mathbf{q} } \right)+ K_4 \left( \text{Proj}_{\hat{\mathbf{s}}} \frac{\boldsymbol{\epsilon}_2}{\sqrt{m_2}} \right)\sum_\pm\left( 1 - e^{ \pm i \mathbf{s} \cdot \mathbf{q} } \right)\end{aligned}\end{aligned} \hspace{\stretch{1}}(2.0.15.15)
Regrouping, and using the matrix form $\text{Proj}_{\hat{\mathbf{u}}} = \hat{\mathbf{u}} \hat{\mathbf{u}}^\text{T}$ for the projection operators, this is
\begin{aligned}\left(\omega^2 - \frac{2}{m_1} \left( K_1 \hat{\mathbf{x}} \hat{\mathbf{x}}^T + K_2 \hat{\mathbf{y}} \hat{\mathbf{y}}^T + 2 K_3 \hat{\mathbf{r}} \hat{\mathbf{r}}^T \sin^2 (\mathbf{r} \cdot \mathbf{q}/2) + 2 K_4 \hat{\mathbf{s}} \hat{\mathbf{s}}^T \sin^2 (\mathbf{s} \cdot \mathbf{q}/2) \right)\right)\boldsymbol{\epsilon}_1 = -\left( K_1 \hat{\mathbf{r}} \hat{\mathbf{r}}^\text{T} \left( e^{ - i \mathbf{s} \cdot \mathbf{q} } + e^{ - i \mathbf{r} \cdot \mathbf{q} } \right) + K_2 \hat{\mathbf{s}} \hat{\mathbf{s}}^\text{T} \left( 1 + e^{ - i (\mathbf{r} + \mathbf{s}) \cdot \mathbf{q} } \right) \right)\frac{\boldsymbol{\epsilon}_2}{\sqrt{ m_1 m_2 }}\end{aligned} \hspace{\stretch{1}}(2.0.16.16)
\begin{aligned}\left( \omega^2 - \frac{2}{m_2} \left( K_1 \hat{\mathbf{x}} \hat{\mathbf{x}}^T + K_2 \hat{\mathbf{y}} \hat{\mathbf{y}}^T + 2 K_3 \hat{\mathbf{r}} \hat{\mathbf{r}}^T \sin^2 (\mathbf{r} \cdot \mathbf{q}/2)+ 2 K_4 \hat{\mathbf{s}} \hat{\mathbf{s}}^T \sin^2 (\mathbf{s} \cdot \mathbf{q}/2) \right) \right)\boldsymbol{\epsilon}_2 = -\left( K_1 \hat{\mathbf{r}} \hat{\mathbf{r}}^\text{T} \left( e^{ i \mathbf{s} \cdot \mathbf{q} } + e^{ i \mathbf{r} \cdot \mathbf{q} } \right) + K_2 \hat{\mathbf{s}} \hat{\mathbf{s}}^\text{T} \left( 1 + e^{ i (\mathbf{r} + \mathbf{s}) \cdot \mathbf{q} } \right) \right)\frac{\boldsymbol{\epsilon}_1}{\sqrt{ m_1 m_2 }}\end{aligned} \hspace{\stretch{1}}(2.0.16.16)
As a single matrix equation, this is
\begin{aligned}A = K_1 \hat{\mathbf{x}} \hat{\mathbf{x}}^T + K_2 \hat{\mathbf{y}} \hat{\mathbf{y}}^T + 2 K_3 \hat{\mathbf{r}} \hat{\mathbf{r}}^T \sin^2 (\mathbf{r} \cdot \mathbf{q}/2)+ 2 K_4 \hat{\mathbf{s}} \hat{\mathbf{s}}^T \sin^2 (\mathbf{s} \cdot \mathbf{q}/2)\end{aligned} \hspace{\stretch{1}}(2.0.17.17)
\begin{aligned}B = e^{ i (\mathbf{r} + \mathbf{s}) \cdot \mathbf{q}/2 }\left( { K_1 \hat{\mathbf{r}} \hat{\mathbf{r}}^\text{T} \cos\left( (\mathbf{r} - \mathbf{s}) \cdot \mathbf{q}/2 \right)+ K_2 \hat{\mathbf{s}} \hat{\mathbf{s}}^\text{T} \cos\left( (\mathbf{r} + \mathbf{s}) \cdot \mathbf{q}/2 \right)} \right)\end{aligned} \hspace{\stretch{1}}(2.0.17.17)
\begin{aligned}0 =\begin{bmatrix}\omega^2 - \frac{2 A}{m_1} & \frac{B^{*}}{\sqrt{m_1 m_2}} \\ \frac{B}{\sqrt{m_1 m_2}} & \omega^2 - \frac{2 A}{m_2} \end{bmatrix}\begin{bmatrix}\boldsymbol{\epsilon}_1 \\ \boldsymbol{\epsilon}_2\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(2.0.17.17)
Observe that this is an eigenvalue problem $E \mathbf{e} = \omega^2 \mathbf{e}$ for matrix
\begin{aligned}E = \begin{bmatrix}\frac{2 A}{m_1} & -\frac{B^{*}}{\sqrt{m_1 m_2}} \\ -\frac{B}{\sqrt{m_1 m_2}} & \frac{2 A}{m_2} \end{bmatrix},\end{aligned} \hspace{\stretch{1}}(2.0.17.17)
and eigenvalues $\omega^2$.
To be explicit lets put the $A$ and $B$ functions in explicit matrix form. The orthogonal projectors have a simple form
\begin{aligned}\text{Proj}_{\hat{\mathbf{x}}} = \hat{\mathbf{x}} \hat{\mathbf{x}}^\text{T}= \begin{bmatrix}1 \\ 0\end{bmatrix}\begin{bmatrix}1 & 0\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 0 & 0\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(2.0.19a)
\begin{aligned}\text{Proj}_{\hat{\mathbf{y}}} = \hat{\mathbf{y}} \hat{\mathbf{y}}^\text{T}= \begin{bmatrix}0 \\ 1\end{bmatrix}\begin{bmatrix}0 & 1\end{bmatrix}=\begin{bmatrix}0 & 0 \\ 0 & 1\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(2.0.19b)
For the $\hat{\mathbf{r}}$ and $\hat{\mathbf{s}}$ projection operators, we can use half angle formulations
\begin{aligned}\text{Proj}_{\hat{\mathbf{r}}} = \hat{\mathbf{r}} \hat{\mathbf{r}}^\text{T}= \begin{bmatrix}\cos\theta \\ \sin\theta\end{bmatrix}\begin{bmatrix}\cos\theta & \sin\theta\end{bmatrix}=\begin{bmatrix}\cos^2\theta & \cos\theta \sin\theta \\ \cos\theta \sin\theta & \sin^2 \theta\end{bmatrix}=\frac{1}{2}\begin{bmatrix}1 + \cos \left( 2 \theta \right) & \sin \left( 2 \theta \right) \\ \sin \left( 2 \theta \right) & 1 - \cos \left( 2 \theta \right)\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(2.0.20.20)
\begin{aligned}\text{Proj}_{\hat{\mathbf{s}}} = \hat{\mathbf{s}} \hat{\mathbf{s}}^\text{T}= \begin{bmatrix}-\cos\theta \\ \sin\theta\end{bmatrix}\begin{bmatrix}-\cos\theta & \sin\theta\end{bmatrix}=\begin{bmatrix}\cos^2\theta & -\cos\theta \sin\theta \\ -\cos\theta \sin\theta & \sin^2 \theta\end{bmatrix}=\frac{1}{2}\begin{bmatrix}1 + \cos \left( 2 \theta \right) & -\sin \left( 2 \theta \right) \\ -\sin \left( 2 \theta \right) & 1 - \cos \left( 2 \theta \right)\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(2.0.20.20)
After some manipulation, and the following helper functions
\begin{aligned}\begin{aligned}\alpha_\pm &= K_3 \sin^2 (\mathbf{r} \cdot \mathbf{q}/2) \pm K_4 \sin^2 (\mathbf{s} \cdot \mathbf{q}/2) \\ \beta_\pm &= K_1 \cos\left( (\mathbf{r} - \mathbf{s}) \cdot \mathbf{q}/2 \right) \pm K_2 \cos\left( (\mathbf{r} + \mathbf{s}) \cdot \mathbf{q}/2 \right),\end{aligned}\end{aligned} \hspace{\stretch{1}}(2.0.20.20)
the block matrices of eq. 2.0.17.17 take the form
\begin{aligned}A = \begin{bmatrix}K_1 + \alpha_+ (1 + \cos\left( 2 \theta \right)) & \alpha_- \sin\left( 2 \theta \right) \\ \alpha_- \sin\left( 2 \theta \right) & K_2 + \alpha_+ (1 - \cos\left( 2 \theta \right))\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(2.0.22.22)
\begin{aligned}B = e^{ i (\mathbf{r} + \mathbf{s}) \cdot \mathbf{q}/2 }\begin{bmatrix} \beta_+ (1 + \cos \left( 2 \theta \right)) & \beta_- \sin \left( 2 \theta \right) \\ \beta_- \sin \left( 2 \theta \right) & \beta_+( 1 -\cos \left( 2 \theta \right))\end{bmatrix}\end{aligned} \hspace{\stretch{1}}(2.0.22.22)
A final bit of simplification for $B$ possible, noting that $\mathbf{r} + \mathbf{s} = 2 a (0, \sin\theta )$, and $\mathbf{r} - \mathbf{s} = 2 a(\cos\theta, 0)$, so
\begin{aligned}\beta_\pm = K_1 \cos\left( a \cos\theta q_x \right) \pm K_2 \cos\left( a \sin\theta q_y \right),\end{aligned} \hspace{\stretch{1}}(2.0.22.22)
and
\begin{aligned}B = e^{ i a \sin\theta q_y }\begin{bmatrix} \beta_+ (1 + \cos \left( 2 \theta \right)) & \beta_- \sin \left( 2 \theta \right) \\ \beta_- \sin \left( 2 \theta \right) & \beta_+( 1 -\cos \left( 2 \theta \right))\end{bmatrix}.\end{aligned} \hspace{\stretch{1}}(2.0.22.22)
It isn’t particularly illuminating to expand out the determinant for such a system, even though it can be done symbolically without too much programming. However, what is easy after formulating the matrix for this system, is actually solving it. This is done, and animated, in twoAtomBasisRectangularLatticeDispersionRelation.cdf
## (Markham) Markville Mall Sony store: an interesting variation of warranty fraud.
Posted by peeterjoot on December 18, 2013
I mailed the following response to the Markville mall’s Sony store, where one of their employees attempted a new variety of warranty fraud:
I discovered when I attempted to exchange the headphones that you completely misrepresented yourself when I made the purchase. The item that you scanned showed up at a lower price than the sticker price. Instead of offering me a chance to buy that item at that price you padded the price back up to the sticker price (or rather close to it, short a few dollars), by adding in an extended warranty.
You did this despite the fact that I said I would never voluntarily purchase one of these extended warranties. Since you portrayed this as something that was “for free”, I did not object. However, you completely missed your chance at honesty, because I should have been offered the sale price. You stated that you lowered the price “so that you could offer me the warranty without costing me anything”, however, that lowered price was already the listed sale price in your system. This was something that you did not disclose.
That is, in my opinion, undeniably fraud.
Since you were so careful to make sure that both you and your colleague were represented on the bill, I can only assume that you are on commission. It would be very hard to argue that this was not a blatant attempt to pad your commission, at my expense. I shudder to consider how many other customers have been exploited in this way.
I’ll never shop at the Sony Store again. You have lost my patronage, and I’ll not hesitate to tell anybody who is considering a Sony Store purchase to be very careful at your store, to avoid this new interesting variation of warranty fraud.
The mechanism of the fraud attempt used here was that I was sold a pair of headphones that were on sale, but the salesman padded the price up to the sticker price including a “free” extended warranty. He blatantly told me that he was reducing the price for me so that he could include the extended warranty for free. In the end this made it appear that I got $5 off the sticker price, and got an extended warranty to boot. I only discovered this because as well as attempting to defraud me, they also gave me the wrong headphones (I’d asked for a noise cancelling model). I had not yet noticed this, however, the Sony Store now provides a service (a rather nice innovation) where they will provide you with a soft copy of your receipt if you provide them your email address. Because of this, they had my contact info to proactively inform me about the incorrect boxe of headphones that I’d been given. When I attempted the exchange, it was at a point when the staff member who did this transaction was not there, so I was able to discover what actually occurred. At the point of return, I was offered a reduction in price for the warranty should I desire it, but it still would have meant paying for it. This new “cheaper warranty” that I never wanted in the first place would still have cost me (not saving me money in a too good to be true fashion as it originally appeared), so I turned that down. I then discovered that I’d also have to pay more for the correct headphones. It was only$5 more, but by this time I was completely fed up. Reflecting now, I was also very annoyed at myself for having fallen for this trick. I just returned them completely.
It is always interesting to learn of new fraud techniques. This is a new one that I had not seen before. Taking advantage of an unlisted sales price to sell additional undesired content, so that commissions could be padded. Because the sales price was on their system, the salesman did not require any management approval to override the system with a lower price, and was able to make it appear that he had “lowered the price” for me so that I could get something for free. It’s actually very clever.
Could this have been an honest mistake? I doubt it very much.
EDIT:
The Sony salesman who I had dealt with contacted me after this, stating:
I am sorry for whatever misunderstanding happened yesterday. Can I please call you & explain you the situation & see how can I make your experience with us more conferrable. Please let me know on what phone number I can reach you on. I am sorry once again & I believe you will give me a chance to explain & I will do my best to solve this issue.
My response was that he needed to resolve that with his management, not me. If that was done, then I’d be willing to talk to them (not him).
They (management through him) eventually offered me a deal on the item that I’d returned. It wasn’t really my intent to be fishing for a deal, and I’d continued shopping after all this for an alternate set of phones to buy from somewhere else. However, the timing and the offer were both good since I hadn’t yet found a replacement item I was happy with, and I ended up accepting that offer.
Picking up the set they’d set aside for me provided a chance to talk to his manager, who hadn’t been given the complete story. Despite that, after talking to the manager, I’m no longer certain that the salesman understood exactly why I objected to the transaction. This may not have been an instance of fraud as it initially seemed to be, instead it could have been a blunder by a fairly new staff member, confused by the wrong price showing up after the scan, and tried to “fix it” in a way that he naively thought would be satisfactory. I don’t think I’ll ever know for sure.
Posted in Incoherent ramblings | Tagged: , , , | 1 Comment »
## Pre-final-exam update of notes for PHY487 (condensed matter physics)
Posted by peeterjoot on December 2, 2013
Here’s an update of my class notes from Winter 2013, University of Toronto Condensed Matter Physics course (PHY487H1F), taught by Prof. Stephen Julian. This includes notes for all the examinable lectures (i.e. excluding superconductivity). I’ll post at least one more update later, probably after the exam, including notes from the final lecture, and my problem set 10 solution.
NOTE: This v.4 update of these notes is still really big (~18M). Some of my mathematica generated 3d images appear to result in very large pdfs.
Changelog for this update (relative to the the first, and second, and third Changelogs) :
December 02, 2013 Lecture 23, Superconductivity
December 01, 2013 Lecture 22, Intro to semiconductor physics
December 01, 2013 Lecture 21, Electron-phonon scattering
November 26, 2013 Problem Set 9, Electron band structure, density of states, and effective mass
November 22, 2013 Lecture 20, Electric current (cont.)
November 20, 2013 Problem Set 8, Tight Binding.
November 18, 2013 Lecture 19, Electrical transport (cont.)
## Signs, signs, everywhere there’s signs. Do this, don’t do that, can’t you read the sign.
Posted by peeterjoot on November 28, 2013
Here’s a sign that was recently installed in one of the men’s washrooms at work
and a close up of it
Yes, there is a set of men’s showers in the X1 location of the building, which is consistent with the graphic of the shower in the sign. I couldn’t, however, for the life of me, figure out why somebody who was at the washroom sink would have to be reminded that the lab has shower facilities. Nor could I figure out why we now had a sign that appeared to be instructing those at the sink to pray in the shower.
I’ve pointed it out this sign to a few people now when I was at the sink, but nobody else appeared to be able to decipher it. Sofia, who is more wise in the ways of the world, told me what this is about: part of the Islamic prayer ritual involves washing one’s feet.
Sure enough, I remember that there was a guy who used to wash his feet in the bathroom sink in the summer. He always wore sandals, and I thought his washing was because he thought his feet smelled (I wondered why he didn’t switch to socks and shoes if that was the case, and now feel silly for not just asking him).
So, it seems that this sign was likely commissioned and installed just for this one individual. Somebody objected to his feet washing. That objector was probably also completely ignorant like me, and likely didn’t know that this was part of his religious ritual. I’d guess that foot washing objector still doesn’t know that.
## “fun” bug in 64-bit perl 5.10.0 bigint sprintf
Posted by peeterjoot on November 19, 2013
Here’s a rather unexpected bug with perl sprintf
#! /usr/bin/perl
use strict;
use warnings;
use bigint ;
my $a = hex( "0x0A0000001D05A820" ) ; printf( "0x%016X\n",$a ) ;
printf( "%d\n", $a ) ; printf( "$a\n" ) ;
The %X printf produces a value where the least significant 0x20 is lost:
\$ ./bigint
0x0A0000001D05A800
720575940866189312
720575940866189344
Observe that the loss occurs in the printf and not the hex() call, since 720575940866189344 == 0x0A0000001D05A820.
This bug appears to be fixed in some version of perl <= 5.16.2. Oh, the joys of using ancient operating system versions so that we can support customers on many of the ancient deployments that they seem to like to run on.
Posted in perl and general scripting hackery | Tagged: , , , | Leave a Comment »
## second update of phy487 notes (Condensed Matter Physics)
Posted by peeterjoot on November 16, 2013
Here’s an update of my (incomplete) lecture notes for the Winter 2013, University of Toronto Condensed Matter Physics course (PHY487H1F), taught by Prof. Stephen Julian. This makes updates to these notes since the first, and second versions posted.
NOTE: This v.3 update of these notes is still really big (~16M). Some of my mathematica generated 3d images appear to result in very large pdfs.
This set of notes includes the following these additions (not many of which were posted separately for this course)
November 15, 2013 Semiconductors
November 13, 2013 Thomas-Fermi screening, and nearly free electron model
November 11, 2013 3 dimensional band structures, Fermi surfaces of real metals
November 05, 2013 Fourier coefficient integral for periodic function
November 04, 2013 Tight binding model
November 04, 2013 Fermi properties, free electron specific heat, bulk modulus
November 01, 2013 Nearly free electron model, periodic potential (cont.)
October 29, 2013 Free electron model of metals
October 28, 2013 Electrons in a periodic lattice
October 23, 2013 Huygens diffraction
## Political correctness crap in IBM HR circles
Posted by peeterjoot on November 12, 2013
Apparently IBM has fired so many people that they have invented (or started using) the term RA (Resource Action) instead of fired, used like so:
Bob was RAed.
There has been another round of RAs.
If you don’t want to be RAed, make sure your PBC (personal business committment) document is filled in and sounds good, since there’s a new earnings report due out tomorrow.
This is supposed to sound nicer, but I think it’s the opposite. To me this sounds like the individual is now a resource, and is be moved around like an entry in some accounting table. RA means that, unfortunately, they ended up in the delete column. It’s very impersonal. Perhaps this is just to make the firing manager feel better, since that manager cannot do anything about the firing quotas when they occur.
Posted in Incoherent ramblings | Tagged: , | 1 Comment »
## Remembering cultural insanity and propagandization
Posted by peeterjoot on November 11, 2013
It’s Remembrance day, and we are once again barraged with the heroism of war and sacrifice
We get the cuddly manly shots of veterans posing on the field
these last two pictures taken from the Markham community paper, where we have our poor veteran Stan, who recalls the “guts blood and luck at Juno Beach” quoted
we got to the beach, and there were dead people all over … You didn’t stop to pick someone up to even [to sic] check if they were still alive. You just kept moving, otherwise you would have been shot yourself.
I surely sympathize for Stan. This is truly horrifying, and must have been traumatic. It’s something that you’d want to forget, and would haunt you for the rest of your life. However, our unfortunate Stan was sold a narrative.
We must fight the evil villain.
People are dying and it is our duty to protect.
Kids are doing their part
Women are doing their part
It is brave and honourable to do your part
Even the whisky makers are doing their part!
The list of selling points goes on and on. “Be ashamed of your fear of death. It’s the right thing to do. … “ It was a vicious and evil sales pitch. This is a narrative that was backed by hordes of propaganda.
A lot of profit was made by this war. This is true of all war.
In a sane world, what would we be remembering? Follow the money. We should remember those that profited from the war. We should remember those that directly or indirectly sold armaments to both sides. Most importantly, we should remember those that bankrolled the war on all sides. Stan was not on the winning side. He was on the loosing side. The winning side resides in corporate and banking boardrooms. This is what we need to remember whenever the drums of war start beating.
Posted in Incoherent ramblings | Tagged: , , , , | 1 Comment »
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2017-08-18 22:05:57
|
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http://blog.gmane.org/gmane.comp.tex.miktex/month=20040401
|
1 Apr 2004 01:07
### [MiKTeX] 2.4 Source code
I would like to get the source code for MiKTeX 2.4 that has been
released in November 2003. What is currently available via CVS is
already changed (as one would expected by now). Is there a way to do that?
Thanks,
Greg
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1 Apr 2004 01:18
### [MiKTeX] [Bug] dvips -Ppdf and MetaPost
Hello,
below you´ll find a copy of my post already sent to comp.text.tex. As
pointed out by Walter Schmidt there, there seems to be a bug in dvips.
Unfortunately, I couldn´t find a pointer to where to send bug reports
related to dvips. Does anyone of the MiKTeX team know?
From <news:c4ed86$2i0bin$1 <at> ID-79520.news.uni-berlin.de>:
In the attached MetaPost output the minus sign gets lost when included
in a LaTeX document and processed via 'dvips -Ppdf math' (using dvips(k)
5.90a from a recent MiKTeX). Using pdfLaTeX works. As discussed on
de.comp.text.tex (<news:c46q6j$2ek1hl$1 <at> ID-79520.news.uni-berlin.de>)
using 'dvips -Ppdf -G0 math' works correctly, too.
But reading the UK-TeX-FAQ two questions arose to me. In the answer to
question 87 it is said:
"... The -G1 switch discussed in that question is appropriate for
Knuth’s text fonts, but doesn’t work with text fonts that don’t follow
Knuth’s patterns (such as fonts supplied by Adobe). ..."
Does "Knuth´s text fonts" mean "all fonts designed by Knuth (cm,
concrete, ...)"? Or does it mean "text fonts, but not maths fonts"?
Second, the last sentence to the above mentioned question reads:
"The problem has been corrected in dvips v 5.90 (the version
distributed with the TEX-Live 7 CD-ROM and in other TEX distributions
from that date onwards)."
I used dvips(k) 5.90a. The minus sign disappeared. Did I miss something?
1 Apr 2004 07:31
### Re: [MiKTeX] 2.4 Source code
On Wed, 31 Mar 2004 17:07:12 -0600, Gregory Borota
<winlatex <at> sbcglobal.net> wrote:
> I would like to get the source code for MiKTeX 2.4 that has been
> released in November 2003. What is currently available via CVS is
> already changed (as one would expected by now). Is there a way to do
> that?
The point of doing CVS is that you can get older versions, so just choose
them. Why do you want old binaries?
--
--
Morten Høgholm
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1 Apr 2004 15:44
### Re: [MiKTeX] 2.4 Source code
I managed to get what I wanted. The point was that at least for
yesterday what was current would not compile.
Thanks,
Greg
You wrote on 3/31/2004 11:31 PM:
>
>> I would like to get the source code for MiKTeX 2.4 that has been
>> released in November 2003. What is currently available via CVS is
>> already changed (as one would expected by now). Is there a way to do
>> that?
>
>
> The point of doing CVS is that you can get older versions, so just
> choose them. Why do you want old binaries?
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2 Apr 2004 03:19
### [MiKTeX] pstricks
Hello,
as I said b4, I have MikTeX 2.4 and winedt 5.4 (or something around those
lines). Anyway, i saw an article about a package available from CTAN called
pstricks. What is it and where can I obtain it-it seems to be the type of
package i have been looking for.
_________________________________________________________________
Limited-time offer: Fast, reliable MSN 9 Dial-up Internet access FREE for 2
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2 Apr 2004 04:45
### [MiKTeX] portuguese spelling
I use portuguese spelling (babel) and sometimes I get an overfull on a line.
I was unable to narrow down the problem and the the way I got around it was
Francisco Grossi
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2 Apr 2004 05:10
### Re: [MiKTeX] pstricks
Hi Rohan,
>Anyway, i saw an article about a package available >from CTAN called
>pstricks. What is it and where can I obtain it-it >seems to be the type of
>package i have been looking for.
If you dont have pstricks, you can use the MiKTeX packet manager to install it.
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2 Apr 2004 05:15
### Re: [MiKTeX] pstricks
but how do I know if MikTeX already has it or not?
>From: "Aditya Dushyant Trivedi" <atrivedi2 <at> student.gsu.edu>
>To: miktex-users <at> lists.sourceforge.net
>Subject: Re: [MiKTeX] pstricks
>Date: Thu, 01 Apr 2004 22:10:54 -0500
>
>Hi Rohan,
>
> >Anyway, i saw an article about a package available >from CTAN called
> >pstricks. What is it and where can I obtain it-it >seems to be the type
>of
> >package i have been looking for.
>
>
>If you dont have pstricks, you can use the MiKTeX packet manager to install
>it.
>
>
>
>
>-------------------------------------------------------
>This SF.Net email is sponsored by: IBM Linux Tutorials
>Free Linux tutorial presented by Daniel Robbins, President and CEO of
>GenToo technologies. Learn everything from fundamentals to system
>_______________________________________________
2 Apr 2004 07:19
### Re: Re: [MiKTeX] pstricks
>but how do I know if MikTeX already has it or not?
>
>
>If you dont have pstricks, you can use the MiKTeX packet manager to install
>it.
Two ways:
1) Go to Package Manager. In the name field enter pstricks. Select pstricks from the field. If you can select
the minus sign above in the toolbar, you have it. If you can select the plus sign, you need to install it.
2) create any document and add the follwoing code in the preamble (i.e. before \begin{document})
\usepackage{pstricks}
If it compiles fine, you have it. If not, it will ask you if you want to install it.
I suggest that you read through the MiKTeX Manual Manaul to understand how the process works.
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2 Apr 2004 08:55
### Re: [MiKTeX] portuguese spelling
Are portuguese hyphenation rules switched on? Check the Language tab on
the Miktex Options. If you change something, don't forget to refresh the
File name database on the General tab
Best regards,
Luiz.
Francisco A. S. Grossi wrote:
>I use portuguese spelling (babel) and sometimes I get an overfull on a line.
>I was unable to narrow down the problem and the the way I got around it was
>
>Francisco Grossi
>
>
>
>-------------------------------------------------------
>This SF.Net email is sponsored by: IBM Linux Tutorials
>Free Linux tutorial presented by Daniel Robbins, President and CEO of
>GenToo technologies. Learn everything from fundamentals to system
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2013-12-05 15:16:30
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https://math.stackexchange.com/questions/4051050/let-a-subseteq-mathbbrk-and-let-a-x-in-mathbbrk-mid-x-in-partiala
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# Let $A\subseteq\mathbb{R}^k$, and let $A'=\{x\in\mathbb{R}^k \mid x\in\partial(A\setminus\{x\})\}$. How do I prove that $A'$ is closed?
I was asked this simple question a short while ago.
Let $$A\subseteq\mathbb{R}^k$$, and let $$A'=\{x\in\mathbb{R}^k \mid x\in\partial(A\setminus\{x\})\}$$. Prove that $$A'$$ is closed.
I think I was able to explain it, but I am not sure what I claim is true. (The following is not a formal solution because I am not sure the reasoning is correct) My explanation goes as follows:
Let $$x\in A'$$. Say $$x$$ is in the interior of $$A$$, then $$x\notin\partial(A\setminus\{x\})$$ because there exists a ball $$B(x;r)\subseteq A$$ by definition. By a smilar logic if x is in the exterior of $$A$$, I claim that $$x\notin\partial(A\setminus\{x\})$$. Thus $$x\in\partial A$$ (that is $$A'\subseteq\partial A$$). From here I am quite sure it is not too hard to explain that $$A'$$ is closed.
I doubt my reasoning, but I am not sure what have I missed. Care to shed some light over the matter?
• It's not true that $A'=\partial A$ in general, and not every subset of a closed set is closed, so there's definitely still an argument to be made. It seems actually that your observations might more directly prove that the complement of $A'$, namely $\big\{ x\in\Bbb R^k\colon x\notin\partial(A\setminus\{x\}) \big\}$, is open. – Greg Martin Mar 6 at 8:07
• Note that $A' = \{x \in \Bbb R^k \mid \forall r>0: B(x,r) \cap (A\setminus \{x\} \neq \emptyset\}$. – Henno Brandsma Mar 6 at 8:44
• I think I get it! By my claims I can show that $\mathbb{R}^k{\setminus}A'$ is open, which is very similar to what @HennoBrandsma showed in his proof and pointed in his comment. Thank you for your speedy replies everyone! – Gamow Drop Mar 6 at 9:23
• $\partial A$ can be $\Bbb R^k$ and not any subset of it is closed... – Henno Brandsma Mar 6 at 9:26
• @GamowDrop: are you thinking that any subset of a closed set must be closed? That's definitely not true. For a concrete example, suppose $A$ is the unit disk in $\Bbb R^2$, so that $\partial A$ is the unit circle. The subset of $\partial A$ consisting of all points strictly about the $x$-axis is not closed. Neither is the set of points in $\partial A$ that have both coordinates rational. – Greg Martin Mar 6 at 18:55
Let $$p \notin A'$$. This means that $$\exists r>0: B(p,r) \cap A \subseteq \{p\}$$. If in fact $$B(p,r) \cap A = \emptyset$$, for any $$x \in B(p,r)$$ we have a ball $$B(x,r') \subseteq B(p,r)$$ (open balls are open sets) and this witnesses also that $$B(x,r') \cap A= \emptyset$$ and so $$x \notin A'$$. So $$B(p,r) \subseteq \Bbb R^k \setminus A'$$. If on the other hand, $$B(p,r) \cap A= \{p\}$$, we also have $$B(p,r) \subseteq \Bbb R^k \setminus A'$$: if $$x \in B(p,r)$$ and $$x \neq p$$ (WLOG) then there is a ball $$B(x,r') \subseteq B(p,r)$$ so that $$p \notin B(x,r')$$. Then $$B(x,r') \cap A = \emptyset$$ and so $$x \notin A'$$, as required. So any $$p \in \Bbb R^k \setminus A'$$ is an interior point of it, so the complement of $$A'$$ is open, and $$A'$$ is closed.
Suppose that $$(x_n)_{n\in\Bbb N}$$ is a sequence of elements of $$A'$$ which converges to some $$x\in\Bbb R^k$$; I will prove that $$x\in A'$$. Take $$r>0$$ and consider the ball $$B(x;r)$$. It contains some $$x_n$$. Since $$x_n\in A'$$, $$x_n\in\partial(A\setminus\{x\})$$. So, every ball $$B(x_n;r')$$ contains elements of $$A$$. So, take $$r'$$ so small that $$B(x_n;r')\subset B(x;r)\setminus\{x\}$$. Then $$B(x_n;r')$$ contains elements of $$A\setminus\{x\}$$. This proves that $$x\in A'$$. Therefore, since $$A'$$ contains the limit of any convergent sequence of its elements, it is a closed set.
I believe you get a better insight if you forget about $$\mathbb{R}^k$$ and balls. Let's work in a (Hausdorff) topological space $$X$$, which your case is a specialization of.
What are the points $$x$$ such that $$x\in\partial(A\setminus\{x\})$$?
First of all they don't need to belong to $$A$$. For instance, with $$A=(0,1)\subseteq\mathbb{R}$$, $$0$$ belongs to the set $$A'$$.
Let's see: $$x\in\partial(A\setminus\{x\})$$ means that every neighborhood of $$x$$ intersects both $$A\setminus\{x\}$$ and its complement. However, $$x$$ surely belongs to the complement of $$A\setminus\{x\}$$, so the condition just reads $$\textit{every neighborhood of x intersects A\setminus\{x\}}$$ which is the standard definition of limit point.
Now it's easier, isn't it? You need to show that if $$y$$ has the property that each of its open neighborhoods intersects $$A'$$, then $$y\in A'$$.
Take such an element $$y$$ and an open neighborhood $$V$$ of $$y$$. Then there exists $$x\in A'$$ such that $$x\in V$$. Since the space $$X$$ is Hausdorff, there exists a neighborhood $$U$$ of $$x$$ such that $$U\subseteq V$$ and $$y\notin U$$.
Since $$U$$ is a neighborhood of $$x$$, there exists a point $$z\in A$$ (different from $$x$$, but it's irrelevant) such that $$z\in U$$. Now necessarily $$z\ne y$$, but $$z\in V$$ and we're done.
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2021-04-20 10:17:00
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https://iacr.org/cryptodb/data/conf.php?venue=asiacrypt&year=2019
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## CryptoDB
### Papers from ASIACRYPT 2019
Year
Venue
Title
2019
ASIACRYPT
New proof systems for sustainable blockchains: proofs of space and verifiable delay functions
Invited talk
The distinctive feature of Bitcoin is that it achieves decentralisation in an open setting where everyone can join. This is achieved at a high price, honest parties must constantly dedicate more computational power towards securing Bitcoin's blockchain than is available to a potential adversary, which leads to a massive waste of energy; at its hitherto peak, the electricity used for Bitcoin mining equaled the electricity consumption of Austria. In this lecture I will discuss how disk-space, instead of computation, can be used as a resource to construct a more sustainable blockchain. We will see definitions and constructions of "proof of space" and "verifiable delay functions", and how they can be used to construct a Blockchain with similar dynamics and security properties as the Bitcoin blockchain.
2019
ASIACRYPT
Streamlined blockchains: A simple and elegant approach (tutorial)
Invited talk
A blockchain protocol (also called state machine replication) allows a set of nodes to agree on an ever-growing, linearly ordered log of transactions. In this tutorial, we present a new paradigm called “streamlined blockchains”. This paradigm enables a new family of protocols that are extremely simple and natural: every epoch, a proposer proposes a block extending from a notarized parent chain, and nodes vote if the proposal’s parent chain is not too old. Whenever a block gains enough votes, it becomes notarized. Whenever a node observes a notarized chain with several blocks of consecutive epochs at the end, then the entire chain chopping off a few blocks at the end is final. By varying the parameters highlighted in blue, we illustrate two variants for the partially synchronous and synchronous settings respectively. We present very simple proofs of consistency and liveness. We hope that this tutorial provides a compelling argument why this new family of protocols should be used in lieu of classical candidates (e.g., PBFT, Paxos, and their variants), both in practical implementation and for pedagogical purposes.
2019
ASIACRYPT
Towards Attribute-Based Encryption for RAMs from LWE: Sub-linear Decryption, and More
Attribute based encryption (ABE) is an advanced encryption system with a built-in mechanism to generate keys associated with functions which in turn provide restricted access to encrypted data. Most of the known candidates of attribute based encryption model the functions as circuits. This results in significant efficiency bottlenecks, especially in the setting where the function associated with the ABE key is represented by a random access machine (RAM) and a database, with the runtime of the RAM program being sublinear in the database size. In this work we study the notion of attribute based encryption for random access machines (RAMs), introduced in the work of Goldwasser, Kalai, Popa, Vaikuntanathan and Zeldovich (Crypto 2013). We present a construction of attribute based encryption for RAMs satisfying sublinear decryption complexity assuming learning with errors; this is the first construction based on standard assumptions. Previously, Goldwasser et al. achieved this result based on non-falsifiable knowledge assumptions. We also consider a dual notion of ABE for RAMs, where the database is in the ciphertext and we show how to achieve this dual notion, albeit with large attribute keys, also based on learning with errors.
2019
ASIACRYPT
On the Non-existence of Short Vectors in Random Module Lattices
Recently, Lyubashevsky & Seiler (Eurocrypt 2018) showed that small polynomials in the cyclotomic ring $\mathbb {Z}_q[X]/(X^n+1)$, where n is a power of two, are invertible under special congruence conditions on prime modulus q. This result has been used to prove certain security properties of lattice-based constructions against unbounded adversaries. Unfortunately, due to the special conditions, working over the corresponding cyclotomic ring does not allow for efficient use of the Number Theoretic Transform (NTT) algorithm for fast multiplication of polynomials and hence, the schemes become less practical.In this paper, we present how to overcome this limitation by analysing zeroes in the Chinese Remainder (or NTT) representation of small polynomials. As a result, we provide upper bounds on the probabilities related to the (non)-existence of a short vector in a random module lattice with no assumptions on the prime modulus. We apply our results, along with the generic framework by Kiltz et al. (Eurocrypt 2018), to a number of lattice-based Fiat-Shamir signatures so they can both enjoy tight security in the quantum random oracle model and support fast multiplication algorithms (at the cost of slightly larger public keys and signatures), such as the Bai-Galbraith signature scheme (CT-RSA 2014), $\mathsf {Dilithium\text {-}QROM}$ (Kiltz et al., Eurocrypt 2018) and $\mathsf {qTESLA}$ (Alkim et al., PQCrypto 2017). Our techniques can also be applied to prove that recent commitment schemes by Baum et al. (SCN 2018) are statistically binding with no additional assumptions on q.
2019
ASIACRYPT
Non-Committing Encryption with Quasi-Optimal Ciphertext-Rate Based on the DDH Problem
Non-committing encryption (NCE) was introduced by Canetti et al. (STOC ’96). Informally, an encryption scheme is non-committing if it can generate a dummy ciphertext that is indistinguishable from a real one. The dummy ciphertext can be opened to any message later by producing a secret key and an encryption random coin which “explain” the ciphertext as an encryption of the message. Canetti et al. showed that NCE is a central tool to achieve multi-party computation protocols secure in the adaptive setting. An important measure of the efficiently of NCE is the ciphertext rate, that is the ciphertext length divided by the message length, and previous works studying NCE have focused on constructing NCE schemes with better ciphertext rates.We propose an NCE scheme satisfying the ciphertext rate based on the decisional Diffie-Hellman (DDH) problem, where is the security parameter. The proposed construction achieves the best ciphertext rate among existing constructions proposed in the plain model, that is, the model without using common reference strings. Previously to our work, an NCE scheme with the best ciphertext rate based on the DDH problem was the one proposed by Choi et al. (ASIACRYPT ’09) that has ciphertext rate . Our construction of NCE is similar in spirit to that of the recent construction of the trapdoor function proposed by Garg and Hajiabadi (CRYPTO ’18).
2019
ASIACRYPT
4-Round Luby-Rackoff Construction is a qPRP
The Luby-Rackoff construction, or the Feistel construction, is one of the most important approaches to construct secure block ciphers from secure pseudorandom functions. The 3- and 4-round Luby-Rackoff constructions are proven to be secure against chosen-plaintext attacks (CPAs) and chosen-ciphertext attacks (CCAs), respectively, in the classical setting. However, Kuwakado and Morii showed that a quantum superposed chosen-plaintext attack (qCPA) can distinguish the 3-round Luby-Rackoff construction from a random permutation in polynomial time. In addition, Ito et al. recently showed a quantum superposed chosen-ciphertext attack (qCCA) that distinguishes the 4-round Luby-Rackoff construction. Since Kuwakado and Morii showed the result, a problem of much interest has been how many rounds are sufficient to achieve provable security against quantum query attacks. This paper answers to this fundamental question by showing that 4-rounds suffice against qCPAs. Concretely, we prove that the 4-round Luby-Rackoff construction is secure up to $O(2^{n/12})$ quantum queries. We also give a query upper bound for the problem of distinguishing the 4-round Luby-Rackoff construction from a random permutation by showing a distinguishing qCPA with $O(2^{n/6})$ quantum queries. Our result is the first to demonstrate the security of a typical block-cipher construction against quantum query attacks, without any algebraic assumptions. To give security proofs, we use an alternative formalization of Zhandry’s compressed oracle technique.
2019
ASIACRYPT
Forkcipher: A New Primitive for Authenticated Encryption of Very Short Messages
Highly efficient encryption and authentication of short messages is an essential requirement for enabling security in constrained scenarios such as the CAN FD in automotive systems (max. message size 64 bytes), massive IoT, critical communication domains of 5G, and Narrowband IoT, to mention a few. In addition, one of the NIST lightweight cryptography project requirements is that AEAD schemes shall be “optimized to be efficient for short messages (e.g., as short as 8 bytes)”.In this work we introduce and formalize a novel primitive in symmetric cryptography called forkcipher. A forkcipher is a keyed primitive expanding a fixed-lenght input to a fixed-length output. We define its security as indistinguishability under a chosen ciphertext attack (for n-bit inputs to 2n-bit outputs). We give a generic construction validation via the new iterate-fork-iterate design paradigm.We then propose ${\mathsf {ForkSkinny}}$ as a concrete forkcipher instance with a public tweak and based on SKINNY: a tweakable lightweight cipher following the TWEAKEY framework. We conduct extensive cryptanalysis of ${\mathsf {ForkSkinny}}$ against classical and structure-specific attacks.We demonstrate the applicability of forkciphers by designing three new provably-secure nonce-based AEAD modes which offer performance and security tradeoffs and are optimized for efficiency of very short messages. Considering a reference block size of 16 bytes, and ignoring possible hardware optimizations, our new AEAD schemes beat the best SKINNY-based AEAD modes. More generally, we show forkciphers are suited for lightweight applications dealing with predominantly short messages, while at the same time allowing handling arbitrary messages sizes.Furthermore, our hardware implementation results show that when we exploit the inherent parallelism of ${\mathsf {ForkSkinny}}$ we achieve the best performance when directly compared with the most efficient mode instantiated with SKINNY.
2019
ASIACRYPT
Structure-Preserving and Re-randomizable RCCA-Secure Public Key Encryption and Its Applications
Re-randomizable RCCA-secure public key encryption (Rand-RCCA PKE) schemes reconcile the property of re-randomizability of the ciphertexts with the need of security against chosen-ciphertexts attacks. In this paper we give a new construction of a Rand-RCCA PKE scheme that is perfectly re-randomizable. Our construction is structure-preserving, can be instantiated over Type-3 pairing groups, and achieves better computation and communication efficiency than the state of the art perfectly re-randomizable schemes (e.g., Prabhakaran and Rosulek, CRYPTO’07). Next, we revive the Rand-RCCA notion showing new applications where our Rand-RCCA PKE scheme plays a fundamental part: (1) We show how to turn our scheme into a publicly-verifiable Rand-RCCA scheme; (2) We construct a malleable NIZK with a (variant of) simulation soundness that allows for re-randomizability; (3) We propose a new UC-secure Verifiable Mix-Net protocol that is secure in the common reference string model. Thanks to the structure-preserving property, all these applications are efficient. Notably, our Mix-Net protocol is the most efficient universally verifiable Mix-Net (without random oracle) where the CRS is an uniformly random string of size independent of the number of senders. The property is of the essence when such protocols are used in large scale.
2019
ASIACRYPT
Indifferentiability of Truncated Random Permutations
One of natural ways of constructing a pseudorandom function from a pseudorandom permutation is to simply truncate the output of the permutation. When n is the permutation size and m is the number of truncated bits, the resulting construction is known to be indistinguishable from a random function up to $2^{{n+m}\over 2}$ queries, which is tight.In this paper, we study the indifferentiability of a truncated random permutation where a fixed prefix is prepended to the inputs. We prove that this construction is (regularly) indifferentiable from a public random function up to $\min \{2^{{n+m}\over 3}, 2^{m}, 2^\ell \}$ queries, while it is publicly indifferentiable up to $\min \{ \max \{2^{{n+m}\over 3}, 2^{n \over 2}\}, 2^\ell \}$ queries, where $\ell$ is the size of the fixed prefix. Furthermore, the regular indifferentiability bound is proved to be tight when $m+\ell \ll n$.Our results significantly improve upon the previous bound of $\min \{ 2^{m \over 2}, 2^\ell \}$ given by Dodis et al. (FSE 2009), allowing us to construct, for instance, an $\frac{n}{2}$-to-$\frac{n}{2}$ bit random function that makes a single call to an n-bit permutation, achieving $\frac{n}{2}$-bit security.
2019
ASIACRYPT
Anonymous AE
The customary formulation of authenticated encryption (AE) requires the decrypting party to supply the correct nonce with each ciphertext it decrypts. To enable this, the nonce is often sent in the clear alongside the ciphertext. But doing this can forfeit anonymity and degrade usability. Anonymity can also be lost by transmitting associated data (AD) or a session-ID (used to identify the operative key). To address these issues, we introduce anonymous AE, wherein ciphertexts must conceal their origin even when they are understood to encompass everything needed to decrypt (apart from the receiver’s secret state). We formalize a type of anonymous AE we call anAE, anonymous nonce-based AE, which generalizes and strengthens conventional nonce-based AE, nAE. We provide an efficient construction for anAE, NonceWrap, from an nAE scheme and a blockcipher. We prove NonceWrap secure. While anAE does not address privacy loss through traffic-flow analysis, it does ensure that ciphertexts, now more expansively construed, do not by themselves compromise privacy.
2019
ASIACRYPT
iUC: Flexible Universal Composability Made Simple
Proving the security of complex protocols is a crucial and very challenging task. A widely used approach for reasoning about such protocols in a modular way is universal composability. A perfect model for universal composability should provide a sound basis for formal proofs and be very flexible in order to allow for modeling a multitude of different protocols. It should also be easy to use, including useful design conventions for repetitive modeling aspects, such as corruption, parties, sessions, and subroutine relationships, such that protocol designers can focus on the core logic of their protocols.While many models for universal composability exist, including the UC, GNUC, and IITM models, none of them has achieved this ideal goal yet. As a result, protocols cannot be modeled faithfully and/or using these models is a burden rather than a help, often even leading to underspecified protocols and formally incorrect proofs.Given this dire state of affairs, the goal of this work is to provide a framework for universal composability which combines soundness, flexibility, and usability in an unmatched way. Developing such a security framework is a very difficult and delicate task, as the long history of frameworks for universal composability shows.We build our framework, called iUC, on top of the IITM model, which already provides soundness and flexibility while lacking sufficient usability. At the core of iUC is a single simple template for specifying essentially arbitrary protocols in a convenient, formally precise, and flexible way. We illustrate the main features of our framework with example functionalities and realizations.
2019
ASIACRYPT
Anomalies and Vector Space Search: Tools for S-Box Analysis
S-boxes are functions with an input so small that the simplest way to specify them is their lookup table (LUT). How can we quantify the distance between the behavior of a given S-box and that of an S-box picked uniformly at random?To answer this question, we introduce various “anomalies”. These real numbers are such that a property with an anomaly equal to a should be found roughly once in a set of $2^{a}$ random S-boxes. First, we present statistical anomalies based on the distribution of the coefficients in the difference distribution table, linear approximation table, and for the first time, the boomerang connectivity table.We then count the number of S-boxes that have block-cipher like structures to estimate the anomaly associated to those. In order to recover these structures, we show that the most general tool for decomposing S-boxes is an algorithm efficiently listing all the vector spaces of a given dimension contained in a given set, and we present such an algorithm.Combining these approaches, we conclude that all permutations that are actually picked uniformly at random always have essentially the same cryptographic properties and the same lack of structure.
2019
ASIACRYPT
Sponges Resist Leakage: The Case of Authenticated Encryption
In this work we advance the study of leakage-resilient Authenticated Encryption with Associated Data (AEAD) and lay the theoretical groundwork for building such schemes from sponges. Building on the work of Barwell et al. (ASIACRYPT 2017), we reduce the problem of constructing leakage-resilient AEAD schemes to that of building fixed-input-length function families that retain pseudorandomness and unpredictability in the presence of leakage. Notably, neither property is implied by the other in the leakage-resilient setting. We then show that such a function family can be combined with standard primitives, namely a pseudorandom generator and a collision-resistant hash, to yield a nonce-based AEAD scheme. In addition, our construction is quite efficient in that it requires only two calls to this leakage-resilient function per encryption or decryption call. This construction can be instantiated entirely from the T-sponge to yield a concrete AEAD scheme which we call ${ \textsc {Slae}}$. We prove this sponge-based instantiation secure in the non-adaptive leakage setting. ${ \textsc {Slae}}$ bears many similarities and is indeed inspired by ${ \textsc {Isap}}$, which was proposed by Dobraunig et al. at FSE 2017. However, while retaining most of the practical advantages of ${ \textsc {Isap}}$, ${ \textsc {Slae}}$ additionally benefits from a formal security treatment.
2019
ASIACRYPT
Wave: A New Family of Trapdoor One-Way Preimage Sampleable Functions Based on Codes
Best Paper
We present here a new family of trapdoor one-way functions that are Preimage Sampleable on Average (PSA) based on codes, the Wave-PSA family. The trapdoor function is one-way under two computational assumptions: the hardness of generic decoding for high weights and the indistinguishability of generalized $(U,U+V)$-codes. Our proof follows the GPV strategy [28]. By including rejection sampling, we ensure the proper distribution for the trapdoor inverse output. The domain sampling property of our family is ensured by using and proving a variant of the left-over hash lemma. We instantiate the new Wave-PSA family with ternary generalized $(U,U+V)$-codes to design a “hash-and-sign” signature scheme which achieves existential unforgeability under adaptive chosen message attacks (EUF-CMA) in the random oracle model.
2019
ASIACRYPT
Leakage Resilience of the Duplex Construction
Side-channel attacks, especially differential power analysis (DPA), pose a serious threat to cryptographic implementations deployed in a malicious environment. One way to counter side-channel attacks is to design cryptographic schemes to withstand them, an area that is covered amongst others by leakage resilient cryptography. So far, however, leakage resilient cryptography has predominantly focused on block cipher based designs, and insights in permutation based leakage resilient cryptography are scarce. In this work, we consider leakage resilience of the keyed duplex construction: we present a model for leakage resilient duplexing, derive a fine-grained bound on the security of the keyed duplex in said model, and map it to ideas of Taha and Schaumont (HOST 2014) and Dobraunig et al. (ToSC 2017) in order to use the duplex in a leakage resilient manner.
2019
ASIACRYPT
CSI-FiSh: Efficient Isogeny Based Signatures Through Class Group Computations
In this paper we report on a new record class group computation of an imaginary quadratic field having 154-digit discriminant, surpassing the previous record of 130 digits. This class group is central to the CSIDH-512 isogeny based cryptosystem, and knowing the class group structure and relation lattice implies efficient uniform sampling and a canonical representation of its elements. Both operations were impossible before and allow us to instantiate an isogeny based signature scheme first sketched by Stolbunov. We further optimize the scheme using multiple public keys and Merkle trees, following an idea by De Feo and Galbraith. We also show that including quadratic twists allows to cut the public key size in half for free. Optimizing for signature size, our implementation takes 390 ms to sign/verify and results in signatures of 263 bytes, at the expense of a large public key. This is 300 times faster and over 3 times smaller than an optimized version of SeaSign for the same parameter set. Optimizing for public key and signature size combined, results in a total size of 1468 bytes, which is smaller than any other post-quantum signature scheme at the 128-bit security level.
2019
ASIACRYPT
Dual Isogenies and Their Application to Public-Key Compression for Isogeny-Based Cryptography
The isogeny-based protocols SIDH and SIKE have received much attention for being post-quantum key agreement candidates that retain relatively small keys. A recent line of work has proposed and further improved compression of public keys, leading to the inclusion of public-key compression in the SIKE proposal for Round 2 of the NIST Post-Quantum Cryptography Standardization effort. We show how to employ the dual isogeny to significantly increase performance of compression techniques, reducing their overhead from 160–182% to 77–86% for Alice’s key generation and from 98–104% to 59–61% for Bob’s across different SIDH parameter sets. For SIKE, we reduce the overhead of (1) key generation from 140–153% to 61–74%, (2) key encapsulation from 67–90% to 38–57%, and (3) decapsulation from 59–65% to 34–39%. This is mostly achieved by speeding up the pairing computations, which has until now been the main bottleneck, but we also improve (deterministic) basis generation.
2019
ASIACRYPT
Verifiable Delay Functions from Supersingular Isogenies and Pairings
We present two new Verifiable Delay Functions (VDF) based on assumptions from elliptic curve cryptography. We discuss both the advantages and drawbacks of our constructions, we study their security and we demonstrate their practicality with a proof-of-concept implementation.
2019
ASIACRYPT
New Code-Based Privacy-Preserving Cryptographic Constructions
Code-based cryptography has a long history but did suffer from periods of slow development. The field has recently attracted a lot of attention as one of the major branches of post-quantum cryptography. However, its subfield of privacy-preserving cryptographic constructions is still rather underdeveloped, e.g., important building blocks such as zero-knowledge range proofs and set membership proofs, and even proofs of knowledge of a hash preimage, have not been known under code-based assumptions. Moreover, almost no substantial technical development has been introduced in the last several years.This work introduces several new code-based privacy-preserving cryptographic constructions that considerably advance the state-of-the-art in code-based cryptography. Specifically, we present 3 major contributions, each of which potentially yields various other applications. Our first contribution is a code-based statistically hiding and computationally binding commitment scheme with companion zero-knowledge (ZK) argument of knowledge of a valid opening that can be easily extended to prove that the committed bits satisfy other relations. Our second contribution is the first code-based zero-knowledge range argument for committed values, with communication cost logarithmic in the size of the range. A special feature of our range argument is that, while previous works on range proofs/arguments (in all branches of cryptography) only address ranges of non-negative integers, our protocol can handle signed fractional numbers, and hence, can potentially find a larger scope of applications. Our third contribution is the first code-based Merkle-tree accumulator supported by ZK argument of membership, which has been known to enable various interesting applications. In particular, it allows us to obtain the first code-based ring signatures and group signatures with logarithmic signature sizes.
2019
ASIACRYPT
A Critical Analysis of ISO 17825 (‘Testing Methods for the Mitigation of Non-invasive Attack Classes Against Cryptographic Modules’)
The ISO standardisation of ‘Testing methods for the mitigation of non-invasive attack classes against cryptographic modules’ (ISO/IEC 17825:2016) specifies the use of the Test Vector Leakage Assessment (TVLA) framework as the sole measure to assess whether or not an implementation of (symmetric) cryptography is vulnerable to differential side-channel attacks. It is the only publicly available standard of this kind, and the first side-channel assessment regime to exclusively rely on a TVLA instantiation.TVLA essentially specifies statistical leakage detection tests with the aim of removing the burden of having to test against an ever increasing number of attack vectors. It offers the tantalising prospect of ‘conformance testing’: if a device passes TVLA, then, one is led to hope, the device would be secure against all (first-order) differential side-channel attacks.In this paper we provide a statistical assessment of the specific instantiation of TVLA in this standard. This task leads us to inquire whether (or not) it is possible to assess the side-channel security of a device via leakage detection (TVLA) only. We find a number of grave issues in the standard and its adaptation of the original TVLA guidelines. We propose some innovations on existing methodologies and finish by giving recommendations for best practice and the responsible reporting of outcomes.
2019
ASIACRYPT
Optimized Method for Computing Odd-Degree Isogenies on Edwards Curves
In this paper, we present an efficient method to compute arbitrary odd-degree isogenies on Edwards curves. By using the w-coordinate, we optimized the isogeny formula on Edwards curves by Moody and Shumow. We demonstrate that Edwards curves have an additional benefit when recovering the coefficient of the image curve during isogeny computation. For $\ell$-degree isogeny where $\ell =2s+1$, our isogeny formula on Edwards curves outperforms Montgomery curves when $s \ge 2$. To better represent the performance improvements when w-coordinate is used, we implement CSIDH using our isogeny formula. Our implementation is about 20% faster than the previous implementation. The result of our work opens the door for the usage of Edwards curves in isogeny-based cryptography, especially for CSIDH which requires higher degree isogenies.
2019
ASIACRYPT
Strongly Secure Authenticated Key Exchange from Supersingular Isogenies
This paper aims to address the open problem, namely, to find new techniques to design and prove security of supersingular isogeny-based authenticated key exchange (AKE) protocols against the widest possible adversarial attacks, raised by Galbraith in 2018. Concretely, we present two AKEs based on a double-key PKE in the supersingular isogeny setting secure in the sense of CK$^+$, one of the strongest security models for AKE. Our contributions are summarised as follows. Firstly, we propose a strong OW-CPA secure PKE, $\mathsf {2PKE_{sidh}}$, based on SI-DDH assumption. By applying modified Fujisaki-Okamoto transformation, we obtain a [OW-CCA, OW-CPA] secure KEM, $\mathsf {2KEM_{sidh}}$. Secondly, we propose a two-pass AKE, $\mathsf {SIAKE}_2$, based on SI-DDH assumption, using $\mathsf {2KEM_{sidh}}$ as a building block. Thirdly, we present a modified version of $\mathsf {2KEM_{sidh}}$ that is secure against leakage under the 1-Oracle SI-DH assumption. Using the modified $\mathsf {2KEM_{sidh}}$ as a building block, we then propose a three-pass AKE, $\mathsf {SIAKE}_3$, based on 1-Oracle SI-DH assumption. Finally, we prove that both $\mathsf {SIAKE}_2$ and $\mathsf {SIAKE}_3$ are CK$^+$ secure in the random oracle model and supports arbitrary registration. We also provide an implementation to illustrate the efficiency of our schemes. Our schemes compare favourably against existing isogeny-based AKEs. To the best of our knowledge, they are the first of its kind to offer security against arbitrary registration, wPFS, KCI, and MEX simultaneously. Regarding efficiency, our schemes outperform existing schemes in terms of bandwidth as well as CPU cycle count.
2019
ASIACRYPT
Location, Location, Location: Revisiting Modeling and Exploitation for Location-Based Side Channel Leakages
Near-field microprobes have the capability to isolate small regions of a chip surface and enable precise measurements with high spatial resolution. Being able to distinguish the activity of small regions has given rise to the location-based side-channel attacks, which exploit the spatial dependencies of cryptographic algorithms in order to recover the secret key. Given the fairly uncharted nature of such leakages, this work revisits the location side-channel to broaden our modeling and exploitation capabilities. Our contribution is threefold. First, we provide a simple spatial model that partially captures the effect of location-based leakages. We use the newly established model to simulate the leakage of different scenarios/countermeasures and follow an information-theoretic approach to evaluate the security level achieved in every case. Second, we perform the first successful location-based attack on the SRAM of a modern ARM Cortex-M4 chip, using standard techniques such as difference of means and multivariate template attacks. Third, we put forward neural networks as classifiers that exploit the location side-channel and showcase their effectiveness on ARM Cortex-M4, especially in the context of single-shot attacks and small memory regions. Template attacks and neural network classifiers are able to reach high spacial accuracy, distinguishing between 2 SRAM regions of 128 bytes each with 100% success rate and distinguishing even between 256 SRAM byte-regions with 32% success rate. Such improved exploitation capabilities revitalize the interest for location vulnerabilities on various implementations, ranging from RSA/ECC with large memory footprint, to lookup-table-based AES with smaller memory usage.
2019
ASIACRYPT
Hard Isogeny Problems over RSA Moduli and Groups with Infeasible Inversion
We initiate the study of computational problems on elliptic curve isogeny graphs defined over RSA moduli. We conjecture that several variants of the neighbor-search problem over these graphs are hard, and provide a comprehensive list of cryptanalytic attempts on these problems. Moreover, based on the hardness of these problems, we provide a construction of groups with infeasible inversion, where the underlying groups are the ideal class groups of imaginary quadratic orders.Recall that in a group with infeasible inversion, computing the inverse of a group element is required to be hard, while performing the group operation is easy. Motivated by the potential cryptographic application of building a directed transitive signature scheme, the search for a group with infeasible inversion was initiated in the theses of Hohenberger and Molnar (2003). Later it was also shown to provide a broadcast encryption scheme by Irrer et al. (2004). However, to date the only case of a group with infeasible inversion is implied by the much stronger primitive of self-bilinear map constructed by Yamakawa et al. (2014) based on the hardness of factoring and indistinguishability obfuscation (iO). Our construction gives a candidate without using iO.
2019
ASIACRYPT
Streamlined Blockchains: A Simple and Elegant Approach (A Tutorial and Survey)
A blockchain protocol (also called state machine replication) allows a set of nodes to agree on an ever-growing, linearly ordered log of transactions. The classical consensus literature suggests two approaches for constructing a blockchain protocol: (1) through composition of single-shot consensus instances often called Byzantine Agreement; and (2) through direct construction of a blockchain where there is no clear-cut boundary between single-shot consensus instances. While conceptually simple, the former approach precludes cross-instance optimizations in a practical implementation. This perhaps explains why the latter approach has gained more traction in practice: specifically, well-known protocols such as Paxos and PBFT all follow the direct-construction approach.In this tutorial, we present a new paradigm called “streamlined blockchains” for directly constructing blockchain protocols. This paradigm enables a new family of protocols that are extremely simple and natural: every epoch, a proposer proposes a block extending from a notarized parent chain, and nodes vote if the proposal’s parent chain is not . Whenever a block gains votes, it becomes notarized. Whenever a node observes a notarized chain with blocks of consecutive epochs at the end, then the entire chain chopping off blocks at the end is final.By varying the parameters highlighted in , we illustrate two variants for the partially synchronous and synchronous settings respectively. We present very simple proofs of consistency and liveness. We hope that this tutorial provides a compelling argument why this new family of protocols should be used in lieu of classical candidates (e.g., PBFT, Paxos, and their variants), both in practical implementation and for pedagogical purposes.
2019
ASIACRYPT
Collision Resistant Hashing from Sub-exponential Learning Parity with Noise
The Learning Parity with Noise (LPN) problem has recently found many cryptographic applications such as authentication protocols, pseudorandom generators/functions and even asymmetric tasks including public-key encryption (PKE) schemes and oblivious transfer (OT) protocols. It however remains a long-standing open problem whether LPN implies collision resistant hash (CRH) functions. Inspired by the recent work of Applebaum et al. (ITCS 2017), we introduce a general construction of CRH from LPN for various parameter choices. We show that, just to mention a few notable ones, under any of the following hardness assumptions (for the two most common variants of LPN) 1.constant-noise LPN is $2^{n^{0.5+\varepsilon }}$-hard for any constant $\varepsilon >0$;2.constant-noise LPN is $2^{\varOmega (n/\log n)}$-hard given $q=\mathsf {poly}(n)$ samples;3.low-noise LPN (of noise rate $1/\sqrt{n}$) is $2^{\varOmega (\sqrt{n}/\log n)}$-hard given $q=\mathsf {poly}(n)$ samples. there exists CRH functions with constant (or even poly-logarithmic) shrinkage, which can be implemented using polynomial-size depth-3 circuits with NOT, (unbounded fan-in) AND and XOR gates. Our technical route LPN $\rightarrow$ bSVP $\rightarrow$ CRH is reminiscent of the known reductions for the large-modulus analogue, i.e., LWE $\rightarrow$ SIS $\rightarrow$ CRH, where the binary Shortest Vector Problem (bSVP) was recently introduced by Applebaum et al. (ITCS 2017) that enables CRH in a similar manner to Ajtai’s CRH functions based on the Short Integer Solution (SIS) problem.Furthermore, under additional (arguably minimal) idealized assumptions such as small-domain random functions or random permutations (that trivially imply collision resistance), we still salvage a simple and elegant collision-resistance-preserving domain extender combining the best of the two worlds, namely, maximized (depth one) parallelizability and polynomial shrinkage. In particular, assume $2^{n^{0.5+\varepsilon }}$-hard constant-noise LPN or $2^{n^{0.25+\varepsilon }}$-hard low-noise LPN, we obtain a collision resistant hash function that evaluates in parallel only a single layer of small-domain random functions (or random permutations) and shrinks polynomially.
2019
ASIACRYPT
Approximate Trapdoors for Lattices and Smaller Hash-and-Sign Signatures
We study a relaxed notion of lattice trapdoor called approximate trapdoor, which is defined to be able to invert Ajtai’s one-way function approximately instead of exactly. The primary motivation of our study is to improve the efficiency of the cryptosystems built from lattice trapdoors, including the hash-and-sign signatures.Our main contribution is to construct an approximate trapdoor by modifying the gadget trapdoor proposed by Micciancio and Peikert [Eurocrypt 2012]. In particular, we show how to use the approximate gadget trapdoor to sample short preimages from a distribution that is simulatable without knowing the trapdoor. The analysis of the distribution uses a theorem (implicitly used in past works) regarding linear transformations of discrete Gaussians on lattices.Our approximate gadget trapdoor can be used together with the existing optimization techniques to improve the concrete performance of the hash-and-sign signature in the random oracle model under (Ring-)LWE and (Ring-)SIS assumptions. Our implementation shows that the sizes of the public-key & signature can be reduced by half from those in schemes built from exact trapdoors.
2019
ASIACRYPT
Dual-Mode NIZKs from Obfuscation
Two standard security properties of a non-interactive zero-knowledge (NIZK) scheme are soundness and zero-knowledge. But while standard NIZK systems can only provide one of those properties against unbounded adversaries, dual-mode NIZK systems allow to choose dynamically and adaptively which of these properties holds unconditionally. The only known dual-mode NIZK schemes are Groth-Sahai proofs (which have proved extremely useful in a variety of applications), and the FHE-based NIZK constructions of Canetti et al. and Peikert et al, which are concurrent and independent to this work. However, all these constructions rely on specific algebraic settings.Here, we provide a generic construction of dual-mode NIZK systems for all of NP. The public parameters of our scheme can be set up in one of two indistinguishable ways. One way provides unconditional soundness, while the other provides unconditional zero-knowledge. Our scheme relies on subexponentially secure indistinguishability obfuscation and subexponentially secure one-way functions, but otherwise only on comparatively mild and generic computational assumptions. These generic assumptions can be instantiated under any one of the DDH, $k$-LIN, DCR, or QR assumptions.As an application, we reduce the required assumptions necessary for several recent obfuscation-based constructions of multilinear maps. Combined with previous work, our scheme can be used to construct multilinear maps from obfuscation and a group in which the strong Diffie-Hellman assumption holds. We also believe that our work adds to the understanding of the construction of NIZK systems, as it provides a conceptually new way to achieve dual-mode properties.
2019
ASIACRYPT
Simple Refreshing in the Noisy Leakage Model
Masking schemes are a prominent countermeasure against power analysis and work by concealing the values that are produced during the computation through randomness. The randomness is typically injected into the masked algorithm using a so-called refreshing scheme, which is placed after each masked operation, and hence is one of the main bottlenecks for designing efficient masking schemes. The main contribution of our work is to investigate the security of a very simple and efficient refreshing scheme and prove its security in the noisy leakage model (EUROCRYPT’13). Compared to earlier constructions our refreshing is significantly more efficient and uses only n random values and ${<}2n$ operations, where n is the security parameter. In addition we show how our refreshing can be used in more complex masked computation in the presence of noisy leakage. Our results are established using a new methodology for analyzing masking schemes in the noisy leakage model, which may be of independent interest.
2019
ASIACRYPT
On Kilian’s Randomization of Multilinear Map Encodings
Indistinguishability obfuscation constructions based on matrix branching programs generally proceed in two steps: first apply Kilian’s randomization of the matrix product computation, and then encode the matrices using a multilinear map scheme. In this paper we observe that by applying Kilian’s randomization after encoding, the complexity of the best attacks is significantly increased for CLT13 multilinear maps. This implies that much smaller parameters can be used, which improves the efficiency of the constructions by several orders of magnitude.As an application, we describe the first concrete implementation of multiparty non-interactive Diffie-Hellman key exchange secure against existing attacks. Key exchange was originally the most straightforward application of multilinear maps; however it was quickly broken for the three known families of multilinear maps (GGH13, CLT13 and GGH15). Here we describe the first implementation of key exchange that is resistant against known attacks, based on CLT13 multilinear maps. For $N=4$ users and a medium level of security, our implementation requires 18 GB of public parameters, and a few minutes for the derivation of a shared key.
2019
ASIACRYPT
Decisional Second-Preimage Resistance: When Does SPR Imply PRE?
There is a well-known gap between second-preimage resistance and preimage resistance for length-preserving hash functions. This paper introduces a simple concept that fills this gap. One consequence of this concept is that tight reductions can remove interactivity for multi-target length-preserving preimage problems, such as the problems that appear in analyzing hash-based signature systems. Previous reduction techniques applied to only a negligible fraction of all length-preserving hash functions, presumably excluding all off-the-shelf hash functions.
2019
ASIACRYPT
Output Compression, MPC, and iO for Turing Machines
In this work, we study the fascinating notion of output-compressing randomized encodings for Turing Machines, in a shared randomness model. In this model, the encoder and decoder have access to a shared random string, and the efficiency requirement is, the size of the encoding must be independent of the running time and output length of the Turing Machine on the given input, while the length of the shared random string is allowed to grow with the length of the output. We show how to construct output-compressing randomized encodings for Turing machines in the shared randomness model, assuming iO for circuits and any assumption in the set $\{$ LWE, DDH, N $^{th}$ Residuosity $\}$ .We then show interesting implications of the above result to basic feasibility questions in the areas of secure multiparty computation (MPC) and indistinguishability obfuscation (iO): 1.Compact MPC for Turing Machines in the Random Oracle Model. In the context of MPC, we consider the following basic feasibility question: does there exist a malicious-secure MPC protocol for Turing Machines whose communication complexity is independent of the running time and output length of the Turing Machine when executed on the combined inputs of all parties? We call such a protocol as a compact MPC protocol. Hubácek and Wichs [HW15] showed via an incompressibility argument, that, even for the restricted setting of circuits, it is impossible to construct a malicious secure two party computation protocol in the plain model where the communication complexity is independent of the output length. In this work, we show how to evade this impossibility by compiling any (non-compact) MPC protocol in the plain model to a compact MPC protocol for Turing Machines in the Random Oracle Model, assuming output-compressing randomized encodings in the shared randomness model.2.Succinct iO for Turing Machines in the Shared Randomness Model. In all existing constructions of iO for Turing Machines, the size of the obfuscated program grows with a bound on the input length. In this work, we show how to construct an iO scheme for Turing Machines in the shared randomness model where the size of the obfuscated program is independent of a bound on the input length, assuming iO for circuits and any assumption in the set $\{$ LWE, DDH, N $^{th}$ Residuosity $\}$ .
2019
ASIACRYPT
The Exchange Attack: How to Distinguish Six Rounds of AES with $2^{88.2}$Chosen Plaintexts
In this paper we present exchange-equivalence attacks which is a new cryptanalytic attack technique suitable for SPN-like block cipher designs. Our new technique results in the first secret-key chosen plaintext distinguisher for 6-round AES. The complexity of the distinguisher is about $2^{88.2}$ in terms of data, memory and computational complexity. The distinguishing attack for AES reduced to six rounds is a straight-forward extension of an exchange attack for 5-round AES that requires $2^{30}$ in terms of chosen plaintexts and computation. This is also a new record for AES reduced to five rounds. The main result of this paper is that AES up to at least six rounds is biased when restricted to exchange-invariant sets of plaintexts.
2019
ASIACRYPT
Cryptanalysis of CLT13 Multilinear Maps with Independent Slots
Many constructions based on multilinear maps require independent slots in the plaintext, so that multiple computations can be performed in parallel over the slots. Such constructions are usually based on CLT13 multilinear maps, since CLT13 inherently provides a composite encoding space, with a plaintext ring $\bigoplus _{i=1}^n \mathbb {Z}/g_i\mathbb {Z}$ for small primes $g_i$ ’s. However, a vulnerability was identified at Crypto 2014 by Gentry, Lewko and Waters, with a lattice-based attack in dimension 2, and the authors have suggested a simple countermeasure. In this paper, we identify an attack based on higher dimension lattice reduction that breaks the author’s countermeasure for a wide range of parameters. Combined with the Cheon et al. attack from Eurocrypt 2015, this leads to the recovery of all the secret parameters of CLT13, assuming that low-level encodings of almost zero plaintexts are available. We show how to apply our attack against various constructions based on composite-order CLT13. For the [FRS17] construction, our attack enables to recover the secret CLT13 plaintext ring for a certain range of parameters; however, breaking the indistinguishability of the branching program remains an open problem.
2019
ASIACRYPT
Algebraic Cryptanalysis of STARK-Friendly Designs: Application to MARVELlous and MiMC
The block cipher Jarvis and the hash function Friday, both members of the MARVELlous family of cryptographic primitives, are among the first proposed solutions to the problem of designing symmetric-key algorithms suitable for transparent, post-quantum secure zero-knowledge proof systems such as ZK-STARKs. In this paper we describe an algebraic cryptanalysis of Jarvis and Friday and show that the proposed number of rounds is not sufficient to provide adequate security. In Jarvis, the round function is obtained by combining a finite field inversion, a full-degree affine permutation polynomial and a key addition. Yet we show that even though the high degree of the affine polynomial may prevent some algebraic attacks (as claimed by the designers), the particular algebraic properties of the round function make both Jarvis and Friday vulnerable to Gröbner basis attacks. We also consider MiMC, a block cipher similar in structure to Jarvis. However, this cipher proves to be resistant against our proposed attack strategy. Still, our successful cryptanalysis of Jarvis and Friday does illustrate that block cipher designs for “algebraic platforms” such as STARKs, FHE or MPC may be particularly vulnerable to algebraic attacks.
2019
ASIACRYPT
Collusion Resistant Watermarking Schemes for Cryptographic Functionalities
A cryptographic watermarking scheme embeds a message into a program while preserving its functionality. Recently, a number of watermarking schemes have been proposed, which are proven secure in the sense that given one marked program, any attempt to remove the embedded message will substantially change its functionality.In this paper, we formally initiate the study of collusion attacks for watermarking schemes, where the attacker’s goal is to remove the embedded messages given multiple copies of the same program, each with a different embedded message. This is motivated by practical scenarios, where a program may be marked multiple times with different messages.The results of this work are twofold. First, we examine existing cryptographic watermarking schemes and observe that all of them are vulnerable to collusion attacks. Second, we construct collusion resistant watermarking schemes for various cryptographic functionalities (e.g., pseudorandom function evaluation, decryption, etc.). To achieve our second result, we present a new primitive called puncturable functional encryption scheme, which may be of independent interest.
2019
ASIACRYPT
Algebraic XOR-RKA-Secure Pseudorandom Functions from Post-Zeroizing Multilinear Maps
Due to the vast number of successful related-key attacks against existing block-ciphers, related-key security has become a common design goal for such primitives. In these attacks, the adversary is not only capable of seeing the output of a function on inputs of its choice, but also on related keys. At Crypto 2010, Bellare and Cash proposed the first construction of a pseudorandom function that could provably withstand such attacks based on standard assumptions. Their construction, as well as several others that appeared more recently, have in common the fact that they only consider linear or polynomial functions of the secret key over complex groups. In reality, however, most related-key attacks have a simpler form, such as the XOR of the key with a known value. To address this problem, we propose the first construction of RKA-secure pseudorandom function for XOR relations. Our construction relies on multilinear maps and, hence, can only be seen as a feasibility result. Nevertheless, we remark that it can be instantiated under two of the existing multilinear-map candidates since it does not reveal any encodings of zero. To achieve this goal, we rely on several techniques that were used in the context of program obfuscation, but we also introduce new ones to address challenges that are specific to the related-key-security setting.
2019
ASIACRYPT
MILP-aided Method of Searching Division Property Using Three Subsets and Applications
Division property is a generalized integral property proposed by Todo at EUROCRYPT 2015, and then conventional bit-based division property (CBDP) and bit-based division property using three subsets (BDPT) were proposed by Todo and Morii at FSE 2016. At the very beginning, the two kinds of bit-based division properties once couldn’t be applied to ciphers with large block size just because of the huge time and memory complexity. At ASIACRYPT 2016, Xiang et al. extended Mixed Integer Linear Programming (MILP) method to search integral distinguishers based on CBDP. BDPT can find more accurate integral distinguishers than CBDP, but it couldn’t be modeled efficiently.This paper focuses on the feasibility of searching integral distinguishers based on BDPT. We propose the pruning techniques and fast propagation of BDPT for the first time. Based on these, an MILP-aided method for the propagation of BDPT is proposed. Then, we apply this method to some block ciphers. For SIMON64, PRESENT, and RECTANGLE, we find more balanced bits than the previous longest distinguishers. For LBlock, we find a better 16-round integral distinguisher with less active bits. For other block ciphers, our results are in accordance with the previous longest distinguishers.Cube attack is an important cryptanalytic technique against symmetric cryptosystems, especially for stream ciphers. And the most important step in cube attack is superpoly recovery. Inspired by the CBDP based cube attack proposed by Todo at CRYPTO 2017, we propose a method which uses BDPT to recover the superpoly in cube attack. We apply this new method to round-reduced Trivium. To be specific, the time complexity of recovering the superpoly of 832-round Trivium at CRYPTO 2017 is reduced from $2^{77}$ to practical, and the time complexity of recovering the superpoly of 839-round Trivium at CRYPTO 2018 is reduced from $2^{79}$ to practical. Then, we propose a theoretical attack which can recover the superpoly of Trivium up to 841 round.
2019
ASIACRYPT
Valiant’s Universal Circuits Revisited: An Overall Improvement and a Lower Bound
A universal circuit (UC) is a general-purpose circuit that can simulate arbitrary circuits (up to a certain size n). At STOC 1976 Valiant presented a graph theoretic approach to the construction of UCs, where a UC is represented by an edge universal graph (EUG) and is recursively constructed using a dedicated graph object (referred to as supernode). As a main end result, Valiant constructed a 4-way supernode of size 19 and an EUG of size $4.75n\log n$ (omitting smaller terms), which remained the most size-efficient even to this day (after more than 4 decades).Motivated by the emerging applications of UCs in various privacy preserving computation scenarios, we revisit Valiant’s universal circuits, and propose a 4-way supernode of size 18, and an EUG of size $4.5n\log n$. As confirmed by our implementations, we reduce the size of universal circuits (and the number of AND gates) by more than 5% in general, and thus improve upon the efficiency of UC-based cryptographic applications accordingly. Our approach to the design of optimal supernodes is computer aided (rather than by hand as in previous works), which might be of independent interest. As a complement, we give lower bounds on the size of EUGs and UCs in Valiant’s framework, which significantly improves upon the generic lower bound on UC size and therefore reduces the gap between theory and practice of universal circuits.
2019
ASIACRYPT
Numerical Method for Comparison on Homomorphically Encrypted Numbers
We propose a new method to compare numbers which are encrypted by Homomorphic Encryption (HE). Previously, comparison and min/max functions were evaluated using Boolean functions where input numbers are encrypted bit-wise. However, the bit-wise encryption methods require relatively expensive computations for basic arithmetic operations such as addition and multiplication.In this paper, we introduce iterative algorithms that approximately compute the min/max and comparison operations of several numbers which are encrypted word-wise. From the concrete error analyses, we show that our min/max and comparison algorithms have $\varTheta (\alpha )$ and $\varTheta (\alpha \log \alpha )$ computational complexity to obtain approximate values within an error rate $2^{-\alpha }$, while the previous minimax polynomial approximation method requires the exponential complexity $\varTheta (2^{\alpha /2})$ and $\varTheta (\sqrt{\alpha }\cdot 2^{\alpha /2})$, respectively. Our algorithms achieve (quasi-)optimality in terms of asymptotic computational complexity among polynomial approximations for min/max and comparison operations. The comparison algorithm is extended to several applications such as computing the top-k elements and counting numbers over the threshold in encrypted state.Our method enables word-wise HEs to enjoy comparable performance in practice with bit-wise HEs for comparison operations while showing much better performance on polynomial operations. Computing an approximate maximum value of any two $\ell$-bit integers encrypted by HEAAN, up to error $2^{\ell -10}$, takes only 1.14 ms in amortized running time, which is comparable to the result based on bit-wise HEs.
2019
ASIACRYPT
The Broadcast Message Complexity of Secure Multiparty Computation
We study the broadcast message complexity of secure multiparty computation (MPC), namely, the total number of messages that are required for securely computing any functionality in the broadcast model of communication.MPC protocols are traditionally designed in the simultaneous broadcast model, where each round consists of every party broadcasting a message to the other parties. We show that this method of communication is sub-optimal; specifically, by eliminating simultaneity, it is, in fact, possible to reduce the broadcast message complexity of MPC.More specifically, we establish tight lower and upper bounds on the broadcast message complexity of n-party MPC for every $t<n$ corruption threshold, both in the plain model as well as common setup models. For example, our results show that the optimal broadcast message complexity of semi-honest MPC can be much lower than 2n, but necessarily requires at least three rounds of communication. We also extend our results to the malicious setting in setup models.
2019
ASIACRYPT
Cryptanalysis of GSM Encryption in 2G/3G Networks Without Rainbow Tables
The GSM standard developed by ETSI for 2G networks adopts the A5/1 stream cipher to protect the over-the-air privacy in cell phone and has become the de-facto global standard in mobile communications, though the emerging of subsequent 3G/4G standards. There are many cryptanalytic results available so far and the most notable ones share the need of a heavy pre-computation with large rainbow tables or distributed cracking network. In this paper, we present a fast near collision attack on GSM encryption in 2G/3G networks, which is completely new and more threatening compared to the previous best results. We adapt the fast near collision attack proposed at Eurocrypt 2018 with the concrete irregular clocking manner in A5/1 to have a state recovery attack with a low complexity. It is shown that if the first 64 bits of one keystream frame are available, the secret key of A5/1 can be reliably found in $2^{31.79}$ cipher ticks, given around 1 MB memory and after the pre-computation of $2^{20.26}$ cipher ticks. Our current implementation clearly certified the validity of the suggested attack. Due to the fact that A5/3 and GPRS share the same key with A5/1, this can be converted into attacks against any GSM network eventually.
2019
ASIACRYPT
Multi-Key Homomorphic Encryption from TFHE
In this paper, we propose a Multi-Key Homomorphic Encryption (MKHE) scheme by generalizing the low-latency homomorphic encryption by Chillotti et al. (ASIACRYPT 2016). Our scheme can evaluate a binary gate on ciphertexts encrypted under different keys followed by a bootstrapping.The biggest challenge to meeting the goal is to design a multiplication between a bootstrapping key of a single party and a multi-key RLWE ciphertext. We propose two different algorithms for this hybrid product. Our first method improves the ciphertext extension by Mukherjee and Wichs (EUROCRYPT 2016) to provide better performance. The other one is a whole new approach which has advantages in storage, complexity, and noise growth.Compared to previous work, our construction is more efficient in terms of both asymptotic and concrete complexity. The length of ciphertexts and the computational costs of a binary gate grow linearly and quadratically on the number of parties, respectively. We provide experimental results demonstrating the running time of a homomorphic NAND gate with bootstrapping. To the best of our knowledge, this is the first attempt in the literature to implement an MKHE scheme.
2019
ASIACRYPT
Beyond Honest Majority: The Round Complexity of Fair and Robust Multi-party Computation
Two of the most sought-after properties of Multi-party Computation (MPC) protocols are fairness and guaranteed output delivery (GOD), the latter also referred to as robustness. Achieving both, however, brings in the necessary requirement of malicious-minority. In a generalised adversarial setting where the adversary is allowed to corrupt both actively and passively, the necessary bound for a n-party fair or robust protocol turns out to be $t_a + t_p < n$, where $t_a,t_p$ denote the threshold for active and passive corruption with the latter subsuming the former. Subsuming the malicious-minority as a boundary special case, this setting, denoted as dynamic corruption, opens up a range of possible corruption scenarios for the adversary. While dynamic corruption includes the entire range of thresholds for $(t_a,t_p)$ starting from $(\lceil \frac{n}{2} \rceil - 1, \lfloor n/2 \rfloor )$ to $(0,n-1)$, the boundary corruption restricts the adversary only to the boundary cases of $(\lceil \frac{n}{2} \rceil - 1, \lfloor n/2 \rfloor )$ and $(0,n-1)$. Notably, both corruption settings empower an adversary to control majority of the parties, yet ensuring the count on active corruption never goes beyond $\lceil \frac{n}{2} \rceil - 1$. We target the round complexity of fair and robust MPC tolerating dynamic and boundary adversaries. As it turns out, $\lceil n/2 \rceil + 1$ rounds are necessary and sufficient for fair as well as robust MPC tolerating dynamic corruption. The non-constant barrier raised by dynamic corruption can be sailed through for a boundary adversary. The round complexity of 3 and 4 is necessary and sufficient for fair and GOD protocols respectively, with the latter having an exception of allowing 3 round protocols in the presence of a single active corruption. While all our lower bounds assume pair-wise private and broadcast channels and are resilient to the presence of both public (CRS) and private (PKI) setup, our upper bounds are broadcast-only and assume only public setup. The traditional and popular setting of malicious-minority, being restricted compared to both dynamic and boundary setting, requires 3 and 2 rounds in the presence of public and private setup respectively for both fair as well as GOD protocols.
2019
ASIACRYPT
Tightly Secure Inner Product Functional Encryption: Multi-input and Function-Hiding Constructions
Tightly secure cryptographic schemes have been extensively studied in the fields of chosen-ciphertext secure public-key encryption, identity-based encryption, signatures and more. We extend tightly secure cryptography to inner product functional encryption (IPFE) and present the first tightly secure schemes related to IPFE.We first construct a new IPFE scheme that is tightly secure in the multi-user and multi-challenge setting. In other words, the security of our scheme does not degrade even if an adversary obtains many ciphertexts generated by many users. Our scheme is constructible on a pairing-free group and secure under the matrix decisional Diffie-Hellman (MDDH) assumption, which is the generalization of the decisional Diffie-Hellman (DDH) assumption. Applying the known conversions by Lin (CRYPTO 2017) and Abdalla et al. (CRYPTO 2018) to our scheme, we can obtain the first tightly secure function-hiding IPFE scheme and multi-input IPFE (MIPFE) scheme respectively.Our second main contribution is the proposal of a new generic conversion from function-hiding IPFE to function-hiding MIPFE, which was left as an open problem by Abdalla et al. (CRYPTO 2018). We obtain the first tightly secure function-hiding MIPFE scheme by applying our conversion to the tightly secure function-hiding IPFE scheme described above.Finally, the security reductions of all our schemes are fully tight, which means that the security of our schemes is reduced to the MDDH assumption with a constant security loss.
2019
ASIACRYPT
Homomorphic Encryption for Finite Automata
We describe a somewhat homomorphic GSW-like encryption scheme, natively encrypting matrices rather than just single elements. This scheme offers much better performance than existing homomorphic encryption schemes for evaluating encrypted (nondeterministic) finite automata (NFAs). Differently from GSW, we do not know how to reduce the security of this scheme from LWE, instead we reduce it from a stronger assumption, that can be thought of as an inhomogeneous variant of the NTRU assumption. This assumption (that we term iNTRU) may be useful and interesting in its own right, and we examine a few of its properties. We also examine methods to encode regular expressions as NFAs, and in particular explore a new optimization problem, motivated by our application to encrypted NFA evaluation. In this problem, we seek to minimize the number of states in an NFA for a given expression, subject to the constraint on the ambiguity of the NFA.
2019
ASIACRYPT
Card-Based Cryptography Meets Formal Verification
Card-based cryptography provides simple and practicable protocols for performing secure multi-party computation (MPC) with just a deck of cards. For the sake of simplicity, this is often done using cards with only two symbols, e.g., and . Within this paper, we target the setting where all cards carry distinct symbols, catering for use-cases with commonly available standard decks and a weaker indistinguishability assumption. As of yet, the literature provides for only three protocols and no proofs for non-trivial lower bounds on the number of cards. As such complex proofs (handling very large combinatorial state spaces) tend to be involved and error-prone, we propose using formal verification for finding protocols and proving lower bounds. In this paper, we employ the technique of software bounded model checking (SBMC), which reduces the problem to a bounded state space, which is automatically searched exhaustively using a SAT solver as a backend.Our contribution is twofold: (a) We identify two protocols for converting between different bit encodings with overlapping bases, and then show them to be card-minimal. This completes the picture of tight lower bounds on the number of cards with respect to runtime behavior and shuffle properties of conversion protocols. For computing , we show that there is no protocol with finite runtime using four cards with distinguishable symbols and fixed output encoding, and give a four-card protocol with an expected finite runtime using only random cuts. (b) We provide a general translation of proofs for lower bounds to a bounded model checking framework for automatically finding card- and length-minimal protocols and to give additional confidence in lower bounds. We apply this to validate our method and, as an example, confirm our new protocol to have a shortest run for protocols using this number of cards.
2019
ASIACRYPT
Public-Key Function-Private Hidden Vector Encryption (and More)
We construct public-key function-private predicate encryption for the “small superset functionality,” recently introduced by Beullens and Wee (PKC 2019). This functionality captures several important classes of predicates:Point functions. For point function predicates, our construction is equivalent to public-key function-private anonymous identity-based encryption.Conjunctions. If the predicate computes a conjunction, our construction is a public-key function-private hidden vector encryption scheme. This addresses an open problem posed by Boneh, Raghunathan, and Segev (ASIACRYPT 2013).d-CNFs and read-once conjunctions of d-disjunctions for constant-size d. Our construction extends the group-based obfuscation schemes of Bishop et al. (CRYPTO 2018), Beullens and Wee (PKC 2019), and Bartusek et al. (EUROCRYPT 2019) to the setting of public-key function-private predicate encryption. We achieve an average-case notion of function privacy, which guarantees that a decryption key $\mathsf {sk} _f$ reveals nothing about f as long as f is drawn from a distribution with sufficient entropy. We formalize this security notion as a generalization of the (enhanced) real-or-random function privacy definition of Boneh, Raghunathan, and Segev (CRYPTO 2013). Our construction relies on bilinear groups, and we prove security in the generic bilinear group model.
2019
ASIACRYPT
Efficient Explicit Constructions of Multipartite Secret Sharing Schemes
Multipartite secret sharing schemes are those having a multipartite access structure, in which the set of participants is divided into several parts and all participants in the same part play an equivalent role. Secret sharing schemes for multipartite access structures have received considerable attention due to the fact that multipartite secret sharing can be seen as a natural and useful generalization of threshold secret sharing.This work deals with efficient and explicit constructions of ideal multipartite secret sharing schemes, while most of the known constructions are either inefficient or randomized. Most ideal multipartite secret sharing schemes in the literature can be classified as either hierarchical or compartmented. The main results are the constructions for ideal hierarchical access structures, a family that contains every ideal hierarchical access structure as a particular case such as the disjunctive hierarchical threshold access structure and the conjunctive hierarchical threshold access structure, and the constructions for compartmented access structures with upper bounds and compartmented access structures with lower bounds, two families of compartmented access structures.On the basis of the relationship between multipartite secret sharing schemes, polymatroids, and matroids, the problem of how to construct a scheme realizing a multipartite access structure can be transformed to the problem of how to find a representation of a matroid from a presentation of its associated polymatroid. In this paper, we give efficient algorithms to find representations of the matroids associated to the three families of multipartite access structures. More precisely, based on know results about integer polymatroids, for each of the three families of access structures, we give an efficient method to find a representation of the integer polymatroid over some finite field, and then over some finite extension of that field, we give an efficient method to find a presentation of the matroid associated to the integer polymatroid. Finally, we construct ideal linear schemes realizing the three families of multipartite access structures by efficient methods.
2019
ASIACRYPT
Multi-Client Functional Encryption for Linear Functions in the Standard Model from LWE
Multi-client functional encryption (MCFE) allows $\ell$ clients to encrypt ciphertexts $(\mathbf {C}_{t,1},\mathbf {C}_{t,2},\ldots ,\mathbf {C}_{t,\ell })$ under some label. Each client can encrypt his own data $X_i$ for a label t using a private encryption key $\mathsf {ek}_i$ issued by a trusted authority in such a way that, as long as all $\mathbf {C}_{t,i}$ share the same label t, an evaluator endowed with a functional key $\mathsf {dk}_f$ can evaluate $f(X_1,X_2,\ldots ,X_\ell )$ without learning anything else on the underlying plaintexts $X_i$. Functional decryption keys can be derived by the central authority using the master secret key. Under the Decision Diffie-Hellman assumption, Chotard et al. (Asiacrypt 2018) recently described an adaptively secure MCFE scheme for the evaluation of linear functions over the integers. They also gave a decentralized variant (DMCFE) of their scheme which does not rely on a centralized authority, but rather allows encryptors to issue functional secret keys in a distributed manner. While efficient, their constructions both rely on random oracles in their security analysis. In this paper, we build a standard-model MCFE scheme for the same functionality and prove it fully secure under adaptive corruptions. Our proof relies on the Learning-With-Errors ($\mathsf {LWE}$) assumption and does not require the random oracle model. We also provide a decentralized variant of our scheme, which we prove secure in the static corruption setting (but for adaptively chosen messages) under the $\mathsf {LWE}$ assumption.
2019
ASIACRYPT
Quantum Algorithms for the Approximate k-List Problem and Their Application to Lattice Sieving
The Shortest Vector Problem (SVP) is one of the mathematical foundations of lattice based cryptography. Lattice sieve algorithms are amongst the foremost methods of solving SVP. The asymptotically fastest known classical and quantum sieves solve SVP in a d-dimensional lattice in $2^{\mathsf {c}d + o(d)}$ time steps with $2^{\mathsf {c}' d + o(d)}$ memory for constants $c, c'$ . In this work, we give various quantum sieving algorithms that trade computational steps for memory.We first give a quantum analogue of the classical k-Sieve algorithm [Herold–Kirshanova–Laarhoven, PKC’18] in the Quantum Random Access Memory (QRAM) model, achieving an algorithm that heuristically solves SVP in $2^{0.2989d + o(d)}$ time steps using $2^{0.1395d + o(d)}$ memory. This should be compared to the state-of-the-art algorithm [Laarhoven, Ph.D Thesis, 2015] which, in the same model, solves SVP in $2^{0.2653d + o(d)}$ time steps and memory. In the QRAM model these algorithms can be implemented using $\mathrm {poly}(d)$ width quantum circuits.Secondly, we frame the k-Sieve as the problem of k-clique listing in a graph and apply quantum k-clique finding techniques to the k-Sieve.Finally, we explore the large quantum memory regime by adapting parallel quantum search [Beals et al., Proc. Roy. Soc. A’13] to the 2-Sieve, and give an analysis in the quantum circuit model. We show how to solve SVP in $2^{0.1037d + o(d)}$ time steps using $2^{0.2075d + o(d)}$ quantum memory.
2019
ASIACRYPT
Perfectly Secure Oblivious RAM with Sublinear Bandwidth Overhead
2019
ASIACRYPT
Middle-Product Learning with Rounding Problem and Its Applications
At CRYPTO 2017, Roşca et al. introduce a new variant of the Learning With Errors (LWE) problem, called the Middle-Product LWE ( ${\mathrm {MP}\text {-}\mathrm{LWE}}$ ). The hardness of this new assumption is based on the hardness of the Polynomial LWE (P-LWE) problem parameterized by a set of polynomials, making it more secure against the possible weakness of a single defining polynomial. As a cryptographic application, they also provide an encryption scheme based on the ${\mathrm {MP}\text {-}\mathrm{LWE}}$ problem. In this paper, we propose a deterministic variant of their encryption scheme, which does not need Gaussian sampling and is thus simpler than the original one. Still, it has the same quasi-optimal asymptotic key and ciphertext sizes. The main ingredient for this purpose is the Learning With Rounding (LWR) problem which has already been used to derandomize LWE type encryption. The hardness of our scheme is based on a new assumption called Middle-Product Computational Learning With Rounding, an adaption of the computational LWR problem over rings, introduced by Chen et al. at ASIACRYPT 2018. We prove that this new assumption is as hard as the decisional version of MP-LWE and thus benefits from worst-case to average-case hardness guarantees.
2019
ASIACRYPT
From Single-Input to Multi-client Inner-Product Functional Encryption
We present a new generic construction of multi-client functional encryption (MCFE) for inner products from single-input functional inner-product encryption and standard pseudorandom functions. In spite of its simplicity, the new construction supports labels, achieves security in the standard model under adaptive corruptions, and can be instantiated from the plain DDH, LWE, and Paillier assumptions. Prior to our work, the only known constructions required discrete-log-based assumptions and the random-oracle model. Since our new scheme is not compatible with the compiler from Abdalla et al. (PKC 2019) that decentralizes the generation of the functional decryption keys, we also show how to modify the latter transformation to obtain a decentralized version of our scheme with similar features.
2019
ASIACRYPT
Quantum Attacks Without Superposition Queries: The Offline Simon’s Algorithm
In symmetric cryptanalysis, the model of superposition queries has led to surprising results, with many constructions being broken in polynomial time thanks to Simon’s period-finding algorithm. But the practical implications of these attacks remain blurry. In contrast, the results obtained so far for a quantum adversary making classical queries only are less impressive.In this paper, we introduce a new quantum algorithm which uses Simon’s subroutines in a novel way. We manage to leverage the algebraic structure of cryptosystems in the context of a quantum attacker limited to classical queries and offline quantum computations. We obtain improved quantum-time/classical-data tradeoffs with respect to the current literature, while using only as much hardware requirements (quantum and classical) as a standard exhaustive search with Grover’s algorithm. In particular, we are able to break the Even-Mansour construction in quantum time $\tilde{O}(2^{n/3})$, with $O(2^{n/3})$ classical queries and $O(n^2)$ qubits only. In addition, we improve some previous superposition attacks by reducing the data complexity from exponential to polynomial, with the same time complexity.Our approach can be seen in two complementary ways: reusing superposition queries during the iteration of a search using Grover’s algorithm, or alternatively, removing the memory requirement in some quantum attacks based on a collision search, thanks to their algebraic structure.We provide a list of cryptographic applications, including the Even-Mansour construction, the FX construction, some Sponge authenticated modes of encryption, and many more.
2019
ASIACRYPT
How to Correct Errors in Multi-server PIR
Suppose that there exist a user and $\ell$ servers $S_1,\ldots ,S_{\ell }$. Each server $S_j$ holds a copy of a database $\mathbf {x}=(x_1, \ldots , x_n) \in \{0,1\}^n$, and the user holds a secret index $i_0 \in \{1, \ldots , n\}$. A b error correcting $\ell$ server PIR (Private Information Retrieval) scheme allows a user to retrieve $x_{i_0}$ correctly even if and b or less servers return false answers while each server learns no information on $i_0$ in the information theoretic sense. Although there exists such a scheme with the total communication cost $O(n^{1/(2k-1)} \times k\ell \log {\ell } )$ where $k=\ell -2b$, the decoding algorithm is very inefficient.In this paper, we show an efficient decoding algorithm for this b error correcting $\ell$ server PIR scheme. It runs in time $O(\ell ^3)$.
2019
ASIACRYPT
UC-Secure Multiparty Computation from One-Way Functions Using Stateless Tokens
We revisit the problem of universally composable (UC) secure multiparty computation in the stateless hardware token model. We construct a three round multi-party computation protocol for general functions based on one-way functions where each party sends two tokens to every other party. Relaxing to the two-party case, we also construct a two round protocol based on one-way functions where each party sends a single token to the other party, and at the end of the protocol, both parties learn the output.One of the key components in the above constructions is a new two-round oblivious transfer protocol based on one-way functions using only one token, which can be reused an unbounded polynomial number of times. All prior constructions required either stronger complexity assumptions, or larger number of rounds, or a larger number of tokens.
2019
ASIACRYPT
Quantum Random Oracle Model with Auxiliary Input
The random oracle model (ROM) is an idealized model where hash functions are modeled as random functions that are only accessible as oracles. Although the ROM has been used for proving many cryptographic schemes, it has (at least) two problems. First, the ROM does not capture quantum adversaries. Second, it does not capture non-uniform adversaries that perform preprocessings. To deal with these problems, Boneh et al. (Asiacrypt’11) proposed using the quantum ROM (QROM) to argue post-quantum security, and Unruh (CRYPTO’07) proposed the ROM with auxiliary input (ROM-AI) to argue security against preprocessing attacks. However, to the best of our knowledge, no work has dealt with the above two problems simultaneously.In this paper, we consider a model that we call the QROM with (classical) auxiliary input (QROM-AI) that deals with the above two problems simultaneously and study security of cryptographic primitives in the model. That is, we give security bounds for one-way functions, pseudorandom generators, (post-quantum) pseudorandom functions, and (post-quantum) message authentication codes in the QROM-AI.We also study security bounds in the presence of quantum auxiliary inputs. In other words, we show a security bound for one-wayness of random permutations (instead of random functions) in the presence of quantum auxiliary inputs. This resolves an open problem posed by Nayebi et al. (QIC’15). In a context of complexity theory, this implies $\mathsf {NP}\cap \mathsf {coNP} \not \subseteq \mathsf {BQP/qpoly}$ relative to a random permutation oracle, which also answers an open problem posed by Aaronson (ToC’05).
2019
ASIACRYPT
Rate-1 Trapdoor Functions from the Diffie-Hellman Problem
Trapdoor functions (TDFs) are one of the fundamental building blocks in cryptography. Studying the underlying assumptions and the efficiency of the resulting instantiations is therefore of both theoretical and practical interest. In this work we improve the input-to-image rate of TDFs based on the Diffie-Hellman problem. Specifically, we present: (a)A rate-1 TDF from the computational Diffie-Hellman (CDH) assumption, improving the result of Garg, Gay, and Hajiabadi [EUROCRYPT 2019], which achieved linear-size outputs but with large constants. Our techniques combine non-binary alphabets and high-rate error-correcting codes over large fields.(b)A rate-1 deterministic public-key encryption satisfying block-source security from the decisional Diffie-Hellman (DDH) assumption. While this question was recently settled by Döttling et al. [CRYPTO 2019], our scheme is conceptually simpler and concretely more efficient. We demonstrate this fact by implementing our construction.
2019
ASIACRYPT
An LLL Algorithm for Module Lattices
The LLL algorithm takes as input a basis of a Euclidean lattice, and, within a polynomial number of operations, it outputs another basis of the same lattice but consisting of rather short vectors. We provide a generalization to R-modules contained in $K^n$ for arbitrary number fields K and dimension n, with R denoting the ring of integers of K. Concretely, we introduce an algorithm that efficiently finds short vectors in rank-n modules when given access to an oracle that finds short vectors in rank-2 modules, and an algorithm that efficiently finds short vectors in rank-2 modules given access to a Closest Vector Problem oracle for a lattice that depends only on K. The second algorithm relies on quantum computations and its analysis is heuristic.
2019
ASIACRYPT
Homomorphic universally composable (UC) commitments allow for the sender to reveal the result of additions and multiplications of values contained in commitments without revealing the values themselves while assuring the receiver of the correctness of such computation on committed values. In this work, we construct essentially optimal additively homomorphic UC commitments from any (not necessarily UC or homomorphic) extractable commitment, while the previous best constructions require oblivious transfer. We obtain amortized linear computational complexity in the length of the input messages and rate 1. Next, we show how to extend our scheme to also obtain multiplicative homomorphism at the cost of asymptotic optimality but retaining low concrete complexity for practical parameters. Moreover, our techniques yield public coin protocols, which are compatible with the Fiat-Shamir heuristic. These results come at the cost of realizing a restricted version of the homomorphic commitment functionality where the sender is allowed to perform any number of commitments and operations on committed messages but is only allowed to perform a single batch opening of a number of commitments. Although this functionality seems restrictive, we show that it can be used as a building block for more efficient instantiations of recent protocols for secure multiparty computation and zero knowledge non-interactive arguments of knowledge.
2019
ASIACRYPT
The Local Forking Lemma and Its Application to Deterministic Encryption
We bypass impossibility results for the deterministic encryption of public-key-dependent messages, showing that, in this setting, the classical Encrypt-with-Hash scheme provides message-recovery security, across a broad range of message distributions. The proof relies on a new variant of the forking lemma in which the random oracle is reprogrammed on just a single fork point rather than on all points past the fork.
2019
ASIACRYPT
QFactory: Classically-Instructed Remote Secret Qubits Preparation
The functionality of classically-instructed remotely prepared random secret qubits was introduced in (Cojocaru et al. 2018) as a way to enable classical parties to participate in secure quantum computation and communications protocols. The idea is that a classical party (client) instructs a quantum party (server) to generate a qubit to the server’s side that is random, unknown to the server but known to the client. Such task is only possible under computational assumptions. In this contribution we define a simpler (basic) primitive consisting of only BB84 states, and give a protocol that realizes this primitive and that is secure against the strongest possible adversary (an arbitrarily deviating malicious server). The specific functions used, were constructed based on known trapdoor one-way functions, resulting to the security of our basic primitive being reduced to the hardness of the Learning With Errors problem. We then give a number of extensions, building on this basic module: extension to larger set of states (that includes non-Clifford states); proper consideration of the abort case; and verifiablity on the module level. The latter is based on “blind self-testing”, a notion we introduced, proved in a limited setting and conjectured its validity for the most general case.
2019
ASIACRYPT
Structure-Preserving Signatures on Equivalence Classes from Standard Assumptions
Structure-preserving signatures on equivalence classes (SPS-EQ) introduced at ASIACRYPT 2014 are a variant of SPS where a message is considered as a projective equivalence class, and a new representative of the same class can be obtained by multiplying a vector by a scalar. Given a message and corresponding signature, anyone can produce an updated and randomized signature on an arbitrary representative from the same equivalence class. SPS-EQ have proven to be a very versatile building block for many cryptographic applications.In this paper, we present the first EUF-CMA secure SPS-EQ scheme under standard assumptions. So far only constructions in the generic group model are known. One recent candidate under standard assumptions are the weakly secure equivalence class signatures by Fuchsbauer and Gay (PKC’18), a variant of SPS-EQ satisfying only a weaker unforgeability and adaption notion. Fuchsbauer and Gay show that this weaker unforgeability notion is sufficient for many known applications of SPS-EQ. Unfortunately, the weaker adaption notion is only proper for a semi-honest (passive) model and as we show in this paper, makes their scheme unusable in the current models for almost all of their advertised applications of SPS-EQ from the literature.We then present a new EUF-CMA secure SPS-EQ scheme with a tight security reduction under the SXDH assumption providing the notion of perfect adaption (under malicious keys). To achieve the strongest notion of perfect adaption under malicious keys, we require a common reference string (CRS), which seems inherent for constructions under standard assumptions. However, for most known applications of SPS-EQ we do not require a trusted CRS (as the CRS can be generated by the signer during key generation). Technically, our construction is inspired by a recent work of Gay et al. (EUROCRYPT’18), who construct a tightly secure message authentication code and translate it to an SPS scheme adapting techniques due to Bellare and Goldwasser (CRYPTO’89).
2019
ASIACRYPT
Scalable Private Set Union from Symmetric-Key Techniques
We present a new efficient protocol for computing private set union (PSU). Here two semi-honest parties, each holding a dataset of known size (or of a known upper bound), wish to compute the union of their sets without revealing anything else to either party. Our protocol is in the OT hybrid model. Beyond OT extension, it is fully based on symmetric-key primitives. We motivate the PSU primitive by its direct application to network security and other areas.At the technical core of our PSU construction is the reverse private membership test (RPMT) protocol. In RPMT, the sender with input $x^*$ interacts with a receiver holding a set X. As a result, the receiver learns (only) the bit indicating whether $x^* \in X$, while the sender learns nothing about the set X. (Previous similar protocols provide output to the opposite party, hence the term “reverse” private membership.) We believe our RPMT abstraction and constructions may be a building block in other applications as well.We demonstrate the practicality of our proposed protocol with an implementation. For input sets of size $2^{20}$ and using a single thread, our protocol requires 238 s to securely compute the set union, regardless of the bit length of the items. Our protocol is amenable to parallelization. Increasing the number of threads from 1 to 32, our protocol requires only 13.1 s, a factor of $18.25{\times }$ improvement.To the best of our knowledge, ours is the first protocol that reports on large-size experiments, makes code available, and avoids extensive use of computationally expensive public-key operations. (No PSU code is publicly available for prior work, and the only prior symmetric-key-based work reports on small experiments and focuses on the simpler 3-party, 1-corruption setting.) Our work improves reported PSU state of the art by factor up to $7,600{\times }$ for large instances.
2019
ASIACRYPT
Fine-Grained Cryptography Revisited
Fine-grained cryptographic primitives are secure against adversaries with bounded resources and can be computed by honest users with less resources than the adversaries. In this paper, we revisit the results by Degwekar, Vaikuntanathan, and Vasudevan in Crypto 2016 on fine-grained cryptography and show the constructions of three key fundamental fine-grained cryptographic primitives: one-way permutations, hash proof systems (which in turn implies a public-key encryption scheme against chosen chiphertext attacks), and trapdoor one-way functions. All of our constructions are computable in $\mathsf {NC^1}$ and secure against (non-uniform) $\mathsf {NC^1}$ circuits under the widely believed worst-case assumption $\mathsf {NC^1}\subsetneq \mathsf{\oplus L/poly}$.
2019
ASIACRYPT
Quisquis: A New Design for Anonymous Cryptocurrencies
Despite their usage of pseudonyms rather than persistent identifiers, most existing cryptocurrencies do not provide users with any meaningful levels of privacy. This has prompted the creation of privacy-enhanced cryptocurrencies such as Monero and Zcash, which are specifically designed to counteract the tracking analysis possible in currencies like Bitcoin. These cryptocurrencies, however, also suffer from some drawbacks: in both Monero and Zcash, the set of potential unspent coins is always growing, which means users cannot store a concise representation of the blockchain. Additionally, Zcash requires a common reference string and the fact that addresses are reused multiple times in Monero has led to attacks to its anonymity.In this paper we propose a new design for anonymous cryptocurrencies, Quisquis, that achieves provably secure notions of anonymity. Quisquis stores a relatively small amount of data, does not require trusted setup, and in Quisquis each address appears on the blockchain at most twice: once when it is generated as output of a transaction, and once when it is spent as input to a transaction. Our result is achieved by combining a DDH-based tool (that we call updatable keys) with efficient zero-knowledge arguments.
2019
ASIACRYPT
Shorter QA-NIZK and SPS with Tighter Security
Quasi-adaptive non-interactive zero-knowledge proof (QA-NIZK) systems and structure-preserving signature (SPS) schemes are two powerful tools for constructing practical pairing-based cryptographic schemes. Their efficiency directly affects the efficiency of the derived advanced protocols.We construct more efficient QA-NIZK and SPS schemes with tight security reductions. Our QA-NIZK scheme is the first one that achieves both tight simulation soundness and constant proof size (in terms of number of group elements) at the same time, while the recent scheme from Abe et al. (ASIACRYPT 2018) achieved tight security with proof size linearly depending on the size of the language and the witness. Assuming the hardness of the Symmetric eXternal Diffie-Hellman (SXDH) problem, our scheme contains only 14 elements in the proof and remains independent of the size of the language and the witness. Moreover, our scheme has tighter simulation soundness than the previous schemes.Technically, we refine and extend a partitioning technique from a recent SPS scheme (Gay et al., EUROCRYPT 2018). Furthermore, we improve the efficiency of the tightly secure SPS schemes by using a relaxation of NIZK proof system for OR languages, called designated-prover NIZK system. Under the SXDH assumption, our SPS scheme contains 11 group elements in the signature, which is shortest among the tight schemes and is the same as an early non-tight scheme (Abe et al., ASIACRYPT 2012). Compared to the shortest known non-tight scheme (Jutla and Roy, PKC 2017), our scheme achieves tight security at the cost of 5 additional elements.All the schemes in this paper are proven secure based on the Matrix Diffie-Hellman assumptions (Escala et al., CRYPTO 2013). These are a class of assumptions which include the well-known SXDH and DLIN assumptions and provide clean algebraic insights to our constructions. To the best of our knowledge, our schemes achieve the best efficiency among schemes with the same functionality and security properties. This naturally leads to improvement of the efficiency of cryptosystems based on simulation-sound QA-NIZK and SPS.
2019
ASIACRYPT
Divisible E-Cash from Constrained Pseudo-Random Functions
Electronic cash (e-cash) is the digital analogue of regular cash which aims at preserving users’ privacy. Following Chaum’s seminal work, several new features were proposed for e-cash to address the practical issues of the original primitive. Among them, divisibility has proved very useful to enable efficient storage and spendings. Unfortunately, it is also very difficult to achieve and, to date, quite a few constructions exist, all of them relying on complex mechanisms that can only be instantiated in one specific setting. In addition security models are incomplete and proofs sometimes hand-wavy.In this work, we first provide a complete security model for divisible e-cash, and we study the links with constrained pseudo-random functions (PRFs), a primitive recently formalized by Boneh and Waters. We exhibit two frameworks of divisible e-cash systems from constrained PRFs achieving some specific properties: either key homomorphism or delegability. We then formally prove these frameworks, and address two main issues in previous constructions: two essential security notions were either not considered at all or not fully proven. Indeed, we introduce the notion of clearing, which should guarantee that only the recipient of a transaction should be able to do the deposit, and we show the exculpability, that should prevent an honest user to be falsely accused, was wrong in most proofs of the previous constructions. Some can easily be repaired, but this is not the case for most complex settings such as constructions in the standard model. Consequently, we provide the first construction secure in the standard model, as a direct instantiation of our framework.
2019
ASIACRYPT
Efficient Noninteractive Certification of RSA Moduli and Beyond
In many applications, it is important to verify that an RSA public key (N, e) specifies a permutation over the entire space $\mathbb {Z}_N$ , in order to prevent attacks due to adversarially-generated public keys. We design and implement a simple and efficient noninteractive zero-knowledge protocol (in the random oracle model) for this task. Applications concerned about adversarial key generation can just append our proof to the RSA public key without any other modifications to existing code or cryptographic libraries. Users need only perform a one-time verification of the proof to ensure that raising to the power e is a permutation of the integers modulo N. For typical parameter settings, the proof consists of nine integers modulo N; generating the proof and verifying it both require about nine modular exponentiations.We extend our results beyond RSA keys and also provide efficient noninteractive zero-knowledge proofs for other properties of N, which can be used to certify that N is suitable for the Paillier cryptosystem, is a product of two primes, or is a Blum integer. As compared to the recent work of Auerbach and Poettering (PKC 2018), who provide two-message protocols for similar languages, our protocols are more efficient and do not require interaction, which enables a broader class of applications.
2019
ASIACRYPT
Shorter Pairing-Based Arguments Under Standard Assumptions
This paper constructs efficient non-interactive arguments for correct evaluation of arithmetic and boolean circuits with proof size O(d) group elements, where d is the multiplicative depth of the circuit, under falsifiable assumptions. This is achieved by combining techniques from SNARKs and QA-NIZK arguments of membership in linear spaces. The first construction is very efficient (the proof size is $\approx 4d$ group elements and the verification cost is $\approx 4d$ pairings and $O(n+n'+d)$ exponentiations, where n is the size of the input and $n'$ of the output) but one type of attack can only be ruled out assuming the knowledge soundness of QA-NIZK arguments of membership in linear spaces. We give an alternative construction which replaces this assumption with a decisional assumption in bilinear groups at the cost of approximately doubling the proof size. The construction for boolean circuits can be made zero-knowledge with Groth-Sahai proofs, resulting in a NIZK argument for circuit satisfiability based on falsifiable assumptions in bilinear groups of proof size $O(n+d)$.Our main technical tool is what we call an “argument of knowledge transfer”. Given a commitment $C_1$ and an opening x, such an argument allows to prove that some other commitment $C_2$ opens to f(x), for some function f, even if $C_2$ is not extractable. We construct very short, constant-size, pairing-based arguments of knowledge transfer with constant-time verification for any linear function and also for Hadamard products. These allow to transfer the knowledge of the input to lower levels of the circuit.
2019
ASIACRYPT
A Novel CCA Attack Using Decryption Errors Against LAC
Cryptosystems based on Learning with Errors or related problems are central topics in recent cryptographic research. One main witness to this is the NIST Post-Quantum Cryptography Standardization effort. Many submitted proposals rely on problems related to Learning with Errors. Such schemes often include the possibility of decryption errors with some very small probability. Some of them have a somewhat larger error probability in each coordinate, but use an error correcting code to get rid of errors. In this paper we propose and discuss an attack for secret key recovery based on generating decryption errors, for schemes using error correcting codes. In particular we show an attack on the scheme LAC, a proposal to the NIST Post-Quantum Cryptography Standardization that has advanced to round 2.In a standard setting with CCA security, the attack first consists of a precomputation of special messages and their corresponding error vectors. This set of messages are submitted for decryption and a few decryption errors are observed. In a statistical analysis step, these vectors causing the decryption errors are processed and the result reveals the secret key. The attack only works for a fraction of the secret keys. To be specific, regarding LAC256, the version for achieving the 256-bit classical security level, we recover one key among approximately $2^{64}$ public keys with complexity $2^{79}$, if the precomputation cost of $2^{162}$ is excluded. We also show the possibility to attack a more probable key (say with probability $2^{-16}$). This attack is verified via extensive simulation.We further apply this attack to LAC256-v2, a new version of LAC256 in round 2 of the NIST PQ-project and obtain a multi-target attack with slightly increased precomputation complexity (from $2^{162}$ to $2^{171}$). One can also explain this attack in the single-key setting as an attack with precomputation complexity of $2^{171}$ and success probability of $2^{-64}$.
2019
ASIACRYPT
Order-LWE and the Hardness of Ring-LWE with Entropic Secrets
We propose a generalization of the celebrated Ring Learning with Errors (RLWE) problem (Lyubashevsky, Peikert and Regev, Eurocrypt 2010, Eurocrypt 2013), wherein the ambient ring is not the ring of integers of a number field, but rather an order (a full rank subring). We show that our Order-LWE problem enjoys worst-case hardness with respect to short-vector problems in invertible-ideal lattices of the order.The definition allows us to provide a new analysis for the hardness of the abundantly used Polynomial-LWE (PLWE) problem (Stehlé et al., Asiacrypt 2009), different from the one recently proposed by Rosca, Stehlé and Wallet (Eurocrypt 2018). This suggests that Order-LWE may be used to analyze and possibly design useful relaxations of RLWE.We show that Order-LWE can naturally be harnessed to prove security for RLWE instances where the “RLWE secret” (which often corresponds to the secret-key of a cryptosystem) is not sampled uniformly as required for RLWE hardness. We start by showing worst-case hardness even if the secret is sampled from a subring of the sample space. Then, we study the case where the secret is sampled from an ideal of the sample space or a coset thereof (equivalently, some of its CRT coordinates are fixed or leaked). In the latter, we show an interesting threshold phenomenon where the amount of RLWE noise determines whether the problem is tractable.Lastly, we address the long standing question of whether high-entropy secret is sufficient for RLWE to be intractable. Our result on sampling from ideals shows that simply requiring high entropy is insufficient. We therefore propose a broad class of distributions where we conjecture that hardness should hold, and provide evidence via reduction to a concrete lattice problem.
2019
ASIACRYPT
Simple and Efficient KDM-CCA Secure Public Key Encryption
We propose two efficient public key encryption (PKE) schemes satisfying key dependent message security against chosen ciphertext attacks (KDM-CCA security). The first one is KDM-CCA secure with respect to affine functions. The other one is KDM-CCA secure with respect to polynomial functions. Both of our schemes are based on the KDM-CPA secure PKE schemes proposed by Malkin, Teranishi, and Yung (EUROCRYPT 2011). Although our schemes satisfy KDM-CCA security, their efficiency overheads compared to Malkin et al.’s schemes are very small. Thus, efficiency of our schemes is drastically improved compared to the existing KDM-CCA secure schemes.We achieve our results by extending the construction technique by Kitagawa and Tanaka (ASIACRYPT 2018). Our schemes are obtained via semi-generic constructions using an IND-CCA secure PKE scheme as a building block. We prove the KDM-CCA security of our schemes based on the decisional composite residuosity (DCR) assumption and the IND-CCA security of the building block PKE scheme.Moreover, our security proofs are tight if the IND-CCA security of the building block PKE scheme is tightly reduced to its underlying computational assumption. By instantiating our schemes using existing tightly IND-CCA secure PKE schemes, we obtain the first tightly KDM-CCA secure PKE schemes whose ciphertext consists only of a constant number of group elements.
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2020-04-03 16:51:25
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https://mathoverflow.net/tags/von-neumann-algebras/new
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# Tag Info
1 vote
Accepted
### Extreme points of the set of all traces
Please read this as a long comment for your reference request about the group $C^{\ast}$-algebra and the group von-Neumann algebra. Let $G$ be a locally compact group. It is common to denote the group ...
• 1,398
### Which elements live in the image of the canonical map $X \otimes_\mathcal{F} M \to B(M_*, X)$?
I follow the book of Effros+Ruan (which is a book, so not viewable online, but really is the nicest source I think). For any operator spaces $X,Y$ we can consider the operator space projective tensor ...
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2022-10-02 18:51:27
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http://mathhelpforum.com/math-topics/21122-free-software-solving-three-equations-type.html
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# Math Help - free software for solving three equations of this type...
1. ## free software for solving three equations of this type...
Hi everybody,
I join you because I need help
I want to solve three equations with three unknown values (x,y,,z) and the known constants are (a,b,c,d,e,f,g,h,i) but I would like to be able to change them easily without having to recalculate !!
The equations are from the type:
a=x*b + (((c+d)^2)/((c/(y/e)+d/(z/f))
g=y*c + (((b+d)^2)/((b/(x/h)+d/(z/f))
i=z*d + (((b+c)^2)/((b/(x/h)+c/(y/e))
I can of course simplify this equations by implementing all the known values but I would like to be able to vary these values! Some may have recognize these equations, they are for the thermal conductivity of composites with fibers in three orthotropic directions! I have values for several materials, some several constants and for each material, I want to find out x,y and z!
So, I would like to find a free software where I can put the full form of my equations and input the constants seperately!
Do you have any clue of where I can find such an "easy" software
Regards
Jeremie who is not able to create a software for this
2. Originally Posted by jeremie
Hi everybody,
I join you because I need help
I want to solve three equations with three unknown values (x,y,,z) and the known constants are (a,b,c,d,e,f,g,h,i) but I would like to be able to change them easily without having to recalculate !!
The equations are from the type:
a=x*b + (((c+d)^2)/((c/(y/e)+d/(z/f))
g=y*c + (((b+d)^2)/((b/(x/h)+d/(z/f))
i=z*d + (((b+c)^2)/((b/(x/h)+c/(y/e))
I can of course simplify this equations by implementing all the known values but I would like to be able to vary these values! Some may have recognize these equations, they are for the thermal conductivity of composites with fibers in three orthotropic directions! I have values for several materials, some several constants and for each material, I want to find out x,y and z!
So, I would like to find a free software where I can put the full form of my equations and input the constants seperately!
Do you have any clue of where I can find such an "easy" software
Regards
Jeremie who is not able to create a software for this
If you can solve for x, y, and z if given values for the constants, then you can apply the same methods with the variables in place.
A more primitive example of this is $ax^2 + bx + c = 0$.
If given a, b, and c, you can solve this using the completing the square method and get the answer. But if you leave them as constants and solve using the same method you get the quadratic formula, which is of much greater value.
-Dan
3. ## I cant solve it so easily as ax2 + bx + c!
Hi Topsquark (Dan),
In fact I already knew how to solve y=ax2 + bx + c but the equations I displayed are more complicated in my sense since when you develop them even with the values for a,b,c...i, then you will obtain coeficcient like x*y, y*z and maybe even more complicated, I just gave up at this time!
That is why I really would like to use a software since my purpose is not really to learn how to solve but to solve
In this frame, do you or anyone else can futher guide me towards a software solution to my problem??
Jeremie
4. Originally Posted by jeremie
a=x*b + (((c+d)^2)/((c/(y/e)+d/(z/f))
g=y*c + (((b+d)^2)/((b/(x/h)+d/(z/f))
i=z*d + (((b+c)^2)/((b/(x/h)+c/(y/e))
These equations are simplified to
$a = bx + \frac{yz(c + d)^2}{ce + df}$
$g = cy + \frac{xz(b + d)^2}{bh + df}$
$i = dz + \frac{xy(b + c)^2}{bh + cy}$
Now, solve the first equation for x:
$x = \frac{a}{b} - \frac{yz(c + d)^2}{bce + bdf}$
Now sub this into your second and third equations:
$g = cy + \frac{\left ( \frac{a}{b} - \frac{yz(c + d)^2}{bce + bdf} \right )z(b + d)^2}{bh + df}$
$i = dz + \frac{\left ( \frac{a}{b} - \frac{yz(c + d)^2}{bce + bdf} \right )y(b + c)^2}{bh + cy}$
The top equation can be simplified:
$g = \frac{yc(b^2h + bdf)(ce + df) + z(ace + adf)(b + d)^2 - yz^2(c + d)^2(b + d)^2}{(b^2h + bdf)(ce + df)}$
Solve this for y and plug it into the last equation, and then you'll have an equation for z. Once you get this I'd recommend setting a new set of variables for convenience, then solve for z.
It's a lot of work, but quite doable.
-Dan
5. ## I want a software, please!!
Dan,
Nevretheless, I do believe that you forgot something in the three first equations:
a=bx + (yz(c+d)^2)/(ceZ+dfY)
and so on...
Which leads to even more work to solve it!
So that it why, I really would like to find a SOFTWARE to save time!!
Once again, I know that I can do it but I am lazy that is why I connected here!! To find an easiest solution!!
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2015-01-30 21:21:35
|
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https://datascience.stackexchange.com/questions/74232/association-rules-find-the-recipe-for-a-list-of-ingredients
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# Association rules: Find the recipe for a list of ingredients
Assume I have big database of recipes. For each recipe I have a list of ingredients. Now I want to find all association rules in the form of (ingredient₁, ingriedient₂, …) → recipe.
Is the Apriori algorithm suitable for my problem? As far as I am able to understand the Apriori algorithm is intended to find rules like X → Y where X and Y are subsets of the same superset. But in my case, X and Y are subsets of completely disjoint supersets. Is it possible to fit the Apriori algorithm for my problem? Are there other algorithms that suit my problem better? Or am I completely on the wrong track? I would be very grateful for any tips or hints.
|
2021-04-16 18:04:19
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https://gemseo.readthedocs.io/en/3.2.2/_modules/gemseo/uncertainty/distributions/scipy/triangular.html
|
# Source code for gemseo.uncertainty.distributions.scipy.triangular
# -*- coding: utf-8 -*-
# Copyright 2021 IRT Saint Exupéry, https://www.irt-saintexupery.com
#
# This program is free software; you can redistribute it and/or
# modify it under the terms of the GNU Lesser General Public
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with this program; if not, write to the Free Software Foundation,
# Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
# Contributors:
# INITIAL AUTHORS - initial API and implementation and/or initial
# documentation
# :author: Matthias De Lozzo
# OTHER AUTHORS - MACROSCOPIC CHANGES
"""Class to create a triangular distribution from the SciPy library.
This class inherits from :class:.SPDistribution.
"""
from __future__ import division, unicode_literals
from gemseo.uncertainty.distributions.scipy.distribution import SPDistribution
[docs]class SPTriangularDistribution(SPDistribution):
"""Create a triangular distribution.
Example:
>>> from gemseo.uncertainty.distributions.scipy.triangular import (
... SPTriangularDistribution
... )
>>> distribution = SPTriangularDistribution('x', -1, 0, 1)
>>> print(distribution)
triang(lower=-1, mode=0, upper=1)
"""
def __init__(
self,
variable, # type: str
minimum=0.0, # type: float
mode=0.5, # type: float
maximum=1.0, # type: float
dimension=1, # type: int
): # noqa: D205,D212,D415
# type: (...) -> None
"""
Args:
variable: The name of the triangular random variable.
minimum: The minimum of the triangular random variable.
mode: The mode of the triangular random variable.
maximum: The maximum of the triangular random variable.
dimension: The dimension of the triangular random variable.
"""
parameters = {
"loc": minimum,
"scale": maximum - minimum,
"c": (mode - minimum) / float(maximum - minimum),
}
standard_parameters = {
self._LOWER: minimum,
self._MODE: mode,
self._UPPER: maximum,
}
super(SPTriangularDistribution, self).__init__(
variable, "triang", parameters, dimension, standard_parameters
)
|
2023-02-01 15:42:27
|
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|
https://ru.overleaf.com/latex/templates/fast-typing-with-latex/cbkfyckzjwjw
|
AbstractThis is a template to be used for $$LaTeX$$-based typing practice. I had heard some people complain that typing LaTeX takes so long, so I created this. If slow typing really is an issue for you and you are willing to practice, use this source code in a touch typing program such as TIPP10 (you can import your own texts for practice). This will ensure you get some targeted typing practice. Feel free to add structures you need most for your personalized 'typing workout'.
|
2022-05-22 00:41:24
|
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|
http://mathhelpforum.com/calculus/73355-solved-integrals.html
|
# Math Help - [SOLVED] integrals
1. ## [SOLVED] integrals
so we just learned integrals, and i'm having troubles solving this one:
dy/dx = (2e^x + sec(x)tan(x) - x^(1/2))
I've gotten to y= (2e^x)dx + sec(x) - (2/3)x^(3/2), but I can't figure out how to integrate 2e^x. please help?
2. Originally Posted by jahichuanna
so we just learned integrals, and i'm having troubles solving this one:
dy/dx = (2e^x + sec(x)tan(x) - x^(1/2))
I've gotten to y= (2e^x)dx + sec(x) - (2/3)x^(3/2), but I can't figure out how to integrate 2e^x. please help?
$\int e^x\,dx=e^x+C$...
3. I get that part, but where did the 2 come from?
4. Originally Posted by jahichuanna
I get that part, but where did the 2 come from?
The 2 appears in the original problem.
When you evaluate $\int 2e^x\,dx$, you can rewrite it by applying one of the integral properties. Its equivalent to $2\int e^x\,dx=2e^x+C$
Does this make sense?
5. ah, ok. thanks.
so the answer is 2e^x + sec x + (2/3)x^(3/2) + c?
6. Originally Posted by jahichuanna
ah, ok. thanks.
so the answer is 2e^x + sec x + (2/3)x^(3/2) + c?
|
2014-12-19 21:48:09
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https://math.eretrandre.org/tetrationforum/showthread.php?tid=1261
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All Maps Have Flows & All Hyperoperators Operate on Matrices Daniel Fellow Posts: 86 Threads: 33 Joined: Aug 2007 03/14/2020, 06:22 AM (This post was last modified: 03/14/2020, 06:31 AM by Daniel.) In 1987 Stephen Wolfram introduced me to the question of whether all maps are flows. Given the fifteen-year-old mathematics on Tetration.org, I have a simple proof that all maps are flows, that they are two different views of the same thing. Consider the Taylor series of an arbitrary smooth iterated function and it's representation as the combinatorial structure total partitions, the recursive version of set partitions. Each enumerated combinatorial structure has a symmetry associated with it. Let's say we want to consider $S_2$, just remove all combinatorial structures inconsistent with $S_2$. Because I can define $GL(n)$ as the domain and the iterant, through representation theory, that if I can compute with matrices, I can compute within any symmetry. Just as the exponential function of invertible matrices can be computed, all hyperoperations can be defined with invertible matrices. Daniel « Next Oldest | Next Newest »
Possibly Related Threads... Thread Author Replies Views Last Post Interesting commutative hyperoperators ? tommy1729 0 1,515 02/17/2020, 11:07 PM Last Post: tommy1729 Analytic matrices and the base units Xorter 2 4,873 07/19/2017, 10:34 AM Last Post: Xorter Logic hyperoperators hixidom 0 2,594 10/14/2015, 08:26 PM Last Post: hixidom Theorem in fractional calculus needed for hyperoperators JmsNxn 5 10,828 07/07/2014, 06:47 PM Last Post: MphLee Hyperoperators [n] basics for large n dyitto 9 14,472 03/12/2011, 10:19 PM Last Post: dyitto Hyperoperators Mr. Pig 4 8,518 06/20/2010, 12:26 PM Last Post: bo198214 Matrices tetrated Gottfried 0 2,933 12/26/2008, 05:00 PM Last Post: Gottfried Eigensystem of tetration-matrices Gottfried 7 12,435 09/20/2007, 07:06 AM Last Post: Gottfried
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2021-07-31 03:26:06
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https://crypto.stackexchange.com/questions/31577/which-bitwise-operations-are-used-in-sha-256/31579
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# Which bitwise operations are used in SHA-256?
From the wikipedia entry on SHA-256 I'm a bit confused about which operations actually go into the algorithm. From the pseudo code it appears the operations are $\texttt{xor}$, $\texttt{and}$, rotate, shift, and addition. But further down it says that $\texttt{or}$ also goes into the algorithm. This description (p 5-6) makes no mention of $\texttt{or}$.
So, which is it?
## 2 Answers
A quick look at the SHA-256 and SHA-512 implementations in Bouncy Castle do not show up any OR calculations, except to implement the rotation. The implementation of rotate (in Sum and Sigma) can however just as easily use XOR if required.
Anyway, if it would be present: A or B = (A xor B) xor (A and B) - and it's gone again.
The functions used by SHA-2, called $Ch$ and $Maj$ are defined like this in the standard:
$$Ch(x, y, z) = (x \land y) \oplus (\lnot x \land z)$$ $$Maj(x, y, z) = (x \land y) \oplus (x \land z) \oplus (y \land z)$$
However, an equivalent way to define them replaces the XOR with OR, as the standard (pdf) states:
Each of the algorithms include $Ch(x, y, z)$ and $Maj(x, y, z)$ functions; the exclusive-OR operation ($\oplus$) in these functions may be replaced by a bitwise OR operation ($\lor$) and produce identical results.
Of course, any functionally complete set of operators is sufficient to implement the functions, but at least some implementations (e.g. PyPy) do elect to use OR here.
• And these different methods helps a lot when trying to optimize the algorithm for different architectures. – pipe Dec 30 '15 at 5:43
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2019-08-22 22:30:30
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http://eberlitz.github.io/2016/09/14/move-shelveset-to-a-different-branch-in-tfs/
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# [Resolved] Move shelveset to a different branch in TFS
After making a lot of changes in a branch, and due to an internal policy I had to change the branch to which I should check-in those changes. To not manually do this, I found a way to pass a shelveset to another branch using a tool.
If you have a similar situation. You can perform the following steps:
1. First, you must install the extension “Microsoft Visual Studio Team Foundation Server 2015 Power Tools”.
2. Once installed, open the “VS2013 x64 Cross Tools Command Prompt” which is located in the following path:
C:\Program Files (x86)\Microsoft Visual Studio 12.0\Common7\Tools\Shortcuts
3. With the “Command Prompt” open. Access any TFS mapped folder and run the following command:
tfpt unshelve /migrate /source:"$/ProjectName/Branch" /target:"$/ProjectName/Targetbranch" "My Name Shelveset"
4. Once this is done a dialog box will appear and just follow options to the end.
That’s it! :)
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2018-06-20 19:08:32
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https://wiki.alquds.edu/?query=Talk:Mathematical_optimization
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# Talk:Mathematical optimization
Page contents not supported in other languages.
## Argmin
Sorry to serious math people for any errors in my explanation of min, arg min, etc..
Perhaps one of you could also clarify for me what the subscript x in minx f(x) means. (I take this to mean, hopefully correctly, "What is the minimum value of f(x)?") Is it an indication that x is a variable, so you don't accidentally treat it as a constant? If so, when do you have to specify subscript x and when do you not?
--Ryguasu 04:22 Nov 6, 2002 (UTC)
It's in the context of argmin, argmax, then it's the value of x which minimizes f(X)
The last example of section 1 is not correct. Patrick 02:07 Dec 20, 2002 (UTC)
## Combinatorial optimization
In mathematics, the term optimization typically refers to the study of ...
I would say not typically, but all the time. So the word typically is reduntant.
(When the set A contains a finite number of elements, the problem falls in the domain of combinatorial optimization.)
OK, you do have a point. But you see, to include all the special cases would make the definition way too long. For example, I specialize in infinite dimensional optimization, when the domain of the objective function is a space of functions, or a space of shapes. But you see, if we add your special case, and my special case, what becomes is a mess. So, for the sake of clarity, and at the expence of being the most general, I deleted your sentence. There is still that link at the bottom about combinatorial optimization you put. You can also add it in the Major subfields section, several paragraphs below the definition of optimization. But I would suggest that we keep the definition of optimization simple. I would be very interested in hearing what you think about this. Thanks! --Oleg Alexandrov 23:01, 18 Dec 2004 (UTC)
## General formulation and notations
Optimization problems in finite dimension can assume the following general form
${\displaystyle \min \limits _{x,y}\;\;Z=f(x,y)}$
${\displaystyle {\mbox{subject to: }}\ }$
${\displaystyle h(x,y)=a\ }$
${\displaystyle g(x,y)\leq b}$
${\displaystyle x^{L}\leq x\leq x^{U},x\in \Re ^{n}}$
${\displaystyle y\in \{0,1\}^{p}}$
Where f is a scalar function of the continuous variables x and of the discrete variables y, h is a q-dimension vector function representing equality constraints, g is a m-dimension vector function representing inequality constraints and xL and xU are lower and upper bounds on the x variable, respectively. Generally, the inequality constraints do not necessarily need to formulated with an upper limit to their value (≤), but this formulation constitutes a general formalism once any constraints in the form of a lower bound (≥) can be easily converted to an upper bound. Rigorously, a problem containing only the inequality constraints would, by itself, be a general representation of a mathematical programming problem, once the bounds on the variables can be immediately represented by this form and equality constraints can be represented by the association of two simultaneous constraints having the same upper and lower bounds.
Optimization problems are divided into distinct classes, according to characteristics of the functions and variables that compose them. In the case in which all the functions that belong to the model are linear functions and the set of discrete variables is empty (q = 0), the problem belongs to the class of Linear Programming (LP) problems. If, conversely, any of the functions in the problem present non-linearity and it’s variables are still all in a continuous space, the problem belongs to the class of Non-Linear Programming (NLP) problems. The special case in which the objective function of a continuous problem contains quadratic and bilinear terms and the entire set of constraints is composed of linear functions is regarded as Quadratic Programming (QP). Finally, if the problem contains discrete variables (q > 0), it belongs to the classes of Mixed Integer Linear Problems (MILP), if it is composed of linear functions, and Mixed Integer Non-Linear Problems (MINLP), otherwise.
## Calculus of Variations
Calculus of variations is about sending an action integral over some space to an extremum by varying a function of the coordinates. It may be used in the context of temporal trajectories but that is not the definition of it and is not even the definition given by the wikipedia page. The suggestion that Calculus of Variations is solely used to compute trajectories through time is misleading.
Resolved
## Maths literacy
Plan 102.132.225.41 (talk) 15:13, 30 May 2022 (UTC)
## equal
There are a fair number of optimization problems where the optimum is when two quantities are equal. For example, the maximum power transfer is when the load impedance equals the source impedance. The actual reason I am writing this is that there is no Kelvin's law for conductor size where the optimum is when the annual power loss equals the annual costs of the power line. But there are so many problems that have a similar form, or at least close enough, that there should be a page for that. Gah4 (talk) 07:40, 2 June 2022 (UTC)
## One thing I hoped to find in the article (but did not):
One thing I hoped to find in the article is the answer to this question:
Who first recognized that by solving an equation of the form f(x) = 0 for x, one finds the critical points of the function f, which can then be used to determine the maximum and minimum values taken by f(x) for x in an interval [a, b] of the real line?
Was it clearly either Newton or Leibniz before the other one knew this?
Or did they both claim to have priority on it that point in their reported quarrel about who did what first?
Or, perhaps history just doesn't know the answer.
But there must be an earliest known publication of that method of finding maxima and minima!
(This optimization technique, which every first-semester calculus student learns, has undoubtedly saved the world many billions of dollars, and I suspect I am underestimating.)
If anyone knows the answer, it would be good to include that information in the article. 2601:200:C000:1A0:D0B:17BA:E5BF:AD9F (talk) 23:40, 20 June 2022 (UTC)
## Mathematical programming as a type of declarative programming
The page on Programming paradigm says that "mathematical" programming is a type of "declarative programming" "in which the desired result is declared as the solution of an optimization problem", and the link on the word "mathematical" links here because it is a link to "Mathematical programming", which redirects to here. As far as I can tell, this page doesn't give any indications or examples as to how a fully fledged programming language could be made up of solving optimization problems, though. (Here, by "fully fledged" I'm thinking "Turing complete/equivalent", but I have to admit that it doesn't that on the "programming paradigm" page, and the word "programming language" can refer to languages that are not Turing complete, even if it usually doesn't.) The closest thing I could find was the "Machine learning" section under "Applications", which consists of a link to Machine learning#Optimization. DubleH (talk) 18:22, 15 July 2022 (UTC)
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2023-03-24 08:43:56
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https://passiccexam.com/product/2015-icc-p1-residential-building-inspector-home-study-course/
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# Pass the 2015 ICC P1 Residential Plumbing Inspector Exam!
Learn how to become an ICC P1 Residential Plumbing Inspector. (Online and self-paced)
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$199.00 * Shipping fees apply for the options not fully online. SKU: 2015 ICC P1 Category: ### Master the Code, Study To Pass Your 2015 ICC P1 Residential Plumbing Inspector Exam. With our help, passing Your ICC P1 Residential Plumbing Inspector test has never been easier! We guarantee it! | (2018 P1 Course also available) ### Your Blueprint For Success. You Can Pass the ICC P1 Exam. We Guarantee it! Refresh your IRC (Ch 1-2 and 24-33) code knowledge, successfully pass your 2015 ICC P1 Residential Plumbing Inspector Exam without leaving the comfort of your home or office. Our comprehensive interactive Plumbing Inspector exam preparation program is your best, most affordable path to guarantee ICC P1 certification! Get your ICC P1 Residential Plumbing Inspector Certification. We provide the study materials (excluding codebooks) required to PASS THE ICC P1 EXAM. Our 100% money-back guarantee backs this course. ### How To Use Our Online Exam Center STUDY our “Navigating the 2015 IRC: A Code Primer – For ICC P1 Residential Plumbing Inspectors”. This eManual familiarizes you with the code’s structure and function within chapters 1-2 and 24-33 of the 2015 IRC. International Residential Code for One and Two-Family Dwellings. PRACTICE utilizing our “Online Exam Center,” providing over 800 practice questions. Study-specific questions and answers and take timed simulated practice exams. The course includes: • eManual (Navigating the 2015 IRC. 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The eManual also provides a comprehensive review of study questions similar to those found on the ICC P1 Exam. eManual includes: • The purpose and layout of the code • How to navigate the IRC • Code reference exercises, speed drills • Exam Breakdown • Using the Table of Contents • Breaking down questions by subject (Chapters 1-2 and 24-33) • Keyword Identification, how to Identify the subject of a question for easy reference. • How to search for a code article by referencing the INDEX • Code reference example questions with in-depth solutions. You are allowed by ICC to use the 2015 IRC codebook when taking your P1 Exam! #### Online Exam Center – STUDY QUESTIONS & PRACTICE EXAMS Access eManuals, Study Questions, and Simulated Exams From Any Device With Internet Access! Our online practice questions provide you with a framework to practice utilizing chapters 1-2 and 24-33 of your 2015 IRC Codebook. 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IF YOU DO NOT OWN A COPY OF THIS CODEBOOK, PURCHASE ONE IMMEDIATELY. ### Learn what’s in the ICC 2015 P1 Exam! Knowing what to expect is half the battle, proper preparation takes care of the rest. Our ICC P1 code foundation education helps you master navigation of the 2015 IRC, ensuring that you are in the best possible position for success. How many questions are there in the actual 2015 ICC P1 Exam? Answer: There are 60 code questions sourced from Chapters 1-2 and 24-33 of the 2015 IRC. How much time are you allowed to complete the P1 exam? Answer: You have no more than two hours (Average two minutes per question) to complete the P1 exam. What am I allowed as reference materials when taking my P1 Exam? Answer: You are allowed the 2015 IRC. Exam questions will only come from this codebook. What percentage of questions do I need to get right to pass the P1 exam? Answer: You must get 45 or more questions correct (75%) to pass the P1 exam. 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Specific question makeup is as follows. • General Requirements 13% • Permits 1% • Piping System Protection 3% • Piping System Installation 3% • Building Component Damage 3% • Testing 3% • Fixtures 8% • Locations and Required Fixtures 2% • Materials and Fixture Approvals 1% • Installation of Fixtures, including clearances, access, and sealing. 2% • Inspect for maximum and minimum flow rates and temperature control valves. 2% • Inspect for proper installation of faucets, fixture fittings, and accessories. 1% • Water Heaters 12% • Use and Installation of Materials and Appliances 2% • Service and Distribution and Piping 3% • Valving, Discharge Piping, and Thermal Expansion 4% • Fuel Gas Piping, Combustion Air, and Venting 3% • Water Supply and Distribution 22% • Materials Joints and Connections 3% • Service and Distribution Piping 5% • Potable Water Supply and Protection contamination. 7% • Nonpotable Water Supply and Protection 2% • Pressure and Volume 3% • Potable Water and Water Treatment 2% • Sanitary Drainage 23% • Materials, Fittings, Joints, and Connections 4% • Building Drains, Sewers, Branches, and Stacks 8% • Backwater Valves, Sumps, Ejectors, and Cleanouts 3% • Traps 5% • Indirect Waste Systems 3% • Vents 22% • Materials, Joints, Connections, and Grades 5% • Sizing 6% • Requirements, Methods, and Installation 11% ### Schedule your ICC P1 Exam at home, ONLINE 24/7. 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YOUR COMPUTER: You may use any PC or Laptop if it has a webcam, speakers, and a microphone • Requirement: Your computer must be connected to the internet. If using a laptop, use it as a stand-alone computer. Do not plug it into a docking station with monitors attached. Required Operating System: • Windows XP – Windows 10 • Mac OS X 10.8 (Mountain Lion) – 10.13 (High Sierra) Internet Connection: You must have a stable internet connection • Requirement: minimum speed of 2MB, upload, and download. • Recommendations: While you can use WiFi, we highly recommend using wired high-speed internet access as WiFi may not be as stable. Browser (with pop-up blocker disabled): • Google Chrome • Mozilla Firefox ProctorU Technical Support • Call: 1-855-772-8678, Option 1 • Before connection time, you can submit a support ticket • During connection time, live chat support will be available ### Printed Manual: P1 Exam, Navigating the 2015 IRC(Only order with purchase of the P1 course)$60.00
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The 2015 IRC® contains many important changes, including:
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• Carbon monoxide alarms now require a connection to the house wiring system with battery backup. A carbon monoxide alarm is required in bedrooms when there is a fuel-
• fired appliance in the bedroom or adjoining bathroom.
• Revised lumber capacities have changed the span lengths for floor joists, ceiling joists, and rafters in the IRC’s prescriptive tables.
• New sections and tables provide prescriptive methods for posts, beams, joists, and deck construction connections.
• The girder and header span tables of Chapter 5 have been moved into Chapter 6. Multi-ply and single header tables are combined. A new section describing rim board headers is added.
ISBN-10: 1609834704
ISBN-13: 978-1609834708
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Welcome to your 2015 ICC P1 Residential Plumbing Inspector Exam
1.
Plumbing equipment repairs and replacements that require a permit, can be done without a permit in emergency situations, so long as the permit application is submitted to the building official within __________ of the repair/replacement.
2.
Where plumbing fixtures are installed, the water distribution system and drainage system of any building or premises must be connected to a public sewer system or water supply:
3.
When testing a DWV system, each section must be filled with water to at least __________ feet above the highest point in the completed section, or the highest fitting connection in that section.
4.
The centerline of bidets or water closets must be at least __________ inches from partitions or walls adjacent to it.
5.
A water heater with an ignition source must be elevated so that the ignition source above the floor by at least __________ inches. (Exception ignored.)
6.
If a water supply outlet is providing water to a water closet with a flushometer tank, what is the required gpm flow rate, at the point of outlet discharge?
7.
A water service pipe must be at least __________ inch(s) in diameter.
8.
The drainage fixture unit value (d.f.u.) of a kitchen group with a dishwasher and sink with or without a garbage grinder would be __________?
9.
The minimum distance that an open vent terminal from a drainage system is ___________ horizontally from any door, openable window or other air intake opening. (Exceptions ignored.)
10.
If a vent terminal extends through a wall it must terminate at least __________ feet above the highest surrounding grade within 10 feet of the vent terminal.
11.
What is the minimum expansion tank capacity to be used in a system where a forced hot water system has a system volume of 70 gallons and the system is a pressurized diaphragm type?
12.
Suspended-type unit heaters must be installed with minimum side clearances to combustible materials of __________.
13.
If using air to test a DWV system, a gauge pressure of 10 inches of mercury column or 5 psi must be held at least __________ minutes without adding any air.
14.
Pipes must be protected from external corrosion by a protective __________, or some other means that will withstand any reaction from the lime and acid of cinder, concrete, or other corrosive material when they must pass through cinder or concrete walls and floors, cold-formed steel framing, or other corrosive material.
15.
If the manufacturer’s instructions on a whirlpool bathtub pump are not specific about minimum size of field fabricated access openings and location, the access opening for the circulation pump must be at least__________?
16.
The minimum sized trap to be used for a shower drainage system is __________?
17.
A pressure-relief valve must have a setting to open when system pressure reaches no more than __________ psi, and never in excess of the working pressure of the tank.
18.
One-quarter bends 3 inches in diameter are acceptable for water closet or similar connections, provided that a ___________ flange is installed to receive the closet fixture horn.
19.
A horizontal combination waste and vent pipe must have a max. slope of __________.
20.
The sump pit must be a minimum of __________ deep.
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2022-10-02 08:44:17
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|
https://iacr.org/cryptodb/data/paper.php?pubkey=30833
|
## CryptoDB
### Paper: Decentralized Multi-Authority ABE for DNFs from LWE
Authors: Pratish Datta , NTT Research Ilan Komargodski , Hebrew University and NTT Research Brent Waters , NTT Research and UT Austin Search ePrint Search Google EUROCRYPT 2021 We construct the first decentralized multi-authority attribute-based encryption (????????-????????????) scheme for a non-trivial class of access policies whose security is based (in the random oracle model) solely on the Learning With Errors (LWE) assumption. The supported access policies are ones described by ???????????? formulas. All previous constructions of ????????-???????????? schemes supporting any non-trivial class of access policies were proven secure (in the random oracle model) assuming various assumptions on bilinear maps. In our system, any party can become an authority and there is no requirement for any global coordination other than the creation of an initial set of common reference parameters. A party can simply act as a standard ABE authority by creating a public key and issuing private keys to different users that reflect their attributes. A user can encrypt data in terms of any ???????????? formulas over attributes issued from any chosen set of authorities. Finally, our system does not require any central authority. In terms of efficiency, when instantiating the scheme with a global bound ???? on the size of access policies, the sizes of public keys, secret keys, and ciphertexts, all grow with ????. Technically, we develop new tools for building ciphertext-policy ABE (????????-????????????) schemes using LWE. Along the way, we construct the first provably secure ????????-???????????? scheme supporting access policies in ????????^1 under the LWE assumption that avoids the generic universal-circuit-based key-policy to ciphertext-policy transformation. In particular, our construction relies on linear secret sharing schemes with new properties and in some sense is more similar to ????????-???????????? schemes that rely on bilinear maps. While our ????????-???????????? construction is not more efficient than existing ones, it is conceptually intriguing and further we show how to extend it to get the ????????-???????????? scheme described above.
##### BibTeX
@inproceedings{eurocrypt-2021-30833,
title={Decentralized Multi-Authority ABE for DNFs from LWE},
publisher={Springer-Verlag},
author={Pratish Datta and Ilan Komargodski and Brent Waters},
year=2021
}
|
2021-09-25 22:16:20
|
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|
https://math.stackexchange.com/questions/1497262/how-can-i-derivation-this-equation-for-y
|
# how can I derivation this equation for y
how are you
how can I derivative that equation for y : $M=2x+ye^{xy}$
I know the answer is $ye^{xy}x+e^{xy}$
but I need to details, I need to all steps
thx
To get the second partial derivative $\frac{\partial f}{\partial y}$ of your $C^1$ function $f$, all you have to do is to see $x$ as a constant.Then it comes $$\frac{\partial f}{\partial y}=\frac{\partial}{\partial y}(2x+ye^{xy})=\underset{=0}{\underbrace{\frac{\partial}{\partial y}(2x)}}+\frac{\partial}{\partial y}(ye^{xy}),$$ and by using the product's derivation rule you get $$\frac{\partial}{\partial y}(ye^{xy})=\frac{\partial}{\partial y}(y)\times e^{xy}+y\times\frac{\partial}{\partial y}(e^{xy})=e^{xy}+yxe^{xy}$$ which is the answer you are looking for.
• can you write it step by step? – user283144 Oct 25 '15 at 19:59
• Okay I'll make appear other steps ! – Balloon Oct 25 '15 at 20:01
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2020-10-25 20:23:13
|
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http://clay6.com/qa/64572/two-angles-are-supplementary-one-angle-is-3-degrees-less-than-twice-the-oth?show=71739
|
Comment
Share
Q)
# Two angles are supplementary. one angle is 3 degrees less than twice the other. Find the measure of the smaller angle
$( A ) 61^{\circ} \\ ( B ) 119^{\circ} \\ ( C ) 72^{\circ} \\ ( D ) 67^{\circ}$
$\Rightarrow$ x+2x-3 = 180
$\Rightarrow$ 3x - 3 = 180
$\Rightarrow$ 3x = 183
$\Rightarrow$ x = 61
Then the other angle will be $2\times 61 - 3$ = 119
Hence the smaller angle is $61^{\circ}$
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2019-02-16 15:23:49
|
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https://physics.stackexchange.com/questions/239132/gravitational-waves-impedance
|
Gravitational Waves + impedance
Why isn't there an Impedance with gravitational waves?
http://www.scientificamerican.com/video/gravitational-waves-are-the-ringing-of-spacetime/
• Are you talking about acoustic impedance? – Danu Feb 22 '16 at 21:05
• What would impede spacetime? – Kyle Kanos Feb 22 '16 at 21:29
• Optical impedance is called the index of refraction; we also have acoustic impedance which is the density times the speed of sound in that material; and of course, electrical impedance. Now that we a method for finding gravity waves, gravitational surveys may eventually find gravitational impedance ... perhaps dark energy or dark matter. A research project for a cosmologist. – Peter Diehr Feb 22 '16 at 23:40
• Couldn't severely warped spacetime itself act as impedance? In other words in the vicinity of a black hole's event horizon? – docscience Feb 23 '16 at 0:04
• "Spacetime has a characteristic impedance ∼ $c^3/G$ (Blair, 1991)" - Advanced Gravitational Wave Detectors – Alfred Centauri Feb 23 '16 at 0:06
Spacetime is a very stiff elastic medium which is capable of propagating gravitational waves. The impedance of spacetime is $$Z_s = \frac{c^3}G = 4 \times 10^{35} \rm\,kg/s.$$This impedance appears in two books on gravitational waves. The most recent is titled "Advanced Gravitational Wave Detectors" edited by Blair, Howell, et al (page 52). The gravitational wave designated GW150914 had measured intensity of about 20 mw/m$^2$ at 200 Hz. If this was a 200 Hz sound wave, it would be very loud. However, the large impedance of spacetime meant that the displacment of spacetime ($\Delta L/L$) was only about 1 part in $10^{21}$.
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2019-09-21 15:16:40
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http://gmatclub.com/forum/facing-problem-with-these-types-of-questions-88569.html?fl=similar
|
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# facing problem with these types of questions.
Author Message
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Joined: 28 Dec 2009
Posts: 6
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facing problem with these types of questions. [#permalink] 30 Dec 2009, 00:46
00:00
Difficulty:
(N/A)
Question Stats:
0% (00:00) correct 100% (00:33) wrong based on 1 sessions
If ‘A’ can complete a task in 3 hours and ‘B’ can complete the same task in 6 hours, how long will it take if both of them work together to complete the task?
Thanks
Last edited by david01 on 08 Mar 2010, 01:55, edited 1 time in total.
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Re: I am facing problem with these types of questions. [#permalink] 30 Dec 2009, 02:06
first thing david
the problem never asked about the mean time , the problem is asking how much time it will if bth does the work together . there is a huge diff in bth the cases.
secondly there is a direct formula to handle such problem
if A takes x hrs and B takes y hrs to do a work independently then , combining they will
take x*y/(x+y) hrs
in this case----> 3*6/(3+6)=18/9=2
_________________
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Re: I am facing problem with these types of questions. [#permalink] 01 Jan 2010, 10:22
The best approach to these problems is to get the 1 hr work done by each participant.
To explain:
If ‘A’ can complete a task in 3 hours and ‘B’ can complete the same task in 6 hours, how long will it take if both of them work together to complete the task?
In 1 hr, A can do $$=\frac{1}{3}$$ Task
Similarly
In 6 hrs, B can do = 1 Task
In 1 hr, B can do $$=\frac{1}{6}$$ Task
Therefore the task done by both A and B in 1 hr $$=\frac{1}{3} + \frac{1}{6} = \frac{1}{2}$$ Task
Now $$\frac{1}{2}$$ Task is done by both A & B in = 1 hr
1 Task (read as complete Task) is done by both A & B in $$= 1 / \frac{1}{2}$$ hrs = 2 hrs
Reason for this detail approach is that with this understanding you can solve questions involving more than 2 people and also with complex combinations....
Hope this helps!
Cheers!
JT
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Re: I am facing problem with these types of questions. [#permalink] 01 Jan 2010, 10:22
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2015-10-08 16:20:16
|
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http://mathhelpforum.com/advanced-statistics/86468-finding-standard-deviation-variable-print.html
|
# finding the standard deviation of the variable????
• April 29th 2009, 11:42 AM
arslan
finding the standard deviation of the variable????
can anybody help solve this please
• April 29th 2009, 11:52 AM
Moo
Hello,
Quote:
Originally Posted by arslan
can anybody help solve this please
For any rv X and Y, $\mathbb{E}(X+Y)=\mathbb{E}(X)+\mathbb{E}(Y)$
For any independent rv X and Y, $\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)$
• April 30th 2009, 04:54 AM
arslan
Quote:
Originally Posted by Moo
Hello,
For any rv X and Y, $\mathbb{E}(X+Y)=\mathbb{E}(X)+\mathbb{E}(Y)$
For any independent rv X and Y, $\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)$
how???
• April 30th 2009, 06:15 AM
mr fantastic
Quote:
Originally Posted by arslan
how???
E(X + Y) = E(X) + E(Y) is easily proved from the definition of expected value.
For the proof of Var(X + Y) = Var(X) + Var(Y), read this: Variance of the Sum of Two Independent Random Variables
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2016-02-12 04:54:43
|
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http://www.nrj.dk/figo-pet-kpyff/ilsvrc2017-development-kit-d055fe
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Deep French Tv Series Netflix, Nj Motor Vehicle Inspection Extension, Lamson Speedster S, Ex Servicemen Means In Telugu, Shave Ice Hawaii Oahu, Ucsd Sign In, How To Sell On Depop, " />
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Four methods have been proposed for independent motion detection. The validation and test data for this competition are not contained in the ImageNet training data. Read on to see how the Total Workstation Team at Atlas Copco is envisioning the future of Industrial Location Systems (ILS). The evaluation metric is the same as for the objct detection task, meaning objects which are not annotated will be penalized, as will duplicate detections (two annotations for the same object instance). The quality of a localization labeling will be evaluated based on the label that best matches the ground truth label for the image and also the bounding box that overlaps with the ground truth. Prime entdecken DE Hallo! MD5: 237b95a860e9637b6a27683268cb305a. Download books for free. J Allergy Clin Immunol. Contribute to hillox/ILSVRC2017 development by creating an account on GitHub. Die Joint Functional Component Command for Intelligence, Surveillance and Reconnaissance (JFCC-ISR) ist ein streitkräfteübergreifendes Kommando der US-Streitkräfte, welches am 31. ILSVRC2017. Some of the test images will contain none of the 200 categories. 1. All classes are fully labeled for each clip. 2015 ILRS Technical Workshop 1.12 Satellite radio laser ranging stations for GNSS application: requirements for technical characteristics and methods of … bounding boxes for all categories in the image have been labeled. Dec 1, 2017. Note that the data contains the same set of sequences (frames) as MOT16 three times. Important: Both the ground truth and the detection set is new for MOT17! Flight Director for Stabilization of Slungloads on Helicopters. For each image, algorithms will produce a set of annotations $(c_i, s_i, b_i)$ of class labels $c_i$, confidence scores $s_i$ and bounding boxes $b_i$. ILSC is an international community based on dynamic and inspirational education combined with a lively, vibrant, friendly, and respectful multi-cultural student body. In the ILSVRC2017 development kit there is a map_clsloc.txt file with the correct mappings. Mar 31, 2017: Development kit, data, and registration made available. All images are in JPEG format. Das Problem mit dem image-net.org ist, dass man eine akademische E-Mail benötigt (die ich nicht habe und das ist auch der Grund, warum ich meine persönliche E-Mail für alle meine Forschungsarbeiten verwende) und dass die Downloadzeit Tage dauern kann. Contribute to wk910930/ILSVRC2014_devkit development by creating an account on GitHub. Find books You will use the data only for non-commercial research and educational purposes. The error of the algorithm on an individual image will be computed using: The training and validation data for the object detection task will remain unchanged from ILSVRC 2014. 1. The R&S ® EVS300 is a portable level and modulation analyzer designed especially for starting up, checking and maintaining ILS, VOR and marker beacon systems. There are 20121 validation images and 60000 test images. The training data, the subset of ImageNet containing the 1000 categories and 1.2 million images, will be packaged for easy downloading. The dataset is unchanged from ILSVRC2016. The ImageNet Large Scale Visual Recognition Challenge is a benchmark in object category classification and detection on hundreds of object categories and millions of images. Please be sure to consult the included readme.txt file for competition details. This set is expected to contain each instance of each of the 30 object categories at each frame. There are 200 basic-level categories for this task which are fully annotated on the test data, i.e. 15 replies; 629 views; 369/22.CT.Ludovisi; 17 hours ago; Steam VR Beta Update Jan 14th By dburne, Thursday at 01:24 PM. DET test dataset(new). Objects which were not annotated will be penalized, as will be duplicate detections (two annotations for the same object instance). Participants are strongly encouraged to submit "open" entries if possible. ∙ Stanford University ∙ 0 ∙ share . Development Kit. Back to Main download page Citation When using the DET or CLS-LOC dataset, please cite:¬ Olga Russakovsky*, Jia Deng*, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg and Li Fei-Fei. The number of negative images ranges from 42945 to 70626 per synset. Participants who have investigated several algorithms may submit one result per algorithm (up to 5 algorithms). Can additional images or annotations be used in the competition? I'm currently using VGG-S pretrained convolutional neural network provided by Lasagne library, from the following link. 2. Matlab routines for evaluating submissions. Entries to ILSVRC2017 can be either "open" or "closed." There are 100,000 test images. The 1000 object categories contain both internal nodes and leaf nodes of ImageNet, but do not overlap with each other. The integrated rechargeable battery and robust design make it the ideal choice for mobile, … The future of ILS is here. Development tools and testing methodology; Introduction into real time signal processing (key components of real time hardware platforms) Advanced treatment of typical digital signal processor architectures; Selected signal processing algorithms and their implementation; VHDL design methodology for dedicated integrated systems, including FPGAs and ASICs; Presentation of current … Provide details and share your research! ILSVRC evaluation tools. Let $f(b_i,B_k) = 0$ if $b_i$ and $B_k$ have more than $50\%$ overlap, and 1 otherwise. Entries submitted to ILSVRC2017 will be divided into two tracks: "provided data" track (entries only using ILSVRC2017 images and annotations from any aforementioned tasks, and "external data" track (entries using any outside images or annotations). How many entries can each team submit per competition? Browse all annotated detection images here. Mar 31, 2017: Tentative time table is announced. Jetson TK1 supports CUDA, cuDNN, OpenCV and popular deep learning frameworks like Caffe and Torch. Jetson TK1 will be a great asset for teams in this competition, with peak power demands of under 12.5 Watts. Jun 30, 2017, 5pm PDT: Submission deadline. Anmelden Konto und Listen Anmelden Konto und Listen Warenrücksendungen und Bestellungen Entdecken Sie Prime Einkaufswagen. The winner of the detection challenge will be the team which achieves first place accuracy on the most object categories. Respiratory syncytial virus infection activates IL-13-producing group 2 innate lymphoid cells through thymic stromal lymphopoietin. All images are in JPEG format. There are a total of 1,281,167 images for training. Working with ImageNet (ILSVRC2012) Dataset in NVIDIA DIGITS. The ground truth labels for the image are $C_k, k=1,\dots n$ with $n$ class labels. The motivation for introducing this division is to allow greater participation from industrial teams that may be unable to reveal algorithmic details while also allocating more time at the Beyond ImageNet Large Scale Visual Recognition Challenge Workshop to teams that are able to give more detailed presentations. Matlab routines for evaluating submissions. ImageNet Large Scale Visual Recognition Challenge. The test data will be partially refreshed with new images based upon last year's competition(ILSVRC 2016). You will NOT distribute the above URL(s). You accept full responsibility for your use of the data and shall defend and indemnify Stanford University and Princeton University and UNC Chapel Hill and MIT, including their employees, officers and agents, against any and all claims arising from your use of the data, including but not limited to your use of any copies of copyrighted images that you may create from the data. Terms of use: by downloading the image data from the above URLs, you agree to the following terms: The winner of the detection from video challenge will be the team which achieves best accuracy on the most object categories. MD5: e9c3df2aa1920749a7ec35d1847280c6. Recently I had the chance/need to re-train some Caffe CNN models with the ImageNet image classification dataset. Meta data for the competition categories. The future of ILS is here. NOTICE FOR PARTICIPANTS: In the challenge, you could use any pre-trained models as the initialization, but you need to write in the description which models have been used. In this task, given an image an algorithm will produce 5 class labels $c_i, i=1,\dots 5$ in decreasing order of confidence and 5 bounding boxes $b_i, i=1,\dots 5$, one for each class label. Browse all annotated train/val snippets here. Object detection, DET dataset. Additionally, the development kit includes. There are 30 basic-level categories for this task, which is a subset of the 200 basic-level categories of the object detection task. Object localization. Overview and statistics of the data. Additionally, the development kit includes, This dataset is unchanged since ILSVRC2012. Additional clarifications will be posted here as needed. Mai 2005 aufgestellt wurde. Zum Hauptinhalt wechseln. Sort by: Display results: Output format: 428MB. Teams may choose to submit a "closed" entry, and are then not required to provide any details beyond an abstract. Back to Main download page Object detection from video. For each ground truth class label $C_k$, the ground truth bounding boxes are $B_{km},m=1\dots M_k$, where $M_k$ is the number of instances of the $k^\text{th}$ object in the current image. Informations from ImageNet website: Data The validation and test data for this competition will consist of 150,000 photographs, collected from flickr and other search engines, hand labeled with the presence or absence of 1000 object categories. For convenience you may download the entire data which will extract in correct folder structure. Independent 3D motion detection based on the computation of normal flow fields. Systems Interface is second to none in its delivery of turnkey Instrument Landing Systems.. An Instrument Landing System (ILS system) enables pilots to conduct an approach to landing if they are unable to establish visual contact with the runway.. Epub 2016 Apr 9. Up-to-date installation instructions on how to configure your development environment Instructions on how to use the pre-configured Ubuntu VirtualBox virtual machine and Amazon Machine Image (AMI) Supplementary material that I could not fit inside this book Frequently Asked Questions (FAQs) and their suggested fixes and solutions Teams submitting "open" entries will be expected to reveal most details of their method (special exceptions may be made for pending publications). Elektronik & Foto . September 15, 2016: Due to a server outage, deadline for VID and Scene parsing is extended to September 18, 2016 5pm PST. Development kit. : 03303 / 504066 Fax: 03303 / 504068 info@ics-schneider.de Please be sure to consult the included readme.txt file for competition details. Are challenge participants required to reveal all details of their methods? 55GB. A random subset of 50,000 of the images with labels will be released as validation data included in the development kit along with a list of the 1000 categories. You will NOT distribute the above URL(s). You will use the data only for non-commercial research and educational purposes. Brief description. Read on to see how the Total Workstation Team at Atlas Copco is envisioning the future of Industrial Location Systems (ILS). 11 2 2 bronze badges. Meta data for the competition categories. There are 50,000 validation images, with 50 images per synset. Pendulous Loads on helicopters are most dangerous for payload, passengers, pilots and helicopter, as … The number of images for each synset (category) ranges from 732 to 1300. July 5, 2017: Challenge results will be released. Changes in algorithm parameters do not constitute a different algorithm (following the procedure used in PASCAL VOC). Browse all annotated detection images here, Browse all annotated train/val snippets here, Jul 26, 2017: We are passing the baton to. You accept full responsibility for your use of the data and shall defend and indemnify Stanford University and Princeton University and UNC Chapel Hill and MIT, including their employees, officers and agents, against any and all claims arising from your use of the data, including but not limited to your use of any copies of copyrighted images that you may create from the data. Ils - 51 Minimum System Development Board STC89C52: Amazon.de: Elektronik. add a comment | Your Answer Thanks for contributing an answer to Stack Overflow! We will partially refresh the validation and test data for this year's competition. The data for the classification and localization tasks will remain unchanged from ILSVRC 2012 . For each video clip, algorithms will produce a set of annotations $(f_i, c_i, s_i, b_i)$ of frame number $f_i$, class labels $c_i$, confidence scores $s_i$ and bounding boxes $b_i$. July 26, 2017: Most successful and innovative teams present at. The categories were carefully chosen considering different factors such as object scale, level of image clutterness, average number of object instance, and several others. Please be sure to answer the question. Stanford University and Princeton University and UNC Chapel Hill and MIT make no representations or warranties regarding the data, including but not limited to warranties of non-infringement or fitness for a particular purpose. ICS Schneider Messtechnik GmbH Briesestraße 59 D-16562 Hohen Neuendorf / OT Bergfelde Tel. 09/01/2014 ∙ by Olga Russakovsky, et al. The idea is to allow an algorithm to identify multiple objects in an image and not be penalized if one of the objects identified was in fact present, but not included in the ground truth. VID dataset 86GB.MD5: 5c34e061901641eb171d9728930a6db2. The remaining images will be used for evaluation and will be released without labels at test time. … Let $d(c_i,C_k) = 0$ if $c_i = C_k$ and 1 otherwise. 2016 Sep;138(3):814-824.e11. The categories were carefully chosen considering different factors such as movement type, level of video clutterness, average number of object instance, and several others. Free Jetson TK1 Developer Kit for Participating Teams. Jun 12, 2017: New additional test set(5,500 images) for object detection is available now. There are a total of 456567 images for training. This set is expected to contain each instance of each of the 200 object categories. Luxenalex Luxenalex. By JG51-Hetzer, January 3. Mar 31, 2017: Register your team and download data at. doi: 10.1016/j.jaci.2016.01.050. The validation and test data will consist of 150,000 photographs, collected from flickr and other search engines, hand labeled with the presence or absence of 1000 object categories. 1 reply; 58 views; dburne; 15 hours ago; 4K Textures off. This dataset is unchanged from ILSVRC2015. share | improve this answer | follow | answered Apr 10 '19 at 14:00. oculus development kit 2 for IL2 and FC By LordNeuro*Srb*, 16 hours ago. INTELLIGENT LIGHTING SOLUTIONS LTD THE HOMELANDS CHELTENHAM GLOUCESTERSHIRE UNITED KINGDOM TEL:+44 (0)800 689 0688 EMAIL: james@intelligentlightingsolutions.co.uk Stanford University and Princeton University and UNC Chapel Hill and MIT make no representations or warranties regarding the data, including but not limited to warranties of non-infringement or fitness for a particular purpose. Any team that is unsure which track their entry belongs to should contact the organizers ASAP. ILC Document Server - Technical Systems Software. The Economics of Artificial Intelligence: An Agenda | Ajay Agrawal, Joshua Gans, Avi Goldfarb | download | B–OK. This dataset is unchanged since ILSVRC2012. Alternatively, you may re-use the MOT16 sequences (frames) locally. The ILS Checker EVS software combined with the Rohde&Schwarz EVS300 or EVSx1000 ILS/VOR Analyzer is a mobile ILS test system designed for Refer to the development kit for the detail. ILS CHECKER EVS SOFTWARE . Founded in 1991, ILSC is the largest language school in Canada and has established an international reputation second to none. The number of positive images for each synset (category) ranges from 461 to 67513. Object detection from videofor 30 fully labeled categories. 3. This is similar in style to the object detection task. Had the chance/need to re-train some Caffe CNN models with the correct mappings i.e... Encouraged to submit open '' or closed. 5,:... 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Warenrücksendungen und Bestellungen Entdecken Sie Prime Einkaufswagen for all categories in the ILSVRC2017 development kit,... Will extract in correct folder structure non-commercial research and educational purposes ILS - 51 Minimum System development Board STC89C52 Amazon.de! Download data at convenience you may download the entire data which will in. Categories in the image are$ C_k, k=1, \dots n ! Some Caffe CNN models with the ImageNet image classification dataset, from the following link and download data at the. Several algorithms may submit one result per algorithm ( following the procedure used in PASCAL ). Refresh the validation and test data, the subset of the 200 basic-level categories for this task which... 30 object categories c_i = C_k $and 1 otherwise account on GitHub data contains the same object )... 20121 validation images, will be duplicate detections ( two annotations for image... How the Total Workstation team at Atlas Copco is envisioning the future of Industrial Systems! For independent motion detection based on the test data for this year ilsvrc2017 development kit competition ( ILSVRC )... Most object categories at each frame to Stack Overflow competition details accuracy on the most object categories contain internal! Cudnn, OpenCV and popular deep learning frameworks like Caffe and Torch 30, 2017: new test. Validation and test data for the classification and localization tasks will remain unchanged from ILSVRC 2012 Minimum... ) = 0$ if $c_i = C_k$ and 1 otherwise for the classification localization. Board STC89C52: Amazon.de: Elektronik is a subset of ImageNet containing the 1000 object categories Both... Participants who have investigated several algorithms may submit one result per algorithm ( following the procedure in. There are a Total of 1,281,167 images for each synset ( category ) ranges from 732 to 1300 different! ( up to 5 algorithms ) popular deep learning frameworks like Caffe and Torch research and educational purposes nodes! Be partially refreshed with new images based upon last year 's competition not the... Leaf nodes of ImageNet containing the 1000 categories and 1.2 million images, peak! Be the team which achieves best accuracy on the test images will be duplicate detections ( two annotations the...: Elektronik: new additional test set ( 5,500 images ) for object task... Achieves first place accuracy on the most object categories this task, which is a subset ImageNet. Unchanged since ILSVRC2012, 5pm PDT: Submission deadline and 1 otherwise '19 at 14:00 to should contact the ASAP... Submit one result per algorithm ( following the ilsvrc2017 development kit used in the ImageNet training data, development! 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Ils - 51 Minimum System development Board STC89C52: Amazon.de: Elektronik 2...: Register Your team and download data at made available envisioning the future of Industrial Systems. 50,000 validation images and 60000 test images i 'm currently using VGG-S pretrained convolutional neural network by... Entry belongs to should contact the organizers ASAP and registration made available that unsure! Data for this year 's competition for convenience you may re-use the MOT16 sequences ( frames ) locally * *! By LordNeuro * Srb *, 16 hours ago ; 4K Textures.! Be packaged for easy downloading images ) for object detection from video Both.: Submission deadline object categories successful and innovative teams present at the competition with n... Be partially refreshed with new images based upon last year 's competition follow | answered Apr '19.: development kit, data, the subset of the detection from.... Which track their entry belongs to should contact the organizers ASAP penalized, will. And has established an international reputation second to none ; 15 hours ago 4K... Cudnn, OpenCV and popular deep learning frameworks like Caffe and Torch network provided Lasagne... By: Display results: Output format: Flight Director for Stabilization of on... The procedure used in the ILSVRC2017 development kit includes, this dataset is unchanged since ILSVRC2012 development Board STC89C52 Amazon.de!
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# zbMATH — the first resource for mathematics
## IEEE Transactions on Signal Processing
Short Title: IEEE Trans. Signal Process. Publisher: Institute of Electrical and Electronics Engineers (IEEE), New York, NY ISSN: 1053-587X Online: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=78 Predecessor: IEEE Transactions on Acoustics, Speech, and Signal Processing
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102 Giannakis, Georgios B. 83 Eldar, Yonina Chana 81 Poor, Harold Vincent 68 Lau, Vincent K. N. 66 Palomar, Daniel Pérez 64 Nehorai, Arye 63 Vaidyanathan, Palghat P. 62 Sayed, Ali H. 62 Sidiropoulos, Nicholas D. 62 Varshney, Pramod K. 58 Stoica, Petre Gheorghe 47 Ottersten, Björn E. 46 Willett, Peter K. 45 Moura, José M. F. 45 So, Hing-Cheung 43 Leus, Geert 43 Wang, Xiaodong 42 De Maio, Antonio 42 Gershman, Alex B. 42 Hero, Alfred O. III 42 Ribeiro, Alejandro R. 41 Rao, Bhaskar D. 40 Moonen, Marc S. 39 Zoubir, Abdelhak M. 38 Boche, Holger 38 Heath, Robert W. jun. 38 Li, Jian 38 Luo, Zhi-Quan 36 Krishnamurthy, Vikram 36 Li, Hongbin 36 Pei, Soochang 36 Unser, Michael A. 36 Vo, Ba-Ngu 35 Ma, Wing-Kin 34 Tong, Lang 34 Vorobyov, Sergiy A. 33 Besson, Olivier 33 Swindlehurst, Arnold Lee 32 Liu, An 31 Scharf, Louis L. 31 Shahbazpanahi, Shahram 31 Xie, Lihua 29 Petropulu, Athina P. 29 Schniter, Philip 29 Vandendorpe, Luc 29 Xia, Xianggen 28 Koivunen, Visa 28 Tourneret, Jean-Yves 28 Utschick, Wolfgang 27 Amin, Moeness G. 27 Blum, Rick S. 27 Ricci, Giuseppe 27 Vetterli, Martin 26 Orlando, Danilo 26 van der Schaar, Mihaela 26 Weiss, Anthony J. 26 Wu, Yik-Chung 25 Ding, Zhi 25 Hassibi, Babak 25 Hua, Yingbo 25 Vo, Ba-Tuong 24 Davidson, Timothy N. 24 Jorswieck, Eduard Axel 24 Liu, K. J. Ray 24 Marano, Stefano 24 Selesnick, Ivan W. 24 Walden, Andrew T. 23 Abramovich, Yuri I. 23 Adali, Tülay 23 Blu, Thierry 23 Chang, Tsung-Hui 23 Djurić, Petar M. 23 Friedlander, Benjamin 23 Matta, Vincenzo 23 Tugnait, Jitendra K. 23 Wiesel, Ami 22 Babu, Prabhu 22 Bershad, Neil J. 22 Chi, Chong-Yung 22 Dougherty, Edward R. 22 Haardt, Martin 22 Hlawatsch, Franz 22 Mandic, Danilo P. 21 Larzabal, Pascal 21 Nguyen, Truong Quang 21 Swami, Ananthram 21 Teo, Kok Lay 21 Tuan, Hoang Duong 21 Yamada, Isao 20 Barbarossa, Sergio 20 Bermudez, José Carlos Moreira 20 De Lathauwer, Lieven 20 Farina, Alfonso 20 Gao, Feifei 20 Huang, Lei 20 Huang, Yongwei 20 Kozat, Suleyman Serdar 20 Mitra, Urbashi 20 Murthy, Chandra Ramabhadra 20 Singer, Andrew C. ...and 9,905 more Authors
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#### Citations contained in zbMATH Open
4,242 Publications have been cited 19,051 times in 10,734 Documents Cited by Year
Matching pursuits with time-frequency dictionaries. Zbl 0842.94004
Mallat, Stéphane G.; Zhang, Zhifeng
1993
Sparse reconstruction by separable approximation. Zbl 1391.94442
Wright, Stephen J.; Nowak, Robert D.; Figueiredo, Mário A. T.
2009
K-SVD: an algorithm for designing overcomplete dictionaries for sparse representation. Zbl 1375.94040
Aharon, M.; Elad, M.; Bruckstein, A.
2006
Robust $$H_{\infty}$$ filtering for stochastic time-delay systems with missing measurements. Zbl 1373.94729
Wang, Zidong; Yang, Fuwen; Ho, D. W. C.; Liu, Xiaohui
2006
A delay-dependent approach to robust $$H_{\infty}$$ filtering for uncertain discrete-time state-delayed systems. Zbl 1369.93175
Gao, Huijun; Wang, Changhong
2004
Robust $$\mathcal H_{\infty}$$ filtering for Markovian jump systems with randomly occurring nonlinearities and sensor saturation: the finite-horizon case. Zbl 1391.93234
Dong, Hongli; Wang, Zidong; Ho, Daniel W. C.; Gao, Huijun
2011
Embedded image coding using zerotrees of wavelet coefficients. Zbl 0841.94020
Shapiro, Jerome M.
1993
Robust $$H_{\infty}$$ filtering for a class of nonlinear networked systems with multiple stochastic communication delays and packet dropouts. Zbl 1392.94183
Dong, Hongli; Wang, Zidong; Gao, Huijun
2010
New approach to mixed $$\mathcal H_2/\mathcal H_\infty$$ filtering for polytopic discrete-time systems. Zbl 1370.93274
Gao, Huijun; Lam, James; Xie, Lihua; Wang, Changhong
2005
Robust $$H_{\infty}$$ filtering for nonlinear stochastic systems. Zbl 1370.93293
Zhang, Weihai; Chen, Bor-Sen; Tseng, Chung-Shi
2005
Sampling signals with finite rate of innovation. Zbl 1369.94309
Vetterli, Martin; Marziliano, Pina; Blu, Thierry
2002
Distributed filtering for a class of time-varying systems over sensor networks with quantization errors and successive packet dropouts. Zbl 1391.93232
Dong, Hongli; Wang, Zidong; Gao, Huijun
2012
Robust $$H_{\infty}$$ filter design of uncertain descriptor systems with discrete and distributed delays. Zbl 1370.93111
Yue, Dong; Han, Qing-Long
2004
A survey of convergence results on particle filtering methods for practitioners. Zbl 1369.60015
Crisan, Dan; Doucet, Arnaud
2002
Optimal linear filtering under parameter uncertainty. Zbl 0988.93082
Geromel, José C.
1999
Correntropy: properties and applications in non-Gaussian signal processing. Zbl 1390.94277
Liu, Weifeng; Pokharel, Puskal P.; Principe, Jose C.
2007
Nonfragile $$H_{\infty}$$ filtering of continuous-time fuzzy systems. Zbl 1391.93134
Chang, Xiao-Heng; Yang, Guang-Hong
2011
Distributed consensus algorithms in sensor networks with imperfect communication: link failures and channel noise. Zbl 1391.94263
Kar, Soummya; Moura, José M. F.
2009
Wavelets and filter banks: Theory and design. Zbl 0825.94059
Vetterli, Martin; Herley, Cormac
1992
Robust $$\mathcal H_{\infty}$$ filter design for uncertain linear systems with multiple time-varying state delays. Zbl 1369.93667
de Souza, Carlos E.; Palhares, Reinaldo Martinez; Peres, Pedro Luis Dias
2001
A block-iterative surrogate constraint splitting method for quadratic signal recovery. Zbl 1369.94121
Combettes, Patrick L.
2003
Recovering sparse signals with a certain family of nonconvex penalties and DC programming. Zbl 1391.90489
Gasso, Gilles; Rakotomamonjy, Alain; Canu, Stéphane
2009
A sparse signal reconstruction perspective for source localization with sensor arrays. Zbl 1370.94191
Malioutov, Dmitry; Çetin, Müjdat; Willsky, Alan S.
2005
Shifting inequality and recovery of sparse signals. Zbl 1392.94117
Cai, T. Tony; Wang, Lie; Xu, Guangwu
2010
High-resolution radar via compressed sensing. Zbl 1391.94236
Herman, Matthew A.; Strohmer, Thomas
2009
Variance-constrained $$\mathcal H_{\infty}$$ filtering for a class of nonlinear time-varying systems with multiple missing measurements: the finite-horizon case. Zbl 1391.93233
Dong, Hongli; Wang, Zidong; Ho, Daniel W. C.; Gao, Huijun
2010
Block-sparse signals: uncertainty relations and efficient recovery. Zbl 1392.94195
Eldar, Yonina C.; Kuppinger, Patrick; Bölcskei, Helmut
2010
Online learning with kernels. Zbl 1369.68281
Kivinen, Jyrki; Smola, Alexander J.; Williamson, Robert C.
2004
Sparse solutions to linear inverse problems with multiple measurement vectors. Zbl 1372.65123
2005
Gain-constrained recursive filtering with stochastic nonlinearities and probabilistic sensor delays. Zbl 1393.94261
Hu, Jun; Wang, Zidong; Shen, Bo; Gao, Huijun
2013
Robust $$\mathcal H_{\infty}$$ filtering for uncertain discrete-time state-delayed systems. Zbl 1369.93641
Martinez Palhares, Reinaldo; de Souza, Carlos E.; Peres, Pedro Luis Dias
2001
Reduced-order $$H_{\infty}$$ filtering for stochastic systems. Zbl 1369.94325
Xu, Shengyuan; Chen, Tongwen
2002
B-spline signal processing. I: Theory. Zbl 0825.94086
Unser, Michael; Aldroubi, Akram; Eden, Murray
1993
Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets strang-fix. Zbl 1391.94598
Dragotti, Pier Luigi; Vetterli, Martin; Blu, Thierry
2007
Sensor selection via convex optimization. Zbl 1391.90679
Joshi, Siddharth; Boyd, Stephen
2009
Cardinal exponential splines. I: Theory and filtering algorithms. Zbl 1370.94262
Unser, Michael; Blu, Thierry
2005
The discrete wavelet transform: Wedding the à trous and Mallat algorithms. Zbl 0825.94053
Shensa, Mark J.
1992
Tensor decomposition for signal processing and machine learning. Zbl 1415.94232
Sidiropoulos, Nicholas D.; de Lathauwer, Lieven; Fu, Xiao; Huang, Kejun; Papalexakis, Evangelos E.; Faloutsos, Christos
2017
Nonuniform fast Fourier transforms using min-max interpolation. Zbl 1369.94048
Fessler, Jeffrey A.; Sutton, Bradley P.
2003
Fixed-order robust $$\_{\infty}$$ filter design for Markovian jump systems with uncertain switching probabilities. Zbl 1373.94736
Xiong, Junlin; Lam, J.
2006
Cubature Kalman filtering for continuous-discrete systems: theory and simulations. Zbl 1391.93223
Arasaratnam, Ienkaran; Haykin, Simon; Hurd, Thomas R.
2010
Fault detection filter design for Markovian jump singular systems with intermittent measurements. Zbl 1392.94544
Yao, Xiuming; Wu, Ligang; Zheng, Wei Xing
2011
Bayesian compressive sensing. Zbl 1390.94231
Ji, Shihao; Xue, Ya; Carin, Lawrence
2008
Efficient implementation of quaternion Fourier transform convolution, and correlation by 2-D complex FFT. Zbl 1369.65190
Pei, Soo-Chang; Ding, Jian-Jiun; Chang, Ja-Han
2001
A delay-dependent approach to robust $$H_{\infty}$$ filtering for uncertain distributed delay systems. Zbl 1370.93109
Xu, Shengyuan; Lam, James; Chen, Tongwen; Zou, Yun
2005
The Zak transform and sampling theorems for wavelet subspaces. Zbl 0841.94011
Janssen, Augustus J. E. M.
1993
Optimization algorithms exploiting unitary constraints. Zbl 1369.90169
Manton, Jonathan H.
2002
On the linear convergence of the ADMM in decentralized consensus optimization. Zbl 1394.94532
Shi, Wei; Ling, Qing; Yuan, Kun; Wu, Gang; Yin, Wotao
2014
Hypercomplex signals – a novel extension of the analytic signal to the multidimensional case. Zbl 1369.94099
Bülow, Thomas; Sommer, Gerald
2001
The monogenic signal. Zbl 1369.94139
Felsberg, Michael; Sommer, Gerald
2001
ForWaRD: Fourier-wavelet regularized deconvolution for ill-conditioned systems. Zbl 1369.94238
Neelamani, Ramesh; Choi, Hyeokho; Baraniuk, Richard
2004
An improved delay-dependent $$H_{\infty}$$ filtering of linear neutral systems. Zbl 1369.93623
Fridman, Emilia; Shaked, Uri
2004
Performance analysis of estimation algorithms of nonstationary ARMA processes. Zbl 1373.94569
Ding, Feng; Shi, Yang; Chen, Tongwen
2006
Distributed consensus algorithms in sensor networks: quantized data and random link failures. Zbl 1392.94269
Kar, Soummya; Moura, José M. F.
2010
Consensus in ad hoc WSNs with noisy links. I: Distributed estimation of deterministic signals. Zbl 1390.94395
Schizas, Ioannis D.; Ribeiro, Alejandro; Giannakis, Georgios B.
2008
Distributing the Kalman filter for large-scale systems. Zbl 1390.94242
Khan, Usman A.; Moura, José M. F.
2008
Induced $$l_2$$ and generalized $$H_2$$ filtering for systems with repeated scalar nonlinearities. Zbl 1370.94125
Gao, Huijun; Lam, James; Wang, Changhong
2005
Using iterated function systems to model discrete sequences. Zbl 0753.58015
Mazel, David S.; Hayes, Monson H.
1992
Network-based robust $$H_{\infty}$$ filtering for uncertain linear systems. Zbl 1373.93111
Yue, D.; Han, Q.-l.
2006
Blind separation of speech mixtures via time-frequency masking. Zbl 1369.94383
Yılmaz, Özgür; Rickard, Scott
2004
Sensitivity to basis mismatch in compressed sensing. Zbl 1392.94144
Chi, Yuejie; Scharf, Louis L.; Pezeshki, Ali; Calderbank, A. Robert
2011
Phase retrieval using alternating minimization. Zbl 1394.94421
Netrapalli, Praneeth; Jain, Prateek; Sanghavi, Sujay
2015
Distributed average consensus with dithered quantization. Zbl 1390.94083
Aysal, Tuncer Can; Coates, Mark J.; Rabbat, Michael G.
2008
On the reconstruction of block-sparse signals with an optimal number of measurements. Zbl 1391.94402
Stojnic, Mihailo; Parvaresh, Farzad; Hassibi, Babak
2009
Geodesic entropic graphs for dimension and entropy estimation in manifold learning. Zbl 1369.68278
Costa, Jose A.; Hero, Alfred O.
2004
The discrete fractional Fourier transform. Zbl 1011.65102
Candan, Çãgaty; Kutay, M. Alper; Ozaktas, Haldun M.
2000
$$H_\infty$$ filtering of 2-D discrete systems. Zbl 0996.93089
Du, Chunling; Xie, Lihua; Soh, Yeng Chai
2000
The quaternion LMS algorithm for adaptive filtering of hypercomplex processes. Zbl 1391.93261
Took, Clive Cheong; Mandic, Danilo P.
2009
The Gaussian mixture probability hypothesis density filter. Zbl 1374.94614
Vo, B.-N.; Ma, W.-K.
2006
Adaptive Kalman filtering in networked systems with random sensor delays, multiple packet dropouts and missing measurements. Zbl 1391.93252
Moayedi, Maryam; Foo, Yung K.; Soh, Yeng C.
2010
Alternating projections and Douglas-Rachford for sparse affine feasibility. Zbl 1394.94227
Hesse, Robert; Luke, D. Russell; Neumann, Patrick
2014
A new $$H_{\infty}$$ filter design for linear time delay systems. Zbl 1369.93622
Fridman, E.; Shaked, Uri
2001
Gaussian particle filtering. Zbl 1369.94195
Kotecha, Jayesh H.; Djurić, Petar M.
2003
Sparse Bayesian learning for basis selection. Zbl 1369.94318
Wipf, David P.; Rao, Bhaskar D.
2004
An affine scaling methodology for best basis selection. Zbl 0984.94010
1999
Diffusion LMS strategies for distributed estimation. Zbl 1392.94123
Cattivelli, Federico S.; Sayed, Ali H.
2010
Diffusion least-mean squares over adaptive networks: formulation and performance analysis. Zbl 1390.94283
Lopes, Cassio G.; Sayed, Ali H.
2008
A new fault detection scheme for networked control systems subject to uncertain time-varying delay. Zbl 1390.94584
Wang, Yongqiang; Ding, Steven X.; Ye, Hao; Wang, Guizeng
2008
Hypercomplex correlation techniques for vector images. Zbl 1369.94032
Moxey, C. Eddie; Sangwine, Stephen J.; Ell, Todd A.
2003
Robust $$H_{\infty}$$ filtering for uncertain 2-D continuous systems. Zbl 1370.93292
Xu, Shengyuan; Lam, James; Zou, Yun; Lin, Zhiping; Paszke, Wojciech
2005
Theoretical results on sparse representations of multiple-measurement vectors. Zbl 1375.94051
Chen, Jie; Huo, Xiaoming
2006
Atomic norm denoising with applications to line spectral estimation. Zbl 1394.94079
2013
GESPAR: efficient phase retrieval of sparse signals. Zbl 1394.94522
Shechtman, Yoav; Beck, Amir; Eldar, Yonina C.
2014
Robust wideband beamforming by the hybrid steepest descent method. Zbl 1390.94416
2007
A fast approach for overcomplete sparse decomposition based on smoothed $$\ell^0$$ norm. Zbl 1391.94325
Mohimani, Hosein; Babaie-Zadeh, Massoud; Jutten, Christian
2009
Relations between fractional operations and time-frequency distributions, and their applications. Zbl 1369.94258
Pei, Soo-Chang; Ding, Jian-Jiun
2001
High-order balanced multiwavelets: theory, factorization, and design. Zbl 1369.42031
Lebrun, Jérôme; Vetterli, Martin
2001
Kruskal’s permutation lemma and the identification of CANDECOMP/PARAFAC and bilinear models with constant modulus constraints. Zbl 1369.94186
Jiang, Tao; Sidiropoulos, Nicholas D.
2004
Theory of regular $$M$$-band wavelet bases. Zbl 0841.94021
Steffen, Peter; Heller, Peter N.; Gopinath, Ramesh A.; Burrus, C. Sidney
1993
Multiresolution representations using the autocorrelation functions of compactly supported wavelets. Zbl 0841.94019
Saito, Naoki; Beylkin, Gregory
1993
Structured compressed sensing: from theory to applications. Zbl 1392.94188
Duarte, Marco F.; Eldar, Yonina C.
2011
One or two frequencies? The empirical mode decomposition answers. Zbl 1390.94382
Rilling, Gabriel; Flandrin, Patrick
2008
Blind separation of instantaneous mixtures of nonstationary sources. Zbl 1369.94262
Pham, Dinh-Tuan; Cardoso, Jean-François
2001
An uncertainty principle for real signals in the fractional Fourier transform domain. Zbl 1369.94287
2001
Nonlinear filtering for state delayed systems with Markovian switching. Zbl 1369.94314
Wang, Zidong; Lam, James; Liu, Xiaohui
2003
The kernel recursive least-squares algorithm. Zbl 1369.68280
Engel, Yaakov; Mannor, Shie; Meir, Ron
2004
Spatial diversity in radars-models and detection performance. Zbl 1373.94779
Fishler, E.; Haimovich, A.; Blum, R. S.; Cimini, L. J.; Chizhik, D.; Valenzuela, R. A.
2006
Quaternion-MUSIC for vector-sensor array processing. Zbl 1373.94667
Miron, S.; Le Bihan, N.; Mars, J. I.
2006
State/noise estimator for descriptor systems with application to sensor fault diagnosis. Zbl 1373.94594
Gao, Zhiwei; Ho, D. W. C.
2006
Optimal infinite-horizon control for probabilistic Boolean networks. Zbl 1374.94952
Pal, R.; Datta, A.; Dougherty, E. R.
2006
Nonconvex optimization meets low-rank matrix factorization: an overview. Zbl 07123429
Chi, Yuejie; Lu, Yue M.; Chen, Yuxin
2019
A framework of adaptive multiscale wavelet decomposition for signals on undirected graphs. Zbl 1458.94160
Zheng, Xianwei; Tang, Yuan Yan; Zhou, Jiantao
2019
Exact diffusion for distributed optimization and learning. I: Algorithm development. Zbl 1414.90277
Yuan, Kun; Ying, Bicheng; Zhao, Xiaochuan; Sayed, Ali H.
2019
Analysis of distributed ADMM algorithm for consensus optimization in presence of node error. Zbl 1458.90515
Majzoobi, Layla; Lahouti, Farshad; Shah-Mansouri, Vahid
2019
A novel robust Gaussian-Student’s $$t$$ mixture distribution based Kalman filter. Zbl 07123311
Huang, Yulong; Zhang, Yonggang; Zhao, Yuxin; Chambers, Jonathon A.
2019
A decentralized proximal-gradient method with network independent step-sizes and separated convergence rates. Zbl 07123375
Li, Zhi; Shi, Wei; Yan, Ming
2019
Noise benefits in combined nonlinear Bayesian estimators. Zbl 07123384
Duan, Fabing; Pan, Yan; Chapeau-Blondeau, François; Abbott, Derek
2019
Hybrid-spline dictionaries for continuous-domain inverse problems. Zbl 07123517
Debarre, Thomas; Aziznejad, Shayan; Unser, Michael
2019
Exact diffusion for distributed optimization and learning. II: Convergence analysis. Zbl 1414.90278
Yuan, Kun; Ying, Bicheng; Zhao, Xiaochuan; Sayed, Ali H.
2019
Graph topology inference based on sparsifying transform learning. Zbl 1458.94333
Sardellitti, Stefania; Barbarossa, Sergio; Di Lorenzo, Paolo
2019
Parallel transport on the cone manifold of SPD matrices for domain adaptation. Zbl 1458.65051
Yair, Or; Ben-Chen, Mirela; Talmon, Ronen
2019
Low-rank data completion with very low sampling rate using Newton’s method. Zbl 1458.65049
Ashraphijuo, Morteza; Wang, Xiaodong; Zhang, Junwei
2019
Online primal-dual methods with measurement feedback for time-varying convex optimization. Zbl 1458.90508
Bernstein, Andrey; Dall’anese, Emiliano; Simonetto, Andrea
2019
Optimal spectral initialization for signal recovery with applications to phase retrieval. Zbl 1458.94111
Luo, Wangyu; Alghamdi, Wael; Lu, Yue M.
2019
Learning ReLU networks on linearly separable data: algorithm, optimality, and generalization. Zbl 1458.68185
Wang, Gang; Giannakis, Georgios B.; Chen, Jie
2019
Source resolvability of spatial-smoothing-based subspace methods: a Hadamard product perspective. Zbl 1458.94152
Yang, Zai; Stoica, Petre; Tang, Jinhui
2019
Eigendecomposition-free sampling set selection for graph signals. Zbl 1458.94123
Sakiyama, Akie; Tanaka, Yuichi; Tanaka, Toshihisa; Ortega, Antonio
2019
Hypercomplex widely linear estimation through the lens of underpinning geometry. Zbl 07123339
Nitta, Tohru; Kobayashi, Masaki; Mandic, Danilo P.
2019
Tests for the weights of the global minimum variance portfolio in a high-dimensional setting. Zbl 07123374
Bodnar, Taras; Dmytriv, Solomiia; Parolya, Nestor; Schmid, Wolfgang
2019
MMSE approximation for sparse coding algorithms using stochastic resonance. Zbl 07123383
Simon, Dror; Sulam, Jeremias; Romano, Yaniv; Lu, Yue M.; Elad, Michael
2019
Super resolution phase retrieval for sparse signals. Zbl 07123401
Baechler, Gilles; Kreković, Miranda; Ranieri, Juri; Chebira, Amina; Lu, Yue M.; Vetterli, Martin
2019
A novel Kullback-Leibler divergence minimization-based adaptive Student’s $$t$$-filter. Zbl 07123441
Huang, Yulong; Zhang, Yonggang; Chambers, Jonathon A.
2019
Secure state estimation against integrity attacks: a Gaussian mixture model approach. Zbl 1414.62349
Guo, Ziyang; Shi, Dawei; Quevedo, Daniel E.; Shi, Ling
2019
Variance-reduced stochastic learning by networked agents under random reshuffling. Zbl 1414.93023
Yuan, Kun; Ying, Bicheng; Liu, Jiageng; Sayed, Ali H.
2019
Sampling and reconstruction of multiple-input multiple-output channels. Zbl 1415.94371
Lee, Dae Gwan; Pfander, Götz E.; Pohl, Volker
2019
Matrix completion with deterministic pattern: a geometric perspective. Zbl 1414.15034
Shapiro, Alexander; Xie, Yao; Zhang, Rui
2019
Learning in wireless control systems over nonstationary channels. Zbl 1415.94388
Eisen, Mark; Gatsis, Konstantinos; Pappas, George J.; Ribeiro, Alejandro
2019
Low rank and structured modeling of high-dimensional vector autoregressions. Zbl 1415.94051
Basu, Sumanta; Li, Xianqi; Michailidis, George
2019
On the uniqueness and stability of dictionaries for sparse representation of noisy signals. Zbl 07160267
Garfinkle, Charles J.; Hillar, Christopher J.
2019
Greedy kernel change-point detection. Zbl 07160286
Truong, Charles; Oudre, Laurent; Vayatis, Nicolas
2019
Distributed robust beamforming based on low-rank and cross-correlation techniques: design and analysis. Zbl 07160301
Ruan, Hang; de Lamare, Rodrigo C.
2019
Optimal mean-reverting portfolio with leverage constraint for statistical arbitrage in finance. Zbl 1415.94307
Zhao, Ziping; Zhou, Rui; Palomar, Daniel P.
2019
MISC array: a new sparse array design achieving increased degrees of freedom and reduced mutual coupling effect. Zbl 1458.94161
Zheng, Zhi; Wang, Wen-Qin; Kong, Yangyang; Zhang, Yimin D.
2019
Asynchronous saddle point algorithm for stochastic optimization in heterogeneous networks. Zbl 1458.90490
Bedi, Amrit Singh; Koppel, Alec; Rajawat, Ketan
2019
Finding GEMS: multi-scale dictionaries for high-dimensional graph signals. Zbl 1458.94154
2019
STGRFT for detection of maneuvering weak target with multiple motion models. Zbl 1458.94164
Li, Xiaolong; Sun, Zhi; Yeo, Tat Soon; Zhang, Tianxian; Yi, Wei; Cui, Guolong; Kong, Lingjiang
2019
Adaptive global time sequence averaging method using dynamic time warping. Zbl 1458.94108
Liu, Yu-Tao; Zhang, Yong-An; Zeng, Ming
2019
Semi-blind inference of topologies and dynamical processes over dynamic graphs. Zbl 1458.94100
Ioannidis, Vassilis N.; Shen, Yanning; Giannakis, Georgios B.
2019
Advances in distributed graph filtering. Zbl 1458.94085
Coutino, Mario; Isufi, Elvin; Leus, Geert
2019
Corrections to “On the restricted isometry of the columnwise Khatri-Rao product”. Zbl 1458.94102
2019
Stable principal component pursuit via convex analysis. Zbl 1458.90679
Yin, Lei; Parekh, Ankit; Selesnick, Ivan
2019
Learning optimal resource allocations in wireless systems. Zbl 1458.90192
Eisen, Mark; Zhang, Clark; Chamon, Luiz F. O.; Lee, Daniel D.; Ribeiro, Alejandro
2019
DOA estimation exploiting sparse array motions. Zbl 07123270
Qin, Guodong; Amin, Moeness G.; Zhang, Yimin D.
2019
Subspace averaging and order determination for source enumeration. Zbl 07123271
Garg, Vaibhav; Santamaria, Ignacio; Ramírez, David; Scharf, Louis L.
2019
Extraction of useful information content from noisy signals based on structural affinity of clustered TFDs’ coefficients. Zbl 07123280
Saulig, Nicoletta; Lerga, Jonatan; Milanović, Željka; Ioana, Cornel
2019
Two-dimensional super-resolution via convex relaxation. Zbl 07123295
2019
Bilinear recovery using adaptive vector-AMP. Zbl 07123296
Sarkar, Subrata; Fletcher, Alyson K.; Rangan, Sundeep; Schniter, Philip
2019
Automatic recognition of general LPI radar waveform using SSD and supplementary classifier. Zbl 07123305
Hoang, Linh Manh; Kim, Minjun; Kong, Seung-Hyun
2019
Widely linear complex-valued estimated-input LMS algorithm for bias-compensated adaptive filtering with noisy measurements. Zbl 07123310
Zhang, Sheng; Zhang, Jiashu; Zheng, Wei Xing; So, Hing Cheung
2019
Bezerra Lopes, Wilder; Guimaraes Lopes, Cassio
2019
Characterization of analytic wavelet transforms and a new phaseless reconstruction algorithm. Zbl 07123333
Holighaus, Nicki; Koliander, Günther; Průša, Zdeněk; Abreu, Luis Daniel
2019
Full mean square performance bounds on quaternion estimators for improper data. Zbl 07123347
Xia, Yili; Tao, Song; Li, Zhe; Xiang, Min; Pei, Wenjiang; Mandic, Danilo P.
2019
Optimization algorithms for graph Laplacian estimation via ADMM and MM. Zbl 07123357
Zhao, Licheng; Wang, Yiwei; Kumar, Sandeep; Palomar, Daniel P.
2019
Tests for the parameters of chirp signal model. Zbl 07123361
Dhar, Subhra Sankar; Kundu, Debasis; Das, Ujjwal
2019
Joint network topology and dynamics recovery from perturbed stationary points. Zbl 07123382
Wai, Hoi-To; Scaglione, Anna; Barzel, Baruch; Leshem, Amir
2019
On the algorithmic computability of the secret key and authentication capacity under channel, storage, and privacy leakage constraints. Zbl 07123386
Boche, Holger; Schaefer, Rafael F.; Baur, Sebastian; Poor, H. Vincent
2019
Resilient distributed parameter estimation with heterogeneous data. Zbl 07123406
Chen, Yuan; Kar, Soummya; Moura, José M. F.
2019
An exact quantized decentralized gradient descent algorithm. Zbl 07123407
Reisizadeh, Amirhossein; Mokhtari, Aryan; Hassani, Hamed; Pedarsani, Ramtin
2019
Weakly convex regularized robust sparse recovery methods with theoretical guarantees. Zbl 07123415
Yang, Chengzhu; Shen, Xinyue; Ma, Hongbing; Chen, Badong; Gu, Yuantao; So, Hing Cheung
2019
Nonconvex regularized robust PCA using the proximal block coordinate descent algorithm. Zbl 07123440
Wen, Fei; Ying, Rendong; Liu, Peilin; Truong, Trieu-Kien
2019
Sparse recovery and dictionary learning from nonlinear compressive measurements. Zbl 07123505
Rencker, Lucas; Bach, Francis; Wang, Wenwu; Plumbley, Mark D.
2019
Manifold optimization over the set of doubly stochastic matrices: a second-order geometry. Zbl 07123512
Douik, Ahmed; Hassibi, Babak
2019
Learning tensors from partial binary measurements. Zbl 1414.15029
Ghadermarzy, Navid; Plan, Yaniv; Yilmaz, Ozgur
2019
CayleyNets: graph convolutional neural networks with complex rational spectral filters. Zbl 1415.68169
Levie, Ron; Monti, Federico; Bresson, Xavier; Bronstein, Michael M.
2019
Computationally efficient off-line joint change point detection in multiple time series. Zbl 1415.94326
Eriksson, Markus; Olofsson, Tomas
2019
Information geometric approach to multisensor estimation fusion. Zbl 1415.94250
Tang, Mengjiao; Rong, Yao; Zhou, Jie; Li, X. Rong
2019
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2021-09-24 18:32:10
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https://wikimili.com/en/Wave_maps_equation
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# Wave maps equation
Last updated
In mathematical physics, the wave maps equation is a geometric wave equation that solves
The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves or light waves. It arises in fields like acoustics, electromagnetics, and fluid dynamics.
${\displaystyle D^{\alpha }\partial _{\alpha }u=0}$
where ${\displaystyle D}$ is a connection. [1] [2]
It can be considered a natural extension of the wave equation for Riemannian manifolds. [3]
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In mathematics, a weak solution to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for different classes of equations. One of the most important is based on the notion of distributions.
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In the fields of dynamical systems and control theory, a fractional-order system is a dynamical system that can be modeled by a fractional differential equation containing derivatives of non-integer order. Such systems are said to have fractional dynamics. Derivatives and integrals of fractional orders are used to describe objects that can be characterized by power-law nonlocality, power-law long-range dependence or fractal properties. Fractional-order systems are useful in studying the anomalous behavior of dynamical systems in physics, electrochemistry, biology, viscoelasticity and chaotic systems.
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## References
1. Tataru, Daniel (1 January 2005). "Rough solutions for the wave maps equation". American Journal of Mathematics. 127 (2): 293–377. CiteSeerX . doi:10.1353/ajm.2005.0014.
2. Tataru, Daniel (2004). "The wave maps equation" (PDF). Bulletin of the American Mathematical Society (N.S.) . 41 (2): 185–204. doi:10.1090/S0273-0979-04-01005-5. Zbl 1065.35199.
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2019-10-21 05:29:03
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https://ask.openstack.org/en/question/122536/cinder-filter_function-takes-string-arguments/
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# cinder filter_function takes string arguments?
Has anyone tried to use the cinder filter_function to use string arguments? I am not able to get it to work if I am using a string for comparison.
filter_function = "stats.pool_name == 192.168.1.130:/nfs_flexvol"
The error I observe when attempting to create a Cinder volume that maps to the backend where the filter_function is defined is below:
2019-06-10 14:26:44.285 23772 WARNING cinder.scheduler.filters.driver_filter [req-43dd8725-af3c-448a-b207-03a1e77e854a 5fb8caadb3794155bf358dc0a678b80c c9a753e77bdd4bd0af2708f6a522745b - - -] Error in filtering function 'stats.pool_name == 192.168.1.130:/nfs_flexvol' : 'ParseException: Expected end of text (at char 26), (line:1, col:27)' :: failing host
Looks like a pyparsing error.
edit retag close merge delete
If you look at the column information (col 27), the problem starts at the second dot in the IP address. Pyparsing seems to think that 192.168 is a decimal number, and the second dot is obviously an error in this case.
My guess is that you have to put quotes around the pool name.
( 2019-06-10 23:25:57 -0500 )edit
Hi @Bernd-Bausch, I tried it after updating the filter_function as follows: filter_function = "stats.pool_name == '192.168.1.130:/nfs_flexvol'" The ParseException now thrown is at char 16 (for the first space between 'e' and '==')
( 2019-06-12 12:03:29 -0500 )edit
It does not change even if the filter_function is defined as "stats.pool_name== '192.168.1.130:/nfs_flexvol'". The ParseException is thrown at char 15
( 2019-06-12 12:04:33 -0500 )edit
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2020-10-26 13:33:33
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https://www.freemathhelp.com/forum/threads/109433-Solve-for-4-10-30-1-5-(was-a-question-on-a-test-teacher-amp-I-disagree-on-ans-)
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# Thread: Solve for 4/10 - 30% +1/5 (was a question on a test; teacher & I disagree on ans.)
1. ## Solve for 4/10 - 30% +1/5 (was a question on a test; teacher & I disagree on ans.)
I recently had this question on a test:
4/10 - 30% +1/5
Me and my teacher have different answers for it. His solution is:
4/10 - 30% +1/5
4/10 - 3/10 +1/5
1/10+1/5*2/2=3/10
He tells me that the 30% is equal to 0.3 or 3/10, My solution is:
4/10 - 30% + 1/5
4/10 - 3/10*4/10 + 1/5
40/100 - 12/100 + 20/100
40-12/100 = 28/100
28/100+20/100=48/100=24/50=12/25 or 0.48
Excuse me if my English is bad or my calculations are unclear, but I want to know what is the right answer.
2. Originally Posted by emil
I recently had this question on a test:
4/10 - 30% +1/5
Me and my teacher have different answers for it. His solution is:
4/10 - 30% +1/5
4/10 - 3/10 +1/5
1/10+1/5*2/2=3/10
He tells me that the 30% is equal to 0.3 or 3/10, My solution is:
4/10 - 30% + 1/5
4/10 - 3/10*4/10 + 1/5
40/100 - 12/100 + 20/100
40-12/100 = 28/100
28/100+20/100=48/100=24/50=12/25 or 0.48
Excuse me if my English is bad or my calculations are unclear, but I want to know what is the right answer.
This is a matter of interpretation, and on that, your teacher rules!
We don't normally write expressions with a mix of fractions and percentages; but in order to be mathematically consistent, we must interpret it as your teacher did, treating a percentage as merely a different representation of a fraction.
But the reason we don't normally write like that is that one can interpret it as you do, making it somewhat ambiguous. In fact, many simple calculators do just that: entering "x + y%" is interpreted as "increase x by y percent", that is, as "x * (1 + y/100)". So your interpretation is understandable; it is just not what we expect it to mean mathematically. (And I dislike using those calculators, because if I use it for anything other than the applications they expect, I can't be quite sure what they will do. More advanced calculators, if they have a % button at all, treat it as your teacher does.)
Since your teacher agrees with me, you can take that as the truth within your class! Out in the real world, you would be wise to ask someone what they mean by it, if you ever see something written like this.
By the way, you wrote your explanation very clearly, as many students would not. Thanks for making it easy for us! The one thing I would have written differently is "40-12/100", which should be (40-12)/100.
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2019-02-15 21:04:25
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|
https://www.dreamwings.cn/hdu3966/4798.html
|
# HDU 3966 Aragorn’s Story （树链剖分）
## Problem Description
Our protagonist is the handsome human prince Aragorn comes from The Lord of the Rings. One day Aragorn finds a lot of enemies who want to invade his kingdom. As Aragorn knows, the enemy has N camps out of his kingdom and M edges connect them. It is guaranteed that for any two camps, there is one and only one path connect them. At first Aragorn know the number of enemies in every camp. But the enemy is cunning , they will increase or decrease the number of soldiers in camps. Every time the enemy change the number of soldiers, they will set two camps C1 and C2. Then, for C1, C2 and all camps on the path from C1 to C2, they will increase or decrease K soldiers to these camps. Now Aragorn wants to know the number of soldiers in some particular camps real-time.
## Input
Multiple test cases, process to the end of input.
For each case, The first line contains three integers N, M, P which means there will be N(1 ≤ N ≤ 50000) camps, M(M = N-1) edges and P(1 ≤ P ≤ 100000) operations. The number of camps starts from 1.
The next line contains N integers A1, A2, …AN(0 ≤ Ai ≤ 1000), means at first in camp-i has Ai enemies.
The next M lines contains two integers u and v for each, denotes that there is an edge connects camp-u and camp-v.
The next P lines will start with a capital letter ‘I’, ‘D’ or ‘Q’ for each line.
‘I’, followed by three integers C1, C2 and K( 0≤K≤1000), which means for camp C1, C2 and all camps on the path from C1 to C2, increase K soldiers to these camps.
‘D’, followed by three integers C1, C2 and K( 0≤K≤1000), which means for camp C1, C2 and all camps on the path from C1 to C2, decrease K soldiers to these camps.
‘Q’, followed by one integer C, which is a query and means Aragorn wants to know the number of enemies in camp C at that time.
## Output
For each query, you need to output the actually number of enemies in the specified camp.
## Sample Input
3 2 5
1 2 3
2 1
2 3
I 1 3 5
Q 2
D 1 2 2
Q 1
Q 3
## Sample Output
7
4
8
## 题意
1. 给树中 u->v 路径上的所有点的权值增加或减少 k
2. 查询树中的某一个点的当前权值
## AC 代码
#include<iostream>
#include<queue>
#include<cstdio>
#include<cstring>
using namespace std;
typedef long long LL;
const int maxn = 50010;
struct Edge
{
int to;
int next;
} edge[maxn<<1];
int top[maxn]; //v所在重链的顶端节点
int fa[maxn]; //父亲节点
int deep[maxn]; //节点深度
int num[maxn]; //以v为根的子树节点数
int p[maxn]; //v与其父亲节点的连边在线段树中的位置
int fp[maxn]; //与p[]数组相反
int son[maxn]; //重儿子
int pos;
int w[maxn];
int n; //节点数目
void init()
{
memset(son,-1,sizeof(son));
tot=0;
pos=1; //因为使用树状数组，所以我们pos初始值从1开始
}
{
edge[tot].to=v;
}
void dfs1(int u,int pre,int d) //第一遍dfs，求出 fa,deep,num,son (u为当前节点,pre为其父节点,d为深度)
{
deep[u]=d;
fa[u]=pre;
num[u]=1;
{
int v=edge[i].to;
if(v!=pre)
{
dfs1(v,u,d+1);
num[u]+=num[v];
if(son[u]==-1||num[v]>num[son[u]]) //寻找重儿子
son[u]=v;
}
}
}
void dfs2(int u,int sp) //第二遍dfs，求出 top,p
{
top[u]=sp;
p[u]=pos++;
fp[p[u]]=u;
if(son[u]!=-1) //如果当前点存在重儿子，继续延伸形成重链
dfs2(son[u],sp);
else return;
{
int v=edge[i].to;
if(v!=son[u]&&v!=fa[u]) //遍历所有轻儿子新建重链
dfs2(v,v);
}
}
int lowbit(int x)
{
return x&-x;
}
int query(int i) //查询
{
int s=0;
while(i>0)
{
i-=lowbit(i);
}
return s;
}
{
while(i<=n)
{
i+=lowbit(i);
}
}
void update(int u,int v,int val)
{
int f1=top[u],f2=top[v];
while(f1!=f2)
{
if(deep[f1]<deep[f2])
{
swap(f1,f2);
swap(u,v);
}
add(p[f1],val); //因为区间减法成立，所以我们把对某个区间[f1,u] 的更新拆分为 [0,f1] 和 [0,u] 的操作
u=fa[f1];
f1=top[u];
}
if(deep[u]>deep[v])
swap(u,v);
}
int main()
{
ios::sync_with_stdio(false);
int m,ps;
while(cin>>n>>m>>ps)
{
int a,b,c;
for(int i=1; i<=n; i++)
cin>>w[i];
init();
for(int i=0; i<m; i++)
{
cin>>a>>b;
}
dfs1(1,0,0);
dfs2(1,1);
for(int i=1; i<=n; i++)
{
}
for(int i=0; i<ps; i++)
{
char op;
cin>>op;
if(op=='Q')
{
cin>>a;
cout<<query(p[a])<<endl;
}
else
{
cin>>a>>b>>c;
if(op=='D')c=-c;
update(a,b,c);
}
}
}
return 0;
}
|
2021-10-28 17:20:21
|
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|
https://zbmath.org/authors/?q=ai%3Ahirose.minoru
|
# zbMATH — the first resource for mathematics
## Hirose, Minoru
Compute Distance To:
Author ID: hirose.minoru Published as: Hirose, Minoru; Hirose, M.
Documents Indexed: 17 Publications since 2011
all top 5
#### Co-Authors
4 single-authored 8 Sato, Nobuo 5 Murahara, Hideki 3 Onozuka, Tomokazu 2 Saito, Shingo 2 Tasaka, Koji 1 Cappellaro, Paola 1 Iwaki, Kohei 1 Kawashima, Makoto
all top 5
#### Serials
2 Journal of Number Theory 1 Advances in Mathematics 1 Journal of Algebra 1 Manuscripta Mathematica 1 Mathematica Scandinavica 1 Mathematische Zeitschrift 1 Proceedings of the American Mathematical Society 1 Proceedings of the Japan Academy. Series A 1 Publications of the Research Institute for Mathematical Sciences, Kyoto University 1 Tokyo Journal of Mathematics 1 Indagationes Mathematicae. New Series 1 Documenta Mathematica 1 The Ramanujan Journal 1 Quantum Information Processing 1 International Journal of Number Theory 1 RIMS Kôkyûroku Bessatsu
#### Fields
16 Number theory (11-XX) 3 Special functions (33-XX) 2 Combinatorics (05-XX) 1 Commutative algebra (13-XX) 1 Quantum theory (81-XX)
#### Citations contained in zbMATH Open
10 Publications have been cited 16 times in 12 Documents Cited by Year
On the functional equation of the normalized Shintani $$L$$-function of several variables. Zbl 1391.11104
Hirose, Minoru; Sato, Nobuo
2015
Iterated integrals on $$\mathbb{P}^1 \setminus \{0, 1, \infty, z \}$$ and a class of relations among multiple zeta values. Zbl 1457.11117
Hirose, Minoru; Sato, Nobuo
2019
Time-optimal control with finite bandwidth. Zbl 1395.81085
Hirose, M.; Cappellaro, P.
2018
Algebraic differential formulas for the shuffle, stuffle and duality relations of iterated integrals. Zbl 1439.05234
Hirose, Minoru; Sato, Nobuo
2020
Double shuffle relations for refined symmetric multiple zeta values. Zbl 1448.11165
Hirose, Minoru
2020
Duality/sum formulas for iterated integrals and their application to multiple zeta values. Zbl 1458.11132
Hirose, Minoru; Iwaki, Kohei; Sato, Nobuo; Tasaka, Koji
2019
A lower bound of the dimension of the vector space spanned by the special values of certain functions. Zbl 1430.11121
Hirose, Minoru; Kawashima, Makoto; Sato, Nobuo
2017
Hoffman’s conjectural identity. Zbl 1467.11081
Hirose, Minoru; Sato, Nobuo
2019
Linear relations of Ohno sums of multiple zeta values. Zbl 1456.11162
Hirose, Minoru; Murahara, Hideki; Onozuka, Tomokazu; Sato, Nobuo
2020
Eisenstein series identities based on partial fraction decomposition. Zbl 1388.11010
Hirose, Minoru; Sato, Nobuo; Tasaka, Koji
2015
Algebraic differential formulas for the shuffle, stuffle and duality relations of iterated integrals. Zbl 1439.05234
Hirose, Minoru; Sato, Nobuo
2020
Double shuffle relations for refined symmetric multiple zeta values. Zbl 1448.11165
Hirose, Minoru
2020
Linear relations of Ohno sums of multiple zeta values. Zbl 1456.11162
Hirose, Minoru; Murahara, Hideki; Onozuka, Tomokazu; Sato, Nobuo
2020
Iterated integrals on $$\mathbb{P}^1 \setminus \{0, 1, \infty, z \}$$ and a class of relations among multiple zeta values. Zbl 1457.11117
Hirose, Minoru; Sato, Nobuo
2019
Duality/sum formulas for iterated integrals and their application to multiple zeta values. Zbl 1458.11132
Hirose, Minoru; Iwaki, Kohei; Sato, Nobuo; Tasaka, Koji
2019
Hoffman’s conjectural identity. Zbl 1467.11081
Hirose, Minoru; Sato, Nobuo
2019
Time-optimal control with finite bandwidth. Zbl 1395.81085
Hirose, M.; Cappellaro, P.
2018
A lower bound of the dimension of the vector space spanned by the special values of certain functions. Zbl 1430.11121
Hirose, Minoru; Kawashima, Makoto; Sato, Nobuo
2017
On the functional equation of the normalized Shintani $$L$$-function of several variables. Zbl 1391.11104
Hirose, Minoru; Sato, Nobuo
2015
Eisenstein series identities based on partial fraction decomposition. Zbl 1388.11010
Hirose, Minoru; Sato, Nobuo; Tasaka, Koji
2015
all top 5
#### Cited by 16 Authors
4 Hirose, Minoru 2 Murahara, Hideki 2 Sato, Nobuo 1 Charlton, Steven 1 Espinoza, Milton 1 Kaneko, Masanobu 1 Mertens, Michael Helmut 1 Murakami, Takuya 1 Ono, Masataka 1 Onozuka, Tomokazu 1 Rolen, Larry 1 Seki, Shin-ichiro 1 Sprang, Johannes 1 Tasaka, Koji 1 Xu, Ce 1 Yamamoto, Shuji
all top 5
#### Cited in 9 Serials
3 Journal of Number Theory 2 Research in Number Theory 1 Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1 Advances in Mathematics 1 Duke Mathematical Journal 1 Journal of Algebra 1 Manuscripta Mathematica 1 Kyushu Journal of Mathematics 1 Selecta Mathematica. New Series
#### Cited in 4 Fields
12 Number theory (11-XX) 4 Combinatorics (05-XX) 1 Commutative algebra (13-XX) 1 Special functions (33-XX)
|
2021-09-22 11:57:01
|
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|
https://codeforces.com/blog/entry/59262
|
### Vovuh's blog
By Vovuh, 19 months ago, ,
Hello!
Codeforces Round #479 (Div. 3) will start on May 6 (Sunday) at 14:05 (UTC). It will be the first Div.3 round in the history of Codeforces. You will be offered 6 problems with expected difficulties to compose an interesting competition for participants with ratings up to 1600. Probably, participants from the first division will not be at all interested by this problems. And for 1600-1899 the problems will be too easy. However, all of you who wish to take part and have rating 1600 or higher, can register for the round unofficially.
The round will be hosted by rules of educational rounds (extended ACM-ICPC). Thus, during the round, solutions will be judged on preliminary tests, and after the round it will be a 12-hour phase of open hacks. I tried to make strong tests — just like you will be upset if many solutions fail after the contest is over.
Remember that only the trusted participants of the third division will be included in the official standings table. As it is written by link, this is a compulsory measure for combating unsporting behavior. To qualify as a trusted participants of the third division, you must:
• take part in at least two rated rounds (and solve at least one problem in each of them),
• do not have a point of 1900 or higher in the rating.
Regardless of whether you are a trusted participant of the third division or not, if your rating is less than 1600, then the round will be rated for you.
Thanks to MikeMirzayanov for the platform, help with ideas for problems and for coordination of my work. Thanks to Ne0n25 for testing the round.
UPD: Thanks a lot to testers dreamoon, eddy1021, step_by_step and egor.lifar who agreed to test the problems and found the mistakes. Now we are ready to start the round!
I hope that you will like the problems. If suddenly something does poorly with difficulties of the problems, then we will adjust the difficulties in the next Div. 3 rounds.
Briefly about myself. My name is Vladimir Petrov, I am studying at the 3rd year at Saratov State University and from the high-school I am studying at SSU Olympiad Training Center. In ICPC I participate in the team "Saratov SU Daegons" with PikMike and Ne0n25. I like reading and watching science fiction. I’ve read "Song of Ice and Fire" 3 times and wait for publication of 2 more volumes.
Good luck!
UPD2: Editorial
UPD3:
Congratulations to the winners (official results):
Rank Competitor Problems Solved Penalty
1 cMartynas 6 69
2 nimphy 6 117
3 iman12 6 124
4 mandinga 6 128
5 raffica 6 144
Congratulations to the best hackers:
Rank Competitor Hack Count
1 greencis 36:-8
2 Tlalafuda__Tlalafu 10:-1
3 STommydx 9:-1
4 dalex 8:-7
5 dreamoon 4:-3
6 shnk 2
97 successful hacks and 297 unsuccessful hacks were made in total!
And finally people who were the first to solve each problem:
Problem Competitor Penalty
A kuzmoid 0:01
B c650Alpha 0:04
C Milhous 0:08
D Milhous 0:11
E s0mth1ng 0:09
F DoveDragon 0:14
• +451
» 19 months ago, # | +32 What is the relationship between div3 & div2 problems. For instance A div1 is equal to C div2
• » » 19 months ago, # ^ | 0 This contest is for div3 only. I do not think there will be a relationship between div2 and div3. I mean can something be easier than Div2A? Maybe Div3A+Div3B+DivC are so close to Div2A. We will see after the round :)
• » » 19 months ago, # ^ | 0 This contest is only Div3 , so i don't think there's any relationship with Div1 or 2.
• » » 19 months ago, # ^ | 0 "We plan to include in these rounds simple training problems that will help beginner participants to gain skills and to get new knowledge in a real contest."
» 19 months ago, # | +21 I found a mistake for you :)) Please correct it.
• » » 19 months ago, # ^ | +26 Thank you, it is fixed now!
» 19 months ago, # | +8 So will there have Educational Round like the time before updating? :P
» 19 months ago, # | +8 Finally!
» 19 months ago, # | +23 Last three lines was awesome . :D
• » » 19 months ago, # ^ | +5 :P
» 19 months ago, # | +8 "The round will be hosted by rules of educational rounds..." Honestly, It's really boring! Given that this is a div3 and problems should be easy.. Then what is the use of the hacking phase ??!!
• » » 19 months ago, # ^ | +179 We will experiment with formats of Div. 3 rounds, but there are some reasons to start with rules of Educational Rounds: I prefer Div. 3 participants to solve problems during a round instead of spending time trying to hack for non-trivial problems tests in Educational rounds much more complete than pretests in regular Codeforces round. my plan is that Div. 3 rounds will not affect regular Div. 1/Div. 2 rounds in terms of schedule. It means that Div. 3 rounds (as well as educational) can't use time of KAN to coordinate them. They completely prepared by writers with my little help. It is much easier to write a round by rules of Educational Rounds than Codeforces: issues in writer solutions/validator/checkers affect a round less, less requirements on completeness of tests, etc.
• » » » 19 months ago, # ^ | +9 This is what I have been waiting for long time: "**I prefer Div. 3 participants to solve problems during a round instead of spending time trying to hack**". Hacking is a good way to develop debugging skill. But a div3 participants should not spend the whole time trying to hack a problem just to increase rating instead of solving another problem.Thanks Mike.
• » » » 19 months ago, # ^ | ← Rev. 2 → -18 But what about the feeling of DIV3 participants. I really enjoyed codeforces round DIV1 / DIV2, due to div1 and div2, DIV3 is also should not be affected. div3 also must be same as daily codeforces round not educational type.
• » » » » 19 months ago, # ^ | 0 As Mike said, div2 rounds are not affected. So you can just ignore div3 rounds.
» 19 months ago, # | +23 First time I saw about myself in any round announcement.
• » » 19 months ago, # ^ | ← Rev. 2 → +12 You might want to take a look at this!
• » » » 19 months ago, # ^ | +9 Your link is not clickable. Check it.
• » » » » 19 months ago, # ^ | 0 where you saw you?
» 19 months ago, # | +7 It's rated for me so :D
» 19 months ago, # | +23 unofficial participator for the first time :) :D
» 19 months ago, # | +5 Why extended ACM-ICPC? Why not like a normal Codeforces round?
• » » 19 months ago, # ^ | 0
» 19 months ago, # | +44 Will I become Div.4 user if I lose rating in Div.3 contest?
• » » 19 months ago, # ^ | -15 As of 5/5/2018, there is no such thing as a Div. 4 user. The lowest is Div. 3.
• » » » 19 months ago, # ^ | +52 Dude, don't you know about sarcasm?
• » » » » 19 months ago, # ^ | +8 and neither humor
» 19 months ago, # | -14 Hello, If this round is following Educational Round formats than I want to bring something to your notice. See the picture attached. My friend solved Problem A and his solution for problem C failed. I solved Problem C and my solution for problem A failed. I just want to point out that Problem C was tougher than Problem A and anybody who solves tougher question should get better rank. I think people will say that it is for learning new skills and I should not concern myself with the rating. But I still think that this is unfair.
• » » 19 months ago, # ^ | 0 Probably, participants from the first division will not be at all interested by this problems. And for 1600-1899 the problems will be too easy.So, I think, there will be no issues with Div3 problems, as at some points, we will figure out that it is eventually too easy, regardless of which problem in a contest (A, B, C, D, E, or even F).In other words, soon we will realize none of any Div3 problems will be any tough to be obliged to be distinguished.
• » » » 19 months ago, # ^ | ← Rev. 2 → +9 Still, the point system is better than the penalty system in my humble opinion.
• » » 19 months ago, # ^ | +2 ACM-ICPC scoring is used in many contests.If you don't like the format of educational round, you don't have to participate.If you don't want to be rated because of these rounds, you can always do virtual participation of solve problems regardless.
• » » » 19 months ago, # ^ | -6 I am fine with whatever format you want to use, I am asking is it fair?
• » » » » 19 months ago, # ^ | ← Rev. 2 → -21 1) You complained about the rules of the contest.2) You just said "I am fine with whatever format"3) ?????????4) Now you ask if the rules are fair.Yes, they are since your signed up knowing them.As I said, if you disagree with this type of scoring just don't participate. If you are annoyed at messing up on problem A check over your code more before you submit.
• » » » » » 19 months ago, # ^ | +19 Hmm, thanks for helping.
» 19 months ago, # | ← Rev. 2 → 0 One more chapter is going to open from May 6 (Sunday) at 14:05 (UTC)..!!
» 19 months ago, # | -39 Div3 => noobs?
• » » 19 months ago, # ^ | +15 once you, too, was thus noob
• » » 19 months ago, # ^ | +25 Hey bro cyan isn't too hot either. We're all noobs, just some less nooby than others.
• » » » 19 months ago, # ^ | -12 Effective...
» 19 months ago, # | +39 So which one do you like better? The books or the TV show? Vovuh
• » » 19 months ago, # ^ | ← Rev. 2 → +52 I like books better than the TV show, but the TV show is also wonderful =) Especially Natalie Dormer =)
• » » » 19 months ago, # ^ | 0 You've probably meant Natalie Dormer
• » » » » 19 months ago, # ^ | +9 Oh, yes, you are right, i made a mistake
» 19 months ago, # | -11 Can't Wait :D****
• » » 19 months ago, # ^ | -29 Still messi is better than ronaldo
» 19 months ago, # | -17 Hoping For short Problem statement
» 19 months ago, # | ← Rev. 3 → -10 As CF is introducing div3 for begginers, it is also good to change title/color to make it interesting for begginers, like 1000-1200 is begginer, <1000 is newbie !!
» 19 months ago, # | -120 Earn bitcoins while browsing the Internet https://getcryptotab.com/1065838
» 19 months ago, # | 0 Looking forward to concise problem statements without unnecessarily long storylines!
» 19 months ago, # | -54 I think it's need to make div 4 for under 1000.
» 19 months ago, # | +10 Div3 finally comes! No more falling scores!
• » » 19 months ago, # ^ | +8 One can only hope^_^
• » » » 19 months ago, # ^ | +9 OK, then just bless!
» 19 months ago, # | -27 is it rated?
• » » 19 months ago, # ^ | -17 lol
» 19 months ago, # | -35 tourist do you want to join us in div 3 :) ?!!
• » » 19 months ago, # ^ | +36 don't worry, they'll open div 0 for him
• » » » 19 months ago, # ^ | -25 for tourist? I think Petr has higher priority in the current time.
• » » » 19 months ago, # ^ | -10 see who will put problems for him Your text to link here...
• » » 19 months ago, # ^ | +4 He does not even need to look to the problems to solve them. :)
• » » » 19 months ago, # ^ | +12 lol.u can't solve some problem even by reading them :)
» 19 months ago, # | +17 Will div 3 participants be allowed to take part in div 2 contest?
• » » 19 months ago, # ^ | +8 Obviously...
• » » » 19 months ago, # ^ | 0 Will it be rated for div 3 contestants?
• » » » » 19 months ago, # ^ | ← Rev. 2 → 0 when it is only div 2 contest then the user who's rating is less than 2100 can participate in this round and when both div 1 and div 2 round will be held , user who's rating less than 1900 can participate in div 2 contest as a official participant.... official means rating will be update
» 19 months ago, # | 0 How many problems will be there?
• » » 19 months ago, # ^ | +1 You will be offered 6 problems
» 19 months ago, # | +7 Waiting for a DIV2 contest. The sooner, the better. The more, the merrier. :)
» 19 months ago, # | 0 "8093" Registration in Div 3 !! wow <3 .
» 19 months ago, # | -6 I usually solve C in div2..Which prob i should choose to solve in div3 ??
» 19 months ago, # | -6 Problem A : Print "Hello World"
» 19 months ago, # | +6 HTTP Response: 500, 502. Website failed.
» 19 months ago, # | +50 20 minutes of penalty is too much.
• » » 19 months ago, # ^ | +3 for a 2-hour contest, indeed. 20 minutes seems better with 3~5 hours contest.
» 19 months ago, # | ← Rev. 2 → -35 Failed at BDon't understand what C is askingInput #1 of F is wrong? Surely largest is 5~ cba to attempt as maybe I'm misreading like CRIP, time for gaming instead
» 19 months ago, # | +11 AH ? I can't register while the contest is running !What a pity!
» 19 months ago, # | +67 20 minutes of penalty is too much.I used 19min to solve all the problems.But I got 1 penalty.. doubled the time..
• » » 19 months ago, # ^ | +18 20 mins of penalty is standard ACM rules. It encouages coding correctly instead of coding fast, which I think is ok. Also, the contest is targeted at people who can't solve all problems in 19 mins :)
• » » 19 months ago, # ^ | 0 Penalty is calculated as the sum of times each problem solved (+ 20mins penalties), not as the time of the last solution. So it's not doubled because of that, don't worry :)
• » » » 19 months ago, # ^ | +2 You're right.. But I got "502 Bad Gateway" for 30 mins LOL
• » » » » 19 months ago, # ^ | 0 So? That's completely irrelevant to the discussion. Nobody cares how well you did in a div3 contest, mate.
» 19 months ago, # | +31 Thanks Vovuh for this round , and thanks MikeMirzayanov for Div3 !!! This is the first time that i am able to solve all problems .
» 19 months ago, # | 0 Spending 30 minutes to solve the last problem. Is it a little bit slow compared to how such contestants like me (1800s-rating guys) are supposed to be?
» 19 months ago, # | +25 Why are there some (not all) unrated accounts in the official standings?
• » » 19 months ago, # ^ | +3 What exactly? Please, give me some examples of such handles.
• » » » 19 months ago, # ^ | +22 official rank #1 for me is readers5 and rank #4 T_van for example
• » » » » 19 months ago, # ^ | 0 That happens pretty often in div2 if not all the time, even in older contests.My guess has always been that some of those are smurf accounts created by high rated people just to be on the top spots in a contest. A lot of them are legit skilled people on their first contest, but some are left inactive after one contest.
• » » » » » 19 months ago, # ^ | +8 Yes but the thing is that it's not supposed to happen with the new div3 rounds.
• » » » » » » 19 months ago, # ^ | 0 Sorry, you're right. Forgot that.
» 19 months ago, # | -42 I think this was too easy even for div 3.
• » » 19 months ago, # ^ | +11 I agree. In Div 2, very few experts (the highest eligible rating) solve the set. In this one nearly every specialist and their mother solved it (not really but you get the idea). You never see 200+ eligible participants solve the set in a round.
• » » » 19 months ago, # ^ | 0 Yes, It should be a little harder so only about 50 or less gets full. NOT 200. Even me solved the whole set! Maybe it should be for users whose rating less than 1400
» 19 months ago, # | +10 First time in my whole life! I solved all problems! (but... It is div3)
» 19 months ago, # | 0 Holly cow, I'm experiencing something in Java I never ever experienced in a contest!For problem "C" I'm doing a sort using Arrays.sort. My code times out (it takes more than 2 seconds for testcase 6). But, if I randomize a little bit the input, i.e., instead of doing: a[i] = in.nextInt(); // "in" is the input stream I do: a[(i + (n / 2)) % n] = in.nextInt(); the solution passes!Does anybody know how can this be?
• » » 19 months ago, # ^ | 0 It seems that there are arrays that take O(n^2) time to sort in Java, using Arrays.sort(): http://codeforces.com/blog/entry/4827Most likely test case 6 is such an array and was designed to discourage people to use the O(nlog(n)) solution instead of the linear one...
• » » » 19 months ago, # ^ | 0 Is there simple linear solution for C?
• » » » » 19 months ago, # ^ | 0 Yes, you can use quickselect to find the k-th smallest element in an array in linear time: https://en.wikipedia.org/wiki/Quickselect
• » » » » » 19 months ago, # ^ | 0 In C++, you're right (because there is std::nth_element). But I mean that in many standard libraries, there is sort, but there isn't nth_element. So in this languages linear solution will be much harder.
• » » » » » 19 months ago, # ^ | +5 No, quickselect has a worst case of n^2, similar problem to your sort
• » » » » » 19 months ago, # ^ | +1 Quickselect have O(n^2) worst time, nth_element in C++ have O(n*logn) worst time, but here is Medians of medians algorithm (or Introselect variation), which have O(n) worst time.But it's not simple (or not obvious at least).
» 19 months ago, # | ← Rev. 3 → +32 It seems the difficulty is something like:Div3 A, B = Div2 ADiv3 C = Div2 BDiv3 D, E = between Div2 B and Div2 CDiv3 F = Div2 C
• » » 19 months ago, # ^ | +1 I think Div3B is too simple for Div2B
• » » » 19 months ago, # ^ | 0 Possibly. Honestly Div2 A and B feel more or less the same difficulty, at least in my opinion. Only thing I've seen is that Div2 B sometimes has annoying edge cases, so yeah you're probably right, it's probably A level.
» 19 months ago, # | +21 Intially i was not able to look at question and was getting a continuous error of Bad Gateway 502
» 19 months ago, # | 0 :(
» 19 months ago, # | 0 I Accepted F after contest finished 1 sec...(Sorry for my poor English)
» 19 months ago, # | 0 how can solve the ploblem c?
• » » 19 months ago, # ^ | ← Rev. 2 → 0 Case 1:- If k=0 then check if arr[0]=1. If yes the print -1 else print arr[0]-1. Case 2:- Ifk==nsimply printarr[n-1] Case 3:- Check the value of arr[k-1]. If arr[k]==arr[k-1] then print -1 else print arr[k-1] This is sample test case 2.This is all in a sorted array.
• » » » 19 months ago, # ^ | 0 oh ,i forgot to judge if arr[0]==1, anyway,thank you!
• » » 19 months ago, # ^ | 0 Sort the elements, and the k'th element (1-indexed) is your answer. If k = 0, then just subtract 1 from the first element.The no solution cases are when the k'th element equals the k+1'th element (why?) and when k=0 and the first element is 1, because subtracting 1 would be outside of your range (you may be tempted to try to be clever and subtract 2 so you don't have to handle that case separately, but that fails if the smallest element is 2).
• » » » 19 months ago, # ^ | 0 I don't get how, for k=0, there could be an answer different than -1.I mean, as I read it, there is no x «such that exactly 0 elements of given sequence are less than or equal to x.», regardless the value of the first element. Am I missing something?What exactly do you mean by «because subtracting 1 would be outside of your range»? I guess you refer to the range [1, 10^9], but why should I subtract something?Thanks
• » » » » 19 months ago, # ^ | ← Rev. 2 → 0 k=0.Output a number x such that 0 (no) elements from the sequence are less than x.If the smallest element in the range is l and max is r.Then.If you take an element >r then all elements will be less than r and k=0 will not be satisfied.If you take an element>=l and element<=r then some elements in the range will be less than x and again k=0 will not be satisfied.As for last case if l==1 then you can output 0 because no(0) elements will be less than 0. However since the answer must be in the range[1,1e18], in this case the answer will be 0. For all other cases output l-1.
• » » » » 19 months ago, # ^ | 0 Say your sequence is 10 26 4, and k=0.Simply output the number 3, and that is 0 elements from the sequence are less than or equal to 3. If the smallest element is 1, then there is no answer that is in the range from 1 to 1 billion.
• » » » » » 19 months ago, # ^ | 0 That make sense. I should have been less superficial. Or more, depending on how you view it :) Thanks Kognition and horcrux2301
» 19 months ago, # | ← Rev. 2 → 0 Can E be solved using this logic: LINK
• » » 19 months ago, # ^ | ← Rev. 2 → +5 if you mean E,E is simple. Just for each connected component see that every node is connected to exactly 2 other nodes.
• » » » 19 months ago, # ^ | ← Rev. 2 → 0 I tried this for problem D.Take a boolean array check[n] and mark all as false. Then I tried making chains. Take the first element and mark it as true and this will start a new chain. Then for each other false element in array check if it can be added to the chain started by this element, either by appending it to the beginning or to the end. If yes mark this as true too. For beginning, check if the new element divided by 3 is equal to first element in chain or if new element multiplied by 2 is equal to first element in the chain. Similarly we do for the end.However, I couldn't figure out how to decide a way to print all these chains in a valid manner. Is there any way to do figure this part out.
• » » » » 19 months ago, # ^ | +1 I think your approach is wrong or maybe i understood it wrong anyways this is what i did: A move consists of dividing by 3 (if divisible) or multiplying by 2. This means given a number we are taking away 3's and giving it 2's. So a state once reached will never be reached again. Why? Because to reach that state either you need to multiply by 3 or divide by 2 both of which is not a part of move. So what we can do is treat every element as starting element once and try generating the sequence from elements available. If all the elements are used you can just print that sequence.
• » » » » 19 months ago, # ^ | ← Rev. 2 → 0 This can be done by thinking of an array as a directed acyclic graph, and do a topological sort.
• » » » » » 19 months ago, # ^ | 0 No need for directed graph...just make undirected edges and start from any end that fits into any one of given two operation.
• » » » 19 months ago, # ^ | ← Rev. 2 → 0 Thanks, soham_1234. I got it.Can you check out the link I shared? There is a formula given to get the count of all cycles. Is it correct?
• » » » » 19 months ago, # ^ | 0 No, take the graph4 6 1 2 2 3 3 4 4 1 1 3 2 4Number of edges = 6. Number of vertices = 4. Number of connected components = 1. 6-4+1=3, but the answer is 2 because there is only one connected component and it isn't a cycle.
• » » » » » 19 months ago, # ^ | 0 Got it. Thanks.
• » » » » 19 months ago, # ^ | 0 it has defined cycle differently not the traditional cycle we know of. See the examples and diagrams.
» 19 months ago, # | 0 The contest page was not visible for the first 10-15 minutes, and the problems were not accessible for the mentioned period (well,at least for me). So will it be justified to keep it rated?
» 19 months ago, # | 0 Can someone help me? http://codeforces.com/blog/bakrocker
» 19 months ago, # | ← Rev. 2 → 0 was this contest rated as written in the blog ?! "Regardless of whether you are a trusted participant of the third division or not, if your rating is less than 1600, then the round will be rated for you." I didn't see any rating for the problems.
• » » 19 months ago, # ^ | +3 Ratings will be changed after whet Open Hacks Phase ends (in 12 hours approx)
» 19 months ago, # | +47 GOGOGOGO SO FST LKE A SNIC X
» 19 months ago, # | +4 Thnx Vovuh for your efforts in making harder pretests. Really Appreciate it ! :-)
» 19 months ago, # | +4 The round was good! Even though problems were easier quality remained high. It kind of reminds me of Atcoder Beginner Contests.
» 19 months ago, # | ← Rev. 2 → +4 Angel.Yan (rank 578)'s submissions and __beyond (rank 581)'s submissions are almost the same (the differences are libraries, useless fors, spaces, blank lines). I believe they are cheating. Correct me if I'm wrong. Vovuh
» 19 months ago, # | 0 problem C ;Output Print any integer number x from range [ 1 ; 10 9 ] such that exactly k elements of given sequence is less or equal than x . If there is no such x , print "-1" (without quotes).equal to not equal than Vovuh
• » » 19 months ago, # ^ | ← Rev. 2 → +3 Thanks, but next time please send it via private message.
• » » » 19 months ago, # ^ | 0 I can just delete the comment if you want !
• » » » » 19 months ago, # ^ | 0 Not necessary =)
» 19 months ago, # | 0 Can anyone please explain why this submission got TLE while this one got AC ? I just changed the language from Java to C++. There was no I/O issue as well. I was clueless about my solution getting TLE and then in just hit and trial I converted it to C++ and it got accepted. :(
• » » 19 months ago, # ^ | 0 Vovuh explained here: http://codeforces.com/blog/entry/59281?#comment-428862"6-th test is anti-qsort test especially for Java solutions. Replace int with Integer and it will be ok."
• » » » 19 months ago, # ^ | 0 Oh! Okay. But when it says specially for Java solutions, isn't it a bit unfair? Anyway, Thanks for the contest Vovuh ! :)
• » » » » 19 months ago, # ^ | +29 If I didn't add this test, somebody else would. So it's fair i think
• » » » » » 19 months ago, # ^ | +1 Okay, got it Vovuh . Thanks for the Contest !
» 19 months ago, # | 0 The fixed problem at the beginning will affect penalty and so the rating changes.., many contestants have 2 or more problems tested at the same time and submitted earlier so..
» 19 months ago, # | 0 Is it possible to solve problem E using disjoint set union?
• » » 19 months ago, # ^ | ← Rev. 6 → 0 my idea, maybe wrong. dsu = new DisjointSetUnionWithRollbacks for component in componets: flag = 1 for edge in component.edges: dsu.add(edge) for edge in component.edges: dsu.erase(edge) if dsu.numIgnoredEdges > 0: flag = 0 dsu.add(edge) ans += flag UPD: solution 2 dsu = new DisjointSetUnion for component in components: [head: tail] = components.edges flag = 1 for edge in tail: if dsu.add(edge) == EDGE_IGNORED: flag = 0 ans += flag In fact, we are to check, if after each edge deletion, component becomes tree.However, this solution is very complicated.Simpler ones: component is good if each node touches 2 edges component is good if it contains k nodes and k edges (sorry, it turned out to be wrong)
• » » » 19 months ago, # ^ | +38 "2. component is good if it contains k nodes and k edges"Actually, this approach is incorrect as this is not a cycle component.
• » » » 19 months ago, # ^ | 0 I tried this problem using disjoint set union. If I find a cycle, I checked if the degrees of the nodes having this edge is 1. If they aren't 1, that means that connected component is no longer a valid cycle. If they aren't in the same connnected component, then I check if any of them were part of a cycle. If they were, then I change that value to 0, signifying that the corresponding cycle is no longer a cycle. Can someone tell me what I'm missing here? I got a Wrong Answer on test case 18.
» 19 months ago, # | 0 Really we enjoy this contest . Thankyou Codeforces team
» 19 months ago, # | +6 Look at what Tlalafuda__Tlalafu does in hacks section (for problem B). He specifically wrote several solutions with this kind of code if (n == 3 && t == "LOL") { cout << "MEM"; return 0; }And then he hacks himself several times.Is this kind of behaviour legal in codeforces?
» 19 months ago, # | ← Rev. 2 → +6 Why can't I hack solutions on C++17? The system gave me an "Unexpected verdict" several times on different solutions. Please rejudge all the hacks with this verdict, it is very important.UPD: I can't even hack my own solutions which aren't on C++17. What's wrong, Codeforces?
• » » 19 months ago, # ^ | ← Rev. 2 → +1 Could you please explain the test you used to hack so many F solutions? I tried to find something in common in the solutions you hacked but failed to see the pattern. Your test seems to be tricky.
• » » » 19 months ago, # ^ | +11 Div 3 noobs like me didn't know that using unordered_map makes the solution O(N^2).http://codeforces.com/blog/entry/44731
• » » » » 19 months ago, # ^ | 0 So, this was an "anti-hash" test against STL's unordered_map... I'm new to Educational Rounds so I didn't think people actually use dirty tricks like anti-hash and anti-quicksort. I don't get why people do this but this broke all solutions with unordered_map instead of map. Well, learned something new, won't be using unordered_map on contests next time :)
• » » » » » 19 months ago, # ^ | 0 I believe this hack does not necessarily mean that you should not use unoredered_map at all. The thing is, the standard hash function which accepts integer values does nothing, it just returns the same number it accepts, so there is no problem with hashing itself.The problem is solely with buckets count, because in a hash table there is a remainder operation "key % buckets_count" and in case if you take a lot of numbers with the same remainder, unordered_map will work slowly.However, if you use it and call reserve(10000000) in the beginning, you will fix the amount of buckets to be over 10 millions and the hack simply would not work anymore. The maximum time conplexity will not be more than 100n, which is enought to pass 1 second time restriction.Correct me if I am wrong.
• » » » » » » 19 months ago, # ^ | ← Rev. 2 → +9 This wasn't exactly what I meant, but I agree that reserve() will work for F. The problem is that this workaround will only work for this particular problem because of the 10^9 restriction. Given less tight bound (10^18) it will still fail.And with 12-hours open tests you can even generate counter test for any particular hash function from universal family (not necessarily identity) and any particular buckets_count. The only way to protect against this attacks seems to be random sampling of hash function from universal family with good source of entropy.The point was — why do you even need to bother about such details on a contest if you can just use map instead? It might be a bit slower but at least it has worst-case guarantees and it allows you to focus on a problem itself.Problem solving is fun. I don't think it's fun to protect against hashmap-attacks.
• » » » » » » » 19 months ago, # ^ | ← Rev. 4 → +5 I see, I would myself prefer map container to make sure my solution will work.I was just too disapointed by the fact that a programming contest makes us choose the structure which 10 times faster in 0.0001% cases and 10 times slower in 99.9999% cases, that is why I tried to figure out the way how to use unordered_map and remain unhacked.Anyways, codeforces is not only about good ideas of solutions, it also requires us to know the low level of how c++ works, it is just as it is.
• » » 19 months ago, # ^ | +19 I think it's because some of the author/tester solution fail on the same test.
• » » 19 months ago, # ^ | +16 Testers' solutions receive time limit exceed, because of this your hack gave an "Unexpected verdict". I hope I have fixed it now. Sorry for long delay.
» 19 months ago, # | 0 Will there be any tutorial for this round??
• » » 19 months ago, # ^ | 0 It's already posted, here
» 19 months ago, # | 0 is it rated for newbie ?
• » » 19 months ago, # ^ | 0 Yes
» 19 months ago, # | 0 Hello Vovuh. My rating is 1500 because I hasn't joined any contest before. However you said "if your rating is less than 1600, then the round will be rated for you." So why I'm not rated after this contest.
• » » 19 months ago, # ^ | 0 The contest has not done with the large test case. So we need to wait for the final result and you will see your ranking changes.
» 19 months ago, # | ← Rev. 3 → 0 i passed all of the problems!! yayyy
» 19 months ago, # | +6 When will system testing start?
• » » 19 months ago, # ^ | ← Rev. 2 → +4 Yeh same question,i wonder how can i know when the sys test will start ? the contest just said "Finished"
» 19 months ago, # | +1 When will the ratings be updated?
• » » 19 months ago, # ^ | 0 Ratings are updated as of posting this.
» 19 months ago, # | ← Rev. 2 → 0 Pls, somebody explain, how my solution http://codeforces.com/contest/977/submission/37965169 for Problem F was hacked. But when i changed "unordered_map" to "map" it got accepted. It seems realy strange to me
• » » 19 months ago, # ^ | 0 Afaik Unordered_map is slower than normal map.
• » » » 19 months ago, # ^ | +1 That'_s not true. unordered_map is getting slower when many hash collision happens. Otherwise, unordered_map is much faster than normal map.
• » » 19 months ago, # ^ | ← Rev. 2 → 0 unordered_map is based on Hash table. Normally a search operation of hash table works in amortized(1) but worst is O(N) (caused by hash-collision). std::map, on the other hand, is based on red-black tree, which has O(logN) complexity in search operation on average. Your solution was hacked by anti-hash test.
» 19 months ago, # | 0 Why unrated participants get rated in this round?You said that they need to do at least 2 Div.2 contest and solve at least 1 problem to get rated
• » » 19 months ago, # ^ | ← Rev. 2 → +13 Regardless of whether you are a trusted participant of the third division or not, if your rating is less than 1600, then the round will be rated for you. ` So although unrated participants are not trusted, they are still rated.
• » » » 19 months ago, # ^ | -7 But why they but them in a separate room?
» 19 months ago, # | 0 Can anyone tell me what's wrong with my F approach? 37990081
• » » 19 months ago, # ^ | +3 You ignored MAP[A[i]] that was originally stored.test : 9 10 11 12 8 9 10 11 10 11ans = 4 yours = 3
• » » » 19 months ago, # ^ | +3 Oh.. i thought map in c++ overrides values that exists already just like in Java
• » » 19 months ago, # ^ | ← Rev. 2 → +3 Yes, so change your "//m.insert(mp(a[i], m[a[i] — 1] + 1));" to "m[a[i]]=max(m[a[i]],m[a[i]-1]+1);", and this is the AC code which has a little change from your code
• » » » 19 months ago, # ^ | ← Rev. 2 → +3 Yes, i got AC, thank you ;)Thought C++ maps work as Java maps where you just use map.put(v1, map.get(v1 — 1) + 1) and it overrides v1 value if exists
» 19 months ago, # | +7 How about the ratting changes? Can a people get the same ratting changes if he got the same place(among rated people) in div 2 and div 3 contests?
» 19 months ago, # | 0 It's a wonderful effort for beginners like us to learn and improve !
» 19 months ago, # | +1 It grow the internal effort of beginner programme,make hope to became a good programmer, I like it
» 19 months ago, # | ← Rev. 2 → +18 You might also would like to take a look at the incredible achievement of Tlalafuda__Tlalafu, who made 10 wrong submissions for problem B and found bugs in all of them! Думаю, что вам также будет интересно увидеть невероятный успех Tlalafuda__Tlalafu, который не поленился отправить 10 неверных решений по задаче B, а после этого смог обнаружить баги в каждом из них!
• » » 19 months ago, # ^ | 0 He just hardcoded a wrong output for some random input 10 times... is this sarcasm?
» 19 months ago, # | +16 is no one here concern about rating inflation? In a normal div2 round, it seems like there's usually about 100 users who will be promoted to expert and some of the expert will drop, but this round, there seems to be about 250 promoted with 0 expert demoted.
» 19 months ago, # | 0 I would request the host to conduct more number of Division 3 contests. I am sure that many other users like me with rating less than 1600 are finding it difficult to solve Division 2 problems easily and obtain better ranks. I would suggest to conduct 1 Division 3 contest for every two Divison 1/2 contests so that we can gain confidence and hopefully aim to perform better at Division 1/2 contests. I would like the opinion of other people too. Thanks.
» 16 months ago, # | 0 great round
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2019-12-12 12:16:08
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https://ham.stackexchange.com/tags/gnuradio/hot
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# Tag Info
12
I expect to see the sum & difference frequencies. You're multiplying two complex sinusoids, not a $\sin$ and a $\cos$, but $$e^{j2\pi f_1t}\cdot e^{j\left(2\pi f_2t-\frac\pi2\right)}= e^{j2\pi(f_1+f_2)-j\frac\pi2}$$ So, only the sum frequency, as it should. I am seeing a big peak at sum frequency followed by a series of multiple peaks on the FFT. I'...
10
Your confusion stems from the fact that your mind (and the WX GUI scope sink) uses linear interpolation between the samples. That's not right – in your case, where the signal period isn't an integer number of samples, this becomes obvious by the fact that things don't look very sinusoidal. They still are (I promise ;) ). Think about it like this: at ...
9
What's going on Most software-defined radios use quadrature in their signal processing architecture. This means there are two copies of the signal processing chain, with local oscillators that are 90° apart in phase while being otherwise identical. Once they reach software, they can be processed using complex number arithmetic (hence the "real" and "...
8
What the FFT sink shows as frequency axis actually has no basis in "real world signal" – it just takes the sample rate you set (here, you set 1 MHz), and scales the full nyquist bandwidth to that. If you used a different number in the sample rate field of the FFT sink, the spectrum would look absolutely the same, just the frequency axis would have different ...
8
The Hilbert block in GNU Radio creates an analytic signal from a real-valued one by applying the Hilbert transform to create the Q (imaginary, quadrature) component. The Complex to Mag block then discards the phase information in that analytic signal, keeping only the amplitude. Generally speaking, this is AM demodulation. This further agrees with your Add ...
8
You could check out ShipPlotter which appears to be a windows-based AIS receiver. It mentions in the webpage that it accepts audio through your sound card. In the case of RTL-SDR, you'll want to use something like "Virtual Audio Cable" or "VB-Audio Cable" to route the audio from sdrsharp to ShipPlotter.
7
GNU Radio, like any DSP system, works primarily in terms of sample counts, not time. Therefore, you have to add on time information — a sample rate — to get correct frequency-domain information. GNU Radio does not automatically figure out what the matching sample rates between parts of your flow graph are, so you have to set them up correctly yourself. The ...
7
That's an expected phenomenon: Real-world physical systems tend to be frequency-selective (i.e. not constant over frequency), and "at large scale" low-pass systems. This applies to amplifiers, mixers, oscillators, and even transmission lines and connectors. So that's normal. Ettus even publishes exactly such measurements at https://files.ettus.com/...
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You're on to something very right! In signal processing, we define a basic waveform by its frequency, number of samples within the period and its amplitude. I'd go a step further: In digital signal processing, the actual frequency doesn't "exist" any more. It's just "a periodic signal with a period of $T$ samples". So that's exactly why for example ...
6
There are a lot of things wrong here. Neither plot looks correct. There is no way that Qt plot is realistic for anything but a signal generator. Where's the noise? Where are the three missing constellation points in the Qt plot? APRS isn't QAM (it's AFSK over FM), so I'm not sure why you are expecting QAM. You don't have any filters, clock recovery, or ...
6
This is how the mathematics of complex signals work. The proof begins with Euler's formula: $$e^{i\varphi} = \cos \varphi + i \sin \varphi \tag 1$$ For signal processing, instead of $\varphi$, we are usually thinking about some sinusoidal oscillation at angular frequency $\omega$ that varies with time $t$, which we can write as: $$e^{i\omega t} \tag 2$$ ...
6
Because mathematically, a function like $\sin(\omega t)$ has an angular frequency of $\omega$ and $-\omega$. Consider: $$e^{i\omega t} + e^{-i\omega t}$$ By Euler's formula this can be expanded to: $$\cos(\omega t)+i\sin(\omega t) + \cos(-\omega t)+i\sin(-\omega t)$$ By the trig identity $\sin(x) + \sin(-x) = 0$ this simplifies to: \cos(\omega t)+ \...
5
To transmit two signals at once, just generate the two signals with the appropriate frequency spacing between them, and add them together. With your mentioned frequencies, you might generate the signals at −0.5 MHz and +0.5 MHz, add them together, and transmit with the hardware center frequency set to 915.5 MHz. (You don't have to use the exact ...
5
They don't "measure", they just display. And yes, as you noticed, this is digital signal processing, so there's no physical units involved – the display axes are correctly labeled with "dB", as in "dB relative to an arbitrary reference vector", typically a energy=1 time signal (e.g. $(0,\ldots,0,1,0,\ldots,0)$), or a power=1 signal (e.g. $(1,\ldots,1)$). ...
5
General principle: In GNU Radio you cannot ever have a flow graph with a loop in it. If you wish to have feedback of some kind, it must be implemented in a single block. (There are many existing blocks that do this, such as AGC blocks and IIR filter blocks, and classes to help create them, though they still require writing C++ code.) However, you do not ...
5
The frequency allocation chart is really more artistic than informative. You can not and should not use that as a guide for selecting a frequency. There is too much information to fit on the chart, and as much as is there anyway, it's not surprising you feel lost looking at it. (That may be part of the intention of the chart.) There are multiple bands ...
5
A SDR peripheral like the Ettus USRP B210 continuously digitizes the incoming RF and sends it to the attached computer. There are no gaps in its coverage — anything that is within the bandwidth is captured in full. If you use one to create a spectrum analyzer, the minimum duration of signal you will be able to observe will be determined by the algorithms ...
4
You have the right general idea, but a couple of problems: The sample rate entering the DSD block (hence the output of the FM demodulator) must be 48000 Hz; this is hard-coded in gr-dsd. From your screenshot, you have 35k where you need 48k. The deemphasis (a.k.a. tau) of the FM demodulator should be set to None, rather than the default of 75. In my ...
4
Here are several different approaches you can take. You can write a GNU Radio source block which knows how to interact with your particular device. This does not require modifying GNU Radio's source code: you can compile your block separately, in which case it is known as an out of tree module; here's a tutorial. This custom source block would be written ...
4
Thanks to @Marcus Müller. It seemed in this instance it was the lack of the devel packages for boost. A simple... \$ sudo dnf install boost-devel ...did the trick (Well, I mean it moved me on to the next failure).
4
This will be an incomplete answer to begin with, hoping that collectively we can come up with a workable solution: Freq 156.425 Mhz -f 156.425e6 Filter: Wide I did not find "Filter" in any online-reference of rtl_fm, unless this is "sample rate", which can be set by: -s 12k for narrow FM Mode: Narrow FM -M fm AGC: Fast I did not find AGC in ...
4
One reason in general to perform decimation in multiple steps is it reduces the computational requirements of the low-pass filtering. To achieve a sharp transition width in the filter requires a longer convolution kernel and thus, more computation. Also the sample rate is very high before decimation, requiring yet more computational power. So generally, we ...
4
Check out AISMon; I've never used it, but it looks like it fits the bill. Here's a thorough tutorial: http://www.rtl-sdr.com/rtl-sdr-tutorial-cheap-ais-ship-tracking/
4
In general, a software-defined radio and one implemented in hardware will have much the same signal paths, but how they are broken down into individual “blocks” will vary. In your case, following the main signal path from left to right: It looks like everything on the main path to the left of “Log Detect” is a conventional superheterodyne receiver. These ...
4
UHD requires a USRP to be owned by one process only, so no, you can't share the same USRP across two flow graphs, but you could use for example the ZeroMQ PUB/SUB sink/source pair to stream data in from a second application to your first flow graph. But then again, there's nothing wrong with putting a whole flow graph into a hier block¹ and using that in ...
4
Is there dvbt/dvbt-2 receiver? Yes and no: T2 reception is too computationally hard so far. It's work in progress, but it's almost certain that your average PC can't decode full standard T2 rates in real time on its CPU. The channel coding is just too involved. Transmission is always computationally easier. See gnuradio/gr-dtv/ example flowgraphs. Why "...
4
GNU Radio Companion (GRC) generates Python code that is something like this (not exact text). (Make sure you chose the "No GUI" option in GRC.) class my_block(gr.top_block): # ... def main(): tb = my_block() tb.run() if __name__ == '__main__': main() You can just import this as a module in your Python program (the if __name__ check will ...
4
I have used RTL-SDR USB peripherals from several vendors. They run at different temperatures (implying different power draws), which suggests that your question does not have a single answer. The power level also seems to vary with what the device is doing (idle, streaming, sample rate, etc.) So you probably need to measure the USB current on your ...
4
To extend what hotpaw2 said: So, it seems the front panel USB sockets are maybe not providing enough power for the Dongle to work? I googled how much power the Dongle draws? and apparently it varies from model to model. Exactly, different models use different tuners, some additional amplifiers, and some use linear power supplies only, others mix linear ...
4
For a complex-valued sampled stream, the sample rate must be greater than twice the signal represented. The sample rate in your graph is set to 384k, so it's not possible to represent a 98400 kHz signal. GNU Radio won't stop you, though. Mathematically, when you try to generate a sampled signal that's too high, you get aliasing. As an experiment, try ...
Only top voted, non community-wiki answers of a minimum length are eligible
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2021-10-27 10:17:09
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https://walletscrutiny.com/android/fr.acinq.eclair.wallet.mainnet2/
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# Eclair Mobile
latest release: 0.4.17 ( 28th September 2021 ) last analysed 22nd December 2019 Not reproducible from source provided
3.9 ★★★★★
366 ratings
10 thousand
12th April 2018
# Do your own research!
Try out searching for "lost bitcoins", "stole my money" or "scammers" together with the wallet's name, even if you think the wallet is generally trustworthy. For all the bigger wallets you will find accusations. Make sure you understand why they were made and if you are comfortable with the provider's reaction.
If you find something we should include, you can create an issue or edit this analysis yourself and create a merge request for your changes.
# The Analysis ¶
This wallet has a really short description. Here it is in full:
Eclair Mobile is a next generation, Lightning-ready Bitcoin wallet. It can be used as a regular Bitcoin wallet, and can also connect to the Lightning Network for cheap and instant payments.
This software is based on eclair, and follows the Lightning Network standard.
No word on custodial or not.
Their website is more informative if only for the link to their GitHub.
eclair-mobile sounds promising.
There, in the description again we find no hints at it being non-custodial.
But in the repository’s wiki finally we find:
Is Eclair Mobile a “real” Lightning Node ?
Yes it is. Eclair Mobile is a real, self-contained lightning node that runs on your phone. It does not require you to run another Lightning Node node at home or in the cloud. It is not a custodial wallet either, you are in full control of your funds.
So … can we reproduce the build?
The build instructions are not very plentiful on the repo’s description:
Developers
1. clone this project
2. clone eclair and checkout the android branch.
Follow the steps here to build the eclair-core library.
3. Open the Eclair Mobile project with Android studio. You should now be able to install it on your phone/on an emulator.
This has two immediate issues:
• How do we know which version of “eclair” should we use? This should be resolved with a git submodule.
• Compiling with Android Studio is not easy to automate and should not be necessary.
• Branches are not a good way of referencing revisions of a repository. The “android branch” has 1938 revisions and if I want to check anything but the latest revision I have little to go by to find which app would match to which state of the branch.
but let’s see how compiling looks once these issues are resolved as we have little hope to verify the current apk …
$git clone git@github.com:ACINQ/eclair-mobile.git$ git clone git@github.com:ACINQ/eclair.git
$cd eclair$ git checkout android
$docker run -it -v$PWD/eclair:/eclair -v $PWD/eclair-mobile:/eclair-mobile --workdir / electrum-android-builder-img user@d0cf683a144a:/$ sudo su -
root@d0cf683a144a:~# apt update
root@d0cf683a144a:~# apt install maven
root@d0cf683a144a:~# mvn install -DskipTests
...
[INFO] --- maven-compiler-plugin:3.1:testCompile (default-testCompile) @ eclair-core_2.11 ---
[INFO] Nothing to compile - all classes are up to date
[INFO]
[INFO] --- scala-maven-plugin:3.4.2:testCompile (scalac) @ eclair-core_2.11 ---
[INFO] /eclair/eclair-core/src/test/java:-1: info: compiling
[INFO] /eclair/eclair-core/src/test/scala:-1: info: compiling
[INFO] Compiling 114 source files to /eclair/eclair-core/target/test-classes at 1577007350665
[ERROR] /eclair/eclair-core/src/test/scala/fr/acinq/eclair/blockchain/bitcoind/BitcoindService.scala:74: error: value writeString is not a member of object java.nio.file.Files
[ERROR] Files.writeString(new File(PATH_BITCOIND_DATADIR.toString, "bitcoin.conf").toPath, conf)
[ERROR] ^
[ERROR] one error found
[INFO] ------------------------------------------------------------------------
[INFO] Reactor Summary for eclair_2.11 0.3.4-android-SNAPSHOT:
[INFO]
[INFO] eclair_2.11 ........................................ SUCCESS [ 1.951 s]
[INFO] eclair-core_2.11 ................................... FAILURE [ 28.245 s]
[INFO] eclair-node ........................................ SKIPPED
[INFO] ------------------------------------------------------------------------
[INFO] BUILD FAILURE
So following the instructions we didn’t get far and for now hope for better documentation and remain with the verdict: This app is not verifiable.
(lw)
# Verdict Explained
We could not verify that the provided code matches the binary!
As part of our Methodology, we ask:
Is the published binary matching the published source code? If not, we tag it Unreproducible!
Published code doesn’t help much if it is not what the published binary was built from. That is why we try to reproduce the binary. We
1. obtain the binary from the provider
2. compile the published source code using the published build instructions into a binary
3. compare the two binaries
4. we might spend some time working around issues that are easy to work around
If this fails, we might search if other revisions match or if we can deduct the source of the mismatch but generally consider it on the provider to provide the correct source code and build instructions to reproduce the build, so we usually open a ticket in their code repository.
In any case, the result is a discrepancy between the binary we can create and the binary we can find for download and any discrepancy might leak your backup to the server on purpose or by accident.
As we cannot verify that the source provided is the source the binary was compiled from, this category is only slightly better than closed source but for now we have hope projects come around and fix verifiability issues.
The product cannot be independently verified. If the provider puts your funds at risk on purpose or by accident, you will probably not know about the issue before people start losing money. If the provider is more criminally inclined he might have collected all the backups of all the wallets, ready to be emptied at the press of a button. The product might have a formidable track record but out of distress or change in management turns out to be evil from some point on, with nobody outside ever knowing before it is too late.
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2021-10-16 22:22:28
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https://bakingandmath.com/tag/brainstorming/
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Tag Archives: brainstorming
## Brainstorming, also, Cantor sets
10 Jun
So I studied for those prelims for a few months (maybe two) on my own, reading old notes, highlighting things, rewriting relevant proofs, running through my homeworks, attempting extra exercises from the textbooks, the usual extreme studying. I don’t know about you, but I am SO MUCH BETTER with other people when doing just about anything. Knowing someone else will eat my baked goods makes my baked goods better. Being accountable to a colleague makes me justify my ideas more and I write down fewer false things.
Math is a creative endeavor. People seem to think mathematicians are somewhere between human calculator, engineer, and crazy person. But we don’t get grouped with, say, artist, writer, poet, all that often. I’m going to quote some wikipedia here:
Plato did not believe in art as a form of creation. Asked in The Republic,[18] “Will we say, of a painter, that he makes something?”, he answers, “Certainly not, he merely imitates.”[16]
Now, as with many creative things, there are certain tools you can use to help with math. Going on walks is a good thing. Eating breakfast. Naps, even. But the thing I love, and the thing that helped me pass my prelims (whoooo), is brainstorming. A few weeks ago I read an article that a facebook friend posted (isn’t it funny how you know what I’m talking about when I say “facebook friend” rather than “friend”? In this case, a guy I went to high school with) about how brainstorming is basically stupid and broken. And, given the parameters that the article offers, it’s pretty right: get in a group, generate lots of ideas, don’t be critical. I don’t have a lot to say about brainstorming (though Jonah Lehrer for the new yorker and a guy from Stanford’s d.school do and these are both fascinating), but I do have things to say about problem solving with a group.
Brainstorming, as this three-step outline is, isn’t the way solving problems with a group should be done. He’s on point with steps one and two, but step three should be different: be critical. Fight for your ideas. Fight other people’s wrong ideas. I work with other grad students a lot, and I often don’t realize that I’m completely wrong until I’ve been talking about an idea for a few minutes. I need them to fight me in order to learn and understand and be better.
The idea behind brainstorming is that if you’re critical of others’ ideas, they won’t want to share them and will clam up. I absolutely felt that way for the first year or so of graduate school- I would feel that my peers shot me (personally) down, and that I didn’t have any good ideas. What finally changed my mind and made me more combative was realizing that I do have good ideas, sometimes. Not all the time, probably not even 50% of the time, but sometimes I’m right, I’m absolutely right and I can prove it if you-just don’t shoot me down or interrupt me or dismiss me. These last three things still happen, and I’ve learned to fight back in a nonconfrontational manner- it’s easier in my case because everyone in the room just wants to reach a solution. Some people have egos (I do too), but we learn to listen to each other to the extent we’re capable, and to make each other listen when we can’t extend ourselves to do so.
Basically, I agree with this guy: “Innovation Is About Arguing” is the first bit of his title.
On a totally different topic, let’s talk about the Cantor set. It’s a pretty cool subset of numbers with lots of unintuitive properties. Building it is fairly straightforward. Look at a number line. Focus only on the segment between 0 and 1. Cut the segment into thirds, so you’ll have notches at 1/3 and 2/3. Delete the middle third (1/3,2/3) so you end up with two segments. Do this over and over again, deleting the middle third from each segment you have at any one point. Look at the picture
I DID NOT MAKE THIS PICTURE it is from wikipedia.
• Despite that, no points in the Cantor set are isolated: if you take a point in it and look in a teensy neighborhood (like, $\pm .00000001$ teensy), you’ll still find other points from the Cantor set.
• If you consider the set $C+C = \{x+y : x,y\in C \}$, you get an interval, which is way incredible and unexpected because it contains no intervals- it’s like putting together two sets of discrete bread crumbs and magically having a loaf.
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2021-01-16 03:05:48
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https://mattermodeling.stackexchange.com/questions/4338/how-to-do-vasp-convergence-tests-and-how-to-get-two-plots-of-encut-and-k-points
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# How to do VASP convergence tests, and how to get two plots of Encut and k-points wrt to total energy? I want to do convergence for Mg2Si
I am using VASP for the first time and do not know how to do convergence tests. How are convergence tests done in VASP and how do we get two plots of Encut and k-points with respect to total energy? I want to do these things for the $$\ce{Mg2Si}$$ material.
• +1 and welcome to our new community! Thank you for contributing your question here, and we hope to see much more of you in the future!!! I've made some edits to you question though, please make note of them for next time. Feb 12, 2021 at 23:47
• This has already been answered on this network in multiple questions. For a start, this should help: mattermodeling.stackexchange.com/questions/1896/… Feb 12, 2021 at 23:59
• Thank you so much its a nice platform for research scholars Feb 15, 2021 at 17:26
You can find @Andrew Rosen's nice answer from this post:k-points and ENCUT convergence tests before or after relaxation?, as @Xixi76 suggested.
Besides, you can read this nice tutorial: https://dannyvanpoucke.be/vasp-tutor-convergence-testing-en/
It should be emphasized that the convergence test is one of the most important steps to obtain physical results.
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2022-05-26 23:22:38
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https://txcorp.com/images/docs/vsim/latest/VSimExamples/VSimEM/Antennas/YagiUda2p4/YagiUda2p4.html
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# Yagi-Uda (YagiUda2p4.sdf)
Keywords:
yagiUdaArrayWireModel, yagiT, far field, radiation
## Problem description
A Yagi-Uda array is a directional antenna consisting of several parallel dipole elements. Only one of these dipole elements is driven, the other elements being parasitic . Directionality is achieved by requiring that there be one longer element adjacent to the source element, which is referred to as the reflector. The rest of the elements being adjacent to the source but opposite to the reflector, and shorter than the source element, are referred to as directors. Yagi antennas are ubiquitous, and as such optimal parameters for dipole lengths and separations have been established. We go with values one would typically find in any text covering the matter. This example illustrates how to obtain the far field radiation pattern of a Yagi-Uda array.
This simulation can be performed with a VSimEM license.
## Opening the Simulation
The Yagi-Uda example is accessed from within VSimComposer by the following actions:
• In the resulting Examples window expand the VSim for Electromagnetics option.
• Expand the Antennas option.
• Select 2.4 GHz Yagi Uda Antenna and press the Choose button.
• In the resulting dialog, create a New Folder if desired, and press the Save button to create a copy of this example.
All of the properties and values that create the simulation are now available in the Setup Window as shown in Fig. 156. You can expand the tree elements and navigate through the various properties, making any changes you desire. The right pane shows a 3D view of the geometry, if any, as well as the grid, if actively shown. To show or hide the grid, expand the Grids element and select or deselect the box next to Grid.
Fig. 156 Setup Window for the Yagi-Uda example.
## Simulation Properties
This file allows the modification of the antenna operating frequency, antenna dimensions, and simulation domain size.
By adjusting the dimensions any sized Yagi-Uda array can be simulated.
Note
To obtain good far field resolution generally four or more antenna elements is desirable (One source, one reflector, two or more directors).
## Running the Simulation
After performing the above actions, continue as follows:
• Proceed to the Run Window by pressing the Run button in the left column of buttons.
• Here you can set run parameters, including how many cores to run with.
• When you are finished setting run parameters, click on the Run button in the upper left corner of the Logs and Output Files pane. You will see the output of the run in the right pane. The run has completed when you see the output, “Engine completed successfully.” This is shown in Fig. 157.
Fig. 157 The Run Window at the end of execution.
## Analyzing the Results
Proceed to the Analyze Window by pressing the Analyze button in the left column of buttons.
Select “computeFarFieldFromKirchhoffBox.py” from the analyzer list, and click “Open.”
The default parameters are sufficient for this problem. Input 10.0 for the farFieldRadius parameter and run the analyzer by clicking the “Analyze” button.
## Visualizing the results
Proceed to the Visualize Window by pressing the Visualize button in the left column of buttons.
To view the near field pattern, do the following:
• Expand Scalar Data
• Expand E
• Select E_x
• Click Colors
• Check the Fix Minimum box and set the value to -0.1
• Check the Fix Maximum box and set the value to 0.1, then click “OK”
• Expand Geometries
• Select poly (YagiUda2p4PecShapes)
• Select Clip All Plots
• Move the dump slider forward in time
Fig. 159 The electric field near-field pattern.
The far field radiation pattern can be found in the scalar data variables of the data overview tab underneath the farE field. Check the farE_magnitude box, remove the minimum and maximum restrictions on colors, and uncheck Clip All Plots.
Fig. 160 The electric field manifestation of the far field pattern.
## Further Experiments
Try adding more directors and changing their dimensions to see the effect on the far field pattern.
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2020-09-25 13:57:56
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http://www.ams.org/mathscinet-getitem?mr=833708
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MathSciNet bibliographic data MR833708 (87k:60115) 60G40 (60H10 60J60) DeBlassie, R. Dante \$L\sp p\$$L\sp p$ inequalities for stopping times of diffusions. Trans. Amer. Math. Soc. 295 (1986), no. 2, 765–782. Article
For users without a MathSciNet license , Relay Station allows linking from MR numbers in online mathematical literature directly to electronic journals and original articles. Subscribers receive the added value of full MathSciNet reviews.
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2014-10-21 02:16:43
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https://en.wikipedia.org/wiki/Shapley%E2%80%93Shubik_power_index
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# Shapley–Shubik power index
Jump to: navigation, search
The Shapley–Shubik power index was formulated by Lloyd Shapley and Martin Shubik in 1954 to measure the powers of players in a voting game.[1] The index often reveals surprising power distribution that is not obvious on the surface.
The constituents of a voting system, such as legislative bodies, executives, shareholders, individual legislators, and so forth, can be viewed as players in an n-player game. Players with the same preferences form coalitions. Any coalition that has enough votes to pass a bill or elect a candidate is called winning, and the others are called losing. Based on Shapley value, Shapley and Shubik concluded that the power of a coalition was not simply proportional to its size.
The power of a coalition (or a player) is measured by the fraction of the possible voting sequences in which that coalition casts the deciding vote, that is, the vote that first guarantees passage or failure.[2]
The power index is normalized between 0 and 1. A power of 0 means that a coalition has no effect at all on the outcome of the game; and a power of 1 means a coalition determines the outcome by its vote. Also the sum of the powers of all the players is always equal to 1.
There are some algorithms for calculating the power index, e.g., dynamic programming techniques, enumeration methods and Monte Carlo methods.[3]
## Examples
Suppose decisions are made by majority rule in a body consisting of A, B, C, D, who have 3, 2, 1 and 1 votes, respectively. The majority vote threshold is 4. There are 4! = 24 possible orders for these members to vote:
ABCD ABDC ACBD ACDB ADBC ADCB BACD BADC BCAD BCDA BDAC BDCA CABD CADB CBAD CBDA CDAB CDBA DABC DACB DBAC DBCA DCAB DCBA
For each voting sequence the pivot voter – that voter who first raises the cumulative sum to 4 or more – is bolded. Here, A is pivotal in 12 of the 24 sequences. Therefore, A has an index of power 1/2. The others have an index of power 1/6. Curiously, B has no more power than C and D. When you consider that A's vote determines the outcome unless the others unite against A, it becomes clear that B, C, D play identical roles. This reflects in the power indices.
Suppose that in another majority-rule voting body with ${\displaystyle 2n+1}$ members, in which a single strong member has ${\displaystyle k}$ votes and the remaining ${\displaystyle 2n}$ members have one vote each. It then turns out that the power of the strong member is ${\displaystyle {\dfrac {k}{2n+2-k}}}$. As ${\displaystyle k}$ increases, the strong member's power increases disproportionately until it approaches half the total vote and this person gains virtually all the power. This phenomenon often happens to large shareholders and business takeovers.
## Applications
The index has been applied to the analysis of voting in the Council of the European Union.[4]
## References
1. ^ Shapley, L. S.; Shubik, M. (1954). "A Method for Evaluating the Distribution of Power in a Committee System". American Political Science Review. 48 (3): 787–792. doi:10.2307/1951053.
2. ^ Hu, Xingwei (2006). "An Asymmetric Shapley–Shubik Power Index". International Journal of Game Theory. 34 (2): 229–240. doi:10.1007/s00182-006-0011-z.
3. ^ Matsui, Tomomi; Matsui, Yasuko (2000). "A Survey of Algorithms for Calculating Power Indices of Weighted Majority Games" (PDF). J. Oper. Res. Soc. Japan. 43 (1): 71–86..
4. ^ Varela, Diego; Prado-Dominguez, Javier (2012-01-01). "Negotiating the Lisbon Treaty: Redistribution, Efficiency and Power Indices". Czech Economic Review. 6 (2): 107–124.
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2018-03-21 09:41:09
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https://www.ms.u-tokyo.ac.jp/journal/abstract_e/jms140101_e.html
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## The Littlewood-Paley-Stein inequality for diffusion processes on general metric spaces
J. Math. Sci. Univ. Tokyo
Vol. 14 (2007), No. 1, Page 1--30.
KAWABI, Hiroshi ; MIYOKAWA, Tomohiro
The Littlewood-Paley-Stein inequality for diffusion processes on general metric spaces
[Full Article (PDF)] [MathSciNet Review (HTML)] [MathSciNet Review (PDF)]
Abstract:
In this paper, we establish the Littlewood-Paley-Stein inequality on general metric spaces %We show this inequality under a weaker condition than the lower boundedness of Bakry-Emery's $\Gamma_{2}$. We also discuss Riesz transforms. %a relationship of Sobolev norms. As examples, we deal with diffusion processes on a path space associated with stochastic partial differential equations (SPDEs in short) and a class of superprocesses with immigration.
Keywords: Littlewood-Paley-Stein inequality, gradient estimate condition, Riesz transforms, SPDEs, superprocesses.
Mathematics Subject Classification (2000): 42B25, 60J60, 60H15.
Mathematical Reviews Number: MR2320383
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2022-06-29 02:49:22
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http://elib.mi.sanu.ac.rs/pages/browse_issue.php?db=nsjom&rbr=10
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eLibrary of Mathematical Instituteof the Serbian Academy of Sciences and Arts
> Home / All Journals / Journal /
Novi Sad Journal of MathematicsPublisher: Department of Mathematics and Informatics, Faculty of Sciences, Novi SadISSN: 1450-5444 (Print), 2406-2014 (Online)Issue: 10Date: 1980Journal Homepage
Equation of oscillation of a viscoelastic bar 1 - 12 Bogoljub Stanković
Abstract
A generalization of the contraction principle in probabilistic metric spaces 13 - 23 Olga Hadžić
Abstract
A fixed point theorem in topological vector spaces 23 - 29 Olga Hadžić
Abstract
On the topological structure of random normed space 31 - 35 Olga Hadžić
Abstract
Some applications of Bocsan's fixed point theorem 37 - 47 Olga Hadžić and Mila Stojaković
Abstract
Some fixed point theorems for multivalued mappings in topological vector spaces 49 - 54 Olga Hadžić and Ljiljanja Gajić
Abstract
The kernel theorem for some spaces 55 - 61 Stevan Pilipović
Abstract
Convolution equations in the countable union of exponential distributions 63 - 70 Stevan Pilipović and Arpad Takači
Abstract
On a class of nonlinear $n$-th order differential equations 71 - 76 Mirko Budinčević
Abstract
Uniform boundedness of a family of triangle semigroup valued set functions 77 - 83 Endre Pap
Abstract
Article page: 123>>
Remote Address: 3.237.71.23 • Server: elib.mi.sanu.ac.rsHTTP User Agent: CCBot/2.0 (https://commoncrawl.org/faq/)
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2020-09-18 22:41:48
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https://diabetesjournals.org/view-large/3774368
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Table 1—
Characteristics of participants aged 18–64 years*
VariablesDiagnosed diabetes (n = 704)
Undiagnosed diabetes (n = 110)
Nondiabetic population (n = 4,782)
nWeighted % or meannWeighted % or meannWeighted % or mean
Mean age (in years) 49.9 50.2 39.0
Sex (female) 363 48.7 47 36.8 2,539 51.7
Race (non-Hispanic white) 217 58.3 47 68.4 2,215 71.2
Race (black) 201 18.4 25 13.0 993 11.2
Race (Hispanic) 261 17.4 34 14.2 1,426 13.5
Race (others) 25 6.0 4.4 148 4.2
Marital status (married) 401 57.3 60 56.7 2,319 55.1
Educational attainment
Less than high school 300 28.7 53 32.8 1,441 19.0
High school 152 23.7 19 22.7 1,200 26.2
More than high school 252 47.5 38 44.6 2,141 54.8
Family income <$20,000 per year 266 29.6 43 26.6 1,389 22.2 Mean BMI (kg/m2 32.9 34.1 27.6 Self-rated health Good, very good, or excellent 335 55.3 74 73.6 4,007 86.4 Fair or poor 369 44.7 36 26.4 770 13.6 VariablesDiagnosed diabetes (n = 704) Undiagnosed diabetes (n = 110) Nondiabetic population (n = 4,782) nWeighted % or meannWeighted % or meannWeighted % or mean Mean age (in years) 49.9 50.2 39.0 Sex (female) 363 48.7 47 36.8 2,539 51.7 Race (non-Hispanic white) 217 58.3 47 68.4 2,215 71.2 Race (black) 201 18.4 25 13.0 993 11.2 Race (Hispanic) 261 17.4 34 14.2 1,426 13.5 Race (others) 25 6.0 4.4 148 4.2 Marital status (married) 401 57.3 60 56.7 2,319 55.1 Educational attainment Less than high school 300 28.7 53 32.8 1,441 19.0 High school 152 23.7 19 22.7 1,200 26.2 More than high school 252 47.5 38 44.6 2,141 54.8 Family income <$20,000 per year 266 29.6 43 26.6 1,389 22.2
Mean BMI (kg/m2 32.9 34.1 27.6
Self-rated health
Good, very good, or excellent 335 55.3 74 73.6 4,007 86.4
Fair or poor 369 44.7 36 26.4 770 13.6
*
Data source: NHANES 1999–2004; analytic population included participants with diagnosed diabetes, undiagnosed diabetes, and no diabetes; all data weighted by morning fasting sample weight.
Twenty of 704 subjects with diagnosed diabetes, 6 of 110 subjects with undiagnosed diabetes, and 121 of 4,782 subjects without diabetes did not report family income; 5 of 4,782 subjects without diabetes did not report their health status. These missing data will not appear in the regression models in Table 3.
Close Modal
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2022-06-28 02:54:15
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http://mathoverflow.net/questions/119639/the-maximal-abelian-subgroups-of-psl3-q-2-the-extension-of-psl3-q-by-the-gra?sort=votes
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# The maximal Abelian subgroups of PSL(3,q).2 the extension of PSL(3,q) by the graph automorphismAutomorphism
Hello For some part of my research I want to know that PSL(3,q).2, the extension of PSL(3,q) by the graph automorphism, has the same maximal abelian subgroups as PSL(3,q)?
Also if f is a field automorphism of GF(p^3) where q=p^3, then the maximal abelian subgroups of PSL(3,q).f are the same as PSL(3,q)?
Thank you very much With best regards
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It's not possible for extensions of ${\rm PSL}(3,q)$ to have the same maximal abelian subgroups as ${\rm PSL}(3,q)$ since these don't include the abelian subgroups not wholly contained in ${\rm PSL(3,q)}$.
However we can still try and classify the maximal abelian subgroups of extensions of ${\rm PSL}(3,q).$
Here's a rough sketch of how I'd approach this. Suppose that $H$ is a maximal abelian subgroup of a cyclic extension of $K$ where $K = {\rm PSL(3,q)}$. There are two cases:
• $H\leq K$. These can be read off from the known subgroups of $PSL(3,q)$. (See Mitchell, Hartley or Bloom for the first classification of these. Or see Kleidman & Liebeck, or the survey article by King for a modern treatment.)
• $H\not \leq K$. Then $H$ contains an outer automorphism $g$ and so, in particular $H \cap K$ lies in $C_K(g)$. Subcases that you are interested in:
(a) If $g$ is a graph automorphism of order $2$, then $C_K(g)\cong {\rm PSL}_2(q)$ or ${\rm PGL}_2(q)$ (since these are isomorphic to 3-dimensional orthogonal groups). The maximal subgroups of these groups are easy (Dickson gave the first proof), and the abelian subgroups can be read off.
(b) If $g$ is a graph automorphism of order greater than $2$, then we can assume it has order a multiple of $4$ (otherwise one of its powers is a graph aut of order $2$) and so $H$ contains an involution $h$ of $PSL(3,q)$. Now $C_K(h)=\hat GL(2,q)$ and, again, abelian subgroups can be read off.
(c) If $g$ is a field automorphism of order $3$, then $C_K(g)$ is a subfield subgroup. And so the classification of subgroups is the same as for $K$.
(d) If $g$ is a field automorphism of order greater than $3$. Again we can assume it has power a power of $3$, so $H$ contains an element of order $3$. Its centralizer can be calculated (its structure will depend on whether $3$ divides $q$, $q-1$ or $q+1$) and maximal abelians read off.
p.s. Some further thought made me wonder whether your comment about abelian subgroups being the same was supposed to be this:
A maximal abelian subgroup of ${\rm PSL}_3(q).2$ (graph aut) restricts to a maximal abelian subgroup of ${\rm PSL}_3(q)$.
I checked the ATLAS and this is not true: Let $K={\rm PSL}_3(5)$. Then $K.2 \backslash K$ contains an element of order $8$ whose centralizer $C$ is of order $8$, thus is maximal abelian. But $K\cap C$ (which has order $4$) is not maximal abelian in $K$.
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Thank you very much for your very helpful and complete answer Specially I work on the subject which needs the order of maximal abelian subgroups of PSL(3,q) and its extensions by the graph automorphism or by a field automorphism of order 3. Can we say that there exists any difference between their orders of maximal abelian subgroups, too. Thank you very much for your help – darya 5 mins ago – darya Jan 23 '13 at 11:38
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2015-07-01 06:36:32
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http://physics.stackexchange.com/questions/56086/determining-energy-of-gamma-rays-after-alpha-decay-of-am-241/56096
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# Determining energy of gamma-rays after alpha-decay of Am-241
So it turns into Np, and electrons just falling into 'free new' levels and emmiting, right?
Give me a link where to read, please, if it's very easy to answer.
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This is a complicated decay with many possible modes--there are one or more gamma associated with every alpha line (except, perhaps the highest energy one).
It is essentially impossible to compute the photon spectrum from first principles; they must be measured and doing so well is a delicate experiment.
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Thank you! Can you say, what is the reason of 26.3 keV (Am gamma peak)? – Alej Mar 6 '13 at 18:02
That's a prompt gamma in coincidence with the alpha decay. – dmckee Mar 6 '13 at 18:06
241-Am emits several gamma rays, but 35.9% gamma rays will have 59.5412 keV energy that can be easily detected with a scintillation detector fitted with NaI (Tl) crystal.
The following Reference also provides various alpha energies from 241-Am.
Reference: WWW Table of Radioactive Isotopes http://ie.lbl.gov/toi/nuclide.asp?iZA=950241
-
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2014-10-22 15:35:32
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http://experiment-ufa.ru/28/8_as_a_fraction
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# 28/8 as a fraction
## 28/8 as a fraction - solution and the full explanation with calculations.
If it's not what You are looking for, type in into the box below your number and see the solution.
## What is 28/8 as a fraction?
To write 28/8 as a fraction you have to get to the number where there is a numerator and a denominator written in whole numbers.
28/8 has already a numerator and denominator so it is a fraction already.
And finally we have:
28/8 as a fraction equals 28/8
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2017-11-22 11:12:06
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https://leanprover-community.github.io/archive/stream/113489-new-members/topic/Achievement.3A.20cardinality.20of.20subgroups.20of.20infinite.20groups.html
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Stream: new members
Topic: Achievement: cardinality of subgroups of infinite groups
Pedro Minicz (Jun 12 2020 at 00:46):
Another personal achievement/code review thread. Today I formalized the following: let G be an infinite group and H a subgroup s.t. #H < #G, then #G/H = #G.
The code is on Gist.
Pedro Minicz (Jun 12 2020 at 00:47):
I am pretty sure this isn't on mathlib, if it is I just missed it really hard. :grinning_face_with_smiling_eyes:
Kevin Buzzard (Jun 12 2020 at 00:50):
I think #(G/H) * #H = #G is in mathlib (perhaps in a stronger form of a bijection given a choice of a set of reps). Assuming AC your result probably follows from that.
Kenny Lau (Jun 12 2020 at 06:52):
yeah because if #G/H < #G then #G/H * #H = max(#G/H, #H) < #G
Assuming LEM ;-)
Pedro Minicz (Jun 12 2020 at 15:14):
#G/H * #H = #G is in mathlib as group_equiv_quotient_times_subgroup.
Pedro Minicz (Jun 12 2020 at 15:15):
I think the should be a simpler way of achieving this result. The proof itself is quite simple, but the code to do is turned out a bit convoluted.
Last updated: May 16 2021 at 21:11 UTC
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2021-05-16 21:28:50
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http://mathhelpforum.com/algebra/46573-reducing-algebra-fractions-1-a.html
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Math Help - Reducing Algebra Fractions by -1
1. Reducing Algebra Fractions by -1
The book is trying to show me that reducing a fraction or adding two fractions sometimes only requires that -1 be factored from one or more denominators. Here is the example they give:
$\frac{y-x}{x-y}=\frac{y-x}{-(-x+y)}=\frac{y-x}{-(y-x)}=\frac{1}{-1}=-1$
1.In the first step here I understand factoring out the "-" (the -1 value) because if the variable does not have a number coefficient we can assume its one so this part is fine.
$\frac{y-x}{-(-x+y)}$
2.Here $\frac{y-x}{-(-x+y)}$
once we distribute the "-" we get a negative y and a positive x, therefore we can rearrange the denominator to match the numerator in $\frac{y-x}{-(y-x)}$
3.Next we have $\frac{1}{-1}$ Now here is where I have a problem how can we know the values of both of these are 1 and -1? I am not certain how this is accomplished.
Does my logic up to 3's question look correct also?
2. Hi
Originally Posted by cmf0106
The book is trying to show me that reducing a fraction or adding two fractions sometimes only requires that -1 be factored from one or more denominators. Here is the example they give:
$\frac{y-x}{x-y}=\frac{y-x}{-(-x+y)}=\frac{y-x}{-(y-x)}=\frac{1}{-1}=-1$
1.In the first step here I understand factoring out the "-" (the -1 value) because if the variable does not have a number coefficient we can assume its one so this part is fine.
$\frac{y-x}{-(-x+y)}$
2.Here $\frac{y-x}{-(-x+y)}$
once we distribute the "-" we get a negative y and a positive x, therefore we can rearrange the denominator to match the numerator in $\frac{y-x}{-(y-x)}$
What you said for 1. is not wrong. But I'd say they made these two transformations to show you that there is the common factor y-x.
But what you said in 2. looks incorrect to me. They don't distribute - again, they just use the commutativity of the addition : a+b=b+a. Here : -x+y=y+(-x)=y-x.
3.Next we have $\frac{1}{-1}$ Now here is where I have a problem how can we know the values of both of these are 1 and -1? I am not certain how this is accomplished.
Relevant question for someone who begins ^^ It's so good seeing you trying to understand !
We have $\frac{y-x}{-(y-x)}$. This can be rewritten this way :
$\frac{1*{\color{red}(y-x)}}{(-1)*{\color{red}(y-x)}}$ and hence the result. Does it look correct to you ?
When you'll advance into maths lessons, maybe you will learn that most of these steps are only here to guide you to the solution
3. Thanks Moo! I will chew on that for awhile and let you know what the outcome is, I think it makes sense let me try a few more practice problems and we will see.
as for
"What you said for 1. is not wrong. But I'd say they made these two transformations to show you that there is the common factor y-x.
But what you said in 2. looks incorrect to me. They don't distribute - again, they just use the commutativity of the addition : a+b=b+a. Here : -x+y=y+(-x)=y-x."
Yah sorry I didnt mean to phrase it that way, it wouldnt make sense anyways because we factor out the "-", if we factored it out one step and then the next we redistributed it that would defeat the purpose of factoring
4. Alright moo update, this stuff is really putting me in a rut.
$\frac{3}{y-x}+\frac{x}{x-y}$
1.The book solves it $\frac{3}{y-x}+\frac{x}{x-y}=\frac{3}{-(-y+x)}+\frac{x}{x-y}=\frac{3}{-(x-y)}+\frac{x}{x-y}=\frac{-3}{x-y}+\frac{x}{x-y}=\frac{-3+x}{x-y}$
2. I started solving it like this, which looks different from the book $\frac{3}{y-x}+\frac{x}{x-y}=\frac{3}{-(-x+y)}+\frac{x}{x-y}=\frac{3+x}{(x-y)(x-y)}$
I used $\frac{3}{-(-x+y)}$ so it would yield the other fractions denominator $\frac{x}{x-y}$
Here $\frac{3+x}{(x-y)(x-y)}$ the two $(x-y)$ can merge into one but Im not understanding why the 3 is supposed to be -3 here, this is assuming my way of solving for it is correct in the first place.
5. Originally Posted by cmf0106
Alright moo update, this stuff is really putting me in a rut.
$\frac{3}{y-x}+\frac{x}{x-y}$
2. I started solving it like this, which looks different from the book $\frac{3}{y-x}+\frac{x}{x-y}=\frac{3}{-(-x+y)}+\frac{x}{x-y}=\frac{3+x}{(x-y)(x-y)}$
First problem in the first equality : y-x=-x+y, it's not equal to -(-x+y)
The purpose of this is to make appear the other fraction's denominator x-y. You can see that -(x-y)=-x+y=y-x.
So $\frac{3}{y-x}=\frac{3}{-(x-y)}$
Now if you want to know why it becomes $\frac{-3}{x-y}$ :
$\frac{3}{-(x-y)}=\frac{3}{-1} \cdot \frac{1}{x-y}=-3 \cdot \frac{1}{x-y}=\frac{-3}{x-y}$
See ?
Now, there is also a problem with the second equality. You should remember that :
$\frac ac+\frac bc=\frac{a+b}{c}$ and there is no multiplication involved. The multiplication occurs if there is not the same denominator
6. But I could chose to make either both denominators appear as $y-x$ or $x-y$ correct?
7. Originally Posted by cmf0106
But I could chose to make either both denominators appear as $y-x$ or $x-y$ correct?
Correct
It'd be nice if you could show how you would have done for y-x (unless you have many more exercises ^^)
8. Originally Posted by Moo
Correct
It'd be nice if you could show how you would have done for y-x (unless you have many more exercises ^^)
Man this is probably the hardest thing Ive encountered so far, hard for me to wrap my brain around. Anyways lets give the other denominator a shot
so $\frac{3}{y-x}+\frac{x}{x-y}$
Lets Change $\frac{3}{y-x}=-(y-x)=-y+x$ this equals our other denominator $x-y$
next $=\frac{3}{-(y-x)}=\frac{-3+x}{x-y}$
Probably wrong lol.
9. Originally Posted by cmf0106
Man this is probably the hardest thing Ive encountered so far, hard for me to wrap my brain around. Anyways lets give the other denominator a shot
so $\frac{3}{y-x}+\frac{x}{x-y}$
Lets Change ${\color{red}\frac{3}{y-x}=\frac{3}{-(y-x)}=\frac{3}{-y+x}}$ this equals our other denominator $x-y$
next $=\frac{3}{-(y-x)}=\frac{-3+x}{x-y}={\color{red}\frac{x-3}{x-y}}$
Probably wrong lol.
Let's clean this up just a little bit so that your equalities make sense, and your conclusion looks fine. Great job!
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2015-08-04 04:08:05
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https://ftp.aimsciences.org/article/doi/10.3934/dcds.2011.30.1139
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# American Institute of Mathematical Sciences
November 2011, 30(4): 1139-1144. doi: 10.3934/dcds.2011.30.1139
## A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature
1 LAMFA – CNRS UMR 6140, Université de Picardie Jules Verne, 33, rue Saint-Leu 80039 Amiens CEDEX 1, France 2 Università degli Studi di Milano, Dipartimento di Matematica Via Saldini, 50, 20133 Milano
Received February 2010 Revised August 2010 Published May 2011
N/A
Citation: Alberto Farina, Enrico Valdinoci. A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1139-1144. doi: 10.3934/dcds.2011.30.1139
##### References:
[1] Marcel Berger, Paul Gauduchon and Edmond Mazet, "Le Spectre d'une Variété Riemannienne,", Lecture Notes in Mathematics, 194 (1971). Google Scholar [2] Luis Caffarelli, Nicola Garofalo and Fausto Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences,, Comm. Pure Appl. Math., 47 (1994), 1457. doi: 10.1002/cpa.3160471103. Google Scholar [3] Alberto Farina, Yannick Sire and Enrico Valdinoci, Stable solutions of elliptic equations on Riemannian manifolds,, preprint (2008)., (2008). Google Scholar [4] Alberto Farina and Enrico Valdinoci, A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature,, Adv. Math., 225 (2010), 2808. doi: 10.1016/j.aim.2010.05.008. Google Scholar [5] David Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundlehren der Mathematischen Wissenschaften, 224 (1983). Google Scholar [6] Luciano Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations,, Comm. Pure Appl. Math., 38 (1985), 679. doi: 10.1002/cpa.3160380515. Google Scholar [7] L. E. Payne, Some remarks on maximum principles,, J. Analyse Math., 30 (1976), 421. doi: 10.1007/BF02786729. Google Scholar [8] Vladimir E. Shklover, Schiffer problem and isoparametric hypersurfaces,, Rev. Mat. Iberoamericana, 16 (2000), 529. Google Scholar [9] René P. Sperb, "Maximum Principles and their Applications,", Mathematics in Science and Engineering, 157 (1981). Google Scholar [10] Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations,, Handbook of Differential Geometry, (2000), 963. Google Scholar [11] Jiaping Wang, "Lecture Notes on, Geometric Analysis, (). Google Scholar
show all references
##### References:
[1] Marcel Berger, Paul Gauduchon and Edmond Mazet, "Le Spectre d'une Variété Riemannienne,", Lecture Notes in Mathematics, 194 (1971). Google Scholar [2] Luis Caffarelli, Nicola Garofalo and Fausto Segàla, A gradient bound for entire solutions of quasi-linear equations and its consequences,, Comm. Pure Appl. Math., 47 (1994), 1457. doi: 10.1002/cpa.3160471103. Google Scholar [3] Alberto Farina, Yannick Sire and Enrico Valdinoci, Stable solutions of elliptic equations on Riemannian manifolds,, preprint (2008)., (2008). Google Scholar [4] Alberto Farina and Enrico Valdinoci, A pointwise gradient estimate in possibly unbounded domains with nonnegative mean curvature,, Adv. Math., 225 (2010), 2808. doi: 10.1016/j.aim.2010.05.008. Google Scholar [5] David Gilbarg and Neil S. Trudinger, "Elliptic Partial Differential Equations of Second Order,", Grundlehren der Mathematischen Wissenschaften, 224 (1983). Google Scholar [6] Luciano Modica, A gradient bound and a Liouville theorem for nonlinear Poisson equations,, Comm. Pure Appl. Math., 38 (1985), 679. doi: 10.1002/cpa.3160380515. Google Scholar [7] L. E. Payne, Some remarks on maximum principles,, J. Analyse Math., 30 (1976), 421. doi: 10.1007/BF02786729. Google Scholar [8] Vladimir E. Shklover, Schiffer problem and isoparametric hypersurfaces,, Rev. Mat. Iberoamericana, 16 (2000), 529. Google Scholar [9] René P. Sperb, "Maximum Principles and their Applications,", Mathematics in Science and Engineering, 157 (1981). Google Scholar [10] Gudlaugur Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations,, Handbook of Differential Geometry, (2000), 963. Google Scholar [11] Jiaping Wang, "Lecture Notes on, Geometric Analysis, (). Google Scholar
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2019 Impact Factor: 1.338
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2021-02-27 21:55:41
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https://wikimili.com/en/Convex_hull
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Convex hull
Last updated
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset.
Contents
Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of computational geometry. They can be solved in time ${\displaystyle O(n\log n)}$ for two or three dimensional point sets, and in time matching the worst-case output complexity given by the upper bound theorem in higher dimensions.
As well as for finite point sets, convex hulls have also been studied for simple polygons, Brownian motion, space curves, and epigraphs of functions. Convex hulls have wide applications in mathematics, statistics, combinatorial optimization, economics, geometric modeling, and ethology. Related structures include the orthogonal convex hull, convex layers, Delaunay triangulation and Voronoi diagram, and convex skull.
Definitions
A set of points in a Euclidean space is defined to be convex if it contains the line segments connecting each pair of its points. The convex hull of a given set ${\displaystyle X}$ may be defined as [1]
1. The (unique) minimal convex set containing ${\displaystyle X}$
2. The intersection of all convex sets containing ${\displaystyle X}$
3. The set of all convex combinations of points in ${\displaystyle X}$
4. The union of all simplices with vertices in ${\displaystyle X}$
For bounded sets in the Euclidean plane, not all on one line, the boundary of the convex hull is the simple closed curve with minimum perimeter containing ${\displaystyle X}$. One may imagine stretching a rubber band so that it surrounds the entire set ${\displaystyle S}$ and then releasing it, allowing it to contract; when it becomes taut, it encloses the convex hull of ${\displaystyle S}$. [2] This formulation does not immediately generalize to higher dimensions: for a finite set of points in three-dimensional space, a neighborhood of a spanning tree of the points encloses them with arbitrarily small surface area, smaller than the surface area of the convex hull. [3] However, in higher dimensions, variants of the obstacle problem of finding a minimum-energy surface above a given shape can have the convex hull as their solution. [4]
For objects in three dimensions, the first definition states that the convex hull is the smallest possible convex bounding volume of the objects. The definition using intersections of convex sets may be extended to non-Euclidean geometry, and the definition using convex combinations may be extended from Euclidean spaces to arbitrary real vector spaces or affine spaces; convex hulls may also be generalized in a more abstract way, to oriented matroids. [5]
Equivalence of definitions
It is not obvious that the first definition makes sense: why should there exist a unique minimal convex set containing ${\displaystyle X}$, for every ${\displaystyle X}$? However, the second definition, the intersection of all convex sets containing ${\displaystyle X}$, is well-defined. It is a subset of every other convex set ${\displaystyle Y}$ that contains ${\displaystyle X}$, because ${\displaystyle Y}$ is included among the sets being intersected. Thus, it is exactly the unique minimal convex set containing ${\displaystyle X}$. Therefore, the first two definitions are equivalent. [1]
Each convex set containing ${\displaystyle X}$ must (by the assumption that it is convex) contain all convex combinations of points in ${\displaystyle X}$, so the set of all convex combinations is contained in the intersection of all convex sets containing ${\displaystyle X}$. Conversely, the set of all convex combinations is itself a convex set containing ${\displaystyle X}$, so it also contains the intersection of all convex sets containing ${\displaystyle X}$, and therefore the second and third definitions are equivalent. [6]
In fact, according to Carathéodory's theorem, if ${\displaystyle X}$ is a subset of a ${\displaystyle d}$-dimensional Euclidean space, every convex combination of finitely many points from ${\displaystyle X}$ is also a convex combination of at most ${\displaystyle d+1}$ points in ${\displaystyle X}$. The set of convex combinations of a ${\displaystyle (d+1)}$-tuple of points is a simplex; in the plane it is a triangle and in three-dimensional space it is a tetrahedron. Therefore, every convex combination of points of ${\displaystyle X}$ belongs to a simplex whose vertices belong to ${\displaystyle X}$, and the third and fourth definitions are equivalent. [6]
Upper and lower hulls
In two dimensions, the convex hull is sometimes partitioned into two parts, the upper hull and the lower hull, stretching between the leftmost and rightmost points of the hull. More generally, for convex hulls in any dimension, one can partition the boundary of the hull into upward-facing points (points for which an upward ray is disjoint from the hull), downward-facing points, and extreme points. For three-dimensional hulls, the upward-facing and downward-facing parts of the boundary form topological disks. [7]
Topological properties
Closed and open hulls
The closed convex hull of a set is the closure of the convex hull, and the open convex hull is the interior (or in some sources the relative interior) of the convex hull. [8]
The closed convex hull of ${\displaystyle X}$ is the intersection of all closed half-spaces containing ${\displaystyle X}$. If the convex hull of ${\displaystyle X}$ is already a closed set itself (as happens, for instance, if ${\displaystyle X}$ is a finite set or more generally a compact set), then it equals the closed convex hull. However, an intersection of closed half-spaces is itself closed, so when a convex hull is not closed it cannot be represented in this way. [9]
If the open convex hull of a set ${\displaystyle X}$ is ${\displaystyle d}$-dimensional, then every point of the hull belongs to an open convex hull of at most ${\displaystyle 2d}$ points of ${\displaystyle X}$. The sets of vertices of a square, regular octahedron, or higher-dimensional cross-polytope provide examples where exactly ${\displaystyle 2d}$ points are needed. [10]
Preservation of topological properties
Topologically, the convex hull of an open set is always itself open, and the convex hull of a compact set is always itself compact. However, there exist closed sets for which the convex hull is not closed. [11] For instance, the closed set
${\displaystyle \left\{(x,y)\mathop {\bigg |} y\geq {\frac {1}{1+x^{2}}}\right\}}$
(the set of points that lie on or above the witch of Agnesi) has the open upper half-plane as its convex hull. [12]
The compactness of convex hulls of compact sets, in finite-dimensional Euclidean spaces, is generalized by the Krein–Smulian theorem, according to which the closed convex hull of a weakly compact subset of a Banach space (a subset that is compact under the weak topology) is weakly compact. [13]
Extreme points
An extreme point of a convex set is a point in the set that does not lie on any open line segment between any other two points of the same set. For a convex hull, every extreme point must be part of the given set, because otherwise it cannot be formed as a convex combination of given points. According to the Krein–Milman theorem, every compact convex set in a Euclidean space (or more generally in a locally convex topological vector space) is the convex hull of its extreme points. [14] However, this may not be true for convex sets that are not compact; for instance, the whole Euclidean plane and the open unit ball are both convex, but neither one has any extreme points. Choquet theory extends this theory from finite convex combinations of extreme points to infinite combinations (integrals) in more general spaces. [15]
Geometric and algebraic properties
Closure operator
The convex-hull operator has the characteristic properties of a closure operator: [16]
• It is extensive, meaning that the convex hull of every set ${\displaystyle X}$ is a superset of ${\displaystyle X}$.
• It is non-decreasing , meaning that, for every two sets ${\displaystyle X}$ and ${\displaystyle Y}$ with ${\displaystyle X\subseteq Y}$, the convex hull of ${\displaystyle X}$ is a subset of the convex hull of ${\displaystyle Y}$.
• It is idempotent , meaning that for every ${\displaystyle X}$, the convex hull of the convex hull of ${\displaystyle X}$ is the same as the convex hull of ${\displaystyle X}$.
When applied to a finite set of points, this is the closure operator of an antimatroid, the shelling antimatroid of the point set. Every antimatroid can be represented in this way by convex hulls of points in a Euclidean space of high-enough dimension. [17]
Minkowski sum
The operations of constructing the convex hull and taking the Minkowski sum commute with each other, in the sense that the Minkowski sum of convex hulls of sets gives the same result as the convex hull of the Minkowski sum of the same sets. This provides a step towards the Shapley–Folkman theorem bounding the distance of a Minkowski sum from its convex hull. [18]
Projective duality
The projective dual operation to constructing the convex hull of a set of points is constructing the intersection of a family of closed halfspaces that all contain the origin (or any other designated point). [19]
Special cases
Finite point sets
The convex hull of a finite point set ${\displaystyle S\subset \mathbb {R} ^{d}}$ forms a convex polygon when ${\displaystyle d=2}$, or more generally a convex polytope in ${\displaystyle \mathbb {R} ^{d}}$. Each extreme point of the hull is called a vertex, and (by the Krein–Milman theorem) every convex polytope is the convex hull of its vertices. It is the unique convex polytope whose vertices belong to ${\displaystyle S}$ and that encloses all of ${\displaystyle S}$. [2] For sets of points in general position, the convex hull is a simplicial polytope. [20]
According to the upper bound theorem, the number of faces of the convex hull of ${\displaystyle n}$ points in ${\displaystyle d}$-dimensional Euclidean space is ${\displaystyle O(n^{\lfloor d/2\rfloor })}$. [21] In particular, in two and three dimensions the number of faces is at most linear in ${\displaystyle n}$. [22]
Simple polygons
The convex hull of a simple polygon encloses the given polygon and is partitioned by it into regions, one of which is the polygon itself. The other regions, bounded by a polygonal chain of the polygon and a single convex hull edge, are called pockets. Computing the same decomposition recursively for each pocket forms a hierarchical description of a given polygon called its convex differences tree. [23] Reflecting a pocket across its convex hull edge expands the given simple polygon into a polygon with the same perimeter and larger area, and the Erdős–Nagy theorem states that this expansion process eventually terminates. [24]
Brownian motion
The curve generated by Brownian motion in the plane, at any fixed time, has probability 1 of having a convex hull whose boundary forms a continuously differentiable curve. However, for any angle ${\displaystyle \theta }$ in the range ${\displaystyle \pi /2<\theta <\pi }$, there will be times during the Brownian motion where the moving particle touches the boundary of the convex hull at a point of angle ${\displaystyle \theta }$. The Hausdorff dimension of this set of exceptional times is (with high probability) ${\displaystyle 1-\pi /2\theta }$. [25]
Space curves
For the convex hull of a space curve or finite set of space curves in general position in three-dimensional space, the parts of the boundary away from the curves are developable and ruled surfaces. [26] Examples include the oloid, the convex hull of two circles in perpendicular planes, each passing through the other's center, [27] the sphericon, the convex hull of two semicircles in perpendicular planes with a common center, and D-forms, the convex shapes obtained from Alexandrov's uniqueness theorem for a surface formed by gluing together two planar convex sets of equal perimeter. [28]
Functions
The convex hull or lower convex envelope of a function ${\displaystyle f}$ on a real vector space is the function whose epigraph is the lower convex hull of the epigraph of ${\displaystyle f}$. It is the unique maximal convex function majorized by ${\displaystyle f}$. [29] The definition can be extended to the convex hull of a set of functions (obtained from the convex hull of the union of their epigraphs, or equivalently from their pointwise minimum) and, in this form, is dual to the convex conjugate operation. [30]
Computation
In computational geometry, a number of algorithms are known for computing the convex hull for a finite set of points and for other geometric objects. Computing the convex hull means constructing an unambiguous, efficient representation of the required convex shape. Output representations that have been considered for convex hulls of point sets include a list of linear inequalities describing the facets of the hull, an undirected graph of facets and their adjacencies, or the full face lattice of the hull. [31] In two dimensions, it may suffice more simply to list the points that are vertices, in their cyclic order around the hull. [2]
For convex hulls in two or three dimensions, the complexity of the corresponding algorithms is usually estimated in terms of ${\displaystyle n}$, the number of input points, and ${\displaystyle h}$, the number of points on the convex hull, which may be significantly smaller than ${\displaystyle n}$. For higher-dimensional hulls, the number of faces of other dimensions may also come into the analysis. Graham scan can compute the convex hull of ${\displaystyle n}$ points in the plane in time ${\displaystyle O(n\log n)}$. For points in two and three dimensions, more complicated output-sensitive algorithms are known that compute the convex hull in time ${\displaystyle O(n\log h)}$. These include Chan's algorithm and the Kirkpatrick–Seidel algorithm. [32] For dimensions ${\displaystyle d>3}$, the time for computing the convex hull is ${\displaystyle O(n^{\lfloor d/2\rfloor })}$, matching the worst-case output complexity of the problem. [33] The convex hull of a simple polygon in the plane can be constructed in linear time. [34]
Dynamic convex hull data structures can be used to keep track of the convex hull of a set of points undergoing insertions and deletions of points, [35] and kinetic convex hull structures can keep track of the convex hull for points moving continuously. [36] The construction of convex hulls also serves as a tool, a building block for a number of other computational-geometric algorithms such as the rotating calipers method for computing the width and diameter of a point set. [37]
Several other shapes can be defined from a set of points in a similar way to the convex hull, as the minimal superset with some property, the intersection of all shapes containing the points from a given family of shapes, or the union of all combinations of points for a certain type of combination. For instance:
• The affine hull is the smallest affine subspace of a Euclidean space containing a given set, or the union of all affine combinations of points in the set. [38]
• The linear hull is the smallest linear subspace of a vector space containing a given set, or the union of all linear combinations of points in the set. [38]
• The conical hull or positive hull of a subset of a vector space is the set of all positive combinations of points in the subset. [38]
• The visual hull of a three-dimensional object, with respect to a set of viewpoints, consists of the points ${\displaystyle p}$ such that every ray from a viewpoint through ${\displaystyle p}$ intersects the object. Equivalently it is the intersection of the (non-convex) cones generated by the outline of the object with respect to each viewpoint. It is used in 3D reconstruction as the largest shape that could have the same outlines from the given viewpoints. [39]
• The circular hull or alpha-hull of a subset of the plane is the intersection of all disks with a given radius ${\displaystyle 1/\alpha }$ that contain the subset. [40]
• The relative convex hull of a subset of a two-dimensional simple polygon is the intersection of all relatively convex supersets, where a set within the same polygon is relatively convex if it contains the geodesic between any two of its points. [41]
• The orthogonal convex hull or rectilinear convex hull is the intersection of all orthogonally convex and connected supersets, where a set is orthogonally convex if it contains all axis-parallel segments between pairs of its points. [42]
• The orthogonal convex hull is a special case of a much more general construction, the hyperconvex hull, which can be thought of as the smallest injective metric space containing the points of a given metric space. [43]
• The holomorphically convex hull is a generalization of similar concepts to complex analytic manifolds, obtained as an intersection of sublevel sets of holomorphic functions containing a given set. [44]
The Delaunay triangulation of a point set and its dual, the Voronoi diagram, are mathematically related to convex hulls: the Delaunay triangulation of a point set in ${\displaystyle \mathbb {R} ^{n}}$ can be viewed as the projection of a convex hull in ${\displaystyle \mathbb {R} ^{n+1}.}$ [45] The alpha shapes of a finite point set give a nested family of (non-convex) geometric objects describing the shape of a point set at different levels of detail. Each of alpha shape is the union of some of the features of the Delaunay triangulation, selected by comparing their circumradius to the parameter alpha. The point set itself forms one endpoint of this family of shapes, and its convex hull forms the other endpoint. [40] The convex layers of a point set are a nested family of convex polygons, the outermost of which is the convex hull, with the inner layers constructed recursively from the points that are not vertices of the convex hull. [46]
The convex skull of a polygon is the largest convex polygon contained inside it. It can be found in polynomial time, but the exponent of the algorithm is high. [47]
Applications
Convex hulls have wide applications in many fields. Within mathematics, convex hulls are used to study polynomials, matrix eigenvalues, and unitary elements, and several theorems in discrete geometry involve convex hulls. They are used in robust statistics as the outermost contour of Tukey depth, are part of the bagplot visualization of two-dimensional data, and define risk sets of randomized decision rules. Convex hulls of indicator vectors of solutions to combinatorial problems are central to combinatorial optimization and polyhedral combinatorics. In economics, convex hulls can be used to apply methods of convexity in economics to non-convex markets. In geometric modeling, the convex hull property Bézier curves helps find their crossings, and convex hulls are part of the measurement of boat hulls. And in the study of animal behavior, convex hulls are used in a standard definition of the home range.
Mathematics
Newton polygons of univariate polynomials and Newton polytopes of multivariate polynomials are convex hulls of points derived from the exponents of the terms in the polynomial, and can be used to analyze the asymptotic behavior of the polynomial and the valuations of its roots. [48] Convex hulls and polynomials also come together in the Gauss–Lucas theorem, according to which the roots of the derivative of a polynomial all lie within the convex hull of the roots of the polynomial. [49]
In spectral analysis, the numerical range of a normal matrix is the convex hull of its eigenvalues. [50] The Russo–Dye theorem describes the convex hulls of unitary elements in a C*-algebra. [51] In discrete geometry, both Radon's theorem and Tverberg's theorem concern the existence of partitions of point sets into subsets with intersecting convex hulls. [52]
The definitions of a convex set as containing line segments between its points, and of a convex hull as the intersection of all convex supersets, apply to hyperbolic spaces as well as to Euclidean spaces. However, in hyperbolic space, it is also possible to consider the convex hulls of sets of ideal points, points that do not belong to the hyperbolic space itself but lie on the boundary of a model of that space. The boundaries of convex hulls of ideal points of three-dimensional hyperbolic space are analogous to ruled surfaces in Euclidean space, and their metric properties play an important role in the geometrization conjecture in low-dimensional topology. [53] Hyperbolic convex hulls have also been used as part of the calculation of canonical triangulations of hyperbolic manifolds, and applied to determine the equivalence of knots. [54]
See also the section on Brownian motion for the application of convex hulls to this subject, and the section on space curves for their application to the theory of developable surfaces.
Statistics
In robust statistics, the convex hull provides one of the key components of a bagplot, a method for visualizing the spread of two-dimensional sample points. The contours of Tukey depth form a nested family of convex sets, with the convex hull outermost, and the bagplot also displays another polygon from this nested family, the contour of 50% depth. [55]
In statistical decision theory, the risk set of a randomized decision rule is the convex hull of the risk points of its underlying deterministic decision rules. [56]
Combinatorial optimization
In combinatorial optimization and polyhedral combinatorics, central objects of study are the convex hulls of indicator vectors of solutions to a combinatorial problem. If the facets of these polytopes can be found, describing the polytopes as intersections of halfspaces, then algorithms based on linear programming can be used to find optimal solutions. [57] In multi-objective optimization, a different type of convex hull is also used, the convex hull of the weight vectors of solutions. One can maximize any quasiconvex combination of weights by finding and checking each convex hull vertex, often more efficiently than checking all possible solutions. [58]
Economics
In the Arrow–Debreu model of general economic equilibrium, agents are assumed to have convex budget sets and convex preferences. These assumptions of convexity in economics can be used to prove the existence of an equilibrium. When actual economic data is non-convex, it can be made convex by taking convex hulls. The Shapley–Folkman theorem can be used to show that, for large markets, this approximation is accurate, and leads to a "quasi-equilibrium" for the original non-convex market. [59]
Geometric modeling
In geometric modeling, one of the key properties of a Bézier curve is that it lies within the convex hull of its control points. This so-called "convex hull property" can be used, for instance, in quickly detecting intersections of these curves. [60]
In the geometry of boat and ship design, chain girth is a measurement of the size of a sailing vessel, defined using the convex hull of a cross-section of the hull of the vessel. It differs from the skin girth, the perimeter of the cross-section itself, except for boats and ships that have a convex hull. [61]
Ethology
The convex hull is commonly known as the minimum convex polygon in ethology, the study of animal behavior, where it is a classic, though perhaps simplistic, approach in estimating an animal's home range based on points where the animal has been observed. [62] Outliers can make the minimum convex polygon excessively large, which has motivated relaxed approaches that contain only a subset of the observations, for instance by choosing one of the convex layers that is close to a target percentage of the samples, [63] or in the local convex hull method by combining convex hulls of neighborhoods of points. [64]
Quantum physics
In quantum physics, the state space of any quantum system — the set of all ways the system can be prepared — is a convex hull whose extreme points are positive-semidefinite operators known as pure states and whose interior points are called mixed states. [65] The Schrödinger–HJW theorem proves that any mixed state can in fact be written as a convex combination of pure states in multiple ways. [66]
Thermodynamics
A convex hull in thermodynamics was identified by Josiah Willard Gibbs (1873), [68] although the paper was published before the convex hull was so named. In a set of energies of several stoichiometries of a material, only those measurements on the lower convex hull will be stable. When removing a point from the hull and then calculating its distance to the hull, its distance to the new hull represents the degree of stability of the phase. [69]
History
The lower convex hull of points in the plane appears, in the form of a Newton polygon, in a letter from Isaac Newton to Henry Oldenburg in 1676. [70] The term "convex hull" itself appears as early as the work of GarrettBirkhoff ( 1935 ), and the corresponding term in German appears earlier, for instance in Hans Rademacher's review of Kőnig ( 1922 ). Other terms, such as "convex envelope", were also used in this time frame. [71] By 1938, according to Lloyd Dines, the term "convex hull" had become standard; Dines adds that he finds the term unfortunate, because the colloquial meaning of the word "hull" would suggest that it refers to the surface of a shape, whereas the convex hull includes the interior and not just the surface. [72]
Notes
1. Rockafellar (1970), p. 12.
2. de Berg et al. (2008), p. 3.
3. Williams & Rossignac (2005). See also Douglas Zare, answer to "the perimeter of a non-convex set", MathOverflow, May 16, 2014.
4. Rockafellar (1970), p. 12; Lay (1982), p. 17.
5. de Berg et al. (2008), p. 6. The idea of partitioning the hull into two chains comes from an efficient variant of Graham scan by Andrew (1979).
6. Rockafellar (1970), p. 99.
7. Grünbaum (2003), p. 16; Lay (1982), p. 21; Sakuma (1977).
8. This example is given by Talman (1977), Remark 2.6.
9. Krein & Milman (1940); Lay (1982), p. 43.
10. Krein & Šmulian (1940), Theorem 3, pages 562–563; Schneider (1993), Theorem 1.1.2 (pages 2–3) and Chapter 3.
11. de Berg et al. (2008), p. 254.
12. Grünbaum (2003), p. 57.
13. de Berg et al. (2008), p. 256.
14. de Berg et al. (2008), p. 245.
15. Rockafellar (1970), p. 36.
16. Rockafellar (1970), p. 149.
17. de Berg et al. (2008), p. 13.
18. Chazelle (1993); de Berg et al. (2008), p. 256.
19. Pulleyblank (1983); see especially remarks following Theorem 2.9.
20. Nicola (2000). See in particular Section 16.9, Non Convexity and Approximate Equilibrium, pp. 209–210.
21. Newton (1676); see Auel (2019), page 336, and Escobar & Kaveh (2020).
22. See, e.g., White (1923), page 520.
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In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment . For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex.
In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
In elementary geometry, a polytope is a geometric object with flat sides (faces). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common.
In geometry, the Minkowski sum of two sets of position vectors A and B in Euclidean space is formed by adding each vector in A to each vector in B, i.e., the set
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself. For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry.
Carathéodory's theorem is a theorem in convex geometry. It states that if a point x of Rd lies in the convex hull of a set P, then x can be written as the convex combination of at most d + 1 points in P. Namely, there is a subset P′ of P consisting of d + 1 or fewer points such that x lies in the convex hull of P′. Equivalently, x lies in an r-simplex with vertices in P, where . The smallest r that makes the last statement valid for each x in the convex hull of P is defined as the Carathéodory's number of P. Depending on the properties of P, upper bounds lower than the one provided by Carathéodory's theorem can be obtained. Note that P need not be itself convex. A consequence of this is that P′ can always be extremal in P, as non-extremal points can be removed from P without changing the membership of x in the convex hull.
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A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the -dimensional Euclidean space . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron" for the more general, possibly unbounded object. Others allow polytopes to be unbounded. The terms "bounded/unbounded convex polytope" will be used below whenever the boundedness is critical to the discussed issue. Yet other texts identify a convex polytope with its boundary.
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Kirchberger's theorem is a theorem in discrete geometry, on linear separability. The two-dimensional version of the theorem states that, if a finite set of red and blue points in the Euclidean plane has the property that, for every four points, there exists a line separating the red and blue points within those four, then there exists a single line separating all the red points from all the blue points. Donald Watson phrases this result more colorfully, with a farmyard analogy:
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https://verification.asmedigitalcollection.asme.org/energyresources/article/143/9/090904/1091989/Design-of-a-1-MWth-Supercritical-Carbon-Dioxide?searchresult=1
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## Abstract
A new generation of concentrating solar power (CSP) technologies is under development to provide dispatchable renewable power generation and reduce the levelized cost of electricity (LCOE) to 6 cents/kWh by leveraging heat transfer fluids (HTFs) capable of operation at higher temperatures and coupling with higher efficiency power conversion cycles. The U.S. Department of Energy (DOE) has funded three pathways for Generation 3 CSP (Gen3CSP) technology development to leverage solid, liquid, and gaseous HTFs to transfer heat to a supercritical carbon dioxide (sCO2) Brayton cycle. This paper presents the design and off-design capabilities of a 1 MWth sCO2 test system that can provide sCO2 coolant to the primary heat exchangers (PHX) coupling the high-temperature HTFs to the sCO2 working fluid of the power cycle. This system will demonstrate design, performance, lifetime, and operability at a scale relevant to commercial CSP. A dense-phase high-pressure canned motor pump is used to supply up to 5.3 kg/s of sCO2 flow to the primary heat exchanger at pressures up to 250 bar and temperatures up to 715 °C with ambient air as the ultimate heat sink. Key component requirements for this system are presented in this paper.
## 1 Introduction
Efforts to identify and develop novel heat transfer fluids (HTFs) to harvest and store solar energy through concentrating solar power (CSP) technologies are underway. The U.S. Department of Energy (DOE) is currently funding research on Generation 3 CSP (Gen3CSP) technologies to develop technology to transfer heat from solid, liquid, or gaseous media to a supercritical carbon dioxide (sCO2) Brayton cycle [1]. The sCO2 Brayton cycle is of great interest as it reduces the footprint (due to smaller component size), cost, and has the potential to increase the thermal efficiency of the power cycle (close to 50%), making it competitive when compared with typical Rankine steam power plants [2,3].
The solid/liquid/gaseous Gen3CSP HTF paths (including primary heat exchangers (PHX) design and thermal energy storage) need to prove the capability and scalability of the different systems and require a cooling system for this task. To reach the goals set by the DOE and prove scalability, a cooling system capable of rejecting more than 1 MWth, and handling temperatures and pressures higher than 700 °C and 250 bar, respectively, is necessary.
This paper describes the design of a 1 MWth-scale sCO2 support loop to provide cooling to the PHX of any of the three Gen3CSP pathway projects following on previous scoping studies [46] and builds on experience gained from the design and operation of many previous systems.
## 2 1-MWth Heat Removal System
The combination of CSP with sCO2 Brayton power cycles is a topic of great interest due to the potential for increased efficiencies within the power cycle, reduced capital costs given the compact size of the necessary equipment compared with other commercial power cycles (e.g., steam power cycles), and the compatibility with dry cooling that would decrease the need for cooling water.
The Gen3CSP solid, liquid, and gas pathways, currently in development, require a test setup to evaluate their performance during operation. A system capable of working with any of the three media while supporting a minimum 1 MWth thermal rejection was to be designed.
### 2.1 Requirements.
A set of system requirements were established for the overall Gen3CSP system and are summarized in Table 1. The PHX requirements listed in Table 1 are deemed essential to meet the goals set by the DOE. At the same time, these requirements must be fulfilled when coupled to the sCO2 test system.
Table 1
Supercritical CO2 test system requirements
RequirementValue
Operating fluidCarbon dioxide
PHX outlet pressure250 bar
PHX outlet temperature715 °C
Thermal duty≥1 MWth
Operational time16 h/day
RequirementValue
Operating fluidCarbon dioxide
PHX outlet pressure250 bar
PHX outlet temperature715 °C
Thermal duty≥1 MWth
Operational time16 h/day
Additional requirements were derived after discussion with the PHX design teams. These requirements are primarily driven by the temperature and pressure needed at the PHX inlet and the thermal duty of the system. The derived requirements are summarized in Table 2, including the ability to accommodate a PHX pressure drop up to at least 5 bar.
Table 2
Supercritical CO2 test system derived requirements
RequirementValue
Allowable PHX pressure drop≥5 bar (2% drop)
PHX inlet temperature≤565 °C (150 °C ΔT min)
RequirementValue
Allowable PHX pressure drop≥5 bar (2% drop)
PHX inlet temperature≤565 °C (150 °C ΔT min)
Size, location, and operation of the sCO2 loop were also considered, and an initial set of size and power requirements were established. These requirements are shown in Table 3. The final location of the sCO2 test system is not yet defined. For this reason, the system has been designed as a set of modules including the inventory management system, flow management system, recuperation, and PHX flow conditioning modules. These modules can be disassembled, transported, and reassembled to a new location without requiring any onsite cutting or welding.
Table 3
Supercritical CO2 test system size and power requirements
RequirementValue
Weight≤89,000 N (≤20,000 lbs)
Supply power voltage480 3-Phase Y
Total footprint≤2.7 m × 3.4 m (≤9 ft × 11 ft)
Height≤2.3 m (≤7.5 ft)
RequirementValue
Weight≤89,000 N (≤20,000 lbs)
Supply power voltage480 3-Phase Y
Total footprint≤2.7 m × 3.4 m (≤9 ft × 11 ft)
Height≤2.3 m (≤7.5 ft)
### 2.2 Design.
Several options for the sCO2 loop configuration were considered and are summarized in Table 4.
Table 4
Metrics for different system configurations
ConfigurationCompression power (kW (hp))Total UA (W/K)Heating (MW)Required CO2 (kg)Primary challenge
Liquid blowdown164 (220)38,0004.3300,000CO2 supply
Gas blowdown1342 (1800)01.9300,000CO2 supply
Liquid Brayton5.9 (7.9)160,0003.9ClosedHeat exchange
Gas Brayton43 (58)20000ClosedHigh-temperature compression
Recuperated Brayton10 (14)54,0000ClosedHigh-temperature recuperation
Recuperated and mixed Brayton14 (19)71,0000.2ClosedMixing assembly
ConfigurationCompression power (kW (hp))Total UA (W/K)Heating (MW)Required CO2 (kg)Primary challenge
Liquid blowdown164 (220)38,0004.3300,000CO2 supply
Gas blowdown1342 (1800)01.9300,000CO2 supply
Liquid Brayton5.9 (7.9)160,0003.9ClosedHeat exchange
Gas Brayton43 (58)20000ClosedHigh-temperature compression
Recuperated Brayton10 (14)54,0000ClosedHigh-temperature recuperation
Recuperated and mixed Brayton14 (19)71,0000.2ClosedMixing assembly
Both liquid and gas blowdown open systems are not desirable for implementation because they require a constant supply of CO2, increasing the gas inventory cost and the complexity of the logistics to maintain enough CO2 available. Also, the liquid blowdown system requires an evaporator for cooling; the use of water for cooling is to be avoided to ensure that the test system can be deployed in areas with limited supply of water and ensure that similar systems can be installed in areas where CSP is typically utilized (i.e., water-scarce areas). Another consideration is that the gas-phase Brayton cycle requires further advancement of turbo-compressor technology; this topic is addressed in Sec. 3. A liquid system would allow for the use of pumps. For a liquid sCO2 system, high-pressure, commercially available water pumps can be used by requiring a minimum pump inlet density. Given the technology readiness of water pumps, the decision to use liquid cycles was taken.
The non-recuperated cycles both require significant investment in compression and heat exchanger equipment in addition to the advancement of low technology readiness level (TRL) components. Recuperated cycle layouts leveraging liquid pumping equipment like that shown in Fig. 1 were preferred because they required smaller equipment and the low TRL equipment has already been demonstrated for a 100 kWth-scale system. The recuperated Brayton cycle was chosen as the path to follow for this test system. Note that this is not an actual recuperated Brayton power conversion cycle, but shares the same flow path as this cycle with a pump instead of a compressor and no turbine. The nickel alloy recuperator is coupled with a stainless steel recuperator to reduce the quantity of nickel alloy needed. The Gen3CSP sCO2 loop includes throttle, recirculation, recuperator bypass, and pump speed control. The system was designed using the most conservative conditions expected (i.e., minimum PHX temperature rise, maximum PHX outlet temperature, and maximum PHX outlet pressure) in order to maximize margin for any given operating condition.
Fig. 1
Fig. 1
Close modal
A summary of the fluid states obtained during the analysis of the recuperated cycle is given in Table 5 and Fig. 2. The model (implemented in Engineering Equation Solver (ees)) includes actual pump performance curves, flowmeter pressure drop behavior, throttle valve Cv, and heat exchanger pressure drops assuming quadratic behavior with mass flowrate. The model assumes 2% pressure drop on PHX, 70% isentropic efficiency, and 27 °C ambient temperature.
Fig. 2
Fig. 2
Close modal
Table 5
Supercritical CO2 fluid states
StateT (°C)P (bar)ρ (kg/m3)
156.7243800
257.6259807
3423256190
4565254153
5715250127
6575248147
7147245416
StateT (°C)P (bar)ρ (kg/m3)
156.7243800
257.6259807
3423256190
4565254153
5715250127
6575248147
7147245416
An off-design study of the Gen3CSP loop was also performed in ees to understand the range of test conditions attainable through the combination of recuperator bypass and mass flow control. Intermediate combinations of temperature rise and PHX inlet temperature can be achieved by modulating the flowrate and recuperator bypass flow. A summary of this work is shown in Fig. 3. PHX inlet temperatures as low as 440 °C can be provided while still operating with 1 MWth heat transfer duty and a PHX outlet temperature (or turbine inlet temperature, TIT) of 715 °C.
Fig. 3
Fig. 3
Close modal
### 2.3 Key Component Details.
The definition of four key components was initiated for the recuperated Brayton cycle as it was expected that lead times for manufacturing would be longer. In addition, final specifications for these key components are needed to complete the design of the rest of the system. The four key components are as follows:
1. supercritical carbon dioxide circulator,
2. nickel alloy recuperator,
3. stainless steel recuperator, and
and are described in this section.
#### 2.3.1 Supercritical Carbon Dioxide Circulator.
The sCO2 circulator is the most critical piece of equipment in the Gen3CSP sCO2 coolant loop because of its high cost, long lead-time, and moderate implementation risk. sCO2 compressors with significant commercial experience are typically rated for subcritical pressures in refrigeration applications or designed to operate on low-density fluid for gas pipeline compression applications. Several activities are underway to design compressors specifically for sCO2 applications [710], but no reliable and compact commercial option yet exists.
Instead, high-pressure liquid pumps used for boiler feedwater injection and other high-pressure applications are the most reliable option to use for the sCO2 circulator. Both canned motor and magnetically coupled configurations operate without any rotating shaft seals eliminating the need for a dry gas seal support system and significantly reducing the complexity of operation and amount of leakage from the system. In addition, these configurations rely on bushings lubricated by the process fluid (sCO2) within the rotor cavity or ball bearings for the coupled motor eliminating the need for a lubricating oil support skid.
The maximum pump head rise required can be estimated from various combinations of primary heat exchanger and individual component pressure drop allowances as shown in Fig. 4. For the baseline assumptions of 1.5% PHX and 1% component pressure drops, a pump head rise of approximately 168 m H2O (550 ft H2O) minimum is required. Varying from the baseline assumption for pressure drop allowances requires balancing decreases in component pressure drop with increases in PHX pressure drop depending on achievable pump performance.
Fig. 4
Fig. 4
Close modal
The maximum required pump flowrate can be determined for a range of mass flowrates based on the pump inlet density as shown in Fig. 5. For a baseline requirement of 5.3 kg/s, a pump flowrate of 341 lpm (90 gpm) is required for 950 kg/m3 conditions. This high-density condition is difficult to maintain on hot days using only dry cooling, so it is desirable to reduce the pump inlet density to the lowest value possible. In addition, higher flowrates would potentially provide more capability for the pump, leading to a desire to increase the pump flowrate up to as much as 454 lpm (120 gpm) at low-density conditions.
Fig. 5
Fig. 5
Close modal
For a nominal operating pressure of 250 bar at the primary heat exchanger outlet, the pump outlet pressure could range from 258 bar for the baseline assumption of 1% component pressure drop and 1.5% PHX pressure drop up to 270 bar as shown in Fig. 6. The most likely peak pressure expected is 265 bar (3896 psig) for a PHX allowable pressure drop of 4% (6% pressure drop from the pump outlet to the PHX outlet).
Fig. 6
Fig. 6
Close modal
A seal-less centrifugal pump suitable for the circulation of supercritical carbon dioxide was selected to serve as the fluid circulator. A high-pressure water pump was determined to be a feasible option to use as a commercially available circulator. The pump must be capable of working with sCO2 at a specific gravity of 0.9 or lower and a viscosity of 71 µPa·s (0.071 cP) at a design point of 278 lpm (100 gpm) and a head rise of 259 m (850 ft) at 0.9 specific gravity. Requirements are based on thermodynamics analysis performed on ees.
For material compatibility, only austenitic stainless steels or nickel alloys can be used for wetted surfaces and polymer and elastomer materials should be limited to Nylon, Buna-N, urethane, and polyethylene (high-density polyethylene). Since no water will be present during operation, the bearings must be able to function with sCO2 as lubricant.
The specifications for a canned motor pump that closely meets all requirements pump are listed in Table 6. This pump has a vertical configuration (radial load expected to be minimal) and includes a radial bearing wear monitoring coil with remote indictor to track bearing wear. The pump includes a variable frequency drive (VFD) for motor control.
Table 6
Centrifugal pump technical specifications
RequirementValue
Inlet temperature/°C (°F)37.7 (100)
BEP head/m H2O (ft H2O)259 (850)
BEP flow/lpm (gpm)378 (100)
Max head/m H2O (ft H2O)287 (942)
Max flow/lpm (gpm)689 (182)
MAWP/barg (psig)282.7 (4100)
MDMT/°C (°F)93.3 (200)
Bearing materialGraphite
Weight/N (lbf)20,000 (4500)
Motor size/kW (hp)64.1 (85.9)
Current/A312
Impellor OD/mm260
Number of stages1
Min flow/lpm (gpm)170 (45)
RequirementValue
Inlet temperature/°C (°F)37.7 (100)
BEP head/m H2O (ft H2O)259 (850)
BEP flow/lpm (gpm)378 (100)
Max head/m H2O (ft H2O)287 (942)
Max flow/lpm (gpm)689 (182)
MAWP/barg (psig)282.7 (4100)
MDMT/°C (°F)93.3 (200)
Bearing materialGraphite
Weight/N (lbf)20,000 (4500)
Motor size/kW (hp)64.1 (85.9)
Current/A312
Impellor OD/mm260
Number of stages1
Min flow/lpm (gpm)170 (45)
The system curves for different pressure drops overlaid with the pump curves (vendor provided) for the selected option, and other pumps evaluated are shown in Fig. 7. The canned motor pump can provide the needed flowrate of 354 lpm (93.5 gpm) of CO2 for the 1 MW power system.
Fig. 7
Fig. 7
Close modal
#### 2.3.2 Nickel Alloy Recuperator.
The nickel alloy recuperator is a printed circuit heat exchanger (PCHE) with both hot and cold sides designed for supercritical carbon dioxide. Key requirements and related fluid properties are listed in Table 7. Note that the hot side inlet and outlet temperature corresponds to a PHX outlet temperature of 715 °C and a PHX temperature drop of 150 °C. Given the location within the system, the nickel recuperator is exposed to the highest temperature and pressure within the system as it is the component in the test system closest to the PHX and therefore must be constructed from nickel alloys to avoid excessive corrosion even in a short time as discussed later. During operation, the recuperator will be insulated to maintain a thermal efficiency of 95% or above (i.e., maximum allowed loss of 50 kWth). The overall heat transfer coefficient-area product (UA) has been approximated to be 6630 W/K.
Table 7
Key parameters of the nickel recuperator
Value
RequirementCold sideHot side
Fluid flowrate (kg/s)5.255.25
Temperature (in/out) (°C)413565715565
Pressure (in) (MPa)25.725.0
Density (in/out) (kg/m3)194153127149
Viscosity (µPa · s)34.738.642.638.5
Value
RequirementCold sideHot side
Fluid flowrate (kg/s)5.255.25
Temperature (in/out) (°C)413565715565
Pressure (in) (MPa)25.725.0
Density (in/out) (kg/m3)194153127149
Viscosity (µPa · s)34.738.642.638.5
The nickel alloys considered for construction were unified numbering system (UNS) N08810, UNS N06625, UNS N06617, UNS N06230, and UNS N07740. The material selected was UNS N06625 as a suitable diffusion bonding procedure had been developed in a previous project. Diffusion bonding procedures for the other alloys are also underway but are not expected to be completed in time for this system.
#### 2.3.3 Stainless Steel Recuperator.
The stainless steel recuperator is a PCHE with both hot and cold sides designed for supercritical carbon dioxide. Key requirements and related fluid properties are listed in Table 8. The temperature drop across the stainless steel recuperator is more dramatic since it interfaces with the nickel recuperator and the pump. Since the pump can only handle sCO2 close to density conditions near to those of water, the stainless steel recuperator must have a much higher heat transfer coefficient (approximated as UA = 26,900 W/K) to handle such different densities.
Table 8
Key parameters of the stainless steel recuperator
Value
RequirementCold sideHot side
Fluid flowrate (kg/s)5.255.25
Temperature (in/out) (°C)37413565123
Pressure (in) (MPa)26.024.8
Density (in/out) (kg/m3)900194149487
Viscosity (µPa · s)91.334.738.538.7
Value
RequirementCold sideHot side
Fluid flowrate (kg/s)5.255.25
Temperature (in/out) (°C)37413565123
Pressure (in) (MPa)26.024.8
Density (in/out) (kg/m3)900194149487
Viscosity (µPa · s)91.334.738.538.7
Also, the nickel recuperator size was to be minimized to decrease the amount of nickel alloy needed, hence reducing the cost of the recuperation subsystem. During operation, the recuperator will be insulated to maintain a thermal efficiency of 95% or above (i.e., maximum allowed loss of 150 kWth).
A 1 MWth cooler/radiator is used as the heat sink to support the full duty of the test system need. For this heat exchanger, the hot side is designed for carbon dioxide and the cold side for ambient air. Key requirements and parameters for the radiator are listed in Table 9. The temperature range needed for the radiator allows for use of stainless steel to minimize the cost of the exchanger. Finally, the calculated “UA” is approximately 27,600 W/K.
Table 9
Value
RequirementHot side (CO2)Cold side (air)
Fluid flowrate (kg/s)5.2535 (62,000 cfm)
Temperature (in/out) (°C)145533867
Pressure (in) (MPa)24.5Ambient
Density (in/out) (kg/m3)4208141.21.0
Viscosity (µPa · s)35742729
Value
RequirementHot side (CO2)Cold side (air)
Fluid flowrate (kg/s)5.2535 (62,000 cfm)
Temperature (in/out) (°C)145533867
Pressure (in) (MPa)24.5Ambient
Density (in/out) (kg/m3)4208141.21.0
Viscosity (µPa · s)35742729
### 2.4 Minor Component Details.
Other important considerations for the design of the Gen3CSP and that are described in this section.
#### 2.4.1 Inventory Management.
The Gen3CSP sCO2 loop will require an inventory control system to provide for filling, pressurization, venting, and inventory recovery. The baseline design for this system is shown in Fig. 8 to leverage commercially available two-phase CO2 dewars, ambient temperature vaporizers, CO2 compression equipment, and remote-actuated regulators and valves for automation. The dewar tanks require only intermittent filling with liquid CO2 and power for minor electrical loads and does not rely on facility compressed air or significant electrical power for vaporization or compression.
Fig. 8
Fig. 8
Close modal
#### 2.4.2 Piping.
Corrosion and high-temperature operation considerations are important to ensure the piping compatibility. Walker et al. [11] compared corrosion rates for several alloys operating in high temperature and pressure sCO2 environments across a range of studies. Based on these results from sample weight gain measurements, stainless steel alloys are expected to have low corrosion rates up to 550 °C and moderate rates up to 600 °C, but will likely have excessive corrosion rates above 600 °C. High nickel alloys, however, appear to have low corrosion rates up to at least 700 °C. Precise corrosion allowance guidance would require sample thickness reduction measurements which are not yet available for sCO2 environments, so these general expectations for corrosion were used with industrial guidance for corrosion allowances ranging from 0 to 2.54 mm (0–0.1 in.) with a value of 0 used for materials with low expected rates of corrosion, 1.27 mm for moderate rates, and 2.54 mm for high corrosion rates given the relatively short operating life of the Gen3CSP system.
The pipe strength was evaluated using Eq. (1) per ASME B31.1 requirements where P is the internal pressure, S is the maximum allowable stress, E is the weld joint efficiency factor, OD is the pipe outer diameter, t is the pipe wall thickness, y is a coefficient based on material and temperature read from tables in the B31.1 piping code, and A is the corrosion allowance. A value of 0.4 was used for the y parameter in Eq. (1) based on the expected temperatures observed for the alloys considered (nickel and austenitic steels):
$PS=2E(t−A)OD−2y(t−A)=2(1)(t−A)OD−2(0.4)(t−A)$
(1)
The pressure containment ratio more intuitively captures the sharp transition for each material from allowable stress ratings based on yield stress at lower temperatures to those based on creep limits and can be directly compared with the geometric strength ratio. Figure 9 shows P/S ratios at different temperatures based on maximum allowed stress for different materials. Most stainless steels and N06625 quickly lose strength or are not allowed above 537 or 593 °C (1000 or 1100 °F) with pressure containment ratios ranging from 0.15 to 0.4 across several materials. N07740 and N06617 are the only viable code-approved options for significantly higher temperatures with pressure containment ratios around 0.28 for N07740 and 0.48 for N06617 at 735 °C (1355 °F).
Fig. 9
Fig. 9
Close modal
Based on the information for the pressure containment ratio presented above, and the pressure containment ratio for different pipe sizes and thicknesses, Table 10 lists pipe schedule sizes that can perform at the high pressures and temperatures required for the Gen3CSP loop. N07740 and N06617 are the only viable B31.1 material options for temperatures above 593 °C (1100 °F), while numerous options are available for temperatures below 315 °C (600 °F) including most common austenitic stainless steels. For temperatures between 315 and 593 °C (600 °F and 1100 °F), N06625, N08800, and S34709 are the most viable options.
Table 10
Pipe size and schedule of different alloys for use in the Gen3CSP loop
Material/MAWP/MDMT/P/SSCHMaximum nominal pipe size
UNSbar°CA = 0 mm (A = 0 in.)A = 1.3 mm (A = 0.05 in.)A = 2.5 mm (A = 0.1 in.)
N077402807350.27Various3.5 SCH160≥6 SCHXXS4 SCHXXS
N066172807260.45XXS32.5NA
N066252805900.21Various≥6 SCH160≥6 SCH160≥6 SCHXXS
N088002805900.32XXS433
S347092805900.31XXS443
S347002803150.28XXS654
S316002803150.32XXS433
S304002803150.33XXS432.5
Material/MAWP/MDMT/P/SSCHMaximum nominal pipe size
UNSbar°CA = 0 mm (A = 0 in.)A = 1.3 mm (A = 0.05 in.)A = 2.5 mm (A = 0.1 in.)
N077402807350.27Various3.5 SCH160≥6 SCHXXS4 SCHXXS
N066172807260.45XXS32.5NA
N066252805900.21Various≥6 SCH160≥6 SCH160≥6 SCHXXS
N088002805900.32XXS433
S347092805900.31XXS443
S347002803150.28XXS654
S316002803150.32XXS433
S304002803150.33XXS432.5
Another important consideration for piping design is the minimum pipe size to limit pressure drop. Assuming a total pressure drop limitation of 0.1% and allowing the high-temperature piping greater pressure drop due to the limited selection of pipe sizes expected the pressure drop per foot of straight pipe must range from 0.005 bar/m (0.02 psi/ft) on the cold end to 0.02 bar/m (0.1 psi/ft) on the hot end of the system. This metric is trivial to meet for straight pipe even at the smallest expected pipe sizes of 2 nominal pipe size (NPS) SCHXXS, but the loss for even a small number of pipe fittings could quickly exceed the relatively low target of 0.31 bard (4.5 psid) pressure drop total in the piping for small pipe sizes.
An analysis was performed with the preliminary piping layout as shown in Fig. 10 and summarized in Table 11. This minimum sizing yields average cross-sectional flow velocities ranging from 1 to 17 m/s depending on the flow temperature which is roughly in line with conventional guidelines of 1–2 m/s for liquids and 10–30 m/s for gasses as the density of sCO2 transitions from liquid-like to gas-like over this temperature range.
Fig. 10
Fig. 10
Close modal
Table 11
Fittings and flow resistance for operation of the Gen3CSP loop
PipingAbove 593.3 °C (1100 °F)Between 593.3 °C and 315.5 °CBelow 315.5 °C (600 °F)
Elbows114
Run tees022
U-bends000
Allowable pressure drop/kPa (psi)5.5 (0.8)10 (1.5)14 (2.0)
Minimum pipe size3 NPS SCHXXS3 NPS SCHXXS3 NPS SCHXXS
Flow velocity (m/s)14–179–142–9
PipingAbove 593.3 °C (1100 °F)Between 593.3 °C and 315.5 °CBelow 315.5 °C (600 °F)
Elbows114
Run tees022
U-bends000
Allowable pressure drop/kPa (psi)5.5 (0.8)10 (1.5)14 (2.0)
Minimum pipe size3 NPS SCHXXS3 NPS SCHXXS3 NPS SCHXXS
Flow velocity (m/s)14–179–142–9
#### 2.4.3 Non-Welded Connections.
Clamp connections are used for all piping at any temperature and pressure required. Conventional components are only rated to 537.7 °C (1000 °F), but higher temperature and pressure ratings are available on request. Several different seal ring sizes from 1 to 6, equivalent to 1 NPS to 6 NPS pipe, are available up to 537.7 °C in standard 316 and 304 grades of stainless steel at both 281.7 and 298.2 barg (4085 and 4325 psig). Vendors recommend the use of high carbon grades or “H-grades” of austenitic stainless steels or nickel alloys for hubs, blinds, and clamps for process temperatures above 537.7 °C where the hubs, blinds, and clamps are fully insulated together with the process piping. In addition, silver-coated UNS N07718 or another suitable seal ring material must be used in order to aid in sealing, prevent seizing of the hub material at high temperatures, and provide sufficient seal rigidity at high temperature.
For uninsulated connections, the clamp material is generally assumed to be at a temperature below 80% of the process temperature, allowing the use of standard 304 or 316 materials up to 676.6 °C (1250 °F). In addition, clamp connections are rated under the ASME code for long-term service such that the short term operation at higher temperatures will likely not cause acute or permanent damage to the joint. The expected failure mode of clamps at higher temperatures is eventual fatigue cracking, but this has not been seen in practical service according to vendor representatives. Threaded connections were not considered because no satisfactory thread sealant material has been found for long-term service in sCO2 at elevated temperatures and threaded connections larger than 1/2; NPS are not allowed under B31.1 piping code.
Inspection ports and instrumentation feedthroughs will be required throughout the system to assess fluid conditions, inspect for corrosion, wear, and contamination, and to provide flexibility to change out connections over time. Due to the small pipe diameters used for this system, standard thermowells are often not available or cannot meet the temperature requirements of a given location and custom thermowells must be designed. Clamp or ferrule connections are utilized for inspection ports depending on their size with ferrule connections limited to temperatures below 537.7 °C (1000 °F) after accounting for thermal standoff effects. Gland-based feedthroughs are used for instrumentation where welded thermowells are not practical. Several sizes provide sufficient pressure ratings with feedthrough ports ranging from 1/8 NPS to 3/8 NPS.
### 2.5 Implementation and Modularity.
The Gen3CSP sCO2 loop is currently being designed as a modular system to provide additional layout options to the teams currently developing the rest of the CSP system and to ease in transportation once final location for the system is selected. Given the size of the components (based on initial size estimates and experience with previous similar systems) and the function they serve, the loop is split into four modules as shown in Figs. 10 and 11.
Fig. 11
Fig. 11
Close modal
## 3 Conclusion
This work summarizes the design of a 1 MWth-scale sCO2 test system to provide up to 5.3 kg/s of sCO2 flow to the primary heat exchanger of any Gen3CSP thermal storage system operating at pressures up to 250 bar and temperatures up to 715 °C. This system is critical to validate design expectations for performance, lifetime, and operability of a CSP PHX with full or near-full scale modules to de-risk their application in commercial plants. A set of high-level requirements based on conservative numbers have been established to ensure delivery of a suitable system, while the potential to accommodate various PHX temperature rises, power levels, and alternative system components has been designed in from the beginning. Finally, the use of ASME codes and standards for design lifetime and reasonable allowances for corrosion provides confidence that this system can be leveraged for future testing after the Gen3CSP program to demonstrate a complete, integrating CSP pilot facility.
## Acknowledgment
Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. SAND2020-13464 J.
## Conflict of Interest
There are no conflicts of interest.
## Funding Data
• U.S. Department of Energy Solar Energy Technologies Office under Award No. DE-EE0001697 34151.
## Nomenclature
• y =
coefficient accounting for temperature
•
• A =
corrosion allowance
•
• E =
weld joint efficiency factor
•
• P =
pressure
•
• S =
maximum allowable stress
•
• T =
pipe wall thickness
•
• OD =
outer diameter
## References
1.
Mehos
,
M.
,
Turchi
,
C.
,
Vidal
,
J.
,
Wagner
,
M.
,
Ma
,
Z.
,
Ho
,
C.
,
Kolb
,
W.
,
Andraka
,
C.
, and
Kruizenga
,
A.
,
2017
, “
Concentrating Solar Power Gen3 Demonstration Roadmap
,” NREL, Golden, CO, NREL/TP-5500-67464.
2.
Crespi
,
F.
,
Gavagnin
,
G.
,
Sánchez
,
D.
, and
Martínez
,
G. S.
,
2017
, “
Supercritical Carbon Dioxide Cycles for Power Generation: A Review
,”
Appl. Energy
,
195
, pp.
152
183
.
3.
Fang
,
L.
,
Li
,
Y.
,
Yang
,
X.
, and
Yang
,
Z.
,
2020
, “
Analyses of Thermal Performance of Solar Power Tower Station Based on a Supercritical CO2 Brayton Cycle
,”
ASME J. Energy Resour. Technol.
,
142
(
3
), p.
031301
.
4.
Carlson
,
M.
,
2019
, “
Design and Implementation of a 1-3 MWth sCO2 Support Loop for Gen3 CSP Primary Heat Exchangers
,”
Presented at the Solar Energy Technologies Office CSP Program Summit 2019
,
Oakland, CA
,
Mar. 19
.
5.
Carlson
,
M. D.
,
2019
, “
Guidelines for the Design and Operation of Supercritical Carbon Dioxide R&D Systems
,”
Proceedings of SolarPACES 2019
,
Daegu, South Korea
,
Oct. 2
.
6.
Carlson
,
M. D.
,
2018
, “
sCO2 Test Loop and Heat Transfer Facility, A 1 MWth-Scale sCO2 System for Any Gen3CSP Heat Transfer Pathway
,”
Presented at the DOE Gen 3 CSP Kickoff Meeting
,
Orlando, FL
,
June 25
.
7.
Pasch
,
J.
,
Conboy
,
T.
,
Fleming
,
D.
,
Carlson
,
M.
, and
Rochau
,
G.
,
2014
, “
Steady State Supercritical Carbon Dioxide Recompression Closed Brayton Cycle Operating Point Comparison With Predictions
,”
Proceedings of the ASME Turbo Expo 2014: Turbine Technical Conference and Exposition
,
Düsseldorf, Germany
,
June 16
, pp.
1
10
.
8.
Rapp
,
L.
, and
Stapp
,
D.
,
2019
, “
Experimental Testing of a 1MW sCO2 Turbocompressor
,”
Presented at the 3rd European sCO2 Conference
,
Paris, France
,
Sept. 20
.
9.
Moore
,
J.
,
Cich
,
S.
,
Day
,
M.
,
Allison
,
T.
,
,
J.
, and
Hofer
,
D.
,
2018
, “
Commissioning of a 1 MWe Supercritical CO2 Test Loop
,”
Presented at the 6th International Supercritical CO2 Power Cycles Symposium
,
Pittsburgh, PA
,
Mar. 27
.
10.
Allison
,
T. C.
,
Smith
,
N. R.
,
Pelton
,
R.
,
Jung
,
S.
, and
Wilkes
,
J. C.
,
2018
, “
Experimental Validation of a Wide-Range Centrifugal Compressor Stage for Supercritical CO2 Power Cycles
,”
Proceedings of ASME Turbo Expo 2018 Turbomachinery Technical Conference and Exposition
,
Oslo, Norway
, Paper No. GT2018–77026.
11.
Walker
,
M.
,
Kruizenga
,
A.
,
Weck
,
P.
, and
Withey
,
E.
,
2016
, “
Progress in Overcoming Materials Challenges With sCO2 RCBCs: Final Report
,”
Sandia National Laboratories, Albuquerque, NM, SAND 2016-9774
.
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# What if there are no Schrödinger's cats?
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1. Jan 5, 2016
### Petr Matas
Experiment description: Let us have a slight modification of the famous thought experiment, in which the cat is killed only if a particle decay is detected in a fixed time interval, say from 0 to 1 s. Let the box with the cat, the deadly device, and the radioactive sample with the detector be an isolated system, which at t = 0 s is in a pure quantum state (the initial state), which is known (although very complicated and practically impossible to determine). The quantum state of the system at t = 1000 s (the final state) is given by the time evolution operator, so it is a known pure state as well. Let us have an observable A indicating, whether the cat is alive or dead, but no measurement is going to be made.
Question: Is it true that for a majority of initial states meeting the experiment requirements (i.e. the radioactive substance amount is with high probability in the "interesting" range, etc.), the expectation value for A in the final state is very close to one of the two discrete values, getting even closer to that discrete value as time advances? Of course, it will be "alive" for some initial states and "dead" for others.
Motivation: I feel that the mutual interaction of the parts of the system will cause some sort of quantum decoherence, so no observer is needed: The isolated system will make the "almost-decision" itself. Am I right?
2. Jan 5, 2016
### StevieTNZ
At t = 1000s, the cat is going to be alive, only if there was no particle decay detected between t=0 and t=1 s.
Quantum decoherence does not turn a superposed system into one state or the other. All it is, is merely entanglement with the environment. It is often claimed due to decoherence the moon is always there, for example. But if that were the case there would be no measurement problem to resolve anymore. As far as I'm aware the measurement problem still exists.
3. Jan 5, 2016
### Petr Matas
That's what I thought, but the expectation value of an observable generally changes with time. For example, the expectation value of the position of an isolated particle moving to the right with v = 1.0 ± 0.1 km/s surely does, and no measurement is needed. The question is: Ignoring the duration of the cat's agony, does the expectation value of A change between t = 2 s and t = 3 s and how?
Maybe decoherence is not the right term to describe my idea. In a double-slit experiment, the photon incidence position probability distribution is influenced by interference. I expect the probability distribution of A at t = 1000 s to be influenced analogously, becoming 1:99 for one initial state, and 99:1 for another. Another analogy: The larger quantum computer you build, the lower the probability that it will work correctly, be the laboratory with the device an isolated system or not.
4. Jan 5, 2016
### Staff: Mentor
Whether the expectation value of an observable changes with time or not depends on whether that observable commutes with the Hamiltonian. If it does commute it will be a constant of the motion (for example, the magnitude of the angular momentum in a closed system); if it does not commute then its expectation value may change over time. Position does not commute with the Hamiltonian, and its expectation value does generally vary with time as you say.
5. Jan 5, 2016
### Staff: Mentor
The title of this thread is a bit misleading in that it asks a question for which the answer is already known: There are no Schrodinger's cats, and no one has ever seriously argued otherwise.
Schrodinger proposed his thought experiment to point out a problem in the then-current understanding of quantum mechanics. Common sense and millennia of observation say that there are no superposed dead/alive cats, so the box must contains either a dead cat or a living cat; there's no more quantum weirdness here than in having tossed a classical coin and knowing that it ended up either heads or tails. However, there was nothing in the 1920's-vintage understanding of QM that led to this outcome - on the contrary, although no one accepted the dead/alive superposition, it seemed to be an inescapable consequence of the theory.
As you suggest, decoherence goes a long ways towards addressing this problem. It doesn't solve the "measurement problem" (at least as mpst people think of it) but it does at least ensure that we will always get some sensible macroscopic outcome.
6. Jan 5, 2016
### Staff: Mentor
There's still a measurement problem even though the moon is always there and even though decoherence ensures that the cat is either dead or alive but not stuck in a superposition of the two. The problem is that although decoherence tells us that the state will evolve into a mixture (as opposed to a superposition) of dead and alive, it still doesn't tell us anything about how or why we eventually end up with exactly one of these outcomes. What happened to that unitarily evolving state to turn it into a particular definite outcome?
7. Jan 6, 2016
### Petr Matas
But in the isolated system it will be a superposition, right?
Is it possible to answer the Question from the first post?
8. Jan 6, 2016
### Petr Matas
The expectation value of A at t = 0 s is "alive". Later it is somewhere between "dead" and "alive". This shows that its expectation value may change, so it cannot commute with the Hamiltonian. But does it actually change between 2 s and 3 s?
9. Jan 6, 2016
### Staff: Mentor
But the cat is not isolated - its entangled with the radioactive nucleus.
Thanks
Bill
10. Jan 6, 2016
### Petr Matas
Yes, but I want to work with the quantum state of the entire composite system from the box to the radioactive nucleus, which is isolated by definition of the experiment. The operator A acts on the state of the composite system.
11. Jan 6, 2016
### Staff: Mentor
Then of course that evolves unitarily.
But I don't understand what such a view gains as far as understanding goes.
Thanks
Bill
12. Jan 6, 2016
### Petr Matas
1. Let us have a composite isolated system H1 × H2. At the start, both subsystems are in a pure state. Then an interaction between them occurs, so they become entangled, and H1 is in a mixed state. Then another interaction between them occurs and I thought that it is not possible to model the process precisely without taking the correlation between the subsystems into account, i.e. to keep track of the inseparable composite state.
2. I am not very familiar with mixed states.
13. Jan 6, 2016
### Staff: Mentor
14. Jan 7, 2016
### Petr Matas
Thanks for the link, I have made it to the end of section 1.2.3, so I think that I have an idea what information I can get from a mixed state. So am I right that precise simulation of an isolated system consisting of repeatedly interacting subsystems is not possible by tracking just each subsystem's density matrix?
15. Jan 7, 2016
### StevieTNZ
If I understand correctly, decoherence only occurs because we cannot fully specify all the information about the systems involved (leading to a density matrix?)? But in principle, if we specify all the information then we have a superposition.
16. Jan 7, 2016
### Petr Matas
Although I don't fully understand what @Nugatory said here, I think that I can answer: Imagine you are doing an experiment on a quantum system H1 and you want it to evolve unitarily in order to be in a superposition just before making the measurement on H1 (otherwise the quantum effects would not be observed). But the system interacts with another system H2 (the environment), so H1 will be in a mixture. Knowing the composite-system state, which is really a superposition, does not help, because you are making your measurement on H1.
I would even say that H1 being in a mixture does not matter; what matters is that its state has been altered by an unwanted interaction.
Last edited: Jan 7, 2016
17. Jan 7, 2016
### StevieTNZ
18. Jan 7, 2016
### Staff: Mentor
Of course not since those subsystems are likely entangled with other subsystems.
Thanks
Bill
19. Jan 8, 2016
### Petr Matas
Did you mean "Of course you are not right" or "Of course it is not possible"?
20. Jan 8, 2016
### Staff: Mentor
Of course the state of the sub-sytems cant tell you the state of the system because they can be entangled.
Thanks
Bill
21. Jan 8, 2016
### Petr Matas
Now it should be clear why I want to work with the quantum state of the entire composite system, so I hope that we can return to the initial question: Is it true that for a majority of (pure) initial states, the expectation value for A in the (pure) final state is very close (but not equal) to one of the two discrete values ("alive" for some initial states and "dead" for others)?
22. Jan 8, 2016
### Staff: Mentor
Its not quite clear cut.
See post 22:
Now look what happens to the composite system when the cat is actually observed. That is the issue I think really concerns you.
Once you get that we can discuss this important issue that doesn't get discussed a lot.
Thanks
Bill
23. Jan 12, 2016
### Petr Matas
Ok, so it is perfectly sufficient to make the measurement on a subsystem A consisting of a single qubit, whose base is { |ai> }, i ∈ {0, 1}. The rest of the composite system is a subsystem B with a base { |bj> }, j = 1..n.
The final (unnormalized) composite pure state is $|\psi\rangle = \sum_{i,j} c_{i,j} |a_i\rangle |b_j\rangle$. Due to (unspecified) properties of the system the coefficients $c_{i,j}$ are obtained using some linear map L from coefficients $\{ d_k \}$, so the possible final states form a subspace of the space of all $|\psi\rangle$ of the mentioned form.
The probability of obtaining result i from measurement on A in the final state is $$p(i) = { \sum_j |c_{i,j}|^2 \over \sum_{i',j} |c_{i',j}|^2 }.$$ Question from post #1 suggests that for e.g. 39% of possible final states is p(1) ≤ 0.01, for another 59% is p(1) ≥ 0.99, and for the remaining 2% is 0.01 < p(1) < 0.99. Thanks to the map L being linear, it seems that the suggestion can't be true, although I have not figured out yet how to prove it. It actually seems that for the majority of states, p(1) will be very near to some constant value in between, like 0.6, given by the radioactive sample activity.
So if my analysis is correct, the answer to the question is "no". Prior to a measurement the box really contains a Schrödinger's cat in an improper mixture of states. What do you think? Of course, after obtaining that single classical bit of information from the measurement, we know pretty much about the cat: Whether it is breathing, its heart beating, etc...
Last edited: Jan 12, 2016
24. Jan 12, 2016
### Staff: Mentor
I cant follow what you are trying to say.
My point was simple. The system is in a mixed state as explained in my link. However if you observe the system there is still 'collapse' because you are observing outside the entangled system.
I think I will have to leave it there.
Thanks
Bill
25. Jan 12, 2016
### Petr Matas
I see the collapse in this: In post #23, after the bit is obtained from the measurement on subsystem A, the subsystem B is no longer in a mixture, but in a pure state $\sum_j c_{i,j} |b_j\rangle$, where $i$ is the result of the measurement on A. The Question from post #1 is basically: What if, due to the mutual interaction of subsystems A and B, the result $i$ is known almost certainly (depending on the initial state of the system) even before the measurement is made. In post #23 I am trying to disprove the answer being "yes".
Last edited: Jan 12, 2016
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2018-07-18 20:47:32
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http://mathematica.stackexchange.com/questions/16524/catching-only-the-first-event-in-ndsolve-eventlocator/16537
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# Catching only the first event in NDSolve EventLocator
I have a system of ODEs that I solve. During the integration process, there's an event that I want to catch, but I want to (a) continue the integration after the event and (b) catch only the first one.
For example, let's say I want to calculate $\pi/2$ by this method:
flag = False;
NDSolve[{y''[t] == -y[t], y[0] == 0, y'[0] == 1}, y, {t, 0, 20},
Method -> {"EventLocator", "Event" -> ((! flag) && y'[t] == 0),
"EventAction" :> (flag = True; piHalf = t;)}]
For some reason this does not work - the events keep being triggered even though the value of flag does change.
In the end of the process I get $\pi/2=17.28$ (this might be disastrous if I'll use this value to build airplanes!) How can I solve this?
-
I think there are two misunderstandings here, both are somewhat subtle. First, giving the "Event" option as you did doesn't work because it is immediately evaluated, so that NDSolve does only see this:
flag = False;
And[! flag, y'[t] == 0]
(*
==> Derivative[1][y][t] == 0
*)
meaning it never even sees that the event should depend on flag at all. This can be circumvented by using RuleDelayed (:>) instead of Rule (->). Unfortunately the result will still not be what you expect. The reason is that y'[t]==0 will only fire if at any step y'[t]==0 evaluates to True, which will only happen by chance and/or if the step size is small enough. It is much better/more reliable to let NDSolve check for changes in sign and the combination of these changes will make your example work as I think you expected:
flag = False; piHalf =.;
NDSolve[{y''[t] == -y[t], y[0] == 0, y'[0] == 1}, y, {t, 0, 20},
Method -> {
"EventLocator",
"Event" :> And[! flag, y'[t] >= 0],
"EventAction" :> (flag = True; piHalf = t;)
}
];
piHalf
1) Nassers WhenEvent basically is equivalent to these "EventLocator" settings:
flag = False; piHalf =.;
NDSolve[{y''[t] == -y[t], y[0] == 0, y'[0] == 1}, y, {t, 0, 20},
Method -> {
"EventLocator",
"Event" -> y'[t],
"EventAction" :> (If[! flag, piHalf = t]; flag = True;)
}
];
piHalf
which is somewhat different: the "EventAction" is in this case evaluated at every sign change of y'[t], but only the first time piHalf is set. The other solution will only evaluate "EventAction" once. For the example at hand the net result (value of piHalf) is of course the same.
2) WhenEvent has the attribute HoldAll, which makes it behave more like the RuleDelayed case of the "Event" setting which solves the evaluation problem. Using a predicate in WhenEvent which is just refering to a global variable seems to cause kernel crashes, but there is a very elegant way to achieve that the event will only be triggered once without even bothering with a global variable:
NDSolve[{y''[t] == -y[t], y[0] == 0, y'[0] == 1,
WhenEvent[y'[t] == 0, piHalf = t; "RemoveEvent"]}, y, {t, 0, 20}
];
piHalf
3) if you decide to go with the "EventLocator" method you might also want to have a look at the option "Direction" which lets you discriminate between sign changes from below vs. from above.
-
@Nasser: this was not meant as critisism of your solution, the original question wasn't very percise about which of the two would be prefered, if any. I just wanted to explain in how the two solutions differ and also to justify that I wrote another answer. In fact your solution made me look closer into how to use WhenEvent and caused me to learn about "RemoveEvent" myself, so thanks for that :-). – Albert Retey Dec 18 '12 at 17:02
I do not know why EventAction is not working as you show. (could be a bug?) but I just want to give a solution using WhenEvent (new in V9) which does the same thing you are trying to do which is to assign a value to the variable piHalf only once (first time the event happens), but continue the NDSolve action.
flag = False;
sol = NDSolve[{y''[t] == -y[t], y[0] == 0, y'[0] == 1,
WhenEvent[y'[t] == 0,
If[Not[flag], flag = True; piHalf = t; Print[flag, " ", " t=", t]]
]
},
y, {t, 0, 20}
]
(* True t=1.5707963620260097
Out[4]= {{y -> InterpolatingFunction[]}} *)
piHalf
(*1.5708*)
Plot[Evaluate[y[t] /. First[sol]], {t, 0, 20},
Frame -> True,
FrameLabel -> {{y[t], None}, {t, "capturing event first time"}},
GridLines -> Automatic,
Epilog -> {Red, PointSize[Large], Point[{piHalf, 0}]}
]
Here is a screen shot of the whole process
-
Thanks @Nasser. Unforutnately, though, I don't have V9, so i can't use that solution. I found an ad-hoc workaround that I'll post soon, unless someone else gives a good solution. – yohbs Dec 18 '12 at 16:18
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2015-04-27 13:45:03
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https://socratic.org/questions/men-have-head-breadths-that-are-normally-distributed-with-a-mean-of-6-0-inches-a#510381
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# Men have head breadths that are normally distributed with a mean of 6.0 inches and a standard deviation of 1.0 inches. If one male is randomly selected, what is the probability that his head breadth is less than 6.2 inches?
## The Safeguard Helmet Company plans an initial production run of 1200 helmets. What is the probability that 100 randomly selected men have a mean head breadth less than 6.2 inches?
Nov 23, 2017
.5793 or 57.93%
.9772 or 97.72%
#### Explanation:
We want to find the probability that, in a normally distributed series, we will encounter a value that is less than the value that is .2 greater than the mean.
The z-table is great for this kind of problem.
The z-score for a head breadth of 6.2 would be .2 because the mean is 6, and the standard deviation is 1.
$\frac{\mathrm{de} v i a t i o n}{\sigma} = z$
$\frac{.2}{1} = .2$ (If the standard deviation were a number other than 1, we would need to do this to find the z-score of a deviation from the mean)
We can now look in our positive z-table (the one I've linked here is from chegg.com) to find the area of a normal curve to the left of $z = 0.20$, and we see that it is .5793, meaning that there is a 57.93% chance that you will stumble upon a man with a head breadth under 6.2 inches.
Edit: I see that this is a two-part question.
For the second part, you want to use this formula:
${\sigma}_{\overline{x}} = \frac{\sigma}{\sqrt{n}}$
with $n$ being the sample size.
$\frac{\sigma}{\sqrt{n}} \to \frac{1.0}{\sqrt{100}}$
${\sigma}_{\overline{100}} = 0.1$
We can now use the central limit theorem.
$z = \frac{\overline{x} - {\mu}_{\overline{x}}}{\sigma} _ \overline{x}$
$z = \frac{6.2 - 6.0}{0.1}$
$z = 2.00$
Going back to our z table:
$z = 2.00 \to .9772$
or a 97.72% chance.
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2022-01-22 02:57:24
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|
https://academy.vertabelo.com/course/standard-sql-functions/aggregate-functions/function-count/count-distinct-expression
|
Only this week, get the SQL Complete Track of 9 courses in a special prize of $330$89!
Introduction
Function COUNT
Function AVG
Functions SUM, MAX, MIN
Revision
## Instruction
Well done. If you didn't know, you can also use COUNT(DISTINCT ...) with an expression. Let's try it out right now.
## Exercise
Count how many distinct rates per word our translation agency uses. To obtain the rate for one word, divide the price by number of words.
Make sure not to use integer division.
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2019-04-19 08:18:35
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https://radars.ac.cn/fileup/HTML/R18061.htm
|
«上一篇
文章快速检索 高级检索
雷达学报 2018, Vol. 7 Issue (5): 557-564 DOI: 10.12000/JR18061 0
### 引用本文
Zhang Pengfei, Li Gang, Huo Chaoying, et al.. Classification of drones based on micro-doppler radar signatures using dual radar sensors[J]. Journal of Radars, 2018, 7(5): 557-564. DOI: 10.12000/JR18061.
### 文章历史
(清华大学电子系 北京 100084)
(北京环境特性研究所 北京 100854)
Classification of Drones Based on Micro-Doppler Radar Signatures Using Dual Radar Sensors
Zhang Pengfei, Li Gang, Huo Chaoying, Yin Hongcheng
(Department of Electronic Engineering, Tsinghua University, Beijing 100084, China)
(Beijing Institute of Environmental Features, Beijing 100854, China)
Foundation Item: Ministry Research Foundation, Ministry Key Laboratory Research Foundation
Abstract: Classification of drones is important due to their increasing popularity and potential threats. The micro-Doppler signatures that depend on the rotation of rotor blades facilitate the classification of drones. To enhance the robustness of micro-Doppler based classification of drones, dual radar sensing classification scheme is proposed in this paper. First, time-frequency spectrograms are obtained by performing a short-time Fourier transform on the radar data collected by two radar sensors that have similar angular diversity. Then, principal components analysis is utilized to extract the features from the time-frequency spectrograms and the features obtained by the two radar sensors are fused together. Finally, the classification results are obtained by using the support vector machine. The experimental results show that the classification accuracy obtained by the fusion of dual radar sensors is 5% higher than that obtained by only using a single radar sensor.
Key words: Micro-Doppler Drones Target classification Multi-angle and multi-band observation
1 引言
2 实验数据采集
图 1 实验场景几何示意图 Fig.1 The experimental geometry
图 2 3类无人机示意图 Fig.2 Appearance of the drones
3 基于微动特征的无人机分类
图 3 典型识别流程 Fig.3 Typical flow chart of radar recognition
图 4 特征提取与融合流程示意图 Fig.4 The flow chart of feature extraction and fusion with dual radar sensors
3.1 时频分析
图 5 3种无人机的时频图 Fig.5 Spectrograms of three types of drones
3.2 特征提取
PCA是一种基于目标统计特性的最佳正交变换。由于它能够有效地消除冗余数据并且本身的计算量小，能很好地用于实时处理，本文采用PCA进行特性提取。首先将2维时频图进行向量化操作，排列为1维向量 ${{x}}$ ，则可以得到训练样本集 ${{X}} = \{ {{{x}}_1}$ ${{x}}_2\; ·\!·\!·\; {{{x}}_n}\}$ 。利用PCA求特征向量：
图 6 实验数据的特征分布 Fig.6 Feature distribution of the experimental data
3.3 SVM分类器
4 实测数据结果分析
4.1 不同训练样本比例下的识别率分析
图 7 识别率随训练样本比例变化情况 Fig.7 Recognition accuracy under different sizes of training set
4.2 不同噪声水平下的识别率分析
图 8 识别率随加性噪声强度变化情况 Fig.8 Recognition accuracies under different levels of additive Gaussian white noise
5 结论
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2023-03-26 22:05:44
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https://gitbook.rootwhois.cn/Alogrithm/Leetcode/414.%20Third%20Maximum%20Number.html
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# 414. Third Maximum Number
## 1. Question
Given an integer array nums, return the third distinct maximum number in this array. If the third maximum does not exist, return the maximum number.
## 2. Examples
Example 1:
Input: nums = [3,2,1]
Output: 1
Explanation:
The first distinct maximum is 3.
The second distinct maximum is 2.
The third distinct maximum is 1.
Example 2:
Input: nums = [1,2]
Output: 2
Explanation:
The first distinct maximum is 2.
The second distinct maximum is 1.
The third distinct maximum does not exist, so the maximum (2) is returned instead.
Example 3:
Input: nums = [2,2,3,1]
Output: 1
Explanation:
The first distinct maximum is 3.
The second distinct maximum is 2 (both 2's are counted together since they have the same value).
The third distinct maximum is 1.
## 3. Constraints
• 1 <= nums.length <= 104
• -231 <= nums[i] <= 231 - 1
Follow up: Can you find an O(n) solution?
## 5. Solutions
class Solution {
public int thirdMax(int[] nums) {
TreeSet<Integer> set = new TreeSet<>();
for (int num : nums) {
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2022-06-27 21:39:09
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http://kalendarzebiurkowe.com/discontinued-kincaid-pfkotbu/6eaed4-the-effective-dielectric-constant-%E2%88%88r-for-a-microstrip-line
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2. Microwave Theory Tech., vol. The basic parameters used in these expres-sions are ε r-relative dielectric constant, h -height of substrate, w-mirostrip line width, Z o 6G: Fantastic, Yes. b) 1.936*108 m/s The frequency dependence of effective dielectric constant and characteristic impedance is also studied for this analysis. Therefore, the propagating mode in the microstrip line is not a pure transverse-electromagnetic (TEM) mode but, rather, a quasi-TEM. 18 March 18, 1982, pp. Microstriplines are used to convey microwave-frequency signals.. On the other hand, the relative dielectric constant of the epoxy-glass substrate was found to be dependent upon frequency. 2). The microstrip line for this type of substrates can be considered lossless at frequencies to several gigahertz. This is a natural way to implement a transmission line on a printed circuit board, and so accounts for an important and expansive range of applications. Bryant and J.A. Some structures contain only part of the electric field within the dielectric material. d) None of the mentioned Given that the transmission line signal carrying inner conductor is fully encased in the substrate, the effective dielectric constant is simply that of the substrate, and the structure is non … b) Decreases b) (∈r+1)/2 + (∈r-1)/2 The Polytechnic Higher Institute of Yefren, Libya, Parallel coupled microstrip lines are defined by the characteristic impedance and the effective permittivity of the even and the odd mode. Abstract— In this work, the effect of dielectric constant on the bandwidth of rectangular microstrip antenna (RMSA) operating at center frequency of 2 GHz is considered. e-mail: firas@ieee.org. Simple and correct formulas in regard to frequency dispersion are derived for the calculation of an isotropic effective dielectric constant for the microstrip line on m‐cut sapphire substrate. Department of Electronic Engineering, c) (∈r+1)/2 (1/√1+12d/w) Return to Calculator Index. 5 and can be rewritten as: Substituting Eq. Where εe = Effective dielectric constant εr = Dielectric constant of substrate h = Height of dielectric substrate W = Width of the patch Consider Fig 1a below, which shows a rectangular microstrip patch antenna of length L, width W resting on a substrate of height h. The co-ordinate axis is selected such that the length is along the x direction, a) True Effective dielectric constant for various values of relative permittivity. In this paper we present the effect of fringing … Formula of the Dielectric Attenuation The schematic geometry of a microstrip line is given in Figure 1 , which is constructed on a substrate with dielectric constant ɛ r and height h . According with , the reflection coefficient for a triangular taper is a function of . The relative dielectric constant of the PTFE substrate was found to be 2.68. In this case, the phase velocity of the wave is related to the velocity of light ( c 0) divided by the square root of the quasi-static effective dielectric constant [ ? The thickness of the substrate is 1.5748mm. c) TM mode View Answer, 5. IEEE Trans MTT, MTT-16 (Dec. 1988) pp 1021-27. The characteristic parameter of a microstrip line tends to change as the frequency of operation is increased. A microstrip transmission line consists of a narrow metallic trace separated from a metallic ground plane by a slab of dielectric material, as shown in Figure $$\PageIndex{1}$$. a) Conductor loss is more significant than di electric loss All the previous work for a conformal rectangular micro-strip antenna assumed that the curvature does not affect the effective dielectric constant and the extension on the length. 1). Unfortunately, high dielectric materials are … Effective Dielectric Constant ε eff): Characteristic Impedance (Z o): Ohms: Custom Cable Creator. c) It is a constant for a certain material The fringing fields are … To enhance the fringing fields from the patch, which accounts for the radiation, the width W of the patch is increased. The effective dielectric constant formula is from: M. V. Schneider, "Microstrip lines for microwave integrated circuits," Bell Syst Tech. The effective dielectric constant ∈r for a microstrip line: Weiss, "MSTRIP (Parameters of d) 209 dielectric constant ε r of the substrate and the effective dielectric constant ε eff of the microstrip are related by a filling factor that weighs the amount of the field in air and the amount of field in the dielectric substrate. When the longitudinal components of the fields for the dominant mode of a microstrip line is much smaller than the transverse components, the quasi-TEM approximation is applicable to facilitate design. This is a natural way to implement a transmission line on a printed circuit board, and so accounts for an important and expansive range of applications. Effective dielectric constant of a microstrip is given by: J., vol. The microstrip has some (usually most) of its field lines in … View Answer, 8. The use of substrate material with higher dielectric constant in microstrip patch antenna design, results degradation of antenna performance but size of the antenna reduces. Physical and effective lengths of rectangular microstrip patch. ISM/DSRC External Antennas. The closed form dispersion expression for comparing is shown below from (1) to (10). Microstrip is a type of electrical transmission line which can be fabricated with any technology where a conductor is separated from a ground plane by a dielectric layer known as the substrate. This estimation procedure was used to determine the relative dielectric constant for two types of substrate materials: Teflon-fiber glass (PTFE) and epoxy-glass boards. The Model . 1. d) TE mode There exist many diffe rent substrates that can be used for microstrip antenna design, and their dielectric constants (Ԑ r) are usually in the range of 2.2 Ԑ r 12. If the wave number of an EM wave is 301/m in air , then the propagation constant β on a micro strip line with effective di electric constant 2.8 is: c) 1 M. Kirschning and R.H. Jansen, "Accurate Model for Effective Dielectric Constant of Microstrip and Validity up in Millimeter-Wave Frequencies," Electron. Effective Dielectric Constant (ε re) and Characteristic Impedance(Z C) Odd-and Even-Mode: The characteristic impedances (Z co and Z ce) and effective dielectric constants (εo re and ε e re) are obtained from the capacitances (C o and C e): Odd-Mode: Even-Mode: Odd mode Even mode aC o and Ca e are even-and odd-mode capacitances for the coupled microstrip line configuration with air as … Microstrip line can support a pure TEM wave. The reader should be aware that microstrip is distinct from View Answer, 10. eff ( f=0)], which is a function of the relative dielectric constant ( ? d) Check the results of (a) and (b) against the simple empirical results for a wide (i.e. All the previous work for a conformal rectangular micro-strip antenna assumed that the curvature does not affect the effective dielectric constant and the extension on the length. In this context, "K" refers to the effective dielectric constant, which might also be called Epsilon_effective (Ee). The physical dimension also makes a contribution for the impedance matching. What follows is a simple and practical method for estimating the relative dielectric constant of a microstrip substrate based on well-known microstrip line empirical equations. Not So Much. 1422-1444, 1969. Closed-form expressions are necessary for the computer-aided design (CAD) and optimization of microstrip circuits. Electrostatic field analysis can be applied to derive relations for the … Strip Lines: This is a planar type of transmission line that lends itself well to microwave integrated circuitry and photolithographic fabrication. This includes the enclosed-TEM case, where Dk=Keff. c) 312 dielectric constant, ε r = 2.2 is used for this design in order to enhance the input impedance matching (inset feed) and the antenna gain. View Answer, 7. Microstrip … © 2020 Endeavor Business Media, LLC. Since it is an open structure, microstrip line has a major fabrication advantage over stripline. The dielectric material serves as a substrate and it is sandwiched between the strip conductor and the ground plane. T.G. All rights reserved. Figure 3 offers an example of a test fixture for this purpose. c) 150 b) Independent of frequency Microwave components such as antennas, couplers, filters, power dividers etc. b) Equal to the permittivity of the material Together, with the increasing number of its users is the continuation of developing the method for more applications. Firas Mohammed Ali Al-Raie This behaviour is commonly described by stating the effective dielectric constant (or effective relative permittivity) of the microstrip; this being the dielectric constant of an equivalent homogeneous medium (i.e., one resulting in the same propagation velocity). In a microstrip line, the wave-length, L, is given by: where: e eff = the effective … REFERENCES Figures 4 and 5 show dispersion characteristics and the effective dielectric constant for various values of wa with a relative permittivity εr =88. 48, pp. The examples of such structures are stripline and the coaxial cable. The custom cable creator enables wiring harness designers to develop solutions that meet exact needs. All Rights Reserved. Simple and correct formulas in regard to frequency dispersion are derived for the calculation of an isotropic effective dielectric constant for the microstrip line on m‐cut sapphire substrate. Considering stripline, research has shown that the structure is a TEM structure. b) 503.669 a) 602 The equations for the single microstrip line presented by E. Hammerstad and Ø. Jensen [] are based upon an equation for the impedance of microstrip in an homogeneous medium and an equation for the microstrip effective dielectric constant.The obtained accuracy gives errors at least less than those caused by physical tolerances and is better than for and for . Calculation of effective Dielectric constant (ε reff) Substituting ε r = 3.4, W = 42.1 mm and h = 1.5 mm we get: ε reff = 3.2 3. a) 200 Pasternack's Microstrip Calculator computes a microstrip's height/width ratio, impedance and relative dielectric constant for a microstrip transmission line. r) and thickness of the substrate material ( h), and the dimensions of the upper strip of the microstrip line ( w, t). d) Depends on the material used to make microstrip Width and Effective Dielectric Constant Equations for Design of Microstrip Transmission Lines 1. 14 can be solved using the Newton-Raphson iterative formula. Cross-sectional view of a microstrip line The height, h of the substrate is chosen to be very smaller than its wavelength. The formulas were verified by comparison to the results of full‐wave analysis based on the equivalent surface impedance approach. This value is approximately constant over a wide range of frequencies. The examples of such structures are stripline and the coaxial cable. The effective dielectric constant of a micro strip line is 2.4, then the phase velocity in the micro strip line is given by: c) Conductor loss is not significant Microstrip is a type of electrical transmission line which can be fabricated using printed circuit board technology, and is used to convey microwave-frequency signals.It consists of a conducting strip separated from a ground plane by a dielectric layer known as the substrate. The temperature dependence of the dielectric permittivity of sapphire is taken … The effective dielectric constant formula is from: M. V. Schneider, "Microstrip lines for microwave integrated circuits," Bell Syst Tech. a) Photo lithographic process d) 2.43 The substrate parameters (ε r and h) and the frequency of interest are … With an increase in the operating frequency of a micro strip line, the effective di electric constant of a micro strip line: These parameters can be obtained from the effective dielectric constant and the characteristic impedance of the corresponding air line. View Answer, 9. A closed-form expression is presented for the effective dielectric constant of single microstrip lines which is valid with high accuracy up to mm-wave frequencies. Step 2: Calculation of effective dielectric constant. b) Di electric loss is more significant than conductor loss These equations can be applied to a number of different substrate materials to estimate the relative dielectric constant of circuit boards with microstrip transmission lines. The effect of curvature on the resonant frequency has been presented previously[9]. The effective dielectric constant satisfies the relation and it is dependent on the substrate thickness h, and conductor width W. ε e can be interpreted as the dielectric constant of a homogeneous medium that replaces the air and dielectric regions of the microstrip, as shown. Important electrical parameters for microstrip line design are the characteristic impedance, Zo, the guide wavelength, λg, and the attenuation constant, ?. Active, expires 2024-04-16 Application number US10/789,931 Other versions US20050190587A1 (en Inventor Roy Greeff Current Assignee (The listed … εeff is the calculated effective dielectric constant of the microstrip line due to the nonhomogeneous nature of the structure (i.e., the structure is made up of two dielectric materials: air and the substrate material). Department Head View Answer, 4. dielectric constant as ε r. Fig. T.G. c) Microstrip supports only TM mode In reality, the thickness of the strip has some bearing on the electrical performance, and the strip thickness, t, affects the microstrip characteristics. Note that … Trivedi, "A Designer's Guide to Microstrip Line" MicroWaves, May 1977, pp.174-182. J., vol. The value of ere is significantly lower than that of er when the fields external to the substrate are taken into account. Dielectric Constant (ε r): Dielectric Height (h): Frequency: GHz: Electrical Parameters : Zo: Ω : Elec. Kirschning and Jansen. c) 3*108 m/s a) Calculate Z o for the line. 4) (11. MTT-25, pp. For coupled lines, the effective dielectric constants for both even and odd modes are given. This free software is available for use or download on our website at http://www.rogerscorporation.com/acm. 1422-1444, 1969. ISM/DSRC external antennas offer high … 10 and equating Eqs. In a microstrip line, the electro- magnetic (EM) fields exist partly in the air above the dielectric substrate and partly within the substrate itself. It was also found that losses in the epoxy-glass board make it impractical for applications above 500 MHz. 1. The effective dielectric constant can be calculated using the formulas above, but the formula to be used depends on the the ratio of the width to the height of the microstrip line (W/H), as well as the dielectric constant of the substrate material. The microstrip line is a transmis-sion-line geometry with a single con-ductor trace on one side of a dielectric substrate and a single ground plane on the opposite side. For microstrip, as the frequency is increased, more and more of the field is confined to the substrate, and the effective dielectric constant increases. The Model . In this first of a series of three media galleries, we present the Top Products of 2020 as chosen by you, our faithful audience. However, for a thickness- to-height ratio (t/h) less than or equal to 0.005, the agreement between experimental and theoretical results obtained by assuming t/h = 0 is acceptable.2. 2. K effective takes into account the geometry of a transmission line, which makes it less than Dk of the substrate in many cases. This relation clearly shows that the effective permittivity is a function of various parameters of a microstrip line, the relative permittivity, effective width and the thickness of the substrate. $$\epsilon_{R}$$ = dielectric constant $$\epsilon_{eff} = \frac{\epsilon_{R}+1}{2} + \frac{\epsilon_{R}-1}{2}\left[\frac{1}{\sqrt{1+12(\frac{h}{W})}}\right]$$ Applications. The model is simulated in the Time Domain Solver in the frequency range of 5 – 15 GHz. Figure 1 shows a cutaway view of a typical microstrip circuit with the three dimensions and the relative dielectric constant (e r). Effective Dielectric Constant: Impedance: A Microstrip is a type of electrical transmission line used to transmit RF signals and are commonly fabricated using printed circuit board (PCB) technology. The effect of curvature on the resonant frequency has been presented previously[9]. The ground plane and the substrate are assumed infinite in width. The reader should be aware that microstrip is distinct from d) 200 Hammerstad and Jensen. 48, pp. 1422-1444, 1969. The more complex formulas for effective dielectric constant, characteristic impedance, surface resistivity of conductor and attenuation on the line are given in [1]. J., vol. While 2020 may not make anyone's top-ten list of years, here's some of the year's Top Products, chosen by our audience. Intuitively, the effective dielectric constant of the line is expected to be greater than the dielectric constant Λ=λε/ () (1)eff 0.5 Effective Dielectric Constant CHAPTER 4 4.8 Effective Dielectric Constant If the material between the signal trace and the return path is “completely” filled by a dielectric material, all the electric field is contained within the dielectric material. MTT-25, pp. 272-273. Measurement of dielectric constant using a microstrip ring resonator Abstract: An approach for measuring the permittivity of dielectric materials by means of a microstrip ring resonator is presented. b) Calculate the effective dielectric constant, phase velocity (in fractions of c, the speed of light), and the wavelength at 10GHz. 14 can be formulated as: A computer program can be written to minimize f (er) according to the algorithm: and where ere is substituted from Eq. 631-647, Aug. 1977. Dielectric Microstrip Lines Achieve Low Loss at THz Frequencies. The results discussed above assume a negligible strip thickness. Participate in the Sanfoundry Certification contest to get free Certificate of Merit. b) 211 The effective dielectric constant equation is from: M. V. Schneider, "Microstrip lines for microwave integrated circuits," Bell Syst Tech. c) Cannot be predicted The classic microstrip-line equations found in the literature were formulated as a root finder algorithm to search for er. View Answer, 11. Relative Dielectric Constant (ε r ): Track Width: mm: Track Thickness: mm: Dielectric Thickness: mm: Output. c = the speed of light and e re = the effective dielectric constant of the microstrip line. Bookmark or "Favorite" this microstrip line impedance calculator page by pressing CTRL + D. © 2011-2020 Sanfoundry. Cylindrical-Rectangular Patch Antenna . Fantasy? These controlled-impedance lines are typically designed with a characteristic impedance of 50 Ω for RF/microwave circuits or 75 Ω for cable television (CATV). Length: deg : Physical Parameters : Width (W): Length (L): Description. View Answer, 12. Microstrip Line Calculator. The phase velocity in the microstrip line is given by: c = the speed of light and ere = the effective dielectric constant of the microstrip line. The more complex formulas for effective dielectric constant, characteristic impedance, surface resistivity of conductor and attenuation on the line are given in [1]. The geometry of a microstrip line is shown in the figure below. Effective dielectric constant of a microstrip is given by: TWT is preferred to multicavity klystron amplifier because it is Microwave tubes are grouped into two categories depending on the type of: The klystron tube used in a klystron amplifier is a _____ type beam amplifier. I.J. The microstrip calculator determines the width and length of a microstrip line for a given characteristic impedance (Zo) and electrical length or vice versa. The wave number in air for EM wave propagating on a micro strip line operating at 10GHz is given by: This paper gives the exact design data for all line parameters for the most important cases. If the substrate thickness, h, of the circuit board is known, then the relative dielectric constant of the board can be determined using a simple experiment. Bryant and J.A.Weiss, "Parameters of Microstrip Transmission Lines and Coupled Pairs of Microstrip Lines". 48, pp. ESTIMATION AND CHARACTERISTIC OF THE EFFECTIVE DIELECTRIC CONSTANT OF MICROSTRIP LINE USING QUASI STATIC ANALYSIS Surya Kumar Pandey1 and Narad Prasad2 1Professor, Department of Physics, *J.P.University, Chapra 2Research Scholar, Department of Physics, *J.P.University, Chapra E-mail : uk7181@gmail.com Abstract- In this paper we propose the estimation of the characteristic … Whitepaper: 5G mmWave (NR) Systems and Noise Consideration. The main formula is attributable to Harold A. Wheeler and was published in, "Transmission-line properties of a strip on a dielectric sheet on a plane", IEEE Tran. Characteristic impedance and effective dielectric constant. Fringing makes the microstrip line look wider electrically compared to its physical dimensions.Since some of the waves travel in the substrate and some in air, an effective dielectric constant is introduced, given as: eq. 5) with (11. This calculator is provided free by Chemandy Electronics in order to promote the FLEXI-BOX. a) 2.6 Sanfoundry Global Education & Learning Series – Microwave Engineering. Once the effective dielectric constant is calculated, the guided wavelength through the microstrip can be calculated by dividing the free space wavelength (which is … The mode of propagation in a microstrip line is: The electromagnetic (EM) field lines in the microstrip line are not contained entirely in the substrate but also propagate outside of the microstrip (Fig. One of the techniques to … Dr. Talal Skaik 2012 IUG 6 Microstrip 2: b) Electrochemical process (11. 48, pp. Highly accurate and computationally efficient equations are presented for the characteristic impedance and the effective dielectric constant of both single and coupled microstrip with a top cover at a non-infinite distance above the substrate. G. Gonzalez, Microwave Transistor Amplifiers Analysis and Design, 2nd ed., Prentice-Hall, Upper Saddle River, NJ, 1997, Chapter 2. The effective dielectric constant equation is from: M. V. Schneider, "Microstrip lines for microwave integrated circuits," Bell Syst Tech. For most of the micro strip substrates: Bahl and D.K. of the microstrip line is well described by its effective dielectric constant, ε eff and its characteristic impedance, Zₒ expressed mathematically as [3, 7]: 0.053 0.9 0.564 1110 3 1 2 2 ( / ) r r a rr eff Wh (1) and, 2 2 ln 1 o 2 // eff F Z W h W h (2) The terms, a and F are computed using the width-height ratio, W/h of the microstrip laminate. 1. d) Depends on the material of the substrate used as the microstrip line Transmission line model is used in this design with an RT DUROID 5888 dielectric substrate such that its dielectric constant values are varied from 2.0 to 3.0 with 0.2 step size. J., vol. The main equation is attributable to Harold A. Wheeler and was published in, "Transmission-line properties of a strip on a dielectric sheet on a plane", IEEE Tran. Cylindrical-Rectangular Patch Antenna . with dielectric constant "r and height h. The strip width is w, and its thickness t is assumed much thinner than the strip width w but still thicker than the skin depth d c in the usual RF range of interest. The ground plane and the substrate are assumed infinite in width. For a microstrip line with W/h ≥ 1, Zo is given by Eq. A microstrip transmission line consists of a narrow metallic trace separated from a metallic ground plane by a slab of dielectric material, as shown in Figure $$\PageIndex{1}$$. Physical and effective lengths of rectangular microstrip patch. E. Hammerstad and O. Jensen, "Accurate Models for Microstrip Computer-aided Design," MTT Symposium Digest, 1980. As in a microstrip transmission line, the electric field is concentrated between the strip and the ground plane and a weak fringing field exists beyond the dielectric substrate. 2: Available numerical methods for the characterization of microstrip lines involve extensive computations. Figure 4 shows the variation of er with frequency for this type of substrate. View Answer, 3. The ground plane and the substrate are assumed infinite in … The y- and S-parameters are depicted in section 9.22. The formula given has been designed for use in MIC and MMIC CAD programs as well as in the indirect measurement of substrate permittivities by resonance techniques. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.) The method is used in conjunction with the variational calculation of the line capacitance of a multilayer microstriplike transmission line to compute the effective permittivity and hence the resonant … We report the impedance and attenuation measurements performed on microstrips. But in looking back over 2020's crop of new products of interest to our audience, we found quite a few bright spots. Fig1: Geometry of inset-fed microstrip patch antenna Table 1: Physical dimensions of inset-fed microstrip patch antenna Operating Frequency, f in GHz 2.4 Dielectric Constant, ε r 4.4 Length of Patch, L p in cm 2.9 Width of Patch, W p in cm 3.839 Inset Distance, d in cm 0.9768 Width of microstrip feed line, W in cm 0.301 effective dielectric constant of a microstrip line of width W. The value of εe is slightly less than the dielectric constant εr of the substrate because the fringing fields from the patch to the ground plane are not confined in the dielectric body, but are also spread in the air. Meet exact needs for er W/h ≥ 1, Zo is given by Eq and loss! Interest to our audience, we found quite a few bright spots for the algorithm to search for.. Propagation velocity is c/SQRT ( Keff ) are depicted in section 9.22 with, the relative constant... Determined by the characteristic impedance ( or reactance ) of the corresponding air line May 1977, pp.174-182 the plane. Physical tolerances and is better than for and for line the effective dielectric constant ∈r for a microstrip line 0.475 0.67 reff r 87 5.98 ln 0.8 o. Ground plane and the coaxial cable account the geometry of a strip conductor a! The height, h of the inherent advantages lines, the propagating mode in the epoxy-glass make. ) False c ) microstrip supports only TM mode d ) Check the of... The simple empirical results for a triangular taper is a two-port microstrip transmission constructed... Mcqs ) focuses on “ Micro-Strip lines ” for all line parameters for the impedance and the characteristic impedance relative. W of the line should be recorded for each sample frequency strip and! At frequencies to several gigahertz and through computational simulations single microstrip lines for microwave circuitry..., power dividers etc K effective takes into account the geometry of a microstrip transmission line model the! A number of methods have been suggested 15 GHz than for and.... Stripline, research has shown that the size and bandwidth of a test fixture for this.. Lines consist of a square or rectangular microstrip patch antenna designers to develop solutions that exact... Answers ( MCQs ) focuses on “ Micro-Strip lines ” in CST Studio Suite 1021-27! The method for more applications 5 and can be solved using the Newton-Raphson iterative formula 0.475 reff. Participate in the frequency range of 5 – 15 GHz frequency of operation determined. Found in the sanfoundry Certification contest to get free Certificate of Merit lines Low. Extensive computations that of e re is significantly lower than that of er close to the wavelength... microstrip lines '' accuracy of the PTFE substrate was found to be 2.68 its! Which might also be called Epsilon_effective ( Ee the effective dielectric constant ∈r for a microstrip line TEM ) mode but rather! Cable Creator enables wiring harness designers to develop solutions that meet exact needs microstrip 's height/width ratio, and. Constant as ε r. Fig equations found in the sanfoundry Certification contest to free..., λ, as shown by Eq er when the fields external to the free-space,... Epoxy-Glass substrate was found to be 2.68 conductor loss at the analyzed/synthesized length by physical tolerances and is better for. C ) microstrip supports only TE mode view Answer, 4 effect of curvature the! Εr =88 and Answers from: M. V. Schneider, K refers! Of developing the method for more applications on the equivalent surface impedance approach Schneider. Microstrip transmission lines and coupled Pairs of microstrip lines which is a TEM structure strip! Reflection coefficient for a microstrip line has a major fabrication advantage over stripline are depicted in 9.22!
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2021-07-24 13:47:37
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https://electronics.stackexchange.com/questions/410173/which-region-is-the-transistor-operating-at
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# Which region is the transistor operating at?
PNP and NPN transistors in the circuit have identical Is values. Based on the PNP transistor is connected to a DC voltage source.
• k = 1.38064852×10-23 J/K
• qe = 1.60217662 ×10-19 C
• Is = 0.69 fA at 17°C
• VA = ?
• β = 100 A/A (in Forward Active and Soft Saturation Regions)
1. Determine VOUT and IC when Vx=0.8V. Which region is the transistor operating at?
2. Determine VOUT and IC when Vx=0.75 V. Which region is the transistor operating at?
3. Determine VOUT and IC when Vx=0.85 V. Which region is the transistor operating at?
Solution:
VT = kT/q, so VT = 25 mV
In PNP, IC = Is eVEB/VT, but how to determine VEB?
You have two active devices, so you're going to have to solve a system of simultaneous equations to resolve the unknowns.
It will probably help if you start by replacing the base bias network of Q1 with its Thévenin equivalent.
There's no requirement to solve simultaneous equations for this problem. It's not so complex as that.
The first step, though, is to Thevenize your resistor pair at the base of $$\Q_1\$$. This will be $$\V_\text{TH}\approx 2.207\:\text{V}\$$ and $$\R_\text{TH}\approx 728\:\Omega\$$. This means things will be saturated if the collector of $$\Q_1\$$ goes below about $$\2.2\:\text{V}\$$. Which means you don't have to worry about any collector current exceeding about $$\1\:\text{mA}\$$. This implies that the base current (for active mode) will be at or under $$\\approx 10\:\mu\text{A} \$$. The drop across $$\R_\text{TH}\$$ is therefore, again in active mode, around $$\7\:\text{mV}\$$ or less. So, given these details we can assume that when in active mode the base voltage will in all such cases be $$\\approx 2.2\:\text{V}\$$.
Now that you know the active mode base voltage, and since you know a-priori that because both saturation currents and beta values are the same in the two BJTs, it follows that $$\V_\text{BE}\$$ for each BJT is one-half of the difference between $$\\approx 2.2\:\text{V}\$$ and $$\V_X\$$, or $$\V_\text{BE}=\frac{2.2\:\text{V}-V_X}{2}\$$. Given your three values to test, this means $$\V_\text{BE}=\left\{725\:\text{mV}, 700\:\text{mV}, 675\:\text{mV}\right\}\$$ for $$\V_X=\left\{750\:\text{mV}, 800\:\text{mV}, 850\:\text{mV}\right\}\$$.
Keep clearly in mind, now, that we are talking only about the case where $$\Q_1\$$ is in active mode. This simply means that the collector voltage of $$\Q_1\$$ doesn't precede below the base voltage of $$\\approx 2.2\:\text{V}\$$ (where $$\I_C\le\left(\frac{3.3\:\text{V}-2.2\:\text{V}}{1\:\text{k}\Omega}=1.1\:\text{mA}\right)\$$.
From here, it's pretty easy to work out. You know the maximum value of $$\I_C\$$ when $$\Q_1\$$ is in active mode. So you know the worst-case values for $$\V_\text{BE}\$$: $$\V_\text{BE}\le \left[V_T\cdot\operatorname{ln}\left(\frac{I_C}{I_\text{SAT}}+1\right)\approx 702.435\:\text{mV}\right]\$$.
Therefore, it is immediately apparent which cases are active mode and which are not.
I've left some details for you to worry about ($$\V_\text{OUT}\$$ and $$\I_C\$$.)
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2019-08-24 11:46:48
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https://socratic.org/questions/how-many-atoms-are-in-each-element-of-6ag-2so-4
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# How many atoms of each type are in SIX formula units of Ag_2SO_4?
May 31, 2018
Silver - Ag = 12 atoms
Sulfur - S = 6 atoms
Oxygen - O = 24 atoms
#### Explanation:
To find the number of atoms in the molecules represented by $6 A {g}_{2} S {O}_{4}$
We begin by determining the atoms in one molecule of $A {g}_{2} S {O}_{4}$
We look at the subscripts of the molecule and determine that there are
2 atoms of Ag - Silver
1 atom of S - Sulfur
and
4 atoms of O - Oxygen
Since the original situation is $6 A {g}_{2} S {O}_{4}$
We know that the coefficient $6$ tells us that there are 6 molecules of Silver Sulfate $\left(A {g}_{2} S {O}_{4}\right)$
So we must multiply the single molecule values by 6
$2 \cdot 6 = 12$ atoms of Ag - Silver
$1 \cdot 6 = 6$ atoms of S - Sulfur
and
$4 \cdot 6 = 24$ atoms of O - Oxygen
()
SMARTERTEACHER YouTube "Chemistry it is all that MATTERS
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2020-05-28 18:44:22
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https://www.quizover.com/online/course/5-6-conservation-of-energy-work-and-energy-by-openstax?page=2
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# 5.6 Conservation of energy (Page 3/10)
Page 3 / 10
Another example of energy conversion occurs in a solar cell. Sunlight impinging on a solar cell (see [link] ) produces electricity, which in turn can be used to run an electric motor. Energy is converted from the primary source of solar energy into electrical energy and then into mechanical energy.
Energy of various objects and phenomena
Object/phenomenon Energy in joules
Big Bang ${\text{10}}^{\text{68}}$
Energy released in a supernova ${\text{10}}^{\text{44}}$
Fusion of all the hydrogen in Earth’s oceans ${\text{10}}^{\text{34}}$
Annual world energy use $4×{\text{10}}^{\text{20}}$
Large fusion bomb (9 megaton) $3\text{.}8×{\text{10}}^{\text{16}}$
1 kg hydrogen (fusion to helium) $6\text{.}4×{\text{10}}^{\text{14}}$
1 kg uranium (nuclear fission) $8\text{.}0×{\text{10}}^{\text{13}}$
Hiroshima-size fission bomb (10 kiloton) $4\text{.}2×{\text{10}}^{\text{13}}$
90,000-ton aircraft carrier at 30 knots $1\text{.}1×{\text{10}}^{\text{10}}$
1 barrel crude oil $5\text{.}9×{\text{10}}^{9}$
1 ton TNT $4\text{.}2×{\text{10}}^{9}$
1 gallon of gasoline $1\text{.}2×{\text{10}}^{8}$
Daily home electricity use (developed countries) $7×{\text{10}}^{7}$
Daily adult food intake (recommended) $1\text{.}2×{\text{10}}^{7}$
1000-kg car at 90 km/h $3\text{.}1×{\text{10}}^{5}$
1 g fat (9.3 kcal) $3\text{.}9×{\text{10}}^{4}$
ATP hydrolysis reaction $3\text{.}2×{\text{10}}^{4}$
1 g carbohydrate (4.1 kcal) $1\text{.}7×{\text{10}}^{4}$
1 g protein (4.1 kcal) $1\text{.}7×{\text{10}}^{4}$
Tennis ball at 100 km/h $\text{22}$
Mosquito $\left({10}^{–2}\phantom{\rule{0.25em}{0ex}}g at 0.5 m/s\right)$ $1\text{.}3×{\text{10}}^{-6}$
Single electron in a TV tube beam $4\text{.}0×{\text{10}}^{-\text{15}}$
Energy to break one DNA strand ${\text{10}}^{-\text{19}}$
## Efficiency
Even though energy is conserved in an energy conversion process, the output of useful energy or work will be less than the energy input. The efficiency $\text{Eff}$ of an energy conversion process is defined as
$\text{Efficiency}\left(\text{Eff}\right)=\frac{\text{useful energy or work output}}{\text{total energy input}}=\frac{{W}_{\text{out}}}{{E}_{\text{in}}}\text{.}$
[link] lists some efficiencies of mechanical devices and human activities. In a coal-fired power plant, for example, about 40% of the chemical energy in the coal becomes useful electrical energy. The other 60% transforms into other (perhaps less useful) energy forms, such as thermal energy, which is then released to the environment through combustion gases and cooling towers.
Efficiency of the human body and mechanical devices
Activity/device Efficiency (%) Representative values
Cycling and climbing 20
Swimming, surface 2
Swimming, submerged 4
Shoveling 3
Weightlifting 9
Steam engine 17
Gasoline engine 30
Diesel engine 35
Nuclear power plant 35
Coal power plant 42
Electric motor 98
Compact fluorescent light 20
Gas heater (residential) 90
Solar cell 10
## Phet explorations: masses and springs
A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energies for each spring.
## Section summary
• The law of conservation of energy states that the total energy is constant in any process. Energy may change in form or be transferred from one system to another, but the total remains the same.
• When all forms of energy are considered, conservation of energy is written in equation form as ${\text{KE}}_{i}+{\text{PE}}_{i}+{W}_{\text{nc}}+{\text{OE}}_{i}={\text{KE}}_{f}+{\text{PE}}_{f}+{\text{OE}}_{f}$ , where $\text{OE}$ is all other forms of energy besides mechanical energy.
• Commonly encountered forms of energy include electric energy, chemical energy, radiant energy, nuclear energy, and thermal energy.
• Energy is often utilized to do work, but it is not possible to convert all the energy of a system to work.
• The efficiency $\text{Eff}$ of a machine or human is defined to be $\text{Eff}=\frac{{W}_{\text{out}}}{{E}_{\text{in}}}$ , where ${W}_{\text{out}}$ is useful work output and ${E}_{\text{in}}$ is the energy consumed.
## Conceptual questions
Consider the following scenario. A car for which friction is not negligible accelerates from rest down a hill, running out of gasoline after a short distance. The driver lets the car coast farther down the hill, then up and over a small crest. He then coasts down that hill into a gas station, where he brakes to a stop and fills the tank with gasoline. Identify the forms of energy the car has, and how they are changed and transferred in this series of events. (See [link] .)
Describe the energy transfers and transformations for a javelin, starting from the point at which an athlete picks up the javelin and ending when the javelin is stuck into the ground after being thrown.
Do devices with efficiencies of less than one violate the law of conservation of energy? Explain.
List four different forms or types of energy. Give one example of a conversion from each of these forms to another form.
List the energy conversions that occur when riding a bicycle.
## Problems&Exercises
Using values from [link] , how many DNA molecules could be broken by the energy carried by a single electron in the beam of an old-fashioned TV tube? (These electrons were not dangerous in themselves, but they did create dangerous x rays. Later model tube TVs had shielding that absorbed x rays before they escaped and exposed viewers.)
Using energy considerations and assuming negligible air resistance, show that a rock thrown from a bridge 20.0 m above water with an initial speed of 15.0 m/s strikes the water with a speed of 24.8 m/s independent of the direction thrown.
Equating ${\text{ΔPE}}_{g}$ and $\text{ΔKE}$ , we obtain $v=\sqrt{2\text{gh}+{{v}_{0}}^{2}}=\sqrt{2\left(\text{9.80 m}{\text{/s}}^{2}\right)\left(\text{20.0 m}\right)+\left(\text{15.0 m/s}{\right)}^{2}}=\text{24.8 m/s}$
If the energy in fusion bombs were used to supply the energy needs of the world, how many of the 9-megaton variety would be needed for a year’s supply of energy (using data from [link] )? This is not as far-fetched as it may sound—there are thousands of nuclear bombs, and their energy can be trapped in underground explosions and converted to electricity, as natural geothermal energy is.
(a) Use of hydrogen fusion to supply energy is a dream that may be realized in the next century. Fusion would be a relatively clean and almost limitless supply of energy, as can be seen from [link] . To illustrate this, calculate how many years the present energy needs of the world could be supplied by one millionth of the oceans’ hydrogen fusion energy. (b) How does this time compare with historically significant events, such as the duration of stable economic systems?
(a) $\text{25}×{\text{10}}^{6}\phantom{\rule{0.25em}{0ex}}\text{years}$
(b) This is much, much longer than human time scales.
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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2018-05-21 16:52:49
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https://cs.stackexchange.com/questions/142560/prove-that-fn-is-omegagn-but-not-ogn
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# Prove that $f(n)$ is $= \Omega(g(n))$ but not $= O(g(n))$
I am trying to prove the following statement.
if $$\displaystyle \lim_{n\rightarrow\infty}\frac{f(n)}{g(n)}= \infty$$, then $$f(n) = \Omega(g(n))$$ but $$f(n) \neq O(g(n))$$
## What I've done so far
Using the definition of limit: $$\forall M>0 \quad \exists n_0 : \forall n \ge n_0 \quad \displaystyle\frac{f(n)}{g(n)} > M$$
So I multiplied both halves by $$g(n)$$ obtaining $$f(n) > M\cdot g(n)$$
I therefore chose a value $$c_1$$ that respect the condition $$0 \le M \le c_1$$ in order to define the relation $$0 \le c_1g(n)\le f(n)$$ that proves that $$f(n) = \Omega(g(n))$$
So now my questions are:
• Is the (first part of the) solution mathematically right?
• How can I mathematically prove that $$f(n) \neq O(g(n))$$?
I'm struggling a bit to prove the second part of the demonstration.
The main idea I had is to do a reductio ad absurdum where I state that the value $$c_2$$ I get (following the definition of $$O(g(n))$$) leads me to a contradiction. But I'm not entirely sure it makes sense.
Your basic idea is good. Here is how you would approach it:
Let us assume towards contradiction that $$\exists c_2,n_0:\forall n>n_0:f(n)\le c_2\cdot g(n)$$
In particular, we can re-write this as:
$$\exists c_2,n_0:\forall n>n_0:\frac{f(n)}{g(n)}\le c_2$$
Now, since $$\lim_{n\rightarrow \infty}\frac{f(n)}{g(n)}=\infty$$, then we know by definition of the limit that $$\forall M>0:\exists n_0:\forall n>n_0:\frac{f(n)}{g(n)}>M$$
In particular, for any $$c_2$$ and $$n_0$$, there is some $$n>n_0$$ with $$f(n)>c_2\cdot g(n)$$. Hence, we get a contradiction to the assumption.
• I wrote almost the same proof as you wrote above. I wasn't sure if that made sense, but now know for sure! Thank you very much! Jul 22 at 19:48
• @LukeTheWolf this proof is directly following the definition of big-O and the limit definition that you see in calculus classes. It is very formal, but not always as understandable :o Jul 22 at 20:00
• Unfortunately, I know it is very formal, but I was asked to make a formal demonstration. Perhaps can you give me a few more tips on how to make that demonstration easier, but just as effective? Jul 22 at 20:04
• @LukeTheWolf sadly, I do not know of another way, except for using the big-O definition using limits. In that definition, $f$ is said to be $O(g)$ if $\limsup_{n\rightarrow \infty} \frac{f(n)}{g(n)}<\infty$ and as you can see, it is exactly what you asked in the question (and hence not an interesting solution since it just assumes the answer) Jul 22 at 20:27
Suppose $$f(n)=n^4,g(n)=n$$, so
$$\lim_{n\to\infty}\frac{f(n)}{g(n)}=\frac{n^4}{n^3}=\frac{n^3}{1}=\infty.$$
That means $$f(n)=\Omega(g(n))$$ and $$f(n)\neq\mathcal{O}(g(n)).$$ As a result, if $$f(n)=\omega(g(n))$$, then we can conclude that, $$f(n)=\Omega(g(n))$$ and $$f(n)\neq \mathcal{O}(g(n)).$$
Note that in a such case that
$$\lim_{n\to\infty}\frac{f(n)}{g(n)}=\infty$$ easily we can conclude that $$f(n)=\omega(g(n))$$.
• Thanks for the reply, but I would like to find the mathematical proof that determines that $f(n) \neq O(g(n))$, could you help me please? Jul 22 at 17:09
• You can take $f(n)=\omega(g(n))$, as a result, $f(n)=\Omega(g(n))$ and $f(n)\neq \mathcal{O}(g(n))$
– Jut
Jul 22 at 17:10
• Ok, thanks for that! Jul 22 at 17:17
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2021-09-19 05:52:49
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https://www.physicsforums.com/threads/function-problem.185005/
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# Function problem
1. Sep 16, 2007
### starchild75
1. The problem statement, all variables and given/known data
A norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window.
2. Relevant equations
2l +2w=30
3. The attempt at a solution
If this were simply a rectangle,
I can solve for l in terms of w.
l=15-w
A=w(15-w)
but I don't know how to handle the semicircle part.
1. The problem statement, all variables and given/known data
2. Relevant equations
(pir^2)/2
3. The attempt at a solution
Last edited: Sep 16, 2007
2. Sep 16, 2007
### neutrino
First, 2l+2w is NOT 30 ft. The perimeter includes the curved (semi-circle) part, so the perimeter is the sum of the lengths of the two sides, the base (the width 'x'), and the semi-circle.
3. Sep 16, 2007
### starchild75
So how do I convert that into function form?
4. Sep 16, 2007
### neutrino
Just add the quantities I stated.
Length = l, width = x, perimeter of semi-circle = ?
5. Sep 16, 2007
### starchild75
But the perimeter of the entire window is 30 ft. The length and width of the rectangle/semicircle are unknown.
6. Sep 16, 2007
### starchild75
OK, here's what I came up with.
A(w)=w(30-w-pir^2)/2
7. Sep 16, 2007
### symbolipoint
You are asked to find a function; not to compute a value. A value of width or length is not needed.
area= L*2*r + 0.5*p*r^2;
perimeter= 2*r + 2*L + p*r = 30
You only have two variables there. Any found value of r will determine the corresponding value for L (the other dimension of the rectangle).. There is enough information to find the function of area based on ONE variable, the radius, r.
I used 2*r as the width, so you may want to rewrite some of the above shown equations based on "x" instead of 2*r.
8. Sep 16, 2007
### starchild75
So in solving for r, I get (30-2l-ps)/2
where ps is perimeter of semicircle.
But I want that in r in terms of l right?
9. Sep 16, 2007
### starchild75
putting the equation in terms of width, I got w(30-r-Ps)
P being perimeter of the semicircle.
How does that look?
Last edited: Sep 16, 2007
10. Sep 16, 2007
### symbolipoint
Maybe changing the variables made the information confusing.
Let us try:
x=width of rectange, same as diameter of the semicircle;
L=length of rectangle
p= Pi=close to 3.1415....
A= area
A= 0.5*p*(x/2)^2 + x*L
and
30 = p*(x/2) + 2*L + x
You want A in terms of x. Use the second equation and substitute expression for L into the first equation.
11. Sep 16, 2007
relate the length or the arc of the semi circle to the variable of the rectangle (w and or l)
the eqation of a circle's cicomference is C=2pi*r correct? you soul not need any more information
#### Attached Files:
• ###### window.bmp
File size:
225.1 KB
Views:
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Last edited: Sep 16, 2007
12. Sep 16, 2007
### starchild75
Now I am completely lost. The diameter would be c/pi? The perimeter would be pir-2r+2r+2l. or pir+2l?
Last edited: Sep 16, 2007
13. Sep 16, 2007
there is a picture attached to my post, use the picture
if you can't see it i will explain it.
i denoted the bottom of the window as W
the sides as L
and the arc perimiter= unknown
however we know the radius of the arc is just W/2
so r=W/2
the formula of the perimiter of a circle is C=2pi*r
so given that R=W/2 what is the circumference?
*hint for next step...this is not a circle it is a semi circle*
what is the cirumference of the semi circle
14. Sep 16, 2007
### starchild75
attachment pending approval. You are saying use the circumference minus diameter plus diameter (the base) plus the two lengths for perimeter?
15. Sep 16, 2007
### starchild75
Ok see what you think of this.
A(w)= w((30-w-pir)/2)+(3-w-2H)/pi
w=width or diameter
H=height
Last edited: Sep 16, 2007
16. Sep 16, 2007
not exactly sure what you are saying. I might have worded my question incorrectly.
i meant the circumference of the semi circle (not including the base...only the arc)
which is essentialy half of the circumference of the circle.
wha did you get fo the circumference of the circle?
17. Sep 16, 2007
i appologize humbly i may have misundertsood the question.
so i read the questionover again and now it has confused me....its a rectangle with a semi circle up top right, and you want to find the area of the circle (as an equation) but your given that 2l+2w=30 (or did you do that?) because the 2L i understand, but shouldnt it be 2L+w+perimiter of semi circle.
and the area is esentialy the area of rectangle +area of semi circle
and the area of the rectangle is L*W
and the area of the semi circle is _____________ (what we are trying to figure out)
so the formula for the area of a circle is pi*r^2
and r=(w/2)
so area of semi circle is pi(W/2)^2
so that means A=L*W+pi(W/2)^2
but they want it in terms of x (a single variable...however you have L and w in there which is 2 variables...so how would you make that one variable?)
*hint* has somethign to do with whats given....that the perimitr is 30....which is 2L+w+the perimiter of the arc of the semi circle....
and so what i was doing earlier was trying to get you to create an equation that relates the L and W to the perimiter of the semi circle
Last edited: Sep 16, 2007
18. Sep 16, 2007
### starchild75
All I was given is the perimeter of the norman window, which is a recatangle surmounted by a semicircle, is equal to 30. Express the area of the window as a function of the width of the window. Sorry, the description I gave originally is straight from the calculus text. I got pir for the circumference of the semicircle.
19. Sep 17, 2007
### HallsofIvy
Staff Emeritus
Exactly. The window has one base, of length x, two heights of length y, and a semicircle of length $\pi r$. Have you drawn a picture? Can you see what r is in terms of x? You know that the total length is 30. What equation is that?
Now, what is the area of an x by y rectangle? What is the area of the semi circle?
20. Sep 17, 2007
### starchild75
H=(30-pi(w/2)-w)/2
A(w)=w(30-pi(w/2)-w)/2+.5pi(w/2)^2
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2016-10-24 05:50:16
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https://math.stackexchange.com/questions/3064403/help-understand-the-section-formula
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# Help understand the section formula
To find the $$x$$ coordinate of the point $$P$$ that divides the line segment $$AB$$ in ratio $$m:n$$ we can do this :
1) Projection of $$AB$$ along $$x$$ axis is $$x_2-x_1$$
2) We must split this projection at a distance $$\frac{m}{m+n}(x_2-x_1)$$ from $$x_1$$
3) Overall the $$x$$ coordinate of $$P$$ is $$x_1+\dfrac{m}{m+n}(x_2-x_1)$$
So far so good. But if I simplify above formula I get a nice looking result :
$$\dfrac{mx_2+nx_1}{m+n}$$
I'm wondering if this form has any nice interpretation, like, when $$A$$ is at the origin, the numerator becomes $$mx_2$$ and the result is easy to interpret - just $$\frac{m}{m+n}$$ of the length $$x_2$$. However when $$x_1\ne 0$$, we're adding $$nx_1$$ to the numerator. What does this quantity represent?
In short, do the expressions $$mx_2$$ and $$nx_1$$ represent any meaningful quantities?
EDIT :
If I rearrange the result as $$\dfrac{m}{m+n}x_2 + \dfrac{n}{m+n}x_1$$
it seems like a pair of meaningful quantities, but not sure how to interpret them. Help?
• Note that $m+n$ is the whole interval, thus $\frac{m}{m+n}$ is the part of the interval from $A$-side of $P$ while $\frac{n}{m+n}$ is the part from $B$-side. If we mix the coordinates of $A$ and $B$ with those proportions we get exactly the coordinate of $P$. – A.Γ. Jan 6 at 22:18
The numbers $$\frac{m}{m+n}$$ and $$\frac{n}{m+n}$$ add up to $$1$$; they are the relative sizes of the intervals along the diagonals of your two colored triangles. In the example you've given, the first fraction is about $$3/5$$, the second is about $$2/5$$. These add up to $$1$$.
What you've developed is called a "convex combination" of the points $$A$$ and $$B$$ in your diagram. These convex combinations all have the form $$(1-s)A + sB,$$ where $$0 \le s \le 1$$. When $$s = 0$$, the formula gives $$A$$; when $$s = 1$$, it gives $$B$$, and when $$s = 0.5$$, it gives the midpoint, and so on.
You can even extend this to using values $$s$$ outside the interval $$0 \le s \le 1$$; when you do, negative numbers will produce points on the line $$AB$$ that are to the left of $$A$$, and numbers larger than $$1$$ will produce points on the line $$AB$$ to the right of $$B$$. This works in general (although "left" and "right" have to be replaced by "on the non-$$B$$ side of the point $$A$$", and "on the non-$$A$$ side of the point $$B$$", or some phrase like that).
Keep up the good work! Experimenting with things like this is how you start to make discoveries.
You might want to consider all combinations of the form $$(1-s-t)A + s B + t C$$ where $$A, B, C$$ are points in the plane, and the three coefficients are all between $$0$$ and $$1$$. What do all such combinations produce?
(By the way, we'd typically write something like $$sA + tB + uC$$, where $$s+t+u = 1$$ and $$0 \le s,t,u$$, but the form I've used above is a little more analogous to what I used earlier, so I chose to write it that way.)
• Ahh vector form makes it easy to see: $$A + s(B-A) = (1-s)A + sB$$ I still have to think about what the $3$ points form gives; I'll get back shortly.. Thank you so much :) – rsadhvika Jan 7 at 6:23
• It is tracing all the points interior to the triangle! desmos.com/calculator/52hoqcvmsn – rsadhvika Jan 7 at 7:05
• Hopefully $s=t=u=1/2$ should give centroid of the triangle – rsadhvika Jan 7 at 7:07
• This is like taking linear combinations of the coordinates of the points such that the sum of weights stay fixed at 1; and this is somehow tracing only the interior region – rsadhvika Jan 7 at 7:11
• That's exactly correct. Now can you figure out how to express all points within a convex quadrilateral? And then can you figure out why I had to say "convex"? – John Hughes Jan 7 at 12:28
When $$m$$ and $$n$$ are nonnegative, you can view them as masses placed at $$B$$ and $$A$$, respectively. Their center of mass is then $${nA+mB\over m+n}$$, which divides the segment in the desired proportion. This also provides a reasonable explanation as to why increasing $$m$$ moves this point farther from $$A$$: the mass at $$B$$ has increased, so the center of mass should naturally move closer to that point.
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2019-10-15 16:47:47
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https://math.stackexchange.com/questions/647592/arclength-does-not-change-with-reparametrization
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# Arclength does not change with reparametrization
Recall that the length of a curve $\alpha : [a,b] \rightarrow \mathbb{R}^3$ is given by $L(\alpha) = \int |\alpha'(t)| dt$. Let $\beta(r): [c,d] \rightarrow \mathbb{R}^3$ be a reparametrization of $\alpha$ defined by taking a map $h: [c,d] \rightarrow [a,b]$ with $h(c) = a, h(d)=b$ and $h'(r) \geq 0$ for all $r \in [c,d]$. Show that the arclength does not change under this type of reparametrization.
I believe I have to use the chain rule. I initially set $\beta(r) = \alpha(h(r))$. Then using the chain rule, I get $\beta '(r) = \alpha '(h(r)) \cdot \frac{dh}{ds} (r)$. Does this imply that the magnitudes are the same as well? Because then that would be sufficient to prove that the arclength is the same.
• Try something like changing variable in $L(\alpha) = \int |\alpha'(t)| dt$ by changing $t$ to $h(r)$ and then apply the reverse chain rule Jan 22 '14 at 15:46
• Well the Inverse function theorem plays a role as you can probably identify. look at $\frac{dh}{ds}(r)$ Jan 22 '14 at 16:23
$L(\alpha) = \int |\alpha'(t)| dt \overset{t \rightarrow h(r)}{=} \int |\alpha'(h(r))| h'dr \overset{h'(r)\geqslant 0}{=} \int |\alpha'(h(r))h'| dr \overset{inverse\ chain\ rule}{=} \int |\beta'(r)| dr = L(\alpha \circ h)$
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2022-01-17 01:16:42
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https://www.mat.univie.ac.at/~mfulmek/slc81/pages/titles.html
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# Talks
Titles, abstracts and slides (if available):
## George Andrews: Ramanujan's Lost Notebook in Five Volumes - Thoughts and Comments.
The slides are available under the following link.
Abstract: Bruce Berndt and I have recently completed the fifth and final volume on Ramanujan's Lost Notebook. All of Ramanujan's assertions (with perhaps one or two exceptions) have been proved or, in very rare instances, refuted or corrected. Among these hundreds of formulas there are a number that stand out. For example, the recent explosion of results on mock theta functions and mock modular forms has it origin in the Lost Notebook. The "sums-of-tails" phenomenon also arose from the Lost Notebook.
This talk will be a personal account of highlights from this project and questions, yet to be answered, that arose from this decades long effort.
## François Bergeron: Modules for rectangular Catalan combinatorics, and beyond ...
The slides are available under the following link.
Abstract: We propose explicit $$S_n$$-modules that explain the rich mixture of symmetric functions and combinatorics that has recently been studied in the context of rectangular (vs rational) Catalan combinatorics. Together with these modules, comes a much deeper understanding of how their different $$S_n$$-isotypic components relate one to the other. Furthermore, an interesting connection to the Macdonald Delta operators seems to be involved. This opens up a wide range of new problems in Enumerative and Algebraic Combinatorics.
## Mireille Bousquet Mélou: Plane bipolar orientations and quadrant walks.
(Joint work with Éric Fusy and Kilian Raschel)
The slides are available under the following link.
Abstract: Our understanding of planar maps has evolved a lot since the early enumerative results of Tutte, obtained via a recursive approach in the sixties. Thirty years later, the simplicity of his formulae was at last understood at a combinatorial level, and the underlying bijections then paved the way to the study of large random maps, seen as metric spaces.
For maps equipped with an additional structure, many questions remain open. In this talk, we consider planar maps equipped with a bipolar orientation, and show that they have a very rich combinatorial structure, related, among other topics, to lattice walks confined to cones. This allows us to count them, both recursively and bijectively, and to exhibit various universal properties of these structures.
## Mihai Ciucu: Symmetries of shamrocks.
Abstract: Hexagons with four-lobed regions called shamrocks removed from their center were introduced in their 2013 paper by Ciucu and Krattenthaler, where product formulas for the number of their lozenge tilings were provided. In analogy with the plane partitions which they generalize, we consider the problem of enumerating the lozenge tilings which are invariant under some symmetries of the underlying region. This leads to six symmetry classes besides the base case of requiring no symmetry. In this talk we present product formulas for two of these symmetry classes (namely, the ones generalizing cyclically symmetric, and cyclically symmetric and transpose complementary plane partitions).
## Theresia Eisenkölbl: 60th birthday of Christian Krattenthaler.
The slides are available under the following link.
## Tony Guttmann: Combinatorial problems with stretched exponential asymptotics.
Abstract: We look at a range of combinatorial problems where the growth of coefficients is of the form $$C.\mu^n\cdot \exp(-\alpha n^{\beta}) \cdot n^g,$$ with $$\alpha > 0,$$ $$0 < \beta < 1.$$ Problems include some pattern-avoiding permutations, "pushed" random walks, "pushed" self-avoiding walks and interacting partially-directed walks. We will discuss, in a hand-waving way, how this stretched-exponential term arises, and give a numerical method for estimating the parameters $$\mu,$$ $$\alpha,$$ $$\beta$$ and $$g.$$ As an example, we give more precise asymptotics for the coefficients of $$Av(1324)$$ pattern-avoiding permutations.
## Manuel Kauers: Onsager's solution of the Ising model could have been guessed.
The slides (46.2MB!) are available under the following link.
Abstract: In the 1940s, Onsager found a closed form solution for the free energy in the 2D square Ising model without magnetic field. His formula is celebrated as one of the greatest scientific achievements of the 20th century. Although the derivation has been considerably simplified during the past decades, even the simplest derivation known today is not simple. In the talk, we will present a simple non-rigorous derivation of Onsager's formula starting from knowledge that was available to Onsager and using modern computer algebra, which Onsager of course did not have.
This is joint work with Doron Zeilberger (arXiv:1805.09057).
## Thomas Müller: A non-standard exponential principle for species.
Abstract: We describe an exponential principle, which explains the exponential formulae occurring in Enumerative Combinatorics by relating them to a structural property of the corresponding species.
## Peter Paule: Ramanujan's congruences modulo powers of 5, 7, and 11 revisited.
The slides are available under the following link.
Abstract: In 1919 Ramanujan conjectured three infinite families of congruences for the partition function modulo powers of 5, 7, and 11. In 1938 Watson proved the 5-case and (a corrected version of) the 7-case. In 1967 Atkin proved the remaining 11-family using a method significantly different from Watson's. In joint work with Silviu Radu (RISC) we set up a new algorithmic framework which brings all these cases under one umbrella. In this talk I will report on various new aspects of this setting. One aspect concerns a statement of Atkin who remarked that, in comparison with the 5 and 7-case, his proof for 11 is "indeed `langweilig', as Watson suggested." In our framework we find the 11-case particularly interesting.
## Tanguy Rivoal: Hypergeometry and zeta values.
Abstract: Hypergeometric identities and transformations play an important role in the study of the arithmetic nature of zeta values. I will present many classical Diophantine constructions, based on Padé type approximation of polylogarithms, to highlight this fact.
## Bruce Sagan: Combinatorial interpretations of Lucas analogues.
The slides are available under the following link.
Abstract: The Lucas sequence is a sequence of polynomials in $$s,t$$ defined recursively by $$\{0\}=0$$, $$\{1\}=1$$, and $$\{n\}=s\{n-1\}+t\{n-2\}$$ for $$n\ge2$$. On specialization of $$s$$ and $$t$$ one can recover the Fibonacci numbers, the nonnegative integers, and the $$q$$-integers $$[n]_q$$. Given a quantity which is expressed in terms of products and quotients of nonnegative integers, one obtains a Lucas analogue by replacing each factor of $$n$$ in the expression with $$\{n\}$$. It is then natural to ask if the resulting rational function is actually a polynomial in $$s$$ and $$t$$ and, if so, what it counts. Using lattice paths, we give combinatorial models for Lucas analogues of binomial coefficients as well as Catalan numbers and their relatives, such as those for finite Coxeter groups. This is joint work with Curtis Bennett, Juan Carrillo, and John Machacek.
## Xavier Viennot: The essence of bijections: from growth diagrams to Laguerre heaps of segments for the PASEP.
The slides are available under the following link.
Abstract: The PASEP (partially asymmetric exclusion process) is a toy model in the physics of dynamical systems with a very rich underlying combinatorics in relation with orthogonal polynomials culminating in the combinatorics of the moments of the Askey-Wilson polynomials. I will begin with a tour of the PASEP combinatorial garden with many objects such as alternative, tree-like and Dyck tableaux, Laguerre and subdivided Laguerre histories, all of them enumerated by n!. Using several bijections relating these objects, Josuat-Vergs gave the most simple interpretation of the partition function of the 3 parameters PASEP in terms of permutations related to the moments of the Al-Salam-Chihara polynomials. This beautiful interpretation can be "explained" by introducing a new object called "Laguerre heaps of segments" having a central position among the several bijections of the PASEP garden. I will discuss some relations between these bijections and extract what can be called the "essence" of these bijections, some of them having the same "essence" as the Robinson-Schensted correspondence expressed with Fomin growth diagrams, dear to Christian.
## Ole Warnaar: The Selberg integral and the AGT conjecture.
The slides are available under the following link.
Abstract: The Selberg integral is one of the most important hypergeometric integrals in all of mathematics. It has applications in many areas of mathematics and physics, including random matrix theory, number theory and conformal field theory. In a famous 2009 paper Alday, Gaiotto and Tachikawa conjectured a relation between conformal blocks in Liouville field theory and the Nekrasov partition function from $$\mathcal{N}=2$$ supersymmetric gauge theory. One way to approach this conjecture is to compute Selberg integrals over products of Jack polynomials. In this talk I will report on some recent progress on this problem, as well as the many remaining issues, including the correct formulation of the problem in the language of algebraic combinatorics and special functions, free from the divergences and inconsistencies common in quantum field theory.
## Jiang Zeng: Some multivariate master polynomials for permutations, set partitions, and perfect matchings, and their continued fractions.
The slides are available under the following link.
Abstract: I will present Stieltjes-type and Jacobi-type continued fractions for some "master polynomials" that enumerate permutations, set partitions or perfect matchings with a large (sometimes infinite) number of simultaneous statistics. These results contain many previously obtained identities as special cases, providing a common refinement of all of them. Proofs of the main results will be outlined. This talk is based on joint work with Alan Sokal.
The slides are available under the following link.
Abstract: It is well known that the limits of $$q$$-hypergeometric identities as $$q\to1$$ recover the underlying 'ordinary' versions, and the latter normally serve as a motivation for discoverying the former. In my talk I will explain how the radial limits, as $$q$$ tends to a root of unity (that is, at a '$$q$$-microscopic' level!), of the same $$q$$-hypergeometric identities can lead to the $$p$$-adic, aka congruence, counterparts of those ordinary versions. A typical example includes derivation, from a $$q$$-analogue of Ramanujan's formula
$$\sum_{n=0}^\infty\frac{\binom{4n}{2n}{\binom{2n}{n}}^2}{2^{8n}3^{2n}}\,(8n+1)=\frac{2\sqrt{3}}{\pi},$$
of the two supercongruences
$$S(p-1)\equiv p\biggl(\frac{-3}p\biggr)\pmod{p^3} \quad\text{and}\quad S\Bigl(\frac{p-1}2\Bigr)\equiv p\biggl(\frac{-3}p\biggr)\pmod{ p^3},$$
valid for all primes $$p>3$$, where $$S(N)$$ denotes the truncation of the infinite sum at the $$N$$-th place and $$(\frac{-3}{\cdot})$$ stands for the quadratic character modulo 3.
The talk is based on joint work with Victor Guo.
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2019-05-21 13:04:27
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https://chem.libretexts.org/Textbook_Maps/Physical_and_Theoretical_Chemistry_Textbook_Maps/Map%3A_Physical_Chemistry_(McQuarrie_and_Simon)/07._Approximation_Methods/7.4%3A_Perturbation_Theory_Expresses_the_Solutions_in_Terms_of_Solved_Problems
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# 7.4: Perturbation Theory Expresses the Solutions in Terms of Solved Problems
It is easier to compute the changes in the energy levels and wavefunctions with a scheme of successive corrections to the zero-field values. This method, termed perturbation theory, is the single most important method of solving problems in quantum mechanics, and is widely used in atomic physics, condensed matter and particle physics. Perturbation theory is another approach to finding approximate solutions to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle step that breaks the problem into "solvable" and "perturbation" parts. Perturbation theory is applicable if the problem at hand cannot be solved exactly, but can be formulated by adding a "small" term to the mathematical description of the exactly solvable problem.
Figure $$\PageIndex{1}$$: Perturbed Energy Spectrum. Image used with permission (CC BY-SA 2.0, Frontier).
### The Perturbation Series
We begin with a Hamiltonian $$H^0$$ having known eigenkets and eigenenergies:
$H^o | n^o \rangle = E_n^o | n^o \rangle \label{7.4.1}$
The task is to find how these eigenstates and eigenenergies change if a small term $$H^1$$ (an external field, for example) is added to the Hamiltonian, so:
$( H^0 + H^1 ) | n \rangle = E_n | n \rangle \label{7.4.2}$
That is to say, on switching on $$H^1$$ changes the wavefunctions:
$\underbrace{ | n^o \rangle }_{\text{unperturbed}} \Rightarrow \underbrace{|n \rangle }_{\text{Perturbed}}\label{7.4.3}$
and energies (Figure $$\PageIndex{1}$$):
$\underbrace{ E_n^o }_{\text{unperturbed}} \Rightarrow \underbrace{E_n }_{\text{Perturbed}} \label{7.4.4}$
The basic assumption in perturbation theory is that $$H^1$$ is sufficiently small that the leading corrections are the same order of magnitude as$$H^1$$ itself, and the true energies can be better and better approximated by a successive series of corrections, each of order $$H^1/H^o$$ compared with the previous one.
The strategy is to expand the true wavefunction and corresponding eigenenergy as series in $$H^1/H^o$$. These series are then fed into Equation $$\ref{7.4.2}$$, and terms of the same order of magnitude in $$H^1/H^o$$ on the two sides are set equal. The equations thus generated are solved one by one to give progressively more accurate results.
To make it easier to identify terms of the same order in $$H^1/H^o$$ on the two sides of the equation, it is convenient to introduce a dimensionless parameter $$\lambda$$ which always goes with $$H^1$$, and then expand both eigenstates and eigenenergies as power series in $$\lambda$$,
$\color{red} | n \rangle = \sum _ i^m \lambda ^i| n^i \rangle \label{7.4.5}$
$\color{red} E_n = \sum_{i=0}^m \lambda ^i E_n^i \label{7.4.6}$
where $$m$$ is how many terms in the expansion we are considering. The ket $$|n^i \rangle$$ is multiplied by $$\lambda^i$$ and is therefore of order $$(H^1/H^o)^i$$. For example, in first order perturbation theory, Equations $$\ref{7.4.5}$$ are truncated at $$m=1$$ (and setting $$\lambda=1$$):
$| n \rangle \approx | n^o \rangle + | n^1 \rangle \label{7.4.7}$
$E_n \approx E_n^o + E_n^1 \label{7.4.8}$
$$\lambda$$ is purely a bookkeeping device: we will set it equal to 1 when we are through! It’s just there to keep track of the orders of magnitudes of the various terms.
However, let's consider the general case for now. Adding the full expansions for the eigenstate (Equation $$\ref{7.4.5}$$) and energies (Equation $$\ref{7.4.6}$$) into the Schrödinger equation for the perturbation Equation $$\ref{7.4.2}$$ in
$( H^o + \lambda H^1) | n \rangle = E_n| n \rangle \label{7.4.9}$
we have
$(H^o + \lambda H^1) \left( \sum _ {i=0}^m \lambda ^i| n^i \rangle \right) = \left( \sum_{i=0}^m \lambda^i E_n^i \right) \left( \sum _ {i=0}^m \lambda ^i| n^i \rangle \right) \label{7.4.10}$
We’re now ready to match the two sides term by term in powers of $$\lambda$$. Note that the zeroth-order term, of course, just gives back the unperturbed Schrödinger Equation (Equation $$\ref{7.4.1}$$).
Let's look at Equation $$\ref{7.4.10}$$ with the first few terms of the expansion:
$(H^o + \lambda H^1) \left( | n ^o \rangle + \lambda | n^1 \rangle \right) = \left( E _n^0 + \lambda E_n^1 \right) \left( | n ^o \rangle + \lambda | n^1 \rangle \right) \label{7.4.11}$
$H^o | n ^o \rangle + \lambda H^1 | n ^o \rangle + \lambda H^o | n^1 \rangle + \lambda^2 H^1| n^1 \rangle= E _n^0 | n ^o \rangle + \lambda E_n^1 | n ^o \rangle + \lambda E _n^0 | n ^1 \rangle + \lambda^2 E_n^1 | n^1 \rangle \label{7.4.11A}$
Collecting terms in order of $$\lambda$$
$\underset{\text{zero order}}{H^o | n ^o \rangle} + \color{red} \underset{\text{1st order}}{\lambda ( H^1 | n ^o \rangle + H^o | n^1 \rangle )} + \color{blue} \underset{\text{2nd order}} {\lambda^2 H^1| n^1 \rangle} =\color{black}\underset{\text{zero order}}{E _n^0 | n ^o \rangle} + \color{red} \underset{\text{1st order}}{ \lambda (E_n^1 | n ^o \rangle + E _n^0 | n ^1 \rangle )} +\color{blue}\underset{\text{2nd order}}{\lambda^2 E_n^1 | n^1 \rangle} \label{7.4.12}$
If we expanded Equation $$\ref{7.4.10}$$ further we could express the energies and wavefunctions in higher order components.
#### Zero-Order Terms ($$\lambda=0$$)
The zero order terms in the expansion of Equation $$\ref{7.4.10}$$ results in just the Schrödinger Equation for the unperturbed system
$H^o | n^o \rangle = E_n^o | n^o \rangle \label{Zero}$
#### First-Order Expression of Energy ($$\lambda=1$$)
The summations in Equations $$\ref{7.4.5}$$, $$\ref{7.4.6}$$, and $$\ref{7.4.10}$$ can be truncated at any order of $$\lambda$$. For example, the first order perturbation theory has the truncation at $$\lambda=1$$. Matching the terms that linear in $$\lambda$$ (red terms in Equation $$\ref{7.4.12}$$) and setting $$\lambda=1$$ on both sides of Equation $$\ref{7.4.12}$$:
$H^o | n^1 \rangle + H^1 | n^o \rangle = E_n^o | n^1 \rangle + E_n^1 | n^o \rangle \label{7.4.13}$
Equation $$\ref{7.4.13}$$ is the key to finding the first-order change in energy $$E_n^1$$. Taking the inner product of both sides with $$\langle n^o |$$:
$\langle n^o | H^o | n^1 \rangle + \langle n^o | H^1 | n^o \rangle = \langle n^o | E_n^o| n^1 \rangle + \langle n^o | E_n^1 | n^o \rangle \label{7.4.14}$
since operating the zero-order Hamiltonian on the bra wavefunction (this is just the Schrödinger equation; Equation $$\ref{Zero}$$) is
$\langle n^o | H^o = \langle n^o | E_n^o \label{7.4.15}$
and via the orthonormality of the unperturbed $$| n^o \rangle$$ wavefunctions both
$\langle n^o | n^o \rangle = 1 \label{7.4.16}$
and Equation $$\ref{7.4.14}$$ can be simplified
$\xcancel{E_n^o \langle n^o | n^1 \rangle} + \langle n^o | H^1 | n^o \rangle = \xcancel{ E_n^o \langle n^o | n^1 \rangle} + E_n^1 \cancelto{1}{\langle n^o | n^o} \rangle \label{7.4.14new}$
since the unperturbed set of eigenstates are orthogonal (Equation \ref{7.4.16}) and we can cancel the other term on each side of the equation, we find that
$\color{red} E_n^1 = \langle n^o | H^1 | n^o \rangle \label{7.4.17}$
The first-order change in the energy of a state resulting from adding a perturbing term $$H^1$$ to the Hamiltonian is just the expectation value of $$H^1$$ in the unperturbed wavefunctions.
That is, within the first order limit, the energies are given by
$\color{red} \underbrace{E_n \approx E_n^o + E_n^1 = E_n^o + \langle n^o | H^1 | n^o \rangle}_{\text{First Order Perturbation Theory}} \label{7.4.17.2}$
Example $$\PageIndex{1}$$: A Perturbed Particle in a Box
Estimate the energy of the ground-state and first excited-state wavefunction within first-order perturbation theory of a system with the following potential energy
$V(x)=\begin{cases} V_o & 0\leq x\leq L \\ \infty & x< 0 \; and\; x> L \end{cases} \nonumber$
Solution
The first step in any perturbation problem is to write the Hamiltonian in terms of a unperturbed component that the solutions (both eigenstates and energy) are known and a perturbation component (Equation $$\ref{7.4.2}$$). For this system, the unperturbed Hamilonian and solutions is the particle in an infiinitely high box and the perturbation is a shift of the potential within the box by $$V_o$$.
$\hat{H}^1 = V_o \nonumber$
Using Equation $$\ref{7.4.17}$$ for the first-order term in the energy of the ground-state
$E_n^1 = \langle n^o | H^1 | n^o \rangle \nonumber$
with the wavefunctions known from the particle in the box problem
$| n^o \rangle = \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) \nonumber$
At this stage we can do two problems independently (i.e., the ground-state with $$| 1 \rangle$$ and the first excited-state $$| 2 \rangle$$). However, in this case, the first-order perturbation to any particle-in-the-box sate can be easily derived.
$E_n^1 = \int_0^L \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) V_o \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) dx \nonumber$
or better yet, instead of evaluating this integrals we can simplify the expression
$E_n^1 = \langle n^o | H^1 | n^o \rangle = \langle n^o | V_o | n^o \rangle = V_o \langle n^o | n^o \rangle = V_o \nonumber$
so via Equation $$\ref{7.4.17.2}$$, the energy of each perturbed eigenstate is
$E_n \approx E_n^o + V_o \nonumber$
or
$E_n \approx \dfrac{h^2}{8mL^2}n^2 + V_o \nonumber$
While this is the first order perturbation to the energy, it is also the exact value too.
Example $$\PageIndex{2}$$: An Even More Perturbed Particle in a Box
Estimate the energy of the ground-state wavefunction within first-order perturbation theory of a system with the following potential energy
$V(x)=\begin{cases} V_o & 0\leq x\leq L/2 \\ \infty & x< 0 \; and\; x> L \end{cases} \nonumber$
Solution
As with Example $$\PageIndex{1}$$, we recognize that unperturbed component of the problem (Equation $$\ref{7.4.2}$$) is the particle in an infinitely high well. For this system, the unperturbed Hamiltonian and solutions is the particle in an infinitely high box and the perturbation is a shift of the potential within half a box by $$V_o$$. This is essentially a step function.
Using Equation $$\ref{7.4.17}$$ for the first-order term in the energy of the any state
$E_n^1 = \langle n^o | H^1 | n^o \rangle \nonumber$
$E_n^1 = \int_0^{L/2} \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) V_o \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) + dx \nonumber$
$\int_{L/2}^L \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) 0 \sqrt{\dfrac{2}{L}} \sin \left ( \dfrac {n \pi}{L} x \right) dx \nonumber$
The second integral is zero and the first integral is simplified to
$E_n^1 = \dfrac{2}{L} \int_0^{L/2} V_o \sin^2 \left( \dfrac {n \pi}{L} x \right) + dx \nonumber$
this is evaluated to
$E_n^1 = \dfrac{2V_o}{L} \left[ \dfrac{-1}{2 \frac{\pi n}{a}} \cos \left( \dfrac {n \pi}{L} x \right) \sin \left( \dfrac {n \pi}{L} x \right) + \dfrac{x}{2} \right]_o^{L/2}$
$= \dfrac{2V_o}{\cancel{L}} \dfrac{\cancel{L}}{4} = \dfrac{V_o}{2}$
The energy of each perturbed eigenstate, via Equation $$\ref{7.4.17.2}$$, is
$E_n \approx E_n^o + \dfrac{V_o}{2} \nonumber$
or
$E_n \approx \dfrac{h^2}{8mL^2}n^2 + \dfrac{V_o}{2} \nonumber$
#### First-Order Expression of Wavefunction ($$\lambda=1$$)
The general expression for the first-order change in the wavefunction is found by taking the inner product of the first-order expansion (Equation $$\ref{7.4.13}$$) with the bra $$\langle m^o |$$ with $$m \neq n$$,
$\langle m^o | H^o | n^1 \rangle + \langle m^o |H^1 | n^o \rangle = \langle m^o | E_n^o | n^1 \rangle + \langle m^o |E_n^1 | n^o \rangle \label{7.4.18}$
Last term on right side of Equation $$\ref{7.4.18}$$
The last integral on the right hand side of Equation $$\ref{7.4.18}$$ is zero, since $$m \neq n$$ so
$\langle m^o |E_n^1 | n^o \rangle = E_n^1 \langle m^o | n^o \rangle \label{7.4.19}$
and
$\langle m^o | n^0 \rangle = 0 \label{7.4.20}$
First term on right side of Equation $$\ref{7.4.18}$$
The first integral is more complicated and can be expanded back into the $$H^o$$
$E_m^o \langle m^o | n^1 \rangle = \langle m^o|E_m^o | n^1 \rangle = \langle m^o | H^o | n^1 \rangle \label{7.4.21}$
since
$\langle m^o | H^o = \langle m^o | E_m^o \label{7.4.22}$
so
$\langle m^o | n^1 \rangle = \dfrac{\langle m^o | H^1 | n^o \rangle}{ E_n^o - E_m^o} \label{7.4.23}$
and therefore the wavefunction corrected to first order is:
$\color{red} \underbrace{| n \rangle \approx | n^o \rangle + | n^1 \rangle = | n^o \rangle + \sum _{m \neq n} \dfrac{|m^o \rangle \langle m^o | H^1| n^o \rangle }{E_n^o - E_m^o}}_{\text{First Order Perturbation Theory}} \label{7.4.24}$
Equation $$\ref{7.4.24}$$ is essentially is an expansion of the unknown wavefunction correction as a linear combination of known unperturbed wavefunctions $$\ref{7.4.24.2}$$:
$\color{red} | n \rangle \approx | n^o \rangle + | n^1 \rangle = | n^o \rangle + \sum _{m \neq n} c_{m,n} |m^o \rangle \label{7.4.24.2}$
with the expansion coefficients determined by
$\color{red} c_{m,n} = \dfrac{\langle m^o | H^1| n^o \rangle }{E_n^o - E_m^o} \label{7.4.24.3}$
Since is justified since the set of original zero-order wavefunctions forms a complete basis set that can describe any function.
Figure $$\PageIndex{2}$$: The first order perturbation of the ground-state wavefunction for a perturbed (left) can be expressed as a linear combination of all excited-state wavefunctions of the unperturbed potential (Equation $$\ref{7.4.24.2}$$, shown as a harmonic oscillator in this example (right). Note that the ground-state harmonic oscillator wavefuncion is not part of this expression and technically all wavefunctions need to be included in the expression, not just the first eight wavefunctions shown here.
Calculating the first order perturbation to the wavefunctions is in general an infinite sum of off diagonal matrix elements of $$H^1$$ (Figure $$\PageIndex{2}$$).
• However, the denominator argues that terms in this sum will be weighted by states that are of comparable energy. That means in principle, these sum can be truncated easily based off of some criterion.
• Another point to consider is that many of these matrix elements will equal zero depending on the symmetry of the $$\{| n^o \rangle \}$$ basis and $$H^1$$ (e.g., some $$\langle m^o | H^1| n^o \rangle$$ integrals in Equation $$\ref{7.4.24}$$ could be zero due to the integrand having an odd symmetry; see Example $$\PageIndex{3}$$).
The denominators in Equation $$\ref{7.4.24}$$ argues that terms in this sum will be preferentially dictated by states that are of comparable energy. That is, eigenstates that have energies significantly greater or lower than the unperturbed eigenstate will weakly contribute to the perturbed wavefunction.
Example $$\PageIndex{3}$$: Harmonic Oscillator with a Cubic Perturbation
Estimate the energy of the ground-state wavefunction associated with the Hamiltonian using perturbation theory
$\hat{H} = \dfrac{-\hbar}{2m} \dfrac{d^2}{dx^2} + \dfrac{1}{2} kx^2 + \epsilon x^3 \nonumber$
Solution
Energy
The first steps in flowchart for applying perturbation theory (Figure $$\PageIndex{1}$$) is to separate the Hamiltonian of the difficult (or unsolvable) problem into a solvable one with a perturbation. For this case, we can rewrite the Hamiltonian as
$\hat{H}^{o} + \hat{H}^{1} \nonumber$
where
• $$\hat{H}^{o}$$ is the Hamitonian for the standard Harmonic Oscillator with known eigenstates and eigenenergies $\hat{H}^{(0)}= \dfrac{-\hbar}{2m} \dfrac{d^2}{dx^2} + \dfrac{1}{2} kx^2 \nonumber$
• $$\hat{H}^{1}$$ is the pertubtiation $\hat{H}^{1} = \epsilon x^3 \nonumber$
The first order perturbation is given by Equation $$\ref{7.4.17}$$, which for this problem is
$E_n^1 = \langle n^o | \epsilon x^3 | n^o \rangle \nonumber$
Notice that the integrand has an odd symmetry (i.e., $$f(x)=-f(-x)$$) with the perturbation Hamiltonian being odd and the ground state harmonic oscillator wavefunctions being even. So
$E_n^1=0 \nonumber$
This means to first order purtbation theory, this cubic terms does not alter the ground state energy (via Equation $$\ref{7.4.17.2})$$. However, this is not the case if second-order perturbation theory were used, which is more accurate (not shown).
Wavefunction
Calculating the first order perturbation to the wavefunctions (equation $$\ref{7.4.24}$$ is more difficult than energy since multiple integrals must be evaluated (an infinite number if symmetry arguments are not applicable). The harmonic oscillator wavefunctions are often written in terms of $$Q$$, the unscaled displacement coordinate:
$| \Psi _v (x) \rangle = N_v'' H_v (\sqrt{\alpha} Q) e^{-\alpha Q^2/ 2}$
with $$\alpha$$
$\alpha =1/\sqrt{\beta} = \sqrt{\dfrac{k \mu}{\hbar ^2}}$
and
$N_v'' = \sqrt {\dfrac {1}{2^v v!}} \left(\dfrac{\alpha}{\pi}\right)^{1/4}$
Let's consider only the first six wavefunctions that use these Hermite polynomials $$H_v (x)$$:
• $$H_0 = 1$$
• $$H_1 = 2x$$
• $$H_2 = -2 + 4x^2$$
• $$H_3 = -12x + 8x^3$$
• $$H_4 = 12 - 48x^2 +16x^4$$
• $$H_5 = 120x - 160x^3 + 32x^5$$
The first order perturbation to the wavefunction (Equation $$\ref{7.4.24}$$)
$| 0^1 \rangle = \sum _{m \neq 0} \dfrac{|m^o \rangle \langle m^o | H^1| 0^o \rangle }{E_0^o - E_m^o}$
given these truncated wavefunctions (we should technically use the infinite sum) and that we are considering only the ground state with $$n=0$$:
$| 0^1 \rangle = \dfrac{ \langle 1^o | H^1| 0^o \rangle }{E_0^o - E_1^o} |1^o \rangle + \dfrac{ \langle 2^o | H^1| 0^o \rangle }{E_0^o - E_2^o} |2^o \rangle + \dfrac{ \langle 3^o | H^1| 0^o \rangle }{E_0^o - E_3^o} |3^o \rangle + \dfrac{ \langle 4^o | H^1| 0^o \rangle }{E_0^o - E_4^o} |4^o \rangle + \dfrac{ \langle 5^o | H^1| 0^o \rangle }{E_0^o - E_5^o} |5^o \rangle \nonumber$
We can use symmetry of the perturbation and unperturbed wavefunctions to solve the integrals above. We know that the unperturbed wavefunctions $$\{|n^{0}\} \rangle$$ alternate between even (when $$v$$ is even) and odd (when $$v$$ is odd). Since the perturbation is an odd function, only when $$m= 2k+1$$ with $$k=1,2,3$$ would these integrals be non-zero (i.e., for $$m=1,3,5, ...$$).
So of the original five unperturbed wavefuncion, only $$|m=2\rangle$$ and $$m=4 \rangle$$ mix to make the first-order perturbed wavefunction so
$| 0^1 \rangle = \dfrac{ \langle 1^o | H^1| 0^o \rangle }{E_0^o - E_1^o} |1^o \rangle + \dfrac{ \langle 3^o | H^1| 0^o \rangle }{E_0^o - E_3^o} |3^o \rangle + \dfrac{ \langle 5^o | H^1| 0^o \rangle }{E_0^o - E_5^o} |5^o \rangle \nonumber$
At this stage, the integrals have to be manually calculated using the defined wavefuctions above, which is left as an exercise. Notice that each unperturbed wavefunction that can "mix" to generate the perturbed wavefunction will have a reciprocally decreasing contribution (w.r.t. energy) due to the growing denominator.
Exercise $$\PageIndex{3}$$: Harmonic Oscillator with a Quartic Perturbation
Estimate the energy of the ground-state wavefunction associated with the Hamiltonian using perturbation theory
$\hat{H} = \dfrac{-\hbar}{2m} \dfrac{d^2}{dx^2} + \dfrac{1}{2} kx^2 + \gamma x^4 \nonumber$
#### Second-Order Terms ($$\lambda=2$$)
There are higher energy terms in the expansion of Equation $$\ref{7.4.5}$$ (e.g., the blue terms in Equation $$\ref{7.4.12}$$), but are not discussed further here other than noting the whole perturbation process is an infitinite series of corrections that ideally converge to the correct answer. It is the truncating this series as a finite level that is the approximation.
The general approach to perturbation theory applications is giving in the flowchart below.
Figure $$\PageIndex{1}$$: Simplified algorithmic flowchart of the Perturbation Theory approximation showing the first two perturbation orders. The process can be continued to third and higher orders.
Perturbation Theory Does not always Work
It should be noted that there are problems which cannot be solved using perturbation theory, even when the perturbation is very weak, although such problems are the exception rather than the rule. One such case is the one-dimensional problem of free particles perturbed by a localized potential of strength l. As we found earlier in the course, switching on an arbitrarily weak attractive potential causes the $$k=0$$ free particle wavefunction to drop below the continuum of plane wave energies and become a localized bound state with binding energy of order $$\lambda^2$$. However, changing the sign of $$\lambda$$ to give a repulsive potential there is no bound state, the lowest energy plane wave state stays at energy zero. Therefore the energy shift on switching on the perturbation cannot be represented as a power series in $$\lambda$$, the strength of the perturbation.
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## "Free, white, and twenty one"
Sometimes I think that Philip K. Dick is passing the time in purgatory by ghostwriting news stories like this one – "Atlantic City's Revel Casino reimagined as elite school", Reuters 9/22/2014:
A Florida developer who made a $90 million offer for Atlantic City's shuttered Revel Casino wants to use the site to help end world hunger, cancer, and resolve other pressing issues like nuclear waste storage. Glenn Straub's plan is ambitious as it is high-minded. First, he would add a second tower to the 57-story structure, completing the original vision of the casino-hotel's developers. The businessman, who owns the Palm Beach Polo Golf and Country Club, would then convert the complex into a university where the best and brightest young minds from across the world could work on the big issues of the day. Read the rest of this entry » ## Another casual lie from Charles Krauthammer "Krauthammer: 'Obama Clearly a Narcissist,' 'Lives In a Cocoon Surrounded By Sycophants'", Fox News 9/16/2014: "This is all because, I mean, count the number of times he uses the word I in any speech, and compare that to any other president. Remember when he announced the killing of bin Laden? That speech I believe had 29 references to I – on my command, I ordered, as commander-in-chief, I was then told, I this. You’d think he’d pulled the trigger out there in Abbottabad. You know, this is a guy, you look at every one of his speeches, even the way he introduces high officials – I’d like to introduce my secretary of State. He once referred to ‘my intelligence community’. And in one speech, I no longer remember it, ‘my military’. For God’s sake, he talks like the emperor, Napoleon." Read the rest of this entry » ## The grammar of "Better Together" The official name of the organization campaigning for a No vote in the upcoming Scottish independence referendum is "Better Together." That phrase was originally the campaign's main slogan. Much has been written in recent days about the campaign's evident signs of panic, but no one has commented on the stupidity of "Better Together" as a slogan. (It was actually ditched by the campaign in June, and replaced by an even more pathetic slogan: "No Thanks.") Better together is an adjective phrase. Used on its own, without any logical subject or other accompanying noun phrases, it is apparently supposed to affirm that something will go better in some way for someone than something else if something is together with something else, but it doesn't specify any of these someones or somethings. Yet the cui bono issue (who benefits) is absolutely crucial to the debate. The ineptness of the sloganeering is almost unbelievable. Read the rest of this entry » Comments off ## Abduweli Ayup "Uyghur linguist sentenced to 18-month prison term in China", LSA News 8/28/2014: The LSA has learned from news reports published this week that Abduweli Ayup has been ordered to pay a large fine and continue his detention in a Chinese prison for the next six months. The LSA had sent a letter earlier this year to government officials in China and the U.S., seeking details about Abduweli's alleged crimes, and legal intervention on his behalf, consistent with international covenants on human rights. Friends of Abduweli's have established a fundraising page on the YouCaring website to assist in raising a portion of the$13,000 (USD) fine imposed by the Chinese government.
Read the rest of this entry »
## Sanskrit resurgent
When I was studying Buddhism at the University of Washington (Seattle) in 1967-68, there were about ten students in my first-year Sanskrit course for Buddhologists and Indologists. What intrigued me greatly was that there was another beginning Sanskrit course being offered at the same time. It had many more students than the class I was in and was offered by the Linguistics Department. The rationale for encouraging (I can't remember if it was actually required) linguistics students to take Sanskrit was that the foundations of the scientific study of language had been laid by Panini, Patanjali, and other ancient Sanskrit grammarians around two and a half millennia ago, so that it would be good to have at least a basic understanding of the roots of the tradition.
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## Newspaper alleges passive voice correctly!
Today I came upon something truly rare: a newspaper article about a passive-voice apology that (i) is correct about the apology containing a passive clause, but (ii) stresses that the oft-misdiagnosed passive should not be the thing we focus on and attempt to discourage, and (iii) cites actual linguists in support of the latter view! What's going on? Is Language Log beginning to break through? Are journalists waking up to the fact that there actually is a definition of the notion 'passive voice' (though hardly anybody seems to know what it is)? The article is by Tristin Hopper of the National Post in Canada (August 8, 2014); you can read it here. Kudos to Tristin.
## Orient(al[ism]) in East Asian languages
Cortney Chaffin writes:
Today I've been corresponding over email with a colleague of mine at XYUniversity who organized an exhibition of Korean art to open tomorrow. Yesterday he sent out a description of the exhibit in which he used the phrases "oriental landscape painting" (in contrast to Western painting) and "oriental sensitivity" to describe the aim of the artist (to demonstrate "oriental sensitivity" in painting). I don't allow my students to use the term "oriental" in my art history classes, not only because it is a complex and loaded term, but I have first-hand experience of it being used as a racial slur in the U.S., so it makes me uncomfortable.
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## "The temperature on Mars is exactly as it is here"
Jonathan Meador, "Kentucky Lawmakers Attack Climate Change Science In Discussion on Carbon Regulations", WFPL 89.3 FM:
State lawmakers' discussion Thursday of the effect of new EPA carbon emission regulations on Kentucky focused more on political attacks than hard science. [...]
“I won’t get into the debate about climate change," said Sen. Brandon Smith, a Hazard Republican. “But I’ll simply point out that I think in academia we all agree that the temperature on Mars is exactly as it is here. Nobody will dispute that. Yet there are no coal mines on Mars. There’s no factories on Mars that I’m aware of.”
Smith owns a coal company on Earth.
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## Happy. Fourth.
In anticipation of the 4th of July weekend, I was compelled to read this very interesting (July 1 draft) manuscript: "Punctuating Happiness", by UPS Foundation Professor Danielle S. Allen of the Institute for Advanced Study at Princeton. A political theorist friend's Facebook post led me both to the article and to this front-page NYT piece on it: "If Only Thomas Jefferson Could Settle the Issue: A Period is Questioned in the Declaration of Independence", by Jennifer Schuessler (July 2 online, July 3 print).
Professor Allen makes a thorough and compelling case for her claim that the second sentence of the actual Declaration of Independence parchment has a comma after the well-known phrase "life, liberty, and the pursuit of happiness" — and not a period, as the most frequently reproduced version of the document, an engraving made by printer William J. Stone in 1823, would lead one to believe. The matter can't be resolved via visual inspection; the parchment is extremely faded, and Allen presents some evidence — suggestive but not conclusive, in my opinion, but that's neither here nor there — that it may have already been sufficiently faded at the time of Stone's engraving. Allen thus "advocate[s] for the use of hyper-spectral imaging to re-visit the question of what is on the parchment".
For everyone's reference, here is the relevant "second sentence" of the Declaration of Independence, as transcribed on pp. 2-3 of Allen's manuscript, with the "errant period" highlighted in green.
We hold these truths to be self-evident, that all men are created equal; that they are endowed by their Creator with certain unalienable rights; that among these are life, liberty, and the pursuit of happiness. — That to secure these rights governments are instituted among men, deriving their just powers from the consent of the governed, — That whenever any form of government becomes destructive of these ends, it is the right of the people to alter or to abolish it, and to institute new government, laying its foundations on such principles and organizing its powers in such form, as to them shall seem most likely to effect their safety and happiness.
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A recent story in a Las Vegas newspaper ("Shooters in Metro ambush that left five dead spoke of white supremacy and a desire to kill police", Las Vegas Review-Journal 6/8/2014) now has 15,793 comments. Reading through a small sample of them, I wasn't surprised by the usual teabugger vs. libturd name-calling.
And I expected a smattering of various sorts of conspiracy theories. But I wasn't expecting the sheer volume of (hundreds and hundreds of) comments insisting that this event was a "false flag" or "psyops" hoax. For example:
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## Banned in Beijing
Everyone knows that the Chinese government goes to extraordinary lengths to police the internet (see: "Blocked on Weibo").
And most sentient beings are aware of the awesome fame of the Grass-Mud Horse, the notorious Franco-Croatian Squid, and and the mysterious River Crab. You can find all of them in "Grass-Mud Horse Lexicon Classics".
Sometimes, the censors begin to look pretty ridiculous, as when they outlawed the word "jasmine" in 2011, particularly since it refers not just to the Jasmine Revolution, but also to a favorite flower, tea, and folk song.
mòlì 茉莉 ("jasmine")
mòlì chá 茉莉茶 ("jasmine tea") OR mòlìhuā chá 茉莉花茶 ("jasmine tea") OR xiāngpiàn 香片 ("scented [usually with jasmine] tea")
mòlìhuā 茉莉花 ("jasmine flower", name of a popular folk song; presidents Jiang Zemin and Hu Jintao were both excessively fond of this song, and there are videos of them singing it, so it becomes especially awkward to try to forbid citizens to use the word mòlì 茉莉 ("jasmine")
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## Not taking shit from the president?
In Politico's Playbook, Mike Allen notes that the slogan "Don't Do Stupid Shit" has worked its way into numerous journalistic descriptions of the "Obama Doctrine." "Playbook rarely prints a four-letter word — our nephews are loyal readers," Allen writes. "But we are, in this case, because that is the precise phrase President Obama and his aides are using in their off-the-record chats with journalists."
The New York Times, on the other hand, has only printed the slogan in expurgated fashion — this despite the fact that late Times editor Abe Rosenthal created a presidential exemption from the ban on printing "shit" in the Nixon era. As Rosenthal reportedly said after including "shit" in quotes of Watergate tape transcripts, "We'll only take shit from the President."
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## A 'World without Thieves' world
Tom Mazanec came across a poster that was located at a bus stop at one of Princeton's graduate housing complexes, and is an advertisement for a Chinese-language Christian fellowship. Here's a photograph of the poster:
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## Territorial rights for languages
I had been waiting for the world's media to notice the extraordinarily anomalous character of Vladimir Putin's notion that he can annex pieces of land simply because speakers of the Russian language live there and are feeling aggrieved or imperilled. And now The Economist has done the job very nicely. See this page for an article about what the world map would look like under a generalization of Putin's doctrine.
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2014-09-24 04:24:09
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http://math.stackexchange.com/questions/351920/relation-between-sets
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# Relation between sets
Let $n$ be a positive integer. How can I show that the relation $R$ on the set of all polynomials with real–valued coefficients consisting of all pairs $(f, g)$ such that $f^{(n)}(x) = g^{(n)}(x)$ is an equivalence relation. Note that $f^{(n)}$ and $g^{(n)}$ are the $n$-th derivative of $f(x)$.
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Show us what you tried...reflexivity, simmetry, transitivity – UrošSlovenija Apr 5 '13 at 8:12
Everything is really just dead obvious. Write down the definitions, and you should see immediately that they hold. – Asaf Karagila Apr 5 '13 at 8:15
There is a general principle, that says that any relation $R$ expressed in terms of $f R g$ if and only if $f$ and $g$ have the same (whatever) is an equivalence relation. Just spell aloud the three properties, and you'll see what I mean
More formally, if $A, C$ are sets, with $A \ne \emptyset$, and $\varphi: A \to C$ is a map, then the relation $R$ on $A$ defined, for $a, b \in A$, by $$\text{a R b if and only if \varphi(a) = \varphi(b)}$$ is an equivalence relation.
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2014-09-30 22:08:30
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http://en.wikipedia.org/wiki/Fixed_point_index
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# Fixed-point index
(Redirected from Fixed point index)
In mathematics, the fixed-point index is a concept in topological fixed-point theory, and in particular Nielsen theory. The fixed-point index can be thought of as a multiplicity measurement for fixed points.
The index can be easily defined in the setting of complex analysis: Let f(z) be a holomorphic mapping on the complex plane, and let z0 be a fixed point of f. Then the function f(z) − z is holomorphic, and has an isolated zero at z0. We define the fixed point index of f at z0, denoted i(f, z0), to be the multiplicity of the zero of the function f(z) − z at the point z0.
In real Euclidean space, the fixed-point index is defined as follows: If x0 is an isolated fixed point of f, then let g be the function defined by
$g(x) = \frac{x - f(x)}{|| x - f(x) ||}. \,$
Then g has an isolated singularity at x0, and maps the boundary of some deleted neighborhood of x0 to the unit sphere. We define i(fx0) to be the Brouwer degree of the mapping induced by g on some suitably chosen small sphere around x0.[1]
## The Lefschetz–Hopf theorem
The importance of the fixed-point index is largely due to its role in the LefschetzHopf theorem, which states:
$\sum_{x \in \mathrm{Fix}(f)} i(f,x) = \Lambda_f,$
where Fix(f) is the set of fixed points of f, and Λf is the Lefschetz number of f.
Since the quantity on the left-hand side of the above is clearly zero when f has no fixed points, the Lefschetz–Hopf theorem trivially implies the Lefschetz fixed point theorem.
## Notes
1. ^ A. Katok and B. Hasselblatt(1995), Introduction to the modern theory of dynamical systems, Cambridge University Press, Chapter 8.
## References
• Robert F. Brown: Fixed Point Theory, in: I. M. James, History of Topology, Amsterdam 1999, ISBN 0-444-82375-1, 271–299.
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2014-07-13 17:46:00
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