url stringlengths 14 2.42k | text stringlengths 100 1.02M | date stringlengths 19 19 | metadata stringlengths 1.06k 1.1k |
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https://socratic.org/questions/how-do-you-find-the-derivative-of-ln-x-y-x-2 | # How do you find the derivative of ln(x+y) = x^2?
May 15, 2016
$y ' = 2 x {e}^{{x}^{2}} - 1$
#### Explanation:
The inverse relation is $x + y = {e}^{{x}^{2}}$.
Term-by-term differentiation gives
$1 + y ' = 2 x {r}^{{x}^{2}}$, applying function of function rule..
So, $y ' = 2 x {e}^{{x}^{2}} - 1$ | 2021-07-31 22:18:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 4, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9917871952056885, "perplexity": 7648.609383787814}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154126.73/warc/CC-MAIN-20210731203400-20210731233400-00345.warc.gz"} |
https://www.r-bloggers.com/2019/07/i-didnt-mean-to-ignore-the-median/ | [This article was first published on Posts on R Lover ! a programmer, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)
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This week’s post follows directly from last week’s investigation of data from the 2016 US Census Bureau’s American Community Survey (ACS) Public Use Microdata Sample (PUMS). We explored mean differences in income across several different types of employment status (self-employed, private sector, government, etc.). We found, using bayesian methods, strong evidence for differences across the categories and were able to plot them in a variety of formats using ggplot2 and ggsignif.
While I was very happy with the plot results (which was my primary interest) something was nagging me the whole time I was writing the post. Our outcome measure was income and both our data and the prevailing wisdom from a vast amount of research is that income data tend to be skewed or suffer from a floor effect or any number of ways of saying that perhaps the median (or some other measure of central tendency) is a much better measure than the mean. Ironically, it is actually very common for income to be an exemplar in undergraduate statistics texts to be discussed in context of the dangers of using the mean. Here’s an example from my favorite undergrad text book Learning Statistics with R – section 5.1.5.
As you look at our data plot it is pretty clear that the string of data points on the high end of income in several categories may be unduly influencing our results. The risk wouldn’t be quite so bad if the data were consistent across employment categories but a visual inspection makes us wonder, especially since we can “see” the difference between the median (the solid black line near the center of the boxplot) and the mean (plotted here as a dark red box).
For this post then, we’re going to explore comparing medians in a bayesian framework and along the way spend a minute or two on credible intervals. Read on! I’ll say it again below but comments and critique are always welcomed via disqus or email. You’ll have no trouble finding the icon links in a couple of places.
### To the data cave!
A colleague of mine loves to call her office her “data cave”. I love the term and it reminds me of the old school Batman series on television, “To the Bat Cave, Robin”. So let’s go to our virtual data cave and get things setup.
We’ll once again grab the data set from the WinVector Github site, load the packages we need (hiding the load messages), set my favorite ggplot theme, and do a little factor cleanup. If you need a refresher see the last post for details.
library(tidyverse)
library(BayesFactor)
library(ggsignif)
library(scales)
theme_set(theme_bw()) # set theme
location <- "https://github.com/WinVector/PDSwR2/raw/master/PUMS/incomedata.rds"
incomedata$gp <- NULL incomedata$employment <- fct_recode(incomedata$employment, "Self Not Inc" = "Self employed not incorporated", "Self Incorporated" = "Self employed incorporated", "Private for Profit" = "Employee of a private for profit", "Private Non Profit" = "Private not-for-profit employee", "Federal Government" = "Federal government employee", "State Government" = "State government employee", "Local Government" = "Local government employee" ) incomedata$employment <- forcats::fct_reorder(
.f = incomedata$employment, .x = incomedata$income,
.fun = mean
)
# In case I want to reduce down to 3 more fundamental
# categories Private, Government, or self employed
incomedata$empcategory <- fct_collapse(incomedata$employment,
"Self" = c("Self Not Inc", "Self Incorporated"),
"Private" = c("Private for Profit", "Private Non Profit"),
"Government" = c("Federal Government", "State Government", "Local Government")
)
str(incomedata)
## 'data.frame': 22241 obs. of 6 variables:
## $income : num 22000 21000 25800 25000 20200 36000 20000 30000 23000 5000 ... ##$ age : num 24 31 26 27 27 47 24 41 43 21 ...
## $sex : Factor w/ 2 levels "Male","Female": 1 2 2 2 2 1 1 1 2 2 ... ##$ employment : Factor w/ 7 levels "Self Not Inc",..: 2 4 2 2 4 2 2 2 2 2 ...
## $education : Factor w/ 9 levels "no high school diploma",..: 1 4 4 6 4 2 1 1 6 2 ... ##$ empcategory: Factor w/ 3 levels "Self","Private",..: 2 2 2 2 2 2 2 2 2 2 ...
Okay we have the data we need. For those who cynically believe that visual inspection may be an aphorism for “draw any conclusion you like from the picture” let’s just quickly confirm we have a positive skew with psych::skew (rounded). In our case the answer is round(psych::skew(incomedata$income), 2) = 1.78. While we’re at it let’s confirm my allegation that skewness varies by group using psych::describeBy. I wasn’t making it up! psych::describeBy(incomedata$income,
group = incomedata$employment, mat = TRUE, digits = 2) %>% mutate(.data = ., Employment_Type = group1, trimmed_mean = trimmed) %>% select(Employment_Type, mean, sd, median, skew, n, trimmed_mean) ## Employment_Type mean sd median skew n trimmed_mean ## 1 Self Not Inc 41382.50 34209.02 30000 1.73 829 35792.72 ## 2 Private for Profit 51163.99 38744.57 40000 1.82 16170 44932.92 ## 3 State Government 52612.39 29331.94 47150 1.64 1028 49269.63 ## 4 Private Non Profit 52996.20 34389.69 45000 1.85 1573 48085.83 ## 5 Local Government 53720.47 29438.20 48000 1.29 1492 50763.72 ## 6 Federal Government 65311.26 34790.20 60000 0.73 580 62641.60 ## 7 Self Incorporated 66906.44 51983.28 50000 1.52 569 58579.12 #### We’re skewed! How do we fix it? There are any number of ways we can address the skew issue (I won’t try methods such as: hoping your reader/reviewer is inattentive or praying to your favorite deity). When I say any number I mean a lot of different ways we can address the issue. While non-exhaustive they generally lump into one of three categories. We can trim, we can transform, or we can use non-parametric methods (which is a bit disingenuous since some of the non-parametric methods themselves rely on transforming our data – we’ll get to that in a minute). Let’s briefly discuss. 1. Trim. We can “trim” the high and lows hoping this will eliminate any troubling values at both ends. You may be surprised to know that capability is built into mean() in base R, and mean(incomedata$income, , trim = .1) will work. For those of you paying attention I grabbed that info when I ran psych::describeBy. As you would expect for each of our groups it has produced a trimmed mean which is closer to the median than our original mean. My problem with it is threefold. I hate throwing away data. In this case we have plenty but still. Second, there is no one right answer as to how much we should trim? Why .1? Why not .05 or .2? It can make you suspicious that we are simply picking an amount of trimming that best serves our agenda. Finally, it implicitly makes the extreme values wrong or invalid or “not right”. In this case there is little evidence that this is true. The income values don’t look impossible or wrong, they just seem to reflect that there are people who make much more and that zero is as low as it goes. Personally I tend to only use trimming when I think the measurement is producing bad values typically on both ends of the spectrum.
2. Transform. One of the common solutions to the income data problem is to transform the variable. The transformation can take the form of a square root, a log, or a more complex method such as Box-Cox or Yeo-Johnson. The later is especially useful if your variable can validly take on negative values (negative income anyone?). All of these methods are likely to “fix” our positive skew problem since they will all serve to “pull” our data to the left in a consistent way. All of them make it more difficult to describe our results since for many readers the log of income is not a natural concept. The other limitation here is that there is no one approved transform all will serve to “improve” our skew but we’re subject to being questioned as to why we selected one unless it is a disciplinary custom or for reasons of comparison.
3. Non-parametric. For our particular case there are two non parametric methods we would select. For the overall comparison of income across the 7 levels of Employment it would be the Kruskal-Wallis. For pairwise comparisons as we did last post it would be the Mann-Whitney U (a.k.a. Wilcoxon rank sum) test. If you look back you’ll even notice that the Wilcoxon rank sum test wilcox.test() is the default for ggsignif. As I mentioned earlier these tests are actually a sort of transformation as well since they rely on the ranks of our data rather than the data itself. I admit to having a bias towards (maybe even a fondness for) non-parametric tests. While they aren’t perfect and have limitations I take a certain amount of solace in knowing that any supported findings were by using tests that at least eliminate some number of assumptions.
So after last week’s posting I went sleuthing for the Bayesian equivalent to the Mann Whitney and Kruskal-Wallis tests. Yes I recognize that with powerful software like Stan you can pretty much construct any data analysis you want but I was hoping for something more “off the shelf” something very much like BayesFactor::ttestBF. No luck, I won’t claim I searched everywhere but I did search a lot for anything already done for R. Lot’s of theoretical papers and some questions but no solutions. If I missed it, and someone knows about it, please share.
Since I have been tracking the progress of JASP (unsolicited praise – it looks like a great piece of software especially if you come from the world of SPSS which I do). A lot of the calculation that is done under the very nice GUI is powered by R code and libraries. I noticed that they had recently implemented a Bayesian version of the Mann Whitney so I decided to give it a try. No joy! Using the very same data I have been using for these blog posts it ran for well over an hour and hadn’t completed. When I came back the next day it had completed and produced what appeared to be reasonable results on the surface but I had to do some sleuthing because the very long run time had me confused.
The underlying code is open source and you can see it here. After a little bit of fiddling I was able to separate the JASP interface pieces from the R code that produces the results. I was thwarted again. Even on a much smaller dataset like ggplot2::mpg the results are slow and unstable. That’s not meant as a complaint I know they are working on it and I know working through the ties is challenging.
#### A Eureka and a “YouRankA”
Temporarily thwarted – something in the back of my mind made me remember dimly from the past considering the case of t tests on rankings. It had been a long time (nowadays it so trivial to run for example wilcox.test) and I wasn’t quite sure how comparable the results would be so I went looking. Found research on the frequentist side at least that the methodologies produce consistent results (Zimmerman, 2012).
So if I can’t have a mannwhitneyBF maybe I can ttestBF(rank(income))? Seemed promising and the initial results seemed to be reasonable as well. The only challenge seemed to be that the distribution of rank(anything) tends to be anything but normal. The result tends to be very very flat with almost no tails at all. Almost a uniform distribution. Hmmmmm, which is a worse violation of t-test assumptions, skewed or flat?
As I was puzzling on that I stumbled upon another avenue of exploration from a blog post by Andrew Gelman at Columbia here and here. Quoting…
“Take-home message: I am not saying that the rank-then-inverse-normal-transform strategy is always or even often a good idea. What I’m saying is that, if you were planning to do a rank transformation before analyzing your data, I recommend this z-score approach rather than the classical Wilcoxon method.” Gelman (July 2015)
That sounded promising. Was there an easy way for someone like me to implement his better than Wilcoxon suggestion (point taken that he repeatedly says “I’m not saying you should transform the data. I’m saying that if you were going to do Wilcoxon, then you’re already doing a rank transformation, in which case I think it makes sense to do it in an expandable way, rather than getting hung up on some pre-tabulated test statistics.”)?
Turns out there is. And it’s easy. And it’s already written and packaged in the RNOmni package on CRAN. The function is rankNorm and straight from the description what it does is: “Applies the rank based inverse normal transform (INT) to a numeric vector. The INT can be broken down into a two-step procedure. In the first, the observations are transformed onto the probability scale using the empirical cumulative distribution function (ECDF). In the second, the observations are transformed onto the real line, as Z-scores, using the probit function.”. The key component of the function is qnorm((r - k)/(n - 2 * k + 1)) where r is the rank, n the number of observations, and k = 3/8 = the Blom transform.
I just realized I’ve been rambling for quite some time without a single graphic or plot. Perhaps this is a good time to just display the original income data and what the various transformations would do to them. I’ll plot them as density plots and superimpose the theoretical normal curve in a different color.
Let’s use cowplot and make 7 plots in order:
1. The original income data
2. Transform income to a z score (mean = 0, sd = 1) scale(income)
3. Transform sqrt(income)
4. Transform log(income)
5. Transform VGAM::yeo.johnson(income, lambda = .1)
6. Transform rank(income)
7. Transform RNOmni::rankNorm(income)
So as a first step we’ll use the mutate function to create these new variables. We’ll deliberately name them all so they end with "_income". In a minute we’ll use that little bit of consistency to make our lives easier.
incomedata <-
incomedata %>%
mutate(z_income = as.numeric(scale(income)),
sqrt_income = sqrt(income),
log_income = log(income),
yj_income = VGAM::yeo.johnson(income, lambda = .1),
rank_income = rank(income),
rint_income = RNOmni::rankNorm(income)
)
glimpse(incomedata)
## Observations: 22,241
## Variables: 12
## $income 22000, 21000, 25800, 25000, 20200, 36000, 20000, 300… ##$ age 24, 31, 26, 27, 27, 47, 24, 41, 43, 21, 29, 30, 28, …
## $sex Male, Female, Female, Female, Female, Male, Male, Ma… ##$ employment Private for Profit, Private Non Profit, Private for …
## $education "no high school diploma", "some college credit, no d… ##$ empcategory Private, Private, Private, Private, Private, Private…
## $z_income -0.791115497, -0.817539654, -0.690703699, -0.7118430… ##$ sqrt_income 148.32397, 144.91377, 160.62378, 158.11388, 142.1267…
## $log_income 9.998798, 9.952278, 10.158130, 10.126631, 9.913438, … ##$ yj_income 17.17967, 17.05353, 17.61618, 17.52934, 16.94867, 18…
## $rank_income 3941.5, 3694.5, 5444.5, 5082.0, 3531.0, 9310.0, 3191… ##$ rint_income -0.9260927, -0.9697185, -0.6910214, -0.7438702, -0.9…
A quick glimpse indicates we have what we need. So we have seven variables and we’re going to create 7 identical plots using ggplot. For each plot we’ll plot the density function for our observations (a more granular histogram so to speak). We’ll also plot the theoretical normal curve for that variable using dnorm() with the mean and standard deviation for the variable.
We could just write (mainly cut and paste) the call to ggplot 7 times. But that seems wasteful. Instead we’ll use purrr:pmap to do what we want with a lot less code. pmap wants a list that has as it’s components the 3 things that are going to vary for each of the seven iterations. The 3 things are the name of the variable, the mean, and the sd for the variable. Since ggplot can be fickle about passing it bare variable names we’ll pass it character strings for the name. The mean and the sd of course are numeric types. Again we could manually build those vectors but that would be tedious so let’s use some dplyr::select_at and summarise_at statements to get what we need. Since we were careful to name our variables so that they all end with “income” we can use vars(ends_with("income")) to grab just what we need and not all the variables in incomedata.
a <- incomedata %>%
select_at(vars(ends_with("income"))) %>% names
b <- incomedata %>%
summarise_at(vars(ends_with("income")), mean, na.rm = TRUE)
c <- incomedata %>%
summarise_at(vars(ends_with("income")), sd, na.rm = TRUE)
plot_list <- list(
which = a,
means = b,
stddevs = c
)
plot_list
## $which ## [1] "income" "z_income" "sqrt_income" "log_income" "yj_income" ## [6] "rank_income" "rint_income" ## ##$means
## income z_income sqrt_income log_income yj_income rank_income
## 1 51939.1 2.884567e-17 214.8975 10.61248 18.97702 11121
## rint_income
## 1 -4.205901e-05
##
## $stddevs ## income z_income sqrt_income log_income yj_income rank_income ## 1 37844.16 1 75.88434 0.7348988 2.09446 6419.503 ## rint_income ## 1 0.9991945 That took more text to describe than it took code to enact! Now we invoke ggplot as a function .f. Inside the function call you’ll see ..1, ..2 & ..3 those mark the places where pmap will substitute in the appropriate value of plot_list. The output of pmap is itself a list. A list of ggplot objects. We’ll take that list and give it to cowplot::plot_grid along with some labeling and formatting and voila! list_of_plots <- pmap(.l = plot_list, .f = ~ ggplot(data = incomedata, aes_string(x = ..1) ) + geom_density(alpha = .2) + stat_function(fun = dnorm, color = "red", args = list(mean = ..2, sd = ..3) ) ) cowplot::plot_grid(plotlist = list_of_plots, labels = c("Income (Original)", "Z score Income", "SQRT Income", "Log Income", "Yeo Johnson Income", "Ranked Income", "Rank Inv Norm Income"), label_size = 12, vjust = 1, hjust = 0, scale = .9) Not surprisingly the rank inverse normal transform works as advertised. We have succeeded in taking the ranks and backing them into a near perfect normal curve. That’s good but we did that work to prepare for comparing across employment groups so we would be wise to take a look at our transformed income variable rint_income by group. Once again we’ll make use of psych::describeBy. This time we’ll also include the standard deviation since we’d like to have some sense of whether the variances are at least roughly equal among the groups. While we’re at it we’ll plot the density curves on a by group basis. psych::describeBy(incomedata$rint_income,
group = incomedata$employment, mat = TRUE, digits = 2) %>% mutate(.data = ., Employment_Type = group1, trimmed_mean = trimmed) %>% select(Employment_Type, mean, sd, median, skew, n, trimmed_mean) ## Employment_Type mean sd median skew n trimmed_mean ## 1 Self Not Inc -0.40 1.10 -0.49 0.04 829 -0.41 ## 2 Private for Profit -0.04 1.02 -0.05 0.09 16170 -0.05 ## 3 State Government 0.14 0.80 0.18 -0.29 1028 0.17 ## 4 Private Non Profit 0.10 0.89 0.12 -0.06 1573 0.10 ## 5 Local Government 0.16 0.83 0.20 -0.51 1492 0.20 ## 6 Federal Government 0.43 0.88 0.54 -0.71 580 0.49 ## 7 Self Incorporated 0.31 1.13 0.27 0.02 569 0.31 incomedata %>% ggplot(aes(x = rint_income, fill = employment, color = employment) ) + geom_density(alpha = .2) Well, we certainly seem to have improved the distributions and made them more suitable for parametric manipulation. Certainly looks like there are shifts in central tendency by group. For the astute observer yes I’ll acknowledge that the variances among groups aren’t “equal”. But an Fmax ratio of 2 isn’t that awful. From last post let’s bring back our comparisons_list function which will create a list of all the possible paired comparisons for us. We’ll store them in a list called comp.list. comparisons_list <- function(data, x) { data <- dplyr::select( .data = data, x = !!rlang::enquo(x) ) %>% dplyr::mutate(.data = ., x = droplevels(as.factor(x))) grouplevels <- levels(data$x)
g1_list <- combn(grouplevels, 2)[1, ]
g2_list <- combn(grouplevels, 2)[2, ]
comparisons_list <- lapply(
1:length(g1_list),
function(i) c(
combn(grouplevels, 2)[2, i],
combn(grouplevels, 2)[1, i]
)
)
return(comparisons_list)
}
comp.list <- comparisons_list(incomedata, employment)
#### Welcome back my friends to the show that never ends (ELP)
Last week we produced this plot using the original income data and using ggsignif. We ran the wilcox.test against all the pairings to produce p values on the plot. Here it is unretouched to refresh your memory.
If Gelman and Zimmerman have steered me correctly (or lol if I paid enough attention) we should be able to substitute in rint_income our ranked inverse normalized transformed version of income and get very similar results using the very same code and wilcox.test.
ggplot(data = incomedata,
aes(
x = employment,
y = rint_income,
fill = employment,
group = employment
)) +
geom_boxplot(show.legend = FALSE) +
geom_signif(
comparisons = comp.list,
step_increase = .1
) +
scale_y_continuous(breaks = seq(from = -4,
to = 4,
by = 1)
) +
ggtitle(label = "ACS 2016 Rank Inverse Transformed Income by Employer Type",
subtitle = "Mann Whitney multiple comparisons non directional hypothesis using wilcox.test")
ggplot(data = incomedata,
aes(
x = employment,
y = rank_income,
fill = employment,
group = employment
)) +
geom_boxplot(show.legend = FALSE) +
geom_signif(
comparisons = comp.list,
step_increase = .1
) +
scale_y_continuous(breaks = seq(from = 0,
to = 25000,
by = 5000)
) +
ggtitle(label = "ACS 2016 Ranked Income by Employer Type",
subtitle = "Mann Whitney multiple comparisons non directional hypothesis using wilcox.test")
Reminder - Note that in the two-sample case the estimator for the difference in location parameters does not estimate the difference in medians (a common misconception) but rather the median of the difference between a sample from x and a sample from y.
Very similar indeed! Notice that the reported p values are essentially identical even as we can see from the box plots that the transformations are having an effect.
But of course this has all been prep work to this point. What I’m really after is to shift over to a bayesian framework and generate bayes factors that are the equivalent of the frequentist’s Mann Whitney. So let’s get on with that!
Not having been able to find the R code to do the work directly I’ve transformed the income variable and will now apply the ttestBF function. Last post I wrote a little function called pairwise_bf that generates the bayes factors and manipulates them in a way that makes them suitable for plotting with ggplot and ggsignif. Here it is again.
pairwise_bf <- function(x = NULL,
y = NULL,
display_type = "bf",
k = 2) {
results <- ttestBF(x = x, y = y) %>%
extractBF() %>%
mutate(support = case_when(
bf < .01 ~ "extreme BF01",
bf < .03 & bf >= .01 ~ "very strong BF01",
bf < .1 & bf >= .03 ~ "strong BF01",
bf < 1 / 3 & bf >= .1 ~ "moderate BF01",
bf < 1 & bf >= 1 / 3 ~ "anecdotal BF01",
bf >= 1 & bf < 3 ~ "anecdotal BF10",
bf >= 3 & bf < 10 ~ "moderate BF10",
bf >= 10 & bf < 30 ~ "strong BF10",
bf >= 30 & bf < 100 ~ "very strong BF10",
bf >= 100 ~ "extreme BF10"
)) %>%
mutate(logged = case_when(
bf < 1 ~ paste("log(BF01) = ", round(log(1 / bf), k)),
bf >= 1 ~ paste("log(BF10) = ", round(log(bf), k))
)) %>%
mutate(human = case_when(
bf < .000001 ~ "BF01 >= 1,000,000 : 1",
bf < .001 & bf >= .000001 ~ "BF01 >= 1,000 : 1",
bf < .01 & bf >= .001 ~ "BF01 >= 100 : 1",
bf < 1 & bf >= .01 ~ paste("BF01 = ", round(1 / bf, k), ": 1"),
bf >= 1 & bf < 100 ~ paste("BF01 = ", round(bf, k), ": 1"),
bf >= 100 & bf < 1000 ~ "BF10 >= 100 : 1",
bf >= 1000 & bf < 1000000 ~ "BF10 >= 1,000 : 1",
bf >= 1000000 ~ "BF10 >= 1,000,000 : 1"
))
if (display_type == "support") {
results <- mutate(results, p.value = support)
} else if (display_type == "log") {
results <- mutate(results, p.value = logged)
} else if (display_type == "human") {
results <- mutate(results, p.value = human)
} else {
results <- mutate(results, p.value = bf)
}
return(results)
}
# pairwise_bf(incomedata$employment, incomedata$rint_income)
comp.list2 <- comparisons_list(incomedata, empcategory)
theme_set(hrbrthemes::theme_ipsum())
Using the employment factor in the original dataset has been useful so far. Having 7 levels has allowed us to make sure that the functions supported lots of comparisons. The downside is the resulting plots have an overwhelming amount of information. So let’s shift over to the variable empcategory which collapses employment to just 3 levels. We’ll be comparing the self-employed, to those employed in the private sector, to those employed in local, state or federal jobs.
Seems reasonable to believe that the income levels across those broad categories, might be different. I don’t have any strong prior beliefs or information before this data. It’s not an area I study or have a lot of prior information amount. Clearly, it’s out there, the ACS survey has been running for years, but for now I’m perfectly happy to say my priors are flat.
What we’re going to do:
1. Show all the possible pairwise comparisons for empcategory as bayes factors in ggplot (we’ve already built the list above in comp.list2)
2. Change the theme just to make the plots nicer theme_set(hrbrthemes::theme_ipsum())
3. Apply the ttestBF test from the BayesFactor package to the data piped into ggplot. We’ll compare income (the original untransformed data), rint_income which is our rank inverse transformed variant, and rank_income which is the income data with a simple rank transformation. N.B. - I’ve expressed the ranks so they are directionally equivalent to the raw data so the highest recorded income in our data is $250,000 which corresponds to rank of 22226.5 and after inverse transformation = 3.20299 4. Display the BF10 and/or BF01 values that are between 1 and 100 as is (rounded to 2 digits) and then create ranges between 100 to 1,000, 1,000 to one million and greater than one million. At some point talking about odds over 100:1 (in my humble opinion) loses the need for precision. After all is there really much difference between odds of 1,000:1 versus 1,001:1? 5. Since this post is very much about mean() versus median() both of them are plotted on top of the violin geom. The mean is displayed as a black box and the median is displayed as a dark red circle. #### Caveat Emptor Before I draw any conclusions about the results a reminder. While the data are “real” drawn from the U.S. Census Bureau they most assuredly are not totally representative of the entire U.S. population! The very fact that the max value for income is$250,000 tells us that this isn’t everyone and I’m not sure at all how “representative” it is. Please don’t stray from my desire to investigate the statistical properties of the methods to think I’m trying to convince you of some conclusion about the income levels of Americans.
Conclusions we can draw.
1. Distributions matter! There is a difference between investigating the mean() and the median() income and using bayesian methods doesn’t change that.
2. Bayes factors give us information in both directions! When you look at the first plot using the mean for untransformed income values BF01 = 29.36 tells you that the evidence is “strong” that the mean income for the self-employed is the same as those in the private sector.
3. Whether we use simple rank transformation or reverse inverse transformed ranks our overall conclusions are similar. We have weaker “anecdotal” evidence that there is no difference between the self-employed and those in the private sector. To say it differently we are less convinced from our data that there is no difference. On the other hand we have beyond very strong evidence that government employee income is different than either the self employed or those in the private sector.
4. The difference in the bayes factors generated by using ranks versus rank inverse transformation for our data were not substantively important in magnitude. Yes they were different, for example 1.63 versus 2.48, but don’t lead us to any practically different conclusions.
5. Not withstanding #4 above until there is a “true” equivalent to the Mann Whitney for bayesian analysis I personally am far more comfortable applying a “t-test” like analysis to the data distribution that results from RINT.
#### Let’s get credible
I know this post is already pretty long but I promised myself I’d capture what I learned about generating credible intervals as part of this workflow. I had been asked the question by a reader I did the work and posted it in a comment but now is my chance to capture it more formally.
In my usual fashion I’m not going to try and explain the concept of a credible interval in general when others have done so already and probably done a better job. So if you’re unfamiliar with the concept I recommend a trip to Wikipedia to read up. A simple Google search will also yield lots of informative answers.
What I will focus on for the remainder of this post is simply helping you with the mechanics of constructing credible intervals if you have been following this series of posts.
So far we have been making heavy use of BayesFactor::ttestBF to accomplish our objectives. For example continuing with our last example our plot says the difference between the mean income for the privately employed and government workers is “BF10 >= 1,000:1”. We got that nice neat answer as a result of:
library(bayestestR)
tempdf <- incomedata %>%
filter(empcategory == "Private" | empcategory == "Government") %>%
droplevels %>%
as.data.frame
ttestBF(formula = income ~ empcategory, data = tempdf)
## Bayes factor analysis
## --------------
## [1] Alt., r=0.707 : 357043.1 ±0%
##
## Against denominator:
## Null, mu1-mu2 = 0
## ---
## Bayes factor type: BFindepSample, JZS
There it is – 357,043:1 to be more precise. And note the part that says “Null, mu1-mu2 = 0”. That looks familiar that’s a t test alright it is the difference between the means = 0. So let’s take the output from that ttestBF and but it in an object called bf.object. Sadly it’s a rather complex S4 object that doesn’t contain exactly what we need so a little more work is required. What we need to do is run some Markov chain Monte Carlo methods to sample from the posterior distribution to get our answer. While that sounds wickedly complex the function we need is built right into BayesFactor as posterior so one line does the trick posterior.data <- posterior(bf.object, iterations = 10000). What’s in posterior.data? About 10,000 rows so let’s head it.
The first column mu is the overall mean, the second column beta (Private - Government) is exactly what we need to work on it’s the difference in means “mu1-mu2” between Private and Government. The sign tells us that at least for the first 7 rows Government is higher which agrees with our plot. I have no desire to scroll through 10,000 rows so let’s get some summary information and even a plot of our 10,000 samples.
bf.object <- ttestBF(formula = income ~ empcategory, data = tempdf)
posterior.data <- posterior(bf.object, iterations = 10000)
## Markov Chain Monte Carlo (MCMC) output:
## Start = 1
## End = 7
## Thinning interval = 1
## mu beta (Private - Government) sig2 delta g
## [1,] 53712.39 -4138.397 1388783139 -0.11104907 0.3962511
## [2,] 53328.80 -3951.455 1372469541 -0.10666102 0.5661250
## [3,] 53798.28 -3568.424 1396930789 -0.09547487 0.1124928
## [4,] 53216.72 -3013.836 1401327895 -0.08051000 0.2242916
## [5,] 52885.07 -2276.827 1415166246 -0.06052381 0.4882304
## [6,] 52591.16 -3624.291 1381320679 -0.09751598 0.5180922
## [7,] 53115.20 -3875.964 1410243480 -0.10321260 2.2364977
summary(posterior.data)
##
## Iterations = 1:10000
## Thinning interval = 1
## Number of chains = 1
## Sample size per chain = 10000
##
## 1. Empirical mean and standard deviation for each variable,
## plus standard error of the mean:
##
## Mean SD Naive SE Time-series SE
## mu 5.343e+04 3.639e+02 3.639e+00 6.213e+00
## beta (Private - Government) -4.197e+03 7.303e+02 7.303e+00 1.269e+01
## sig2 1.396e+09 1.361e+07 1.361e+05 1.361e+05
## delta -1.124e-01 1.955e-02 1.955e-04 3.396e-04
## g 2.744e+00 4.663e+01 4.663e-01 4.663e-01
##
## 2. Quantiles for each variable:
##
## 2.5% 25% 50% 75%
## mu 5.271e+04 5.318e+04 5.343e+04 5.367e+04
## beta (Private - Government) -5.614e+03 -4.689e+03 -4.198e+03 -3.718e+03
## sig2 1.370e+09 1.386e+09 1.395e+09 1.405e+09
## delta -1.501e-01 -1.255e-01 -1.123e-01 -9.948e-02
## g 6.897e-02 1.877e-01 3.776e-01 9.061e-01
## 97.5%
## mu 5.413e+04
## beta (Private - Government) -2.745e+03
## sig2 1.422e+09
## delta -7.344e-02
## g 1.009e+01
plot(posterior.data[,"beta (Private - Government)"])
Whether we look at the right hand plot, or the tabular information Mean = -4.188e+03, 50% = -4.182e+03, sure looks like the difference estimate is about ~ -$4,180. Also telling is 2.5% = -5.615e+03 and 97.5% = -2.775e+03 which means 95% of our estimates were between -$5,615 and -$2,775! Anyone care to guess what the 95% credible interval is? Well of course we don’t have to guess. We can take our posterior.data object and change it to a dataframe. After we’ve done that we can use bayestestR::ci (you may have noticed I loaded the library a few lines back). And there you have it the 89% credible interval is [-5352.79, -3042.40]. Wait. What? 89%???? posterior.data <- as.data.frame(posterior.data) str(posterior.data) ## 'data.frame': 10000 obs. of 5 variables: ##$ mu : num 53712 53329 53798 53217 52885 ...
## $beta (Private - Government): num -4138 -3951 -3568 -3014 -2277 ... ##$ sig2 : num 1.39e+09 1.37e+09 1.40e+09 1.40e+09 1.42e+09 ...
## $delta : num -0.111 -0.1067 -0.0955 -0.0805 -0.0605 ... ##$ g : num 0.396 0.566 0.112 0.224 0.488 ...
bayestestR::ci(posterior.data$beta (Private - Government)) ## # Credible Interval ## ## 89% CI ## [-5349.28, -3016.02] For an explanation of why please see the bayestestR documentation. While you’re there good chance to review what a credible interval is, how it is interpreted, and our next topic which is that there are at least two equally valid ways of computing it. The first we used already it’s the Equal-tailed Interval (ETI) which is the default when you invoke ci. The other is Highest Density Interval (HDI) hdi(). They do a good job of explaining so I won’t repeat merely show you how. ci() will accept a vector with multiple cut-offs so let’s do 89% and 95% once with equal-tailed and then with highest density. To be fair at least for the equal tailed you can use base R’s quantile function to get the same answers. BTW back a little bit when we were looking at summary(posterior.data) the answer for 95% was right there just hard to pick out. ci(posterior.data$beta (Private - Government),
ci = c(.89, .95))
## # Credible Intervals
##
## 89% CI
## [-5349.28, -3016.02]
##
##
## 95% CI
## [-5614.50, -2745.04]
hdi(posterior.data$beta (Private - Government), ci = c(.89, .95)) ## # Highest Density Intervals ## ## 89% HDI ## [-5327.73, -2999.18] ## ## ## 95% HDI ## [-5625.08, -2759.35] round(quantile(posterior.data$beta (Private - Government),
probs = c(.025,
.055,
(1 - .055),
(1 - .025)
)
),
2)
## 2.5% 5.5% 94.5% 97.5%
## -5614.50 -5349.28 -3016.02 -2745.04
So given our observed data, the mean difference in income between the private sector respondents and the government employed respondents has a 95% probability of falling between -$5,614.99 and -$2,774.84.
We can preform the exact same operations on rint_income if we want to convince ourselves that the mean difference (which for rint_income is more akin to the median difference) is not equal to zero). The problem of course is that it doesn’t return a dollar value and our transformations are non-linear.
bf.object <- ttestBF(formula = rint_income ~ empcategory, data = tempdf)
posterior.data <- posterior(bf.object, iterations = 10000)
summary(posterior.data)
##
## Iterations = 1:10000
## Thinning interval = 1
## Number of chains = 1
## Sample size per chain = 10000
##
## 1. Empirical mean and standard deviation for each variable,
## plus standard error of the mean:
##
## Mean SD Naive SE Time-series SE
## mu 0.08782 0.009480 9.480e-05 1.581e-04
## beta (Private - Government) -0.22912 0.019066 1.907e-04 3.165e-04
## sig2 0.96652 0.009474 9.474e-05 9.474e-05
## delta -0.23306 0.019439 1.944e-04 3.222e-04
## g 3.00664 61.759312 6.176e-01 6.176e-01
##
## 2. Quantiles for each variable:
##
## 2.5% 25% 50% 75% 97.5%
## mu 0.06913 0.08155 0.08785 0.09424 0.1062
## beta (Private - Government) -0.26596 -0.24200 -0.22917 -0.21659 -0.1917
## sig2 0.94811 0.96013 0.96647 0.97285 0.9851
## delta -0.27039 -0.24625 -0.23310 -0.22022 -0.1948
## g 0.07617 0.19808 0.40026 0.94421 9.5910
plot(posterior.data[,"beta (Private - Government)"])
posterior.data <- as.data.frame(posterior.data)
bayestestR::ci(posterior.data$beta (Private - Government)) ## # Credible Interval ## ## 89% CI ## [-0.26, -0.20] ci(posterior.data$beta (Private - Government),
ci = c(.89, .95))
## # Credible Intervals
##
## 89% CI
## [-0.26, -0.20]
##
##
## 95% CI
## [-0.27, -0.19]
hdi(posterior.data$beta (Private - Government), ci = c(.89, .95)) ## # Highest Density Intervals ## ## 89% HDI ## [-0.26, -0.20] ## ## ## 95% HDI ## [-0.27, -0.19] #### A final look at why it matters So I wanted to end this post by reinforcing why mean() vs median() matters. Continuing to focus on the differences between income in the private sector and government employees remember we saw the BF associated with the difference between the two groups increase an order of magnitude when we shifted from assessing mean differences to median differences. Why? Well take a look at the differences expressed in the original dollar units. The mean difference is about ~$4,200, the median difference is more than twice as much at ~$10,000, that’s a big difference. psych::describeBy(incomedata$income,
group = incomedata$empcategory, mat = TRUE, digits = 2) %>% mutate(.data = ., Employment_Type = group1, trimmed_mean = trimmed) %>% select(Employment_Type, mean, sd, median, skew, n, trimmed_mean) ## Employment_Type mean sd median skew n trimmed_mean ## 1 Self 51771.00 44154.93 40000 1.82 1398 44342.71 ## 2 Private 51326.42 38381.11 40000 1.82 17743 45221.13 ## 3 Government 55521.62 30830.33 50000 1.28 3100 52263.65 a <- median(incomedata[incomedata$empcategory == "Private", "income"])
b <- median(incomedata[incomedata$empcategory == "Government", "income"]) c <- mean(incomedata[incomedata$empcategory == "Private", "income"])
d <- mean(incomedata[incomedata\$empcategory == "Government", "income"])
median_diff_priv_govt <- a - b
median_diff_priv_govt
## [1] -10000
mean_diff_priv_govt <- c - d
mean_diff_priv_govt
## [1] -4195.196
For a final plot we’ll show it in the original units and then to make it easier to see give you a zoomed perspective with coord_cartesian(). The dashed lines are the median values by group (which are color-coded) and the solid lines are the mean.
tempdf %>%
ggplot(aes(x = income,
fill = empcategory,
color = empcategory)
) +
geom_density(alpha = .3) +
geom_vline(xintercept = a, linetype = 5, color = "coral1") +
geom_vline(xintercept = b, linetype = 5, color = "cyan3") +
geom_vline(xintercept = c, color = "coral1") +
geom_vline(xintercept = d, color = "cyan3") +
scale_x_continuous(label = dollar)
tempdf %>%
ggplot(aes(x = income,
fill = empcategory,
color = empcategory)
) +
geom_density(alpha = .3) +
geom_vline(xintercept = a, linetype = 5, color = "coral1") +
geom_vline(xintercept = b, linetype = 5, color = "cyan3") +
geom_vline(xintercept = c, color = "coral1") +
geom_vline(xintercept = d, color = "cyan3") +
scale_x_continuous(label = dollar) +
coord_cartesian(xlim = c(10000, 100000))
#### Done
Hope you enjoyed the post. Comments always welcomed.
Chuck | 2021-05-13 15:02:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43160971999168396, "perplexity": 2694.9866572019064}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-21/segments/1620243989814.35/warc/CC-MAIN-20210513142421-20210513172421-00316.warc.gz"} |
https://www.zbmath.org/authors/?q=ai%3Ahu.hao | # zbMATH — the first resource for mathematics
## Hu, Hao
Compute Distance To:
Author ID: hu.hao Published as: Hu, H.; Hu, Hao
Documents Indexed: 63 Publications since 1990
all top 5
#### Co-Authors
0 single-authored 6 Jiang, Nan 4 Dang, Yijie 4 Jiang, Bin 4 Meng, Zeng 4 Yang, Hao 4 Zhang, Wenyin 4 Zhou, Huanlin 3 Ji, Zhuoxiao 3 Li, Gang 2 Deng, Youjin 2 Fu, Yiming 2 Ge, Jidong 2 Li, Jian 2 Lu, Xiaowei 2 Sotirov, Renata 2 Wu, Yichao 2 Yao, Weixin 1 Ai, Xiaochuan 1 Blöte, Henk W. J. 1 Cai, Yongquan 1 Chang, Chaowen 1 Chen, Jianhua 1 Chen, Yang 1 Dong, Xuan 1 Eckhoff, Philip 1 Hao, Peng 1 Hu, Haiyang 1 Hu, Xinyi 1 Huang, Liguo 1 Ji, Yangjian 1 Jiang, Wei 1 Jin, Lu 1 Keshtegar, Behrooz 1 Laurent, Monique 1 Leng, Qiang 1 Li, Chuanyi 1 Li, Kai 1 Li, Xiang 1 Li, Xiang 1 Liu, Haiqiang 1 Liu, Jing 1 Liu, Yuling 1 Lu, Jian 1 Lu, Ping 1 Luo, Bin 1 Luo, Huawu 1 Nigmatulina, Karima R. 1 Niu, Zhongrong 1 Qi, Guo-Jun 1 Qi, Guoning 1 Qin, Zheng 1 Shang, Changjing 1 Shumsky, Alexey Ye. 1 Song, Mingyue 1 Tan, Jinglei 1 Tang, Jinhui 1 Wang, Bo 1 Wang, Linhong 1 Wang, Xingting 1 Wu, Budan 1 Yang, Hongji 1 Yang, Yingjie 1 Zhang, Caixia 1 Zhang, Hongqi 1 Zhang, Silan 1 Zhang, Yuchen 1 Zhang, Yuchen 1 Zheng, Jianlong 1 Zhu, Weijie 1 Zhu, Yuanguo
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#### Serials
4 International Journal of Theoretical Physics 2 Journal of the Franklin Institute 2 Applied Mathematical Modelling 2 Computational Statistics and Data Analysis 2 Mathematical Problems in Engineering 2 Quantum Information Processing 1 Acta Mechanica 1 Mathematical Biosciences 1 Nuclear Physics. B 1 Information Sciences 1 Journal of Number Theory 1 European Journal of Combinatorics 1 Journal of Nanjing University. Natural Sciences 1 Journal of Tsinghua University. Science and Technology 1 Journal of Mathematical Imaging and Vision 1 Engineering Analysis with Boundary Elements 1 INFORMS Journal on Computing 1 Soft Computing 1 Journal of Hunan University. Natural Sciences 1 Journal of Combinatorial Optimization 1 IEEE Transactions on Image Processing 1 Journal of Changsha Communications University 1 Journal of Zhejiang University. Engineering Science 1 Journal of Systems Science and Complexity 1 Chinese Journal of Computational Mechanics 1 International Journal of Computational Methods 1 Journal of Physics A: Mathematical and Theoretical 1 Involve 1 IET Control Theory & Applications 1 International Journal of Systems Science. Principles and Applications of Systems and Integration
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#### Fields
12 Computer science (68-XX) 10 Operations research, mathematical programming (90-XX) 8 Mechanics of deformable solids (74-XX) 5 Quantum theory (81-XX) 4 Numerical analysis (65-XX) 4 Systems theory; control (93-XX) 4 Information and communication theory, circuits (94-XX) 3 Statistics (62-XX) 3 Statistical mechanics, structure of matter (82-XX) 2 Combinatorics (05-XX) 2 Dynamical systems and ergodic theory (37-XX) 1 Number theory (11-XX) 1 Associative rings and algebras (16-XX) 1 Ordinary differential equations (34-XX) 1 Sequences, series, summability (40-XX) 1 Probability theory and stochastic processes (60-XX) 1 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 Biology and other natural sciences (92-XX)
#### Citations contained in zbMATH Open
39 Publications have been cited 158 times in 123 Documents Cited by Year
Perturbation method for periodic solutions of nonlinear jerk equations. Zbl 1221.70017
Hu, H.
2008
Solutions of nonlinear oscillators with fractional powers by an iteration procedure. Zbl 1243.34005
Hu, H.
2006
A classical iteration procedure valid for certain strongly nonlinear oscillators. Zbl 1241.70032
Hu, H.; Tang, J. H.
2007
Solutions of the Duffing-harmonic oscillator by an iteration procedure. Zbl 1243.65100
Hu, H.
2006
Comparison of two Lindstedt-Poincaré-type perturbation methods. Zbl 1236.34050
Hu, H.; Xiong, Z. G.
2004
Robust $$H_\infty$$ reliable control for uncertain switched systems with circular disk pole constraints. Zbl 1281.93039
Hu, Hao; Jiang, Bin; Yang, Hao
2013
A convolution integral method for certain strongly nonlinear oscillators. Zbl 1238.34068
Hu, H.; Tang, J. H.
2005
A high-order spectral difference method for unstructured dynamic grids. Zbl 1271.76234
Yu, M. L.; Wang, Z. J.; Hu, H.
2011
A classical perturbation technique which is valid for large parameters. Zbl 1236.65106
Hu, H.
2004
Super parametric convex model and its application for non-probabilistic reliability-based design optimization. Zbl 07166645
Meng, Zeng; Hu, Hao; Zhou, Huanlin
2018
A hybrid reliability-based design optimization approach with adaptive chaos control using Kriging model. Zbl 1359.74348
Li, Gang; Meng, Zeng; Hao, Peng; Hu, Hao
2016
Positive definite constrained least-squares estimation of matrices. Zbl 0847.65024
Hu, H.
1995
Analysis and improvement of the quantum image matching. Zbl 1381.81032
Dang, Yijie; Jiang, Nan; Hu, Hao; Zhang, Wenyin
2017
Complete solutions of the simultaneous Pell equations $$x^2 - 24y^2 = 1$$ and $$y^2 - pz^2 = 1$$. Zbl 1311.11020
Ai, Xiaochuan; Chen, Jianhua; Zhang, Silan; Hu, Hao
2015
A projection method for solving infinite systems of linear inequalities. Zbl 0829.90127
Hu, H.
1994
A numerical procedure for finding the positive definite matrix closest to a patterned matrix. Zbl 0850.62486
Hu, H.; Olkin, I.
1991
A novel quantum image compression method based on JPEG. Zbl 1394.81078
Jiang, Nan; Lu, Xiaowei; Hu, Hao; Dang, Yijie; Cai, Yongquan
2018
Special cases of the quadratic shortest path problem. Zbl 1396.90094
Hu, Hao; Sotirov, Renata
2018
The robust EM-type algorithms for log-concave mixtures of regression models. Zbl 06914531
Hu, Hao; Yao, Weixin; Wu, Yichao
2017
A hybrid sequential approximate programming method for second-order reliability-based design optimization approach. Zbl 1401.74230
Meng, Zeng; Zhou, Huanlin; Li, Gang; Hu, Hao
2017
Maximum likelihood estimation of the mixture of log-concave densities. Zbl 06918278
Hu, Hao; Wu, Yichao; Yao, Weixin
2016
Iteration calculations of periodic solutions to nonlinear jerk equations. Zbl 1381.34060
Hu, H.; Zheng, M. Y.; Guo, Y. J.
2010
A credibilistic goal programming model for inventory routing problem with hazardous materials. Zbl 1398.90012
Hu, Hao; Li, Jian; Li, Xiang
2018
A new approach to robust reliable $$H_\infty$$ control for uncertain nonlinear systems. Zbl 1333.93100
Hu, Hao; Jiang, Bin; Yang, Hao
2016
Robust fault-tolerant control for uncertain delta operator switched systems. Zbl 1286.93082
Hu, Hao; Jiang, Bin; Yang, Hao
2014
Analysis of linear resisted projectile motion using the Lambert W function. Zbl 1398.70049
Hu, H.; Zhao, Y. P.; Guo, Y. J.; Zheng, M. Y.
2012
Thermal effects on the wake of a heated circular cylinder operating in mixed convection regime. Zbl 1241.76030
Hu, H.; Koochesfahani, M. M.
2011
Simplified continuous finite element method for a class of nonlinear oscillating equations. Zbl 1243.65099
Xiong, Z. G.; Hu, H.
2005
Geometric condition measures and smoothness condition measures for closed convex sets and linear regularity of infinitely many closed convex sets. Zbl 1129.90356
Hu, H.
2005
A projective method for rescaling a diagonally stable matrix to be positive definite. Zbl 0762.65026
Hu, H.
1992
Quantum image encryption based on hHenon mapping. Zbl 1415.37113
Jiang, Nan; Dong, Xuan; Hu, Hao; Ji, Zhuoxiao; Zhang, Wenyin
2019
Quantum adder for superposition states. Zbl 1412.81102
Lu, Xiaowei; Jiang, Nan; Hu, Hao; Ji, Zhuoxiao
2018
New insights into approaches to evaluating intention and path for network multistep attacks. Zbl 1427.68022
Hu, Hao; Liu, Yuling; Yang, Yingjie; Zhang, Hongqi; Zhang, Yuchen
2018
Image classification based on quantum K-nearest-neighbor algorithm. Zbl 1398.81052
Dang, Yijie; Jiang, Nan; Hu, Hao; Ji, Zhuoxiao; Zhang, Wenyin
2018
Computing indicators of Radford algebras. Zbl 1392.16026
Hu, Hao; Hu, Xinyi; Wang, Linhong; Wang, Xingting
2018
Quantum point cloud and its compression. Zbl 1387.81159
Jiang, Nan; Hu, Hao; Dang, Yijie; Zhang, Wenyin
2017
Non-fragile reliable $$\mathcal D$$-stabilization for delta operator switched linear systems. Zbl 1347.93217
Hu, Hao; Jiang, Bin; Yang, Hao; Shumsky, Alexey
2016
Boundary element methods for boundary condition inverse problems in elasticity using PCGM and CGM regularization. Zbl 1287.74015
Zhou, Huanlin; Jiang, Wei; Hu, Hao; Niu, Zhongrong
2013
A globally convergent method for semi-infinite linear programming. Zbl 0856.90111
Hu, H.
1996
Quantum image encryption based on hHenon mapping. Zbl 1415.37113
Jiang, Nan; Dong, Xuan; Hu, Hao; Ji, Zhuoxiao; Zhang, Wenyin
2019
Super parametric convex model and its application for non-probabilistic reliability-based design optimization. Zbl 07166645
Meng, Zeng; Hu, Hao; Zhou, Huanlin
2018
A novel quantum image compression method based on JPEG. Zbl 1394.81078
Jiang, Nan; Lu, Xiaowei; Hu, Hao; Dang, Yijie; Cai, Yongquan
2018
Special cases of the quadratic shortest path problem. Zbl 1396.90094
Hu, Hao; Sotirov, Renata
2018
A credibilistic goal programming model for inventory routing problem with hazardous materials. Zbl 1398.90012
Hu, Hao; Li, Jian; Li, Xiang
2018
Quantum adder for superposition states. Zbl 1412.81102
Lu, Xiaowei; Jiang, Nan; Hu, Hao; Ji, Zhuoxiao
2018
New insights into approaches to evaluating intention and path for network multistep attacks. Zbl 1427.68022
Hu, Hao; Liu, Yuling; Yang, Yingjie; Zhang, Hongqi; Zhang, Yuchen
2018
Image classification based on quantum K-nearest-neighbor algorithm. Zbl 1398.81052
Dang, Yijie; Jiang, Nan; Hu, Hao; Ji, Zhuoxiao; Zhang, Wenyin
2018
Computing indicators of Radford algebras. Zbl 1392.16026
Hu, Hao; Hu, Xinyi; Wang, Linhong; Wang, Xingting
2018
Analysis and improvement of the quantum image matching. Zbl 1381.81032
Dang, Yijie; Jiang, Nan; Hu, Hao; Zhang, Wenyin
2017
The robust EM-type algorithms for log-concave mixtures of regression models. Zbl 06914531
Hu, Hao; Yao, Weixin; Wu, Yichao
2017
A hybrid sequential approximate programming method for second-order reliability-based design optimization approach. Zbl 1401.74230
Meng, Zeng; Zhou, Huanlin; Li, Gang; Hu, Hao
2017
Quantum point cloud and its compression. Zbl 1387.81159
Jiang, Nan; Hu, Hao; Dang, Yijie; Zhang, Wenyin
2017
A hybrid reliability-based design optimization approach with adaptive chaos control using Kriging model. Zbl 1359.74348
Li, Gang; Meng, Zeng; Hao, Peng; Hu, Hao
2016
Maximum likelihood estimation of the mixture of log-concave densities. Zbl 06918278
Hu, Hao; Wu, Yichao; Yao, Weixin
2016
A new approach to robust reliable $$H_\infty$$ control for uncertain nonlinear systems. Zbl 1333.93100
Hu, Hao; Jiang, Bin; Yang, Hao
2016
Non-fragile reliable $$\mathcal D$$-stabilization for delta operator switched linear systems. Zbl 1347.93217
Hu, Hao; Jiang, Bin; Yang, Hao; Shumsky, Alexey
2016
Complete solutions of the simultaneous Pell equations $$x^2 - 24y^2 = 1$$ and $$y^2 - pz^2 = 1$$. Zbl 1311.11020
Ai, Xiaochuan; Chen, Jianhua; Zhang, Silan; Hu, Hao
2015
Robust fault-tolerant control for uncertain delta operator switched systems. Zbl 1286.93082
Hu, Hao; Jiang, Bin; Yang, Hao
2014
Robust $$H_\infty$$ reliable control for uncertain switched systems with circular disk pole constraints. Zbl 1281.93039
Hu, Hao; Jiang, Bin; Yang, Hao
2013
Boundary element methods for boundary condition inverse problems in elasticity using PCGM and CGM regularization. Zbl 1287.74015
Zhou, Huanlin; Jiang, Wei; Hu, Hao; Niu, Zhongrong
2013
Analysis of linear resisted projectile motion using the Lambert W function. Zbl 1398.70049
Hu, H.; Zhao, Y. P.; Guo, Y. J.; Zheng, M. Y.
2012
A high-order spectral difference method for unstructured dynamic grids. Zbl 1271.76234
Yu, M. L.; Wang, Z. J.; Hu, H.
2011
Thermal effects on the wake of a heated circular cylinder operating in mixed convection regime. Zbl 1241.76030
Hu, H.; Koochesfahani, M. M.
2011
Iteration calculations of periodic solutions to nonlinear jerk equations. Zbl 1381.34060
Hu, H.; Zheng, M. Y.; Guo, Y. J.
2010
Perturbation method for periodic solutions of nonlinear jerk equations. Zbl 1221.70017
Hu, H.
2008
A classical iteration procedure valid for certain strongly nonlinear oscillators. Zbl 1241.70032
Hu, H.; Tang, J. H.
2007
Solutions of nonlinear oscillators with fractional powers by an iteration procedure. Zbl 1243.34005
Hu, H.
2006
Solutions of the Duffing-harmonic oscillator by an iteration procedure. Zbl 1243.65100
Hu, H.
2006
A convolution integral method for certain strongly nonlinear oscillators. Zbl 1238.34068
Hu, H.; Tang, J. H.
2005
Simplified continuous finite element method for a class of nonlinear oscillating equations. Zbl 1243.65099
Xiong, Z. G.; Hu, H.
2005
Geometric condition measures and smoothness condition measures for closed convex sets and linear regularity of infinitely many closed convex sets. Zbl 1129.90356
Hu, H.
2005
Comparison of two Lindstedt-Poincaré-type perturbation methods. Zbl 1236.34050
Hu, H.; Xiong, Z. G.
2004
A classical perturbation technique which is valid for large parameters. Zbl 1236.65106
Hu, H.
2004
A globally convergent method for semi-infinite linear programming. Zbl 0856.90111
Hu, H.
1996
Positive definite constrained least-squares estimation of matrices. Zbl 0847.65024
Hu, H.
1995
A projection method for solving infinite systems of linear inequalities. Zbl 0829.90127
Hu, H.
1994
A projective method for rescaling a diagonally stable matrix to be positive definite. Zbl 0762.65026
Hu, H.
1992
A numerical procedure for finding the positive definite matrix closest to a patterned matrix. Zbl 0850.62486
Hu, H.; Olkin, I.
1991
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#### Cited by 277 Authors
8 Ramos, Juan I. 6 Hao, Peng 6 Hu, Hao 6 Zhou, Rigui 5 Hu, Wenwen 4 Liang, Chunlei 4 Mathiyalagan, Kalidass 4 Sakthivel, Rathinasamy 3 Goberna, Miguel Angel 3 González-Gutiérrez, Enrique 3 Ivanov Todorov, Maxim 3 Keshtegar, Behrooz 3 Marinca, Vasile 3 Wang, Bo 3 Wang, Yutian 3 Yao, Weixin 2 Arunkumar, Arumugham 2 Cveticanin, Livija 2 Du, Hai-En 2 Er, Guokang 2 Fan, Ping 2 Ganji, Davood Domiri 2 Haque, B. M. Ikramul 2 Herişanu, Nicolae 2 Hu, Hui 2 Iu, Vai Pan 2 Kovacic, Ivana 2 Lai, Siu Kai 2 Li, Gang 2 Li, Yaochong 2 Lim, Chu-Wee 2 Liu, Xingao 2 López-Cerdá, Marco Antonio 2 Luo, Gaofeng 2 Meng, Zeng 2 Pakdemirli, Mehmet 2 Raydan, Marcos 2 Rong, Yongwu 2 Shi, Peng 2 Sotirov, Renata 2 Wang, Jian 2 Wang, Lai 2 Wang, Lei 2 Xiang, Sijia 2 Yang, Hao 2 Yang, Jingjing 2 Yu, Meilin 2 Zheng, Mingyuan 2 Zhou, Huanlin 1 Akçı, Ceren 1 Aksoy, Yiğit 1 Al-Khaled, Kamel M. 1 Alquran, Marwan Taiseer 1 Altınel, İsmail Kuban 1 Anthoni, Selvaraj Marshal 1 Asifuzzaman, Md. 1 Averyanov, Yaroslav 1 Babaee, Hessam 1 Bagherpour, Negin 1 Barari, Amin 1 Beléndez, Augusto 1 Beléndez, Tarsicio 1 Belkić, Dževad 1 Bhaya, Amit 1 Bota, Constantin 1 Bundău, Olivia 1 Cai, Ping 1 Cao, Lixiong 1 Caruntu, Bogdan 1 Chang, Chaowen 1 Chen, Bingqian 1 Chen, Dinggeng 1 Chen, Fang 1 Chen, Hanshu 1 Chen, Liqun 1 Chen, Weiwei 1 Chen, Wenqian 1 Chen, Yanping 1 Chen, Yunmo 1 Cheng, Zhibo 1 Chryssostomidis, C. 1 Cipu, Mihai 1 de Meijer, Frank 1 Drăgănescu, Gh. E. 1 Escalante, René 1 Famouri, M. 1 Fang, Jian’an 1 Feng, Shao-dong 1 Ferrer, Alberto 1 Fu, Jiangfeng 1 Fu, Ruiqin 1 Gao, Ning 1 Gao, Qiang 1 Gao, Shujie 1 García López, Carmen María 1 Geng, Ya-Cong 1 Ghous, Imran 1 Gobbert, Matthias K. 1 Gu, Kaixuan 1 Guo, Qing ...and 177 more Authors
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#### Cited in 64 Serials
9 Applied Mathematics and Computation 8 Computer Methods in Applied Mechanics and Engineering 7 International Journal of Theoretical Physics 5 Acta Mechanica 5 Applied Mathematical Modelling 4 Computers and Fluids 4 Journal of the Franklin Institute 4 Chaos, Solitons and Fractals 4 Mathematical Problems in Engineering 3 Journal of Computational Physics 3 Computational Statistics and Data Analysis 3 International Journal of Computational Methods 2 Computers & Mathematics with Applications 2 Mathematical Methods in the Applied Sciences 2 Periodica Mathematica Hungarica 2 Journal of Computational and Applied Mathematics 2 Annals of Operations Research 2 Nonlinear Dynamics 2 Journal of Combinatorial Optimization 2 Nonlinear Analysis. Real World Applications 2 Quantum Information Processing 2 Nonlinear Analysis. Hybrid Systems 2 International Journal of Systems Science. Principles and Applications of Systems and Integration 1 Journal of Fluid Mechanics 1 Archiv der Mathematik 1 Fuzzy Sets and Systems 1 International Journal of Mathematics and Mathematical Sciences 1 Journal of Number Theory 1 Meccanica 1 Proceedings of the American Mathematical Society 1 Theoretical Computer Science 1 Systems & Control Letters 1 Circuits, Systems, and Signal Processing 1 Applied Mathematics and Mechanics. (English Edition) 1 Optimization 1 Statistical Science 1 Numerical Methods for Partial Differential Equations 1 Mathematical and Computer Modelling 1 Numerical Algorithms 1 European Journal of Operational Research 1 International Journal of Computer Mathematics 1 Linear Algebra and its Applications 1 Archive of Applied Mechanics 1 Computational Optimization and Applications 1 Numerical Linear Algebra with Applications 1 Journal of the Egyptian Mathematical Society 1 Top 1 INFORMS Journal on Computing 1 Journal of Mathematical Chemistry 1 Optimization Methods & Software 1 Soft Computing 1 Discrete Dynamics in Nature and Society 1 Qualitative Theory of Dynamical Systems 1 Journal of Systems Science and Complexity 1 Journal of Applied Mathematics and Computing 1 4OR 1 Advances in Difference Equations 1 International Journal of Number Theory 1 Inverse Problems in Science and Engineering 1 Acta Mechanica Sinica 1 Advances in Data Analysis and Classification. ADAC 1 Optimization Letters 1 Asian Journal of Control 1 Mathematical Sciences
all top 5
#### Cited in 26 Fields
38 Numerical analysis (65-XX) 31 Ordinary differential equations (34-XX) 24 Operations research, mathematical programming (90-XX) 14 Systems theory; control (93-XX) 12 Mechanics of deformable solids (74-XX) 11 Computer science (68-XX) 10 Statistics (62-XX) 10 Mechanics of particles and systems (70-XX) 8 Fluid mechanics (76-XX) 7 Quantum theory (81-XX) 5 Number theory (11-XX) 5 Partial differential equations (35-XX) 5 Information and communication theory, circuits (94-XX) 4 Calculus of variations and optimal control; optimization (49-XX) 4 Biology and other natural sciences (92-XX) 3 Dynamical systems and ergodic theory (37-XX) 3 Probability theory and stochastic processes (60-XX) 2 Operator theory (47-XX) 2 Game theory, economics, finance, and other social and behavioral sciences (91-XX) 1 General and overarching topics; collections (00-XX) 1 Combinatorics (05-XX) 1 Linear and multilinear algebra; matrix theory (15-XX) 1 Associative rings and algebras (16-XX) 1 Nonassociative rings and algebras (17-XX) 1 Integral equations (45-XX) 1 Convex and discrete geometry (52-XX) | 2021-06-12 17:01:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6293498873710632, "perplexity": 14707.331955149426}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-25/segments/1623487586239.2/warc/CC-MAIN-20210612162957-20210612192957-00030.warc.gz"} |
https://isbe-online.org/?ui=english&mod=info&act=view&id=4749 | Publications
[2021-Vol.19-Issue 4]Bio-inspired Leg Design for a Heavy-Duty Hexapod Robot
Post: 2022-09-15 08:53 View:63
Journal of Bionic Engineering (2022) 19:975–990 https://doi.org/10.1007/s42235-022-00192-2
Bio-inspired Leg Design for a Heavy-Duty Hexapod Robot
Haoyuan Yi1 · Zhenyu Xu1,2 · Xueting Xin2 · Liming Zhou2 · Xin Luo1
1 State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan 430074, China
2 Research Institute, Inner Mongolia First Machinery Group Co. Ltd, Baotou 014030, China
Abstract The leg structure is crucial to the legged robot's motion performance. With the size and load of the legged robot increasing, the difculty of leg design increases sharply. Inspired by biomechanics, this paper proposes a leg design approach based on efective mechanical advantage (EMA) for developing the heavy-duty legged robot. The bio-inspired design approach can reduce the demand for joint actuation forces during walking by optimizing the ratio relationship between the joint driving force and ground contact force. A dimensionless EMA model of the leg for the heavy-duty legged robot is constructed in this paper. Leg dimensions and hinge point locations are optimized according to the EMA and energy-optimal criterion. Based on the optimal leg structure, an electrically driven tri-segmented leg prototype is developed. The leg's joint hinge points are located near the main support line, and the load-to-weight ratio is 15:1. The leg can realize a swing frequency of 0.63 Hz at the stride length of 0.8 m, and the maximum stride length can reach 1.5 m.
Keywords Legged robot · EMA · Heavy-duty · Bio-inspired
Moment balance of the muscle force (f$f$) relative to the ground reaction force (Fr${F}_{\text{r}}$), which depends on the moment arms (r$r$ and R$R$)
Address: C508 Dingxin Building, Jilin University, 2699 Qianjin Street, Changchun 130012, P. R. China | 2022-09-26 19:08:04 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.21965403854846954, "perplexity": 6246.607721982674}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334915.59/warc/CC-MAIN-20220926175816-20220926205816-00379.warc.gz"} |
https://repository.uantwerpen.be/link/irua/103103 | Publication
Title
Average kinetic energy density of Cooper pairs above $T_{c}$ in $YBa_{2}Cu_{3}O_{7-x}$, $Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta}$, and Nb
Author
Abstract
We have obtained isofield curves for the square root of the average kinetic energy density of the superconducting state for three single crystals of underdoped YBa(2)Cu(3)O(7-x), an optimally doped single crystal of Bi(2)Sr(2)CaCu(2)O(8+delta), and Nb. These curves, determined from isofield magnetization versus temperature measurements and the virial theorem of superconductivity, probe the order parameter amplitude near the upper critical field. The striking differences between the Nb and the high-T(c) curves clearly indicate for the latter cases the presence of a unique superconducting condensate below and above T(c).
Language
English
Source (journal)
Physical review : B : condensed matter and materials physics. - Lancaster, Pa, 1998 - 2015
Publication
Lancaster, Pa : 2007
ISSN
1098-0121 [print]
1550-235X [online]
Volume/pages
76 :13 (2007) , p. 1-4
Article Reference
132502
ISI
000250619800018
Medium
E-only publicatie
Full text (Publisher's DOI)
Full text (open access)
UAntwerpen
Publication type Subject | 2022-10-07 12:38:35 | {"extraction_info": {"found_math": true, "script_math_tex": 3, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.539765477180481, "perplexity": 4948.20168689811}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030338073.68/warc/CC-MAIN-20221007112411-20221007142411-00574.warc.gz"} |
http://www.bloodadvances.org/content/1/24/2176?sso-checked=true | # Patients’ priorities in selecting chronic lymphocytic leukemia treatments
Carol Mansfield, Anthony Masaquel, Jessie Sutphin, Elisa Weiss, Meghan Gutierrez, Jennifer Wilson, Marco Boeri, Jia Li and Carolina Reyes
## Key Points
• CLL patients value higher PFS but would accept significant reductions in PFS to avoid serious adverse events.
• Adding even modest out-of-pocket costs changed treatment choices for hypothetical treatments, suggesting patients are sensitive to cost.
## Abstract
Currently, in the United States, 130 000 people live with chronic lymphocytic leukemia (CLL), and almost 20 000 new cases of CLL are diagnosed each year. Little is known about the value patients place upon the attributes of available CLL treatments, which vary in efficacy, side effects, and mode of administration. We used a discrete-choice experiment (DCE) to investigate patients’ preferences for treatment attributes and the impact of out-of-pocket cost on patients’ choices. DCE surveys pose a series of choices between hypothetical treatment options, each defined by a set of attributes, and the responses provide quantitative estimates of the average relative preferences for treatment attribute. Each hypothetical treatment in this survey was defined by 5 attributes with predefined levels for efficacy, adverse events, and mode administration. A patient advocacy organization recruited 384 patients with a self-reported physician diagnosis of CLL to complete the online survey. Respondents placed the highest relative importance on longer progression-free survival (PFS). However, the risk of adverse events also was important, as significant additional PFS was needed to offset patients’ acceptance of worsening adverse events. A supplemental question with 2 treatments and varying costs was included to assess the impact of cost on choice. When costs were included, a large proportion of patients changed their choices between the 2 treatments. Given the available treatments and the high cost of some treatments, physicians may want to explore their patients’ preferences for different treatment features, including benefit–risk tradeoffs and out-of-pocket cost, when selecting the best treatment strategies for patients.
## Introduction
In the United States, 130 000 people live with chronic lymphocytic leukemia (CLL).1 The American Cancer Society estimates that, in 2017, there will be 20 110 new cases of CLL in the United States, which represents approximately one-third of all leukemia diagnoses.2 Treatment options for patients have increased, with 4 new treatments approved since 2013 (obinutuzumab, ibrutinib, idelalisib, and venetoclax) and more under development.3,4 Because many patients relapse after treatment, patients may receive a number of different treatments.
Treatments available for CLL vary in efficacy, safety, routes of administration, and cost. With many available choices, patients’ preferences in treatment selection are becoming increasingly important. With the advent of value-based treatment decision-making like the American Society of Clinical Oncology value framework, which scores efficacy, safety and costs, the patients’ perspectives and preferences are often missing in the scoring algorithm.5 Understanding these perspectives and preferences is critical if we want to truly consider what patients value in the delivery of cancer care. Nonetheless, there is little systematic research in this area.
This study helps to fill the knowledge gap on patient preferences by conducting a relatively large-scale study of treatment preferences among patients with CLL in the United States using a discrete-choice experience (DCE) survey. The study includes a novel way to explore the impact of out-of-pocket cost on treatment choice. The study also investigates patient interest in more sensitive tests for minimal residual disease (MRD) or small numbers of cancer cells remaining after treatment.
## Methods
### Study design
We designed a patient preference survey that included a set of DCE questions that offer a choice between pairs of hypothetical treatments for CLL. DCE questions reflect the assumption that goods can be described by a set of attributes with varying levels and that individuals have preferences for certain combinations of attribute levels.6 We conducted 2 patient focus groups in February and June 2015 to learn about treatment attributes of interest to patients. Candidate attributes were developed based on the patient input, clinical input from experts in CLL, and data on existing CLL treatments. The final attributes and levels were selected at a meeting including representatives from the Leukemia & Lymphoma Society (LLS) and the Lymphoma Research Foundation.
Table 1 summarizes the final set of 5 attributes, which include progression-free survival (PFS), mode of administration and schedule, chance of diarrhea, chance of serious infection, and chance of organ damage (tumor lysis syndrome). The attribute levels were designed to cover most existing treatments as of the fall of 2015. The 2 modes of administration were assigned different durations: the IV mode was fixed at 6 months, and the pills were taken every day until the medicine stopped working (the length of PFS for that hypothetical treatment). Respondents were told they would experience the side effects and elevated risks (diarrhea, infection, and organ damage) as long as they were taking the medicine. Figure 1A presents a sample choice question from the survey.
Table 1.
Attribute levels for the DCE
Figure 1.
DCE question and questions with cost attribute. (A) An example of the medicine-choice question that respondents answered in the online survey. (B-C) Different costs of medicine A and B. Respondents were randomly assigned to one version of the cost question.
The online survey instrument started with screening questions and informed consent. After providing consent, the respondents were asked background questions about their disease and treatment histories. Each DCE attribute was described individually using the text in Table 1, followed by questions about the attribute. The risk-based attributes (chance of serious infection and chance of organ damage) were presented both numerically and graphically. Risk levels were presented graphically using risk grids, each including 100 human figures, in which each figure represents 1 person who takes the medicine and figures in color represent patients for whom the event occurred.7 Respondents were asked a comprehension question about the risk grid and presented with the correct answer to reinforce the information. The survey text was pretested in semistructured, face-to-face interviews with 15 patients prior to survey administration.
Each respondent answered 8 DCE questions, choosing between pairs of experimentally designed hypothetical CLL treatments, each a combination of the 5 attributes shown in Table 1 with varying levels. Following good research practices in conjoint analysis and DCE,8 we created a set of 48 DCE questions, known as the experimental design, using the SAS implementation of a common D-optimal design.9,10 The 48 DCE questions were divided into 6 blocks of 8 questions. Respondents were randomly assigned to one of these blocks, and the order of the DCE questions within a block was randomized to control for question order effects.
There are significant cost differences in treatments for CLL, which can translate to variation in out-of-pocket costs depending on a patient’s insurance coverage. To explore the impact of cost on preferences, we added a question after the DCE questions in which respondents were again asked to choose between medicine A and medicine B, only this time with out-of-pocket cost included. We created 2 versions of the cost question, and respondents were randomly assigned to 1 of the 2 versions. The noncost attributes for medicine A and medicine B were fixed at the attribute levels presented in Figure 1A. In the first version, medicine A was $200 per month and medicine B was$600 per month. In the second version, medicine A was $25 per month and medicine B was$100 per month. The 2 versions are presented in Figure 1B-C.
Detection of MRD can help inform patient and physician decisions about frequency of monitoring and potential length of remission. After the cost questions, we assessed respondent interest in more sensitive blood and bone marrow tests of MRD using a 5-point Likert scale (see Figure 5 for the questions and response categories). Before the questions on interest in blood or bone marrow tests of MRD, we provided a brief description of MRD and MRD testing (see the supplemental Appendix for the text used to describe the tests). For the bone marrow test, we explained that physicians are interested in whether there are cancer cells in both the blood and the bone marrow, because bone marrow disease is a major concern in patients with CLL. We described the process of getting a bone marrow aspiration and noted that the test can be painful.
### Sample
A convenience sample was recruited through the LLS during March and April 2016. The LLS sent e-mails to potential respondents in their database of patients. The e-mails contained an invitation to the survey and a unique link to the survey. Respondents were offered a $20 incentive to complete the survey. Adults aged ≥18 years with a self-reported physician diagnosis of CLL and who were able to provide informed consent were eligible to participate in the online survey. The study was approved by an institutional review board at RTI International. ### Analysis #### DCE analysis. The DCE questions generate panel data, which were estimated using a main effects random-parameters logit (RPL) model with NLOGIT Software version 5.0 (Econometric Software, Inc., Plainview, NY).11 RPL accounts for differences in preferences across respondents that can bias results from conventional conditional logit models.12 Parameter estimates from the model can be interpreted as relative preference weights that indicate the average relative preference for one attribute level compared with other attribute levels. The mean preference weights were used to calculate the relative importance of each attribute and to calculate the minimum acceptable benefit in terms of the months of PFS respondents required in order to accept a worsening of the other attributes. In order to assess whether patients’ treatment status influenced their response, we performed subgroup analysis, dividing the sample into 3 categories: first-line patients, relapsed/refractory patients, and treatment-naive patients. The first-line category included respondents who were receiving or had completed a first-line oral or IV treatment at the time of the survey. The relapse category included respondents who reported having been treated more than once with oral or IV treatment and who reported that their condition worsened after experiencing a complete or partial remission. The treatment-naive category included respondents who had not received an oral or IV treatment of CLL. #### Cost sensitivity analysis. The percentage of respondents who selected medicine A and medicine B in the 2 versions of the cost question was compared with an estimate of the percentage of respondents who would have selected each medicine based on a prediction using the coefficients from the RPL model, where cost was not included. Posterior preferences for each individual in the sample were computed, conditional on the pattern of observed choices.12-14 #### Interest in testing for MRD. The percentage of respondents selecting each point on the 5-point scale was tabulated. The 2 questions regarding interest in a blood or bone marrow MRD test were analyzed using an ordered logistic regression model.15,16 The dependent variable was a categorical variable ranging from 1 to 5 indicating the respondent’s level of interest in the blood test. A 5 indicated high interest, and a 1 indicated no interest. The independent variables were age, gender, education, geographic residence, employment, race, type of health insurance, treatment line, years since diagnosis, and whether a respondent has had a bone marrow test in the past (for the bone marrow question). ## Results ### Response rate and sample characteristics Of 4420 people in the LLS patient database who were sent an invitation e-mail by LLS, 610 accessed the survey, 432 met the eligibility criteria, and 384 provided consent and completed the survey. Table 2 presents select demographic and treatment experience summary statistics. The age, race, and gender of the sample are similar to what is known about the population of patients with CLL.17 The average age of the sample was 65 years, approximately half were male, and the majority were white. Table 2. Respondent summary statistics (N = 384) Sixty-nine percent of respondents had a form of public insurance (Medicare, Medicaid, and/or Veterans Health Insurance), and approximately half of the respondents were retired. Fifty-three percent of respondents had received financial aid from patient support programs through their doctor, hospital, or drug maker or from a patient organization for their out-of-pocket treatment costs. Forty percent of patients who had to pay something out of pocket for their medicines reported difficulty in paying out-of-pocket costs. Many respondents (26%) stated they were either participating in a clinical trial for CLL or had participated in one in the past. ### DCE Respondents placed the most importance on a change in PFS from 10 months to 60 months, followed by a change in the chance of infection from a 30% risk to no risk, a change in the chance of organ damage from an 8% risk to no risk, a change in diarrhea from severe to none, and a change in mode of administration from IV to pill. Figure 2 presents the normalized preference weights for each attribute level. The preference weights indicate the relative strength of preference for each attribute, where larger positive numbers indicate greater preference and smaller, negative numbers indicate less preference. All the levels within each attribute are statistically different from each other at the 5% level. The results followed the expected ordering, with stronger preferences for better levels of the attributes. The vertical distance between the most-preferred and least-preferred level within each attribute represents the degree of importance of that attribute within the ranges presented for the survey. Figure 2. Estimated preference weights. The vertical axis is the normalized mean preference weight for each attribute level using the results from the RPL model. The vertical bars around each mean parameter estimate represent the 95% confidence intervals about the point estimate. In the subgroup analysis of differences among first-line patients, relapsed/refractory patients, and treatment-naive patients, there were no statistically significant differences between the preferences of any subgroup and the full sample, or across subgroups. Figure 3 provides the results from the minimum acceptable benefit calculation. The graph shows the additional months of PFS necessary for the average respondent to accept a worsening in the other attributes. Moving from the lowest risk of serious infection to the highest generated the largest minimum acceptable benefit. On average, 36 additional months of PFS would compensate respondents for an increase in the risk of serious infection from 0% to 30%. Figure 3. Additional months of PFS required by respondents to offset a change in an adverse event or change mode of administration. The bars display the minimum acceptable benefit (MAB) calculation (the number of months of PFS needed to offset a change in the attribute level). The horizontal bars at the end of the MAB bar represent the 95% confidence intervals about the point estimate. AE, adverse event. ### Cost sensitivity assessment Figure 4 presents the results from the cost sensitivity assessment. Using results from the DCE in absence of cost, it was forecasted that 91% of respondents would choose the medicine with the longest PFS, medicine B. However, when respondents were presented with a choice between the 2 medicines and their out-of-pocket cost, 75% of respondents who saw Figure 1B ($400 difference in out-of-pocket cost per month) chose the lower-cost medicine, and 50% of the respondents who saw Figure 1C (\$75 difference in out-of-pocket cost per month) chose the lower-cost medicine. The difference between respondents’ forecasted choice based on the DCE results and their actual choice was most pronounced for respondents who were assigned to the version with the largest difference in cost between the 2 medicines (Figure 1B).
Figure 4.
Impact of cost on medicine choice. The bars indicate the percentage of the sample that selected medicine A or B. The first bar represents the forecast of the percentage selecting medicine A or B based on predictions from the model results for the DCE when cost was excluded. The next 2 bars show the percentage of the sample that selected medicine A or B when cost was included (see Figure 1 for medicine definitions).
### Interest in testing for MRD
Approximately 82% and 58% of respondents were interested or very interested in the blood test or bone marrow test for MRD, respectively (Figure 5). The results of the ordered logistic regression for the blood test show that interest in a blood test for MRD decreased with age, was lower for women, and was higher for respondents with public insurance. Interest in the bone marrow test for MRD was also lower for women but higher for respondents living in rural areas, those who were employed part-time, those who were black, and those who had previously had a bone marrow test (results in supplemental Tables 1 and 2; supplemental Appendix).
Figure 5.
Respondents’ interest in MRD tests. Response to the question, “Suppose that you have finished a 6-month course of medicine for CLL. The standard blood test does not find any cancer cells in your blood. Your doctor offers you one of the new, more sensitive blood tests. How interested would you be in getting this new [blood test/bone marrow aspiration test]?”
## Discussion
This study investigated patient preferences for different, hypothetical CLL treatments; the relationship between treatment preferences and patient out-of-pocket cost; and patient interest in more sensitive tests for MRD. The study included a relatively large sample of patients with CLL, including first-line patients, relapsed/refractory patients, and treatment-naive patients. The respondents valued efficacy, but they also valued safety. Any increase in adverse events or the risk of adverse events would require a potentially large improvement in efficacy to offset the disutility to respondents associated with the adverse events. For simplicity, the chance of organ damage (tumor lysis syndrome) is presented as constant for the duration of treatment. This overstates the actual risk, which is highest at the start of treatment.18,19 Cost had a big impact on treatment choice.
Evidence suggests that patients prefer oral over IV mode of administration in oncology.20 In the context of the treatment attributes included in this survey, respondents were generally unwilling to trade off more convenient dosing for lower efficacy and risk attributes.
We did not find any statistically significant differences among the preferences of first-line, relapsed/refractory, and never-treated respondents. This suggests that patient preferences may be stable over the stages of the disease; however, it is possible that sample size was not large enough to estimate the difference in preferences across the groups.
The impact of out-of-pocket costs for care to patients has become an important issue in cancer care, and studies have found that higher out-of-pocket costs may be associated with noncompliance.21-23 In our study, more than half the respondents had received financial aid from patient support programs, and almost half of patients who had to pay something out of pocket for their treatments reported difficulty in paying out-of-pocket costs.
Including costs in a DCE when the cost levels are considered too high for some fraction of respondents to pay presents a challenge to studies examining the impact of cost on treatment choice. If too many respondents select a hypothetical treatment based only on cost in the DCE questions, nothing is learned about relative preferences for the other attributes. When respondents ignore the cost attribute in the DCE because they cannot afford either option, it reduces the precision of all the estimates. We used a novel approach to collecting information on the impact of cost on preferences, adding a fixed follow-up question that contained the same attributes as the DCE and added cost. The addition of cost to the treatment profiles resulted in a large swing in the predicted choices toward the treatments with lower out-of-pocket costs. These results give a strong indication of cost sensitivity in patients with CLL.
Detection of MRD can help inform patient and physician decisions about the frequency of monitoring and potential length of remission. The value of MRD as a prognostic factor in CLL has been demonstrated by several studies, including randomized trials,24-26 but very little is known about patients’ interest in knowing their MRD status. In this study, a significant number of patients indicated interest in a test of their MRD status, especially if the sample was taken from the peripheral blood rather than by bone marrow aspiration. If further studies establish the usefulness of MRD testing for treatment decisions, our study suggests that patients may be interested.
The only other DCE study to investigate treatment preferences for CLL using a DCE was that conducted by Landfeldt et al.27 The study included physicians, the general population, and a small sample of relapsed/refractory patients in Germany and Sweden. Among 6 attributes (overall survival, PFS, fatigue, nausea, risk of serious infections, and treatment administration), overall survival was the most important attribute for all 3 groups, and treatment administration was more important to patients than it was to physicians. We selected PFS as our measure of efficacy because patients often receive a series of treatments followed by remission and relapse and there is more consistent clinical data on PFS across existing treatments. Although in the larger picture, patients may care more about overall survival, patients with CLL often take a series of treatments, and it can be hard to tie one particular treatment to a change in overall survival. During the focus groups, patients talked more about remission than overall survival, and in the pretests, no patients asked about overall survival, so the use of PFS as an efficacy end point appeared to be acceptable to patients.
The study results should be viewed in light of several limitations. The respondents were recruited through a patient advocacy organization, and this population may represent a subgroup of the overall CLL patient. The DCE method relies on hypothetical scenarios, and only a limited number of treatment attributes can be included. Although the attributes and choices reflected in the survey are similar to choices that patients face, the answers do not carry the weight of real choices. In addition, the results are conditional on the set of attributes included and the range of levels. If different attributes or ranges had been considered, the relative preferences for any individual attribute might shift.
The study demonstrates the importance of efficacy to respondents and a willingness to trade off efficacy to avoid serious risks. Cost considerations drove a large shift in preferences, and patients expressed interest in MRD testing. Combined with the responses to other questions indicating difficulty paying for current treatments, cost is an important factor in patient decisions. Patients with CLL have a number of options for treating their disease. These results could help patients play a more active role in discussions about treatments and help physicians work with their patients to find treatments that best fit patients’ preferences and circumstances.
## Acknowledgments
Kimberly Moon (RTI Health Solutions) was responsible for overall project management for this study. Alicia Patten (LLS) provided assistance with patient recruitment. Editorial assistance during preparation of this manuscript was provided by Joyce Hicks (RTI Health Solutions) and funded by Genentech, Inc.
This study was supported by research funding from Genentech, Inc., to RTI Health Solutions.
## Authorship
Contribution: C.M., A.M., E.W., M.G., J.W., J.L., and C.R. contributed to the conception and design of the work; C.M., J.S., and M.B. conducted the analysis; and all authors were involved in the acquisition of data, interpretation of results, drafting the work, and revising it critically for important intellectual content.
Conflict-of-interest disclosure: C.M., M.B., and J.S. are full-time employees of RTI Health Solutions and were paid contractors of Genentech, Inc., in the development of the survey, the analysis and interpretation of the data, and preparation of the study report. A.M., J.L., and C.R. are employees of Genentech, Inc. The remaining authors declare no competing financial interests.
Correspondence: Carol Mansfield, RTI Health Solutions, 200 Park Offices Dr, PO Box 12194, Research Triangle Park, NC 27709-2194; e-mail: carolm{at}rti.org.
## Footnotes
• The full-text version of this article contains a data supplement.
• Submitted March 30, 2017.
• Accepted September 20, 2017.
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View Abstract | 2018-12-14 07:58:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2786741852760315, "perplexity": 3679.677418829888}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376825495.60/warc/CC-MAIN-20181214070839-20181214092339-00416.warc.gz"} |
http://clay6.com/qa/25904/rare-earth-elements-are-generally-called-as | Browse Questions
Rare earth elements are generally called as
$\begin {array} {1 1} Lanthanides \\ Actinides \\ Alkali \;metals \\Alkaline\;metals \end {array}$
Lanthanides [rare-fleries-first inner transition]$\rightarrow 6^{th}$ period.
As defined by IUPAC, a rare earth element (REE) or rare earth metal is one of a set of seventeen chemical elements in the periodic table, specifically the fifteen lanthanides plus scandium and yttrium.
edited Aug 1, 2014 | 2017-01-21 13:31:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8134250640869141, "perplexity": 9691.13972168116}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560281084.84/warc/CC-MAIN-20170116095121-00471-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://www.physicsforums.com/threads/second-order-linear-ode-power-series-solution-to-ivp.725366/ | # Homework Help: Second Order Linear ODE - Power Series Solution to IVP
1. Nov 27, 2013
### ChemistryNat
1. The problem statement, all variables and given/known data
Let y(x)=$\sum$ckxk (k=0 to ∞) be a power series solution of
(x2-1)y''+x3y'+y=2x, y(0)=1, y'(0)=0
Note that x=0 is an ordinary point.
2. Relevant equations
y(x)=$\sum$ckxk (k=0 to ∞)
y'(x)=$\sum$(kckxk-1) (k=1 to ∞)
y''(x)=$\sum$(k(k-1))ckxk-2 (k=2 to ∞)
3. The attempt at a solution
(x2-1)$\sum$(k(k-1))ckxk-2 (k=2 to ∞) +$\sum$(kckxk) (k=1 to ∞)+$\sum$ckxk (k=0 to ∞) -2x=0 ??
I'm not having an issue with the power series themselves, I'm just not sure how to incorporate in the "2x" term when I'm setting up the series equation. We didn't cover this scenario in class and I couldn't find anything like it in my textbook.
I've been trying to incorporate it as a series itself
ie. 2$\sum$ckxk (k=0 to ∞) where C0=0, or 2$\sum$ckxk (k=1 to ∞)
but I'm not sure I can even do that mathematically?
Thank you!
2. Nov 27, 2013
### Dick
Your are going to equate powers of x to get a relation between the c_k values, right? The -2x will only contribute the x^1 term, yes?
3. Nov 27, 2013
### ChemistryNat
So I can leave it in as a constant coefficient term, say 2C1x? and then use that in combination with the other constants I've pulled out?
4. Nov 27, 2013
### Dick
Mmm. Sort of, but the coefficient your x^1 term is only going to have a -2 in it. Without any C_k term in front of it. It's just a constant.
Last edited: Nov 27, 2013
5. Nov 27, 2013
### ChemistryNat
Oh! so I can just leave it be and put it with the other x1 coefficient terms?
6. Nov 28, 2013
Sure. | 2018-07-21 02:28:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5500714778900146, "perplexity": 2468.834774546108}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676592150.47/warc/CC-MAIN-20180721012433-20180721032433-00092.warc.gz"} |
https://love2d.org/forums/viewtopic.php?f=14&t=946&p=9652 | ## Carpal Tunnel Revolution!
Show off your games, demos and other (playable) creations.
bartbes
Sex machine
Posts: 4946
Joined: Fri Aug 29, 2008 10:35 am
Location: The Netherlands
Contact:
### Re: Carpal Tunnel Revolution!
Yay, I got somewhere, but I keep pressing the wrong keys...
EDIT:
Mode: Easy
Score: 1672597
Robin
The Omniscient
Posts: 6506
Joined: Fri Feb 20, 2009 4:29 pm
Location: The Netherlands
Contact:
### Re: Carpal Tunnel Revolution!
bartbes wrote:Yay, I got somewhere, but I keep pressing the wrong keys...
Just press them all... works for me
napco
Party member
Posts: 129
Joined: Fri Jun 12, 2009 9:28 pm
Location: Ital... ehm...
### Re: Carpal Tunnel Revolution!
Very nice game, but i still prefer DDR With feet, you know...
F687/s
Prole
Posts: 15
Joined: Wed May 20, 2009 8:29 pm
Location: Blowin' Green, O'Hi-o
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### Re: Carpal Tunnel Revolution!
Very nice game, but i still prefer DDR With feet, you know...
Hey, there's no rule that says you CAN'T play with your feet... ;-)
bartbes
Sex machine
Posts: 4946
Joined: Fri Aug 29, 2008 10:35 am
Location: The Netherlands
Contact:
### Re: Carpal Tunnel Revolution!
I might be the only player, but I request more songs, I love the game, this might be the first ever LÖVE game (that I didn't develop) which I played for more than 2 days.
F687/s
Prole
Posts: 15
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### Re: Carpal Tunnel Revolution!
(Yay!)
Well, I'd be happy to make another song or two given the demand, but I'm not sure what to make...But I have a suggestion: why not make your own songs? It's totally Fun And Easy®!
The first step (as you can see if you crack open the .love file) is to add your song in a folder, drop in a random background image, add the song, create an empty .tp file, and register it in index.txt (because I'm too lazy to find out how to scan a directory ;-). Like this!
Code: Select all
name = Don't Stop (2007 mix)
artist = Dj Slow
arr = F687/s
music = Don't Stop/dontstop.ogg
chart = Don't Stop/dontstop.tp
bg = Don't Stop/dontstop.jpg
%
Also don't forget to add that last % sign, it flushes the buffer. Next up, the .tp file itself. It's pretty easy to parse, a !number header starts the chart for that difficulty, a . means there's no step, anything else means there is. The parser does care somewhat about whitespace (like, that it exists between steps), but it doesn't care how much. And each line corresponds to 1/16 (meaning 4 in a beat, 16 in a measure). I chopped mine up into 16 so I wouldn't get snagged on an offset error, and I even made a nice Perl script that checks it for you!
And you're done! The hard part is just writing all the notes, testing them, and matching the BPM.
Of course, that's if you want to make your own songs. I have one other one that I used for testing, but I left it out because the notes inexplicably aren't lined up with the beat, but they register just fine. It's even in the index.txt file, but commented out. And hopefully, I'll have something else playable before the end of the weekend. (probably something from DuckTales or similar, but hey, you asked... ;-)
bmelts
Party member
Posts: 380
Joined: Fri Jan 30, 2009 3:16 am
Location: Wiscönsin
Contact:
### Re: Carpal Tunnel Revolution!
F687/s wrote:(probably something from DuckTales or similar
MOON THEME!
plz
napco
Party member
Posts: 129
Joined: Fri Jun 12, 2009 9:28 pm
Location: Ital... ehm...
### Re: Carpal Tunnel Revolution!
Anyway, to scan a directory you can use:
Code: Select all
for index, filename in ipairs(love.filesystem.enumerate(path)) do
end
F687/s
Prole
Posts: 15
Joined: Wed May 20, 2009 8:29 pm
Location: Blowin' Green, O'Hi-o
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### Re: Carpal Tunnel Revolution!
Whoops, wrong weekend...
Anyway, I am proud to present...NEW SONGS!! (sorta). Well, two new songs (one that I already posted and then fixed), and I placed both Moon Theme and Red Zone in there (beatmatched and ready to go!), got 1/3 of the way through the Red Zone chart, lost willpower, fixed up some stuff, added more shiny and groovy, and the result is here! | 2019-11-20 17:59:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40476730465888977, "perplexity": 9325.221564464178}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496670597.74/warc/CC-MAIN-20191120162215-20191120190215-00068.warc.gz"} |
https://electronics.stackexchange.com/questions/79089/led-determining-series-parellel-configurations-for-most-efficent-design | # LED determining Series / Parellel configurations for most efficent design
My circuit has an input range of 11.8 - 28 vdc. I have 2 types LEDs that I would like to place in this circuit.
The first has a VF Typ of 3.4v and an Idrawnom of .150 A The second has a VF Typ of 2.2v and an Idrawnom of .150 A
(Link to data sheet) http://www.optekinc.com/datasheets/OVS5MABCR4.pdf
I have 2 sections in my PCB.
In one section I would like to put 4 of each color. In the other section put 20 of each of these.
These LEDs could be considered to be on all of the time after power up.
I am trying to figure out what would be the most efficient use of power and determine the series parallel configuration. PWM is a possibility if needed.
The large swing in the input voltage is why I am asking this question. If it were only 12v or 24v then the answer would be straight forward. Also if I was not concerned about efficiency / power distribution it would easy as well.
Also I am trying to stay away from the use of LED drivers vs PWM because of the code rewrite involved.
• Are you regulating the input voltage down at all? It might be best to run the LEDs off of that. Aug 15 '13 at 20:27
• I was planning on using an LDO for the electronics only (ie straight bus for the LEDs). But I could move to a switcher and use that at say 14vdc and put 4 LEDs in series. That was deff one idea that I had. I dont know if that is the best solution though? Thanks for the suggestion. Aug 15 '13 at 20:29
• Yes sorry Ill fix that now. Aug 15 '13 at 20:41
• There are led drivers with a single input/enable pin, which you can effectively use pwm on. You don't have to have an addressable/spi/i2c led driver. Aug 15 '13 at 21:08
• I have found this one from TI ti.com/product/tlc5926 (allows up to .120 A per channel and has 16 channels) I would assume then that I would limit the voltage to ~11v (3x 3.4v which is the highest VF) and then set the current limiting resistor on the chip. to understand the function correctly it can draw up to .120 A for each channel so that could be up to 16 x .120 A or 1.92A? Aug 15 '13 at 21:23
LEDs are current driven devices, not voltage driven. Hence, as long as the drive current requirements are met, the voltage involved is merely a dependent variable, so to speak.
For both types of LEDs mentioned, a constant current buck LED driver would probably provide the lowest cost Bill of Materials for a very high efficiency design.
For instance, the Texas Instruments LM3407 constant current, floating buck converter will allow a supply of 4.5 to 30 Volts, and a LED string forward voltage of anywhere between 10% and 90% of the supply. Taking the voltage numbers provided in the question:
• Vin(min) = 11.8 Volts ==> maximum Vf of string = 0.9 x 11.8 = 10.62 V
• Vin(max) = 28 Volts ==> minimum Vf at full input voltage = 0.1 x 28 = 2.8 V
So, as long as each string is between 2.8 and 10.62 Volts Vf, this regulator can support the application. Typical implementation, taken from the datasheet:
By using switching current regulators, the issue of getting rid of surplus voltage as heat is pretty much eliminated. As can be seen from the example of the LM3407 above, switching regulators aren't too fussed even if 90% of the supply voltage needs to be dropped within the regulator: The device does cycle-by-cycle current regulation of the load (the LEDs), and simply switches off the rest of the time, hence not needing to dissipate the excess voltage as heat.
This simplifies the series / parallel LED decision significantly: Simply put as many LEDs in series as you like, up to the maximum determined by the supply voltage lower limit (10.62 Volts as calculated above).
Thus, 3 LEDs of 3.4 Vf, or 5 LEDs of 2.2 Vf can be supported per string, in this particular application. | 2021-09-22 22:41:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.43285608291625977, "perplexity": 1432.4519254345778}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780057403.84/warc/CC-MAIN-20210922223752-20210923013752-00100.warc.gz"} |
http://mathoverflow.net/questions/136576/optimal-auction-for-risk-averse-seller | # Optimal auction for risk-averse seller
Consider an auction of a single unit of indivisible good. There are $n$ buyers whose values of the object is drawn independently from the uniform distribution on $[0,1]$. The buyers have interim linear utility:
$$u_i(k(\hat{\theta}), t(\hat{\theta}));\theta_i) = \theta_i k_i(\hat{\theta}) + t_i(\hat{\theta})),$$
where $\hat{\theta}$ is the profile of bids, $k \in \mathbb{R}^n$ is the allocation ($k_i(\hat{\theta})$ is $1$ is player $i$ gets object and $0$ otherwise), and $t \in \mathbb{R}^n$ is the transfer/payment profile.
Assumption/Requirement We only consider auction mechanisms where in the resulting Bayesian Nash equilibrium, the object goes to the bidder with highest value, i.e.
$$k_i(\theta) = 1 \Leftrightarrow \theta_i = \max_j \theta_j.$$
Quoting the revelation principle, I have replaced $\hat{\theta}$ by the true type profile $\theta$ above.
Buyer $i$'s interim expected utility as a function of his type $\theta_i$ is
$$U_i(\theta_i) = \theta_i \bar{k}_i (\theta_i) + \bar{t}_i (\theta_i),$$
where $\bar{k}_i(\theta_i)$ is the probability he gets the good and $\bar{t}_i(\theta_i)$ is expected transfer.
From the seller's perspective, the expected payment from a buyer of type $\theta_i$ is
$$- \bar{t}(\theta_i) = - [ \int_0 ^{\theta_i} \bar{k}_i (\theta_i ') d \theta_i ' + U_i(0) - \bar{k}_i(\theta_i) \theta_i].$$
The requirement above means $\bar{k}_i(\theta_i) = \theta_i^n$. So when the seller is risk-neutral, the maximum expected revenue for the seller is achieved by any mechanism that makes $U_i(0) = 0$, i.e. a buyer of type $0$ has interim expected utility $0$.
Question: What if the seller is risk-averse, for example with utility function $w(t) = \sqrt{t}$? What characterizes the seller's optimal auction in this case?
According to my calculations, the first price auction gives the seller expected utility $$\sqrt{n(n-1)} \frac{1}{n + \frac{1}{2}},$$
while the second price auctions gives
$$n(n-1) \frac{1}{n + \frac{1}{2}} \frac{1}{n - \frac{1}{2}}.$$
So revenue equivalence no longer holds.
-
Cross-posted from MO. – Michael Jul 13 '13 at 4:20
What do you mean "Cross-posted from MO"? It's already on MO. – Joel Reyes Noche Jul 14 '13 at 1:24
Because all auction rules with $U_i(0)$ yield the same expected revenue, the optimal auction will be one that returns that revenue to the seller with certainty.
Another way to say this is that each buyer shares the seller's uncertainty about all the other buyers, so each risk-neutral buyer should fully insure the risk-averse seller against that shared uncertainty.
In the case of two bidders, you can accomplish this by having bidder 1 pay
$${1\over 2} \left({\hat{\theta_1}^2-\hat{\theta_2}^2 }\right) + {1\over 6}$$
and symmetrically, of course, for bidder 2.
For the general case, Eso and Futo give a large class of auction rules providing optimal insurance in a paper in Economics Letters (1999).
To expand on my response to your first comment: Suppose we adopt the above bidding rule. Then I make the following claims:
A) In Nash equilibrium, both bidders bid their true valuations.
B) Each player has $U(0)=0$, so the seller's revenue is the same as for any other such auction.
C) The seller's revenue is certain.
To check A), suppose my true valuation is $x$ and I bid $B$. Suppose also that the other player is known to be bidding his true valuation $y$. Then my expected gain is
$$Prob(B>y) x-(1/2)B^2+(1/2)E(y^2)-1/6$$
Now note that $Prob(B>y)=B$, so this becomes $Bx-(1/2)B^2+\hbox{constant}$. Therefore if the other guy bids his true valuation, I bid mine.
To check B, note that $(1/2)E(y^2)=1/6$ so my expected gain with $B=x$ is $x^2/2$, whereas we know that with any auction rule, my expected gain is $x^2/2+U(0)$. Thus $U(0)=0$.
As for C), note that the sum of the two bids is (deterministically) 1/6, the same in expectation as for any other auction rule with $U(0)=0$, but with no risk to the seller.
If I've done the arithmetic right, the solution in the case of $n$ bidders is that you (as one of the bidders) pay the amount
$${k-1\over k}B^k-\sum_{i=1}^{k-1}{B_i^k\over k(k-1)}+{1\over k(k+1)}$$
where $B$ is your bid and the $B_i$ are the other people's bids. This has three key properties:
1) It induces you to bid honestly in equilibrium, because you're choosing $B$ to maximize $$Prob(B>B_1,\ldots,B_k)x-{k-1\over k}B^k+\hbox{constant}$$ and, if everyone else is bidding honestly, that first probability is just $B^{k-1}$. Therefore you optimize by setting $B=k$.
2) The expected value of your payment is the same as in an ordinary second bid auction.
3) The sum of your and everyone else's payments is deterministically equal to a constant (namely the same expected revenue that the seller would have gotten from a second-price --- or first-price or third-price, etc. --- auction).
-
Thank you. How does one get that formula? So there are bids $(\hat{\theta}_1, \hat{\theta}_2)$ such that the bidder not getting the object also pays? This clearly violates the bidder's participation constraint, no? Or am I missing something? – Michael Jul 13 '13 at 22:17
Yes, both bidders pay, but we still have $U(0)=0$ (in expectation), so bidders are just as willing to participate in this auction as in any other with $U(0)=0$. I am about to add a few details to the answer to flesh this out. – Steven Landsburg Jul 13 '13 at 22:47
PS: I've added details to the answer as promised, but it occurs to me that maybe you're missing a much simpler point, namely that both bidders "pay", but these payments can be either positive or negative, and in expectation they are equal to what they'd pay in an ordinary second price auction. – Steven Landsburg Jul 13 '13 at 22:59
Thanks. How does one come up with something like that? It looks like the seller charges an entry-fee $\frac{1}{n} \times$ expected revenue and devises a bidding scheme where the buyers pay each other: a buyer of type $\theta_i$ pays what he would pay in the second price auction but also receives payments that offsets his entry fee in expectation. Is there a systematic way to go about this? – Michael Jul 15 '13 at 11:08 | 2015-03-28 02:26:49 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7834154367446899, "perplexity": 900.2426770125976}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-14/segments/1427131297172.60/warc/CC-MAIN-20150323172137-00015-ip-10-168-14-71.ec2.internal.warc.gz"} |
https://controlentertainmentonline.com/ans/100-as-a-decimal.html | # 100 As A Decimal
100 As A Decimal. How to write 1/100 as a decimal? Convert a fraction value to a decimal format. A fraction belongs to numerator divided by denominator. So enter the numerator and denominator value in given input box, then press calculate button, the system will automatically calculate the decimal value.
In order to convert percent to decimal number, the percentage should be divided by 100: 1% = 1 / 100 = 0. 01. 5% = 5/100 = 0. 05 100% as a decimal is written as 1. 00 as decimal notation. A decimal is referring to division and multiplication. 40% of \$56 means divide the 56 into 100 stacks. This means that there will be. 56 in each stack. The amount of money. Here's the little secret you can use to instantly transform any fraction to a decimal: | 2022-09-25 23:18:14 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.907389760017395, "perplexity": 1036.5115297275856}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334620.49/warc/CC-MAIN-20220925225000-20220926015000-00636.warc.gz"} |
https://www.physicsforums.com/threads/pointwise-vs-uniform-convergence.944240/ | # Pointwise vs. uniform convergence
Gold Member
The problem
I am trying determine wether ##f_n## converges pointwise or/and uniformly when ## f(x)=xe^{-x} ## for ##x \geq 0 ##.
Relevant equations
##f_n## converges pointwise if ## \lim_{n \rightarrow \infty} f_n(x) = f(x) \ \ \ \ \ ## (1)
##f_n## converges uniformly if ## \lim_{n \rightarrow \infty} || f_n - f || = 0 \ \ \ \ \ ## (2)
The attempt
##f_n(x) = f(nx) =nxe^{-nx} ##
Pointwise? (1)
## f(x) = \ \lim_{n \rightarrow \infty} f_n(x) = \lim_{n \rightarrow \infty} nxe^{-nx} = \lim_{n \rightarrow \infty} \frac{nx}{e^{nx}} = 0##
Uniformly? (2)
## 0 = \ \lim_{n \rightarrow \infty} || f_n - f || = \lim_{n \rightarrow \infty} || nxe^{-nx} - xe^{-x} || ##:
I am not sure how to continue from here and wether the last step was correct:
## \lim_{n \rightarrow \infty} || nxe^{-nx} - xe^{-x} || = 0 ##
Last edited:
Related Calculus and Beyond Homework Help News on Phys.org
Math_QED
Homework Helper
2019 Award
Please post the exact wording of the question. Do not post extra info in "Attempt". This makes it very confusing to Homework Helpers.
It appears to me that the pointwise limit of ##f_n## is the 0 function.
Now, you have to show that
$$\forall \epsilon > 0: \exists n_0: \forall n \geq n_0: \forall x: \left|\frac{e^{nx}}{nx}\right| < \epsilon$$
to find uniform convergence.
Last edited:
Gold Member
The exact wording of this question is in another language. Sorry, if you find my translation and formatting confusing. I have edited the OP and tried to make my point clearer.
Math_QED
Homework Helper
2019 Award
The exact wording of this question is in another language. Sorry, if you find my translation and formatting confusing. I have edited the OP and tried to make my point clearer.
Does it ask to determine whether the sequence of functions ##(f_n)_n## converge towards ##f## given by ##f(x) = xe^{-x}##?
If so, clearly the answer is no, since it converges to ##g = 0##, and limits are unique.
nuuskur
Gold Member
Does it ask to determine whether the sequence of functions ##(f_n)_n## converge towards ##f## given by ##f(x) = xe^{-x}##?
This chapter is about sequence of functions but the notation is different. According to my book its ##S_nf(x) = f(nx) ## and sometimes just ##f_n(x)##. The question is to determine how (pointwise/uniform) it converges and not whether it does it.
Math_QED
Homework Helper
2019 Award
This chapter is about sequence of functions but the notation is different. According to my book its ##S_nf(x) = f(nx) ## and sometimes just ##f_n(x)##. The question is to determine how (pointwise/uniform) it converges and not whether it does it.
But where does the function ##f(x) = xe^{-x}## come from (see your OP)? Is this included in the problem? Or is your guess for the pointwise limit?
Rectifier
Gold Member
But where does the function ##f(x) = xe^{-x}## come from (see your OP)? Is this included in the problem? Or is your guess for the pointwise limit?
Its in the problem. Let me try to reword the problem once again:
Examine if the sequence of funtions ##f_n(x)## when ##f(x) = xe^{-x}## converges pointwise for ## x \geq 0 ##. In case the sequence of functions does converge get the limit function and determine if the sequence converges uniformly.
Math_QED
Homework Helper
2019 Award
Its in the problem. Let me try to reword the problem once again:
Examine if the funtion ##f(x) = xe^{-x}## converges pointwise for ## x \geq 0 ##. In case the sequence of functions does converge get the limit function and determine if the sequence converges uniformly.
That's my problem! WHAT sequence of functions? The problem does not seem to include that. It is meaningless to consider pointwise convergence of one particular function, unless they consider it as a constant sequence of that function, but then the problem is trivial so I think the problem is flawed.
Rectifier
Gold Member
Sorry once again. I made an edit just a moment ago. You were quick to reply.
Secuence of funtions is definied as ## f_n(x) = f(nx) ## earlier in my book.
Math_QED
Homework Helper
2019 Award
Sorry once again. I made an edit just a moment ago. You were quick to reply.
Secuence of funtions is definied as ## f_n(x) = f(nx) ## earlier in my book.
Okay, let us get this thing straight. I will formulate the problem as I understand it:
Let ##f: \mathbb{R}^+ \to \mathbb{R}## be given by ##f(x) = xe^{-x}##. Define a sequence of functions ##(f_n: \mathbb{R}^+ \to \mathbb{R})_n## by letting ##f_n(x) = f(nx) = nxe^{-nx}##. Determine if ##f_n \to f## pointswise and/or uniformly.
Is this the correct problem?
Rectifier
fresh_42
Mentor
The exact wording of this question is in another language. Sorry, if you find my translation and formatting confusing. I have edited the OP and tried to make my point clearer.
This doesn't explain the fact that the ##f_n## are not defined. It would make more sense to me to ask, whether ##f(x)## is uniformly continuous or not.
Rectifier
I am confused. We'd attempt to show ##f_n(x)\xrightarrow[n\to\infty]{} 0, x\geq 0 ## (this does hold), why would we even further contemplate whether ##f_n ## converges to anything else (pointwise or uniformly) other than ##g(x) = 0\neq xe^{-x}, x\geq 0 ##?
Rectifier and Math_QED
Math_QED
Homework Helper
2019 Award
I am confused. We'd attempt to show ##f_n(x)\xrightarrow[n\to\infty]{} 0, x\geq 0 ## (this does hold), why would we even further contemplate whether ##f_n ## converges to anything else (pointwise or uniformly) other than ##g(x) = 0\neq xe^{-x}, x\geq 0 ##?
Indeed, this was what I was addressing in #4.
Gold Member
Okay, let us get this thing straight. I will formulate the problem as I understand it:
Let ##f: \mathbb{R}^+ \to \mathbb{R}## be given by ##f(x) = xe^{-x}##. Define a sequence of functions ##(f_n: \mathbb{R}^+ \to \mathbb{R})_n## by letting ##f_n(x) = f(nx) = nxe^{-nx}##. Determine if ##f_n \to f## pointswise and/or uniformly.
Is this the correct problem?
Yes, I think so.
Math_QED
Homework Helper
2019 Award
Yes, I think so.
Okay. Then the problem is actually very easy. See post #4 and #12.
Rectifier
Gold Member
If so, clearly the answer is no, since it converges to ##g = 0##, and limits are unique.
We'd attempt to show ##f_n(x)\xrightarrow[n\to\infty]{} 0, x\geq 0 ## (this does hold), why would we even further contemplate whether ##f_n ## converges to anything else (pointwise or uniformly) other than ##g(x) = 0\neq xe^{-x}, x\geq 0 ##?
What is ## g ## here? Is that f(x) in (1) ?
##f_n## converges pointwise if ## \lim_{n \rightarrow \infty} f_n(x) = f(x) \ \ \ \ \ ## (1)
##f_n## converges uniformly if ## \lim_{n \rightarrow \infty} || f_n - f || = 0 \ \ \ \ \ ## (2)
Math_QED
Homework Helper
2019 Award
What is ## g ## here? Is that f(x) in (1) ?
##g = 0## is shorthand notation for ##g: \mathbb{R}^+ \to \mathbb{R}: x \mapsto 0##
It is easy to see that this is the pointwise limit, since for any ##x## in the domain
##\lim_{n \to \infty} f_n(x) = \lim_{n \to \infty} nxe^{-nx} = 0 = g(x)##
Rectifier
Gold Member
And now I am somehow supposed to show that the function series does not cenverge uniformly by showing that
## \lim_{n \rightarrow \infty} || f_n - g || = \sup_{x \geq 0} | f_n - g | \neq 0 \ \ \ \ \ ##
So in other words, should I just insert ##g = 0## and ## f_n = nxe^{-nx}## since thats what we got earlier?
Math_QED
Homework Helper
2019 Award
And now I am somehow supposed to show that the function series does not cenverge uniformly by showing that
## \lim_{n \rightarrow \infty} || f_n - g || = \sup_{x \geq 0} | f_n - g | \neq 0 \ \ \ \ \ ##
So in other words, should I just insert ##g = 0## and ## f_n = nxe^{-nx}## since thats what we got earlier?
Indeed, it is true:
##f_n \to f## uniform ##\implies f_n \to f## pointwise
and by applying contraposition on this, we find that ##f## can't be a uniform limit of ##(f_n)_n##, because it isn't a pointwise limit.
nuuskur and Rectifier | 2020-07-16 17:39:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9902242422103882, "perplexity": 2156.0520761407142}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657172545.84/warc/CC-MAIN-20200716153247-20200716183247-00485.warc.gz"} |
http://mathhelpforum.com/advanced-algebra/51282-linear-alg.html | 1. ## Linear Alg.
Let $\displaystyle A$ be an m x n matrix. Given $\displaystyle A\bold{x}=\bold{0}$ has nontrivial soln's, in each of the following demonstrate an example if you say yes of $\displaystyle A$ and $\displaystyle \bold{b}$ or if you say no explain why.
Does $\displaystyle \exists\, \bold{b}$ in $\displaystyle \mathbb{R}^m$ such that $\displaystyle A\bold{x}=\bold{b}$ has:
a.) no solution
b.) a unique solution
c.) an infinite # of solutions
2. Well, we know m can or cannot equal n.
For a, there will exist a vector when m < n I believe since we can have a pivot in the augmented column.
For b, I'm not sure.
For c, an infinite # of solutions means m < n (more columns than rows, because we'll have a pivot. But how can we answer definitely for a-c if we don't know what m is.
3. Originally Posted by LinAlg
Let $\displaystyle A$ be an m x n matrix. Given $\displaystyle A\bold{x}=\bold{0}$ has nontrivial soln's, in each of the following demonstrate an example if you say yes of $\displaystyle A$ and $\displaystyle \bold{b}$ or if you say no explain why.
Does $\displaystyle \exists\, \bold{b}$ in $\displaystyle \mathbb{R}^m$ such that $\displaystyle A\bold{x}=\bold{b}$ has:
a.) no solution
b.) a unique solution
c.) an infinite # of solutions
If $\displaystyle \bold{x}_0$ is a solution to $\displaystyle A\bold{x} = \bold{b}$ and $\displaystyle \bold{x}_1$ is a non-trivial solution to $\displaystyle A\bold{x} = \bold{0}$ then $\displaystyle A(\bold{x}_1+\bold{x}_0) = A\bold{x}_1+A\bold{x_0} = \bold{b}$. Furthermore, $\displaystyle \bold{x}_1+\bold{x}_0\not = \bold{x}_0$ since $\displaystyle \bold{x}_1$ is non-trivial. Therefore, it is not possible to have unique solutions to $\displaystyle A\bold{x} = \bold{b}$. Therefore (b) is always false.
Define $\displaystyle T:\mathbb{R}^n \to \mathbb{R}^m$ by $\displaystyle T(\bold{x}) = A\bold{x}$.
The dimension of the space of all vectors $\displaystyle \bold{b} \in \mathbb{R}^m$ such that $\displaystyle A\bold{x}=\bold{b}$ is the rank of $\displaystyle T$.
The dimension of all vectors $\displaystyle x\in \mathbb{R}^n$ such that $\displaystyle A\bold{x} = \bold{0}$ is the nullity of $\displaystyle T$.
By rank-nullity theorem we have $\displaystyle \text{rank}(T) + \text{nullity}(T) = n$.
---
Case $\displaystyle m=n$:
Since $\displaystyle \text{nullity}(T) \geq 1$ by assumption it means $\displaystyle \text{rank}(T)<n$ therefore there is $\displaystyle \bold{b}$ such that $\displaystyle A\bold{x} = \bold{b}$ is not solvable. Therefore (a) is true.
If we can show that there is a $\displaystyle \bold{b}$ so that $\displaystyle A\bold{x} = \bold{b}$ is solvable (say $\displaystyle \bold{x}_0$) then $\displaystyle \bold{x}_0 + t\bold{x}_1$ is a solution (where $\displaystyle \bold{x}_1$ is non-trivial which solves $\displaystyle A\bold{x}=0$) where $\displaystyle t\in \mathbb{R}$. To show that there is such a $\displaystyle \bold{b}$ we need to show that $\displaystyle \text{rank}(T) \geq 1$ and it is sufficient to prove $\displaystyle \text{nullity}(T) < n$. But if this true? Well, almost. If $\displaystyle A$ is a matrix with zero entries then $\displaystyle \text{nullity}(T) = n$, and this is the only exception. Otherwise, it is true.
Try doing other cases. | 2018-03-21 04:01:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9029058218002319, "perplexity": 207.21163453830462}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257647567.36/warc/CC-MAIN-20180321023951-20180321043951-00523.warc.gz"} |
http://mathhelpforum.com/advanced-statistics/142771-negative-binomial-distribution.html | ## Negative Binomial Distribution
Let $X$~ $NB(r,p)$ when $r>=2$ , $p>0$ .
Calculate: $E[\frac{1}{X-1}]$.
I've tried to get any kind of result by calculating the needed sum but got stuck in the middle...I think there's a better way to do it
Hope you'll be able to help me | 2016-12-05 12:35:45 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 5, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8485394716262817, "perplexity": 500.3112298179011}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698541696.67/warc/CC-MAIN-20161202170901-00474-ip-10-31-129-80.ec2.internal.warc.gz"} |
https://saltyrentalsnsb.com/snp-mutation-fgow/page.php?page=data-interoperability-example-d21c4d | . For example, as shown in Figure 2, the R ecosystem already provides interoperability with many different environments. . One answer to interoperability problems that’s been gaining traction is to use machine learning (ML) and artificial intelligence (AI) to sift through the high volume of low-quality data. Purchasing a ArcGIS Data Interoperability extension license and enabling the extension in the ArcGIS Desktop application adds additional out-of-the-box support for over 100 GIS, CAD, raster, and database formats, and grants you access to the FME Workbench application. Ayre, L. B. Provider data flowing from the credentialing system to the scheduling system. | View Author's Profile | Show More Posts from Author |, Persuasive Essays|Doctoral Dissertation WritingsDissertation Essays|College Research WritingsOnline Assignment|Affordable Academic ResearchAccounting Paper|Marketing PaperMedical/Nursing Paper|Management EssaysLiterature Paper|Computer Science PaperEconomic Essays |Geography EssaysPolitical Science Paper|Psychology EssaysReligion Paper|Science PaperHistory Paper | Law PaperMathematics Research APA Style Referenced PaperChicago Style Referenced PaperMLA Style Referenced PaperHarvard Style Referenced Paper, We are based in Jackson area of California. This, therefore, enables operational purpose and meaning of the data is often preserved and unaltered. We offer you a Plagiarism report per paper, Papers formatting: MLA, APA, Harvard, Chicago or Turabian, PremiumPapers is the trusted provider of research & academic solutions that matches customers with highly qualified specialist for assignment sample writing and academic editing. Our Office Countries: UK,USA, Canada and Australia, Customer: "(Jefferson, CA)"Topic title:"The Mayans Adventure Essay"Discipline: "History"Pages: 4, (MLA)"What a great piece of article, i highly applaud the writer who wrote my essay, it was great. Meteorologists section the atmosphere into small blocks and apply analytics models to each block, using big data techniques to keep track of changes that hint at the future. This tool is used to import nonnative data … It is divided into three sections that guide you through direct-read formats and interoperability connections, quick conversion tools, and the fundamentals of transforming data using FME Workbench. Content2Share allows patients to pick what data they want to make available, thereby helping protect physicians. Population health technology makes use of risk scores, which are calculations of many clinical data points to predict, for example how likely a patient is to be readmitted to a hospital in the near future. XML and SQL are examples of common data formats and protocols. For example, the utility has 3.5 million power connections, ... Data interoperability in Estonia. Call Us On: +1 (209) 348 9544 Interoperability (pronounced IHN-tuhr-AHP-uhr-uh-BIHL-ih-tee ) is the ability of a system or a product to work with other systems or products without special effort on the part of the customer. Medication orders flowing from the EHR to the pharmacy system. Interoperability describes the extent to which different systems and devices can be able to exchange and interpret shared data. Achieving interoperability in critical IT and communication systems. To get data from workbooks, you will need to loop over the sheets. Chicago, IL: ALA TechSource. Data normalization allows the ability to map the different and diverse terminologies and conflicting standards (Benson, 2010). Boston: Artech House. The most popular articles on Simplicable in the past day. Data interoperability in Estonia. Therefore, it is critical that common, clean approaches to working with data are resilient, meet market needs and support growth. Installing the ArcGIS Data Interoperability extension gives you immediate access to WFS and GML-SF (simple features) datasets. Add the data to ArcMap Prerequisite: The Data Interoperability extension must be enabled. At the Data for Development Festival in Bristol in March 2017, the Collaborative agreed to produce guidance on data interoperability for development practitioners. Example: The object array is directly usable in the C# language. Data and information interoperability are key enablers for the creation of scalable IoT systems, particularly as the systems evolve and change over time to accommodate new components and applications. It is our hope that it provides a useful starting point for … Except where otherwise noted, this site and its content are licensed by the the Data Interoperability Standards Consortium under the Creative Commons Attribution 3.0 United States License. PremiumPapers.net is an experienced service with over 8 years experience having delivered over 79,500 essays over the years.Get Your Essay Done by a Specialist. Interoperability can be achieved if tools agree on a standardized API to read, write, and manipulate data. It is the ability of different information systems, devices and applications (systems) to access, exchange, integrate and cooperatively use data in a coordinated manner, within and across organizational, regional and national boundaries, to provide timely and seamless portability of information and optimize the health of individuals and populations globally. Report violations, 10 Examples of the Entertainment Industry, 18 Characteristics of Renaissance Architecture. The ArcGIS Data Interoperability extension is a simple solution for complex integrations. I loved the English and i scored highly on the research since it was worth 200 points". Within this model, interoperability is explicitly referenced as part of the processing stage of data collection; for example, ensuring that the right classifications and standards are used to collect and record data from the outset or that the individuals tasked with collecting data have liaised with counterparts in other organizations to define how they will capture and store it. Further, data normalization allows the data to be understood in many different places by different technologies. For example, it is common for the internal operating model (and supporting interoperability model) to differ from the one used for the extended enterprise. http://premiumpapers.net What is Interoperability? For example, if a data scientist who prefers R can easily call the Python script developed by a colleague from their preferred language, they avoid reimplementing the same work twice. The mental health industry can be said to be one of the last industries to automate. If two or more systems use a common data formats and communication protocols and are capable of communicating with each other, they exhibit syntactic interoperability. 10 Patient Data Sharing, Interoperability Principles for Providers A group of healthcare stakeholders and professionals aimed to reduce conflicting patient data … A list of common creative thinking techniques. It can ensure that data is exchanged in an effective manner and that it can be described at the data field level. Interoperability refers to the ability of two or more systems or components to exchange information and to use the information that has been exchanged. Available with Data Interoperability license. 11 Examples of Interoperability posted by John Spacey, February 02, 2016 updated on January 22, 2017. Interoperability describes the extent to which different systems and devices can be able to exchange and interpret shared data. Rudimentary UI capabilities were also supported allowing windows to be grouped with other windows. Translations of the phrase DATA INTEROPERABILITY from english to finnish and examples of the use of "DATA INTEROPERABILITY" in a sentence with their translations: ...services are used to achieve data interoperability . The medication formula is programmed into the pump to ensure the correct dosage and duration of teh med formula. The structural interoperability can be described as an intermediate level that defines the structure or format of data exchange. Three examples are: Patient data flowing from a fitness tracking device to a clinic’s EHR. The definition of analysis paralysis with examples. This level basically focuses on the packaging of the data via message format standards. But is it really as hard as it is being made out to be? Within this model, interoperability is explicitly referenced as part of the processing stage of data collection; for example, ensuring that the right classifications and standards are used to collect and record data from the outset or that the individuals tasked with collecting data have liaised with counterparts in other organizations to define how they will capture and store it. Steps: 1. Lower-level data formats also contribute to syntactic interoperability, ensuring that alphabetical characters are stored in the same ASCII or a Unicode … Click the Catalog Window button on the Standard toolbar. Data Interoperability extension can be used in the same way. By clicking "Accept" or by continuing to use the site, you agree to our use of cookies. RFID in libraries: A step toward interoperability. This document is the first attempt at producing such guidance. In addition, the volume of datahealthcare IT systems are producing exacerbate these problems. Over time, open-system ‘containers’ were developed to create a virtual desktop environment in which these applications could be registered and then communicate with each other using simple pub/sub patterns. ONC has a ten year roadmap to its achievement…but a recent survey says providers don't believe that even ten years will be enough if we stay on the current course. Benson, T. (2010). Possibility of data breaches – Protected health information (PHI) is a high-end commodity on the dark web. Example Use Case #4 – Data Validation • Create your own validation tests • Examples:-Consider the . The calculations used to be written on a printed order, then programmed into the device by a nurse. It is the structural interoperability that can effectively define the syntax that exists in data exchange. These include, for example, registries, coordinate reference systems, identifier management, metadata and maintenance, to name just a few. In particular, there are the participants at the first Data Catalog Interoperability Workshop in Edinburgh in May 2011 (organized under the auspices of the LOD2 project), as well as the members of the data-catalogs group and mailing list where this specification has been discussed on an ongoing basis. The ArcGIS Data Interoperability extension for Desktop is an integrated spatial ETL (extract, transform, and load) toolset that runs within the geoprocessing framework using Safe Software's FME technology. This is similar to Visual Basic but not the C# language. This material may not be published, broadcast, rewritten, redistributed or translated. (2012). Elering, the Estonian independent electricity and gas transmission system operator (TSO) created Estfeed in September 2017 – a solution for data access and exchange on Elering’s smart grid platform. Types. Structural interoperability defines the structure or format of the data exchange (i.e., the message format standards) where there is uniform movement of healthcare data from one system to another. Expanding and fine turning the healthcare system’s health data interoperability capabilities won’t just address the symptoms of a sick and staggering industry, argue groups like the CommonWell Health Alliance, The Sequoia Project, Carequality, EHNAC, and a handful of successful state-level health information exchange (HIEs) profiled in a new Government Accountability Office (GAO) report. Sector has been exchanged clear that data blocking '' …but it is that... By many as a consequence, tools rarely can share an interface and its implementation is a high-end on. Described at the data via message format standards ( 7 ) of Directive 2007/2/EC ( INSPIRE address... Has 3.5 million power connections,... data interoperability extension into feature classes interoperability that can define... 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## data interoperability example
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https://www.maxland.pl/0z3e1/takstar-sgc-598-australia-06721e | The SI unit of velocity is m/s. invariable. Walker, Jearl Fundamentals of physics / Jearl Walker, David Halliday, Robert Resnick—10th edition. metre/second is a derived unit. These units are part of the metric system, which uses powers of 10 to relate quantities over the vast ranges encountered in nature. ISBN 978-1-118-23072-5 (Extended edition) Binder-ready version ISBN 978-1-118-23061-9 (Extended edition) 1. Both fundamental units and derived units are parts of International System of Units (SI) and are standardized. Fundamental Physics Terms. Click here to learn the concepts of Fundamental and Derived Units from Physics It is defined by taking the fixed numerical value of the cesium frequency ∆ν Cs, the unperturbed ground-state hyperfine transition frequency of the cesium 133 atom, to be 9,192,631,770 when expressed in the unit Hz, which is equal to s −1. On one hand, due to the fascinating accuracy of atomic clocks, the traditional macroscopic standards of metrology (second, metre, kilogram) are giving way to standards based on fundamental units of nature: velocity of light c and quantum of action h. Note: In physics, understanding fundamental and derived units are very important because the properties of physics are represented only by their units in equations. It is used as pillars for other quantities aka Derived Quantities. Time is defined as the duration between two events. Q2: What are the Advantages of the SI System? For more info, see the NIST Guide for the use of SI. The units of fundamental quantities are called “fundamental units”. In Physics, Length, Mass, Time, Electric Current, Thermodynamic Temperature, etc are … In physics, there are seven fundamental physical quantities that are measured in base or physical fundamental units: length, mass, time, electric current temperature, amount of substance, and … Time – Second (s) Q4: What System of Measurement is Used in the US? YOU are the protagonist of your own life. As the speed of light in a vacuum is 3 x 108 m/s, and, Therefore, one light-year = 3 x 108 x 365 x 24 x 60 x 60 meter. "Unit" refers to 1. It is derived by multiplying or dividing one basic physical quantity with another basic physical quantity. The Steradian - It is the unit of solid angle. These are basic units upon which most units depends. length. You can read on Molecular Mass explained with worked examples here. It is the distance between the tip of the smallest finger and the tip of the thumb. Can someone please explain how fundamental units in physics work? uses a combination of fundamental units that result in a new u…. Physical quantities can be divided into two categories – base quantities and derived quantities. In the SI system, there are seven fundamental units: kilogram, meter, candela, second, ampere, kelvin, and mole. Derived Units. metre/second is a derived unit. m. time. The units for these quantities are called fundamental units and derived units. In this paper we discuss the various aspects of the interrelations among the SI units of physics and fundamental constants. The problem of fundamental units is discussed in the context of achievements of both theoretical physics and modern metrology. The c.g.s or Centimeter-Gram-Second System: A Gaussian system that uses centimeter, gram, and second as the three basic units for length, mass, and time respectively. Share on google. Unlock your Fundamentals Of Physics PDF (Profound Dynamic Fulfillment) today. Pro Lite, Vedantu There are other relationships between physical quantities which can be expressed by means of fundamental constants, and to some extent it is an arbitrary decision whether to retain the fundamental constant as a quantity with dimensions or simply to define it as unity or a fixed dimensionless number, and reduce the number of fundamental constants by one. So I am posting this in this forum. SI unit for mass. Take your favorite fandoms with you and never miss a beat. (i) Astronomical Unit (AU) : It is the average distance of the centre of the sun from the centre of the earth. The four fundamental units are abbreviated as follows: meter, m; kilogram, kg; second, s; and ampere, A. 4 c;h;G { units From the point of view of the future \theory of everything" it is natural to use The International System of Units (abbreviated as SI Units from its French name, Système International d'unités) is an internationally agreed metric system of units of measurement that has been in existence since 1960. Fundamental Quantities. When a physical quantity expresses itself in terms of two or more fundamental physical quantities. In order to measure something, you need to define a unit of measurement. Units of mass are commonly defined in terms of ounces and pounds, rather than the SI unit of kilograms.Other commonly used units from the United States customary system include the fluid volume units of the teaspoon, tablespoon, fluid ounce, US cup, pint, quart, and gallon, as well as the degrees Fahrenheit used to measure temperature. s. temperature. One could eliminate any two of the meter, kilogram and second by setting c and h to unity or to a fixed dimensionless number. In macrocosm measurements, i.e., measurement of very large distances: It is the average distance of the center of the sun from the center of the earth. kg. Before humans created a standardized system of measurement, many cultures utilized local traditions for measuring objects. A set of fundamental units is a set of units for physical quantities from which every other unit can be generated. One radian is the angle subtended by the center of a circle by an arc and is equal in length to the radius of a circle. Similarly, one could eliminate the candela as that is defined in terms of other physical quantities. In this way, all measurements are multiples of that unit. Each system is named with reference to fundamental units on which it is based. I to III) the progress in precision for the fundamental constants achieved in the last ten years will be given. We still use this to measure the height of horses. fundamental units. For example: Newton (N) is a derived unit because it cannot be expressed in the absence of fundamental unit (meter) and can be broken down to multiple units (Newton equals to kg*m /s 2). Ө = 1/60 min = 1/60 x 60 degree = 1/60 x 60 x π/180 radian, Since the radius of an arc, r = length of an arc (l)/angle subtended (Ө), Therefore, 1 parsec = 1 A.U./1 sec = (1.496 x 1011) x (60 x 60 x 180)/(π). long arc subtends an angle of 1”. The M.K.S or Meter-Kilogram-Second System: The fundamental units of length, mass, and time are meter, kilogram, and second respectively. Specific gravity is the ratio of the density of any substance to the density of what? Units of Measurement Physics Class 11 Download in pdf Measurement of physical quantities. The c.g.s or Centimeter-Gram-Second System: A Gaussian system that uses centimeter, gram, and second as the three basic units for length, mass, and time respectively. Halliday - Fundamentals of Physics Extended 9th-HQ.pdf - Google Drive. It is a unit of length in the imperial and the U.S. customary systems equal to 6 feet (1.8288 m). It is the unit of a plane angle. One could then eliminate the ampere either by setting the permittivity of free space to a fixed dimensionless number or by setting the electronic charge to such a number. Pro Lite, CBSE Previous Year Question Paper for Class 10, CBSE Previous Year Question Paper for Class 12. Thus, the measurement of mass is in multiples—or fractions—of 1 kilogram. One steradian is the solid angle subtended at the center of a sphere, by the surface of a sphere which is equal in area to the square of its radius. The Fundamental Quantity is independent Physical Quantity that is not possible to express in other Physical Quanitity. Title: Fundamental units: physics and metrology. Download PDF Abstract: The problem of fundamental units is discussed in the context of achievements of both theoretical physics and modern metrology. In theory, a system of fundamental quantities (or sometimes fundamental dimensions) would be such that every other physical quantity (or dimension of physical quantity) can be generated from them. There is innumerable physical quantifies in physics each of which has its own unit. One steradian is the solid angle subtended at the center of a sphere, by the surface of a sphere which is equal in area to the square of its radius. What is the mass per unit volume of a substance called? The U.S. The Fathom - It is a unit of length in the imperial and the U.S. customary systems equal to 6 feet (1.8288 m). Physics is a quantitative science, based on measurement of physical quantities.Certain physical quantities have been chosen as fundamental or base quantities. Derived units. The creation of the metric system following the complete destruction of the traditional-imperial French unit system marks the beginning of a series of events that eventually lead to the currently accepted International System of Units. The fundamental physical constants are shown in the table below with their current values (CODATA 2014), calculated using energy wave constants.This section details the calculations and provides an explanation for most of the physical constants and why they appear in equations. Mechanics, branch of physics concerned with the motion of bodies under the action of forces, including the special case in which a body remains at rest. Find the fundamental units involved in the following derived units. It was used to build pyramids. Fundamental units: Physical quantities which are independent and cannot be derived from other physical quantities are called fundamental quantities and their units are called the fundamental or base units. www.physicstoday.org July 2014 Physics Today 35 A more fundamental David B. Newell System of Units International The universally accepted method of expressing physical measurements for world commerce, industry, and science is about to get a facelift, thanks to our improved knowledge of fundamental constants. The M.K.S or Meter-Kilogram-Second System: The fundamental units of length, mass, and time are meter, kilogram, and second respectively. At its 24 th meeting in October 2011, the General Conference on Weights and Measures (CGPM) approved possible changes to the International System of Units, including new definitions for the kilogram, ampere, kelvin and mole. It is the unit of solid angle. A base quantity is one of a conventionally chosen subset of physical quantities, where no subset quantity can be expressed in terms of the others. Derived units are those units which cannot be expressed in the absence of fundamental units. The purpose of the CODATA Task Group on Fundamental Physical Constants is to periodically provide the scientific and technological communities with a self-consistent set of internationally recommended values of the basic constants and conversion factors of physics and chemistry based on all of the relevant data available at a given point in time. In theoretical physics it is customary to use such units (natural units) in which c = 1 and $\hbar$ = 1. The unit for volume is a unit derived from the SI unit of length and is not a fundamental SI measurement. Fundamental Units. I was looking at the chapter "Units and Measurement". The system itself is based on the concept of seven fundamental base units of quantity, from which all other units of quantity can be derived. To get a precise answer to these questions, measuring the quantities like distance, velocity, and time becomes essential. Time . The current system of units has three standard units: The meter, kilogram, and second. 2.2. Pro Lite, Vedantu Those that cannot be so expressed can be regarded as "fundamental" in this sense. There are base units and derived units. For example, distance, displacement, momentum, etc. We still use this to measure the height of horses. (i) Newton(N)(ii) Watt (W)(iii) Joule (1)(iv) Pascal (Pa)(v) Cubic metre - the answers to estudyassistant.com Eite Tiesinga, Peter J. Mohr, David B. Newell, and Barry N. Taylor This database gives values of the basic constants and conversion factors of physics and chemistry resulting from the 2018 least-squares adjustment of the fundamental physical constants as published by the CODATA Task Group on Fundamental Constants and recommended for international … The density of a material is defined as its mass per unit volume. fundamental equations of physics were written in four dimensional form (utilizing length as the unit for all dimensions), the unit of time and all factors containing that unit would be eliminated. Sign in NOW is the time to make today the first day of the rest of your life. Sign in. Values of Fundamental Physical Constants. Authors: L. B. Okun. Walker, Jearl Fundamentals of physics / Jearl Walker, David Halliday, Robert Resnick—10th edition. Physics is a quantitative science, based on measurement of physical quantities.Certain physical quantities have been chosen as fundamental or base quantities. Electric current could be used instead of charge or speedcould be used instead of length. A set of fundamental dimensions of physical quantity is a minimal set of units such that every physical quantity can be expressed in terms of this set. Fundamental units are the units of fundamental quantities. Parsec is the distance at which 1 A.U. In the language of measurement, quantities are quantifiable aspects of the world, such as time, distance, velocity, mass, momentum, energy, and weight, and units are used to describe their measure. In order to solve numerical problems in case of unit conversion questions in physics, consulting a tabulated form of unit conversions is highly important especially for beginners. The four fundamental units we will use in this text are the meter (for length), the kilogram (for mass), … Historically, mechanics was among the first of the exact sciences to be developed. Customary Units (The U.S. standard, British Imperial Units, or English standard). It is the unit of long distances and represents the parallactic seconds. The SI unit of heat is the unit joule. These are as follows: The Cubit - This measurement originated in Egypt about 3000 B.C. ‘Those men also divided the complete circle into 360 degrees by taking the angle of the triangle as their fundamental unit and dividing this into 60 sub-units.’ ‘The final surprise I'll mention is that Leibniz's system of doing physics, which is based on fundamental units called monads, has got a few things in common with the modern notion of computational physics, or ‘it from bit.’’ Learn about Fundamental and Derived Units of Measurement topic of Physics in details explained by subject experts on Vedantu.com. Fundamentals of Physics 10th edition Halliday and Resnick pdf A British engineering system of units that uses the foot as the unit of measurement of length, and pound as the unit of mass and second as the unit of time. In the language of measurement, quantities are quantifiable aspects of the world, such as time, distance, velocity, mass, momentum, energy, and weight, and units are used to describe their measure. In order to solve numerical problems in case of unit conversion questions in physics, consulting a tabulated form of unit conversions is highly important especially for beginners. Examples- density, volume, force, power, velocity, acceleration, etc. Q1: Calculate the Number of Astronomical Units in Two Meters. Base units of the International System; quantity unit definition; time: second: s: The second, symbol s, is the SI unit of time. A Complete Unit Conversion Table. In the international system of measurement (S.I.) (iv) there are seven fundamental units. With the help of the German physicist Wilhelm Weber (1804–189… Derived units are those units which cannot be expressed in the absence of fundamental units. What are Fundamental Units? What is the smallest subdivision of a substance? Another way of saying this is that. Fundamentals of Physics 10th edition Halliday and Resnick pdf share it on. So, the number of astronomical units in 2 meters is 1.333 x 10-11A.U. The units of measurement are defined as standard and do not vary. The physical quantities that depend upon the fundamental quantities are known as the derived quantities. Fundamental Physical Constants – Summary of Calculations. $\begingroup$ AFAIK, there is no such thing as a "fundamental unit" defined in SI. As the definition was "the amount of current which passing through a 1 m length of two infinitely long wires 1m apart give a force of 2 * 10 ^ -7 N. Following the … Shed the societal and cultural narratives holding you back and let step-by-step Fundamentals Of Physics textbook solutions reorient your old paradigms. 2.1.1 Characteristics of fundamental units: (i) they are well defined and are of a suitable size (ii) they are easily reproducible at all places (iii) they do not vary with temperature, time pressure etc. The great German mathematician Carl Friedrich Gauss (1777–1855) was the first to promote the idea of combining metric units with the second to form a complete and consistent unit system for mechanics. For instance, time and distance are related to each other by the speed of light, c, which is a fundamental constant. In theory, a system of fundamental quantities (or sometimes fundamental dimensions) would be such that every other physical quantity (or dimension of physical quantity) can … Read formulas, definitions, laws from Fundamental and Derived Units here. meter. Choose from 500 different sets of fundamental units physics flashcards on Quizlet. A Complete Unit Conversion Table. The SI base units are the standard units of measurement defined by the International System of Units for the seven base quantities of what is now known as the International System of Quantities: they are notably a basic set from which all other SI units can be derived. You might think that the number of fundamental units would be well-defined, but even that's not true.. Take electric charge for example. In the first part (Sects. Similar considerations apply to Planck's constant, h, which relates energy (with dimensions of mass, length and time) to frequency (dimensions of time). Here, you can notice that metre and second are fundamental units while the unit of speed, i.e. Units are standards for expressing and comparing the measurement of physical quantities. In the international system of measurement (S.I.) In mechanics we treat length, mass and time as basic or fundamental quantities. Eite Tiesinga, Peter J. Mohr, David B. Newell, and Barry N. Taylor This database gives values of the basic constants and conversion factors of physics and chemistry resulting from the 2018 least-squares adjustment of the fundamental physical constants as published by the CODATA Task Group on Fundamental Constants and recommended for international … Differences between fundamental unit and derived unit I have been in my physics crash course for my entrance exam. Fundamental quantities are those that are defined directly by the process of measurement only. 1 AU = 1.5 × 10 11 metre (ii) Light year : It is defined as the distance traveled by light in vaccum in one year 1 light year = 3 × 10 8 × (365 × 24 × 60 × 60) metre 3. Answer: 2 question 6. The system included clear explanations of how to realize the base units through measurement, and the coherent derived units were products of powers of the base units with a prefactor of 1. Fundamental concepts of the Physics start from this chapter. I to III) the progress in precision for the fundamental constants achieved in the last ten years will be given. The fundamental quantities that are chosen are Length, Mass, Time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. viz— (1) Fundamental unit. The four fundamental units we will use in this text are the meter (for length), the kilogram (for mass), the second (for time), and the ampere (for electric current). Units The fixed and definite quantity taken as the standard of reference with which other quantities of the same kind are measured is defined as a “unit”. This measurement originated in Egypt about 3000 B.C. The New International System of Units based on Fundamental Constants Beat Jeckelmann, Chief Science Officer, Federal Office of Metrology METAS, Bern-Wabern . Students are suggested to be thorough with these aforementioned important physics SI units lists. One could similarly eliminate the mole as a fundamental unit by reference to, One could eliminate the kelvin as it can be argued that temperature simply expresses the energy per particle per degree of freedom which can be expressed in terms of energy (or mass, length, and time). That leaves every physical quantity expressed simply as a dimensionless number, so it is not surprising that there are also physicists who have cast doubt on the very existence of incompatible fundamental quantities.[1]. Some physicists have not recognized temperature as a fundamental dimension of physical quantity sin… We all know that physics is a branch of science which deals with the study of nature and natural phenomena. Values of Fundamental Physical Constants. volumes cm Includes index. In Physics, Length, Mass, Time, Electric Current, Thermodynamic Temperature, etc are examples of Fundamental Quantities. Share on twitter. Fundamental Physical Constants. One radian is the angle subtended by the center of a circle by an arc and is equal in length to the radius of a circle. They are not defined in terms of other quantities; their units are not defined in terms of other units. The Seven Base Units Of Measurement. All units can be expressed as combinations of four fundamental units. Fundamental and Derived Unit. Units of Measurement Wiki is a FANDOM Lifestyle Community. Row 8, and 9: Two supplementary units on the SI system are: The Radian - It is the unit of a plane angle. Density is another measurement derived from SI basic units. The Hand-Span - It is the distance between the tip of the smallest finger and the tip of the thumb. Derived units in Physics:– Except the three fundamental units, all other units in the mechanics are derived in terms of these fundamental units and so they are called as ‘‘Derived Units’’. The common system of units utilized in mechanics are as follows: The f.p.s or Foot-Pound System: A British engineering system of units that uses the foot as the unit of measurement of length, and pound as the unit of mass and second as the unit of time. Length (meter) Mass (kilogram) Time (second) Electric current (ampere) The system of units is the complete set of units, both fundamental units, and derived units, for all kinds of physical quantities. Some Practical units: For the measurement of very large distance, the following three units are used. What Is the Fundamental Metric Units of Physics? A System of Units The system of units is the complete set of units, both fundamental units, and derived units, for all kinds of physical quantities. Unit I Value in another unit; 1 Inch: 2.54 centimeter: 1 Foot: 0.3048 meter: 1 Foot: 30.48 centimeter: 1 Yard: 0.9144 meter: 1 Mile: 1609.34 meter: 1 Chain: 20.1168 meter The quantities which we can measure directly or indirectly are known as physical quantities. The Physical Quantities are Divided into Two Categories: The physical quantities that do not depend upon the other quantities are the fundamental quantities. 1. But almost all the units of physical quantities can be expressed in terms of three units. i.e. Fundamental units . Identification of fundamental quantities - definition. ISBN 978-1-118-23072-5 (Extended edition) Binder-ready version ISBN 978-1-118-23061-9 (Extended edition) 1. It is must to measure all Physical quantities so that we can use them. The ampere and the second are base units, and 1 coulomb is equal to 1 amp-second. the standard unit used for measurement of a quantity in the me…. A widely used choice is the so-called Planck units, which are defined by setting $\hbar$ = c = G = 1. A fundamental unit is a unit adopted for measurement of a base quantity. I got a doubt in this chapter and I didn't get it cleared by my teacher. TABLE II: SI Examples of Derived Quantities and Their Units Property Symbol Unit Dimension Force F newton (N) kg m s 2 = kg m /s2 Speed v meter per second (m/s) m s 1 = m/s Pressure P pascal (Pa) (force per unit area) kg m 1 s 2 Energy E joule (J) kg m2 s 2 Power W watt (W) (energy per unit time) kg m2 s 3 A physical quantity can be expressed with a unique combination of 7 base quanti- These three units form the mks-system or the metric system.A meter is a unit of length, currently defined as the Is the velocity of a body directly related to its mass? Both fundamental unit and derived unit are overwhelmingly used in common purposes to express corresponding physical quantities. Share on facebook. kilogram. Sorry!, This page is not available for now to bookmark. fundamental units. It may be divided into three branches: statics, kinematics, and kinetics. Ans: As 1 ly = 9.46 x 1015m, and 1 parsec = 3.1 x 1016m, Now, 1 parsec/1 ly = 3.1 x 1016/ 9.46 x 1015. Fundamental quantity : quantities which are independent on other physical quantity . It is used as pillars for other quantities aka Derived Quantities. the metre (m) ('meter' in the US) is the basic unit of length and is defined as the distance travelled by light in a vacuum in 1/299,792,458 second. Fundamental units: physics and metrology 3 Planck also considered as universal the Planck temperature TP = mPc2=k. A set of fundamental units is a set of units for physical quantities from which every other unit can be generated.. Learn fundamental units physics with free interactive flashcards. molecule. Water. For example, the unit of mass is the kilogram. Read Also: Physical quantities. SI International system. Length – Meter (m) It is defined as the length of the path travelled by light in an interval of exactly $$\frac {1}{299 792 458}$$ s. It is based on the fundamental quantity, the speed of light in a vacuum which is c=299 792 458 m/s. the second (s) is the basic unit of time and it is defined as the time it takes a cesium (Cs) atom to perform 9,192,631,770 complete oscillations. Basically the terms & concepts which are illustrated in this topic will be used in so many ways because all Physical quantities have units. Density. $\endgroup$ – The Photon Apr 23 '15 at 2:01 Derived Units Table: The Table Shows the List of Derived Units, So, the number of astronomical units in 2 meters is, NCERT Book Class 11 Geography - Fundamental of Physical Geography PDF, MCQ’s on 2nd and 3rd Law of Thermodynamics and Entropy, Systems of Particles and Rotational Motion, Wave Nature of Matter and De Broglie’s Equation, Vedantu These relationships are discussed in dimensional analysis. It was used to build pyramids. It is possible to use this relationship to eliminate either the fundamental unit of time or that of distance. Being a physics enthusiast to understand this natural phenomenon; I will search for answers to the following questions: How much will it take for a ball to reach the ground? For example: Newton (N) is a derived unit because it cannot be expressed in the absence of fundamental unit (meter) and can be broken down to multiple units (Newton equals to kg*m /s 2). second. Each system is named with reference to fundamental units on which it is based. The units and their physical quantities are the second for time, the metre for measurement of length, the kilogram for … The common system of units utilized in mechanics are as follows: In the first part (Sects. The coulomb is a derived unit. The history of the metre and the kilogram, two of the fundamental units on which the system is based, goes back to the French Revolution. https://units.fandom.com/wiki/Fundamental_unit?oldid=5155. In the SI system, there are seven fundamental units: kilogram, meter, candela, second, ampere, kelvin, and mole. Abstract. Which units are fundamental and which are derived is pretty much a matter of arbitrary convention, not an objective fact about the world. This physics textbook is designed to support my personal teaching activities at Duke University, in particular teaching its Physics 141/142, 151/152, or 161/162 series (Introduc-tory Physics for life science majors, engineers, or potential physics majors, respectively). volumes cm Includes index. This definition establishes that the speed of light in a vacuum is … If two water samples have different volumes, they still share a common measurement: the density. Vedantu academic counsellor will be calling you shortly for your Online Counselling session. What are the fundamental metric unit of physics? ... Now there can be argument about whether the ampere is a fundamental unit. Ans: The US uses two major systems of measurement, they are: Metric systems (International system of units or SI), and. How many pounds are in a kilogram? Here, you can notice that metre and second are fundamental units while the unit of speed, i.e. Register free for online tutoring session to clear your doubts. But Boltzman’s k is not a universal unit, it is a conversion factor: k =8:6 10−5 eV/K (hint: h!=kT). The fundamental physical quantities are mass, time, length, temperature, electric current, intensity of light and quantity of … International System of Units (SI Units) Meter. a unit that cannot be expressed in terms of other measurable q…. 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https://tex.stackexchange.com/questions/385649/a-latex-course-for-intermediate-users-stuck-on-the-learning-curve | # A LaTeX course for intermediate users stuck on the learning curve
Background: This question is aimed primarily at those of us who have been using LaTeX for quite some time but feel like they do not have a very "principled understanding" of it even though they would like to (i.e. those of us who want to go from intermediate to advanced but feel lost at sea).
TL;DR: Does anyone know of a course like structure (MOOC or otherwise...Udemy and Coursera have nothing of value) for intermediate LaTeX users to learn LaTeX in a "principled fashion"? Those of you who teach LaTeX at your universities, how are your courses structured (if you don't mind sharing)?
To keep this question relatively short, I'd simply like to say I've been using LaTeX now for a number of years and have never ceased to be amazed by its capabilities and even more so by the incredible support group here on TeX.SE. I've read several of the other questions somewhat related to this:
The frustrating thing is that most educational LaTeX material seems to be geared towards either those who are just starting out or those who are much more advanced. For example, I look at the content of that Udemy course (and this is not meant to be a knock against it) and am rather surprised by the "advanced content" (all things I am fully comfortable with doing and have done before). But I look at something like the computer science behind TeX/LaTeX by Eijkhout and immediately feel a bit out of my depth.
So far my approach to learning has been to simply learn what's necessary as I encounter different problems (and post here on TeX.SE when I can't progress). Even when I have seen questions on here that I feel like I may know how to answer, I never do because I know several of the gurus here have a "proper" way of going about it all. I also know some members teach LaTeX at their universities (egreg and gonzalo come to mind).
In sum, I am eager to learn LaTeX in a "principled way" (understand the code for packages, be able to write my own, etc etc) but am lost as to how to do it effectively without a course like structure. Simply reading through the TeXbook, LaTeX and Friends, The LaTeX Companion, etc. etc. is not really cutting it anymore (it's so easy to get overwhelmed by the sheer volume of different books that one has great difficulty proceeding in a linear fashion). A well-defined structure would help a great deal.
• Flagged this to make it community wiki...if it's more appropriate for meta, please let me know and I'll post it there. – Daniel W. Farlow Aug 10 '17 at 1:01
• It doesn't seem a Meta question to me. I think this is the correct place where to ask. (BTW, +1). – CarLaTeX Aug 10 '17 at 4:12
• The problem with courses beyond 'basic'/'beginners' is always what does one cover: different users have very different ideas of what counts as intermediate. – Joseph Wright Aug 10 '17 at 7:13
• I agree with @JosephWright. The intermediate LaTeX courses I've tried to teach in the past had too wide a range of abilities across the participants that it was impossible to pitch it at the right level to all of them, so for my third LaTeX book I marked up basic, intermediate and advanced in the table of contents as a rough guide. – Nicola Talbot Aug 10 '17 at 13:42
• Thanks for mentioning my book "The computer science of .... " but that book is not really about LaTeX. It's about the science that lies behind the internals. It's a CS topics course, using TeX/LaTeX as a hook, but it's of basically zero use if you want to learn more LaTeX. – Victor Eijkhout Aug 12 '17 at 0:43
The problem: different users have different interests (and needs) and therefore different ideas of what counts as "intermediate" (beyond the basics), as pointed out by Joseph Wright in a comment above. This was perfectly elaborated by Christian Feuersänger in this answer from 2011: he categorized three kinds of things that post-beginner (La)TeX users may like to learn (which I'm rephrasing), and only you can decide what is important to you:
1. TeX's features for typesetting specifically: the box-glue-penalty model, alignments, a few parameters and features built into the program like \parshape and the like. Basically: viewing TeX's output (each page) as a giant box made up of boxes laid out in sequences vertically (e.g. lines of text) and horizontally (e.g. characters within a line) with appropriate glue between them, and learning how to make a page look like some predetermined ideal you already have in mind.
2. TeX's facility for programming: the intricacies of macro expansion: what do terms like \xdef, \expandafter, protected and fragile, \csname mean? How do you write macros that do the work you want them to do? How can you use the features provided by expl3 to make this easier?
3. The package ecosystem: the facilities provided by LaTeX itself, and by the most important of the thousands of packages available on CTAN. For example, LaTeX itself provides features like automatic numbering (you just type \section and somewhere behind the scenes a counter is incremented and it comes out numbered), cross-referencing (\label and \ref), and so on. There are packages for specific fields (mathematics, chemistry, linguistics), for specific languages, for specific kinds of documents (presentations, drawings), etc.
Now the fact is, for most people who are users of LaTeX, the thing that is probably most useful to them, and in that way most worth knowing, is (3). “How to draw X in TikZ” is something about the features present in the TikZ package. The knowledge that package Y exists for drawing chemical diagrams or grammatical parse trees is something specific to the domain you work in. The knowledge that the “recommended” way to set page margins to use package geometry, or whether to use beamer or powerdot or prosper for presentations, or that \usepackage{hyperref} should come towards the end of your list of included packages, are all of this nature.
In short, a lot of this is knowledge about the “state of the world”: about what packages people happen to have written, what features (and bugs!) those packages happen to have, even how good and up-to-date those packages are — all of these are “contingent” and will even change with time. There is no “core” to this, and not necessarily much consistency because they were written by many different people. You can look at the “banner ad” at https://www.ctan.org/pkg to see a bit of the menagerie of packages that exist. This is the core of the difficulty you mention in the question (learn LaTeX in a "principled way" […] great difficulty proceeding in a linear fashion … A well-defined structure would help), and it is unavoidable as far as I can tell.
A post-beginner (“intermediate” or “advanced”) LaTeX course that goes deeper into any of this will be restricted to some people's uses for LaTeX, and possibly go out of date. And yet this is what most people want and need, so you can consider specific books targeting specific users' needs: the linked LaTeX for Administrative Work by Nicola Talbot above (see its table of contents), or Grätzer's More Math into LaTeX (not free, but see what's available on CTAN to get an idea) are particularly good examples of this.
That leaves categories (2) (macros) and (1) (typesetting primitives). Some people, by "intermediate" or "advanced", mean (2): being able to write and understand macros. This is essentially a new way to program, compared to any other form of programming you may be familiar with. See Notes on Programming in TeX for a (very useful) flavour of things. People have written a BASIC interpreter, the incomparable xii.tex, an Othello game (named reverxii as a tribute to the former), a controller for a Mars Rover… so it's definitely possible to do things in it. Much of LaTeX3 work hitherto has apparently gone into expl3 which makes this easier. This is something one can probably pursue, trying harder and harder exercises until they're proficient at writing macros. (I'm not aware of such a book or resource.)
Personally I find programming in macros rather perverse. :-) (Not just xii.tex and the like, but even the code of typical packages, and of LaTeX itself.) Whatever its other merits, TeX does not have a great programming language in it (see the answers to this question). (There are some comments of Knuth indicating this was intentional, but he did not foresee that people would go to great lengths to program in it anyway.) So personally (and this is a minority opinion in the TeX community, perhaps I am even a minority of one, simply out of not having thought this through well enough): I would prefer a style of TeX usage that involves as few macros as possible. (This is not specific to LaTeX: it would apply to ConTeXt, to plain-TeX packages like opmac, probably every way in which TeX is used currently.)
Nevertheless it (proficiency in TeX macros) is something learnable, and may be a concrete answer to your question.
Finally, that leaves (1) (typesetting). At heart TeX has a startlingly simple and unexpectedly versatile model of typesetting: breaking paragraphs into lines using the box-glue-penalty model. It sounds like something that's only about normal paragraphs of text (and that's what it was designed for), but with appropriate choices of your boxes, glues and penalties, you can achieve practically everything you've ever seen in a document typeset with (La)TeX (and more). The power of this model, despite its unifying simplicity (which was something noted by early reviewers of TeX, like Kernighan who worked on troff), was something of a surprise even to Knuth and he says it took him years to learn more and more things that could be done with it. (Of course from a certain perspective you could call these things hacks, but I find them very elegant.)
This is the part I personally am most interested in (somewhat unusually I think); this is also the thing that is directly against LaTeX's philosophy, which is to focus on the content / document structure, and leave the formatting to the packages you use. And if you're happy with that, you probably don't have to learn deeper here (unless you want to write your own packages that provide formatting, and suddenly you have to learn all the things that your packages were helpfully trying to hide from you). If you do want to learn more about what I consider the core of TeX typesetting, I can recommend what I think are the two best resources (even better to learn from than The TeXbook, which you can read next):
• The book A Beginner's Book of TeX by Seroul and Levy. This is such a lovely book: it is gentle and assumes nothing, focuses on only a subset of TeX, yet everything is introduced so appropriately and made to seem natural (including making some typical mistakes and explaining why they go wrong), that the design of TeX will appear to make sense.
• The paper Breaking Paragraphs into Lines by Knuth and Plass. A tour de force: discusses the problem, formulates it mathematically, defines desirability criteria, compares approaches, shows the power of this model (lots of sophisticated examples), then goes into the algorithm with its “bells and whistles” before concluding with an inspiring history.
For more on this aspect of TeX you can read The TeXbook, some other stuff that came out of the Stanford TeX project by Knuth and his associates, some talks and articles by Frank Mittelbach, some stuff by people who have written TeX-like or TeX-inspired systems (there are a few in existence, like SILE and Patoline, and a bigger graveyard of unfinished systems but some of whose authors wrote something useful before their projects died), etc.
But I'll say this upfront: even though this is a lot of fun to learn, it won't help you much when actually using LaTeX. Existing packages have reached a power and complexity beyond what you can easily arrive at from first principles (e.g. LaTeX's float/figure placement is implemented entirely in the macro layer). More on this below…
You mentioned
understand the code for packages, be able to write my own, etc etc
This unfortunately will take some work for anything nontrivial, because the entire system isn't set up for you to succeed at it (it optimizes for the end users, not package writers). The macros of LaTeX or major packages tend to be frightfully complicated because they are the outcome of decades (30+ years in LaTeX's case) of development: starting with something simple, they have grown to encompass all conceivable cases, be robust against various failure modes, etc. (Do you want \section{...} to not break when you stick a macro into its argument? Do you want the code of \section to be simple and readable? Pick one.)
Moreover, most packages are written assuming various functionality of other packages (and, if they have something to do with formatting, the exact formatting of other packages), they may use advanced macro techniques to be robust and comprehensive, these macros in turn assume aspects of other packages, etc.
In short, getting from “I know to use a few packages” to “I understand most of the stuff about most packages and if not I can read and understand them, or write my own” (like some of the top users on this site), involves some combination of all of (1), (2) and (3): you need to know how macros work, know what typesetting abilities exist, and both of these for all other packages you use or which typical users may use along with yours. (You can find packages whose documentation mentions “this package has been tested in combination with the following packages…”.)
So to conclude/reiterate: there are these three different categories, and if you really care about all of them (and not how to be a more skilful LaTeX user who knows how to use which packages, which is what is ultimately useful in real life), then you may have to pursue all three in parallel instead of in any structured manner.
My own preference (“learning style” :P) is to learn things bottom-up, from simplest onwards (without caring for utility), and this usually happens to be the historical order too (things generally get more complex with time: they get easier but rarely simpler), so I'd recommend the following path:
• Learn (or think about) the main reason TeX was written: to set type (in other words: a not horribly inconvenient way to specify how characters should be picked up from certain fonts and placed on a page), the box-glue model, why its macros exist, simple macros (e.g. \def\line{\hbox to\hsize} and \def\centerline#1{\line{\hss#1\hss}}), building up to more complex things like page layout, etc. Learn enough about plain TeX, even if you're not going to use it. (The book I mentioned, by Seroul & Levy, takes basically this approach.) Read The TeXbook. Read historical material from Digital Typography. Skim through early issues of TUGboat.
• The big jump that happens with LaTeX: look at some of LaTeX source code (open texdoc source2e and jump to a random page), learn whatever you need till it makes sense. :-)
• Look at some sample packages which interest you, or answers on this site that seem to have hairy code, understand how they work, etc.
This path is of course very different from learning to do useful things with LaTeX, so I won't recommend it to anyone (also I haven't even got far enough in the first step!). Still, this is one approach that may be somewhat “structured” in the sense you mean. In my case, it has led to fewer things I am completely mystified by and feel “out of my depth”: I may not understand everything (or much), but at least now I can usually get some idea of what's going on and get what I'd need to learn first, when looking at any given situation.
• Very thoughtful answer and greatly appreciated. You mentioned a few things concerning my personal situation that I did not expand on in my original questions (largely to keep it short and more applicable to everyone else), but I have definitely departed from the whole "I want to learn this because it is useful" approach. I want to get into the nitty gritty, but my main issue has been trying to find an entry point that provides some traction (I am more than willing to put in the time...just don't want to waste it unnecessarily). I have all of Knuth's Selected Papers as well as [...] – Daniel W. Farlow Aug 16 '17 at 3:15
• [...] all of the Computers and Typesetting volumes and the LaTeX Companions volumes as well (along with a smattering of other (La)TeX books. Your answer really helped me get a better sense of how to more effectively use my own resources and possibly adopt a few others to more fruitfully approach my goals. Thanks again for taking the time to write such a thoughtful answer. – Daniel W. Farlow Aug 16 '17 at 3:17
• @DanielW.Farlow Look at the package code, but don't assume it is a good way to do it, safe or even sane. It may be. Or it may be not. – cfr Aug 16 '17 at 3:37
• @DanielW.Farlow Glad to have typed it all out if someone found it helpful; nice to see someone else similarly curious. :-) (And +1 to what cfr said: package code may be ancient and doing things in bad ways.) I forgot to mention another resource that you may be interested in: this set of videos (classes by Knuth about TeX: the program has changed but some ideas are still relevant) (also this set of videos: autobiography). – ShreevatsaR Aug 16 '17 at 3:45
Since you asked how people who teach this in universities structure courses, here's mine. I teach a series of workshops to postgraduate students from all disciplines. (Generally - occasionally, I've taught a group for a particular school.)
The structure is as follows:
I also distribute a bunch of handouts, including a LaTeX 'cheat sheet', list of basic maths stuff etc. (This is all the standard stuff - not by me.)
• LaTeX II - intermediate, was a day-long workshop, but recently reorganised into 4 short workshops on particular topics (materials at https://github.com/cfr42/latex-2):
1. writing macros, classes & packages
2. bibliographies with Biblatex/Biber
3. Beamer
4. float management, graphs of functions and using external data, drawing with TikZ
What these workshops do, for the most part, is say 'here's a useful package or tool and a couple of very simple uses, so you get the general idea'. Each has an appendix with some further information for the students who are racing ahead and directions to further resources. Different students need different things and have different interests, even though these topics are general.
I also provide a handout with package recommendations (general on one side; subject-specific on the other), a Biblatex 'cheat sheet', a font sampler and a list of font packages. (These I had to write as I couldn't find them already done.)
I highly doubt this is what you had in mind, though.
• Ha, I wondered what cfr meant! Are you related to Sarah Rees? (I stumbled across this posted as I am putting together a course on latex...) – user30471 Aug 15 '18 at 13:31
• @Andrew Not that I know of, no. 'Rees' is an extremely common name, after all. – cfr Aug 15 '18 at 14:42
Joseph Wright has it. LaTeX started out strictly within the university environment for computer science and mathematics streams, but now it's adapted in all directions according to the individual requirement of the user.
I think the majority of us bumble along, referencing different resources, and reading package descriptions when we require a new function. I employ LaTeX on a very basic level creating templates for business correspondence and other paperwork. I'm also a registered trainer/assessor, but if I were to structure a LaTeX course based on my knowledge of LaTeX it would be too limited for many, and contain too much extraneous material for the rest.
• not directed toward educational goals, but a comment on history, the basic latex document classes may be suitable foir computer science publications, but they are not suitable for math. ams-tex was developed for that purpose, and only later adapted for use with latex as amsmath; additional changes were required to make the latex document paradigm suitable for use by at least one math publisher with (at the time) more than 100 years of a publication track record. – barbara beeton Aug 20 '17 at 1:28
• Basic math requirement is answered to with LaTeX. What I related was relevant to TeX, but that is accessible by way of \usepackage{amsmath} and \usepackage{mathtools} in the LaTeX environment now. – David Crosswell Aug 21 '17 at 7:29
• not referring to the math. the top matter. that which goes into a later bibliographic reference to the current document, is insufficiently defined in the basic latex document classes. that is why every publisher has their input rules. if the top matter had been more carefully subdivided in the beginning, at least the ams would not have felt the necessity to make drastic changes, but would have modified only the appearance of the output, leaving the input conventions intact. – barbara beeton Aug 21 '17 at 14:09 | 2021-03-06 07:55:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8058205246925354, "perplexity": 1419.1227557693958}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178374616.70/warc/CC-MAIN-20210306070129-20210306100129-00373.warc.gz"} |
https://domestic-engineering.com/projects/neural/neural.html | # My first successful neural network project
Photo credit to Colin Behrens
## Origins
This is a (lightly edited) conference paper I published in 2022 at the Digital Pipelines Solution Forum [1]. It describes the first commercially useful neural network I produced. If you were to drop a magnetometer down a steel pipeline and measure the magnetic flux at 200 hz, this neural network would be able to interpret that data and produce a list of joint locations in time. Obviously this problem has a narrow application. We might be the only company in the world that could benefit from solving this particular problem but for us it is incredibly valuable. Prior to the development of this algorithm, the company paid data analysts like me to manually identify joint locations in a GUI by clicking on them one at a time. In a line with 3,000 joints, this might take two to four days. Running on a modern laptop, my neural network produces a result with false positive and false negative rates below 5% in under three minutes. Manually reviewing the joint list with these error rates is between three and ten times faster than manual identification from scratch.
As we collect more training data, we expect false positive and false negative rates to drop below 1%. For context, the discrepancy between two different manual labelers is around 0.5-2%. We expect to beat human level performance in the next two to five years.
## Introduction
In 2016, pipeline failures in the North America resulted in the uncontrolled release of 2,000 metric tons of hydrocarbons (both liquid and gas) [2]. Pigs a vital part of pipeline maintenance. The term pig refers to any device placed inside a pipeline, pushed through from entrance to exit usually with fluid pressure. There are many varieties of pig; some are made of steel and scrape along the wall removing deposits; some are foam and do the same thing but less aggressively and with the ability to traverse tighter bends and diameter changes; some are equiped with tracking devices making them easier to locate if stuck in the pipe; some are filled with sensors designed to detected leaks or corrosion. The primary tools of pipeline integrity management since the 1980s are magnetic flux leakage (MFL) and ultra sound devices. These are "smart pigs." However, the problem of unpiggable pipelines, due to tight bends, non-circular valves, diameter changes or unknown geometry continues to cause problems in the pipeline inspection industry [2,3]. Approximately 70% of US gas lines were built before modern inline inspection (ILI) was a viable technology [4]. As of 2012, industry surveys indicate ~40% of gas pipelines in North America are unpiggable [5]. More recent technologies to address unpiggable lines began in the early 2000s [6,7].
Regardless of the technology, determining location within the pipe has always been a challenge. Traditional ILI tools use odometer wheels but slippage remains a problem, particularly in lines with heavy deposits or rough surfaces [8]. Inaccuracies in distance can be minimized by using external above ground markers (AGMs). AGMs come in magnetic, acoustic, or geophone array varieties [9,10,11]. However, AGM systems are expensive, often cannot be used in crowded urban areas, and are limited in accuracy depending on the depth of the pipeline. Multisensor free-floating devices solve the localization problem with an alternative method which has its own set of challenges and strengths. Welds identified from the magnetometer data are a primary component of the conversion from measurement time to measurement distance [12]. The neural network (NN) we are presenting here successfully automates the detection of pipe welds in steel pipes from residual magnetometer data with near human level performance. Automation of this critical bottleneck dramatically improves the scalability of multisensor free-floating devices.
## One dimensional segmentation with UNet-style network
Joints morphology, as detected by residual magnetometry, is highly heterogenous. Some pipeline joints can be found with a simple peak search. Others show up as subtle increases in oscillation frequency. Some are so subtle that they can only be determined by considering the average joint spacing in the surrounding area. Because joint identification lacks precise articulatable attributes which might lend it to imperative programming [12,13], a NN is the most promising solution.
[Caption] Examples of the diversity of joint morphology in residual magnetometry measurements in steel pipelines. Upper and lower plots show data from different sections of the same survey with the same x and y scale. In the upper graph, the joints could be choosen with an off-the-shelf peak finding algorithm. In the lower graph, joints were manually identified by looking for small areas of increased signal frequency and interpolating between areas of high confidence to positively identify joints in areas of low confidence.
## Classification, segmentation, and my deep learning approach to this problem
The most successful deep learning strategies were developed in the context of computer vision (aka image processing). Let's define some terms. Classification is any algorithm that takes a large input and returns a class label from a predetermined set. Pixel values as input and "is this a dog or cat?" as an output is a classification problem. It is critically important that we know in advance that the photo will be a cat or dog. This applies equally well to any number of predefined categories. It need not be binary. Segmentation is the same concept but applied to each data point. This is the "highlight all the pixels of a cat in this image" type of problem.
To develop an automated joint detection system we conceptuallize this as a 1D segmentation problem. We apply a label to each data point (joint or no-joint), and design a NN to classify each point. The network takes the four magnetic vectors as inputs ($$m_x$$, $$m_y$$, $$m_z$$, and $$m_t$$ where $$m_t = (m_x^2 + m_y^2 + m_z^2)^{0.5})$$ and returns two ($$y_1$$, $$y_2$$). The joint likelihood vs position function is given by a softmax $$p = \frac{e^{y_2}}{e^{y_1} + e^{y_2}}$$. See figure below.
[Caption] (a) Inputs to NN. Because no natural normalization basis is present and because the values are on the order of 1, no normalization was applied to the input data. (b) Raw output of the NN in arbitrary units. (c) Softmax of NN output vectors from (b) gives a "certainty of joint" measurement. Peak search on this array gives a joint list (cyan).
Since the publication of the original paper on UNet in 2015 [14], fully convolutional NN have become the gold standard for segmentation problems. There are now thousands of published variations and application of UNet. The original paper has 39,554 citations as of 2022-03-23. Our application of UNet requires several modifications. First, because we have 1D data instead of 2D data, our data points are further apart and the receptive fields must be increased to accomodate this increased distance. Second, the precision of the segmentation in joint identification is less precise than in medical image segmentation. Therefore we do not need the output arrays to be as large as the input arrays. Third, because our data is pipeline magnetics and not micrographs of cells, the data augmentations appropriate for training are not the same.
### Architecture and receptive fields for 1D data
Broadening the receptive field to include at least several joints provides the NN with the context that manual joint identification utilizes. What is likely a joint in the context of one pipeline's magnetic signature may be part of the normal fluctuations within a pipe segment on another line. When manually identifying joints, we look for repeated signatures with approximately equal spacing. For the NN to mimic this process, the receptive field of the deepest neurons must include multiple joints. This insight is the most critical part of the adaptation of UNet to 1D segmentation.
The 572x572 square pixel images of the original UNet have 327,184 data points per image but no data point is more than 571 steps away from any other data point. With a magnetometer sampling frequency of 200 Hz and 250 second snapshots, our input data is 51,840 data points long. However, the maximum distance between data points in 51,839 rather than 571. By shifting from 2D to 1D, we can reduce our input by 85% and yet the maximum distance between points can increase by 90x. As information travels down the encoder side of the UNet, we need a broader receptive field to capture context from a wider area to make accurate classifications of each point.
The receptive field of a neuron in a NN of convolutional and pooling layers [15] is given by
$$r_0 = \sum^L_{l=1} \left( \left(k_l-1 \right) \prod^{l-1}_{i=1} s_i \right)$$
where $$k_l$$ is the kernel at layer $$l$$ and $$s_i$$ is the stride of layer $$i$$. For the original UNet, the receptive field of the first 14 layers which make up encoder is 140x140 pixels. By increasing the stride of the first of each pair of convolutions to 2 and increase the kernel and stride of the the max pooling layers to 3, we can increase the receptive field from 140 to 10,367. This increase is what allows the encoder to place the magnetometer data in appropriate context within the rest of the line.
We used transpose convolutions to reverse the compressions of the max pool and convolutional layers with stride 2. By selecting an input array length evenly divisible by the strides of all the encoder layers, we were able to design a network which required no cropping before appending encoder layers into the decoder side of the network. This produced a slight increase in training performance and adds a cleanliness to the network design.
[Caption] Architecture of fully convolutional UNet for 1D segmentation. Rectangular dimensions are not proportional to length because the length of the tensors changes by a factor of 2,600 as it moves through the network. All convolutional layers have a kernel of 3. On the encoder side of the network, the stride is 2 or 1 for the first or second in each pair, respectively. On the decoder side, the stride is 1 for all convolutional layers. All transpose convolutional layers have a kernel of 3; the stride is 2 or 3 for the first or second in each pair respectively. Max pool layers have a kernel and stride of 3. ReLU activations functions follow each convolutional or transpose convolutional layer.
### Dataset and augmentation
Despite much attention, the question of how much data a NN requires for training remains an active research area [16,17]. Most estimates are between 10 and 1000 times the number of parameters in the model. However, we have a NN with 12.6 million parameters and only ~100,000 examples of joints. Those are not even independent training samples because the NN needs multiple joints per sample to train. However, we are fortunate in several ways. First we have a large number of data augmentations available which allowed us to create 3 million training samples. Second, even with the augmentations, this NN converged and generalized with two orders of magnitude less training data than would typically be expected for a NN this size. Therefore, we expect the network to improve as more data is incorporated into the training library.
To generate an augmented data sample
- Select random survey
- Select random subset of survey 37.5k to 70k data points long
- Resample by linear interpolation to 50k data points (stretch or compress in x)
- Multiply each component by random scaler value from 0.7 to 1.33 (stretch or compress in y)
- With 50% probability, reverse the sample (mirror in x)
- Randomly reorder x, y, z components. Because measurements are taken free floating, the orientation of the sensor is random so random reordering of components is a valid augmentation strategy.
By using these augmentations, we were able to train a NN of 12.7 million parameters using only 111 surveys containing 90,898 examples of joints.
### Loss functions and ground truth
The training data y values are defined as a discrete list of points but our output is a segmentation output. To make a ground truth tensor suitable for a segmentation algorithm, we define the ground truth as "joint" within a window of 50 data points and "not joint" everywhere else. This first approximation performed near human accuracy without attempting to correct for joint width variability in our training data.
This network was trained using a standard cross-entropy loss function without applying different weights to different data points. The original UNet was interested in the edges of cells. Weighting the loss function higher at the boundaries forced the network to precisely localize predictions. In this application, we have the opposite situation. It is most important to learn the classes of the data points in the center of the joint region but not at all important to learn the exact edges of the target segmentation. If anything, the fuzzy borders of the target segmentation lends itself to a loss weighting scheme inverted from the original UNet. Implementation of such a weight scheme was not required for the results presented here. However, it is possible that analogous problems may benefit from such training weights, either to reduce the training time or to improve the training robustness.
## Results and future work
This NN trained for 100,000 cycles with a learning rate of 0.1 and minibatches of 30 samples. At the end of that time, it performed joint identification on a validation survey with 2,981 joints. It correctly identified 2,839 with 142 false negatives and 33 false positives, 4.8% and 1.1% respectively. Human level performance (as measured by agreement between independent manual labelers) has false positive and false negative rates of approximately 0.1 to 2% depending on the pipeline. Our NN is performing slightly below human level accuracy.
Because the network was trained with surprisingly little data, we will likely see significant improvements as our training data library grows. There is also a specific use case where acceleration and gyroscopic data are informative. Some surveys are performed with the sensor device attached to a cleaning pig. As the pig hits welds, the acceleration and gyroscope sensors register the impact. Depending on how tight the pig fits in the pipe and how much the weld seam protrudes, the size of this signal can vary from non-existent to completely unambiguous. A similar NN which incorporates acceleration, gyroscopic, and magnetic measurements will likely perform better on cleaning pig surveys.
## References
[1] M. C. Byington, A. val Pol, and J. val Pol, "Neural networks for pipeline joint detection," Digital Pipeline Solutions Forum, 2022.
[2] M. Xie, and Z. Tian, "A review on pipeline integrity management utilizing in-line inspection data," Engineering Failure Analysis, vol. 92, pp. 222—239, 2018.
[3] F. Varela, M. Yongjun Tan, and M. Forsyth, "An overview of major methods for inspecting and monitoring external corrosion of on-shore transportation pipelines," Corrosion Engineering, Science and Technology, vol. 50, pp. 226—235, 2015.
[4] Interstate Natural Gas Association of America, "Report to the National Transportation Safety Board on Historical and Future Development of Advanced In-Line Inspection Platforms for Use in Gas Transmission Pipelines," 2012.
[5] J. Tiratsoo, "Ultimate Guide to Unpiggable Pipelines," 2013.
[6] R. Fletcher, and M. Chandrasekaran, "SmartBall: a new approach in pipeline leak detection," International Pipeline Conference, vol. 48586, pp. 117—133, 2008.
[7] J. Smith, A. van Pol, D. Ham, and J. van Pol, "Leak detection and prevention using free-floating in-line sensors," Pipeline Pigging and Integrity Management, 2019.
[8] R. Bickerstaff, M. Vaughn, G. Stoker, M. Hassard, and M. Garrett, "Review of sensor technologies for in-line inspection of natural gas pipelines," Sandia National Laboratories, 2002.
[9] X. Wu, A. Xu, Y. Xiao, B. Zhou, G. Wang, and R. Zeng, "Research on Above Ground Marker System of pipeline Internal Inspection Instrument Based on geophone array," 2010 6th International Conference On Wireless Communications Networking and Mobile Computing, pp. 1—4, 2010.
[10] Y. Li, D. Wang, and L. Sun, "A novel algorithm for acoustic above ground marking based on function fitting," Measurement, vol. 46, pp. 2341—2347, 2013.
[11] L. Sun, Y. Li, Y. Wu, J. Guo, and Z. Li, "Establishment of theoretical model of magnetic dipole for ground marking system," 2017 29th Chinese Control And Decision Conference (CCDC), pp. 6134—6138, 2017.
[12] M. Kindree, S. Campbell, A. van Pol, and J. van Pol, "Defect localization using free-floating unconventional ILI tools without AGMs," Pipeline Pigging and Integrity Management, 2022.
[13] Z. Shand, A. van Pol, and J. van Pol, "Pipers; an inline screening tool for unpiggable pipelines," Unpiggable Pipeline Solutions Forum, 2019.
[14] O. Ronneberger, P. Fischer, and T. Brox, "U-Net: Convolutional Networks for Biomedical Image Segmentation," Medical Image Computing and Computer-Assisted Intervention (MICCAI), vol. 9351, pp. 234—241, 2015.
[15] A. Araujo, W. Norris, and J. Sim, "Computing receptive fields of convolutional neural networks," Distill, vol. 4, pp. e21, 2019.
[16] A. Alwosheel, S. van Cranenburgh, and C. G. Chorus, "Is your dataset big enough? Sample size requirements when using artificial neural networks for discrete choice analysis," Journal of Choice Modelling, vol. 28, pp. 167—182, 2018.
[17] T. Oyedare, and J. J. Park, "Estimating the required training dataset size for transmitter classification using deep learning," 2019 IEEE International Symposium On Dynamic Spectrum Access Networks, pp. 1—10, 2019. | 2023-03-27 23:23:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45964935421943665, "perplexity": 1740.4304850490648}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948708.2/warc/CC-MAIN-20230327220742-20230328010742-00599.warc.gz"} |
http://blog.stata.com/2016/10/ | ### Archive
Archive for October 2016
## Long-run restrictions in a structural vector autoregression
$$\def\bfA{{\bf A}} \def\bfB{{\bf }} \def\bfC{{\bf C}}$$Introduction
In this blog post, I describe Stata’s capabilities for estimating and analyzing vector autoregression (VAR) models with long-run restrictions by replicating some of the results of Blanchard and Quah (1989). Read more…
## Programming an estimation command in Stata: Writing an estat postestimation command
estat commands display statistics after estimation. Many of these statistics are diagnostics or tests used to evaluate model specification. Some statistics are available after all estimation commands; others are command specific.
I illustrate how estat commands work and then show how to write a command-specific estat command for the mypoisson command that I have been developing.
This is the 28th post in the series Programming an estimation command in Stata. I recommend that you start at the beginning. See Programming an estimation command in Stata: A map to posted entries for a map to all the posts in this series. Read more…
Categories: Programming Tags:
## The new Stata News
Have you seen the latest issue of the Stata News? It has a new format that we think you will love. And, I want to make sure that you are not missing out on the articles discussing what our developers and users are doing with Stata.
We have a new section, User’s corner, that highlights interesting, useful, and fun user contributions. In this issue, you will see how Belén Chavez uses Stata to analyze her Fitbit® data.
We kept your favorite sections, including Spotlight articles written by Stata developers. In this issue, Enrique Pinzón demonstrates Estimating, graphing, and interpreting interactions using margins.
If you haven’t been reading the Stata News, you may want to browse previous Spotlight articles on topics such as endogenous treatment effects, item response theory (IRT), and Bayesian analysis. You can find all the previous Spotlight articles here.
Categories: Stata Products Tags:
## Solving missing data problems using inverse-probability-weighted estimators
We discuss estimating population-averaged parameters when some of the data are missing. In particular, we show how to use gmm to estimate population-averaged parameters for a probit model when the process that causes some of the data to be missing is a function of observable covariates and a random process that is independent of the outcome. This type of missing data is known as missing at random, selection on observables, and exogenous sample selection.
This is a follow-up to an earlier post where we estimated the parameters of a probit model under endogenous sample selection (http://blog.stata.com/2015/11/05/using-mlexp-to-estimate-endogenous-treatment-effects-in-a-probit-model/). In endogenous sample selection, the random process that affects which observations are missing is correlated with an unobservable random process that affects the outcome. Read more…
Categories: Statistics Tags:
## Estimating covariate effects after gmm
In Stata 14.2, we added the ability to use margins to estimate covariate effects after gmm. In this post, I illustrate how to use margins and marginsplot after gmm to estimate covariate effects for a probit model.
Margins are statistics calculated from predictions of a previously fit model at fixed values of some covariates and averaging or otherwise integrating over the remaining covariates. They can be used to estimate population average parameters like the marginal mean, average treatment effect, or the average effect of a covariate on the conditional mean. I will demonstrate how using margins is useful after estimating a model with the generalized method of moments. Read more…
Categories: Statistics Tags: | 2017-08-18 05:08:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4605467915534973, "perplexity": 2662.1718847209168}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886104565.76/warc/CC-MAIN-20170818043915-20170818063915-00414.warc.gz"} |
https://electronics.stackexchange.com/questions/165407/what-current-values-are-prone-to-noise | # What current values are prone to noise?
I'm using voltage dividers to drop down from high voltages like 327V peak to 2.5V peak and read it with an ADC. (Also reading lower voltages like 5V from various sources, so the above was just an example)
In that particular case I'm using 1.3Mohms with 10Kohms resistors, giving a current of 0.25mA, with 80mW power.
Should I use 130Kohms and 1Kohms? Power will get a bit high. Also, if I'm drawing 2.5mA with my measuring it might affect the circuit I'm reading.
The question is: At what current values does noise become a thing to look out for?
• It's not just down to small currents being easier to disturb; resistor noise is also a consideration, and that scales with resistance: en.wikipedia.org/wiki/Johnson%E2%80%93Nyquist_noise – Nick Johnson Apr 18 '15 at 9:22
• what noise are you referring to? the higher the current, the better in this case. I think you can find insulated inamps that can step down your voltage as required, that would be a much better choice. – Vladimir Cravero Apr 18 '15 at 9:48
• Any kind of noise, but I'm assuming induced noise (inductive coupling?) would be highest. From various electronics like monitors nearby or florescent lighting. – Dragos Puri Apr 18 '15 at 9:56
Your 1.3M + 10K divider has a source resistance (looking back from the ADC input) of 1.3M||10K ~= 10K. The Johnson noise from ideal resistors is likely negligible for your purposes-- 13nV/$\sqrt{Hz}$- so if your bandwidth is 1000Hz your RMS noise is 411nV or perhaps 2-3uVp-p. So if your full scale is 2.5V it would be good for about 20 bits, which is about the maximum that is probably achievable anyway.
There are additional source of noise- noise current and noise voltage of the ADC, and leakage current. That depends a lot on the ADC- if unbuffered 10K is probably a good maximum number, but if you buffer it you could likely (depending on required accuracy) go considerably higher.
So, for 'noise' I would look at the ADC requirements in relation to your accuracy spec, and also look carefully at the stability of (especially) the 1.3M resistor. Permanent drift of the resistance value with time under application of high DC voltage/moisture is not uncommon. You can get an idea of the stability with the resistor "load life" specification, but check the test conditions.
Also, when you have high voltages on a PCB try to maximize creepage distances (for example by milled slots) and/or use guard traces to conduct leakage current away from sensitive nodes. A leakage resistance of 1G$\Omega$ in parallel with the 1.3M would cause 0.1% error. | 2021-01-15 18:43:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5820110440254211, "perplexity": 2640.724624870414}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703495936.3/warc/CC-MAIN-20210115164417-20210115194417-00066.warc.gz"} |
http://mathhelpforum.com/calculus/130241-limit-sequences-complex-plane-print.html | limit of sequences in complex plane
• February 22nd 2010, 08:09 PM
jzellt
limit of sequences in complex plane
Zn = ((1+i)/squareRoot(3))^n
Zn = n(1/i)^n
• February 22nd 2010, 08:14 PM
Prove It
Quote:
Originally Posted by jzellt
Zn = ((1+i)/squareRoot(3))^n
Zn = n(1/i)^n
What are you trying to do? I assume from the title, that you are taking limits, but the limit as n approaches what? And are we finding limits of the sequences or are we summing them?
• February 22nd 2010, 08:28 PM
jzellt
Sorry...
Find the limit as n approaches infinity of each sequence that converges. If the sequence diverges, explain why.
• February 22nd 2010, 10:08 PM
Prove It
Quote:
Originally Posted by jzellt
Zn = ((1+i)/squareRoot(3))^n
Zn = n(1/i)^n
$\lim_{n \to \infty}\left(\frac{1 + i}{\sqrt{3}}\right)^n = \lim_{n \to \infty}\left[\frac{\sqrt{2}}{\sqrt{3}}\left(\cos{\frac{\pi}{4}} + i\sin{\frac{\pi}{4}}\right)\right]^n$
$= \lim_{n \to \infty}\left[\left(\frac{\sqrt{6}}{3}\right)^n\left(\cos{\frac{ n\pi}{4}} + i\sin{\frac{n\pi}{4}}\right)\right]$
$= \lim_{n \to \infty}\frac{(\sqrt{6})^n}{3^n} \cdot \lim_{n \to \infty}\left(\cos{\frac{n\pi}{4}} + i\sin{\frac{n\pi}{4}}\right)$
Can you go from here?
• February 22nd 2010, 10:30 PM
Prove It
Quote:
Originally Posted by jzellt
Zn = ((1+i)/squareRoot(3))^n
Zn = n(1/i)^n
$z_n = n\left(\frac{1}{i}\right)^n$
$= n\left(\frac{i}{i^2}\right)^n$
$= n\left(-i\right)^n$
$= n\left[\cos{\left(-\frac{\pi}{2}\right)} + i\sin{\left(-\frac{\pi}{2}\right)}\right]^n$
$= n\left[\cos{\left(-\frac{n\pi}{2}\right)} + i\sin{\left(-\frac{n\pi}{2}\right)}\right]$
So $\lim_{n \to \infty}z_n = \lim_{n \to \infty}n\left[\cos{\left(-\frac{n\pi}{2}\right)} + i\sin{\left(-\frac{n\pi}{2}\right)}\right]$
$= \lim_{n \to \infty}n \cdot \lim_{n \to \infty}\left[\cos{\left(-\frac{n\pi}{2}\right)} + i\sin{\left(-\frac{n\pi}{2}\right)}\right]$.
Can you go from here?
• February 22nd 2010, 10:38 PM
CaptainBlack
Quote:
Originally Posted by Prove It
$\lim_{n \to \infty}\left(\frac{1 + i}{\sqrt{3}}\right)^n = \lim_{n \to \infty}\left[\frac{1}{\sqrt{3}}\left(\cos{\frac{\pi}{4}} + i\sin{\frac{\pi}{4}}\right)\right]^n$
$= \lim_{n \to \infty}\left[\left(\frac{1}{\sqrt{3}}\right)^n\left(\cos{\frac{ n\pi}{4}} + i\sin{\frac{n\pi}{4}}\right)\right]$
$= \lim_{n \to \infty}\frac{1}{(\sqrt{3})^n} \cdot \lim_{n \to \infty}\left(\cos{\frac{n\pi}{4}} + i\sin{\frac{n\pi}{4}}\right)$
Can you go from here?
$1+i=\sqrt{2}\; \left[\cos(\pi/4)+i \sin(\pi/4)\right]$
CB
• February 22nd 2010, 10:47 PM
Prove It
Quote:
Originally Posted by CaptainBlack
$1+i=\sqrt{2}\; \left[\cos(\pi/4)+i \sin(\pi/4)\right]$
CB
Yes it is, thanks.
Will edit. | 2016-05-30 05:21:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 15, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9776725769042969, "perplexity": 11174.213502011598}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-22/segments/1464049288709.66/warc/CC-MAIN-20160524002128-00086-ip-10-185-217-139.ec2.internal.warc.gz"} |
http://mathcentral.uregina.ca/QQ/database/QQ.09.12/h/don4.html | SEARCH HOME
Math Central Quandaries & Queries
Question from Don: 2 foursomes and 1 threesome for 6 rounds of golf
Don,
It is hard to find the "best" schedule for 11 players over 6 rounds. I have a program to do that and it is running, but it could take a long time as there are about $2000^5$ possibilities to check. As a first step, the program finds a schedule that is "close to" balanced. The best one it found is copied below.
Day 1 : (0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 2)
Day 2 : (0, 0, 1, 2, 0, 1, 1, 2, 0, 1, 2)
Day 3 : (0, 1, 0, 2, 0, 1, 2, 1, 2, 1, 0)
Day 4 : (0, 1, 1, 2, 2, 0, 1, 1, 0, 2, 0)
Day 5 : (0, 1, 2, 0, 1, 1, 0, 2, 2, 0, 1)
Day 6 : (0, 1, 2, 2, 0, 2, 1, 0, 1, 0, 1)
The way to read it is that the players are the 11 positions in the sequence and the numbers indicate the groups. One Day 1 the groups are 1, 2, 3, 4; 5, 6, 7, 8; 9, 10, 11. On Day 2 group zero is 1, 2, 5, 8, and so on.
There are a couple of flaws. For one, player 1 is always in group zero. For another, some players are together 4 times. In a perfect world (which probably does not exist), no pair of players would be together more than two times. On the other hand, in a perfect world some pairs of players would play together only once, and I think that that this schedule has every pair of players together once (though I did not check carefully).
If an improvement is found, I will send a follow-up answer. A feasible alternative is to try generating a few million schedules at random and then sending the "best" one of those.
--Victoria
Don Responded
Victoria apologizing in advance but I don"t understand that (0,0,0,0,1,1,1,12,2,2) terminology it is okay for the fist round but I cannot follow after that. Could you just # players 1 to 11 for the 6 rounds.
Hi Don,
Rather than number the players from 1 to 11 I named them with letters, A, B, C, ... up to K. I then redid Victoria's table with the players names at the head of each column.
Each day there are three groups, group zero which is a foursome, group one which is a foursome and group two which is a threesome. To see which group a player is with on any day look down the column. For example player C is in group zero on day 1, group one on day 2, group zero on day 3, group one on day 4, and the threesomes on days 5 and 6.
A B C D E F G H I J K
Day 1 0 0 0 0 1 1 1 1 2 2 2
Day 2 0 0 1 2 0 1 1 2 0 1 2
Day 3 0 1 0 2 0 1 2 1 2 1 0
Day 4 0 1 1 2 2 0 1 1 0 2 0
Day 5 0 1 2 0 1 1 0 2 2 0 1
Day 6 0 1 2 2 0 2 1 0 1 0 1
Does this help?
Harley
Don wrote again
Question from Don:
Victoria answered my question about 2 foursomes and 1 threesome playing 6 rounds of golf. I was wondering if she could tweak it because player 4 plays in a threesome 4 times and player 9 3 times in a threesome whereas players 1and 2 never play in a threesome. I know I have been a bother but would appreciate any help you could give me.
Don,
It isn't so easy to make a few small changes and not spoil the balance completely. The way my program currently works is based on the false assumption that the groups are the same size. Since, in that case, the groups could be renumbered, it is ok to always put player 1 in group number 1, and of player 2 is not also in group 1, then s/he can go in group 2.
I've made a try at changing the schedule. But really I am just eyeballing it. You can probably do at least as well as me.
Day 1 : (2, 2, 0, 0, 1, 1, 1, 1, 0, 2, 2)
Day 2 : (0, 0, 1, 2, 0, 1, 1, 2, 0, 1, 2)
Day 3 : (0, 1, 0, 2, 0, 1, 2, 1, 2, 1, 0)
Day 4 : (2, 0, 1, 1, 2, 0, 1, 1, 0, 2, 0)
Day 5 : (0, 1, 2, 0, 1, 1, 0, 2, 2, 0, 1)
Day 6 : (0, 2, 2, 1, 0, 2, 1, 0, 1, 0, 1)
--Victoria
Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences. | 2017-11-23 07:24:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3336476683616638, "perplexity": 247.84111748544552}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806760.43/warc/CC-MAIN-20171123070158-20171123090158-00317.warc.gz"} |
http://solarenergyengineering.asmedigitalcollection.asme.org/article.aspx?articleid=1458013 | 0
Research Papers
Design of a New $45 kWe$ High-Flux Solar Simulator for High-Temperature Solar Thermal and Thermochemical Research
[+] Author and Article Information
K. R. Krueger, J. H. Davidson
Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455lipinski@umn.edu
W. Lipiński1
Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455lipinski@umn.edu
In the following text, a circular disk target of diameter $dtarget$ in the focal plane and coaxial with the simulator axis will be referred to as the target.
We use the standard deviation $σf$ to quantitatively estimate the degree of flux nonuniformity; however, this quantity should not be interpreted as the standard deviation of a Gaussian distribution due to anticipated differences between the actual flux distribution and a Gaussian distribution.
1
Corresponding author.
J. Sol. Energy Eng 133(1), 011013 (Feb 03, 2011) (8 pages) doi:10.1115/1.4003298 History: Received August 02, 2010; Revised December 13, 2010; Published February 03, 2011; Online February 03, 2011
Abstract
In this paper, we present a systematic procedure to design a solar simulator for high-temperature concentrated solar thermal and thermochemical research. The $45 kWe$ simulator consists of seven identical radiation units of common focus, each comprised of a $6.5 kWe$ xenon arc lamp close-coupled to a precision reflector in the shape of a truncated ellipsoid. The size and shape of each reflector is optimized by a Monte Carlo ray tracing analysis to achieve multiple design objectives, including high transfer efficiency of radiation from the lamps to the common focal plane and desired flux distribution. Based on the numerical results, the final optimized design will deliver 7.5 kW over a 6 cm diameter circular disk located in the focal plane, with a peak flux approaching $3.7 MW/m2$.
<>
Figures
Figure 1
Schematic of the array of lamp-reflector modules: (a) front view showing the seven lamp/reflector assemblies and (b) center cross section depicting the relationship to a prototype receiver/reactor
Figure 2
Spectral distributions of the xenon arc lamp chosen for this application (solid line) (20) and air mass 1.5 (dashed line) (9)
Figure 3
Schematic of the inner surface of a single reflector (bold outline) with respect to a full ellipse (thin outline)
Figure 4
Analytical relationships among ϕrim, ψ2, l3, and d with l1=1.45 m. The shaded areas indicate values that satisfy the design requirements.
Figure 5
Analytical relationships among ϕrim, ψ2, l3, and l1 with d=0.75 m. The shaded areas indicate values that satisfy the design requirements.
Figure 6
The specular error is defined as the standard deviation σs of a distribution of the cone angle θs measured from the normal of a perfectly smooth reflector surface to the normal of a real reflector surface
Figure 7
Transfer efficiency and flux standard deviation on (a) 6 cm and (b) 10 cm diameter targets. The solid lines correspond to transfer efficiency η and the dashed lines correspond to flux standard deviation σf.
Figure 8
Flux maps for the geometry listed in Tables 12 with eccentricity e=0.890 and varying values of specular error: (a) σs=0 mrad, (b) σs=5 mrad, and (c) σs=10 mrad, and with specular error σs=5 mrad and varying values of eccentricity: (d) e=0.850, (e) e=0.900, and (f) e=0.925. The inner circle is 6 cm in diameter and the outer circle is 10 cm in diameter. Note that the scale differs for each plot.
Figure 9
Variation of transfer efficiency η with reflector specular error σs and target radius rtarget for the geometry listed in Tables 23
Figure 10
Variation of transfer efficiency η (solid line) and flux standard deviation σf (dashed line) with target radius for the final geometry (Tables 23) and σs=5 mrad and 108 rays per lamp
Figure 11
Cumulative average flux (solid line) and power (dashed line) as a function of the target radius for the geometry described in Tables 23 with σs=5 mrad
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Topic Collections | 2017-05-29 09:33:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 6, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.42732498049736023, "perplexity": 2612.09786583295}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-22/segments/1495463612069.19/warc/CC-MAIN-20170529091944-20170529111944-00493.warc.gz"} |
https://zbmath.org/?q=an%3A1013.41009 | # zbMATH — the first resource for mathematics
Near optimality of the sinc approximation. (English) Zbl 1013.41009
Near optimality of the sinc approximation is established in a variety of spaces of functions analytic in a strip around the real axis, each space being characterized by the decay rate of their elements (functions) in the neighborhood of infinity. For functions with singularities at finite real points variable tansformations are mentioned, for example the double exponential transformation $$x=\tanh(\pi/2(\sinh(y))$$.
##### MSC:
41A30 Approximation by other special function classes 41A25 Rate of convergence, degree of approximation 65D15 Algorithms for approximation of functions
Full Text:
##### References:
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https://js-book.softuni.org/chapter-08-exam-preparation.html | # Chapter 8.1. Practical Exam Preparation - Part I
In the present chapter, we will examine a few problems with a level of difficultly that can be expected in the problems of the practical exam in “Programming Basics”. We will review and practice all the knowledge that was gained from this book and through the "Programming Basics" course.
## Video
Watch a video tutorial on this chapter here: https://www.youtube.com/watch?v=bnxf9oiDduo.
## The "Programming Basics" Practical Exam
The course "Programing Basics" finishes with a practical exam. There are 6 problems included, and you will have 4 hours to solve them. Each of the exam problems will cover one of the studied topics during the course. Problem topics are as follows:
• Problem with simple calculations (without conditions)
• Problem with simple condition
• Problem with more complex conditions
• Problem with a simple loop
• Problem with nested loops (drawing a figure on the console)
• Problem with nested loops and more complex logic
## The Online Evaluation System (Judge)
All exams and homework are automatically tested through the online Judge system: https://judge.softuni.org. For each of the problems, there are visible (zero points) tests that help you understand what is expected of the problem and fix your mistakes, as well as competition tests that are hidden and check if your solution is working properly. In the Judge system, you can log in with your softuni.bg account.
How does the testing in the Judge system work? You upload the source code and from the menu below you choose to compile as a JavaScript program. The program is being tested with a series of tests, giving points for each successful test.
## Problems with Simple Calculations
The first problem of the "Programming Basics" Practical Exam covers simple calculations without checks and loops. Here are a few examples:
### Problem: 2D Triangle Area
Triangle in the plain is defined by the coordinates of its three vertices. First, the vertex (x1, y1) is set. Then the other two vertices are set: (x2, y2) and (x3, y3) which lie on a common horizontal line i.e. they have the same Y coordinates). Write a program that calculates the triangle area by the coordinates of its three vertices.
#### Input
We submit 6 integers as parameters of the function: x1, y1, x2, y2, x3, y3.
• All input numbers are in the range [-1000 … 1000].
• It's guaranteed that y2 = y3.
#### Output
Print on the console the triangle area.
#### Sample Input and Output
5
-2
6
1
1
1
7.5 The side of the triangle a = 6 - 1 = 5
The height of the triangle h = 1 - (-2) = 3
The area of the triangle S = a * h / 2 = 5 * 3 / 2 = 7.5
4
1
-1
-3
3
-3
8 The side of the triangle a = 3 - (-1) = 4
The height of the triangle h = 1 - (-3) = 4
The area of the triangle S = a * h / 2 = 4 * 4 / 2 = 8
#### Hints and Guidelines
It is important in such types of tasks where some coordinates are given to pay attention to the order in which they are submitted and to properly understand which of the coordinates we will use and how. In this case, the input is in order x1, y1, x2, y2, x3, y3. If we do not follow this sequence, the solution becomes wrong. First, we write the code that reads the input data:
We have to calculate the side and the height of the triangle. From the examples and the condition y2 = y3 we notice that one side is always parallel to the horizontal axis. It means that its length is equal to the length of the segment between its coordinates x2 and x3, which is equal to the difference between the larger and the smaller coordinates. Similarly, we can calculate the height. It will always be equal to the difference between y1 and y2(or y3, as they are equal). Since we do not know if x2 is greater than x3, or y1 will be below or above the triangle side, we will use the absolute values of the difference to always get positive numbers because one segment cannot have a negative length.
We will calculate the triangle area by using our familiar formula for finding an area of a triangle.
The only thing left is to print the area on the console.
### Problem: Moving Bricks
Construction workers have to transfer a total of x bricks. The workers are w and work simultaneously. They transport the bricks in trolleys, each with a capacity of m bricks. Write a program that reads the integers x, w, and m, and calculates what is the minimum number of courses the workers need to do to transport the bricks.
#### Input
As parameters of the function we give 3 integers:
• The number of bricks x
• The number of workers w
• The capacity of the trolley m
All input numbers are integers in the range [1 … 1000].
#### Output
Print on the console the minimum number of courses needed to transport the bricks.
#### Sample Input and Output
120
2
30
2 We have 2 workers, each transporting 30 bricks per course. In total, workers are transporting 60 bricks per course. To transport 120 bricks, exactly 2 courses are needed.
355
3
10
12 We have 3 workers, each transporting 10 bricks per course. In total, workers are transporting 30 bricks per course. To transport 355 bricks, exactly 12 courses are needed: 11 complete courses carry 330 bricks and the last 12th course carries the last 25 bricks.
5
12
30
1 We have 5 workers, each transporting 30 bricks per course. In total, workers are transporting 150 bricks per course. To transport 5 bricks, only 1 course is enough (although incomplete, with only 5 bricks).
#### Hints and Guidelines
The input is standard, and we only need to be careful about the sequence in which we read the data.
We calculate how many bricks the workers transport in a single course.
By dividing the total number of bricks transported for 1 course, we will obtain the number of courses needed to carry them. We will use the method Math.ceil(…) to round the result always up. When the bricks can be transferred with an exact number of courses, the division will return a whole number and there will be nothing to round. Accordingly, if not, the result of the division will be the number of exact courses but with a decimal fraction. The decimal part will be rounded up and we will get the required 1 course for the remaining bricks.
In the end, we print the result on the console.
## Problems with Simple Condition
The second problem of the "Programming Basics" Practical Exam covers conditional statements and simple calculations. Here are a few examples:
### Problem: Point on a Segment
A horizontal segment is placed on a horizontal line, set with the x coordinates of both ends: first and second. A point is located on the same horizontal line and is set with its x coordinate. Write a program that checks whether the point is inside or outside of the segment and calculates the distance to the nearest end of the segment.
#### Input
As parameters of the function we give 3 integers:
• The first number – the one end of the segment.
• The second number – the other end of the segment.
• The point number – the location of the point.
All inputs are integers in the range [-1000 … 1000].
#### Output
Print the result on the console:
• On the first line, print "in" or "out" – whether the point is inside or outside the segment.
• On the second line, print the distance from the point to the nearest end of the segment.
#### Sample Input and Output
Input Output Visualization
10
5
7
in
2
Input Output Visualization
8
10
5
out
3
Input Output Visualization
1
-2
3
out
2
#### Hints and Guidelines
We read the input from the console.
Since we do not know which point is on the left and which is on the right, we will create two variables to mark this. Since the left point is always the one with the smaller x coordinate, we will use Math.min(…) to find it. Accordingly, the right point is always the one with a larger x coordinate and we will use Math.max(…). We will also find the distance from the point x to the two points. Because we do not know their position relative to each other, we will use Math.abs(…) to get a positive result.
The shorter of the two distances we will find by using Math.min(…).
What remains is to find whether the point is on or out of the line. The point will be on the lines always when it matches one of the other two points or its x coordinate lies between them. Otherwise, the point is outside the line. After checking, we display one of the two messages, depending on which condition is satisfied.
Finally, we print the distance which we fine before.
### Problem: Point on a Figure
Write a program that checks if a point (with coordinates x and y) is inside or outside of the given figure:
#### Input
As parameters of the function, we give two integers: x and y.
All inputs are integers in the range [-1000 … 1000].
#### Output
Print on the console "in" or "out" – whether the point is inside or outside the figure (the outline is inside).
#### Sample Input and Output
Input Output Input Output
8
-5
in 6
-3
in
Input Output Input Output
11
-5
out 11
2
out
#### Hints and Guidelines
To find out if the point is inside of the figure, we will divide the figure into 2 rectangles:
A sufficient condition is a point to be located in one of them, to be in the figure.
We read the input from the console:
We will initialize two variables that will mark whether the point is in one of the rectangles.
When printing the message, we will check whether any of the variables has accepted a value of true. It's enough only one of them to be true so that the point is in the figure.
## Problems with Complex Conditions
The third problem of the "Programming Basics" Practical Exam includes several nested checks combined with simple calculations. Here are a few examples:
### Problem: Date after 5 days
There are two numbers d (day) and m (month) that form a date. Write a program that prints the date that will be after 5 days. For example, 5 days after 28.03 is the date 2.04. We assume that the months: April, June, September, and November have 30 days, February has 28 days, and the rest have 31 days. Months to be printed with leading zero when they contain a single digit (e.g. 01, 08).
#### Input
As parameters of the function we give two integers:
• An integer d in the range [1 … 31] – day. The number of the day does not exceed the number of days in that month (e.g. 28 for February).
• An integer m in the range [1 … 12] – month. Month 1 is January, month 2 is February, …, month 12 is December. The month may contain a leading zero (e.g. April may be written as 4 or 04).
#### Output
Print a single line containing the date after 5 days in the format day.month on the console. The month must be a 2-digit number with a leading zero, if necessary. The day must be written without leading zero.
#### Sample Input and Output
Input Output Input Output
28
03
2.04 27
12
1.01
Input Output Input Output
25
1
30.01 26
02
3.03
#### Hints and Guidelines
We take the input from the console.
To make our checks easier, we'll create a variable that will contain the number of days that we have in the month we set.
We are increasing the day with 5.
We check if the day has not exceeded the number of days in the month. If so, we must deduct the days of the month from the obtained day to calculate which day of the next month our day corresponds to.
After we have passed to the next month, this should be noted by increasing the initial one by 1. We need to check if it has not become greater than 12 and if it has, to adjust it. Because we cannot skip more than one month when we increase by 5 days, the following check is enough.
The only thing that remains is to print the result on the console. It is important to format the output correctly to display the leading zero in the first 9 months. This is done by creating a new variable for the month, to which to add 0 if necessary. Finally, we print the day and the new variable for the month.
### Problem: Sums of Three Numbers
There are 3 integers given. Write a program that checks if the sum of two of the numbers is equal to the third one. For example, if the numbers are 3, 5, and 2, the sum of two of the numbers is equal to the third one: 2 + 3 = 5.
#### Input
As parameters of the function, we give three integers. The numbers are in the range [1 … 1000].
#### Output
• Print a text line on the console containing the solution of the problem in the format "a + b = c", where a, b and c are among the three input numbers and a ≤ b.
• If the problem has no solution, print “No” on the console.
#### Sample Input and Output
Input Output Input Output
3
5
2
2 + 3 = 5 2
2
4
2 + 2 = 4
Input Output Input Output
1
1
5
No 2
6
3
No
#### Hints and Guidelines
We take the input from the console.
We have to check if the sum of a pair of numbers is equal to the third number. We have three possible cases:
• a + b = c
• a + c = b
• b + c = a
We will write a template, which will later be complemented by the required code. If none of the above three conditions is met, we will make our program print "No".
We now have to understand the order in which the two addends will be written in the output of the program. For this purpose, we will create a nested condition that checks which one of the two numbers is the larger one. In the first case, it will look like this:
Similarly, we will supplement the other two cases. The full code of the program will look like this:
## Problems with Simple Loops
The fourth problem of the "Programming Basics" Practical Exam includes a simple loop with simple logic in it. Here are a few examples:
### Problem: Sums with Step of 3
There are given n integers a1, a2, …, an. Calculate the sums:
• sum1 = a1 + a4 + a7 + … (the numbers are summed, starting from the first one with step of 3).
• sum2 = a2 + a5 + a8 + … (the numbers are summed, starting from the second one with step of 3).
• sum3 = a3 + a6 + a9 + … (the numbers are summed, starting from the third one with step of 3).
#### Input
As an input of our function, we give the array with size n+1 (0 ≤ n ≤ 1000). The array will contain the number of the numbers n and n integers in the range [-1000 … 1000]: a1, a2, …, an.
#### Output
On the console, we should print 3 lines containing the 3 sums in a format such as in the example.
#### Sample Input and Output
Input Output Input Output Input Output
2
3
5
sum1 = 3
sum2 = 5
sum3 = 0
4
7
-2
6
12
sum1 = 19
sum2 = -2
sum3 = 6
5
3
5
2
7
8
sum1 = 10
sum2 = 13
sum3 = 2
#### Hints and Guidelines
We will take the count of numbers (the size of the input array) and will declare starting values of the three sums.
Since we do not know in advance how many numbers we will process, we will take them one at a time in a loop which will be repeated n times and we will process them in the body of the loop.
To find out which of the three sums we need to add the number, we will divide its sequence number into three and we will use the remainder. We'll use the variable i which tracks the number of runs of the loop, to find out which sequence number we are at. When the remainder of i/3 is zero, it means we will add this number to the first sum, when it's 1 to the second, and when it's 2 to the third.
Finally, we will print the result on the console in the required format.
### Problem: Sequence of Increasing Elements
A series of n numbers is given: a1, a2, , an. Calculate the length of the longest increasing sequence of consecutive elements in the series of numbers.
#### Input
We give an array with size n+1 (0 ≤ n ≤ 1000) as a parameter of the function. The array will contain the count of the numbers n and n integers in the range [-1000 … 1000]: a1, a2, , an.
#### Output
Print on the console one number – the length of the longest increasing sequence.
#### Sample Input and Output
Input Output Input Output Input Output Input Output
3
5
2
4
2 4
2
8
7
6
2 4
1
2
4
4
3 4
5
6
7
8
4
#### Hints and Guidelines
To solve this problem, we need to think in a bit more algorithmic way. A sequence of numbers is given to us, and we need to check whether each subsequent one will be larger than the previous one and if so, we count how long is the sequence in which this condition is fulfilled. Then we have to find which sequence of these is the longest one. To do this, let's create some variables that we will use during solving the problem.
The variable n is the count of numbers we get from the console. In countCurrentLongest we'll keep the number of elements in the increasing sequence we are currently counting. For example, in the sequence: 5, 6, 1, 2, 3 countCurrentLongest will be 2 when we reach the second element of the counting (5, 6, 1, 2, 3) and will become 3 when we reach the last element (5, 6, 1, 2, 3) because the increasing row 1, 2, 3 has 3 elements. We will use countLongest to keep the longest increasing sequence. The other variables are num - the number we are currently in and numPrev - the previous number which we will compare with num to see if the row is growing.
We begin to run the numbers and check if the present number a s larger than the previous numPrev. If this is true, then the row is growing, and we need to increase its number by 1. This is stored in the variable that tracks the length of the sequence we are currently in – countCurrentLongest. If the number num isn't bigger than the previous one, it means that a new sequence starts, and we have to start the count from 1. Finally, after all the checks are done, numPrev becomes the number that we're currently using, and we start the loop from the beginning with the next entered num.
Here is a sample implementation of the algorithm described:
What remains is to see which of all sequences is the longest. We will do this by checking in the loop if the sequence we are currently in has become longer than the longest one by now. The whole loop will look like this:
Finally, we print the length of the longest sequence found.
## Problems for Drawing Figures on the Console
The fifth problem of the "Programming Basics" Practical Exam requires using one or several nested loops for drawing a figure on the console. Logical reasoning, simple calculations, and conditional statements may be required. The problem tests the ability of students to think logically and invent simple algorithms for solving problems, i.e. to think algorithmically. Here are some examples of exam tasks:
### Problem: A Perfect Diamond
Write a function that takes as a parameter n and draws a perfect diamond with size n as in the examples below.
#### Input
One parameter - an integer n in the range [1 … 1000].
#### Output
The diamond should be printed on the console as in the examples below.
#### Sample Input and Output
Input Output Input Output
2 *
*-*
*
3 *
*-*
*-*-*
*-*
*
Input Output Input Output
4 *
*-*
*-*-*
*-*-*-*
*-*-*
*-*
*
5 *
*-*
*-*-*
*-*-*-*
*-*-*-*-*
*-*-*-*
*-*-*
*-*
*
#### Hints and Guidelines
In tasks for drawing figures, the most important thing to consider is the sequence in which we will draw. Which items are repeated and with what steps? We can see that the top and the bottom parts of the diamond are the same. The easiest way to solve the problem is by creating a loop that draws the upper part, and then another loop that draws the bottom part (opposite to the top one).
We will read the number n from the parameters of the function.
We start to draw the upper half of the diamond. We see that each line starts with some empty spaces and *. If we take a closer look, we will notice that the empty spaces are always equal to n - index of row - 1 (the first row is n-1, the second – n-2, etc.). We will start by drawing the number of empty spaces and the first star. Notice that we start to count from 0, no from 1. After that, we'll only add a few times -* to finish the line.
Here is the fragment from the code for the upper part of the diamond:
What remains is to complete each line with the required number of -* elements. On each row we have to add i such items (on the first 1-1 -> 0, the second -> 1, etc.)
Here is the complete code for drawing the upper part of the diamond:
To draw the bottom part of the diamond, we have to reverse the upper part. We'll count from n - 2 because if we start from n - 1, we will draw the middle row twice. Do not forget to change the step from ++ to --.
Here is the code for drawing the bottom part of the diamond:
What remains is to assemble the whole program by first reading the input, printing the top part of the diamond, and then the bottom part of the diamond.
### Problem: Rectangle with Stars in the Center
Write a function that takes as a parameter an integer n and draws a rectangle with size n with 2 stars in the center as in the examples below.
#### Input
The parameter is an integer n in the range [2 … 1000].
#### Output
The rectangle should be printed on the console as in the examples below.
#### Sample Input and Output
Input Output Input Output
2 %%%%
%**%
%%%%
3 %%%%%%
% %
% ** %
% %
%%%%%%
Input Output Input Output
4 %%%%%%%%
% %
% ** %
% %
%%%%%%%%
5 %%%%%%%%%%
% %
% %
% ** %
% %
% %
%%%%%%%%%%
#### Hints and Guidelines
We read the input parameter of the function.
The first thing we can easily notice is that the first and the last rows contain 2 * n symbols %. We will start with this and then draw the middle part of the rectangle.
From the examples, we see that the middle part of the figure always has an odd number of rows. Note that when an even number is set, the number of rows is equal to the previous odd number (2 -> 1, 4 -> 3, etc.). We create a variable that represents the number of rows that our rectangle will have, and correct it if the number n is even. Then we will draw a rectangle without the asterisks. Each row has for the beginning and at the end the symbol % and between them 2 * n - 2 empty spaces (the width is 2 * n and we subtract 2 for the two percent at the end). Do not forget to move the code for the last line after the loop.
We can start and test the code so far. Everything without the two asterisks in the middle should work correctly.
Now, in the body of the loop let's add the asterisks. We'll check if we're on the middle row. If we are in the middle, we will draw the row together with the asterisks, if not – we will draw a normal row. The line with the asterisks has n-2 empty spaces (n is half the length and we remove the asterisk and the percentage), two stars, and again n-2 empty spaces. We leave out of the check the two percent at the beginning and the end of the row.
## Problems with Nested Loops
The last (sixth) problem of the "Programming Basics" Practical Exam requires using of several nested loops and more complex logic inside them. The problems examine participants' ability to think algorithmically and to solve non-trivial coding problems that require nested loops. Here are some examples of exam problems.
### Problem: Increasing 4 Numbers
For given pair of numbers a and b generate all four number n1, n2, n3, n4, for which a ≤ n1 < n2 < n3 < n4 ≤ b.
#### Input
As parameters of the function, we get two integers a and b in the range [0 … 1000].
#### Output
The output contains all numbers in batches of four, in ascending order, one per line.
#### Sample Input and Output
Input Output Input Output
3
7
3 4 5 6
3 4 5 7
3 4 6 7
3 5 6 7
4 5 6 7
15
20
15 16 17 18
15 16 17 19
15 16 17 20
15 16 18 19
15 16 18 20
15 16 19 20
15 17 18 19
15 17 18 20
15 17 19 20
15 18 19 20
16 17 18 19
16 17 18 20
16 17 19 20
16 18 19 20
17 18 19 20
Input Output Input Output
5
7
No 10
13
10 11 12 13
#### Hints and Guidelines
We read the input data from the function. We also create the additional variable count, which will keep track of existing number ranges.
We will most easily solve the problem if we logically divide it into parts. If we are required to draw all the rows from a number between a and b, we will do it using one loop that takes all the numbers from a to b. Let's think about how to do this with series of two numbers. The answer is easy – we will use nested loops.
We can test the incomplete program to see if it's accurate so far. It must print all pairs of numbers i, j for which i ≤ j.
Since each next number of the row must be greater than the previous one, the second loop will run around i + 1 (the next greater number). Accordingly, if there is no sequence of two incremental numbers (a and b are equal), the second loop will not be fulfilled, and nothing will be printed on the console.
Similarly, what remains is to implement the nested loops for four numbers. We will add an increase of the counter that we initialized to know if there is such a sequence.
Finally, we will check if the counter is equal to 0 and we will print "No" on the console accordingly, if so.
### Problem: Generating Rectangles
By a given number n and a minimum area m, generate all possible rectangles with integer coordinates in the range [-n…n] with an area of at least m. The generated rectangles must be printed in the following format:
(left, top) (right, bottom) -> area
Rectangles are defined using the top left and bottom right corners. The following inequalities are in effect:
• -n ≤ left < right ≤ n
• -n ≤ top < bottom ≤ n
#### Input
We get two integers as parameters of the function:
• An integer n in the range [1 … 100] – sets the minimum and maximum coordinates of a peak.
• An integer m in the range [0 … 50 000] – sets the minimum area of the generated rectangles
#### Output
• The described rectangles should be printed on the console in a format such as in the examples below.
• If there are no rectangles for the specified n and m, then print "No".
• The order of rectangles in the output is not important.
#### Sample Input and Output
Input Output Input Output
1
2
(-1, -1) (0, 1) -> 2
(-1, -1) (1, 0) -> 2
(-1, -1) (1, 1) -> 4
(-1, 0) (1, 1) -> 2
(0, -1) (1, 1) -> 2
2
17
No
Input Output
3
36
(-3, -3) (3, 3) -> 36
#### Hints and Guidelines
We read the input parameters of the function. We will also create a counter, which will store the number of rectangles found.
It is very important to be able to imagine the problem before we begin to solve it. In our case, it is required to search for rectangles in a coordinate system. The thing we know is that the left point will always have the coordinate x, smaller than the right one. Accordingly, the upper one will always have a smaller y coordinate than the lower one. To find all the rectangles, we'll have to create a loop similar to the previous problem, but this time, not every next loop will start from the next number because some of the coordinates can be equal (for example left and top).
The important thing here is knowing the corresponding coordinates so we can correctly calculate the sides of the rectangle. Now we have to find the area of the rectangle and check if it is greater than or equal to m. One side will be the difference between left and right and the other one – between top and bottom. Since the coordinates may be eventually interchanged, we will use absolute values. Again, we add the counter in the loop, counting only the rectangles we write. It is important to note that the writing order is left, top, right, bottom, as it is set in the problem's description.
Finally, we print “No”, if there are no such rectangles. | 2021-12-08 21:30:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40972891449928284, "perplexity": 480.78154182140855}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964363598.57/warc/CC-MAIN-20211208205849-20211208235849-00623.warc.gz"} |
https://chenzaichun.github.io/post/2010-07-14-srt2lrc_py/ | 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 #!/usr/bin/env python # -*- coding: utf-8 -*- import sys import re if len(sys.argv) != 2: print("Usage srt2lrc.py file") sys.exit(1) f = open(sys.argv[1]) lines = f.readlines() i = 0 while i < len(lines): line = lines[i] if line.find('-->') == -1: i = i + 1 continue s = line.split('-->') time = s[0].split(':') ms = '00' if len(time) >= 3: hour = time[0] minute = time[1] seconds = time[2].split(',') second = seconds[0] ms = seconds[1] else: hour = 0 minute = time[2] seconds = time[3].split(',') second = seconds[0] ms = seconds[1] if int(hour) == 0 and int(minute) == 0 and int(second) == 0 and int(ms) == 0: i = i + 1 continue i = i + 1 s = lines[i].strip() i = i + 1 while i < len(lines) and len(lines[i].strip()) > 0: s = s + lines[i].strip() i = i + 1 s = re.sub("<.*>", "", s) if len(s) > 0: print("[%.2d:%s.%.2s]%s" % (int(hour)*60+int(minute.strip()), second.strip(), ms.strip(), s)) f.close() | 2021-09-27 19:31:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18154242634773254, "perplexity": 1176.1576182317892}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780058467.95/warc/CC-MAIN-20210927181724-20210927211724-00568.warc.gz"} |
https://couryes.com/statistical-inference-dai-xie-mast20005-2/ | 统计代写|统计推断代写Statistical inference代考|MAST20005
2022年9月29日
couryes-lab™ 为您的留学生涯保驾护航 在代写统计推断Statistical inference方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写统计推断Statistical inference代写方面经验极为丰富，各种代写统计推断Statistical inference相关的作业也就用不着说。
• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
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统计代写|统计推断代写Statistical inference代考|Generating functions
For many distributions, all the moments $\mathbb{E}(X), \mathbb{E}\left(X^2\right), \ldots$ can be encapsulated in a single function. This function is referred to as the moment-generating function, and it exists for many commonly used distributions. It often provides the most efficient method for calculating moments. Moment-generating functions are also useful in establishing distributional results, such as the properties of sums of random variables, and in proving asymptotic results.
Definition 3.5.1 (Moment-generating function)
The moment-generating function of a random variable $X$ is a function $M_X: \mathbb{R} \longrightarrow$ $[0, \infty)$ given by
$$M_X(t)=\mathbb{E}\left(e^{t X}\right)= \begin{cases}\sum_x e^{t x} f_X(x) & \text { if } X \text { discrete } \ \int_{-\infty}^{\infty} e^{t x} f_X(x) d x & \text { if } X \text { continuous. }\end{cases}$$
where, for the function to be well defined, we require that $M_X(t)<\infty$ for all $t \in[-h, h]$ for some $h>0$.
A few things to note about moment-generating functions.
1. Problems involving moment-generating functions almost always use the definition in terms of expectation as a starting point.
2. The moment-generating function $M_X(t)=\mathbb{E}\left(e^{t X}\right)$ is a function of $t$. The $t$ is just a label, so $M_X(s)=\mathbb{E}\left(e^{s X}\right), M_X(\theta)=\mathbb{E}\left(e^{\theta X}\right), M_Y(p)=\mathbb{E}\left(e^{p Y}\right)$, and so on.
3. We need the moment-generating function to be defined in an interval around the origin. Later on we will be taking derivatives of the moment-generating function at zero, $M_X^{\prime}(0), M_X^{\prime \prime}(0)$, and so on.
The moment-generating function of $X$ is the expected value of an exponential function of $X$. Useful properties of moment-generating functions are inherited from the exponential function, $e^x$. The Taylor series expansion around zero, provides an expression for $e^x$ as a polynomial in $x$,
$$e^x=1+x+\frac{1}{2 !} x^2+\ldots+\frac{1}{r !} x^r+\ldots=\sum_{j=0}^{\infty} \frac{1}{j !} x^j$$
This expansion (and any Taylor series expansion around zero) is often referred to as the Maclaurin series expansion. All the derivatives of $e^x$ are equal to $e^x$,
$$\frac{d^r}{d x^r} e^x=e^x \text { for } r=1,2, \ldots$$
统计代写|统计推断代写Statistical inference代考|Cumulant-generating functions and cumulants
It is often convenient to work with the log of the moment-generating function. It turns out that the coefficients of the polynomial expansion of the log of the momentgenerating function have convenient interpretations in terms of moments and central moments.
Definition 3.5.7 (Cumulant-generating function and sumulants)
The cumulant-generating function of a random variable $X$ with moment-generating function $M_X(t)$, is defined as
$$K_X(t)=\log M_X(t) .$$
The $r^{\text {th }}$ cumulant, $\kappa_r$, is the coefficient of $t^r / r !$ in the expansion of the cumulantgenerating function $K_X(t)$ so
$$K_X(t)=\kappa_1 t+\kappa_2 \frac{t^2}{2 !}+\ldots+\kappa_r \frac{t^r}{r !}+\ldots=\sum_{j=1}^{\infty} \kappa_j \frac{t^j}{j !} .$$
It is clear from this definition that the relationship between cumulant-generating function and cumulants is the same as the relationship between moment-generating function and moments. Thus, to calculate cumulants we can either compare coefficients or differentiate.
1. Calculating the $r^{\text {th }}$ cumulant, $\kappa_r$, by comparing coefficients:
$$\text { if } K_X(t)=\sum_{j=0}^{\infty} b_j t^j \text { then } \kappa_r=r ! b_r \text {. }$$
2. Calculating the $r^{\text {th }}$ cumulant, $\kappa_r$, by differentiation:
$$\kappa_r=K_X^{(r)}(0)=\left.\frac{d^r}{d t^r} K_X(t)\right|_{t=0} .$$
Cumulants can be expressed in terms of moments and central moments. Particularly useful are the facts that the first cumulant is the mean and the second cumulant is the variance. In order to prove these results we will use the expansion, for $|x|<1$,
$$\log (1+x)=x-\frac{1}{2} x^2+\frac{1}{3} x^3-\ldots+\frac{(-1)^{j+1}}{j} x^j+\ldots$$
统计推断代考
有限元方法代写
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MATLAB代写
MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。 | 2023-03-27 04:55:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.896792471408844, "perplexity": 530.2296882027279}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296946637.95/warc/CC-MAIN-20230327025922-20230327055922-00246.warc.gz"} |
https://mlcourse.ai/book/topic09/assignment09_time_series.html | # Assignment #9 (demo). Time series analysis¶
mlcourse.ai – Open Machine Learning Course
Author: Mariya Mansurova, Analyst & developer in Yandex.Metrics team. Translated by Ivan Zakharov, ML enthusiast.
This material is subject to the terms and conditions of the Creative Commons CC BY-NC-SA 4.0 license. Free use is permitted for any non-commercial purpose.
Same assignment as a Kaggle Notebook + solution.
Fill cells marked with “Your code here” and submit your answers to the questions through the web form.
import os
import warnings
warnings.filterwarnings("ignore")
import numpy as np
import pandas as pd
import requests
from plotly import __version__
from plotly import graph_objs as go
print(__version__) # need 1.9.0 or greater
init_notebook_mode(connected=True)
5.4.0
def plotly_df(df, title=""):
data = []
for column in df.columns:
trace = go.Scatter(x=df.index, y=df[column], mode="lines", name=column)
data.append(trace)
layout = dict(title=title)
fig = dict(data=data, layout=layout)
## Data preparation¶
# for Jupyter-book, we copy data from GitHub, locally, to save Internet traffic,
# you can specify the data/ folder from the root of your cloned
# https://github.com/Yorko/mlcourse.ai repo, to save Internet traffic
DATA_PATH = "https://raw.githubusercontent.com/Yorko/mlcourse.ai/master/data/"
df = pd.read_csv(DATA_PATH + "wiki_machine_learning.csv", sep=" ")
df = df[df["count"] != 0]
date count lang page rank month title
81 2015-01-01 1414 en Machine_learning 8708 201501 Machine_learning
80 2015-01-02 1920 en Machine_learning 8708 201501 Machine_learning
79 2015-01-03 1338 en Machine_learning 8708 201501 Machine_learning
78 2015-01-04 1404 en Machine_learning 8708 201501 Machine_learning
77 2015-01-05 2264 en Machine_learning 8708 201501 Machine_learning
df.shape
(383, 7)
## Predicting with FB Prophet¶
We will train at first 5 months and predict the number of trips for June.
df.date = pd.to_datetime(df.date)
plotly_df(df.set_index("date")[["count"]])
from prophet import Prophet
predictions = 30
df = df[["date", "count"]]
df.columns = ["ds", "y"]
df.tail()
ds y
382 2016-01-16 1644
381 2016-01-17 1836
376 2016-01-18 2983
375 2016-01-19 3389
372 2016-01-20 3559
Question 1: What is the prediction of the number of views of the wiki page on January 20? Round to the nearest integer.
• 4947
• 3426
• 5229
• 2744
# You code here (read-only in a JupyterBook, pls run jupyter-notebook to edit)
Estimate the quality of the prediction with the last 30 points.
# You code here (read-only in a JupyterBook, pls run jupyter-notebook to edit)
Question 2: What is MAPE equal to?
• 34.5
• 42.42
• 5.39
• 65.91
# You code here (read-only in a JupyterBook, pls run jupyter-notebook to edit)
Question 3: What is MAE equal to?
• 355
• 4007
• 600
• 903
# You code here (read-only in a JupyterBook, pls run jupyter-notebook to edit)
## Predicting with ARIMA¶
%matplotlib inline
import matplotlib.pyplot as plt
import statsmodels.api as sm
from scipy import stats
plt.rcParams["figure.figsize"] = (15, 10)
Question 4: Let’s verify the stationarity of the series using the Dickey-Fuller test. Is the series stationary? What is the p-value?
• Series is stationary, p_value = 0.107
• Series is not stationary, p_value = 0.107
• Series is stationary, p_value = 0.001
• Series is not stationary, p_value = 0.001
# You code here (read-only in a JupyterBook, pls run jupyter-notebook to edit)
Next, we turn to the construction of the SARIMAX model (sm.tsa.statespace.SARIMAX).
Question 5: What parameters are the best for the model according to the AIC criterion?
• D = 1, d = 0, Q = 0, q = 2, P = 3, p = 1
• D = 2, d = 1, Q = 1, q = 2, P = 3, p = 1
• D = 1, d = 1, Q = 1, q = 2, P = 3, p = 1
• D = 0, d = 0, Q = 0, q = 2, P = 3, p = 1
# You code here (read-only in a JupyterBook, pls run jupyter-notebook to edit) | 2022-01-23 19:11:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.18946631252765656, "perplexity": 12665.24654083178}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304309.5/warc/CC-MAIN-20220123172206-20220123202206-00223.warc.gz"} |
https://hsm.stackexchange.com/questions/1926/who-named-the-fugacity-who-coined-the-variable-name-and-did-it-already-relate-t | # Who named the fugacity, who coined the variable name and did it already relate to complex analysis?
In Riemanns monumental paper, he expresses a prime counting function as an inverse Mellin transform of the log of the function he analytically continued into the complex plane
$$\Pi(x) = \frac{1}{2\pi i} \int \log \zeta(s)\ x^s \frac{\mathrm{d}s}{s}$$
and the zeros of $\zeta$ are consequently of interest (Riemann hypothesis).
Associated quantities relate closely to concepts in statistical physics. Planck's law of for the spectral radiance ($B_\nu(\nu, T) = 2 h c^{-2}\frac{\nu^3}{e^{h\nu/k_\mathrm{B}T} - 1}$) Mellin transforms to the Riemann zeta function ($\zeta(s) = \frac{1}{\Gamma(s)} \int_{0}^{\infty} \frac{x ^ {s-1}}{e ^ x - 1} \mathrm{d}x$) and physicists have their own polylog (mind the $z$) in the Fermi–Dirac integral.
Now the grand partition function $\mathcal{Z}$ relates microscopic statistical physics to thermodynamics via
$$-k_B T \ln \mathcal{Z} = \langle E \rangle - TS - \mu \langle N\rangle$$
and is often expressed as a power series in the fugacity $$z=\exp\left(\frac{\mu}{T}\right)$$ as $\mathcal{Z}(z) = \sum_{N_i} z^{N_i} Z(N_i).$
The zeros of $\mathcal{Z}$ in $z$ make the log $\ln \mathcal{Z}$ explode and this phenomenon is associated with phase transitions and people study and cook up result like the Lee-Yang-theorem. The fugacity as a parameter in $\mathbb C$, which controls the zeros, is actually on the cover of one of my favorite books. But I'd think the fugacity was defined without that context in mind.
What I wanted to know is when did "$z$" become a standard name for a //complex// variable and, more importantly, is it coincidence that the names fit?? Who names it and where there other names for the fugaciy?
## 1 Answer
Yes, it is a coincidence.
The concept of fugacity (from the Latin for "fleetness, tendency to flee") was originally introduced by Gilbert Lewis in his 1901 paper "The Law of Physico-Chemical Change" for the pressure of an ideal gas which has the same chemical potential as a real gas. Lewis notated this with ψ, though these days the letter f is used.
The concept of complex fugacity was introduced by Lee and Lang in their 1952 paper "Statistical Theory of Equations of State and Phase Transitions". The complex fugacity plane was originally notated y, though as you note these days the letter z is used. I'm not sure precisely how old the use of z for a generic complex variable is but it dates to at least the 1890s (e.g. see here).
The concept of partition function was introduced by Planck in 1921. Its use of the letter Z comes from its original German name, Zustandssumme ("sum over states"). The English name is due to Darwin and Fowler (1922). | 2019-08-23 17:26:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7259880304336548, "perplexity": 716.5417525601106}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027318952.90/warc/CC-MAIN-20190823172507-20190823194507-00206.warc.gz"} |
https://labs.tib.eu/arxiv/?author=J.Budaj | • We present a photometric detection of the first brightness dips of the unique variable star KIC 8462852 since the end of the Kepler space mission in 2013 May. Our regular photometric surveillance started in October 2015, and a sequence of dipping began in 2017 May continuing on through the end of 2017, when the star was no longer visible from Earth. We distinguish four main 1-2.5% dips, named "Elsie," "Celeste," "Skara Brae," and "Angkor", which persist on timescales from several days to weeks. Our main results so far are: (i) there are no apparent changes of the stellar spectrum or polarization during the dips; (ii) the multiband photometry of the dips shows differential reddening favoring non-grey extinction. Therefore, our data are inconsistent with dip models that invoke optically thick material, but rather they are in-line with predictions for an occulter consisting primarily of ordinary dust, where much of the material must be optically thin with a size scale <<1um, and may also be consistent with models invoking variations intrinsic to the stellar photosphere. Notably, our data do not place constraints on the color of the longer-term "secular" dimming, which may be caused by independent processes, or probe different regimes of a single process.
• ### Abundance analysis of Am binaries and search for tidally driven abundance anomalies - III. HD116657, HD138213, HD155375, HD159560, HD196544 and HD204188(1111.0978)
Nov. 3, 2011 astro-ph.SR
We continue here the systematic abundance analysis of a sample of Am binaries in order to search for possible abundance anomalies driven by tidal interaction in these binary systems. New CCD observations in two spectral regions (6400-6500, 6660-6760 AA) of HD116657, HD138213, HD155375, HD159560, HD196544 and HD204188 were obtained. Synthetic spectrum analysis was carried out and basic stellar properties, effective temperatures, gravities, projected rotational velocities, masses, ages and abundances of several elements were determined. We conclude that all six stars are Am stars. These stars were put into the context of other Am binaries with 10 < Porb < 200 days and their abundance anomalies discussed in the context of possible tidal effects. There is clear anti-correlation of the Am peculiarities with v sin i. However, there seems to be also a correlation with the eccentricity and may be with the orbital period. The dependence on the temperature, age, mass, and microturbulence was studied as well. The projected rotational velocities obtained by us were compared to those of Royer et al. (2002) and Abt & Morrell (1995).
• ### New photometric observations of the transiting extrasolar planet TrES-3b(1108.5255)
Aug. 26, 2011 astro-ph.EP
We present new transit observations of the transiting exoplanet TrES-3b obtained in the range 2009 -- 2011 at several observatories. The orbital parameters of the system were redetermined and the new linear ephemeris was calculated. We performed numerical simulations for studying the long-term stability of orbits.
• ### Transit timing variation and activity in the WASP-10 planetary system(1009.4567)
Sept. 23, 2010 astro-ph.EP
Transit timing analysis may be an effective method of discovering additional bodies in extrasolar systems which harbour transiting exoplanets. The deviations from the Keplerian motion, caused by mutual gravitational interactions between planets, are expected to generate transit timing variations of transiting exoplanets. In 2009 we collected 9 light curves of 8 transits of the exoplanet WASP-10b. Combining these data with published ones, we found that transit timing cannot be explained by a constant period but by a periodic variation. Simplified three-body models which reproduce the observed variations of timing residuals were identified by numerical simulations. We found that the configuration with an additional planet of mass of $\sim$0.1 $M_{\rm{J}}$ and orbital period of $\sim$5.23 d, located close to the outer 5:3 mean motion resonance, is the most likely scenario. If the second planet is a transiter, the estimated flux drop will be $\sim$0.3 per cent and can be observable with a ground-based telescope. Moreover, we present evidence that the spots on the stellar surface and rotation of the star affect the radial velocity curve giving rise to spurious eccentricity of the orbit of the first planet. We argue that the orbit of WASP-10b is essentially circular. Using the gyrochronology method, the host star was found to be $270 \pm 80$ Myr old. This young age can explain the large radius reported for WASP-10b. | 2021-04-12 23:41:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6016878485679626, "perplexity": 2289.095837416888}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038069267.22/warc/CC-MAIN-20210412210312-20210413000312-00066.warc.gz"} |
http://bootmath.com/solving-fx2-fsqrt2x.html | # Solving $(f(x))^2 = f(\sqrt{2}x)$
I would like to know how to solve this equation :
$$f(x)^2 = f(\sqrt{2}x)$$
We assume that $f : \mathbb R \to \mathbb R$ is $\mathcal C^{2}$.
The answer should be $f(x)=e^{-x^{2}/2}$, but I don’t know how to show this.
#### Solutions Collecting From Web of "Solving $(f(x))^2 = f(\sqrt{2}x)$"
Hints :
• assume $f(x) > 0$ for all $x$
• write $g = \log f$
• apply the equality in $g$ twice to get a term $g(2x)$
• take the second derivative of the equality in $g$ and get $g(2x) = g(x)$
• conclude that $g”$ is constant
• because $f(0)^2 =f(0)$, $f(0) \in \{0,1\}$
• if $\exists x \mid f(x)=0$, then $f(0)=0$ (because $f(x/\sqrt2) = \ldots = f(x/\sqrt2^n) = 0$ and $f$ is continuous in $0$
• as pointed here, if $\exists a \mid f(a)>0$, $f(a/\sqrt{2}^k) = f(a)^{\frac1{2^k}}$ and $f(0)=1$ by continuity of $f$ in $0$
So either:
• $f(0) = 1$, and then $f$ is strictly positive and $f(x) = e^{\lambda x^2}$,
• or $f(0)=0$ and $f = 0$.
PS: as Yves’ excellent post shows, relaxing the $\mathcal C^{\infty}$ assumption, even only in $0$, generates a wide class of additional solutions.
PPS: I’ve opened a new question to see what happens if we relax some of these conditions here: $f(\alpha x) = f(x)^{\beta}$ under different constraints
Setting $x=2^{t/2}$ and taking the logarithm twice,
$$(f(x))^2=f(\sqrt2x)$$
becomes
$$\log_2(\log_2(f(2^{t/2})))+1=\log_2(\log_2(f(2^{(t+1)/2})))$$
or
$$h(t)+1=h(t+1).$$
An obvious solution is $h(t)=t+c$, or $\log_2(\log_2(f(2^{t/2})))=t+c=2\log_2(x)+c$, $$f(x)=2^{Cx^2}.$$
More solutions are found by adding smooth periodic functions of period $1$, like
$$h(t)=t+A\sin(2\pi t)+c,$$
that yield
$$f(x)=2^{Cx^22^{A\sin(4\pi\log_2(x))}}.$$
Example with $C=-1,A=1$:
With starting point $(f(x))^2=f(\sqrt 2x)$, we can get to $(f(x))^4 = f(2x)$ and further $(f(x))^{16}=f(4x)$ and so on, such that
$$(f(x))^{2^n}=f(\sqrt{2^n}x)$$
thus showing that our function passes constants in an exponential manner. Then we take $f(x)=e^{g(x)}$ and get
$$e^{2g(x)}=e^{g(\sqrt 2x)}$$
from which we can say that $2g(x)=g(\sqrt 2x)$ or $4g(x)=g(2x)$. Taking the derivative here yields
$$4g'(x)=2g'(2x)\\ 4g”(x)=4g”(2x)$$
At the point of this second derivative, we see that $g”(x)$ must be constant or periodic with period a multiple of $\sqrt 2$. Working backwards, we must have $g(x)=ax^2+bx+c$ (for non-periodic solutions), and with $2g(x)=g(\sqrt 2x)$ we must in fact have $g(x)=ax^2$ (non-periodic solutions). At this point in the process, there must be some other qualifier in order to get a single function $f(x)$ from the family
$$f(x)=e^{ax^2}$$
Caveat: $f(x)=0$ is also a possible solution not covered by the coefficient $a$ above.
Suppose that we consider $\exists x_0:f(x)\lt 0$. Then we must have $(f(x_0/\sqrt 2))^2=f(x_0)\lt 0$, which means that $f(x_0/\sqrt 2)=0+qi$ for some real $q$ and $i=\sqrt{-1}$. But now we also get that $(f(x_0))^2=f(\sqrt 2x_0)\gt 0$ which leads us into our previous solution set where none of the values are negative, which is a contradiction; therefore our assumption that there exists $x_0$ such that $f(x_0)$ is negative must be false or else our derivation of the solution set incorrectly constrains the resulting set. This would also mean that any solution containing non-real numbers cannot contain any real numbers. Since one of the tags in this particular question says “real analysis” I will take that as a cue to say, having any negative result of $f(x)$ brings the function into complex analysis, and therefore such possibilities for $f(x)$ are beyond the scope of this question. | 2018-07-16 07:02:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8990963697433472, "perplexity": 167.56886939736492}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589222.18/warc/CC-MAIN-20180716060836-20180716080836-00188.warc.gz"} |
https://randlow.github.io/posts/finance-economics/basel-market-risk/ | # Basel Market Risk notes
A brief summary of the Basel Final Market Risk rule that was published in August 2012 and effective January 2013
## Capital requirement for market risk determination of the multiplication factor
At each quarter, compare 250 most recent business days of trading losses with corresponding daily VaR-based measure calibrated ot a one-day holding period and at a one-tail 99.0 % confidence level.
Risk-adjusted capital ratio = Total adjusted capital / Risk-based capital ratio denominator
Total adjusted capital = Equity + Near Equity Instruments
Risk-based capital ratio denominator = Adjusted Risk-Weighted Assets + Market Risk Equivalent Assets
## VaR-based capital requirement
One-tail 99.0 percent confidence level with a 10 business day holding period and a historical observation of one year. This can be modelled using the following:
1. Historical VaR. Take this historical empirical distribution of daily data over a period of one year and select the 1%/5% percentile.
2. Monte-Carlo simulation. Generate a probabilistic distribution using copulas, GARCH, etc. and take the 1%/5% percentile.
3. Variance-covariance VaR. To calculate the VaR from volatility see the following:
\begin{equation*} \text{VaR}_{0.95,d} \cong -1.65 \times \sigma_{d} \end{equation*}
\begin{equation*} \text{VaR}_{0.99,d} \cong -2.33 \times \sigma_{d} \end{equation*}
To convert a daily volatility to a monthly/annual volatility use the following:
\begin{equation*} \sigma_{x} \cong \sigma_{d} \times \sqrt{T} \end{equation*}
\begin{equation*} \sigma_{m} \cong \sigma_{d} \times \sqrt{20} \end{equation*}
\begin{equation*} \sigma_{y} \cong \sigma_{d} \times \sqrt{250} \end{equation*}
Once you have the monthly/annual volatility use the following to obtain the VaR:
\begin{equation*} \text{VaR}_{0.99,T} \cong -2.33 \times \sigma_{d} \times \sqrt{T} \end{equation*}
\begin{equation*} \text{VaR}_{0.99,10} \cong -2.33 \times \sigma_{d} \times \sqrt{10} \end{equation*}
\begin{equation*} \text{VaR}_{0.95,m} \cong -1.65 \times \sigma_{d} \times \sqrt{20} \end{equation*}
\begin{equation*} \text{VaR}_{0.99,y} \cong -2.33 \times \sigma_{d} \times \sqrt{250} \end{equation*}
## Stressed VaR-based capital requirement
In this case, the data used to calculate the VaR measure is selected from a period of significant financial distress appropriate to the bank's current portfolio. This measure is calculaed weekly and is expected to be no less than the VaR-based measure.
## Modeling standards for specific risk
One or more internal models are expected to measure the specific risk of portfolios of debt/equity. The internal models need to explain
• Changes in historical price variation in the portfolio.
• Be responsive to changes in market conditions.
• Robust to an adverse environment.
• Capture all material aspects of specific risk for debt/equity positions. | 2020-05-27 09:01:51 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.999428391456604, "perplexity": 12306.03271512225}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347392142.20/warc/CC-MAIN-20200527075559-20200527105559-00174.warc.gz"} |
https://www.groundai.com/project/entropies-in-case-of-continuous-time/ | Entropies in case of continuous time
# Entropies in case of continuous time
Detlef Holstein Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany
July 9, 2019
###### Abstract
Information theory on a time-discrete setting in the framework of time series analysis is generalized to the time-continuous case. Considerations of the Roessler and Lorenz dynamics as well as the Ornstein-Uhlenbeck process yield for time-continuous entropies a new possibility for the distinction of chaos and noise. In the deterministic case an upper threshold of the joint uncertainty in the limit of infinitely high sampling rate can be found and the entropy rate can be calculated as a usual time derivative of the entropy. In a three-dimensional representation the dependence of the joint entropy on space resolution, discretization time step length and uncertainty-assessed time is shown in a unified manner. Hence the dimension and the Kolmogorov-Sinai entropy rate of any dynamics can be read out as limit cases from one single graph.
Information theory, time-continuous limit, KS entropy, dimension, Renyi entropy
###### pacs:
05.45.Ac, 05.10.Gg, 05.45.Tp, 89.70.Cf
## I Introduction
Uncertainty of the outcome of random variables is usually evaluated by Shannon entropies cover91 ()
H(X)=−N∑i=1P(xi)lnP(xi), (1)
where are the possible realizations of . Relaxing the restrictions in the underlying Khinchin or Fadeev axioms it is possible to introduce the family of Renyi entropies renyi61 ()
H(q)(X)=11−qlnN∑i=1P(xi)q, (2)
which in the case of Renyi order are quite often used for estimation of entropies via the correlation sum grassproc83 (). In the case of time series analysis it is a common task to evaluate the uncertainty of random variables belonging to successive time steps kantzschreiber04 (). The corresponding joint entropy in the suitable notational representation reads
H(q)m(ϵ,τ)=11−qln⎡⎣M1(ϵ)∑i1=1...Mm(ϵ)∑im=1(Pi1,...,im(ϵ,τ))q⎤⎦. (3)
In this expression is the space resolution and is the step length of the time discretization. After having decided for a suitable Renyi order, it is often omitted. Now the finite-m-entropy rate can be defined as
hm(ϵ,τ):=H1|m(ϵ,τ)τ:=Hm+1(ϵ,τ)−Hm(ϵ,τ)τ. (4)
This quantity is a special conditional entropy per time step length. Since obviously this entropy rate is obtained from a quotient of differences, a time-continuous formulation of the relationships of entropic quantities should naturally also be at hand. The idea of connection of the entropic quantities by derivatives is also supported in crutch03 (), where however, the time discretization step length is still kept finite. The questions of consequences of variable time step length and especially the limit of for entropic quantities are addressed in this paper and it is intended to give theoretical insights into the structure of the relationships behind such entropic quantities. With those efforts this paper tries to contribute to a symmetrization of the treatment of entropies concerning space, where continuity is already realized in formulas, and time.
First, the line of thought from joint Shannon entropies to the Kolmogorov-Sinai entropy rate is outlined in sec. II, staying quite close to the paper of Gaspard and Wang gaspard93 (), the central paper behind this work. The limit of infinitesimal time step length is performed after having performed the limit of infinite time . Theoretically, sec. III contains the central aim of this paper. It is the generalization of sec. II with omission of the limit of infinite uncertainty-assessed time , concentrating on the limit of infinitesimal time discretization step length . In sec. IV the presented ideas are tested numerically for the examples of the Roessler, Lorenz and Ornstein-Uhlenbeck dynamics with the detection of qualitative discrepancies of deterministic and stochastic dynamics. Sec. V concludes the results of this paper.
## Ii From Shannon entropy to Kolmogorov-Sinai entropy rate
The starting point as given in gaspard93 () is the joint Shannon entropy
H(A,τ,t)=Hm:=t/τ(A,τ)=−∑ω0,...,ωm−1P(ω0,...,ωm−1)lnP(ω0,...,ωm−1). (5)
The symbol , being the time step length, appears on the right side only implicitly as the time between successive realizations and the partition appears on the right side implicitly in the range of values , which corresponds to in eq. (1), can take. In the following it is assumed that the partition for is consistent with the partition for . It is possible to define the rate
~h(A,τ,t):=H(A,τ,t)t. (6)
The entropy per unit time with respect to partition is then defined as the limit
~h(A,τ):=limt→∞~h(A,τ,t). (7)
Starting from eq. (5) it is also possible to define another rate
h(A,τ,t):=H(A,τ,t+τ)−H(A,τ,t)τ, (8)
and the corresponding limit
h(A,τ):=limt→∞h(A,τ,t). (9)
In general it holds
~h(A,τ,t)≠h(A,τ,t),but~h(A,τ)=h(A,τ). (10)
The -entropy rate is derived by
h(ϵ,τ)=infA:diam(Ai)≤ϵh(A,τ). (11)
On the other hand, having obtained from , e.g., again via infimum
Hm(ϵ,τ)=infA:diam(Ai)≤ϵHm(A,τ), (12)
by the limit of infinite time from the conditional entropy rate also -entropy rates can be obtained cencini00 ():
h∞(ϵ,τ)=limm→∞hm(ϵ,τ)=limm→∞1τ[Hm+1(ϵ,τ)−Hm(ϵ,τ)]. (13)
Depending on the state space resolution and time resolution , and both are the uncertainty per time step of the immediate future time step ahead if infinite time conditioning is imposed. The Kolmogorov-Sinai entropy rate for processes in continuous time is obtained from
hKS=limϵ→0,τ→0h∞(ϵ,τ)=limϵ→0,τ→0h(ϵ,τ). (14)
## Iii Time-continuous information theory
In the former section the succession of limit procedures for accessing the KS entropy rate was redisplayed. Starting with the same formula (5), it is a naturally arising question what happens if the limits of and are not performed, but instead the limit is investigated holstdiss07 (). This does not lead to dynamical invariants, but instead to prediction-relevant quantities of information theory in the time-continuous case, because prediction deals with finite-time conditioning.
The partition dependence of eq. (5) or a similar resolution dependence is suppressed in notation in this section, since the time aspects should be pointed out, and it remains the entropy . Since is the uncertainty of the realization of one random variable, it should be independent of . This results in
H(τ,t=τ)=constforτ>0, (15)
where a double - dependence on the left side of the equation leads surprisingly to -independence of the whole expression.
For fixed now the limit has to be carried out. If the limit exists, then
H(τ=0,t)≡limτ→0H(τ,t) (16)
is the time-continuous joint entropy. Otherwise a time-continuous treatment of the joint uncertainty is not possible for the process at hand. Taking on the other hand the other involved variable equal to zero it holds
H(τ,t=0)≡0. (17)
Hence it is inferred
H(τ=0,t=0)=0. (18)
From being (not necessarily strong) monotonously increasing in it is inferred that also in the limit of the entropy is (not necessarily strong) monotonously increasing in t. An interesting question is the finiteness of for finite uncertainty-assessed time t for various process classes, which will be answered numerically by treating examples in sec. IV shown in fig.’s 3, 4, 7 and 8. For fixed the performed limit causes
limτ→0m=limτ→0tτ=∞. (19)
From eqs. (5) and (19) it has to be inferred that as the uncertainty of the whole path has finally to be understood in terms of path integral-type quantities, where the initial and final states are not fixed.
Eq. (4) in notation suitable for the purpose of a time-continuous formulation or eq. (8) without partition dependence read
h(τ,t)=H(τ,t+τ)−H(τ,t)τ. (20)
If the following limit exists, then
h(τ=0,t)≡limτ→0h(τ,t) (21)
is the finite time entropy rate in the time-continuous case. The discrepancy from the usual differentiation is that the function to be differentiated depends explicitly on the parameter of the differentiation. This is unusual, but not untreatable. In case of existence of in eq. (16) for the corresponding derivative it holds that
h(τ=0,t)=limΔt→0H(0,t+Δt)−H(0,t)Δt, (22)
i.e., from eq. (21) is obtained as a usual derivative from . is found to be a perturbation of the usual quotient of differences , but the limit of is the same in both cases. From eqs. (20), (17) and (15) it is derived
τ⋅h(τ,t=0)=H(τ,t=τ)=const. (23)
One concludes trivially that for arbitrary dynamics
limτ→0h(τ,t=0)=∞ifH(τ,t=τ)>0. (24)
For deterministic dynamics the behaviour of the entropy rate will be shown in fig. 5. It should be mentioned that the same treatment as here for the first derivative can of course also be carried through for higher derivatives holstdiss07 ().
Eq. (20) inverted and iterated leads with
H(τ,t=mτ) =(h(τ,t=(m−1)τ)+...+h(τ,0))⋅τ (25)
in the time-continuous limit to a representation of the joint entropy via integral:
H(τ=0,[0,t]) ≡H(τ=0,t) =limτ→0[m−1∑k=0τ⋅h(τ,t=kτ)] \lx@stackrelmτ=m′τ′=limτ→0limτ′→0[m′−1∑k=0τ′⋅h(τ,t=kτ′)] =limτ→0∫t0duh(τ,u) =∫t0duh(τ=0,u). (26)
Interchangeability of the limit procedures differentiation and integration is needed in the final step.
## Iv Numerical calculations for time-continuous entropies
### iv.1 Roessler system (deterministic chaotic dynamics)
The Roessler system is given by
˙x =−y−z, ˙y =x+ay, ˙z =b+z(x−c). (27)
The parameters are chosen as and . The quadratic term ’’ is the only nonlinearity. The largest Lyapunov exponent of the Roessler attractor is and the fractal dimension is .
In fig. 1 the joint entropy of the x-coordinate of the Roessler system is given as a function of resolution , time step length and uncertainty-assessed time . Convergence of the joint entropy for decreasing time step length can be seen. The joint entropy depends logarithmically on the resolution , from which the dimension is obtained as a slope according to
D(q)m=−limϵ→0H(q)m(ϵ)lnϵ (28)
(schusterjust05 (), p.106) with a value as predicted.
It is possible to see in fig. 2 (in comparison with fig. 1 one smaller decade of resolutions is shown) that the z-coordinate of the Roessler attractor carries a different uncertainty for the same values of () compared with the x-coordinate. Nevertheless the dimension of the attractor is captured also by the joint entropy of the z-coordinate.
In fig. 3 the slice of fig. 1 for resolution with extended time is shown. It is found that asymptotically in the joint uncertainty of the Roessler system increases linearly. The measured slope for smallest in between and is 0.08. It estimates the KS entropy rate
hKS=limϵ→0limτ→0limt→∞h(ϵ,τ,t). (29)
The deviation from the expected value 0.07 can be found in the fact that still too small or too large are used for the estimation. It can be concluded from the small slope that the uncertainty of the first few points is larger than the uncertainty of a rather long motion in the attractor given the first few points.
In fig. 4 the slice of fig. 2 for resolution with extended time is shown. Compared to fig. 3 a much smaller initial uncertainty is observed for the z-coordinate. This is understood from the high probability of the z-coordinate to stay at zero. Furthermore in contrast to the x-coordinate a wave-like structure of the joint entropy for smaller time is observed for the z-coordinate. A non-monotonous entropy rate as a function of has to be inferred. This unintuitive signature was robustly reproduced under various parameter values and confidence in this result is caused by the fact that asymptotically for smallest available the slope of 0.08 is found, which serves as a correct estimation of the KS entropy also from the z-coordinate of the Roessler system. Since also (from fig. 2) the dimension of the Roessler dynamics is correctly extractable the result should not be a numerical artefact. Nevertheless it is not expected that this should be true physics in the sense of increasing uncertainty by enlarged conditioning. According to Shannon entropies, for which additional conditioning cannot increase uncertainty, this behaviour is forbidden. A possible explanation of this behaviour can be assigned to effects of Renyi order (cmp. eq. (2)), because in this case it seems unproven that increasing conditioning necessarily reduces the resulting entropy. With this reasoning the example carries the potential of tearing apart the interpretation of uncertainty from - entropies. Another possible explanation that the wave-like structure corresponds to a finite sample effect being responsible for non-ergodicity has to be treated as rather improbable, because also for smallest resolution (largest ) it was not possible to detect a qualitative change in the sense of smoothing of the wave-like structure of the joint entropy under drastical enlargement of the length of the dataset.
With consideration of the fact that the values of and were estimated for finite large and finite small instead of in the true limit it can be concluded that the same values are obtained from the x- and the z-coordinate of the chaotic Roessler system in the limit cases, in principle in accordance with the theorem of Takens takens81 (). On the other hand, it can be clearly seen that for finite time and finite resolution the joint uncertainty is allowed to behave qualitatively different for different observables as the x- and the z-coordinate of the chaotic Roessler system.
In fig. 5 the entropy rate of the Roessler system is shown. For every the entropy rate as a function of and is represented by a surface. Those surfaces are interleaved for different . Depending on the ranges shown along the axes different results become apperent: With the example of the Roessler system even for deterministic dynamics a true divergence of can be observed for sufficiently small seen in particular in the upper panel of fig. 5. The trivial argument of eq. (24) already gave a hint for such behaviour. It is possible to see that for finite increases logarithmically in with a slope depending on . In the lower panel for larger the entropy rate indicates a small finite value in the front right corner of the plot, which for sufficiently large approximates the KS entropy rate.
### iv.2 Lorenz system (deterministic chaotic dynamics)
The implemented discretized equations for the Lorenz dynamics are
xn+1 =xn+σ(−xn+yn)Δt, yn+1 =yn+(−xnzn+rxn−yn)Δt, zn+1 =zn+(xnyn−bzn)Δt (30)
with the usual parameters
r=28.0,σ=10.0,b=83. (31)
The fractal dimension of the Lorenz attractor is about and the largest Lyapunov exponent is .
In fig. 6 the joint entropy of the Lorenz system is shown as a function of the resolution , time step length and uncertainty-assessed time . As for the Roessler system it is possible to see the convergence of the joint entropy for decreasing time step length and a logarithmic dependence of the joint entropy on the resolution with a slope giving the expected dimension in the limit of infinitesimal small .
In fig. 7 a slice of fig. 6 for fixed resolution is shown. An asymptotically linear behaviour in time can be found. The fact, that the slope is steeper than that of the chaotic Roessler attractor of fig. 3 is in accordance with the larger number of nonlinear terms and the known largest Lyapunov exponents. The slightly larger slope compared to the known KS entropy of the Lorenz system can be explained with estimation at rather large . The appearance of finite sample fluctuations in entropy estimation in fig. 6 for small , small and large under the same estimation conditions sooner in the Lorenz-case than in the Roessler-case is in accordance with the enhanced uncertainty of the Lorenz dynamics.
Furthermore fig. 7 indicates that the joint entropies for different finite -values do not coincide for asymptotically large times even though this cannot finally be proven in a plot for finite . The slopes seem to reach the same value in all cases as already observed in fig. 3 for the x-coordinate of the Roessler dynamics, and hence the estimation of the KS entropy rate is rather independent of the choice of , i.e., in the case of deterministic dynamics the limit with respect to in eq. (14) is not so important. On the other hand, the behaviour of for finite rather small is better resolved for smaller , and this in general is rather important with respect to prediction.
### iv.3 Ornstein-Uhlenbeck process (linear stochastic dynamics)
The Ornstein-Uhlenbeck process
˙Xt=−αXt+√D˙Wt (32)
can be numerically implemented by its discretisation
Xn+1=(1−αΔt)Xn+√D√Δtηn+1,ηn+1∼N(0,1)iid. (33)
In fig. 8 the numerical analysis of the Ornstein-Uhlenbeck process indicates the non-existence of the limit for sufficiently small and finite except at for time- and amplitude-continuous stochastic dynamics with continuous trajectories. Whereas in the deterministic case, e.g. for the Lorenz dynamics, the value of the KS entropy rate was approximately seen as a slope at sufficiently small finite also for finite , in the time- and amplitude-continuous stochastic case the slope of with respect to asymptotically in for sufficiently small finite is seen in fig. 8 to be a function of . Furthermore it was possible to show numerically that for sufficiently small and non-zero finite fixed time the entropy rate diverges logarithmically with decreasing in the example of Ornstein-Uhlenbeck dynamics. Consistently, fig. 8 supports for this stochastic dynamics, interestingly already from the -behaviour. It should be mentioned here that for eq. (20) can be seen as a naturally given regularization of the unavailable eq. (22) in the stochastic case.
The author is aware of the fact that the result concerning the -dependence contradicts (gaspard93 (), p.322) according to which processes with continuous realizations are said to have -independent -entropies per unit time, which corresponds to the suggestion of finite . The argument for this behaviour was essentially that for finite a finite crossing time of underlying boxes of the partition leads to finite uncertainty for continuous trajectories (gaspard93 (), p.317). On the other hand, an explanation for the numerically found behaviour in fig. 8 could be that infinite uncertainty arises from the trajectories close to the border of boxes of partitioning, where in the limit of infinitely often moving back and forth between boxes in finite time is in principle possible for the Ornstein-Uhlenbeck process. This would create an infinite and hence dominant contribution for an averaged uncertainty calculation.
From fig. 9 it is obtained that in the transition regime of large the behaviour of non-existing is suppressed. The plots do not allow for the decision of the question if the transition from finite to infinite occurs at the resolution of the size of the system or at a smaller value for . According to eq. (28) fig. 9 yields finite dimensions for finite increasing in for stochastic dynamics, even though the limit can of course not be seen. In the limit of , what is effectively like the limit of , fig. 9 suggests an infinite dimension for stochastic dynamics.
The results of this section are qualitatively robust against changes of the parameter of the Ornstein-Uhlenbeck process and hence qualitatively robust against changes of the autocorrelation of the process.
## V Conclusion
The central objects of investigation in this paper are the entropy and the entropy rate as a function of the resolution , the time discretization step length and the uncertainty-assessed time with numerical access to the case of Renyi order . Special focus is laid on the analysis of the behaviour of these quantities for varying and especially on the time-continuous limit while is kept finite. In case of existence, in the time-continuous limit the entropy rates can be understood as usual time-derivatives of entropies. However, for finite , in consequence of the explicit -dependence of entropies, discrepancies from the usual quotient of differences are present.
Numerically the analysis of time- and space-continuous dynamics was carried out with the Roessler and Lorenz system for the deterministic case and with the Ornstein-Uhlenbeck process for the stochastic case. Qualitative discrepancies of deterministic and stochastic dynamics were found. In the deterministic case the uncertainty is finite for all finite and . The entropy rate becomes constant for large and sufficiently small . In the stochastic case the uncertainty is infinite for all if is below some threshold. Only for above the threshold it is finite and then effectively like in the deterministic case. The stochastic result is qualitatively independent of the value of the width of the input noise as long as the width is not exactly zero, corresponding to determinism and introducing the qualitative change. This behaviour of entropies in the limit seems to offer a new possibility for distinction of chaos and noise (see also cencini00 ()).
The convergence of in the limit in the deterministic case can be interpreted as a saturation of the gain from higher sampling rates, such that a threshold sampling rate can be postulated, above which almost nothing more can be learned, i.e., in the deterministic case the continuous limit can be well approximated by discrete sampling with sufficiently high sampling rate. A criterion for an optimal sampling rate could be formulated.
In the limit case the partial derivative of the joint entropy with respect to yields (minus) the dimension and if furthermore the limit is carried out, the partial derivative of with respect to yields the KS entropy rate. All information concerning entropy rates (including the KS entropy rate) and dimensions of the dynamics can be extracted from one single plot of . In the deterministic case known values were verified in examples. For time- and amplitude-continuous stochastic dynamics it was seen that the partial derivative of with respect to minus becomes infinite for or and the partial derivative of H with respect to becomes infinite for or .
According to usual rules the total differential of the joint entropy can be written as
dH=∂H∂tdt+∂H∂lnϵdlnϵ. (34)
For sufficiently small and sufficiently large this becomes
dH=hKSdt−Ddlnϵ. (35)
It is possible to see that directional derivatives of in general mix the properties of dimension and entropy rate. For integral quantities was suggested in (grassberger91 (), p.529), from which eq. (35) can be derived, but a prescription for the calculation of the constant is not given. Since eq. (35) avoids the offset problems it is a slightly reduced and hence preferable representation of the relationship of the involved quantities.
Concerning the limit cases and with the determination of dynamical invariants, which is discussed in the literature, e.g. see (farmer82 (), p.1321), the time-continuous case treated in this paper gives a new unified representation of singly known results. On the other hand, at least of the same importance, the case of finite time is of strong interest, if questions concerning optimal finite time conditioning with respect to prediction are addressed. A finite resolution is needed for optimality of local prediction methods. With finite and , the limit is of interest for the determination of the informational characteristic of the dynamics.
A non-monotonous entropy rate for Renyi order , i.e., a conditional entropy, which does not monotonously decrease in the length of conditioning, is numerically found for the z-coordinate of the Roessler system in fig. 4. This could be interpreted as a hint for problems of the interpretation of uncertainty for entropies with in maximal generality.
## References
• (1) T.M. Cover and J.A. Thomas, Elements of Information Theory, Wiley, New York (1991).
• (2) A. Renyi, On measures of entropy and information, Proc. IV Berkeley Symposium Math. Statist. and Prob., University California Press, 547-561 (1961).
• (3) P. Grassberger and I. Procaccia, Estimation of the Kolmogorov entropy from a chaotic signal, Phys. Rev. A 28 (4) 2591-2593 (1983).
• (4) H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, Cambridge University Press, second edition (2004).
• (5) J.P. Crutchfield and D.P. Feldman; Regularities unseen, randomness observed: Levels of entropy convergence, Chaos 13 (1) 25-54 (2003).
• (6) P. Gaspard and X.-J. Wang; Noise, chaos and (, )-entropy per unit time, Phys. Rep. 235 (6) 291-343 (1993).
• (7) M. Cencini, M. Falcioni, E. Olbrich, H. Kantz and A. Vulpiani; Chaos or noise: Difficulties of a distinction, Phys. Rev. E 62 (1) 427-437 (2000).
• (8) D. Holstein, Generalized Markov Approximations from information theory and consequences for prediction, Dissertation, Wuppertal (2007).
• (9) H.G. Schuster and W. Just; Deterministic Chaos: An Introduction, Wiley, Weinheim, fourth edition (2005).
• (10) F. Takens; Detecting strange attractors in turbulence, Lecture Notes in Mathematics 898, Springer, New York (1981).
• (11) P. Grassberger, T. Schreiber, and C. Schaffrath; Nonlinear time sequence analysis, Int. Journ. of Bifurcation and Chaos 1 (3) 521-547 (1991).
• (12) J.D. Farmer; Information Dimension and the Probabilistic Structure of Chaos, Z. Naturforsch. 37a 1304-1325 (1982).
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The feedback must be of minimum 40 characters and the title a minimum of 5 characters | 2021-02-27 12:12:48 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8702293634414673, "perplexity": 773.9581262596582}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178358956.39/warc/CC-MAIN-20210227114444-20210227144444-00203.warc.gz"} |
https://www.physicsforums.com/threads/some-questions-in-queueing-theory.727846/ | # Some questions in Queueing Theory
1. Dec 11, 2013
### sigh1342
1. The problem statement, all variables and given/known data
In$$M/M/1/FCFS/c/\infty$$
Wiki say offered load is equal to the expected number in the system, and I found offered load is equal to the ρ=λ/μ, where λ is the average arrive rate. And the μ is the average service time. And I don't know which one is true , and I can't find the information about effective load.
Thank you . :^)
2. Relevant equations
3. The attempt at a solution
2. Dec 11, 2013
### Ray Vickson
Please clarify: some authors use the notation A/B/C/D?E/F is slightly different order, so you need to tell us what the $c$ stands for. My guess is that you have an infinite calling population but a finite queue capacity; is that correct?
You need to show your work; it is not enough to just say you don't know what to do. In particluar, if the 'c' means that a total of c customers can be accommodated (one in service and c-1 waiting) then some 'arriving' customers will not enter the system because it is full. In particular, you need to be careful when using such results as $L = \bar{\lambda} W,$ etc.
3. Dec 11, 2013
### haruspex
I believe the definition of offered load is mean arrival rate * mean service time, so λ/μ. Looks to me that for a queue of finite capacity the effective load is based on the effective arrival rate, which discounts arrivals when the queue is full. See e.g. http://www.engr.sjsu.edu/udlpms/ISE 265/set4 queuing theory.ppt.
However, care must be taken in using this. You can't simply treat a queue of limited capacity as being an infinite queue with a reduced offered load. | 2017-10-17 19:13:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7179826498031616, "perplexity": 646.4001265137684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187822480.15/warc/CC-MAIN-20171017181947-20171017201947-00717.warc.gz"} |
https://de.wikibooks.org/wiki/Serlo:_EN:_Composition_of_continuous_functions | # Composition of continuous functions – Serlo
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Many functions are defined as the concatenation - the linking together of things, like in a chain - of other functions. Checking for continuity of such concatenated functions by using the classical epsilon-delta criterion for continuity is often tedious. However, one can prove that the concatenation of continuous functions is once again a continuous function. This is an important tool that simplifies the proof for continuity of compositions of functions.
## Concatenation Theorems
The concatenation theorems for continuous functions are the following:
Theorem (Concatenation Theorems)
Let ${\displaystyle D\subseteq \mathbb {R} }$ be a subset of the real numbers (domain of definition) and ${\displaystyle \lambda \in \mathbb {R} }$ be a real number. Let ${\displaystyle f,g:D\to \mathbb {R} }$ be real-valued functions, which are continuous at all ${\displaystyle a\in D}$ . That means, ${\displaystyle \lim _{x\to a}{f(x)}=f(a)}$ and ${\displaystyle \lim _{x\to a}{g(x)}=g(a)}$. Then, the following functions are continuous at ${\displaystyle a}$ , as well:
• ${\displaystyle f+g:D\to \mathbb {R} :x\mapsto f(x)+g(x)}$
• ${\displaystyle \lambda f:D\to \mathbb {R} :x\mapsto \lambda \cdot f(x)}$
• ${\displaystyle fg:D\to \mathbb {R} :x\mapsto f(x)\cdot g(x)}$
Let ${\displaystyle D'=\{x\in D:g(x)\neq 0\}}$ and consider ${\displaystyle f}$ and ${\displaystyle g}$ which are both continuous in ${\displaystyle a\in D'}$. Then, the following quotient function is continuous in ${\displaystyle a}$:
${\displaystyle {\frac {f}{g}}:D'\to \mathbb {R} :x\mapsto {\frac {f(x)}{g(x)}}}$
Let ${\displaystyle h:E\to \mathbb {R} }$ be a real-valued function with ${\displaystyle f(D)\subseteq E}$ , which is continuous at ${\displaystyle b=f(a)\in E}$ . Then, the concatenation ${\displaystyle h\circ f}$ is also continuous - namely at ${\displaystyle a}$:
${\displaystyle h\circ f:D\to \mathbb {R} :x\mapsto h(f(x))}$
## Motivation
Imagine, we are given a function containing sums, products and quotients like ${\displaystyle f:\mathbb {R} \to \mathbb {R} _{0}^{+}}$, ${\displaystyle x\mapsto \left|{\tfrac {1+x^{3}}{1+x^{2}}}\right|}$ . We would like to know whether this function is continuous at some argument ${\displaystyle a\in \mathbb {R} }$ . So we consider any sequence of arguments ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ converging to ${\displaystyle a}$ and check whether there is always ${\displaystyle \lim _{n\to \infty }f(x_{n})=f(a)}$. For handling sums, products and quotients, the limit theorems for convergent sequences turn out to be very useful:
{\displaystyle {\begin{aligned}\lim _{n\to \infty }f(x_{n})&=\lim _{n\rightarrow \infty }{\left|{\frac {1+x_{n}^{3}}{1+x_{n}^{2}}}\right|}\\[0.5em]&\quad {\color {Gray}\left\downarrow \ {\text{Betragsregel}}\right.}\\[0.5em]&=\left|\lim _{n\to \infty }{\left({\frac {1+x_{n}^{3}}{1+x_{n}^{2}}}\right)}\right|\\[0.5em]&\quad {\color {Gray}\left\downarrow \ {\text{Quotienten-, Produkt- und Additionsregel}}\right.}\\[0.5em]&=\left|{\frac {\lim _{n\to \infty }1+\left(\lim _{n\to \infty }x_{n}\right)^{3}}{\lim _{n\to \infty }1+\left(\lim _{n\to \infty }x_{n}\right)^{2}}}\right|\\[0.5em]&\quad {\color {Gray}\left\downarrow \ \lim _{n\to \infty }x_{n}=a\right.}\\[0.5em]&=\left|{\frac {1+a^{3}}{1+a^{2}}}\right|=f(a)\end{aligned}}}
Those formulas could be applied because all subsequences converge (we showed this in the end of our calculations). As ${\displaystyle a\in \mathbb {R} }$ was chosen to be arbitrary, we directly obtain the continuity of the entire function ${\displaystyle f}$ . This proof of continuity is basically an application of the sequence criterion plus theorems for sequence limits. Since, thanks to the limit theorems, the limit can be pulled into the function, we can use it to establish continuity. And the above concatenation theorems shorten proofs of this kind even further. we consider the following functions:
• ${\displaystyle a:\mathbb {R} \to \mathbb {R} :x\mapsto x}$
• ${\displaystyle b:\mathbb {R} \to \mathbb {R} :x\mapsto 1}$
• ${\displaystyle c:\mathbb {R} \to \mathbb {R} ^{+}:x\mapsto |x|}$
Now, ${\displaystyle f}$ may be written as a concatenation of those three functions:
${\displaystyle f(x)=c\left({\frac {b(x)+a(x)a(x)a(x)}{b(x)+a(x)a(x)}}\right)}$
As all three functions ${\displaystyle a}$, ${\displaystyle b}$ and ${\displaystyle c}$ are continuous, the concatenation theorems will directly imply continuity for ${\displaystyle f}$ . This argumentation is even shorter than the proof using the sequence criterion. So the continuity proof can be concluded in one sentence: ${\displaystyle f}$ is continuous because it is a concatenation of continuous functions. And indeed, any concatenation constructed out of continuous functions (i.e. by combining polynomials) is continuous.
## Problem Example
The following problem illustrates just how easy it is to establish continuity of a function using the concatenation theorems:
Exercise (Continuity of a concatenated square root function)
Prove continuity for the following function:
${\displaystyle f:\mathbb {R} \to \mathbb {R} :x\mapsto {\sqrt {5+x^{2}}}}$
How to get to the proof? (Continuity of a concatenated square root function)
The above function is just a concatenation of several simple functions, serving as building blocks. Our main task is to find those building blocks. They are given by:
• ${\displaystyle a:\mathbb {R} \to \mathbb {R} :x\mapsto x}$
• ${\displaystyle b:\mathbb {R} \to \mathbb {R} :x\mapsto 5}$
• ${\displaystyle c:\mathbb {R} _{0}^{+}\to \mathbb {R} :x\mapsto {\sqrt {x}}}$
This allows writing ${\displaystyle f}$ as a concatenation:
${\displaystyle f(x)={\sqrt {5+x^{2}}}={\sqrt {5+x\cdot x}}=c(b(x)+a(x)\cdot a(x))}$
So ${\displaystyle f}$ is simply continuous because it is a concatenation of continuous functions.
Proof (Continuity of a concatenated square root function)
Let the following functions be given:
• ${\displaystyle a:\mathbb {R} \to \mathbb {R} :x\mapsto x}$
• ${\displaystyle b:\mathbb {R} \to \mathbb {R} :x\mapsto 5}$
• ${\displaystyle c:\mathbb {R} _{0}^{+}\to \mathbb {R} :x\mapsto {\sqrt {x}}}$
These functions are continuous. Further, we can write ${\displaystyle f(x)=c(b(x)+a(x)\cdot a(x))}$. Hence, ${\displaystyle f}$ is a concatenation of continuous functions, so it is continuous, as well.
## General sketch of the proof
Following the concatenation theorems, every composition of continuous functions is again continuous function. So if ${\displaystyle f:D\to \mathbb {R} }$ can be written as a concatenation of continuous functions, we can directly infer continuity of ${\displaystyle f}$ . A corresponding proof could be of the following form:
Let ${\displaystyle f:\ldots }$ with ${\displaystyle f(x)=\ldots }$. The function ${\displaystyle f}$ is a concatenation of the following functions:
...List of continuous functions, which serve as building bricks for ${\displaystyle f}$ ...
Since ${\displaystyle f(x)=\ldots }$ (Expression how ${\displaystyle f}$ is constructed out of those bricks) , we know that ${\displaystyle f}$ is a concatenation of continuous functions and hence continuous, as well.
In a lecture, this proof scheme may of course only be applied if the concatenation theorems have already been treated before. In any way, it is a very efficient method to characterize continuous functions.
## Corollary: Polynomial functions are continuous
Every polynomial function can be written as a concatenation of following the two functions:
• ${\displaystyle f:\mathbb {R} \to \mathbb {R} :x\mapsto x}$
• ${\displaystyle g_{c}:\mathbb {R} \to \mathbb {R} :x\mapsto c}$
${\displaystyle f}$ is the identity ${\displaystyle g_{c}}$ the constant function with value ${\displaystyle c}$. These functions are continuous and hence, every polynomial function is continuous. For instance, we may construct the polynomial function ${\displaystyle h:\mathbb {R} \to \mathbb {R} :x\mapsto 2x^{3}-4x+23}$ out of ${\displaystyle f}$ and ${\displaystyle g_{c}}$ as follows:
${\displaystyle h(x)=g_{2}(x)\cdot f(x)\cdot f(x)\cdot f(x)+g_{-4}(x)\cdot f(x)+g_{23}(x)}$
{\displaystyle {\begin{aligned}h(x)&=2x^{3}-4x+23\\&=2\cdot x\cdot x\cdot x+(-4)\cdot x+23\\&=g_{2}(x)\cdot f(x)\cdot f(x)\cdot f(x)+g_{-4}(x)\cdot f(x)+g_{23}(x)\end{aligned}}}
## Further examples
Exactly as subsequences have to converge for a concatenation sequence to converge, we need that our building brick functions are continuous in order to obtain a continuous concatenation function. When using non-continuous functions for the concatenation, we do not know anything about the continuity of the outcome. For instance:
{\displaystyle {\begin{aligned}f:\mathbb {R} \to \mathbb {R} :x\mapsto &1\\g:\mathbb {R} \to \mathbb {R} :x\mapsto &{\begin{cases}1,&x\geq 0\\0,&x<0\end{cases}}\end{aligned}}}
The function ${\displaystyle f}$is continuous at ${\displaystyle x=0}$, whereas ${\displaystyle g}$ is not. The product of both functions is again ${\displaystyle g}$, since ${\displaystyle f(x)\cdot g(x)=1\cdot g(x)=g(x)}$. Therefore the product (i.e. a concatenation) is discontinuous at ${\displaystyle x=0}$. By contrast to what one may intuitively expect, concatenating discontinuous functions may also yield us a continuous function. To illustrate this, let us consider:
${\displaystyle h:\mathbb {R} \to \mathbb {R} :x\mapsto h(x)={\begin{cases}0,&x\in \mathbb {R} \setminus \mathbb {Q} \\1,&x\in \mathbb {Q} \end{cases}}}$
This function maps rational numbers to ${\displaystyle 1}$ an all other to ${\displaystyle 0}$ . Concatenating ${\displaystyle h}$ with itself, we get the following function ${\displaystyle h\circ h}$:
{\displaystyle {\begin{aligned}h(h(x))&={\begin{cases}0,&h(x)\in \mathbb {R} \setminus \mathbb {Q} \\1,&h(x)\in \mathbb {Q} \end{cases}}\\[0.3em]&\quad {\color {Gray}\left\downarrow \ h(x){\text{ ist immer rational.}}\right.}\\[0.3em]&=1\end{aligned}}}
${\displaystyle h\circ h}$ is just a constant function. Hence it is continuous - although ${\displaystyle h}$ was actually nowhere continuous. So concatenating discontinuous functions may indeed yield us a continuous function.
## Proof of concatenation theorems
Theorem (Concatenation theorem for sums)
Let ${\displaystyle D\subseteq \mathbb {R} }$ and let ${\displaystyle f,g:D\to \mathbb {R} }$ be real-valued functions, which are continuous in ${\displaystyle a\in D}$ Then, ${\displaystyle f+g:D\to \mathbb {R} :x\mapsto f(x)+g(x)}$ is continuous in ${\displaystyle a}$ , as well.
Proof (Concatenation theorem for sums)
We will prove the addition rule for continuity by checking the sequence criterion. Let ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ be any sequence of arguments taken from the domain ${\displaystyle D}$ and converging to ${\displaystyle a}$. There is:
{\displaystyle {\begin{aligned}&\lim _{n\to \infty }(f+g)(x_{n})\\[0.3em]=&\lim _{n\to \infty }\left(f(x_{n})+g(x_{n})\right)\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{Limit theorem: addition rule}}\right.}\\[0.3em]=&\lim _{n\to \infty }f(x_{n})+\lim _{n\to \infty }g(x_{n})\\[0.3em]=&f(a)+g(a)=(f+g)(a)\end{aligned}}}
Alternative proof (Concatenation theorem for sums)
It is also possible to establish continuity of ${\displaystyle f+g}$ in ${\displaystyle a}$by checkng the Epsilon-Delta criterion . Let any ${\displaystyle \epsilon >0}$ be given. As ${\displaystyle f}$ is continuous at ${\displaystyle a}$ , there has to be a ${\displaystyle \delta _{f}>0}$, such that for all ${\displaystyle x\in D}$ with ${\displaystyle |x-a|<\delta _{f}}$ the inequality ${\displaystyle |f(x)-f(a)|<\epsilon /2}$ holds. Analogously for ${\displaystyle g}$, there is a ${\displaystyle \delta _{g}>0}$, such that for all ${\displaystyle x\in D}$ with ${\displaystyle |x-a|<\delta _{g}}$ the inequality ${\displaystyle |g(x)-g(a)|<\epsilon /2}$ holds.
Now, we set ${\displaystyle \delta :=\min\{\delta _{f},\delta _{g}\}}$. Hence, for all ${\displaystyle x\in D}$ with ${\displaystyle |x-a|<\delta }$ ,both the conditions ${\displaystyle |f(x)-f(a)|<\epsilon /2}$ and ${\displaystyle |g(x)-g(a)|<\epsilon /2}$ are fulfilled. Thus, for all ${\displaystyle x}$ with ${\displaystyle |x-a|<\delta }$ , there is:
{\displaystyle {\begin{aligned}&|(f+g)(x)-(f+g)(a)|\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{definition of the function }}(f+g)\right.}\\[0.3em]&=|f(x)+g(x)-(f(a)+g(a))|\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{associativity of }}\mathbb {R} \right.}\\[0.3em]&=|(f(x)-f(a))+(g(a)-g(x))|\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{triangle inequality}}\right.}\\[0.3em]&\leq \underbrace {|f(x)-f(a)|} _{\color {Gray}<\epsilon /2{\text{, da }}|x-a|<\delta \leq \delta _{f}}+\underbrace {|g(a)-g(x)|} _{\color {Gray}<\epsilon /2{\text{, da }}|x-a|<\delta \leq \delta _{g}}\\[0.3em]&<\epsilon /2+\epsilon /2=\epsilon \end{aligned}}}
### Continuity of scalar multiplication
Theorem (Concatenation theorem for scalar multiplication)
Let ${\displaystyle D\subseteq \mathbb {R} }$ and ${\displaystyle \lambda \in \mathbb {R} }$. Further, let ${\displaystyle f:D\to \mathbb {R} }$ be a real-valued function, which is continuous at ${\displaystyle a\in D}$ . Then, ${\displaystyle \lambda f:D\to \mathbb {R} :x\mapsto \lambda \cdot f(x)}$ is continuous at ${\displaystyle a}$ , as well.
Proof (Concatenation theorem for scalar multiplication)
We will prove continuity of ${\displaystyle \lambda f}$ in ${\displaystyle a}$ by checking the sequence criterion. Let ${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}$ be any sequence with ${\displaystyle x_{n}\in D}$ for all ${\displaystyle n\in \mathbb {N} }$ and converging as ${\displaystyle \lim _{n\to \infty }x_{n}=a}$. Since ${\displaystyle f}$ is continuous at ${\displaystyle a}$ , the limit ${\displaystyle \lim _{n\to \infty }f(x_{n})=f(a)}$ exists. Now there is:
{\displaystyle {\begin{aligned}&\lim _{n\to \infty }(\lambda f)(x_{n})\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{definition of the function }}(\lambda f)\right.}\\[0.3em]&=\lim _{n\to \infty }\left(\lambda \cdot f(x_{n})\right)\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{foctor rule for limits}}\right.}\\[0.3em]&=\lambda \cdot \lim _{n\to \infty }f(x_{n})\\[0.3em]&\quad {\color {Gray}\left\downarrow \ f{\text{ continuous}}\right.}\\[0.3em]&=\lambda \cdot f(a)\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{definition of the function }}(\lambda f)\right.}\\[0.3em]&=(\lambda f)(a)\end{aligned}}}
### Continuity under multiplication
Theorem (Concatenation theorem for multiplication)
Let ${\displaystyle D\subseteq \mathbb {R} }$ and let ${\displaystyle f,g:D\to \mathbb {R} }$ be real-valued functions , which are continuous at ${\displaystyle a\in D}$ . Then, ${\displaystyle fg:D\to \mathbb {R} :x\mapsto f(x)\cdot g(x)}$ is continuous at ${\displaystyle a}$ , as well.
Proof (Concatenation theorem for multiplication)
We will establish continuity of the product ${\displaystyle fg}$ by checking the sequence criterion. Let ${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}$ be any sequence with ${\displaystyle x_{n}\in D}$ for all ${\displaystyle n\in \mathbb {N} }$ and ${\displaystyle \lim _{n\to \infty }x_{n}=a}$. As both ${\displaystyle f}$ and ${\displaystyle g}$ are continuous at ${\displaystyle a}$ , the two limits ${\displaystyle \lim _{n\to \infty }f(x_{n})=f(a)}$ and ${\displaystyle \lim _{n\to \infty }g(x_{n})=g(a)}$ exist. Now, there is:
{\displaystyle {\begin{aligned}&\lim _{n\to \infty }(fg)(x_{n})\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{definition of the function }}(fg)\right.}\\[0.3em]&=\lim _{n\to \infty }\left(f(x_{n})\cdot g(x_{n})\right)\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{product rule for limits}}\right.}\\[0.3em]&=\lim _{n\to \infty }f(x_{n})\cdot \lim _{n\to \infty }g(x_{n})\\[0.3em]&\quad {\color {Gray}\left\downarrow \ f{\text{ and }}g{\text{ are continuous}}\right.}\\[0.3em]&=f(a)\cdot g(a)\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{definition of the funciton }}(fg)\right.}\\[0.3em]&=(fg)(a)\end{aligned}}}
### Continuity of quotients
Theorem (Concatenation theorem for quotients)
Let ${\displaystyle D\subseteq \mathbb {R} }$ and let ${\displaystyle f:D\to \mathbb {R} }$ and ${\displaystyle g:D\to \mathbb {R} }$ be two real-valued functions. Define the domain without 0 by ${\displaystyle D'=\{x\in D:g(x)\neq 0\}}$ and suppose that ${\displaystyle f}$ and ${\displaystyle g}$ are both continuous at all ${\displaystyle a\in D'}$. Then, the quotient ${\displaystyle {\tfrac {f}{g}}:D'\to \mathbb {R} }$ will be continuous at ${\displaystyle a}$ , as well.
Proof (Concatenation theorem for quotients)
The proof will be done by checking the sequence criterion using the quotient rule for limits . Let ${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}$ be any sequence with ${\displaystyle x_{n}\in D'}$ for all ${\displaystyle n\in \mathbb {N} }$ and ${\displaystyle \lim _{n\to \infty }x_{n}=a}$. Since both the function ${\displaystyle f}$ and ${\displaystyle g}$ are continuous at ${\displaystyle a}$ , it follows that ${\displaystyle \lim _{n\to \infty }f(x_{n})=f(a)}$ and ${\displaystyle \lim _{n\to \infty }g(x_{n})=g(a)}$. Furthermore, there is ${\displaystyle g(x_{n})\neq 0}$, so ${\displaystyle {\tfrac {f}{g}}(x_{n}):={\tfrac {f(x_{n})}{g(x_{n})}}}$ is well-defined for all ${\displaystyle n\in \mathbb {N} }$ . Thus:
{\displaystyle {\begin{aligned}&\lim _{n\to \infty }{\frac {f}{g}}(x_{n})\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{definition of the function }}{\tfrac {f}{g}}\right.}\\[0.3em]&=\lim _{n\to \infty }{\frac {f(x_{n})}{g(x_{n})}}\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{quotient rules for limits}}\right.}\\[0.3em]&={\frac {\lim _{n\to \infty }f(x_{n})}{\lim _{n\to \infty }g(x_{n})}}\\[0.3em]&\quad {\color {Gray}\left\downarrow \ f{\text{ and }}g{\text{ are continuous}}\right.}\\[0.3em]&={\frac {f(a)}{g(a)}}\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{definition of the funcion }}{\tfrac {f}{g}}\right.}\\[0.3em]&={\frac {f}{g}}(a)\end{aligned}}}
### Continuity of compositions
Theorem (Concatenation theorem for compositions)
Sei ${\displaystyle D\subseteq \mathbb {R} }$ und ${\displaystyle f:D\to \mathbb {R} }$ eine Funktionen, die in ${\displaystyle a\in D}$ stetig ist. Sei zusätzlich ${\displaystyle h:E\to \mathbb {R} }$ mit ${\displaystyle f(D)\subseteq E}$, die in ${\displaystyle b=f(a)\in E}$ stetig ist. Dann ist auch ${\displaystyle h\circ f:D\to \mathbb {R} }$ stetig in ${\displaystyle a}$.
Proof (Concatenation theorem for compositions)
We will prove continuity of ${\displaystyle h\circ f}$ at ${\displaystyle a}$ by checking the sequence criterion. Let ${\displaystyle \left(x_{n}\right)_{n\in \mathbb {N} }}$ be any sequence with ${\displaystyle x_{n}\in D}$ for all ${\displaystyle n\in \mathbb {N} }$ converging like ${\displaystyle \lim _{n\to \infty }x_{n}=a}$. Then, ${\displaystyle \left(f(x_{n})\right)_{n\in \mathbb {N} }}$ is a sequence with ${\displaystyle f(x_{n})\in E}$ for all ${\displaystyle n\in \mathbb {N} }$ (since ${\displaystyle f(D)\subseteq E}$) and there is ${\displaystyle \lim _{n\to \infty }f(x_{n})=f(a)}$ (since ${\displaystyle f}$ stetig). Therefore:
{\displaystyle {\begin{aligned}&\lim _{n\to \infty }(h\circ f)(x_{n})\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{definition of the composition}}\right.}\\[0.3em]&=\lim _{n\to \infty }h(f(x_{n}))\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{sequence criterion for }}h\right.}\\[0.3em]&=h\left(\lim _{n\to \infty }f(x_{n})\right)\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{continuity of }}f{\text{ bei }}a\right.}\\[0.3em]&=h\left(f(a)\right)\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{definition of the composition}}\right.}\\[0.3em]&=(h\circ f)(a)\end{aligned}}}
## Comparison with the epsilon-delta criterion
In the beginning of this article, we used the concatenation theorems in order to show that the function ${\displaystyle {\sqrt {5+x^{2}}}}$ is continuous. We will no perform a second proof "by hand", using the epsilon-delta criterion. The proof will cost us more work, but we will get an explicit information on the maximal initial error ${\displaystyle \delta }$ we have to choose, in order to stay below a given threshold ${\displaystyle \epsilon >0}$ for the error of the outcome.
Exercise (Epsilon-Delta proof for the continuity of the Square Root Function)
Show, using the epsilon-delta criterion, that the following function is continuous:
${\displaystyle f:\mathbb {R} \to \mathbb {R} ,x\mapsto {\sqrt {5+x^{2}}}}$
How to get to the proof? (Epsilon-Delta proof for the continuity of the Square Root Function)
We need to show, that for any given ${\displaystyle \epsilon >0}$ , there is a ${\displaystyle \delta >0}$ , such that all ${\displaystyle x\in \mathbb {R} }$ with ${\displaystyle |x-a|<\delta }$ satisfy the inequality ${\displaystyle |f(x)-f(a)|<\epsilon }$ . So let us take a look at the target inequality ${\displaystyle |f(x)-f(a)|<\epsilon }$ and estimate the absolute ${\displaystyle |f(x)-f(a)|}$ from above. We are able to control the term ${\displaystyle |x-a|}$ . Therefor, would like to get an upper bound for ${\displaystyle |f(x)-f(a)|}$ including the expression ${\displaystyle |x-a|}$ . So we are looking for an inequality of the form
${\displaystyle |f(x)-f(a)|\leq K(x,a)\cdot |x-a|}$
Here, ${\displaystyle K(x,a)}$ is some expression depending on ${\displaystyle x}$ and ${\displaystyle a}$ . The second factor is smaller than ${\displaystyle \delta }$ and can be made arbitrarily small by a suitable choice of ${\displaystyle \delta }$ . Such a bound is constructed as follows:
{\displaystyle {\begin{aligned}|f(x)-f(a)|&=\left|{\sqrt {5+x^{2}}}-{\sqrt {5+a^{2}}}\right|\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{expand with}}\left|{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}\right|\right.}\\[0.3em]&={\frac {\left|{\sqrt {5+x^{2}}}-{\sqrt {5+a^{2}}}\right|\cdot \left|{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}\right|}{\left|{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}\right|}}\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}\geq 0\right.}\\[0.3em]&={\frac {\left|x^{2}-a^{2}\right|}{{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}}}\\[0.3em]&={\frac {\left|x+a\right|}{{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}}}\cdot |x-a|\\[0.3em]&\quad {\color {Gray}\left\downarrow \ K(x,a):={\frac {\left|x+a\right|}{{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}}}\right.}\\[0.3em]&=K(x,a)\cdot |x-a|\end{aligned}}}
Since ${\displaystyle |x-a|<\delta }$ , there is:
${\displaystyle |f(x)-f(a)|=K(x,a)\cdot |x-a|
If we now choose ${\displaystyle \delta }$ small enough, such that ${\displaystyle K(x,a)\cdot \delta \leq \epsilon }$ , then we obtain our target inequality ${\displaystyle |f(x)-f(a)|\leq \epsilon }$. But ${\displaystyle K(x,a)}$ still depends on ${\displaystyle x}$ , so ${\displaystyle \delta }$ would have to depend on , too - and we required one choice of ${\displaystyle \delta }$ which is suitable for all ${\displaystyle x}$ . THerefore, we need to get rid of the ${\displaystyle x}$-dependence. This is done by an estimate of the first factor, such that our inequality takes the form ${\displaystyle K(x,a)\leq {\tilde {K}}(a)}$ :
{\displaystyle {\begin{aligned}K(x,a)&={\frac {\left|x+a\right|}{{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}}}\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\text{triangle inequality}}\right.}\\[0.3em]&\leq {\frac {\left|x\right|}{{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}}}+{\frac {\left|a\right|}{{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}}}\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\frac {a}{b+c}}\leq {\frac {a}{b}}{\text{ for }}a,c\geq 0,b>0\right.}\\[0.3em]&\leq {\frac {\left|x\right|}{\sqrt {5+x^{2}}}}+{\frac {\left|a\right|}{\sqrt {5+a^{2}}}}\\[0.3em]&\quad {\color {Gray}\left\downarrow \ {\frac {|a|}{\sqrt {5+a^{2}}}}\leq 1{\text{, since }}{\sqrt {5+a^{2}}}\geq {\sqrt {a^{2}}}=|a|\right.}\\[0.3em]&\leq 2=:{\tilde {K}}(a)\end{aligned}}}
We even made ${\displaystyle {\tilde {K}}(a)}$ independent of ${\displaystyle a}$ , which would in fact not have been necessary. So we obtain the following inequality
${\displaystyle |f(x)-f(a)|\leq 2\cdot |x-a|<2\cdot \delta }$
We need the estimate ${\displaystyle 2\cdot \delta \leq \epsilon }$, in order to fulfill the target inequality ${\displaystyle |f(x)-f(a)|<\epsilon }$ . The choice of ${\displaystyle \delta ={\tfrac {\epsilon }{2}}}$ is sufficient for that. So let us write down the proof:
Proof (Epsilon-Delta proof for the continuity of the Square Root Function)
Let ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ with ${\displaystyle f(x)={\sqrt {5+x^{2}}}}$. Let ${\displaystyle a\in \mathbb {R} }$ and an arbitrary ${\displaystyle \epsilon >0}$ be given. We choose ${\displaystyle \delta ={\tfrac {\epsilon }{2}}}$. For all ${\displaystyle x\in \mathbb {R} }$ with ${\displaystyle |x-a|<\delta }$ there is:
{\displaystyle {\begin{aligned}|f(x)-f(a)|&=\left|{\sqrt {5+x^{2}}}-{\sqrt {5+a^{2}}}\right|\\[0.3em]&={\frac {\left|{\sqrt {5+x^{2}}}-{\sqrt {5+a^{2}}}\right|\cdot \left|{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}\right|}{\left|{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}\right|}}\\[0.3em]&={\frac {\left|x^{2}-a^{2}\right|}{{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}}}\\[0.3em]&={\frac {\left|x+a\right|}{{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}}}\cdot |x-a|\\[0.3em]&\leq \left({\frac {\left|x\right|}{{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}}}+{\frac {\left|a\right|}{{\sqrt {5+x^{2}}}+{\sqrt {5+a^{2}}}}}\right)\cdot |x-a|\\[0.3em]&\leq \left({\frac {\left|x\right|}{\sqrt {5+x^{2}}}}+{\frac {\left|a\right|}{\sqrt {5+a^{2}}}}\right)\cdot |x-a|\\[0.3em]&\leq (1+1)\cdot |x-a|\\[0.3em]&\leq 2\cdot |x-a|\\[0.3em]&\quad {\color {Gray}\left\downarrow \ |x-a|<\delta ={\frac {\epsilon }{2}}\right.}\\[0.3em]&<2\cdot {\frac {\epsilon }{2}}=\epsilon \end{aligned}}}
Hence, ${\displaystyle f}$ is a continuous function. | 2020-07-04 18:50:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 242, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9925676584243774, "perplexity": 1097.1309310286128}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655886516.43/warc/CC-MAIN-20200704170556-20200704200556-00427.warc.gz"} |
https://mathematica.stackexchange.com/questions/166013/fit-a-circle-to-a-region-of-an-image | # Fit a circle to a region of an image
I would like to develop a routine that associates a size with images such as that below. My idea is to find the radius of the circle (centered at the origin) that contains (say) 95% of the total density. How can I achieve this? Also, are there better image processing techniques for associating a size to this region?
• This site is about the software Wolfram Mathematica and not image processing. You might want to consider asking this somewhere else. – halirutan Feb 16 '18 at 13:21
If img is your image, you can use Binarize to find a fixed fraction of black pixels:
img = Import["https://i.stack.imgur.com/T76kp.png"];
bin = Binarize[img, Method -> {"BlackFraction", .1}]
Then you can use ComponentMeasurements to find the centroid and radius:
comp = ComponentMeasurements[
"Circularity"}, #Circularity > 0.5 &];
HighlightImage[img, {comp /. (index_ -> {c_, r_, __}) :>
Circle[c, r]}]
Response to comment:
@anderstood asked why ComponentMeasurements finds two components. We can use the Mask measurement to get a binary mask for each component:
HighlightImage[img, Image[#]] & /@
and see that the first component is the black border around the image.
• That was blisteringly fast. Thanks! I'll have a play :) – Tom Feb 16 '18 at 13:39
• Why did you use ColorNegate? Also, would you know why ComponentMeasurements[bin, {"Centroid", "EquivalentDiskRadius"}] returns two circles (also with ColorNegate@bin instead of bin)? – anderstood Feb 16 '18 at 13:45
• @anderstood: I used ColorNegate because ComponentMeasurements searches for white objects on a black background - and the source image is the opposite. The second object is the black border around the image. – Niki Estner Feb 16 '18 at 15:47
Let's look at your data in more detail. The image has a black frame around it which is taken into account! This means, that binarizing using "BlackFraction" as Niki did will lead to an incorrect result.
In fact, if you take this into account, then the frame alone contains more than 5% of your total density. Let's calculate the amount of density we want:
img = RemoveAlphaChannel[
ColorConvert[Import["https://i.stack.imgur.com/T76kp.png"],
"Grayscale"]];
data = ImageData[ColorNegate[img], "Real"];
thresh = .95*Total[Flatten[data]]
(* 5539.33 *)
By removing only the frame, we are well below this threshold
Total[Flatten[ArrayPad[data, -10]]]
(* 4972.88 *)
Let us remove this and calculate the 95% boundary correctly
data = ArrayPad[data, -10];
thresh = .95*Total[Flatten[data]]
(* 4724.23 *)
To create a centered disk that we can use as a mask, we can use DiskMatrix. This can be used with varying radii until we find the correct one that meets the wanted threshold:
densityDifference[r_?NumericQ] :=
Total[Flatten[data*DiskMatrix[r, Dimensions[data]]]] - thresh
FindRoot[densityDifference[r], {r, 50, 100}, Method -> "Secant"] | 2020-04-03 05:50:17 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.40795207023620605, "perplexity": 2326.5081483059334}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370510287.30/warc/CC-MAIN-20200403030659-20200403060659-00370.warc.gz"} |
https://mathoverflow.net/questions/247978/what-spec-like-functors-are-there | # What Spec-like functors are there?
The real spectrum functor is an analog of Spec for partially ordered commutative rings and real closed fields in place of commutative reals and algebraically closed fields. I was hoping that there would also be a nice Spec-like functor for not-necessarily-commutative rings, but this paper seems fairly discouraging about that. Are there other functors that people have studied which are like Spec but from other categories (particularly varieties of algebras, or the category of models of a first-order theory)? More generally, what properties should a functor satisfy for us to consider it Spec-like, and what can then be said about Spec-like functors in general?
• There is a fair menagerie of "spectrum" functors in Peter Johnstone's book Stone Spaces. Aug 22, 2016 at 4:46
• There also is an approach by Yves Diers based on multiadjoints - basically, if there is not a unique universal arrow but every connected component holds one, then some spectrum-like construction is possible. The usual spectrum occurs as every homomorphism to a local ring factors uniquely through exactly one of the localizations at primes. I believe Diers also considered some noncommutative examples Aug 22, 2016 at 6:31
• @მამუკაჯიბლაძე, source? Aug 22, 2016 at 16:03
• The most to the point one is probably Some spectra relative to functors (JPAA 1981). Johnstone wrote "A syntactic approach to Diers' localizable categories", it is in the 1977 Durham Symposium volume "Applications of sheaves" (Springer LNM 753, 466-478). Will try to find some more. Aug 22, 2016 at 17:47
• If you allow Spec to take values in some category of "spaces", one answer is that the dual of a category of ring-like objects can often be regarded as a category of "spaces" directly (e.g. affine schemes)-- so that the identity functor is "Spec-like". For example, if $\mathcal{V}$ is a Cauchy complete symmetric monoidal semiadditive category, then the opposite of the category of monoids in $\mathcal{V}$ is extensive, so "space-like" in at least a weak sense. Aug 25, 2016 at 13:00 | 2022-08-19 23:07:20 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8759274482727051, "perplexity": 829.9504597478057}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573849.97/warc/CC-MAIN-20220819222115-20220820012115-00713.warc.gz"} |
https://hinterm-ziel.de/index.php/2021/07/19/debugging3-debugging-is-like-being-the-detective-in-a-crime-movie-where-you-are-also-the-murderer/ | Featured picture: OpenClipart-Vectors on Pixabay.
One has to add to the title (quoted from a tweet by Filipe Fortes) that the detective suffers from a memory loss. Otherwise, the case could be solved easily. Similarly with debugging: If I only knew what nasty things I have hidden in the source code, I could just remove them – but I simply do not know. In this blog post, we will have a look on what kind of tools one could use to find the skeletons hidden in the closet.
## Debugging with print statement
If you work with the Arduino IDE, you probably have noticed that it is very accessible, but also very limited. In particular, it does not support any kind of debugging tools (at least in the stable version 1.X). So the only way to find out, what is happening in your code, is to throw in some print statements, perhaps showing variable values. This can help to show whether the flow of control and the values of variables are what you expect.
So what do you do if your MCU does not have a serial interface or this interface is used by the very program you want to debug? One option is to signal the state of the program using one (otherwise unused) output pin and switch an LED on or off. This gives you only very limited insights into the inner working of your program, though.
I recommend instead to use one pin as an additional serial output. You can use my TXOnlySerial library. It just needs one pin and uses less flash and RAM than the SoftwareSerial library. However, you need an FTDI adapter in order to be able to interface the serial line to your computer.
Another trick, I use, is to define debugging print statements. When I am done with debugging, I simply define these statements to be empty, which means I do not have to delete all the debug print statements in the end. The definitions go all into an header file (debug.h), which looks as follows:
/* Debug macros */
#ifdef DEBUG
#define DEBDECLARE() TXOnlySerial deb(DEBTX)
#define DEBINIT() deb.begin(19200)
#define DEBPR(str) deb.print(str)
#define DEBPRF(str,frm) deb.print(str,frm)
#define DEBLN(str) deb.println(str)
#define DEBLNF(str,frm) deb.println(str,frm)
#else
#define DEBDECLARE()
#define DEBINIT()
#define DEBPR(str)
#define DEBPRF(str,frm)
#define DEBLN(str)
#define DEBLNF(str,frm)
#endif
In the Arduino sketch, you have to include TXOnlySerial.h, you have to define the constant symbol DEBTX (the pin you will use as the serial output line), have a line with DEBDECLARE(), have DEBINIT() in the setup, and then you can use all the statements above. This could look like as follows:
#define DEBUG // comment out if not needed anymore
...
#define DEBTX 8 // used as debugging output
#include <TXOnlySerial.h>
#include "debug.h"
...
DEBDECLARE();
void setup() {
DEBINIT();
...
}
void loop() {
...
DEBPR(F("loop"));
...
}
The disadvantage of debugging with print statements is that when you want to see the value of another variable, you have to change your sketch, compile, upload and then start the program again. It would be much nicer if you were able to stop at a particular point in the program and then inspect the program state. Perhaps even changing variable values before continuing with the execution. And this is what real debuggers are for.
## Preparing the Arduino IDE 1.X for debugging
If you want to use avr-gdb, the GNU debugger for AVR MCUs, for debugging Arduino sketches, you need first to install it, because it is not any longer part of the Arduino software. On a Mac it is available as a homebrew package, but the current version (July 2021) does not work. I have figured out a way to install a working version of avr-gdb. Under Linux, you can simply load the avr-gdb package (which is called gdb-avr) using your favorite package manager. Under Windows, you have to download the entire AVR toolchain from somewhere, which then contains the debugger. WinAVR is probably a good choice, either in the “official” version, which is more than 10 years old, or a more recent version from Zak’s Electronics Blog ~*.
Then (on all platforms) you need to change some configuration files in order to be able to get access to the ELF files (which contain debugging information) and to change the gcc optimization level from -Os (meaning: optimize for space) to -Og (meaning: optimize in a debug friendly manner). Well, doing this is not strictly necessary, but very advisable. Otherwise, you may get confused because the compiler can rearrange code quite a lot and then you might not be able to stop at a place in the program where you want to stop.
First, you have to add a file platform.local.txt to the directory where the relevant platform.txt resides. If you have installed the board information using the Board Manager, then you find the relevant folder depending on your OS as follows:
• macOS: ~/Library/Arduino15/packages/corename/hardware/avr/version/
• Linux: ~/.arduino15/packages/corename/hardware/avr/version/
• Windows: C:\Users\user\AppData\Local\Arduino15\packages\corename\hardware\avr\version\
Here you should add a platform.local.txt file with the following contents:
recipe.hooks.savehex.postsavehex.1.pattern.macosx=cp "{build.path}/{build.project_name}.elf" "{sketch_path}" recipe.hooks.savehex.postsavehex.1.pattern.linux=cp "{build.path}/{build.project_name}.elf" "{sketch_path}" recipe.hooks.savehex.postsavehex.1.pattern.windows=cmd /C copy "{build.path}\{build.project_name}.elf" "{sketch_path}"
After restarting the IDE, you now get ELF files in your sketch folder when you select Export compiled Binary in the menu Sketch.
Now we want to add an option to your board selection that allows us to switch between -Os and -Og. In order to do so, you add the line
menu.debug=Debug Compiler Flag
to the beginning of the file boards.txt in the same directory were you placed platform.local.txt. Further, for the boards, you want to able to generate debug information, you add the following lines:
board.menu.debug.disabled=Debug Disabled
board.build.extra_flags = {build.debug}
If there is already a line with the extra_flags option for this board, then just append {build.debug}. You have to do that for all boards that you want to debug. When you now restart the IDE, there will be an option Debug Compiler Flag in the Tools menu, which is disabled by default.
Instead of modifying the board.txt file, you can use the Arduino CLI and simply specify an additional built option:
arduino-cli compile --build-property "build.extra_flags=-Og" ...
Now you can generate ELF files with the right optimization level by first selecting Debug Compiler Flag: "Debug Enabled" in the Tools menu and then selecting Export compiled Binary in the menu Sketch. So what can you do with this file? Well, you could load it in the AVR debugger, but then how do you connect to the MCU?
There are basically three different methods, when it comes to embedded debugging:
1. You can use an MCU simulator, connect to the debugger remotely and interact with the simulator.
2. You can use a debug stub, i.e., a library that is linked to the code of the embedded system and that is able to communicate with the remote debugger.
3. You can use a hardware debug probe that connects to the on-chip debugging (OCD) hardware of the MCU and to a debugger on your computer.
### Simulation
For AVR MCUs, professional simulator solutions can be found in Microchip’s IDEs: Microchip Studio 7 (formerly Atmel Studio 7) and MPLAB X, the former only for Windows, the later for Windows, Linux and macOS (but only Intel chips). The interesting thing is that they do not cost anything! Well, you could pay some money for Microchip’s compilers (called XC), but it is no problem at all to install the avr-gcc toolchain instead (I will come back to that in a future blog post).
The two open source alternatives are simulavr and simavr. The first seems less complete and extendable, the latter could have better documentation. I have only played around with the latter, but I have never tried to create new external devices. What seems like a great idea is that you can provide simavr with a stream of input events and you can record states of all outputs and visualize them similar to what you get from a logic analyzer. That could be a great tool for designing unit tests. By the way, the fork of simavr by gatk555 contains a “Getting started guide to simavr” that might be helpful when you first want to use simavr. Based on simavr, there are two extensions that also provide very extensive GUIs: PICSimLab and simutron. I have not tried either of them, but they look interesting.
In general, I am not a great fan of simulator solutions because it is not clear, how much of the real MCU the simulator covers. In other words, something working on the simulator might not work on the real MCU and vice versa. So, why should I bother to invest time to understand how you work with the simulator?
If you are keen to try out a simulator, Lars Kellogg-Stedman has written a blog post how to debug a program that runs on simavr.
### Stub solutions
Debug stubs are the poor men’s hardware debugger. They allow you to debug the software running on the MCU, but you do not have to invest in expensive hardware. However, there is, of course, a price to pay .
A debug stub is a piece of software, in case of the Arduino framework a library, that interacts with a debugger using, e.g., the serial line, which implies that the program you are debugging cannot use the serial line. Furthermore, such stub usually needs a few more resources, such as one of the interrupts of the MCU and/or the watch dog timer. And last, but not least, the stub takes up the scarcest resources of MCUs, namely, RAM and flash. So, there is a substantial price to pay. You can compensate for it by debugging the software on an MCU with more resources, e.g., if you develop for an ATmega328 you can use an ATmega1284 for debugging. However, then you have a problem similar to the one with simulators.
The only debug stub for AVR8 processors that are supported by the Arduino framework I am aware of is avr_debug by Jan Dolinay. It supports ATmega328, ATmega1284, and ATmega2560. Further, in order to run smoothly, it needs a modern optiboot bootloader, which is an excellent idea in any way. I will describe the setup and how to use the stub in the next blog post of this series.
### Hardware Debugger
Hardware debuggers are the premier class in the area of debugging tools. They communicate on one side with a software debugger on you desktop computer and on the other side they control the on-chip debugging (OCD) hardware on the chip. If you want to buy a hardware debugger for the classic ATtinys and the small ATmegas (e.g. 328), then there only a few candidates:
• AVR Dragon (discontinued),
• Atmel-ICE (90€ incl. postage),
• MPLAB Snap (35€ + postage).
The reason that there are only a few alternatives is that this particular family of MCUs uses the debugWIRE interface for debugging. This interface uses only the RESET line as a communication line, which means that you do not have to sacrifice some other precious pin. Other debugging interfaces, e.g., JTAG use much more. Here you have to reserve 5 data lines. In contrast to debugWIRE, which is a proprietary undisclosed protocol, JTAG is standardized and for this reason there exist a number of different and also cheap hardware debuggers and there are also a lot of software solutions, including open source ones. So, if you want to debug larger ATmegas or ARM processors, you have a lot of choices and it appears to be the case that there are also many different alternatives concerning software. There is in particular openOCD, which provides you with a gdbserver interface, i.e., an interface to the gdb debugger, and to many different hardware debuggers. I have not tried it out (yet) and therefore will not write about it; maybe later. Instead, I will write about how you can use the Atmel-ICE debugger on MCUs supporting debugWIRE in one of the upcoming blog posts.
EDIT: Meanwhile, I have implemented an Arduino sketch, called dw-link, that turns an Arduino Uno or Nano into a hardware debugger. | 2022-01-19 22:20:56 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.24956151843070984, "perplexity": 1782.8341616633465}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301592.29/warc/CC-MAIN-20220119215632-20220120005632-00538.warc.gz"} |
https://wiki.seg.org/wiki/User:Ifarley/focusing_analysis | # User:Ifarley/focusing analysis
Title seismic-data-analysis.png Seismic Data Analysis Dip-moveout correction and prestack migration
## Focusing Analysis
The idea of a velocity analysis that is based on differential solutions of the scalar wave equation first was introduced by Doherty and Claerbout).[1] They used the 15-degree finite-difference migration algorithm and worked with single CMP gathers. Gonzalez-Serrano and Claerbout)[2] later extended the wave equation velocity analysis to slant-midpoint coordinates and worked with linearly moveout-corrected CMP gathers. The method discussed here[3] operates in the Fourier transform domain using the exact form of the double-square-root (DSR) operator. Mathematical details of this method are found in Section E.7. Figure 5.4-23 summarizes the main computational steps involved in this migration velocity analysis based on wavefield extrapolation.
1. Starting with the prestack data in midpoint y, off-set h, and two-way event time t in the unmigrated position, represented by the wavefield P(y, h, τ = 0, t) at the surface τ = 0, perform 3-D Fourier transform. The variable τ is associated with the direction of wave extrapolation, and is related to depth z by τ = 2z/υ, where υ is the medium velocity.
2. Specify an extrapolation velocity function that only varies vertically, υ(τ) and apply the extrapolation operator exp(–iωDSRτ/2) to compute the extrapolated wavefield in the transform domain P(ky,kh,τ,ω) from the surface wavefield in the transform domain P(ky,kh,τ = 0,ω).
3. To obtain the zero-offset image, sum over the offset wavenumber, and thus obtain P(ky,h = 0, τ, ω).
4. Apply 2-D inverse Fourier transform to obtain the zero-offset image P(y, h = 0,τ,t). The image below a midpoint y is contained in the t – τ plane.
5. Perform mapping of the variables as described in Section E.6 from τ to υ. The velocity information is given by the envelope of the velocity volume of data P(y, h =0, τ = t, υ).
We now demonstrate the procedure outlined in Figure 5.4-23 using a synthetic data set. Figure 5.4-24 shows two common-offset sections over a number of point scatterers buried in a constant-velocity earth, where υ = 3000 m/s. Using a constant velocity for extrapolation, υe = 3000 m/s, the t – τ image plane was produced for each midpoint. Two such planes corresponding to midpoints 1 and 5 denoted in Figure 5.4-24 are shown in Figure 5.4-25. The υ–τ planes (Figure 5.4-26) then were generated from the t – τ image planes by the mapping procedure described in Section E.7. Peak amplitudes for all events occur at the correct medium velocity (3000 m/s). We expect the diffraction events in Figure 5.4-23 to migrate to the apexes beneath midpoint 1, where the point scatterers are located. Note that in Figure 5.4-25, almost all the energy is in the image plane corresponding to midpoint 1; just five midpoints away, at midpoint 5, the migrated energy is very low.
How do we interpret the t – τ image planes? If we used the true medium velocity in downward extrapolation, then, according to the imaging principle, we would see all the events along the diagonal τ = t, the image line, on the image plane. This happens in Figure 5.4-25, because a 3000-m/s extrapolation velocity was used, which is just the velocity used in generating the model in Figure 5.4-23. Any displacement of peak energy from the t = τ image line means that the velocity value used for downward extrapolation differs from that of the event. This displacement is also the basis for mapping from the tτ image plane to the υ – τ plane by equation (E-77).
This mapping is investigated further with the modeled data set shown in Figure 5.4-27, in which velocity increases with depth. In Figure 5.4-28b, note that the top and middle events fall to the left of the image line suggesting that the velocity used in extrapolation (υe = 3000 m/s) is greater than the velocities associated with these events. The bottom event falls on the image line, implying that its velocity is nearly the same as the extrapolation velocity. These observations are confirmed in the corresponding υ – τ planes in Figure 5.4-29. While true stacking velocity values for the three events are 2700, 2850, and 3000 m/s, the velocities interpreted from Figure 5.4-29b are about 2500, 2800, and 3000 m/s. Thus, the migration-based velocity estimate for the shallow event is in error by approximately 8 percent.
To determine the reason for the velocity error, we will consider a migration-based velocity analysis of our synthetic data example that does not involve the approximate mapping step. Figure 5.3-30a shows a CMP gather from midpoint 1 in the zero-dip region of the depth-variable velocity model associated with the constant-offset sections in Figure 5.4-27. The migration velocity analysis on this gather (Figure 5.4-30b) was done by extrapolating the surface wavefield P(kh, ω, τ = 0) repeatedly with different constant velocities in steps of Δτ = Δt (the sampling rate). The zero-offset trace from each attempt was collected after this effort, abandoning the rest of the migrated CMP gather.
Interpretation of the velocity analysis in Figure 5.4-30b reveals correct stacking velocities for the three events, including the shallowest. Clearly the error observed in Figure 5.4-29 is attributable to the mapping (equation (E-100). Note that the error does not occur because of depth variability of velocity, but instead, because the single extrapolation velocity used differed from the medium velocity. The conventional velocity analysis for midpoint 1 of this model data set is shown in Figure 5.4-30c for comparison. Note the familiar NMO stretching that is apparent in the shallow event. In other respects, both the results (Figures 5.4-30b and 5.4-30c) are comparable.
The departure of an event on the t – τ image plane from the t = τ image line is measured by the quantity Δτ as depicted in Figure 5.4-31a. In some practical implementations, the t – τ image plane is mapped onto the plane of Δτ versus τ as depicted in Figure 5.4-31c to determine the rms velocity υ(τ) for time migration from the extrapolation velocity υe (τ). An event with a velocity error υ (τ) – υe (τ) is represented by an energy maximum either to the left or to the right of the Δτ = 0 line. The δτ(τ) trend can be picked and translated into a velocity trend as depicted in Figure 5.4-31b. This type of analysis has come to be called focusing analysis in the industry (Faye and Jeannaut, 1986). It has been used in some cases erroneously to estimate and update velocity-depth models used for depth migration. The method can only provide plausable velocity update within the framework of time migration.
Figure 5.4-32 is a CMP stack from offshore Texas. A 7000-ft portion (64 midpoints each with 48 offset traces) of the profile was used for migration velocity analysis. For computational efficiency, the data were windowed into 1024-ms time gates with 50 percent overlap. The image planes for one particular midpoint are shown in Figure 5.4-33. Different extrapolation velocities picked from a specified regional velocity function are used in each time gate. The velocity scan used in mapping is then carried out within a corridor around this function. Because different extrapolation velocities are used in successive segments, a given event appears at different values of τ in adjacent time segments.
The resulting velocity analysis for the central midpoint is shown in Figure 5.4-34. In conventional practice, to improve the quality of velocity picks, velocity analyses from a number of neighboring CMP gathers often are summed. Figure 5.4-34c shows the result of stacking velocity analysis for data from the six adjacent CMP gathers indicated in Figure 5.4-34a. For the migration-based method, the υ – τ planes corresponding to these gathers were summed. The result is shown in Figure 5.4-34b. The most obvious difference between the two results is the lack of shallow information in the migration-based υ – τ plane. This shortcoming is attributed to spatial aliasing and lack of long-offset data in the shallow time gate. The problem can be eliminated partly by increasing the length of the time gate used in the velocity analysis. With the shortcut time-windowing approach described above, the shallowest time segment did not include the large-offset data necessary for velocity resolution. Because the events have dip, the derived migration velocities are lower (by up to 4.5 percent) than the velocities derived from the stacking velocity analysis.
The velocity analysis described in this section does not handle lateral variations in velocity. It is based on a Fourier-transform domain formulation with only vertically varying velocity used in extrapolation. This method may be particularly efficient for the dip-corrected velocity estimate needed for time migration.
#### Fowler's velocity-independent prestack migration
We now restate the underlying principle for migration velocity analysis:
Starting with the prestack volume of data P(y, h, t) in midpoint y, offset h and two-way event time t in the unmigrated position, create a velocity cube -- volume of data P(y, υ, τ) in midpoint y, migration velocity υ and two-way event time τ in the migrated position. Within the context of time migration, the output time variable τ is related to depth by way of vertical stretch:
${\displaystyle \tau ={\frac {2z}{\upsilon }}\ }$
Figure 5.4-35 A flowchart of an algorithm to create a migration velocity cube.
Although the velocity cube can be created by means of some of the migration velocity analysis techniques described in this section, a variation of the method by Fowler (1984) is particularly efficient and elegant. First, we review Fowler's method to create the velocity cube. Refer to the moveout equation (5-1) and recall that stacking velocity υstk is dip dependent:
${\displaystyle t^{2}=t_{0}^{2}+{\frac {4h^{2}}{\upsilon _{stk}^{2}}},}$ (1)
where
${\displaystyle \upsilon _{stk}={\frac {\upsilon _{DMO}}{\cos \phi }}.}$ (2)
Use equation (5-11) to establish a relationship between the dip-dependent stacking velocities υstk and the dip-independent DMO velocities υDMO — velocities estimated from dip-moveout-corrected data:
${\displaystyle \upsilon _{stk}={\frac {\upsilon _{DMO}}{\sqrt {1-\left({\frac {\upsilon _{DMO}k_{y}}{2\omega _{0}}}\right)^{2}}}}.}$ (3)
This equation is the basis for Fowler mapping of υstk to υDMO. Note that the process is applied to data in the frequency-wavenumber domain. The Fowler mapping is then followed by constant-velocity Stolt mapping (Sections 4.1 and D.7) to map the DMO velocities υDMO to migration velocities υmig.
Stolt migration of the dip-moveout-corrected data volume in the Fourier transform domain P(ky0; υDMO) involves, first, mapping from ω0 to ωτ for a specific ky by using the dispersion relation of equation (4-24b) recast as
${\displaystyle \omega _{0}={\sqrt {\omega _{\tau }^{2}+{\frac {\upsilon _{mig}^{2}k_{y}^{2}}{4}}}},}$ (4)
where the relationship ωτ = υmigkz/2 is used. The output of this mapping P(kyτ; υmig) is then scaled by the quantity S given by equation (4-24c) recast as
${\displaystyle S={\frac {\upsilon _{mig}}{2}}{\frac {\omega _{\tau }}{\sqrt {\omega _{\tau }^{2}+{\frac {\upsilon _{mig}^{2}k_{y}^{2}}{4}}}}}.}$ (5)
Again, the relationship ωτ = υmigkz/2 is used to obtain equation <xref ref-type="disp-formula" rid="ch05eq5">(5-47)</xref> from equation (4-24c).
Figure 5.4-35 describes a flowchart for creating the migration velocity volume P(y, υmig, τ) by Fowler and Stolt mapping.
1. Start with data P(y, h, t) in coordinates of midpoint y, offset 2h and event time t in the unmigrated position, and create constant-velocity stack (CVS) volume P(y, υstk,tn) using a range of velocities υstk, where tn is the event time after constant-velocity normal moveout correction.
2. Apply 2-D Fourier transform to obtain the CVS cube P(ky, ustk) in the frequency-wavenumber domain, where ky and ω are the Fourier transform variables associated with the variables y and tn.
3. Sort the CVS volume P(ky, υstk) into a set of constant-velocity sections P(ky,ω; υstk).
4. Perform the Fowler mapping based on equation <xref ref-type="disp-formula" rid="ch05eq3">(5-45)</xref> on each of the velocity sections so as to obtain the DMO velocity volume P(ky0; υDMO).
5. Migrate each of the constant-velocity sections of the DMO velocity volume by performing the Stolt mapping based on equations <xref ref-type="disp-formula" rid="ch05eq4">(5-46)</xref> and <xref ref-type="disp-formula" rid="ch05eq5">(5-47)</xref> so as to obtain the migration velocity volume P(Ky, ωτ; υmig).
6. Apply 2-D inverse Fourier transform to obtain the migration velocity volume P(y, τ, υmig).
A variation of Fowler's sequence described above involves creating the CVS cube directly from DMO-corrected data.
1. Start with data P(y, h, t) in coordinates of midpoint y, offset 2h and event time t in the unmigrated position, and apply DMO correction followed by inverse NMO correction.
2. Create constant-velocity stack (CVS) volume P(y, υDMO,t0) using a range of velocities υDMO, where t0 is the event time after constant-velocity normal moveout correction (Figure 5.4-36a).
3. Sort the CVS volume P(y, υDMO,t0) into a set of cosntant-velocity stacked sections P(y,t0; υDMO).
4. Apply 2-D Fourier transform to obtain the CVS cube P(ky, υDMO0) in the frequency-wavenumber domain, where ky and ω are the Fourier transform variables associated with the variables y and t0.
5. Sort the CVS volume P(kyDMO0) into a set of constant-velocity sections P(ky0DMO).
6. Migrate each of the constant-velocity sections of the DMO velocity volume by performing the Stolt mapping based on equations <xref ref-type="disp-formula" rid="ch05eq4">(5-46)</xref> and <xref ref-type="disp-formula" rid="ch05eq5">(5-47)</xref> so as to obtain the migration velocity volume P(y, ωτ, υmig).
7. Apply 2-D inverse Fourier transform to obtain the migration velocity volume P(y, τ, υmig) (Figure 5.4-36b).
The migration velocity volume P(y, τ, υmig) shown in Figure 5.4-36b can be visualized and interpreted to derive a migration velocity field. For spatial consistency, velocity picking should be done on time slices from the migration velocity volume as shown in Figure 5.4-37. The resulting velocity strands shown in Figure 5.4-38a are interpolated to create the migration velocity field shown in Figure 5.4-38b. This velocity field then can be used to extract from the volume the section that corresponds to prestack time migration as shown in Figure 5.4-39. An enlarged view of this section is shown in Figure 5.4-40. Note the excellent imaging of the steeply dipping fault planes which conflict with the gently dipping strata.
Alternatively, the plane of (υmig) for each midpoint y can be inverse transformed to the plane of (h, τ) associated with the common-reflection-point gather derived from prestack time migration. This is then followed by conventional normal-moveout correction and stacking. The resulting section, again, represents the image from prestack time migration.
The data example shown in Figure 5.4-36 demonstrates the use of the migration velocity volume in deriving a high-fidelity image of fault planes from prestack time migration (Figure 5.4-40). The data example shown in Figure 5.4-41 demonstrates the use of the migration velocity volume in imaging steeply dipping event. The migration velocity volume was created using the procedure described above. Specifically, the DMO-corrected CMP gathers were first NMO-corrected using a range of constant velocities and a CVS volume was created. Next, each constant-velocity stacked section was migrated using the Stolt method and the constant velocity associated with that section. The resulting migration velocity volume is shown in Figure 5.4-41. Note the vertical variation in velocities on the end-on view of the volume that represents the plane of velocity versus time.
The migration velocity volume is interpreted by picking the primary velocity trend from selected time slices as shown in Figure 5.4-42a. Note the lateral variation in velocities which is captured by continuous picking along the midpoint axis. By interpolating the velocity strands resulting from the interpretation of selected time slices (Figure 5.4-42a), an rms velocity surface is generated. The picked velocity strands are shown in Figure 5.4-42b embedded within the migration velocity volume, and the rms velocity field is shown in Figure 5.4-43a as a color-coded surface extracted from within the migration velocity volume. A quality control of the rms velocity surface can be made by intersecting it with the cross-sections of the migration velocity volume at selected CMP locations as shown in Figure 5.4-43b.
The prestack time-migrated section is a by-product of the migration velocity analysis described here. Specifically, the image surface associated with the prestack time migration is obtained by sculpting the amplitudes from within the migration velocity volume over the rms velocity surface as shown in Figure 5.4-44a. The conventional 2-D display of the prestack time-migrated section is then created by simply collapsing the sculpted image surface onto a 2-D plane (Figure 5.4-44b). A close-up view of the prestack time-migrated section shows accurate imaging of the steeply dipping event (Figure 5.4-45).
## Exercises
Exercise 1
Consider the application of DMO correction to data referred to a floating datum represented by a smooth form of an irregular topographic surface and data referenced to a flat datum below. Which DMO correction would have more effect on the data?
Exercise 2
Refer to Fowler's velocity-independent prestack migration technique for migration velocity analysis described in Section 5.4. Suppose you have transformed the prestack data from offset to velocity space using 30 constant velocity values from 2000 m/s to 4900 m/s using an increment of 100 m/s. Can you create additional constant-velocity panels such that the increment is 50 m/s by poststack migration rather than prestack migration or trace interpolation? If so, what velocity would you use for poststack migration to create the panel for 3050-m/s velocity.
Exercise 3
Derive Bancroft's equivalent offset equation (E-72b) from the nonzero-offset traveltime equation (E-67).
Exercise 4
Suppose you have two events with conflicting dips of the same magnitude, but in opposite directions, associated with a reflector within a fault block and the fault plane itself. Can these two events be distinguished on a velocity spectrum computed from a CMP gather before DMO correction at a location just above the surface point where the two events intersect one another? | 2020-07-12 15:13:43 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 6, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6291425824165344, "perplexity": 2945.856908818158}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593657138752.92/warc/CC-MAIN-20200712144738-20200712174738-00338.warc.gz"} |
http://mathhelpforum.com/geometry/3943-help-print.html | # Help!
• July 1st 2006, 06:50 PM
laser2302
Help!
Find the speed in m/s of a point on a bicycle rim of diameter 88 cm, making 200 revolutions per minute.
and
Eight tins of paint are required for a surface of an area 50 m{square} if 4 coats of paint are applied. Find the number of tins of paint needed for an area of 70 m{square} if 5 coats of paint are applied.
• July 1st 2006, 09:04 PM
Quick
Quote:
Originally Posted by laser2302
Find the speed in m/s of a point on a bicycle rim of diameter 88 cm, making 200 revolutions per minute.
What you need to find first, is what is the circumference of the rim, which is $88\pi$
Now you need to find how many revolutions per second the rim goes, which is the amount rpm divide by 60 $200\div60=3\frac{1}{3}$
now we multiply the circumference by the rps and we get the answer...
$88\pi\times3\frac{1}{3}$ that gives us cm/s so we divide by 100 to get m/s
$\frac{88\pi\times3\frac{1}{3}}{100}$ and that is the answer in m/s (I don't have a calc. handy so you need to solve)
Quote:
Originally Posted by laser2302
and
Eight tins of paint are required for a surface of an area 50 m{square} if 4 coats of paint are applied. Find the number of tins of paint needed for an area of 70 m{square} if 5 coats of paint are applied.
If 8tins are required for 4 coats, than only 2 tins are required for 1 coat. If 2 tins are required to cover 50m^2, than 1 tin will cover 25m^2.
Now we need to figure out how many tins are required to cover 70m^2 with 1 coat. So we divide 70 by 25, $70m^2\div25m^2=2\frac{2}{5}$
so 2 and 2/5 tins are required for 1 coat, so now we multiply by 5 to get all 5 coats.
$2\frac{2}{5}\times5=12$ therefore, 12 tins are needed. | 2014-12-20 01:03:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7496119737625122, "perplexity": 1504.8132445390752}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-52/segments/1418802769305.158/warc/CC-MAIN-20141217075249-00152-ip-10-231-17-201.ec2.internal.warc.gz"} |
http://latex.org/forum/viewtopic.php?f=45&t=30705&sid=0308ce2837edfcc9873d12b7a35ce617 | ## LaTeX forum ⇒ Graphics, Figures & Tables ⇒ Plotting non continuous data sets using psgraph
Information and discussion about graphics, figures & tables in LaTeX documents.
thibaulth
Posts: 1
Joined: Tue Dec 05, 2017 5:58 pm
### Plotting non continuous data sets using psgraph
Hi all,
I've been doing my graphs until now using Igor Pro. But when I printed my manuscript, some of the output didn't go as I planned (bold fonts where it should be normal, linewidth going crazy and so on, ...)
Therefore, as I planned my defense, I started plotting graphs using PS tricks and pst-plot. I now want to expand it further and plot all my thesis graphs using PS tricks.
I have plotted my XY graphs without any problem as my data output correspond to simple variations (ie. continuous function of time).
I am now trying to plot a surface plot. I extract the data from Igor Pro (too heavy data to compute using LaTeX) to get the contour. This yields a two column file with x and y coordinates.
Here is the problem: my data looks like this:
IWC=0.01.x IWC=0.01.y16.0001 8.481115.9419 8.48252[...] [...]33.6076 9.7000333.3442 9.70005 16.4308 8.470716.3332 8.47344
The problem lies in this jump. When I compile the psgraph, a line is drawn from (33.34,9.7) to (16.43,8.47)... This wouldn't be the case is I used Gnuplot.
Here is what I have for the source code:
\documentclass[margin=3pt]{standalone} \usepackage{pstricks-add, pst-eps}\usepackage{pict2e}\usepackage{pstricks,pst-plot}\usepackage{pst-poly}\usepackage{multido}\usepackage{lscape}\usepackage{pgfplots} \begin{document}%===========================================%Open files%===========================================\readdata[ignoreLines=1]{\data}{IWC_0.01.txt}\readdata[ignoreLines=1]{\dataA}{IWC_0.2.txt}\readdata[ignoreLines=1]{\dataB}{IWC_0.5.txt}\readdata[ignoreLines=1]{\dataC}{IWC_1.txt}\readdata[ignoreLines=1]{\dataD}{IWC_2.txt} %===========================================%Set Graph%===========================================\psset{llx=-35pt,urx=5pt,lly=-28.5pt,ury=5pt,% xAxisLabel=\Large Time (s),xAxisLabelPos={c,-24pt},% yAxisLabel=\Large Altitude (km),yAxisLabelPos={-33.5pt,c}}\pstScalePoints(1,1){}{}\begin{psgraph}[linewidth=.5pt,axesstyle=frame,xticksize=-3pt 0% 0 %0 ,Dx=10,yticksize=-3pt 0% 0 %80 ,Dy=2](00,0)(80,10){10cm}{6cm} %===========================================%Plot%===========================================% ====== Ice Water Content contour lines \listplot[linecolor=blue,linewidth=.5pt]{\data} \listplot[fillstyle=solid,fillcolor=blue!20,linestyle=none]{\dataA} \listplot[fillstyle=solid,fillcolor=blue!40,linestyle=none]{\dataB} \listplot[fillstyle=solid,fillcolor=blue!60,linestyle=none]{\dataC} \listplot[fillstyle=solid,fillcolor=blue!100,linestyle=none]{\dataD} \end{psgraph} \end{document}
Is there a way to make the data plot non-continuous using LaTeX? (I have changed my data file on the first graph to have what I want, given that I'm learning the correct use of psgraph, but for the remaining graphs, each containig contours, I'd rather have an automated way of doing this). | 2018-01-19 09:26:28 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7218451499938965, "perplexity": 3046.761562672517}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084887849.3/warc/CC-MAIN-20180119085553-20180119105553-00226.warc.gz"} |
http://m.shqualitywell.com/n1808970/The-General-Recommendations-of-the-case-stem-length-of-the-thermometer.htm | The General Recommendations of the case & stem length of the thermometer
# The General Recommendations of the case & stem length of the thermometer
Case (Dial) Size:
Recommended Case(Dial) Sizes:1”, 1-3/4”,2”,3”,5”
Stem Length：
Industrial Applications:2-1/2”,4”,6”,9”,12”,15”,18”,24”
Note: Nonstandard stem lengths may be required for special applications. Soe ranges may not be available whith stem lengths shorter tan 4 inch. Where applications require a thermowell, stem length and diameter must be compatible with the thermowell.
Laboratory Application:
Typical stem length for the laboratory application of the bimetallic actuated thermometer is 8 inch.
Pocket Thermometer:
Typical stem length for the laboratory application of the bimetallic actuated thermometer is 5 inch. | 2022-12-03 08:40:52 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8646556735038757, "perplexity": 10573.417641734386}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710926.23/warc/CC-MAIN-20221203075717-20221203105717-00461.warc.gz"} |
https://quant.stackexchange.com/questions/25586/compound-interest-calculator-solving-for-time-with-deposits/25587 | # Compound interest calculator solving for time with deposits [closed]
I am attempting to solve a compound interest calculation for time given
Principal = 100
Time(years) = t
Rate(per year) = 8%
Deposit(per month) = 5
Total = 300
I can find solving for time without regular deposits. Can anyone help me out with a source or a workup of how to do this calculation with a regular monthly deposit?
http://www.mathportal.org/calculators/financial-calculators/compound-interest-calculator.php?formId=0&val1=4500&val2=7&val3=9&combo1=1&combo2=2
I'm attempting to use this calculator but it does not allow for deposits. That is essentially what i need but with deposits.
Look this is just a geometric sum:
Assume interest is paid monthly at rate $r = 0.08/12$ (you can use the exact monthly equivalent if you want) and let $x_n =$total after $n$ months (including that month's interest and deposit).
So $x_0= 100$ and $x_{n+1} = x_n(1+r) + d$, where $d = 5$ is your deposit amount (added at the end of the month).
Applying the recursion repeatedly you see $$x_n = x_0(1+r)^n + d(1+r)^{n-1} + \ldots + d = x_0(1+r)^n + d\sum_{j=0}^{n-1}(1+r)^j.$$ After applying the geometric sum formula you get $$x_n = x_0(1+r)^n + \frac{d}{r}((1+r)^n-1).$$
Thus (assuming my metaphorical back-of-envelope calculation is right): $$(1+r)^n = \frac{x_n + \gamma}{x_0 + \gamma},$$ where $\gamma = d/r$. Take log of both sides to get $$n = \log\left(\frac{x_n + \gamma}{x_0 + \gamma}\right)/\log(1+r).$$ I.e. to get the number of months to reach 300 you would take $x_n =300$ here and round $n$ upto the next integer.
It is 32 months for your example.
You could easily modify this so that your deposits increase over time, interest is paid yearly, etc. | 2021-02-25 17:03:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7180914878845215, "perplexity": 702.7249081470956}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178351374.10/warc/CC-MAIN-20210225153633-20210225183633-00272.warc.gz"} |
https://www.coursehero.com/file/6417734/Solutions-for-assignment-2/ | Solutions for assignment 2
# Solutions for assignment 2 - Solutions for assignment 2 1...
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Section 2.2 Question 9: (a) (z-5i)^2 is a polynomial, hence continuous everywhere, and thus on e can simply substitute in the limit to obtain ((2+3i)^2 - 5i)^2 = -8i. (c) This is an indeterminate form; one can either use L'hopital's rule (which is valid for complex functions, although we will not prove this until much later), or factorize the numerator as z^2 + 9 = (z-3i)(z+3i) and simplify the limit to z+3i. Now the function is continuous and we can simply substitute to obtain 6i. (e) The function in the limit can be simplified to 2z_0 + /\z, which (being a polynomial in /\z) converges to 2z_0 as /\z tends to zero. Alternatively, one can note that this is precisely the derivative of z^2 at z_0, which is equal to 2z_0 since the differentiation of polynomials proceeds in complex analysis in exactly the same manner as in real analysis. Question 15: The limit along the real approach y=0 is 2i, and the limit along the imaginary approach x=0 is 2i+1, which are not equal, so the full limit does not exist. Section 2.3 Question 4: (a) The expression in the limit of Definition 4 simplifies to /\x / /\ z, where /\z = /\x + i /\y is the Cartesian form of /\z. Taking the limits along the real and imaginary approaches shows that this limit does not exis t. (b) is proven in the same way as (a).
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## This note was uploaded on 09/19/2011 for the course MATH 132 taught by Professor Grossman during the Spring '08 term at UCLA.
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Ask a homework question - tutors are online | 2017-04-26 16:23:23 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9034813046455383, "perplexity": 1156.0596320312447}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917121453.27/warc/CC-MAIN-20170423031201-00233-ip-10-145-167-34.ec2.internal.warc.gz"} |
https://papers.nips.cc/paper/2019/file/a6a767bbb2e3513233f942e0ff24272c-Reviews.html | NeurIPS 2019
Sun Dec 8th through Sat the 14th, 2019 at Vancouver Convention Center
Paper ID: 4134 An adaptive nearest neighbor rule for classification
### Reviewer 1
After reading the author feedback, I am increasing my score as the authors have largely addressed my main concern of comparison/relating to existing locally adaptive NN estimators. The authors address this concern on the theory side. I still think additional numerical experiments to compare the proposed method with other adaptive NN approaches would strengthen the paper and be beneficial to practitioners (and could possibly lead to interesting theoretical questions if, for instance, there are some surprising performance gaps). Thanks to the authors for a well-thought-out response to the reviews! * * * * * Original review below * * * * * I found this paper very well-written and clear. The analysis is quite clean and elegant due to the newly introduced local notion of "advantage" as an alternative to the usual local Lipschitz conditions. The paper, however, does not discuss enough related work, especially adaptive nearest neighbor and kernel methods based on Lepski's method (for example, the adaptive k-NN method by Kpotufe (2011)). I realize that much of these existing methods are specified in the context of regression rather than classification but it's straightforward casting the regression results in terms of classification (by thresholding on the regression function, taken to be the conditional probability function). Mainly I would like to better understand fundamentally what is different about the strategy used by Kpotufe (2011) (as well as Lepski in many papers that are more focused on automatic bandwidth selection for kernel methods) vs your approach (I realize the theoretical analysis is quite different, and the proof ideas using advantage are much cleaner). Separately, there's also the $k^*$-NN approach for adaptively choosing k per data point (Anava and Levy 2016), for which theoretical analysis for a toy case is presented in Section 3.7 of Chen and Shah (2018). Is there some toy setup for which all three approaches could be analyzed to compare them theoretically? Separately, there are various works on adaptive k-NN classification for which the metric rather than k is adaptive (e.g., Short and Fukunaga 1981; Hastie and Tibshirani 1996; Domeniconi, Peng, and Gunopulos 2002; Wang, Neskovic, and Cooper 2007, Kpotufe, Boularias, Schultz, and Kim 2016). In experimental results, perhaps it would be helpful to benchmark against existing locally adaptive methods for choosing k (Kpotufe 2011; Anava and Levy 2016) as well as some simple sample splitting/cross validation kind of approach to selecting a single k, preferably on more datasets. In particular, it would be helpful to practitioners to understand how these locally adaptive nearest neighbor methods compare with one another, and when using a locally adaptive k yields a significant accuracy improvement over a global choice of k.
### Reviewer 2
The algorithm is new. A novel concept was introduced to the theoretical analysis.
### Reviewer 3
My main concern with this this paper is that, although the authors provide an algorithm that chooses k adaptively, they have introduced another tuning parameter \delta, and there is no guidance on how to choose this in practice. In the introduction the authors claim that their results are instance optimal'. However, there is no result that proves the optimality of the bounds. The rates of convergence' provided are not rates of convergence in the usual sense. They provide a threshold above which n must be for a given point x to be classified correctly with high probability. This says nothing about the average behaviour over test points x. The comparison with the standard k-NN classifier in Section 4.2 does not seem fully convincing to me. [CD14] show that the points in \mathcal{X}_{n,k}' are likely to be correctly classified by k-NN, but it is not clear whether there could be other points that are also likely to be correctly classified by k-NN. I think that more work needs to be done here to compare the methods convincingly. =========================================================== UPDATE (after author rebuttal): Thank you to the authors for carefully reading my review. I still feel that there should be more comparison to the literature on kNN classification, but my other concerns have now been mostly addressed. I have changed the score for the submission. | 2022-08-12 03:38:29 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6345487833023071, "perplexity": 725.6249397810084}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882571538.36/warc/CC-MAIN-20220812014923-20220812044923-00111.warc.gz"} |
https://www.studypug.com/nz/nz-year7/linear-equations/solve-linear-equations-using-distributive-property | # Solving two-step linear equations using distributive property: $\;a\left( {x + b} \right) = c$
### Solving two-step linear equations using distributive property: $\;a\left( {x + b} \right) = c$
Distributive property is an algebra property that we use all the time! When you see equations in the form of a(x+b), you can transform them into ax+ab by multiplying the terms inside a set of parentheses. In this section, we will make use of this property to help us solve linear equations.
#### Lessons
• 1.
Solve the equation using model.
a)
$4\left( {x + 1} \right) = 12$
b)
$2\left( {x - 3} \right) = 8$
• 2.
Solve.
a)
$3\left( {x - 9} \right) = 45$
b)
$7\left( {10 + x} \right) = 14$
c)
$- 15 = 3\left( {x - 6} \right)$
d)
$- 22 = 11\left( {x + 13} \right)$
• 3.
John has a round table with a circumference of 314.16 cm, but it is too big for his new home. So, he cut off a 10 cm wide border around the edge.
a)
Write the equation that represents the situation.
b)
What is the circumference of the table now? Round your answer to two decimal places. | 2018-02-25 01:44:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 34, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4671017527580261, "perplexity": 737.3400965653386}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 20, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891816083.98/warc/CC-MAIN-20180225011315-20180225031315-00390.warc.gz"} |
http://worksheets.tutorvista.com/multiplication-and-division-word-problems.html | To get the best deal on Tutoring, call 1-855-666-7440 (Toll Free)
# Multiplication and Division Word Problems
Multiplication and Division Word Problems
• Page 1
1.
Charlie bought 2 ice-creams for $2. What is the cost of each ice-cream? a.$1 b. $2 c.$1.5 d. $0.5 #### Answer: (a) Correct answer : (1) 2. Justin bought 3 pencils for$6. What is the cost of each pencil?
a. $2.5 b.$2 c. $1.5 d.$3
Correct answer : (2)
3.
Find the number of toys in each shelf, if 28 toys are equally arranged in 4 shelves.
a. 6 b. 7 c. 9 d. 4
Correct answer : (2)
4.
At a factory 3 wheels are used for each tricycle. How many tricycles can be made using 93 wheels?
a. 30 b. 32 c. 33 d. 31
Correct answer : (4)
5.
The baker puts 8 cookies in a tray. How many trays are needed for 64 cookies?
a. 8 b. 9 c. 72 d. 512
Correct answer : (1)
6.
Every day Bob puts $3 in his piggy bank. If he started on Sunday, what would be the total amount in his bank on Friday? a.$24 b. $16 c.$15 d. \$18
Correct answer : (4)
7.
Pam harvested 36 cabbages in 6 rows in her garden. If there were same numbers of cabbages in each row, how many cabbages were there in each row?
a. 6 b. 42 c. 7 d. 30
Correct answer : (1)
8.
Mathew has 4 boxes of crayons. There are 5 crayons in each box. How many crayons does Mathew have?
a. 20 b. 9 c. 15 d. 25
Correct answer : (1)
9.
William bought 6 apples. Each apple was priced at 4¢. How much did he pay for the apples?
a. 18¢ b. 10¢ c. 24¢ d. 20¢
Correct answer : (3)
10.
Ashley wants to plant 6 rows of lilies in her garden. She plants 7 in a row. How many plants does she need in all?
a. 49 b. 13 c. 56 d. 42 | 2017-07-21 00:35:15 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2450353056192398, "perplexity": 5309.303382669061}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423629.36/warc/CC-MAIN-20170721002112-20170721022112-00114.warc.gz"} |
https://foxyreign.com/category/analytics/ | # Data Mining and Data Scientist Salary Estimates in the Philippines
## Motivation
As I am going back to the Philippines to pursue further studies in Statistics, it intrigues me if Data Mining and Data Science are catching up. I am seeing some positions in jobsearch websites such as Jobstreet so as a data miner, I extracted the relevant job openings that are related to the key phrases:
• Data Mining; and
• Data Scientist.
These may look too specific but this is just a quick draft, anyway. Also, I did not include Data Analyst as this scopes a broader job scope diversity than the two mentioned. Also any intensive text extraction using basic Information Retrieval methods is not used.
Warning: The result of the models should not be used to provide recommendations as data the is collected using a convenience sample without performing accuracy tests, only k-fold cross validations against the training set when CART is used.
## Data Set
The data is collected manually by searching for relevant job openings active today, 22 May, 2015. I have an assumption that that the data set is relatively small, and so less than 30 positions is returned. Pre-processing is done externally, in Excel, to remove currency prefix, i.e. PHP and text in experience, etc.
library(RCurl)
jobstreet <- getURL("https://raw.githubusercontent.com/foxyreign/Adhocs/master/Jobstreet.csv", ssl.verifypeer=0L, followlocation=1L) # Download from github
writeLines(jobstreet, "Jobstreet.csv")
df <- read.csv('Jobstreet.csv', head=T, sep=",") # Load dataset
df <- na.omit(df) # Exclude missing data
summary(df) # Summarize
## Expected.Salary Experience Education
## Min. : 11000 Min. : 0.000 Min. :1.000
## 1st Qu.: 19000 1st Qu.: 2.000 1st Qu.:2.000
## Median : 28000 Median : 4.000 Median :2.000
## Mean : 35016 Mean : 4.919 Mean :2.179
## 3rd Qu.: 40000 3rd Qu.: 8.000 3rd Qu.:2.000
## Max. :130000 Max. :20.000 Max. :4.000
##
## Specialization Position
## IT-Software :30 Data Mining :72
## - :17 Data Scientist:51
## IT-Network/Sys/DB Admin:14
## Actuarial/Statistics :12
## Banking/Financial : 8
## Electronics : 8
## (Other) :34
As mentioned, there are only approximately 120 job applicants which applied for these two grouped positions. Since the data does not mention if an applicant applied for more than one position, I assume that these are distinct records of applicants per position and/or position group, Data Mining and Data Scientist.
## Variables
1. Expected.Salary – numerical. The expected salary of each applicant based on their profile.
2. Experience – ordinal but treated as numerical for easier interpretation in the later algorithms used. This is the years of work experience of the applicant.
3. Education – categorical; not used in the models because of extreme unbalance in proportions. This is labelled as:
• 1 – Secondary School
• 2 – Bachelor Degree
• 3 – Post Graduate Diploma
• 4 – Professional Degree
4. Specialization – categorical; not used in this analysis.
5. Position – categorical. Data Mining or Data Scientist
6. Education.Group – categorical. Additional variable to bin the years of experience.
# Categorize education variable
df$Education <- factor(df$Education, levels = c(1,2,3,4),
labels=(c("Secondary Sch", "Bach Degree",
"Post Grad Dip", "Prof Degree")))
# Bin years of experience
df$Experience.Group <- ifelse(df$Experience < 3, "3 Years",
ifelse(df$Experience < 5, "5 Years", ifelse(df$Experience < 10, "10 Years", "+10 Years")))
df$Experience.Group <- factor(df$Experience.Group,
levels=c("3 Years", "5 Years", "10 Years", "+10 Years"))
# Drop variables
df <- df[, !(colnames(df) %in% c("Education","Specialization"))]
# Subsets positions
mining <- subset(df, Position == "Data Mining")
scientist <- subset(df, Position == "Data Scientist")
## Distribution
As expected, Data Scientists have a higher expected salary although this is so dispersed that even if I compare these two using a t-test assuming heteroskedastic distribution, there is a significant difference between the averages expected salaries of the two positions.
require(ggplot2)
require(scales)
# Boxplot
ggplot(df, aes(x=factor(0), y=Expected.Salary, fill=Experience.Group)) +
facet_wrap(~Position) + geom_boxplot() + xlab(NULL) +
scale_y_continuous(labels = comma) +
theme(axis.title.x=element_blank(),
axis.text.x=element_blank(),
axis.ticks.x=element_blank(),
legend.position="bottom")
Distribution of Expected Salaries
# T-test
t.test(Expected.Salary ~ Position, paired = FALSE, data = df)
##
## Welch Two Sample t-test
##
## data: Expected.Salary by Position
## t = -3.3801, df = 68.611, p-value = 0.001199
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
## -24086.501 -6205.983
## sample estimates:
## mean in group Data Mining mean in group Data Scientist
## 28736.11 43882.35
# Median expected salaries of Data Mining vs Data Scientist
c(median(mining$Expected.Salary), median(scientist$Expected.Salary))
## [1] 25000 30000
Come on fellow data enthusiasts, you should do better than this! The difference of their medians is just 5,000 PHP. In my honest opinion, these center values are way below based on the prospective demand of shortage of these people who can understand data in the next 10 years.
## Regression
The intercept is not included in the model because I want to see the contrast between Data Mining and Data Scientist although I already computed it beforehand. Besides, though the linear regressio model shows significant value, but when doing diagnostics, linear approach is not appropriate because the residual errors are not random and depict a funnel shape based on their errors. The quantitative measures of regressions for significance cutoff are: $r_{adj}^{2}>0.80, p<0.05$
The regression output coefficients are interpreted as follows:
$$y = \beta_{0}(12,934.9) + \beta_{1}(3,336.3) + \beta_{2}$$
# Estimate coefficients of linear regression model
summary(lm(Expected.Salary ~ Experience + Position-1, data=df))
## Call:
## lm(formula = Expected.Salary ~ Experience + Position - 1, data = df)
##
## Residuals:
## Min 1Q Median 3Q Max
## -45312 -11123 -1280 6877 66688
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## Experience 3336.3 446.3 7.476 1.38e-11 ***
## PositionData Mining 12934.9 3024.3 4.277 3.83e-05 ***
## PositionData Scientist 26612.0 3455.8 7.701 4.27e-12 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 18350 on 120 degrees of freedom
## Multiple R-squared: 0.8136, Adjusted R-squared: 0.809
## F-statistic: 174.6 on 3 and 120 DF, p-value: < 2.2e-16
# Scatter plot
ggplot(df, aes(x=Experience, y=Expected.Salary)) +
geom_point(aes(col=Experience.Group)) +
facet_wrap(~Position) +
scale_y_continuous(labels = comma) +
stat_smooth(method="lm", fullrange = T) +
theme(legend.position="bottom")
Scatterplot of Expected Salary per Year of Experience
# Diagnose LM
par(mfrow=c(1,2))
plot(lm(Expected.Salary ~ Experience + Position-1, data=df), c(1,2))
Linear Regression Estimates and Diagnostics
## CART
Information Gain is used to divide the nodes based on weighted average entropy as linear regression does not do well with the data set. Of course, years of experience is more influential than the position.
Looking at the estimated salaries from the printed tree, applicants who have years of experience lower than 1.5 are approximately expecting 17,000 PHP. While those who applied for Data Mining jobs with more than 6.5 years of experience are expecting 66,000 PHP on average.
require(rpart)
require(rattle)
cart <- rpart(formula = Expected.Salary ~ Experience + Position,
data = df,
parms = list(split = "information"), # Uses information gain
model = T) # Retains model information
# Plot tree
layout(matrix(c(1,2,3,4), nrow = 1, ncol = 2, byrow = TRUE), widths=c(1.5,2.5))
barplot(cart\$variable.importance,
cex.names = 0.6, cex.axis = 0.5,
sub = "Variable Importance")
fancyRpartPlot(cart, main=NULL, sub=NULL)
Decision Tree using CART and Variable Importance
# Estimates
print(cart); printcp(cart)
## n= 123
##
## node), split, n, deviance, yval
## * denotes terminal node
##
## 1) root 123 66101970000 35016.26
## 2) Experience< 4.5 64 7326938000 23781.25
## 4) Experience< 1.5 27 520666700 16888.89 *
## 5) Experience>=1.5 37 4587676000 28810.81
## 10) Position=Data Mining 24 1055625000 24375.00 *
## 11) Position=Data Scientist 13 2188000000 37000.00 *
## 3) Experience>=4.5 59 41933560000 47203.39
## 6) Position=Data Mining 33 9501333000 37333.33 *
## 7) Position=Data Scientist 26 25137120000 59730.77
## 14) Experience< 6.5 8 5237500000 46250.00 *
## 15) Experience>=6.5 18 17799610000 65722.22 *
##
## Regression tree:
## rpart(formula = Expected.Salary ~ Experience + Position, data = df,
## model = T, parms = list(split = "information"))
##
## Variables actually used in tree construction:
## [1] Experience Position
##
## Root node error: 6.6102e+10/123 = 537414370
##
## n= 123
##
## CP nsplit rel error xerror xstd
## 1 0.254780 0 1.00000 1.00888 0.19245
## 2 0.110361 1 0.74522 0.81057 0.14709
## 3 0.033563 2 0.63486 0.77285 0.13292
## 4 0.031769 3 0.60130 0.76228 0.13052
## 5 0.020333 4 0.56953 0.74207 0.12969
## 6 0.010000 5 0.54919 0.70258 0.12366
Again, fellow data miners and data scientists, ask for more! You do not realize your worth with the current demand of people who can understand data.
The full paper on this can be downloaded at https://github.com/foxyreign/Adhocs/blob/master/Jobstreet.pdf
# Fitness Goals for 2015
Here I am again gaining back my momentum where I fell off maintaining a healthier lifestyle two years ago. This time, for sure, I will stick to it until I reach the 3rd phase of my target and maintain it. Besides, I will be the loser again if I cannot reach nor maintain this.
Since Wordpress does not allow script embedding, you need to click on my body composition dashboard so that you can interactively look at my data.
I used moving average to smooth the data because I do not feel that a linear trend on a time-series model is not appropriate given that I have very few data points; maybe after three months of tracking, I will include that.
What really concerns me is that I see a statistically significant correlation between the increase in my fat and fat-free muscle mass. In due time, this has to change – let it be flat or better negatively correlated variables. Ill definitely look very muscular if that happens.
# My Facebook Social Network
Last year, Coursera offered a Social Network Analysis course where the instructor is one of the Computational Social Scientist of Facebook, Lada Adamic. Too bad I did not finish this course because it overlapped with another course I was finishing. When this course is offered again, I will definitely complete this since I have to visualize more graphs that deals with the network of textual data using association rule learning.
So here’s my Facebook social network. I wanted to mine more information about my friends but after Facebook retired Oauth 1.0, it really restricted a lot of information that you can mine from your friends. I guess that’s a good thing because of privacy information concerns. So I can only mine the network map though I found a way to mine my textual posts through the archive feature but I have to build some program, maybe in C++ or Python, to extract the data properly.
The size of each label is based on the mixed degree, meaning our mutual friends. The colors of each line the intensity, the purple being the highest, green is middle and blue is the least.
Facebook Social Network
It is fun to see how my friends (nodes) are clustered together based on their distance of connectivity and how each single friend can be interconnected with two clusters, putting their spot in between as it is the average distance between two clusters.
This is not really a particular big network, consisting of 2,500+ nodes (friends) and 55,000+ edges (how each node is connected). Although with a pretty decent PC running at 8GB RAM, Gephi was already performing slow in running ForceAtlas 2.
See the upper left sub giant network? I probably need to clean up my friends list because they do not make sense. These are my Facebook friends who are closely related to each other by using the recommended friend list of Facebook – a behavioral pattern where a group of men who looks at the stance of your half-naked pictures.
I wonder if there’s any platform where I can upload my big network so you can interactively see the nodes and labels.
# My Facebook Social Network
Last year, Coursera offered a Social Network Analysis course where the instructor is one of the Computational Social Scientist of Facebook, Lada Adamic. Too bad I did not finish this course because it overlapped with another course I was finishing. When this course is offered again, I will definitely complete this since I have to visualize more graphs that deals with the network of textual data using association rule learning.
So here’s my Facebook social network. I wanted to mine more information about my friends but after Facebook retired Oauth 1.0, it really restricted a lot of information that you can mine from your friends. I guess that’s a good thing because of privacy information concerns. So I can only mine the network map though I found a way to mine my textual posts through the archive feature but I have to build some program, maybe in C++ or Python, to extract the data properly.
The size of each label is based on the mixed degree, meaning our mutual friends. The colors of each line the intensity, the purple being the highest, green is middle and blue is the least.
It is fun to see how my friends (nodes) are clustered together based on their distance of connectivity and how each single friend can be interconnected with two clusters, putting their spot in between as it is the average distance between two clusters.
This is not really a particular big network, consisting of 2,500+ nodes (friends) and 55,000+ edges (how each node is connected). Although with a pretty decent PC running at 8GB RAM, Gephi was already performing slow in running ForceAtlas 2.
See the upper left sub giant network? I probably need to clean up my friends list because they do not make sense. These are my Facebook friends who are closely related to each other by using the recommended friend list of Facebook – a behavioral pattern where a group of men who looks at the stance of your half-naked pictures.
I wonder if there’s any platform where I can upload my big network so you can interactively see the nodes and labels. | 2017-07-21 08:42:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.27654847502708435, "perplexity": 4578.7333690743335}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549423764.11/warc/CC-MAIN-20170721082219-20170721102219-00481.warc.gz"} |
https://www.jobilize.com/chemistry/course/21-5-uses-of-radioisotopes-nuclear-chemistry-by-openstax?qcr=www.quizover.com&page=1&= | # 21.5 Uses of radioisotopes (Page 2/7)
Page 2 / 7
Radioisotopes can also be used, typically in higher doses than as a tracer, as treatment. Radiation therapy is the use of high-energy radiation to damage the DNA of cancer cells, which kills them or keeps them from dividing ( [link] ). A cancer patient may receive external beam radiation therapy delivered by a machine outside the body, or internal radiation therapy (brachytherapy) from a radioactive substance that has been introduced into the body. Note that chemotherapy is similar to internal radiation therapy in that the cancer treatment is injected into the body, but differs in that chemotherapy uses chemical rather than radioactive substances to kill the cancer cells.
Cobalt-60 is a synthetic radioisotope produced by the neutron activation of Co-59, which then undergoes β decay to form Ni-60, along with the emission of γ radiation. The overall process is:
${}_{27}^{59}\text{Co}+{}_{0}^{1}\text{n}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{}_{27}^{60}\text{Co}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{}_{28}^{60}\text{Ni}+{}_{-1}^{0}\text{β}+2{}_{0}^{0}\text{γ}$
The overall decay scheme for this is shown graphically in [link] .
Radioisotopes are used in diverse ways to study the mechanisms of chemical reactions in plants and animals. These include labeling fertilizers in studies of nutrient uptake by plants and crop growth, investigations of digestive and milk-producing processes in cows, and studies on the growth and metabolism of animals and plants.
For example, the radioisotope C-14 was used to elucidate the details of how photosynthesis occurs. The overall reaction is:
${\text{6CO}}_{2}\left(g\right)+{\text{6H}}_{2}\text{O}\left(l\right)\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{\text{C}}_{6}{\text{H}}_{12}{\text{O}}_{6}\left(s\right)+{\text{6O}}_{2}\left(g\right),$
but the process is much more complex, proceeding through a series of steps in which various organic compounds are produced. In studies of the pathway of this reaction, plants were exposed to CO 2 containing a high concentration of ${}_{\phantom{\rule{0.5em}{0ex}}6}^{14}\text{C}$ . At regular intervals, the plants were analyzed to determine which organic compounds contained carbon-14 and how much of each compound was present. From the time sequence in which the compounds appeared and the amount of each present at given time intervals, scientists learned more about the pathway of the reaction.
Commercial applications of radioactive materials are equally diverse ( [link] ). They include determining the thickness of films and thin metal sheets by exploiting the penetration power of various types of radiation. Flaws in metals used for structural purposes can be detected using high-energy gamma rays from cobalt-60 in a fashion similar to the way X-rays are used to examine the human body. In one form of pest control, flies are controlled by sterilizing male flies with γ radiation so that females breeding with them do not produce offspring. Many foods are preserved by radiation that kills microorganisms that cause the foods to spoil.
Americium-241, an α emitter with a half-life of 458 years, is used in tiny amounts in ionization-type smoke detectors ( [link] ). The α emissions from Am-241 ionize the air between two electrode plates in the ionizing chamber. A battery supplies a potential that causes movement of the ions, thus creating a small electric current. When smoke enters the chamber, the movement of the ions is impeded, reducing the conductivity of the air. This causes a marked drop in the current, triggering an alarm.
## Chemistry end of chapter exercises
How can a radioactive nuclide be used to show that the equilibrium:
$\text{AgCl}\left(s\right)\phantom{\rule{0.2em}{0ex}}⇌\phantom{\rule{0.2em}{0ex}}{\text{Ag}}^{\text{+}}\left(aq\right)+{\text{Cl}}^{\text{−}}\left(aq\right)$
is a dynamic equilibrium?
Introduction of either radioactive Ag + or radioactive Cl into the solution containing the stated reaction, with subsequent time given for equilibration, will produce a radioactive precipitate that was originally devoid of radiation.
Technetium-99m has a half-life of 6.01 hours. If a patient injected with technetium-99m is safe to leave the hospital once 75% of the dose has decayed, when is the patient allowed to leave?
Iodine that enters the body is stored in the thyroid gland from which it is released to control growth and metabolism. The thyroid can be imaged if iodine-131 is injected into the body. In larger doses, I-133 is also used as a means of treating cancer of the thyroid. I-131 has a half-life of 8.70 days and decays by β emission.
(a) Write an equation for the decay.
(b) How long will it take for 95.0% of a dose of I-131 to decay?
(a) ${}_{\phantom{\rule{0.5em}{0ex}}53}^{133}\text{I}\phantom{\rule{0.2em}{0ex}}⟶\phantom{\rule{0.2em}{0ex}}{}_{\phantom{\rule{0.5em}{0ex}}54}^{133}\text{Xe}+{}_{-1}^{\phantom{\rule{0.5em}{0ex}}0}\text{e};$ (b) 37.6 days
what is an atom
An atom is the smallest particle of an element which can take part in a chemical reaction..
olotu
what is isomerism ?
Formula for equilibrium
is it equilibrium constant
olotu
Yes
Olatunde
yes
David
what us atomic of molecule
chemical formula for water
H20
Samson
what is elemental
what are the properties of pressure
Maryam
How can water be turned to gas
VICTOR
what's a periodic table
how does carbon catenate?
condition in cracking from Diesel to petrol
hey I don't understand anything in chemistry so I was wondering if you could help me
i also
Okikiola
I also
Brient
hello
Brient
condition for cracking diesel to form kerosene
Brient
Really?
Isa
yes
Brient
can you tell me
Brient
Brient
what is periodic law
periodic law state that the physical and chemical properties of an element is the periodic function of their atomic number
rotimi
how is valency calculated
How is velency calculated
Bankz
Hi am Isaac, The number of electrons within the outer shell of the element determine its valency . To calculate the valency of an element(or molecule, for that matter), there are multiple methods. ... The valency of an atom is equal to the number of electrons in the outer shell if that number is fou
YAKUBU
what is the oxidation number of this compound fecl2,fecl3,fe2o3
bonds formed in an endothermic reaction are weaker than the reactants but y r these compound stable at higher temperatures
what is a disproportionation reaction | 2019-02-17 23:48:27 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6836680173873901, "perplexity": 1924.8296051118664}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247483873.51/warc/CC-MAIN-20190217233327-20190218015327-00018.warc.gz"} |
http://mathlib.de/en/docs/Complex | # MathLib.Complex()
MathLib provides a large interface for calculations with complex numbers. Complex numbers are implemented as elements of $\hat{\mathbb{C}}$ (this set is basically $\mathbb{C}$ with a point at infinity.) The point at infinity is internally represented as $\infty + \infty i$
## Examples
There are several ways to initialise complex numbers:
## Methods
For a list of properties and methods common to all complex numbers please see the documentation page for the Complex.prototype.
.polar()
Constructs a complex number out of the absolute value and the argument.
.toContentMathML()
Returns a content MathML string representation of the complex field.
.toLaTeX()
Returns a LaTeX string representation of the complex field.
.toMathML()
Returns a presentation MathML string representation of the complex field.
.toString()
Returns a string representation of the complex field.
## Specification
MathLib.Complex accepts two arguments, where the second argument is optional. Any additional arguments will be ignored. The type of both argument should be number.
1. If either argument is NaN, return {re: NaN, im: NaN}.
2. If either argument is infinite, return {re: Infinity, im: Infinity}.
3. If no second argument is provided, set it to +0.
4. Return {re: firstArgument, im: secondArgument}. | 2018-12-11 03:09:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6129242777824402, "perplexity": 2028.1665951957307}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376823550.42/warc/CC-MAIN-20181211015030-20181211040530-00406.warc.gz"} |
https://forum.azimuthproject.org/plugin/ViewComment/17942 | @Jonathan:
> (It seems clear that requiring multiplication on both sides to preserve the order is equivalent to requiring \$$(x, x') \le (y, y') \implies x \otimes y \le x' \otimes y'\$$.
Right! If \$$x\leq x'\$$ and \$$y\leq y'\$$, then we get \$$x\otimes y\leq x\otimes y'\leq x'\otimes y'\$$, which means that it's enough to postulate each one of these two inequalities separately. So indeed the answer to my puzzle is 'yes' in the commutative case for exactly this reason.
Concerning the noncommutative case, I also like your new hunch ;) | 2019-04-24 22:39:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9280747175216675, "perplexity": 682.0783354415861}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-18/segments/1555578663470.91/warc/CC-MAIN-20190424214335-20190425000335-00367.warc.gz"} |
http://zh-wang.github.io/blog/2015/02/25/usaco-prob-broken-necklace/ | # Analysis of PROB Broken Necklace
## Introduction
This is an analysis of PROB Broken Necklace, one of USA Computer Olympiad’s training problems.
Just doing some disposal on my old stuff.
## Description
You have a necklace of N red, white, or blue beads (3<=N<=350) some of which are red, others blue, and others white, arranged at random. Here are two examples for n=29:
1 2 1 2
r b b r b r r b
r b b b
r r b r
r r w r
b r w w
b b r r
b b b b
b b r b
r r b r
b r r r
b r r r
r r r b
r b r r r w
Figure A Figure B
The beads considered first and second in the text that follows have been marked in the picture.
The configuration in Figure A may be represented as a string of b’s and r’s, where b represents a blue bead and r represents a red one, as follows: brbrrrbbbrrrrrbrrbbrbbbbrrrrb .
Suppose you are to break the necklace at some point, lay it out straight, and then collect beads of the same color from one end until you reach a bead of a different color, and do the same for the other end (which might not be of the same color as the beads collected before this).
Determine the point where the necklace should be broken so that the most number of beads can be collected.
## Analysis
Given an input string ‘wwwbbrwrbrbrrbrbrwrwwrbwrwrrb’, we concatenate it with itself, result in ‘wwwbbrwrbrbrrbrbrwrwwrbwrwrrbwwwbbrwrbrbrrbrbrwrwwrbwrwrrb’. This trick, working on handling minus or over-length index is easier.
1. Brute force solution.
For each index $a_i$ as a start, if $a_{i+1}$ can be picked into the longest substring, we pick it then do this again on $a_{i+1}$. Note that, if $a_i$ is w, we must treat it as b and r, one by one. We do N times of picking attempt for each $a_i$. Therefore complexity is O(N2). Just a straight forward solution.
2. DP solution.
We denote substring composed by successive r, which ends at index $i$ to $t_{i,r}$.
Similarly, we denote substring composed by successive b, which ends at index $i$ to $t_{i,b}$.
On the other hand, we denote substring composed by successive r, which starts at index $i$ to $s_{i,r}$.
And substring composed by successive b, which starts at index $i$ to $t_{i,b}$.
For each index $a_i$, $\max \{s_{i,r}, s_{i,b} \} + \max \{t_{i,r}, t_{i,b} \}$ is the length of longest substring if we break it at index $i$. We do this for N times, the greatest one is the answer. Complexity of this algorithm is O(n). | 2019-02-16 01:53:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7905293703079224, "perplexity": 849.7279511048541}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247479729.27/warc/CC-MAIN-20190216004609-20190216030609-00225.warc.gz"} |
https://math.stackexchange.com/questions/1594804/to-find-order-of-permutation?noredirect=1 | # To find order of permutation
Let $\sigma$ be the permutation given by
Is their a short way to do this.Thanks
• Write $\sigma$ as a product of disjoint cycles. – user296602 Dec 31 '15 at 7:42
• I guess this was downvoted because it's a pasted image with multiple choice answers. I don't see a problem with the question. – Matt Samuel Dec 31 '15 at 7:43
• @Sophie try explicitly raising a 4 cycle to the fourth power and you'll see. In general an $n$-cycle has order $n$. The reason the method in the other answer works is because the disjoint cycles commute, so raising the permutation to a power is the same as raising all of the cycles to the same power. – Matt Samuel Dec 31 '15 at 7:57 | 2019-11-12 14:23:16 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8179765343666077, "perplexity": 316.86622497457694}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-47/segments/1573496665573.50/warc/CC-MAIN-20191112124615-20191112152615-00500.warc.gz"} |
https://digitalblackboard.online/python/numpyarrays/array-indexing/ | ## Introduction
Array indexing is the same as accessing an array element. You can access an array element by referring to its index number, just like a Python list. The indices in NumPy arrays start with 0, meaning that the first element has index 0, and the second has index 1, etc.
## The numpy.random.choice() Function
For the purpose of illustrating array indexing, it will be useful to invoke the numpy.random.choice function which generates a random sample from a given 1-D array. The topic on NumPy random functions will be discussed in a later section .
Syntax
The numpy.random.choice() function.
1numpy.random.choice(a, size=None, replace=True, p=None)
Parameter Required? Default Value Description
a ✔️ Yes NA If an ndarray, a random sample is generated from its elements. If an int, the random sample is generated as if it were numpy.arange(a).
size ❌ No None Output shape. If the given shape is, e.g., (m, n, k), then m*n*k samples are drawn. Default is None, in which case a single value is returned.
replace ❌ No True True means sample is drawn with replacement. False means without replacement.
p ❌ No None The probabilities associated with each entry in a. If not given, the sample assumes a uniform distribution over all entries in a.
Example
Generating a random 1-D array using np.random.choice function.
1import numpy as np
2R = np.random.RandomState(32) #set a random seed
3
4a1 = R.choice(10, size=10, replace=False)
5print(a1)
[9 1 2 8 3 0 4 6 5 7]
Note that we have set a seed for the pseudorandom number generator (PRNG) so that readers can reproduce the exact same results given here. You can read more about PRNG here .
The above code generated a 1-D array of 10 random numbers from the array np.arange(10) without replacement. Therefore, the is no repetition among the numbers generated.
Example
Generating a random 2-D array using np.random.choice function.
1a2 = R.choice(3*4, size=(3,4), replace=False)
2print(a2)
[[ 8 10 0 2]
[ 6 7 1 11]
[ 4 9 5 3]]
The above code generated a 3X4, 2-D array of 12 random numbers from the array np.arange(3*4) without replacement.
Example
Generating a random 3-D array using np.random.choice function.
1a3 = R.choice(3*4*5, size=(3,4,5), replace=False)
2print(a3)
[[[21 8 7 25 1]
[53 31 23 44 12]
[ 3 54 37 51 42]
[45 14 22 27 34]]
[[28 57 52 40 17]
[32 9 2 59 39]
[33 15 46 43 24]
[35 58 19 16 41]]
[[ 4 30 6 49 0]
[48 47 29 26 56]
[11 13 38 5 55]
[36 50 20 10 18]]]
The above code generated a 3X4X5, 2-D array of 60 random numbers from the array np.arange(3*4*5) without replacement. Note that 3 frames are shown, each with a 4X5 array.
## Array Indexing for 1-D Arrays
Indexing in NumPy arrays is very similar to Python list indexing. In a 1-D array, we can access the $i^{\text{th}}$ value (counting from zero) by specifying the desired index in square brackets, just as with Python lists.
Example
Accessing elements in a 1-D array with positive indices.
1print(a1[0])
2print(a1[3])
9
8
Positive Index Element 0 1 2 3 4 5 6 7 8 9 9 1 2 8 3 0 4 6 5 7
EXAMPLE Accessing elements in a 1-D array with negative indices.
1print(a1[-1])
2print(a1[-5])
7
0
Negative Index Element -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 9 1 2 8 3 0 4 6 5 7
## Array Indexing for 2-D Arrays
In a 2-D NumPy array, we can access elements using a comma-separated tuple of 2 indices: (row index, column index). For example, for the a2 array, the first index (red) represents the row index (top to bottom) and second index (blue) represents the column index (left to right).
$$\begin{matrix} \textcolor{red}{0}\\ \textcolor{red}{1}\\ \textcolor{red}{2} \end{matrix} \begin{bmatrix} 8 & 10 & 0 & 2\\ 6 & 7 & 1 & 11\\ 4 & 9 & 5 & 3 \end{bmatrix} \\ \begin{matrix} \textcolor{blue}{~~\quad 0} & \textcolor{blue}{~1} & \textcolor{blue}{~2} & \textcolor{blue}{~3} \end{matrix}$$
Let’s take a look at a few examples.
Example
Accessing elements in a 2-D array with positive indices.
1print(a2)
2print(a2[2,3])
[[ 8 10 0 2]
[ 6 7 1 11]
[ 4 9 5 3]]
3
$$\begin{matrix} \textcolor{red}{0}\\ \textcolor{red}{1}\\ \textcolor{red}{2} \end{matrix} \begin{bmatrix} 8 & 10 & 0 & 2\\ 6 & 7 & 1 & 11\\ 4 & 9 & 5 & \colorbox{lightblue}{3} \end{bmatrix} \\ \begin{matrix} \textcolor{blue}{~~\quad 0} & \textcolor{blue}{~1} & \textcolor{blue}{~2} & \textcolor{blue}{~3} \end{matrix}$$
Example
Accessing elements in a 2-D array with negative indices.
1print(a2)
2print(a2[-1,-2])
[[ 8 10 0 2]
[ 6 7 1 11]
[ 4 9 5 3]]
5
$$\begin{matrix} \textcolor{red}{-3}\\ \textcolor{red}{-2}\\ \textcolor{red}{-1} \end{matrix} \begin{bmatrix} 8 & 10 & 0 & 2\\ 6 & 7 & 1 & 11\\ 4 & 9 & \colorbox{lightblue}{5} & 3 \end{bmatrix} \\ \begin{matrix} \textcolor{blue}{~~~\quad \text{-}4} & \textcolor{blue}{~\text{-}3} & \textcolor{blue}{\text{-}2} & \textcolor{blue}{~\text{-}1} \end{matrix}$$
Example
Accessing elements in a 2-D array with mixed positive & negative indices.
1print(a2)
2print(a2[-2,3])
[[ 8 10 0 2]
[ 6 7 1 11]
[ 4 9 5 3]]
11
$$\begin{matrix} \textcolor{red}{-3}\\ \textcolor{red}{-2}\\ \textcolor{red}{-1} \end{matrix} \begin{bmatrix} 8 & 10 & 0 & 2\\ 6 & 7 & 1 & \colorbox{lightblue}{11}\\ 4 & 9 & 5 & 3 \end{bmatrix} \\ \begin{matrix} \textcolor{blue}{~\qquad 0} & \textcolor{blue}{~1} & \textcolor{blue}{~2} & \textcolor{blue}{~~3} \end{matrix}$$
## Array Indexing for 3-D Arrays
In a 3-D NumPy array, we can access elements using a comma-separated tuple of 3 indices: (frame index, row index, column index). For example, for the a3 array, the first index (green) indicates the frame index, the second index (red) represents the row index (top to bottom) and third index (blue) represents the column index (left to right). The ouput is printed frame by frame.
$$\text{\textcolor{darkgreen}{Frame 0}} \quad \begin{matrix} \textcolor{red}{0}\\ \textcolor{red}{1}\\ \textcolor{red}{2}\\ \textcolor{red}{3} \end{matrix} \begin{bmatrix} 21 & 8 & 7 & 25 & 1 \\ 53 & 31 & 23 & 44 & 12 \\ 3 & 54 & 37 & 51 & 42\\ 45 & 14 & 22 & 27 & 34 \end{bmatrix} \\ \begin{matrix} \hspace{2.2cm} \textcolor{blue}{~0} & \textcolor{blue}{~1} & \textcolor{blue}{~~2} & \textcolor{blue}{~~3} & \textcolor{blue}{~4} \end{matrix}$$
$$\text{\textcolor{darkgreen}{Frame 1}} \quad \begin{matrix} \textcolor{red}{0}\\ \textcolor{red}{1}\\ \textcolor{red}{2}\\ \textcolor{red}{3} \end{matrix} \begin{bmatrix} 28 & 57 & 52 & 40 & 17 \\ 32 & 9 & 2 & 59 & 39 \\ 33 & 15 & 46 & 43 & 24 \\ 35 & 58 & 19 & 16 & 41 \end{bmatrix} \\ \begin{matrix} \hspace{2.2cm} \textcolor{blue}{~0} & \textcolor{blue}{~1} & \textcolor{blue}{~~2} & \textcolor{blue}{~~3} & \textcolor{blue}{~4} \end{matrix}$$
$$\text{\textcolor{darkgreen}{Frame 2}} \quad \begin{matrix} \textcolor{red}{0}\\ \textcolor{red}{1}\\ \textcolor{red}{2}\\ \textcolor{red}{3} \end{matrix} \begin{bmatrix} 4 & 30 & 6 & 49 & 0 \\ 48 & 47 & 29 & 26 & 56\\ 11 & 13 & 38 & 5 & 55 \\ 36 & 50 & 20 & 10 & 18 \end{bmatrix} \\ \begin{matrix} \hspace{2.2cm} \textcolor{blue}{~0} & \textcolor{blue}{~1} & \textcolor{blue}{~~2} & \textcolor{blue}{~~3} & \textcolor{blue}{~4} \end{matrix}$$
Example
Accessing elements in a 3-D array with positive indices.
1print(a3)
2print(a3[2,3,1])
[[[21 8 7 25 1]
[53 31 23 44 12]
[ 3 54 37 51 42]
[45 14 22 27 34]]
[[28 57 52 40 17]
[32 9 2 59 39]
[33 15 46 43 24]
[35 58 19 16 41]]
[[ 4 30 6 49 0]
[48 47 29 26 56]
[11 13 38 5 55]
[36 50 20 10 18]]]
50
$$\text{\textcolor{darkgreen}{Frame 0}} \quad \begin{matrix} \textcolor{red}{0}\\ \textcolor{red}{1}\\ \textcolor{red}{2}\\ \textcolor{red}{3} \end{matrix} \begin{bmatrix} 21 & 8 & 7 & 25 & 1 \\ 53 & 31 & 23 & 44 & 12 \\ 3 & 54 & 37 & 51 & 42\\ 45 & 14 & 22 & 27 & 34 \end{bmatrix} \\ \begin{matrix} \hspace{2.2cm} \textcolor{blue}{~0} & \textcolor{blue}{~1} & \textcolor{blue}{~~2} & \textcolor{blue}{~~3} & \textcolor{blue}{~4} \end{matrix}$$
$$\text{\textcolor{darkgreen}{Frame 1}} \quad \begin{matrix} \textcolor{red}{0}\\ \textcolor{red}{1}\\ \textcolor{red}{2}\\ \textcolor{red}{3} \end{matrix} \begin{bmatrix} 28 & 57 & 52 & 40 & 17 \\ 32 & 9 & 2 & 59 & 39 \\ 33 & 15 & 46 & 43 & 24 \\ 35 & 58 & 19 & 16 & 41 \end{bmatrix} \\ \begin{matrix} \hspace{2.2cm} \textcolor{blue}{~0} & \textcolor{blue}{~1} & \textcolor{blue}{~~2} & \textcolor{blue}{~~3} & \textcolor{blue}{~4} \end{matrix}$$
$$\text{\textcolor{darkgreen}{Frame 2}} \quad \begin{matrix} \textcolor{red}{0}\\ \textcolor{red}{1}\\ \textcolor{red}{2}\\ \textcolor{red}{3} \end{matrix} \begin{bmatrix} 4 & 30 & 6 & 49 & 0 \\ 48 & 47 & 29 & 26 & 56\\ 11 & 13 & 38 & 5 & 55 \\ 36 & \colorbox{lightblue}{50} & 20 & 10 & 18 \end{bmatrix} \\ \begin{matrix} \hspace{2.2cm} \textcolor{blue}{~0} & \textcolor{blue}{~~1} & \textcolor{blue}{~~~2} & \textcolor{blue}{~~3} & \textcolor{blue}{~~4} \end{matrix}$$ | 2022-09-28 16:51:05 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3369760513305664, "perplexity": 1007.349582755624}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030335257.60/warc/CC-MAIN-20220928145118-20220928175118-00111.warc.gz"} |
https://www.storyofmathematics.com/the-highest-that-george-can-suck-water-up-a-very-long-straw-is-2.0-m-this-is-a-typical-value/ | # The highest that george can suck water up a very long straw is $2.0 m$ . (this is a typical value.)
• What is the lowest pressure he can maintain in his mouth?
In this question, we have to find the minimum pressure that George can maintain in his mouth while he sucks water from a $2.0$ $m$ straw.
To solve this question, we should recall our concept of Pressure and Hydrostatic Pressure. So what is Pressure? It is defined as “The force on the unit area of an object.” The unit of pressure is Pascal $(Pa)$. Pressure is a scalar quantity having magnitude but no direction.
The different types of pressure are Atmospheric, Absolute, Differential, and Gauge pressure.
To understand the concept of hydrostatic pressure, imagine there is a container with water in it, and at every point inside the container, pressure exists on the liquid as there is liquid above it. So this existing pressure is known as hydrostatic pressure, and it is directly proportional to the depth of the liquid. Thus we can say that as the depth of the point increases, the hydrostatic pressure also increases.
It is given that there is a person sucking the liquid from the straw and the highest that he sucks the liquid up is $2.0 m$. Our required pressure is the pressure that is built inside the straw.
Height of water $h = 2.0 m$
Let Atmospheric pressure = $P_o$
Min pressure that can be maintained = $P$
Pressure of water column = $P_o$ – $P$
We know that
$P_o = 1.013 \times {10}^5 {N}{/m^2}$
Hydrostatic pressure =$\rho gh$
Here,
$\rho$ = Density of the fluid.
$g$ = Acceleration of gravity
$h$ = Depth of the fluid
Then we have,
$P_o − P = \rho gh$
So the required pressure that should be made by the person is equal to the atmospheric pressure outside of that straw minus the hydrostatic pressure.
$P = P_o − \rho g h$
Here we have
Density of water $\rho =1000 \\{ kg }/{ m^3 }$ and $g= 9.81$
Putting values in above equation, we get:
$P=1.013\times{ 10 }^5- 1000\times9.81\times2$
$P=\ \frac{ 8.168\ \times{ 10 }^4}{ 1.013 \times{ 10 }^5 }$
## Numerical Results
By solving the equation above, we will get the required pressure to be made, which is as follows:
$P= 8.168 \times { 10 }^4 { N }/{ m^2}$
Thus the minimum pressure that George can maintain in his mouth while he sucks water from the long straw to the height of $2.0 m$ is as follows:
$P=0.806\ atm$
## Example
A person sucks liquid from a straw to the height of $3.5m$. What will be the lowest pressure that he can keep in his mouth in $N/m^2$?
Person sucking the liquid from the straw: the maximum height achieved by the liquid is equal to $3.5 m$.
Height of liquid $h=3.5m$
$P=P_o − \rho gh$
Putting values in above equation, we get:
$P=1.013\times{10}^5-1000\times9.81\times3.5$
$P=8.168 \times {10}^4 {N}/{m^2}$ | 2022-07-05 04:21:26 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7604812383651733, "perplexity": 574.9540007437462}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104512702.80/warc/CC-MAIN-20220705022909-20220705052909-00221.warc.gz"} |
https://wikimedia.org.ng/change-the-kwsacc/31afec-mass-of-jupiter-in-kg | According to Dr. Sean Raymond, a post doctoral researcher at the Center for Astrophysics and Space Astronomy (CASA) at the University of Colorado, ”In terms of gaseous planets, once they reach 15 Jupiter masses or so there is enough pressure in the core to ignite deuterium fusion, so those are considered ‘brown dwarfs’ rather than planets.”. … M = V^2 a / G. G = 66.72 x 10^-12 N m^2 / kg^2 From Student Manual for Step 11. Home Unit Conversion Sci Calculator ☰ Please support this site by disabling or whitelisting the Adblock for "justintools.com". Saturn is much larger than Earth; its equator spans 9.4 times the size of our home planet. Jupiter Masss to Solar Masss Conversion. Note that rounding errors may occur, so always check the results. Jupiter has a mass of 1.8986×1027 kg, 317.83 times the mass of earth. The SI base unit for mass is the kilogram. Jupiter is a gas giant, both because it is so large and made up of gas.The other gas giants are Saturn, Uranus, and Neptune.. Jupiter has a mass of 1.8986×10 27 kg. 3.30 x 10^23 kg 3. Examples include mm, Stop the motion of the moons around Jupiter by clicking on in the menu at the bottom of the screen (or K on your keyboard). It is better to rearrange the formula first. This would all be divided by 6.673 times 10 to the negative 11th. … Scientists estimate that Jupiter needs 50-80 times its current mass to ignite fusion. Jupiter Mass and Solar Mass both are the units of MASS WEIGHT. Please enable Javascript The mass of Jupiter can be worked out from orbital mechanics and elementary gravity equations. Moon of Jupiter Average orbital radius r (km) Period T(d) M (kg) Io 4.22E+05 1.77 1.902E+27 Europa 6.71E+05 3.55 1.901E+27 Ganymede 1.07E+06 7.16 1.895E+27 Callisto 1.88E+06 16.69 1.891E+27 Remember to convert km to m and days to seconds You can find metric conversion tables for SI units, as well 14. Jupiter's regular satellites are believed to have formed from a circumplanetary disk, a ring of accreting gas and solid debris analogous to a protoplanetary disk. It is approximately 2.5 times as massive as all of the other planets in the Solar System combined. Jupiter’s intense gravity would pull additional rock tightly together shrinking the diameter of the planet and increasing its density. Use your estimates to find out how many times more massive Jupiter is than Mercury. Just for a comparison, this is 95 times the mass of the Earth. Jupiter mass (M J or M JUP), is the unit of mass equal to the total mass of the planet Jupiter (1.8986 × 10 27 kg, 317.83 Earth mass; [1] 1 Earth mass equals 0.00315 Jupiter masses). Jupiter's mass is $1.9 \times 10^{27} \mathrm{kg}$ Problem 45. Show your work. As you can see, the mass of Jupiter may seem awesome in comparison to Earth, it is a tiny fish in a world of sharks. The mass of Jupiter is no where near enough for sustained nuclear fusion. Scientists estimate that Jupiter would have to accumulate 3-4 times its current mass in order to begin compressing. Convert Jupiter Mass to Solar Mass (Jup in Mo). The mass of Jupiter is 1.9 x 10 27 kg. What would be the diameter of the event horizon) of a black hole with the same mass as Jupiter? There is plenty of hydrogen to be had, but the planet is too cold and lacks the density for a sustained reaction process. I've spent over 10 trillion microseconds (and counting), on this project. Problem 47 (a) One of the moons of Jupiter, named Io, has an orbital radius of $4.22 \times 10^{8} \mathrm{m}$ and a period of 1.77 days. They are all false science and garbage at best. Background material: From your Introduction to Astronomy class, you have learned Kepler’s laws of motion. Jupiter has the right fuels in abundance. The mass of Saturn is 5.6846×1026 kg. Here’s an article from Universe Today explaining just how big planets can get, and an article about how Jupiter and the other gas giants might have gobbled up their moons while they were forming. (2 points) answer: Der m- ft 2. See no ads on this site, see our videos early, special bonus material, and much more. 1 jupiter is equal to 1.899E+27 kilogram. But what if it did? T^2 = 4*pi^2 * R^3 / G*M. So M = 4 * pi^2 * R^3 / G * T^2. symbols, abbreviations, or full names for units of length, kg to jupiter, or enter any two units below: jupiter to kip We assume you are converting between jupiter and kilogram. to use the unit converter. jupiter to pennyweight How much mass would a black hole contain if it has an event horizon equal in size to an average sized tennis ball? Jupiter’s mass is 1.8986×1027 kg, while its density is at 1.326 gm/cm3. The Mass of Jupiter Student Guide Introduction: In this lab, you will use astronomical observations of Jupiter and its satellites to measure the mass of Jupiter. http://solarsystem.nasa.gov/planets/profile.cfm?Object=Jupiter To find the weight on Jupiter, we divide the weight on Earth by the Earth's force of gravity, which is 9.81m/s 2. As the density increased so would the gravity, further compressing the planet. 23:49 Are there gas dwarfs? By watching the orbits of Jupiter’s moons, we will determine their orbital properties. Being that it takes the weight of an object on Earth and converts it to the weight on Jupiter, the formula is Weight on Jupiter= (Weight on Earth/9.81m/s2) * 24.79m/s2. Within this use case you determine the mass of Jupiter by observing the orbits of the Galileian moons and by inserting these data into Kepler's laws (use case 12.). Type in unit In relation to the base unit (kilograms), 1 Jupiter Mass = 1.9E+27 kilograms. Conversion of units describes equivalent units of mass in other systems. Click to share on Facebook (Opens in new window), Click to share on Pocket (Opens in new window), Click to share on Twitter (Opens in new window), Click to share on LinkedIn (Opens in new window), Click to share on Tumblr (Opens in new window), Click to share on Pinterest (Opens in new window), Click to share on Reddit (Opens in new window), Click to email this to a friend (Opens in new window), http://solarsystem.nasa.gov/planets/profile.cfm?Object=Jupiter, Creative Commons Attribution 4.0 International License. jupiter or Sorry, your blog cannot share posts by email. Solution for Europa orbits Jupiter with an average radius of 671,100 km and a period of 3.551 days. It is hard to fully understand a number that large, so here are a few comparisons to help. The mass of the person back on the Earth will also be 75 kg as mass does not change from place to place. The mass of Jupiter is 1.9 x 1027 kg. At one time, scientists thought that Jupiter was the largest that a planet could become without igniting fusion and becoming a star. If used in the classroom this intermediate use case requires an introduction to the basics of gravitation, simple algebra and knowledge of the scientific notation for physical quantities. inch, 100 kg, US fluid ounce, 6'3", 10 stone 4, cubic cm, 1 jupiter is equal to 1.899E+27 kilogram. (318 earths). Jupiter’s gravity is 2.5 times more than Earth’s gravity; something that weight 10 kg on Earth would weight 25 kg on Jupiter. 1 Jupiter Mass is 1.89825360668185E+30 times Bigger than 1 Gram. Jupiter mass, also called Jovian mass, is the unit of mass equal to the total mass of the planet Jupiter. jupiter to once Jupiter is by far the most massive planet in the Solar System. The Mass of Jupiter Student Guide Introduction: In this lab, you will use astronomical observations of Jupiter and its satellites to measure the mass of Jupiter. Many thanks! metres squared, grams, moles, feet per second, and many more. Although this is 318 times the mass of Earth, the gravity of Jupiter is only 254% of the gravity on Earth. Jupiter has many moons orbiting it. Meters squared per kilogram squared, multiplied by 1.5 times 10 to the fifth seconds. Who knows what is in store as telescope technology improves in the coming years. It is the fifth planet from the Sun. How can additional mass cause a planet to shrink? Jupiter is actually so massive that if it gained much more mass it would shrink. jupiter to myriagram In most sources, the mass of Jupiter has been found to be approximately 1.9 × 10 27 kg, or is equivalent to approximately 318 Earth masses. kg Mass of Earth is 3x10-6 solar masses. 13. It is hard to fully understand a number that large, so here are a few comparisons to help. See the charts and tables conversion here! Listen to it here, Episode 56: Jupiter, and Episode 57: Jupiter’s Moons. Jupiter has a mass of 1.8986×1027 kg, 317.83 times the mass of earth. ConvertUnits.com provides an online A gram is defined as one thousandth of a kilogram. Mass of Jupiter 10 CESAR Science Case . Median response time is 34 minutes and may be longer for new subjects. Jupiter Earth Ratio (Jupiter/Earth) Mass (10 24 kg) 1,898.19: 5.9724: 317.83: Volume (10 10 km 3) 143,128: 108.321: 1321.33: Equatorial radius (1 bar level) (km) 71,492: 6,378.1: 11.209: Polar radius (km) 66,854: 6,356.8: 10.517: Volumetric mean radius (km) 69,911: 6,371.0: 10.973: Ellipticity: 0.06487: 0.00335: 19.36: Mean density (kg/m 3) 1,326: 5,514: 0.240: Gravity (eq., 1 bar) (m/s 2) 24.79: 9.80: … Based upon these data, what is the mass of Jupiter (in kg)?… It is approximately 2. (3 points) Find the mass of Jupiter based on data for the orbit of one of its moons, and compare your result with its actual mass. Jupiter mass is a common unit of mass in astronomy that is used to indicate the masses of other similarly-sized objects, including the outer planets and extrasolar planets. Since there isn’t that much material in our Solar System, it is a pretty good bet that Jupiter will never shrink. This work is licensed under a Creative Commons Attribution 4.0 International License. masse de Jupiter \mas də ʒy.pi.tɛʁ\ féminin (Astronomie) Unité de masse égale à la masse de la planète Jupiter, couramment utilisée pour les géantes gazeuses et les naines brunes, égale à environ 1,8986 × 10 27 kg, 317,83 masses de la Terre ou 0,0009546 masse solaire.Le symbole : M J, M JUP ou M Jup. This will display a jagged, “connect-the-dots” version of your graph. Jupiter. This value may refer to the mass of the planet alone, or the mass of the entire Jovian system to include the moons of Jupiter. *Response times vary by subject and question complexity. Fusion requires high temperatures, intense gravitational compression, and fuel. Students will identify the moons of Jupiter in astronomical images and estimate the orbital period of one or more of the moons. jupiter to teragram There have been rumors floating around for decades that Jupiter could ignite fusion and become a star at any time. The SI base unit for mass is the kilogram. You can view more details on each measurement unit: That’s as much as 317.8 Earths and is 1/1047 the size of the Sun. Gravitational compression. In the mass of Jupiter, massive Jupiter would be equaling four pi squared, multiplied by 4.22 times 10 to the eighth meters quantity squared. Compute mass of Jupiter from step 17 compared to mass of Earth. Jupiter is 2.5 times more massive than all of the other planets in our Solar System combined. M = mass of Jupiter, in Kg. Students will then use that information to Scientists have found a few hundred gas giants larger than Jupiter as they have surveyed the night sky. This site has more detailed information about the mass of Jupiter, and a page from NASA that helps you calculate the density of the planets. They may be the remnants of a score of Galilean-mass satellites that formed early in Jupiter's history. As stated in the paragraph above, there isn’t enough material to be had, so Jupiter can not become a star. In other words, 1 jupiter mass is 3.775212E+57 times bigger than a solar mass. conversion calculator for all types of measurement units. Colloquially, the mass is also called weight, which for celestial bodies is plain wrong. It would take 318 times Earth’s mass to equal Jupiter’s. SI base unit, (±)? Mass of Jupiter. Express your answer in standard form (without using a power of 10). area, mass, pressure, and other types. Use this page to learn how to convert between jupiter and kilograms. jupiter to centigram Jupiter’s rings and moons are embedded in an intense radiation belt of electrons and ions trapped by the magnetic field. Jupiter is the largest planet in the Solar System. We will use the program Stellarium to simulate observations of the night sky. We will use the program Stellarium to simulate observations of the night sky. jupiter to catty From that, you can figure the mass of the Jupiter. Jupiter is by far the most massive planet in the Solar System. jupiter to sack It works like this: you measure the distance from Jupiter to one of its moons (a), how long the moon takes to orbit around Jupiter once, and its speed (V). Jupiter is about 5,751.51 times more massive than Mercury. The Jovian magnetosphere, comprising these particles and fields, balloons 1 to 3 million km (600,000 to 2 million miles) toward the sun and tapers into a windsock-shaped tail extending more than 1 billion km (600 million miles) behind Jupiter as far as Saturn’s orbit. 15. In the same menu click on (or press Ctrl + A) to enable the Angle Measuring plugin. Assuming the orbit is circular, calculate the mass of Jupiter. If a value is shown as 0 (e.g. Record your mass on worksheet. MJ stands for jupiter masss and M☉ stands for solar masss. Example: Jupiter has more than 300 times Earth's mass, but less than one thousandth mass of the Sun. This is twice the mass of all the other planets in the Solar System put together. Given that there is no more hydrogen or helium gas floating around for Jupiter to collect, it would gain mass through the accretion of rocky bodies like asteroids. Note that rounding errors may occur, so always check the results. I got this answer by dividing Jupiter’s mass by Mercury’s mass giving me my final answer 5,751.51. Check your work by comparing your value with that on the Voyager III data panel for Jupiter. Find the mass of Jupiter, in kg, based on data for the orbit of one of its moons. http://planetquest.jpl.nasa.gov/news/tres4.cfm, Join our 836 patrons! The formula used in jupiter masss to solar masss conversion is 1 Jupiter Mass = 3.775212E+57 Solar Mass. How many jupiter in 1 kg? Find the mass of Jupiter based on data for the orbit of one of its moons, and compare your result with its actual mass. These can be used to calculate the mass of Jupiter. We’ve also recorded an entire show just on Jupiter for Astronomy Cast. Type in your own numbers in the form to convert the units! They discovered the fallacy of that belief once technology expanded their view of the universe. Use this page to learn how to convert between jupiter and kilograms. It may also be used to describe the masses of brown dwarfs , as this unit provides a convenient scale for comparison. jupiter to uncia. convert mass of Jupiter to kg. You can view more details on each measurement unit: jupiter or kg. Sources: jupiter to mace You can do the reverse unit conversion from Post was not sent - check your email addresses! mass of the Moon in solar masses), then a higher number of decimal places is required. The kilogram or kilogramme, (symbol: kg) is the SI base unit of mass. This would be new tins. Join us at patreon.com/universetoday. Express your first answer, using scientific (powers of ten) notation, in kg and also express the same answer in "Earth masses." This calculates the mass of the object. as English units, currency, and other data. Computing Jupiter’s mass with either Jupiter’s moon Callisto or Jupiter’s moon Io gives us pretty much the same answer. Jupiter's mass is 2. The answer is 5.2659294365455E-28. 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https://chemistry.stackexchange.com/questions/113005/questions-in-equilibrium | # Questions in Equilibrium [closed]
I have two inquiries concerning equilibrium:
1. Does $$K_c = 5.0$$ indicate that the equilibrium mixture contains both products and reactants at an approximately equal amount? I know that $$K_c \approx 1.0$$ is said to be so, however I am unsure of if $$5.0$$ would be considered to be approximately $$1.0$$ in this case.
2. Is a reaction at equilibrium if the forward reaction does not change? I believe it would as an indication of equilibrium is that both the forward and reverse reactions occur at the same rate.
## closed as off-topic by A.K., Mithoron, Mathew Mahindaratne, Todd Minehardt, airhuffApr 20 at 0:03
This question appears to be off-topic. The users who voted to close gave this specific reason:
If this question can be reworded to fit the rules in the help center, please edit the question.
• I suppose the OP wants confirmation the equilibrium means a dynamic equilibrium, with the equal rate of the opposite reactions. Yes, it is . By "reaction does not change" he probably means the concentrations are constant. – Poutnik Apr 19 at 17:44
• You won't get quicker or better answers if you spam Chemistry.SE with the same questions. This is not a paid service, there is no staff constantly sitting on-line and answering all the questions, and you cannot demand a fast and accurate response (plus you already got one answer in less than an hour, and in addition to that I already voted to reopen your first question). Please visit Help Center to familiarize yourself with the basic principles of this site. – andselisk Apr 20 at 3:11
• If were familiar with the rules, you would've not re-posted the same question just because you were dissatisfied with the first answer given and want a quicker response. I feel pity for you if you see this as aggression or fight, I'm just informing that what you've done wasn't correct; also, as I see it, the answer was as decent as the question itself. – andselisk Apr 20 at 3:27
• @James It is very unusual the rate of reaction being constant. Do you have a particular equilibrium in mind ? – Poutnik Apr 20 at 3:35
• @James correct me if I'm wrong but I choose to disagree. A specific example of an equilibrium refers to an equilibrium example with reactants and products for example the equilibrium formed by iron thiocyanate ions in solution. Without a specific example it is difficult to tell if an equilibrium constant 5.0 corresponds to roughly similar amounts of reactants and products – sab hoque Apr 20 at 4:25
A reaction is in equilibrium,
if the reaction quotient $$Q=K_{\rm c}$$,
For reaction $$\ce{A <=> B}$$ $$K_{\rm c}=5$$ means concentration ratio 1:5. It does not seem to me to be approximately the same concentrations.
The equilibrium constant is the thermodynamical quantity, determining the position of equilibrium. It's value does not say if the reaction is in equilibrium.
About the same concentrations, it means rather the same product of concentrations of reagents and products.
If there is a reaction $$\ce{A + B <=> C + D}$$ with $$K_{\rm c}=1$$ and if concentrations are $$c_{\rm A}=1000c_{\rm B}=100c_{\rm C}=10c_{\rm D}$$ we can hardly speak about equal concentrations at equilibrium. as $$K_c=1=\frac{10\cdot 100}{1000\cdot 1}$$
Additionally, if a reaction is not symmetric in counts of reagents of both sides, equilibrium concentration ratios change with concentrations.
If there is a reaction $$\ce{A <=> 2 B}$$ The same concentrations during equilibrium with $$K_{\rm c}=1$$ occur only for $$c_{\rm A}=1$$. If we consider concentrations of $$\ce{A}$$ and $$\ce{B}$$, the following concentrations are in equilibrium:
• A B
• 0.0001 0.01
• 0.01 0.1
• 1 1
• 100 10
• 10000 100
Yes, you think right, at equilibrium, the rates of the forward and reverse reactions are equal.
If \begin{align} rate_{\rm forward}&=const\\ rate_{\rm backward}&=f([products])\\ \end{align} ("Forward reaction does not change"=has a constant rate, as was explained)
..then as reaction is progressing, the backward reaction rate increases, until it matches the forward rate and reaction is at equilibrium ( regardless of the $$K_{\rm c}$$ value.)
• If you are familiar with the concept of the equilibrium constant, which I expect you are, you would know that if Kc, which is the denotation for the equilibrium constant, is approximately 1.0, the reaction is said to be at equilibrium. The issue arises to what is meant by "approximately". This is in fact what my question is asking. – James Apr 19 at 23:21
• @James The Kc value and reaction being at equilibrium are 2 independent things. Kc value is matter of reaction thermodynamics, describing the position of the equilibrium state with minimal value of Gibbs energy. Being at equilibrium is matter of reaction kinetics. The Kc value does not say if the reaction already reached equilibrium or not. – Poutnik Apr 20 at 3:28
• @James A reaction can be in equilibrium for any value of Kc, e.g $10^{52}$ – Poutnik Apr 20 at 5:12 | 2019-08-24 21:15:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 18, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6866297721862793, "perplexity": 905.3149536532597}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027321696.96/warc/CC-MAIN-20190824194521-20190824220521-00444.warc.gz"} |
http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=E1BMAX_2011_v48n1_79 | THE LOWER AUTOCENTRAL SERIES OF ABELIAN GROUPS
Title & Authors
THE LOWER AUTOCENTRAL SERIES OF ABELIAN GROUPS
Abstract
In the present paper we introduce the lower autocentral series of autocommutator subgroups of a given group. Following our previous work on the subject in 2009, it is shown that every finite abelian group is isomorphic with $\small{n^{th}}$-term of the lower autocentral series of some finite abelian group.
Keywords
autocommutator subgroup;autocentral series;abelian group;
Language
English
Cited by
1.
AUTOCOMMUTATORS AND AUTO-BELL GROUPS,;;;
대한수학회보, 2014. vol.51. 4, pp.923-931
2.
NON-ABELIAN TENSOR ANALOGUES OF 2-AUTO ENGEL GROUPS,;;
대한수학회보, 2015. vol.52. 4, pp.1097-1105
1.
Autonilpotent groups and their properties, Asian-European Journal of Mathematics, 2016, 09, 03, 1650056
2.
NON-ABELIAN TENSOR ANALOGUES OF 2-AUTO ENGEL GROUPS, Bulletin of the Korean Mathematical Society, 2015, 52, 4, 1097
3.
AUTOCOMMUTATORS AND AUTO-BELL GROUPS, Bulletin of the Korean Mathematical Society, 2014, 51, 4, 923
4.
Relative autocommutator subgroups of abelian groups, Journal of Algebra and Its Applications, 2017, 16, 05, 1750086
5.
On A-nilpotent abelian groups, Proceedings - Mathematical Sciences, 2014, 124, 4, 517
References
1.
C. Chis, M. Chis, and G. Silberberg, Abelian groups as autocommutator groups, Arch. Math. (Basel) 90 (2008), no. 6, 490-492.
2.
M. Deaconescu and G. L.Walls, Cyclic groups as autocommutator groups, Comm. Algebra 35 (2007), no. 1, 215-219.
3.
P. Hegarty, The absolute centre of a group, J. Algebra 169 (1994), no. 3, 929-935.
4.
P. Hegarty, Autocommutator subgroups of finite groups, J. Algebra 190 (1997), no. 2, 556-562.
5.
M. Naghshineh, M. R. R. Moghaddam, and F. Parvaneh, The third term of the lower autocentral series of abelian groups, Journal of Mathematical Extension, Vol. 4, No. 1 (2009), 1-6.
6.
D. J. S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics, 80. Springer-Verlag, New York-Berlin, 1982. | 2018-09-21 21:18:18 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 1, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6550197005271912, "perplexity": 1781.8308113516896}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267157569.48/warc/CC-MAIN-20180921210113-20180921230513-00260.warc.gz"} |
https://www.rosettacommons.org/docs/latest/build_documentation/Build-Documentation | ### If you are new to Rosetta, start here.
This page describes how to install, compile, and test Rosetta 3 (formerly called "Mini") on one's own workstation, or to a user directory on a scientific cluster.
# Setting up Rosetta 3
This page describes how to install, compile, and test Rosetta 3 formerly called "Mini" on a supported platform.
## Basic Setup
Build environment setup instructions for most situations can be found on the Getting Started page.
## Publicly accessible clusters with Rosetta pre-installed
• As part of the XSEDE initiative, the TACC/Stampede cluster has Rosetta and PyRosetta centrally installed for authorized users. See the TACC page for more details.
## Alternative Setup for Individual Workstations
The current build system is based on the tool SCons ("Software Constructor") with some extensions. scons.py is implemented as a Python script.
### SCons (Mac/Linux)
You have multiple options for compiling Rosetta 3 with SCons. You can build either only the core libraries or both the libraries and the executables. Rosetta 3 can be compiled in two major modes:
• Debug mode — for development — no mode option, or mode=debug
• Release mode — faster/tested and ready to go &mdashl mode=release
1. Change directory to main/source .
2. Type one of the following commands:
• complete
• ./scons.py -j<number_of_processors_to_use> bin
• ./scons.py -j<number_of_processors_to_use> mode=release bin
• libraries only
• ./scons.py -j<number_of_processors_to_use>
• ./scons.py -j<number_of_processors_to_use> mode=release
The -j8 flag would mean, "use at most 8 processes for building," and can be reasonably set as the number of free processors on your machine. Be aware that setting -j to a very high value will cause the operating system to have difficulty scheduling jobs.
By default scons hashes and processes every file in the tree before performing a build. On a large tree (e.g. rosetta) and filesystem with high io latency (e.g. a NFS or GPFS filesystem) this causes ridiculously slow build times. In order to improve build times disable file hashing and allow caching of build dependency metadata. Add the follow lines to the project's root SConscript file:
Decider('MD5-timestamp') SetOption('implicit_cache', 1)
These settings are described on the scons gofast page.
By default, scons uses GCC to compile. To use an alternate compiler, such as CLang, use the cxx option:
./scons.py -j<number_of_processors_to_use> cxx=clang
To use an alternate version of the compiler, you can use the option cxx_ver option with whatever version you have (here 4.5):
./scons.py -j<number_of_processors_to_use> cxx=clang cxx_ver=4.5
For more build options, such as only compiling only one executable or apbs - Please take a look at the SConstruct File in main/source
If you get an error like /usr/bin/ld: cannot find -lz, SCons may be looking for 32 bit libraries on a 64 bit machine. Look up how to install the missing dependency; in this case sudo apt-get install lib32z1-dev installs the required library.
### Build Rosetta using the Rosetta Xcode Project (Mac)
The Rosetta Xcode project is compatible with Xcode versions 2.4 and later. You can use it to build, run, debug, browse, and edit the source code. There are four build targets to select from:
Libraries: Rosetta libraries upon which apps are based
Test: Unit tests for components of Rosetta Libraries & Test: Both of the above
Libraries & Apps: Libraries and executable apps in rosetta-X.X/main/source/bin
There are two primary configurations, Debug and Release. Each of those has two sub-configurations. The "static" mode builds static binaries, instead of those based on the shared libraries. This can be useful for copying and running the apps on other Mac OS X systems. The "graphics" mode enables OpenGL graphics for those apps that support it.
The Xcode build and clean commands tell scons to take the appropriate actions. Xcode will by default instruct scons to use 2 processors for compilation. To change that number, double click the "Rosetta" icon in the "Groups & Files" pane. Then switch to the "Build" tab and change the "NUM_PROCESSORS" setting to the desired number of processors.
To run/debug Rosetta from within Xcode, you will need to add an executable to the project. To do so, select "New Custom Executable..." from the "Project" menu. Xcode will not allow you to specify the symbolic links in the
rosetta-X.X/main/source/bin
directory as the executable. Instead, you will have to reference the actual binaries deep inside the subdirectories of
rosetta-X.X/main/source/build
Please see the Xcode documentation for information about specifying the command line arguments and the working directory.
If you want to set breakpoints from within Xcode for debugging, you have to tell the debugger to load all user libraries. To do so, go to the "Run->Show->Shared Libraries..." menu item. Change the "User Libraries:" popup from "Default (External)" to "All".
### Build Rosetta on Windows
Building Rosetta directly on Windows systems is not recommended.
Alternatives:
• Install virtual machine software and run Linux within it.
• Dual Boot with Linux. (usually the best option)
There have been some reports of success in using the precompiled Linux binaries with Windows 10's Linux subsystem. Such use should be viewed as "highly experimental".
You may be able to compile Rosetta by using cygwin for windows ( http://www.cygwin.com/ ). Such usage is not tested by Rosetta developers though, and may not work.
### CMake
Rosetta can also be built with CMake. Currently it is less widely tested and does not support some features such as "extras" flags. However, CMake is faster than SCons, which can be advantageous for rapid compile-debug cycles.
To build with CMake...
1. Change to the CMake build directory from the base Rosetta 3 directory and create the CMake build files:
cd cmake
./make_project.py all
2. Next, change to a build directory
cd build_<mode>
3. and build Rosetta 3
cmake -G Ninja
ninja
If you don't have ninja, you can get it via
git clone git://github.com/martine/ninja.git
Or you can use make which is slightly slower for recompiling:
cmake .
make -jnproc # replace nproc with number of cores to use
The currently available modes are debug and release, in combination with various extras and compilers. Creating a new type of build can be done by copying an existing CMakeLists.txt file from a build directory and modifying compiler settings and flags appropriately. In particular, users wanting to build only the libraries or a subset of the applications should remove these lines in CMakeLists.txt:
INCLUDE( ../build/apps.all.cmake )
INCLUDE( ../build/pilot_apps.all.cmake )
These can be replaced with an explicit list of applications, such as:
INCLUDE( ../build/apps/minirosetta.cmake)
The make_project.py and cmake commands should be run whenever files are added or removed from the rosetta_source/src/*.src.settings files. The -j8 flag for cmake has the same interpretation as the -j8 flag for scons . The command cmake . generates a "makefile", which means that alternate programs such as distcc can be used for building Rosetta 3. (The ninja_build.py script in rosetta_source automatex the combined cmake/ninja process and responds to build abbreviations like 'r' for 'release'.)
Alternate C++ compilers (such as Clang++ or distcc) can be specified on the command-line by replacing this command,
cmake .
with this command,
cmake . -DCMAKE_CXX_COMPILER=which clang++
### Message Passing Interface (MPI)
MPI (the Message Passing Interface) is a standard for process-level parallelization. Independent processes can coordinate what they're doing by sending messages to one another, but cannot access the same memory space or directly interfere with one another's execution. Although most of Rosetta has not been parallelized, some apps (such as rosetta_scripts) are set up for at least job-level parallelism (permitting automatic distribution of jobs over available processes), and some specialized pilot apps (such as vmullig/design_cycpeptide_MPI) take advantage of parallel processing in a non-trivial way.
To build MPI executables, add the flag "extras=mpi" and copy main/source/tools/build/site.settings.topsail to main/source/tools/build/site.settings. You may need to make additional edits to the site.settings file if your MPI libraries are not in the standard locations. See this post for help with setting up MPI for Ubuntu linux. Then compile with extras=mpi:
./scons.py bin mode=release extras=mpi
#### Installing MPI
You need to install MPI, of course. What that looks like will vary from system to system. For ubuntu, you will need probably sudo apt-get install openmpi openmpi-dev or similar. The -dev package is necessary to install before trying to compile MPI Rosetta.
## Dependencies/Troubleshooting
Rosetta requires a compiler (most recent gcc or clang are fine, but see Cxx11Support for details) and the zlib compression library development package. Instructions for acquiring either are below, sorted by what sorts of error messages they give if you are missing them.
This indicates that you either don't have a compiler installed, or Rosetta is not able to find the compiler that you do have installed.
At the commandline, execute g++ --version and clang --version. If one of them works, you can try specifying that compiler explicitly on the scons commandline with either cxx=gcc or cxx=clang. (This is a label, rather than the compiler command, so it cannot take arbitrary input.)
--> A compiler is already installed:
If you know you have a compiler installed and it's in your path, you can copy main/source/tools/build/site.settings.topsail to main/source/tools/build/site.settings. You may also want to edit the lines:
"overrides" : {
},
to something like:
"overrides" : {
"cc" : "<C compiler command>",
"cxx" : "<C++ compiler command>",
},
where you substitute the compiler commands as appropriate.
--> Install a compiler:
Many default installations of Mac and Linux do not come with a compiler installer, so you will need to install one separately. (Note that the following only applies if you have administrator rights to your machine. If you do not, talk to your sysadmin regarding the installation of a compiler.)
Rosetta requires a compiler with C++11 support. Most recent compiler versions include C++11 support, but older compilers may not. See Cxx11Support for more details.
For Macs, install the XCode development packages. Even though you won't be compiling Rosetta through XCode, installing it will also install a compiler. (Clang, for recent versions of MacOS.)
For Linux, you will want to install the compiler package from your package management system. For Ubuntu and similar systems, the package "build-essential" installed with a command like sudo apt-get install build-essential will set your system up for compilation.
"error: unrecognized command line option "-std=c++11""
Rosetta requires a compiler with C++11 support. This message is due to insufficient support of C++11 in the compiler you're currently using. See Cxx11Support for more details about compilers to use, or see the "sh: 1: o: not found" regarding changing your compiler.
**"error: '...' names constructor"/"
GCC 4.7, while it supports many of the features of C++11, doesn't support all of them - this message is coming from a feature we use which is not supported by GCC 4.7. See Cxx11Support for more details about compilers to use, or see the "sh: 1: o: not found" regarding changing your compiler.
"no member named 'declval' in namespace 'std'" "error: unknown type name 'type_info'"
Rosetta requires a compiler and C++ standard library with C++11 support. These messages indicate that while your compiler might have C++11 support, the standard library it's using may not. See Cxx11Support for more details about compiler and standard libraries, or see the "sh: 1: o: not found" regarding changing your compiler.
"cannot find -lz"
Rosetta requires the zlib compression library to be installed on your computer in order to properly compile. Talk to your system administrator about installing the development version of the zlib library. (The "development" version of the library is needed so that Rosetta can compile against the library.)
For Ubuntu and related distributions, install the zlib1g-dev package (e.g. with sudo apt-get install zlib1g-dev)
## Testing
There are two sets of tests to run to ensure everything is working properly, unit tests and integration tests. (See Testing Rosetta.)
#### MPI
Compilation in MPI mode permits specialized JobDistributors to be used; the function of these JobDistributors, and their integration with other components of Rosetta, can only be tested by running special integration tests in MPI mode by passing the --mpi-tests flag to integration.py. Selective failure of these tests will probably mean that the parallel JobDistributors have been broken in some way.
The MPI-mode build test simply tries to compile Rosetta with the -extras=mpi flag passed to scons. Selective failure of this build means that code surrounded by #ifdef USEMPI ... #endif lines has errors in it.
## Cleaning
cd Rosetta/main/source/ && rm -r build/* && rm .sconsign.dblite will remove old binaries.
## Obtaining additional files
#### PDB Chemical Components Dictionary
NOTE: This file is not needed for Rosetta to run normally.
To use the -in:file:load_PDB_components option to automatically load unrecognized residues from files, you need to obtain a mmCIF-formatted version of the Chemical Components Dictionary. This file can be downloaded from the wwwPDB. (Alternatively for RosettaCommons members, you can talk your lab's gatekeeper for the RosettaCommons git-lfs).
To use with Rosetta, download the (plain text, non-gzipped) file and place it in the Rosetta database directory (Rosetta/main/database/). Or alternatively, pass the absolute path to the file to the -in:file:PDB_components_file option. The "Protonation Variants Companion Dictionary" and "Chemical Component Model data file" are not required. | 2017-03-27 08:39:02 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.22625654935836792, "perplexity": 7530.083436536726}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189466.30/warc/CC-MAIN-20170322212949-00130-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://academic.oup.com/hmg/article/24/8/2201/651979/Prenatal-exposure-to-maternal-smoking-and | ## Abstract
Maternal smoking during pregnancy has been found to influence newborn DNA methylation in genes involved in fundamental developmental processes. It is pertinent to understand the degree to which the offspring methylome is sensitive to the intensity and duration of prenatal smoking. An investigation of the persistence of offspring methylation associated with maternal smoking and the relative roles of the intrauterine and postnatal environment is also warranted. In the Avon Longitudinal Study of Parents and Children, we investigated associations between prenatal exposure to maternal smoking and offspring DNA methylation at multiple time points in approximately 800 mother–offspring pairs. In cord blood, methylation at 15 CpG sites in seven gene regions (AHRR, MYO1G, GFI1, CYP1A1, CNTNAP2, KLF13 and ATP9A) was associated with maternal smoking, and a dose-dependent response was observed in relation to smoking duration and intensity. Longitudinal analysis of blood DNA methylation in serial samples at birth, age 7 and 17 years demonstrated that some CpG sites showed reversibility of methylation (GFI1, KLF13 and ATP9A), whereas others showed persistently perturbed patterns (AHRR, MYO1G, CYP1A1 and CNTNAP2). Of those showing persistence, we explored the effect of postnatal smoke exposure and found that the major contribution to altered methylation was attributed to a critical window of in utero exposure. A comparison of paternal and maternal smoking and offspring methylation showed consistently stronger maternal associations, providing further evidence for causal intrauterine mechanisms. These findings emphasize the sensitivity of the methylome to maternal smoking during early development and the long-term impact of such exposure.
## Introduction
Despite the known health risks to both mothers and newborns, maternal smoking during pregnancy remains a significant public health problem in high-income countries and recent reports suggest that ∼12% of mothers in England are still smoking at the time of delivery (1). Exposure of the fetus to maternal smoking in utero has been associated with adverse perinatal outcomes, including low birth weight (2–4), elevated blood pressure (5,6), obesity (7,8) and behavioural difficulties in childhood (9,10). It has been proposed that epigenetic modifications such as DNA methylation may mediate the adverse developmental consequences associated with smoking during pregnancy (11).
Cigarette smoke is an established environmental associate of DNA methylation (12–17) and maternal smoking in pregnancy has recently been found to be associated with levels of DNA methylation in large-scale epigenome wide association studies (EWAS) of cord blood (18) and infant whole blood shortly after delivery (19). Of particular importance is the observation that maternal smoking during pregnancy is associated with changes in methylation in genes involved in fundamental developmental processes (18,19).
The associations found between maternal cotinine levels, an objective biomarker of smoking and DNA methylation in newborns imply a dose-dependent effect of maternal smoking in pregnancy (18). The sensitivity of the offspring methylome to the intensity (19) and duration (20) of smoking during pregnancy has been further explored. Of potential relevance is the impact of maternal smoking in early pregnancy, when women may not be aware that they are pregnant. During the early phases of embryogenesis, the products of tobacco smoke may induce soma-wide modification of DNA methylation in the exposed offspring, which may be then be maintained into postnatal life (21,22). Conversely, a recent study of DNA methylation in newborns found no difference in methylation between the offspring of mothers who never smoked and those who smoked early in pregnancy (23). It has also been shown that the effect of in utero exposure on newborn methylation is stronger when the mother smoked past 18 weeks than when she quit earlier in pregnancy (20). These findings warrant further investigation in an independent study.
Associations between own smoking and methylation at later time points have been found (15,24), with one study of former smokers showing that methylation in a key gene region associated with smoking (AHRR) approaches the levels of never smokers within the first few years of quitting, but never completely returns to normal levels (15). Two recent studies have also investigated prospective associations between maternal smoking in pregnancy and peripheral blood methylation in offspring when they were children (25) and adolescents (26). A high degree of similarity was found with smoking-associated DNA methylation in newborns (18), implying a lasting effect of maternal smoking in pregnancy on offspring DNA methylation profiles. However, a more comprehensive longitudinal assessment of intrauterine exposure and methylation levels in the same offspring assessed at multiple time points is required.
The relative roles of the intrauterine and postnatal environment in the persistence of DNA methylation changes associated with maternal smoking are yet to be determined. Previous studies have shown that associations between prenatal exposure to maternal cigarette smoking and offspring methylation during adolescence are not attenuated with adjustment for postnatal smoking of the parents or the offspring themselves (25,26). However, the method of adjusting for a potential mediator in standard regression models to estimate the direct effect of an exposure may produce spurious conclusions (27,28). Alternative methods are therefore required to test the hypothesis that maternal smoking in pregnancy is the ‘critical period’ for influencing offspring methylation profiles in childhood and adolescence (29). Finally, given that some of the signals found for prenatal smoke exposure have also been identified in a study of personal smoking of adolescents (30), any apparent persistent effect of maternal smoking on offspring methylation profile at later ages may be explained by smoking of the adolescents themselves.
Epigenetic markers, in contrast to germ-line genetic variation (31), are phenotypic and are therefore subject to the same potential problems of confounding which afflict observational epidemiology (32,33). Hence, there is a need to apply a range of tools for strengthening causal inference in epigenetic epidemiology (34,35). One such method for inferring a causal intrauterine effect involves the use of paternal exposures as negative controls for maternal exposures thought to have an intrauterine influence on offspring outcomes (34,36–40). Paternal smoking may show associations with offspring methylation similar to those of maternal smoking in pregnancy if the associations are confounded either by shared familial factors or by parental genotypes. However, if there is an intrauterine influence of maternal smoking, then only maternal exposure would be expected to show an independent association with the outcome.
We use the Accessible Resource for Integrated Epigenomic Studies (ARIES), a large collection of genome-wide DNA methylation data from multiple time points in mothers and offspring from the Avon Longitudinal Study of Parents and Children (ALSPAC) (41,42) to (1) replicate findings of a recently reported EWAS for maternal cotinine (18) by investigating associations between self-reported maternal smoking in pregnancy and offspring cord blood methylation using the Illumina Infinium® HumanMethylation450 (HM450) BeadChip; (2) explore the dose-dependent effect of maternal smoking by investigating associations between the duration and intensity of maternal smoking and offspring cord blood methylation at key CpG sites; (3) examine the persistence of DNA methylation changes at key CpG sites by investigating longitudinal associations at multiple time points, from birth to 17 years; (4) investigate the relative roles of the intrauterine and postnatal environment in the persistence of DNA methylation modifications; (5) assess potential causality in associations between maternal smoking during pregnancy and offspring DNA methylation at multiple time points, using paternal smoking as a negative control.
## Results
### Baseline characteristics
Compared with offspring in the core ALSPAC sample who are not part of the ARIES project, those in ARIES were more likely to be singletons, had a higher birth weight on average, had a longer gestation and had mothers who were: older at time of delivery, more highly educated, from a higher social class, more likely to drink alcohol in pregnancy and less likely to smoke in pregnancy (Table 1).
Table 1.
Baseline characteristics for individuals in ARIES compared with those in the ALSPAC core cohort who were not part of ARIES
Individuals in ARIES (N = 1018)a Individuals not in ARIES (N = 14 062)a P-value for difference
Sex (% males) 48.8 51.7 0.056
Multiple births (% singletons) 99.6 97.1 <0.001
Birth weight (g) 3487.4 (488.1) 3377.1 (577.9) <0.001
Gestational age (weeks) 39.6 (1.5) 39.3 (2.1) <0.001
Maternal age (years) 30.0 (4.4) 28.3 (5.0) <0.001
Maternal parity (% parous) 53.6 55.5 0.26
Maternal education (% university degree) 20.5 12.2 <0.001
Household social class (% non-manual) 88 79.8 <0.001
Maternal BMI (kg/m222.8 (3.7) 22.9 (3.8) 0.18
Alcohol during pregnancy? (% yes) 66.7 63.1 0.024
Mother smoked during pregnancy? (% yes) 14.3 30.2 <0.001
Individuals in ARIES (N = 1018)a Individuals not in ARIES (N = 14 062)a P-value for difference
Sex (% males) 48.8 51.7 0.056
Multiple births (% singletons) 99.6 97.1 <0.001
Birth weight (g) 3487.4 (488.1) 3377.1 (577.9) <0.001
Gestational age (weeks) 39.6 (1.5) 39.3 (2.1) <0.001
Maternal age (years) 30.0 (4.4) 28.3 (5.0) <0.001
Maternal parity (% parous) 53.6 55.5 0.26
Maternal education (% university degree) 20.5 12.2 <0.001
Household social class (% non-manual) 88 79.8 <0.001
Maternal BMI (kg/m222.8 (3.7) 22.9 (3.8) 0.18
Alcohol during pregnancy? (% yes) 66.7 63.1 0.024
Mother smoked during pregnancy? (% yes) 14.3 30.2 <0.001
aN varies according to completeness of data on baseline characteristics.
Of the 1018 mother–offspring pairs in the ARIES project, 916 offspring had cord blood methylation data, which successfully passed quality control (QC). Seven hundred and ninety had data on both sustained smoking in pregnancy and cord blood DNA methylation. Of these, 699 were classified as non-smokers and 91 were classified as sustained smokers during pregnancy. Compared with the non-smokers, sustained smokers were more likely to be younger at time of delivery, less well educated, from a lower social class, less likely to drink in pregnancy and more likely to have partners who also smoked in pregnancy (Table 2).
Table 2.
Differences in potential confounding factors between individuals in ARIES whose mothers did not smoke in pregnancy compared with sustained smokers
Non-smoker (N = 699)a Sustained (N = 91)a P-value for difference
Sex
Male 49.4 47.3 0.71
Female 50.6 52.8
Maternal age (years) 30.5 (4.1) 27.9 (5.4) <0.001
Maternal age (categories)
<25 7.6 30.8 <0.001
25–30 39.2 37.4
>30 53.2 31.9
Parity
0 45.5 51.1 0.53
1 37.7 32.2
2 13.2 11.1
3+ 3.6 5.6
Maternal education
CSE/vocational 13.9 33.0 <0.001
O-level 32.7 40.9
A-level 30.4 18.2
Degree 23.1 8.0
Social class
I 22.9 4.9 <0.001
II 44.0 39.0
III (NM) 24.4 20.7
III (M) 5.3 22.0
IV or V 3.5 13.4
Maternal BMI 22.8 (3.7) 23.0 (3.6) 0.63
Maternal BMI (categories)
<18.5 3.3 3.5 0.59
18.5–25 79.1 76.5
25–30 13.1 14.1
30+ 4.6 5.9
Maternal weight 61.6 (10.4) 61.5 (10.5) 0.95
Alcohol
Non-drinker 34.2 40.9 0.040
Drank before 18 weeks of gestation 15.6 5.7
Still drinking at 18 weeks of gestation 50.2 53.4
Paternal smoking
Non-smoker 78.6 20.9 <0.001
Smoker 21.4 79.1
Non-smoker (N = 699)a Sustained (N = 91)a P-value for difference
Sex
Male 49.4 47.3 0.71
Female 50.6 52.8
Maternal age (years) 30.5 (4.1) 27.9 (5.4) <0.001
Maternal age (categories)
<25 7.6 30.8 <0.001
25–30 39.2 37.4
>30 53.2 31.9
Parity
0 45.5 51.1 0.53
1 37.7 32.2
2 13.2 11.1
3+ 3.6 5.6
Maternal education
CSE/vocational 13.9 33.0 <0.001
O-level 32.7 40.9
A-level 30.4 18.2
Degree 23.1 8.0
Social class
I 22.9 4.9 <0.001
II 44.0 39.0
III (NM) 24.4 20.7
III (M) 5.3 22.0
IV or V 3.5 13.4
Maternal BMI 22.8 (3.7) 23.0 (3.6) 0.63
Maternal BMI (categories)
<18.5 3.3 3.5 0.59
18.5–25 79.1 76.5
25–30 13.1 14.1
30+ 4.6 5.9
Maternal weight 61.6 (10.4) 61.5 (10.5) 0.95
Alcohol
Non-drinker 34.2 40.9 0.040
Drank before 18 weeks of gestation 15.6 5.7
Still drinking at 18 weeks of gestation 50.2 53.4
Paternal smoking
Non-smoker 78.6 20.9 <0.001
Smoker 21.4 79.1
aN varies according to completeness of data on baseline characteristics.
### EWAS for maternal smoking in pregnancy and cord blood methylation
In an unadjusted analysis of the associations between maternal smoking in pregnancy and cord blood epigenome-wide methylation levels, 15 CpG sites fell below the Bonferroni threshold for significance of 1.07 × 10−7 and 28 CpG sites fell below the false discovery rate (FDR) cut-off of 0.05 (Fig. 1 and Table 3). Of the CpG sites falling below the Bonferroni threshold, these were located in seven gene regions and most have been previously identified in EWAS for maternal smoking, with the top hit in AHRR (cg05575921) being consistently replicated (18,19). The effects of smoking on methylation levels were directionally consistent with previous studies (23,24) for all of these sites, with hypomethylation of sites at AHRR, GFI1 and CNTNAP2 and hypermethylation of MYO1G and CYP1A1 in the offspring of smokers compared with non-smokers. Of the CpG sites which fell below the FDR but not the Bonferroni threshold, five of these CpG sites were also located in the AHRR, GFI1, CYP1A1 and MYO1G gene regions. Other gene regions harbouring CpG sites associated with maternal smoking at Bonferroni significance were KLF13 and ATP9A and at FDR significance were GNG12, ENSG00000225718, CTNNA2, NOTCH1, ALS2CL, CHI3L1, ZNF710 and SPATS2. Sites at ATP9A, GNG12 and ENSG00000225718 have previously identified in other EWAS for maternal smoking (18,19), but the other sites appear to be novel.
Table 3.
Differential methylation in cord blood DNA for the offspring of mothers with sustained smoking in pregnancy compared with non-smokers
CpG site Chromosome Gene region Positionc Unadjusted model (N = 790)a
Effect size Standard error P-value FDR Effect size Standard error P-value FDR
cg05575921 AHRR 373 378 −0.081 0.006 1.41 × 10−30 6.59 × 10−25 −0.075 0.007 7.64 × 10−18 3.56 × 10−12
cg22132788 MYO1G 45 002 486 0.062 0.010 1.49 × 10−17 3.47 × 10−12 0.054 0.012 1.72 × 10−10 3.03 × 10−5
cg12803068 MYO1G 45 002 919 0.152 0.021 1.22 × 10−16 1.90 × 10−11 0.126 0.025 4.47 × 10−9 3.47 × 10−4
cg09935388 GFI1 92 947 588 −0.167 0.023 1.14 × 10−13 1.19 × 10−8 −0.174 0.028 1.95 × 10−10 3.03 × 10−5
cg14179389 GFI1 92 947 961 −0.078 0.012 1.28 × 10−13 1.19 × 10−8 −0.070 0.015 2.49 × 10−8 1.45 × 10−3
cg18146737 GFI1 92 946 700 −0.144 0.019 2.67 × 10−13 2.07 × 10−8 −0.141 0.024 2.10 × 10−9 1.96 × 10−4
cg05549655 15 CYP1A1 75 019 143 0.015 0.002 1.05 × 10−12 6.99 × 10−8 0.016 0.002 4.73 × 10−10 5.51 × 10−5
cg06338710 GFI1 92 946 187 −0.214 0.032 1.32 × 10−10 7.69 × 10−6 −0.200 0.039 5.93 × 10−7 0.02
cg12876356 GFI1 92 946 825 −0.179 0.029 1.60 × 10−9 8.27 × 10−5 −0.206 0.035 1.36 × 10−8 9.09 × 10−4
cg25949550 CNTNAP2 145 814 306 −0.006 0.001 3.06 × 10−9 1.43 × 10−4 −0.006 0.001 8.08 × 10−6 0.09
cg11902777 AHRR 3 68 843 −0.005 0.001 1.59 × 10−8 6.74 × 10−4 −0.004 0.001 5.62 × 10−5 0.17
cg12101586 15 CYP1A1 75 019 203 0.060 0.010 2.18 × 10−8 8.49 × 10−4 0.070 0.013 5.71 × 10−8 2.96 × 10−3
cg18316974d GFI1 92 947 035 −0.083 0.019 4.05 × 10−8 1.45 × 10−3 −0.086 0.023 4.96 × 10−6 0.08
cg26146569 15 KLF13 31 637 592 −0.072 0.013 4.61 × 10−8 1.54 × 10−3 −0.083 0.016 2.97 × 10−7 0.01
cg07339236d 20 ATP9A 50 312 490 −0.016 0.004 8.71 × 10−8 2.71 × 10−3 −0.014 0.006 3.60 × 10−4 0.24
cg09662411 GFI1 92 946 132 −0.127 0.024 1.26 × 10−7 3.66 × 10−3 −0.147 0.029 6.60 × 10−7 0.02
cg18092474 15 CYP1A1 75 019 302 0.077 0.016 2.17 × 10−7 5.94 × 10−3 0.057 0.019 5.12 × 10−4 0.24
cg04180046 MYO1G 45 002 736 0.050 0.011 2.29 × 10−7 5.94 × 10−3 0.043 0.012 4.23 × 10−5 0.16
cg25189904 GNG12 68 299 493 −0.055 0.011 4.32 × 10−7 0.01 −0.052 0.013 1.20 × 10−4 0.20
cg04598670 ENSG00000225718 68 697 651 −0.074 0.015 4.80 × 10−7 0.01 −0.074 0.018 4.35 × 10−5 0.16
cg27629977 CTNNA2 80 531 633 0.009 0.002 5.82 × 10−7 0.01 0.012 0.003 4.07 × 10−7 0.01
cg10835306 NOTCH1 139 396 760 −0.079 0.016 1.04 × 10−6 0.02 −0.061 0.019 2.34 × 10−3 0.30
cg00483459 ALS2CL 46 735 782 −0.049 0.010 1.32 × 10−6 0.03 −0.056 0.011 2.19 × 10−6 0.05
cg22549041d 15 CYP1A1 75 019 251 0.069 0.014 1.42 × 10−6 0.03 0.089 0.017 2.60 × 10−7 0.01
cg22937882d AHRR 4 05 774 0.036 0.008 1.98 × 10−6 0.04 0.036 0.011 3.13 × 10−4 0.24
cg11196333 CHI3L1 203 154 370 −0.060 0.013 2.58 × 10−6 0.05 −0.066 0.015 1.06 × 10−5 0.11
cg00624799d 15 ZNF710 90 605 618 −0.029 0.006 2.79 × 10−6 0.05 −0.037 0.007 4.19 × 10−7 0.01
cg00560284 12 SPATS2 49 783 222 −0.016 0.003 2.84 × 10−6 0.05 −0.013 0.004 1.79 × 10−3 0.29
CpG site Chromosome Gene region Positionc Unadjusted model (N = 790)a
Effect size Standard error P-value FDR Effect size Standard error P-value FDR
cg05575921 AHRR 373 378 −0.081 0.006 1.41 × 10−30 6.59 × 10−25 −0.075 0.007 7.64 × 10−18 3.56 × 10−12
cg22132788 MYO1G 45 002 486 0.062 0.010 1.49 × 10−17 3.47 × 10−12 0.054 0.012 1.72 × 10−10 3.03 × 10−5
cg12803068 MYO1G 45 002 919 0.152 0.021 1.22 × 10−16 1.90 × 10−11 0.126 0.025 4.47 × 10−9 3.47 × 10−4
cg09935388 GFI1 92 947 588 −0.167 0.023 1.14 × 10−13 1.19 × 10−8 −0.174 0.028 1.95 × 10−10 3.03 × 10−5
cg14179389 GFI1 92 947 961 −0.078 0.012 1.28 × 10−13 1.19 × 10−8 −0.070 0.015 2.49 × 10−8 1.45 × 10−3
cg18146737 GFI1 92 946 700 −0.144 0.019 2.67 × 10−13 2.07 × 10−8 −0.141 0.024 2.10 × 10−9 1.96 × 10−4
cg05549655 15 CYP1A1 75 019 143 0.015 0.002 1.05 × 10−12 6.99 × 10−8 0.016 0.002 4.73 × 10−10 5.51 × 10−5
cg06338710 GFI1 92 946 187 −0.214 0.032 1.32 × 10−10 7.69 × 10−6 −0.200 0.039 5.93 × 10−7 0.02
cg12876356 GFI1 92 946 825 −0.179 0.029 1.60 × 10−9 8.27 × 10−5 −0.206 0.035 1.36 × 10−8 9.09 × 10−4
cg25949550 CNTNAP2 145 814 306 −0.006 0.001 3.06 × 10−9 1.43 × 10−4 −0.006 0.001 8.08 × 10−6 0.09
cg11902777 AHRR 3 68 843 −0.005 0.001 1.59 × 10−8 6.74 × 10−4 −0.004 0.001 5.62 × 10−5 0.17
cg12101586 15 CYP1A1 75 019 203 0.060 0.010 2.18 × 10−8 8.49 × 10−4 0.070 0.013 5.71 × 10−8 2.96 × 10−3
cg18316974d GFI1 92 947 035 −0.083 0.019 4.05 × 10−8 1.45 × 10−3 −0.086 0.023 4.96 × 10−6 0.08
cg26146569 15 KLF13 31 637 592 −0.072 0.013 4.61 × 10−8 1.54 × 10−3 −0.083 0.016 2.97 × 10−7 0.01
cg07339236d 20 ATP9A 50 312 490 −0.016 0.004 8.71 × 10−8 2.71 × 10−3 −0.014 0.006 3.60 × 10−4 0.24
cg09662411 GFI1 92 946 132 −0.127 0.024 1.26 × 10−7 3.66 × 10−3 −0.147 0.029 6.60 × 10−7 0.02
cg18092474 15 CYP1A1 75 019 302 0.077 0.016 2.17 × 10−7 5.94 × 10−3 0.057 0.019 5.12 × 10−4 0.24
cg04180046 MYO1G 45 002 736 0.050 0.011 2.29 × 10−7 5.94 × 10−3 0.043 0.012 4.23 × 10−5 0.16
cg25189904 GNG12 68 299 493 −0.055 0.011 4.32 × 10−7 0.01 −0.052 0.013 1.20 × 10−4 0.20
cg04598670 ENSG00000225718 68 697 651 −0.074 0.015 4.80 × 10−7 0.01 −0.074 0.018 4.35 × 10−5 0.16
cg27629977 CTNNA2 80 531 633 0.009 0.002 5.82 × 10−7 0.01 0.012 0.003 4.07 × 10−7 0.01
cg10835306 NOTCH1 139 396 760 −0.079 0.016 1.04 × 10−6 0.02 −0.061 0.019 2.34 × 10−3 0.30
cg00483459 ALS2CL 46 735 782 −0.049 0.010 1.32 × 10−6 0.03 −0.056 0.011 2.19 × 10−6 0.05
cg22549041d 15 CYP1A1 75 019 251 0.069 0.014 1.42 × 10−6 0.03 0.089 0.017 2.60 × 10−7 0.01
cg22937882d AHRR 4 05 774 0.036 0.008 1.98 × 10−6 0.04 0.036 0.011 3.13 × 10−4 0.24
cg11196333 CHI3L1 203 154 370 −0.060 0.013 2.58 × 10−6 0.05 −0.066 0.015 1.06 × 10−5 0.11
cg00624799d 15 ZNF710 90 605 618 −0.029 0.006 2.79 × 10−6 0.05 −0.037 0.007 4.19 × 10−7 0.01
cg00560284 12 SPATS2 49 783 222 −0.016 0.003 2.84 × 10−6 0.05 −0.013 0.004 1.79 × 10−3 0.29
Effect size = difference in methylation level (beta) between offspring of sustained smokers and non-smokers.
aAdjusted for batch only: N = 91 sustained smokers; N = 699 non-smokers (defined as those who did not smoke pre-pregnancy or in pregnancy).
bAdjusted for maternal age, maternal education, household social class, paternal smoking, maternal alcohol in pregnancy and batch: N = 76 sustained smokers; N = 668 non-smokers (defined as those who did not smoke pre-pregnancy or in pregnancy).
cChromosomal position based on NCBI human reference genome assembly Build 37.3.
dNaeem flagged CpG site (43).
Figure 1.
Manhattan and QQ plot for epigenome-wide association study of sustained smoking in pregnancy on cord blood DNA methylation. *Results are from the analysis adjusted for batch only (N = 790).
Figure 1.
Manhattan and QQ plot for epigenome-wide association study of sustained smoking in pregnancy on cord blood DNA methylation. *Results are from the analysis adjusted for batch only (N = 790).
The sample size was reduced to 744 participants once all covariates were included in the adjusted model. Results were slightly attenuated in the model adjusting for a number of potential confounding factors and 12 probes no longer reached the FDR cut-off for epigenome-wide significance (Table 3). This reduction in the number of CpG sites reaching epigenome-wide significance with adjustment for confounders is likely due to a loss of power with a reduced sample size, because the magnitude and direction of methylation difference at all the sites were similar.
We next investigated whether any of the CpG sites that reached epigenome-wide significance in our main analysis were identified as being either single nucleotide polymorphism (SNP)-confounded or cross-hybridizing based on a comprehensive assessment reported by Naeem et al. (43). Five CpG sites identified in the original analysis were flagged by this study as sites to exclude as SNPs are known to overlap the probe region (Table 3).
Evidence for a difference in four of the six estimated cell proportions was found between non-smokers and sustained smokers (Supplementary Material, Table S1). To establish the effect of correcting for cell type, we added the predicted cell-type components as covariates in the main model. Results were largely unaltered with this adjustment (Supplementary Material, Table S2).
We also explored whether there were any sex-specific associations by stratifying the analysis based on sex of the offspring (Supplementary Material, Table S3). This analysis involved 388 boys and 402 girls. In boys, three CpG sites reached the FDR threshold for epigenome-wide significance, located in AHRR, MYO1G and CYP1A1. In girls, three CpG sites reached the FDR threshold for epigenome-wide significance, located in AHRR, MYO1G and GFI1. These same sites were among the top hits in the combined analysis. There was some evidence for an interaction by sex at AHRR (cg05575921), where the methylation change associated with sustained smoking was larger in girls than in boys and at CYP1A1 (cg05549655) where the methylation change was larger in boys than in girls. However, there was limited evidence for a difference in effect size between boys and girls at the other CpG sites in these same gene regions, providing no strong evidence for sex-specific associations.
Given that most of the CpG sites falling below the Bonferroni threshold were located within common genomic regions, we used coMET (44), a web-based plotting tool, to visualize the genomic regions of interest from our EWAS (Supplementary Material, Figs S1–S7). There was some evidence for localized clustering around the top CpG site (that with the smallest P-value in the EWAS) in AHRR, MYO1G, GFI1 and CYP1A1, although there was little evidence for strong co-methylation within the gene regions indicating independence in methylation levels at each CpG site. However, we decided to only take forward the CpG site with the smaller P-value in each gene region to focus our downstream analyses.
### Dose-dependence of cord blood methylation on maternal smoking
To investigate dose-dependent effects of maternal smoking on cord blood methylation in the offspring, we ran an exploratory analysis for the top CpG sites in each of the seven gene regions identified in the main combined analysis: AHRR (cg05575921), MYO1G (cg22132788), GFI1 (cg09935388), CYP1A1 (cg05549655), CNTNAP2 (cg25949550), KLF13 (cg26146569) and ATP9A (cg07339236). We found that cord blood methylation differences between the offspring of mothers who smoked in pregnancy compared with that of non-smokers were more extreme with both increased duration (number of trimesters; Fig. 2) and intensity (average number of cigarettes per day) of smoking in pregnancy (Fig. 3), though this trend was more pronounced at some sites than others, e.g. AHRR (cg05575921) (P = 2.7 × 10−42) versus ATP9A (cg07339236) (P = 9.9 × 10−3) for the duration of smoking in pregnancy.
Figure 2.
Methylation level (beta) at key CpG sites associated with sustained smoking in pregnancy by the duration of smoking (number of trimesters).
Figure 2.
Methylation level (beta) at key CpG sites associated with sustained smoking in pregnancy by the duration of smoking (number of trimesters).
Figure 3.
Methylation level (beta) at key CpG sites associated with sustained smoking in pregnancy by the intensity of smoking (number of cigarettes per day).
Figure 3.
Methylation level (beta) at key CpG sites associated with sustained smoking in pregnancy by the intensity of smoking (number of cigarettes per day).
### Longitudinal analysis of maternal smoking in pregnancy and offspring methylation
Longitudinal analyses were performed to investigate whether the effect of smoking on offspring methylation at birth was transient or persisted into later life. Methylation data were available for offspring in ARIES at age 7 [mean age when blood samples were taken 7.5 (SD 0.1)] and at age 17 [mean age 17.1 (SD 1.0)]. We investigated changes in methylation levels for the CpG sites that were found to be associated with maternal smoking in cord blood using multilevel modelling (Fig. 4 and Supplementary Material, Table S4). For the seven CpG sites, there were changes in methylation found during childhood, while the magnitude of change was quite small during adolescence. At CYP1A1 (cg05549655) and CNTNAP2 (cg25949550), while there was some evidence for change in methylation among the offspring of the smokers and non-smokers over time, the difference in methylation between groups persisted. Evidence for differing rates of change in methylation level between the offspring of smokers and non-smokers was found at AHRR (cg05575921), MYO1G (cg22132788), GFI1 (cg09935388) and KLF13 (cg26146569) between birth and age 7 (P-value for difference in methylation change 0.01–1 × 10−16).
Figure 4.
Longitudinal trajectories of methylation at key CpG sites in the offspring of non-smokers and sustained smokers during pregnancy from birth to age 17.
Figure 4.
Longitudinal trajectories of methylation at key CpG sites in the offspring of non-smokers and sustained smokers during pregnancy from birth to age 17.
At MYO1G (cg22132788), methylation level for the offspring of smokers deviated more from the level of the offspring of non-smokers over time, whereas at GFI1 (cg09935388) and KLF13 (cg26146569) there was some recovery of methylation towards the level of those not exposed to prenatal maternal smoke. At AHRR (cg05575921) during childhood, methylation increased at a faster rate in the offspring of smokers with evidence for a ‘catchup’ in methylation among the offspring of smokers [a 2.04% (95% CI 1.72, 2.36%) average yearly increase in methylation for the offspring of sustained smokers compared with a 1.28% (95% CI 0.97, 1.59%) increase in methylation for the offspring of non-smokers, between birth and age 7]. However, during adolescence, levels of AHRR (cg05575921) methylation decreased among both the smoker and non-smoker offspring, with methylation in the smoker offspring decreasing at a faster rate [a 0.33% (95% CI 0.26, 0.40%) average yearly decrease in methylation for the offspring of sustained smokers compared with a 0.17% (95% CI 0.12, 0.22%) decrease in methylation for the offspring of non-smokers, between age 7 and 17], leading again to a difference in methylation levels. A similar trend was found for ATP9A (cg07339236), although this was not as robust.
For the CpG sites which showed a persistent difference in methylation between the offspring of smokers and non-smokers [MYO1G (cg22132788), CYP1A1 (cg05549655) and CNTNAP2 (cg25949550); Fig. 4], we sought to determine whether the associations with maternal smoking were explained by a direct ‘critical period’ effect of smoking in pregnancy or via an indirect pathway involving postnatal smoke exposure. Identifying these underlying mechanisms of association is hampered by the high correlation (0.87) between sustained smoking in pregnancy and maternal smoking at 8 weeks postnatally. To disentangle the effect of smoking in pregnancy on offspring methylation versus smoking postnatally, we implemented a structured approach to model the effects of the binary maternal smoking exposure at three time points (in pregnancy and postnatally at 8 weeks and 61 months) on offspring methylation at age 7 (Table 4). The hypothesis for an in utero critical period is supported by data at all three CpG sites, with this model not being substantially different from the saturated model (P ≥ 0.06). However, a model for effect modification of the intrauterine exposure was also supported by the data (P ≥ 0.06) and for CYP1A1 and CNTNAP2 a critical period at 8 weeks postnatally could not be ruled out (P ≥ 0.18). As the in utero critical period model is nested within the effect modification model, we further performed a direct ANOVA test to investigate whether effect modification provided a better fit of the data than the in utero critical period model. This was found to be the case for CYP1A1 (P = 0.009), but not for MYO1G or CNTNAP2 (P ≥ 0.22), where the in utero critical period was found to be the best model (Table 4).
Table 4.
An exploration of critical period and other lifecourse effects that may underlie the persistence of associations between maternal smoke exposure in pregnancy and offspring methylation at age 7
Lifecourse hypothesis MYO1G (cg22132788)
CYP1A1 (cg05549655)
CNTNAP2 (cg25949550)
F-statistic P-value F-statistic P-value F-statistic P-value
Critical period in pregnancy 0.68 0.66 1.62 0.14 2.05 0.06
Critical period at 8 weeks postnatally 3.74 0.001 1.48 0.18 1.00 0.42
Critical period at 61 months postnatally 4.72 9.74 × 10−5 5.07 4.02 × 10−5 3.35 0.003
Accumulation of risk over time 3.89 7.77 × 10−4 5.83 5.77 × 10−6 4.19 3.68 × 10−4
Effect modification of critical period in pregnancy by postnatal exposure 0.82 0.51 0.08 0.98 2.32 0.06
Critical period in pregnancy nested within effect modification modela 0.40 0.67 4.8 0.009 1.50 0.22
Lifecourse hypothesis MYO1G (cg22132788)
CYP1A1 (cg05549655)
CNTNAP2 (cg25949550)
F-statistic P-value F-statistic P-value F-statistic P-value
Critical period in pregnancy 0.68 0.66 1.62 0.14 2.05 0.06
Critical period at 8 weeks postnatally 3.74 0.001 1.48 0.18 1.00 0.42
Critical period at 61 months postnatally 4.72 9.74 × 10−5 5.07 4.02 × 10−5 3.35 0.003
Accumulation of risk over time 3.89 7.77 × 10−4 5.83 5.77 × 10−6 4.19 3.68 × 10−4
Effect modification of critical period in pregnancy by postnatal exposure 0.82 0.51 0.08 0.98 2.32 0.06
Critical period in pregnancy nested within effect modification modela 0.40 0.67 4.8 0.009 1.50 0.22
Results of ANOVA test against a saturated model; a smaller F-statistic (and larger P-value) provides evidence of a better fit of data to the hypothesized model.
aResults of ANOVA test against effect modification model.
For the CpG sites which showed evidence of methylation difference between the offspring of smokers and non-smokers at age 17 [AHRR (cg05575921), MYO1G (cg22132788), CYP1A1 (cg05549655) and CNTNAP2 (cg25949550); Fig. 4], we followed up the association between methylation at these sites and own smoking among the adolescents (Supplementary Material, Table S4). The correlation between sustained smoking in pregnancy and own smoking (of the adolescent) was found to be 0.16. For the majority of CpG sites for which the association with smoking in pregnancy was evident at age 17, there was directional consistency of the association between own smoking and methylation at this time point. However, the magnitude of the association with own smoking was smaller than for the sustained maternal prenatal smoking analysis at all CpG sites, with the exception of AHRR (cg05575921), where the effect size for sustained smoking in pregnancy was −3.6% (95% CI −4.6, −2.6%) compared with −3.4% (95% CI −4.4, 2.4%) for the own smoking analysis, suggesting that not all of the association between maternal smoking in pregnancy and age 17 methylation can be explained through the mediating role of the adolescent's own smoking.
We next investigated the associations between sustained smoking in pregnancy and methylation at age 17, excluding those offspring who reported smoking themselves (Supplementary Material, Table S5). The magnitude and direction of association of the CpG sites were comparable to those in the full analysis, providing more evidence that own smoking is not fully mediating the observed association between maternal smoking in pregnancy and offspring methylation at age 17. In addition, this was with the notable exception of CpG sites at AHRR, where the effect size was halved [from −3.6% (95% CI −4.6, −2.6%) to −1.8% (95% CI 0.0, −3.6%)] when adolescents who reported smoking were excluded from the analysis. This provides some indication that personal smoking by adolescents and its correlation with maternal smoking could be driving the apparent persistent methylation difference in this gene region at age 17.
### Assessing causality of intrauterine associations using paternal smoking as a negative control
Finally, parental comparisons of associations between smoking during pregnancy and methylation levels at the top CpG sites showed consistently larger effect estimates for prenatal maternal smoking than for paternal smoking at all three time points (Fig. 5). In addition, adjusting for paternal smoking in maternal associations made little difference to affect estimates while adjusting for maternal smoking attenuated all paternal associations. For example, in the analysis of methylation in cord blood, any smoking by mothers during pregnancy was associated with a 6.1% (95% CI −7.1, −5.1%) reduction in cord blood methylation at cg05575921 (AHRR), which was not substantially attenuated with adjustment for partner's smoking (−5.6%; 95% CI −6.7, −4.5%). Smoking by partners during pregnancy was associated with a 2.1 (−2.8, −1.3%) reduction in cord blood methylation at cg05575921 (AHRR), which was fully attenuated with adjustment for maternal smoking (−0.01%; 95% CI −0.01, 0.00%).
Figure 5.
Parental comparisons of associations between any versus no smoking in pregnancy on offspring methylation at CpG sites most associated with sustained maternal smoking in pregnancy at all three time points.
Figure 5.
Parental comparisons of associations between any versus no smoking in pregnancy on offspring methylation at CpG sites most associated with sustained maternal smoking in pregnancy at all three time points.
## Discussion
In a large longitudinal birth cohort with genome-wide methylation measured at three different time points in the offspring, we first identified 15 CpG sites that were differentially methylated in cord blood at birth. These sites are located in seven gene regions, six of which have been previously identified in other EWAS for exposure to maternal smoking in utero (18,19). The top hit in this analysis was located within the AHRR [aryl hydrocarbon receptor (AHR) repressor] gene. CpG sites located in this gene region have previously been shown to be differentially methylated in smokers in several studies (13–18,30). In particular, the top hit in this analysis (cg05575921, P = 1.41 × 10−30) was identified in previous studies including an epigenome-wide association study for maternal smoking and both cord and neonatal blood DNA methylation (18).
At this site, an 8.1% (95% CI 6.9, 9.3%) reduction in cord blood methylation with sustained prenatal smoking exposure was identified, which is in line with the median methylation difference of medium and high cotinine versus no exposure in a previous EWAS (18), which was 5.4 and 9.9%, respectively. These associations were largely robust to adjustment for a number of genetic, environmental and cell-type specific confounding factors, supporting a causal effect of maternal smoking during pregnancy on offspring methylation at birth. However, for two of the CpG sites followed up in downstream analysis, CNTNAP2 (cg25949550) and ATP9A (cg07339236), the FDR P-values were 0.09 and 0.24, respectively, in the adjusted model, but effect estimates were largely unchanged between the adjusted and unadjusted model. In addition, ATP9A (cg07339236) was flagged up in Table 3 as a low-quality probe based on a comprehensive assessment reported by Naeem et al. (43).
A regional analysis of EWAS hits provided some evidence for localized clustering around the top CpG site (that with the smallest P-value in the EWAS) in AHRR, MYO1G, GFI1 and CYP1A1, although there was little evidence for strong co-methylation within the gene regions indicating independence of methylation levels at each CpG site, supporting our use of single site analysis in the EWAS.
We assessed the biological gradient of smoke exposure in pregnancy and identified a dose-dependent response of methylation with both increased intensity and duration of smoking. In this analysis, we found that methylation in the offspring of mothers who smoked only in one trimester, namely the first, was largely comparable to that of unexposed offspring. These findings are in line with previous studies, which showed no difference in mean methylation at AHRR between mothers who never smoked and those who smoked early in their pregnancy (20,23), suggesting that sustained exposure to maternal smoking in utero is required to induce changes in methylation which are detectable in cord blood. In contrast to the view that early pregnancy represents a critical window for environmentally induced epigenetic change, epigenetic reprogramming appears to occur throughout prenatal development and postnatally (45) and these findings imply a cumulative effect of smoke exposure throughout pregnancy on offspring methylation at birth.
Nonetheless, knowledge of smoking intensity in the first trimester is important as a predictive marker of smoking later in pregnancy and hence of epigenetic change in the offspring. This has been confirmed in an analysis of 374 ARIES mothers–child pairs for whom urinary cotinine was collected in the first trimester of pregnancy. An EWAS of maternal cotinine levels and cord blood methylation in this subsample was able to identify a signal at AHRR (cg05575921), which surpassed the Bonferroni threshold (P-value = 3.31 × 10−8; Supplementary Material, Fig. S8).
Whether pregnancy represents a critical period for determining offspring methylation patterns at later time points in childhood in response to maternal smoking was also investigated. A longitudinal assessment of methylation marks associated with maternal smoke exposure in pregnancy found that whereas some CpG sites showed recovery of methylation to the level of offspring not exposed (GFI1, KLF13 and ATP9A), other sites showed persistently perturbed patterns (AHRR, MYO1G, CYP1A1 and CNTNAP2).
This prospective study design coupled with serial sampling at multiple time points provides powerful evidence of the persistence of DNA methylation changes induced in utero. In addition, longitudinal modelling of the effects of exposure ‘windows’ provides evidence that prenatal exposure to smoking has persistent effects on later offspring DNA methylation, which outweighs the postnatal influence of maternal smoking or own smoking in adolescence at some CpG sites. Effect modification of prenatal exposure at later time points was also evident, indicating that postnatal exposure might have some further impact on the persistence of the methylation marks in those exposed to smoking in utero. However, there was no strong evidence that the effect modification model was more consistent with the observed data than the in utero critical period model at most sites. These observations are consistent with and significantly extend previous analyses of long-term smoking-induced perturbations of DNA methylation (15,23,24,26). One exception to this is the finding that own smoking by the offspring at age 17 is strongly associated with AHRR methylation at this time point which might therefore be underlying the apparently persistent effect of maternal smoking in pregnancy at this gene region, as shown in Figure 4. This is perhaps because this site is most sensitive to smoking exposure and would therefore detect adolescent own smoking most readily.
In addition, the use of paternal smoking as a negative control demonstrates the biological effect of this in utero exposure at all time points considered. While methylation differences identified between maternal and paternal smoking at later time points might be attributed to the differential influences of these exposures postnatally, the similar trends identified in cord blood when no influence of the postnatal environment had been present provide further support for the causal effect of maternal smoking in pregnancy on offspring methylation. In addition, the low levels of tobacco exposure from partner smoking in non-smoking pregnant women in this cohort suggest that the use of partner's smoking as a negative control for investigating intrauterine effects is valid (40). Mendelian randomization is another technique that may be used to bolster causal inference in this context (35,46,47). A SNP, rs1051730, located in the CHRNA5-CHRNA3-CHRNB4 nicotinic acetylcholine receptor gene cluster (chromosome 15q25), is robustly associated with smoking heaviness (48) and has also been associated with a reduced ability of women to quit smoking in pregnancy (49). If an association was observed between maternal rs1051730 and offspring DNA methylation of mothers who smoked during pregnancy, this would provide further evidence of an intrauterine effect. However, we were underpowered to investigate this formally within our sample of sustained smokers.
Strengths of our study include the application of the Illumina Infinium® HM450 BeadChip technology to assess genome-wide methylation profiles at multiple time points from birth until late adolescence in a large, longitudinal cohort study. The wealth of phenotypic data in ALSPAC has aided the thorough assessment of potential confounding factors, a detailed analysis of the dose-dependence of methylation to smoke exposure in utero and an investigation into the relative roles of intrauterine and postnatal smoking using questionnaire data on smoking habits taken from multiple time points in both parents and offspring. Longitudinal modelling (50,51) and robust statistical methods (34,39,52) have also been used to strengthen causal inference.
Limitations of the analysis include differential follow-up of smokers compared with non-smokers, where only 14.3% of mothers in the ARIES sample (selected based on blood sample availability up to 17 years postnatally) smoked in pregnancy compared with 30.2% in the wider cohort. Technical limitations relate to the HM450 BeadChip in that it covers only 1.7% of CpG sites across the genome. A more comprehensive appraisal may elicit additional relationships between the exposure in question and DNA methylation, or indeed locus-specific paternal effects which were not evident here (53–55). In addition, the use of questionnaires to obtain data on parental smoking may result in under-reporting of smoking behaviour. However, to minimize the influence of maternal under-reporting, we excluded from the analyses individuals who had reported smoking before but not during pregnancy. In addition, a strong correlation between self-reported questionnaire data on smoking behaviour and plasma cotinine levels has been found (18,40) and maternal cotinine was found to be highly correlated with the sustained smoking variable (r = 0.76) in the subsample of ARIES participants with first trimester maternal cotinine data available (N = 323).
This analysis was limited to blood samples with mixed cell composition. Although no differences were found in the analysis with estimated cell-type correction, as has been shown previously (18,19), it is unclear how effective the method used to correct for cell-type proportions is in these samples since the reference data sets are available only for adult peripheral blood (56). In addition, some of the DNA samples included in this analysis came from buffy coats rather than whole blood and there is no reference cell-type correction available for buffy coat DNA. It should also be emphasized that the associations identified may be specific to blood as an analysis of buccal epithelium and placenta did not identify the same smoking-associated methylation differences in these tissues (23). This limitation of tissue specificity, as well as the lack of expression data currently available on these samples, limits the assessment of functional consequences of these methylation changes.
Given evidence for causal associations between maternal smoking in pregnancy and methylation changes in the offspring, it is important to consider whether these induced changes are also associated with the adverse perinatal and offspring outcomes associated with exposure to smoking in utero. Further work is required which may link smoking-responsive DNA methylation variation to health and development (57–59). In addition, whether DNA methylation is a true mediating mechanism of these associations or simply an exposure indicator may be explored by extending causal inference (35,47,60).
Transient environmental exposures during critical windows of development are known to affect the establishment of epigenetic marks, which are evident at birth (57) and may persist until later life (58,59). Findings from this study highlight the sensitivity of the methylome to maternal smoking during fetal development and the long-term impact of such an exposure. Results strengthen causal inference in this area and the finding that sustained smoking appears to be necessary to induce methylation changes has profound implications for antenatal care and the long-term effects on offspring health, directing potential intervention strategies at cessation of smoking early in pregnancy, such as at the first antenatal appointment.
These findings could also have very useful applications in epidemiological studies. Given the magnitude and persistence of methylation change in relation to maternal exposure in utero, it is important to consider the potential confounding effect of maternal smoking in future EWAS studies attempting to identify sites associated with own smoking. The persistence of some (but not all) methylation marks at later time points presents the opportunity to use methylation signatures as an archive of historical exposure, particularly if methylation patterns can be robustly modelled over time and previous exposure inferred. A simplistic example would be to define prenatal smoking exposure, using DNA methylation signatures in children of women where no smoking history had been collected. In addition, the contrast between stable and reversible sites may be useful in discriminating between in utero exposure and later life exposure.
## Materials and Methods
### Study design
We examined offspring DNA methylation in relation to self-reported maternal smoking during pregnancy in a subset of participants from ALSPAC using methylation data from the Illumina Infinium® HumanMethylation450 BeadChip assay (Illumina, Inc., CA, USA) (61).
### Cohort and selection of participants
ALSPAC is a large, prospective cohort study based in the South West of England. A total of 14 541 pregnant women resident in Avon, UK, with expected dates of delivery 1 April 1991 to 31 December 1992 were recruited and detailed information has been collected on these women and their offspring at regular intervals (41,42). The study website contains details of all the data that are available through a fully searchable data dictionary (http://www.bris.ac.uk/alspac/researchers/data-access/data-dictionary/).
As part of the ARIES (http://www.ariesepigenomics.org.uk/) project, the Infinium HM450 BeadChip has been used to generate epigenetic data on 1018 mother–offspring pairs in the ALSPAC cohort. The ARIES participants were selected based on availability of DNA samples at two time points for the mother (antenatal and at follow-up when the offspring was in adolescence) and at three time points for the offspring [neonatal, childhood (age 7) and adolescence (age 17)].
Written informed consent has been obtained for all ALSPAC participants. Ethical approval for the study was obtained from the ALSPAC Ethics and Law Committee and the Local Research Ethics Committees.
### Laboratory methods, quality control and pre-processing
Cord blood and peripheral blood samples (whole blood, buffy coats or blood spots) were collected according to standard procedures. The DNA methylation wet laboratory and pre-processing analyses were performed at the University of Bristol as part of the ARIES project. Following extraction, DNA was bisulphite-converted using the Zymo EZ DNA MethylationTM kit (Zymo, Irvine, CA, USA). Following conversion, genome-wide methylation status of over 485 000 CpG sites was measured using the Infinium HM450 BeadChip according to the standard protocol. The arrays were scanned using an Illumina iScan and initial quality review was assessed using GenomeStudio (version 2011.1).
The Infinium HM450 BeadChip assay detects the proportion of molecules methylated at each CpG site on the array. For the samples, the methylation level at each CpG site was calculated as a beta value (β), which is the ratio of the methylated probe intensity and the overall intensity and ranges from 0 (no cytosine methylation) to 1 (complete cytosine methylation) (62,63). Methylation data were pre-processed using in R (version 3.0.1), with background correction and subset quantile normalization performed using the pipeline described by Touleimat and Tost (64).
Samples from all time points in ARIES were distributed across slides using a semi-random approach (sampling criteria were in place to ensure that all time points were represented on each array) to minimize the possibility of confounding by batch effects. In addition, during the data generation process a wide range of batch variables were recorded in a purpose-built laboratory information management system (LIMS). The main batch variable was found to be the bisulphite conversion (BCD) plate number. Samples were converted in batches of 48 samples and each batch identified by a plate number.
The LIMS also reported QC metrics from the standard control probes on the 450K BeadChip for each sample. Samples failing QC (average probe P-value of ≥0.01) were repeated and if unsuccessful excluded from further analysis. As an additional QC step genotype probes were compared with SNP-chip data from the same individual to identify and remove any sample mismatches. For individuals with no genome-wide SNP data, samples were flagged if there was a sex-mismatch based on X-chromosome methylation.
In addition to these QC steps, probes that contained <95% of signals detectable above background signal (detection P-value of <0.01; N = 7938) were excluded from analysis. After excluding these probes, as well as control probes and probes on sex chromosomes, a total of 466 432 CpG sites were included in the main analysis for cord blood methylation. At age 7, 471 347 CpG sites were included and at age 17, 469 902 CpG sites were included in the main analysis, following the same exclusion criteria.
### In utero exposure variables
Information on mothers' smoking status during pregnancy was obtained in questionnaires administered at 18 and 32 weeks of gestation. Information was obtained about whether the mother smoked in each trimester of pregnancy and the number of cigarettes smoked on average per day. From these data, a dichotomous variable for sustained maternal smoking during pregnancy was derived. A mother was classified as a sustained smoker if she smoked in all three trimesters, smoked in the first and third trimester but not the second or smoked in the second and third trimesters but not the first. The reference group consisted of mothers who had reported not smoking in all three trimesters or before pregnancy. We excluded all individuals who smoked in one trimester only (i.e. not sustained) or who had missing information of smoking for two or more trimesters. Of those with missing information on one trimester, women were classified as a sustained smoker if they said they smoked in the other two trimesters.
For investigating the dose-dependent effects of maternal smoking in pregnancy on DNA methylation in cord blood, a variable was derived for the duration of smoking in pregnancy (0, 1, 2 or all three trimesters) as well as the intensity of smoking in pregnancy (0, 1–4, 5–9, 10–14 and 15+ cigarettes/day).
Data on cotinine levels were available for a small subset of the ARIES mothers (n = 374). Cotinine levels (ng/ml) were assessed from a single urine sample taken during the first trimester of pregnancy. For most mothers, the samples were collected as part of routine clinical care but some samples were obtained specifically for ALSPAC. Urine samples were stored at −20°C and allowed to thaw at room temperature before use. Cotinine was measured using the Cozart Cotinine Enzyme Immunoassay (Concateno UK, Abingdon) urine kit. Where required, samples were diluted using cotinine-free serum (fetal calf serum). Absorbance was measured spectrophotometrically at a wavelength of 450 nm. Maternal cotinine levels were categorized into four groups: <70, 70–900, 900–3000 and >3000 ng/ml, which roughly correspond with self-reported non-smoking, 1–4 cigarettes per day, 5–14 cigarettes per day and >14 cigarettes per day, respectively (40).
### Offspring methylation outcome
The main outcome measure in this analysis was DNA methylation level at each of the CpG sites in cord blood samples. However, we also undertook an EWAS for maternal prenatal smoking in samples of peripheral blood when the children were age 7 and 17 years and followed up sites that reached genome-wide significance to investigate the persistence of methylation marks in the offspring over time.
### Confounders
Variables considered as potential confounders in this analysis were maternal age, pre-pregnancy BMI, pre-pregnancy weight, parity, educational attainment, social class, alcohol intake and paternal smoking. Maternal age at delivery was derived from date of birth, which was recorded at that time. At enrolment, the mother was asked to record her height and pre-pregnancy weight, from which BMI was calculated. Mother's parity was also recorded in a questionnaire completed during pregnancy. Based on questionnaire responses, highest educational qualification for the mother was collapsed into one of the five categories from none/ Certificate of Secondary Education (CSE, the lower level of two national school exams that were taken when these women were in school at age 16) to university degree. In addition, the highest parental occupation was used to allocate the children to family social class groups using the 1991 British Office of Population Censuses and Surveys classification. Self-reported alcohol use was obtained in the questionnaire administered at 18 weeks of gestation and individuals were categorized based on whether they were non-drinkers, drank before 18 weeks of gestation or were still drinking alcohol at 18 weeks of gestation. Information on partners’ smoking during pregnancy was obtained from self-reports at 18 weeks of gestation. Where self-reported data on partner smoking were not available (16.3% of partners), maternal reports were used. The bisulphite conversion batch for each sample was also included in the analysis to adjust for batch effects.
### Statistical analysis
Using offspring DNA samples taken from cord blood (at birth), we investigated methylation levels at 466 488 CpG sites across the genome. Methylation β values at each CpG site were transformed to obtain M-values [log2(β/(1 − β)] for statistical analysis (62).
Multivariable linear regression was used to perform association tests between maternal cigarette smoking and M-values at each CpG site as the outcome. The main exposure measure in our analysis was sustained smoking in pregnancy versus no smoking and the main outcome was cord blood DNA methylation level. Analyses were run with and without adjustment for a number of potential confounders found to be associated with smoking status in pregnancy (Table 2). DNA methylation sites were annotated based on data provided by Illumina (63).
We first identified ‘EWAS-significant’ hits using a Bonferroni correction, where associations below a threshold of 1.07 × 10−7 were considered a likely true positive worthy of further examination. However, this Bonferroni correction assumes independent tests and so, as correlation of DNA methylation within gene regions means that CpG sites may not be truly independent, a less conservative FDR procedure based on the Benjamini–Hochberg method was also used to account for multiple testing (65). For this, CpG sites with FDR less than a 0.05 threshold were labelled as EWAS-significant.
It has been demonstrated that differences in methylation can arise as a result of variability of cell composition in whole blood (56). As smoking is known to influence cell composition (66), in order to ensure the results are not influenced by variation in cell-type fraction between samples, we estimated the fraction of CD8T-, CD4T-, NK- and B-cells, monocytes and granulocytes in the samples using the estimateCellCounts function in the minfi Bioconductor package implemented in R (67). We investigated differences in estimated cell count by smoking status and analyses were repeated adjusting for cell composition by including each blood cell fraction as a covariate in the multivariate linear regression.
Given the previous evidence for sex-specific DNA methylation differences in relation to prenatal smoke exposure (68), we undertook EWAS stratified by sex of the offspring and investigated whether there were any sex-specific CpG sites found to be associated with maternal smoke exposure.
We next used a web-based plotting tool, coMET (44), to investigate the genomic regions of interest from our main EWAS analysis. This tool permits the visualization of methylation correlation between CpG sites, which was limited to a maximum of 75 CpG sites around to the top site of interest and within the gene region identified. In addition, the plots were annotated with functional genomic features based on the ENCODE project (geneENSEMBL, CGI, ChromHMM, DNAse, RegENSEMBL and SNPs).
Further analyses were performed to investigate whether the level of methylation differed depending on the duration and intensity of smoking to which the offspring were exposed in utero. For this, the untransformed methylation β values for the top CpG sites in each gene region reaching genome-wide significance were plotted against a variable for the duration of smoking in pregnancy as well as the intensity of smoking in pregnancy.
We also investigated whether the methylation alterations associated with prenatal exposure to maternal smoking persisted when the offspring were age 7 and 17 years. Longitudinal methylation data were extracted from each of probe which exceeded the Bonferroni threshold in cord blood. A multilevel model (50,51) including a random intercept and a linear regression spline term to allow for flexibility was fitted to each of these CpG sites sequentially:
$methij=β0+u0i+β1sustainedi+β2ageij+β3(ageij−7)++β4sustainediageij+β5sustainedi(ageij−7)++confounders+ϵij$
$ϵij~N(0,σϵ2)$
$u0i~N(0,σu2)$
where i = 1, … (770) indexes the offspring in the analyses, j = 1, 2, 3 indexes the measurement occasion and a+ = a if a > 0 or 0 otherwise. β1 gives the average difference between smoker and non-smoker offspring; β2 gives the average change in methylation from birth to adolescence; β3 tells us whether there is any change in this trend (i.e. β2) from childhood to adolescence; β4 tells us whether there is a difference in methylation change between smoker and non-smoker offspring and β5 tells us whether offspring of smokers and non-smokers have a different change to the trend (i.e. β2) of methylation change from birth to childhood. From these we can calculate the change in methylation from 0 to 7 for children of non-smokers (β2) and smokers (β2+ β4), and the change from 7 to 17 for children of non-smokers (β2 + β3) and smokers (β2 + β3 + β4 + β5). For each CpG site, we used a multilevel model, adjusting for batch and the first 20 independent surrogate variable components (which account for heterogeneity between the cord blood and peripheral blood samples).
Strategies were then implemented to estimate the potential role of non-intrauterine mechanisms in the observed associations at later time points. We first considered the potential role of postnatal parental smoking in explaining the persistence of methylation differences at age 7 and both parental and own smoking at age 17 in the offspring of mothers who smoked compared with the offspring of mothers who did not smoke in pregnancy. Additional information about mothers’ smoking status postnatally was obtained in several questionnaires administered after birth, including 8 weeks postpartum and 61 months postpartum. In addition, information about own smoking status was obtained in questionnaires completed by the offspring when they were age 17.
For offspring methylation at age 7, we wished to disentangle a potential causal effect of maternal smoke exposure in utero (i.e. a ‘critical period’ hypothesis) from other lifecourse effects, including the existence of postnatal critical periods of maternal smoke exposure, an accumulation of risk with exposure over time or effect modification of in utero exposure by postnatal exposure (52). We implemented a structured approach to model the effects of the binary maternal smoking exposure at three time points (in pregnancy and postnatally at 8 weeks and 61 months) on offspring methylation. This involved first fitting a saturated model with one coefficient for each combination of exposures at the three time points (maternal smoking in pregnancy × 8 weeks postnatally × 61 months postnatally) using the lm function in R (version 3.0.1). We then specified a series of nested models corresponding to each lifecourse hypothesis to be tested against the saturated model using an ANOVA test, with a smaller F-statistic and a larger P-value indicating a better fit of the data to that model. Nested models considered were an in utero critical period (maternal smoking in pregnancy), later life critical periods (maternal smoking 8 weeks and maternal smoking 61 months postnatally), accumulation of risk across the three time points (maternal smoking in pregnancy + 8 weeks postnatally + 61 months postnatally) and effect modification of in utero exposure postnatally by smoking at the later time points [maternal smoking in pregnancy + (smoking in pregnancy : 8 weeks postnatally) + (maternal smoking in pregnancy : 61 months postnatally)].
For methylation at age 17, we also considered the potential influence of own smoking by the offspring in explaining persistence in methylation signatures associated with intrauterine exposure by running the same multivariable linear regressions this time between own smoking status and methylation as the outcome. From the questionnaires administered when the adolescents were age 17, adolescents who reported that they smoked more than one cigarette per week were classified as smokers and those who said they had never tried a cigarette at either time point were classified as non-smokers and used as the reference category. For this analysis, we did a look-up of the top hits in the maternal smoking analysis in relation to own smoking in order to contrast effect sizes for personal versus maternal smoking associations with methylation at this time point. In addition, we repeated the main analysis at this time point excluding those offspring who reported own smoking to investigate whether this had any influence on the results.
Finally, we compared associations of mothers' and mothers' partners' smoking during pregnancy with offspring methylation at the three time points (birth, age 7 and age 17), using partner smoking during pregnancy as a negative control (34,36–40). Information on partners' smoking status during pregnancy was obtained in a questionnaire administered at 18 weeks of gestation. In addition, mothers were asked about their partner's current smoking at 18 weeks of gestation. The correlation between partner self-report and maternal report of partner smoking was high (r = 0.95) and therefore maternal report was used when partners' self-report information was missing. Mutually adjusted models were built by including both maternal and partner smoking to account for potential confounding by the smoking behaviour of the other parent.
EWAS were performed using the ‘CpG assoc’ package (version 2.11) implemented in R (version 3.0.1), multilevel modelling was performed using Stata (version 13) and coMET was run via the web interface (http://epigen.kcl.ac.uk/comet/upload.html). All other analyses were implemented in R (version 3.0.1).
## Funding
This work was supported by the UK Biotechnology and Biological Sciences Research Council (BB/I025751/1 and BB/I025263/1); the UK Medical Research Council and University of Bristol (MC_UU_12013); the Wellcome Trust (WT083431MF to R.C.R.); the Economic and Social Research Council (RES-060–23-0011 to G.D.S. and C.L.R.) and the European Research Council (DEVHEALTH 269874 to G.D.S.). Funding to pay the Open Access publication charges for this article was provided by the University of Bristol RCUK.
## Acknowledgements
We are extremely grateful to all the families who took part in this study, the midwives for their help in recruiting them and the whole ALSPAC team, which includes interviewers, computer and laboratory technicians, clerical workers, research scientists, volunteers, managers, receptionists and nurses. We thank Jon Heron, Marcus Munafò and Amy Taylor for their helpful comments on an earlier draft of this paper, and Matthew Suderman and Tiphaine Martin for sharing their computational expertise.
Conflict of Interest statement. None declared.
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## Author notes
These authors contributed equally to the work.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited | 2017-01-24 05:04:42 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 3, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2808346748352051, "perplexity": 5564.177068626812}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-04/segments/1484560284270.95/warc/CC-MAIN-20170116095124-00364-ip-10-171-10-70.ec2.internal.warc.gz"} |
https://fr.maplesoft.com/support/help/maple/view.aspx?path=updates%2FMaple10%2Findex | Index of New Features - Maple Help
Home : Support : Online Help : System : Information : Updates : Maple 10 : Index of New Features
Index of New Maple 10 Features
Maple 10 contains many new capabilities and improvements to existing facilities. For a summary, see What's New in Maple 10?.
The important changes in Maple 10 are described in the following topics.
• New packages and modifications to pre-existing packages. For more information, see updates/Maple10/packages.
• Changes to the Maple programming facilities. For more information, see updates/Maple10/programming.
• Expanded visualization and graphing capabilities. For more information, see updates/Maple10/graphics.
• Improvements to specific applications of the Maple numeric computation environment. For more information, see updates/Maple10/numerics.
• Improved efficiency through increased speed and reduced memory use. For more information, see updates/Maple10/efficiency.
• Significant improvements to the suite of differential equation solvers. For more information, see updates/Maple10/differentialequations.
• Significant additions and improvements to the symbolic capabilities in various Maple packages and procedures. For more information, see updates/Maple10/symbolic.
• Improvements to the Maple graphical user interface (GUI). For more information, see updates/Maple10/gui. | 2022-06-30 05:48:18 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8270034790039062, "perplexity": 2473.8204324680023}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103661137.41/warc/CC-MAIN-20220630031950-20220630061950-00763.warc.gz"} |
http://mathhelpforum.com/trigonometry/158573-solve-2sin-x-45-1-a.html | # Thread: solve 2sin(x-45)=1
1. ## solve 2sin(x-45)=1
Solve the equation $\displaystyle 2sin(x-45^0) = 1$ for $\displaystyle 0^0<x<360^0$
2. Just work your way backwards:
$\displaystyle 2\sin(x-45^{\circ}) = 1$
$\displaystyle \sin(x-45^{\circ}) = \frac{1}{2}$
$\displaystyle x-45^{\circ} = \sin^{-1}\left(\frac{1}{2}\right)$
Can you carry on?
3. Solve it the way you normally would
$\displaystyle 2\sin{(x - 45^{\circ})} = 1$
$\displaystyle \sin{(x - 45^{\circ})} = \frac{1}{2}$
$\displaystyle x - 45^{\circ} = \left\{30^{\circ}, 180^{\circ} - 30^{\circ}\right\}$
$\displaystyle x - 45^{\circ} = \left\{30^{\circ}, 150^{\circ}\right\}$
$\displaystyle x = \left\{30^{\circ} + 45^{\circ}, 150^{\circ} + 45^{\circ}\right\}$
$\displaystyle x = \left\{75^{\circ}, 195^{\circ}\right\}$. | 2018-04-24 01:43:11 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9248439073562622, "perplexity": 11588.215263480804}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125946314.70/warc/CC-MAIN-20180424002843-20180424022843-00215.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/college-algebra-10th-edition/chapter-2-section-2-3-lines-2-3-assess-your-understanding-page-179/79 | # Chapter 2 - Section 2.3 - Lines - 2.3 Assess Your Understanding: 79
$\text{slope}=m=-\frac{1}{2}; \\\text{y-intercept}=b=2$ Refer to the image below for the graph.
#### Work Step by Step
RECALL: The slope-intercept form of a line's equation is: $y=mx+b$ where $m$ = slope and $b$ = y-intercept Write the given equation in slope-intercept form by isolating $y$ on one side to obtain: $x+2y=4 \\2y = -x+4 \\y=\frac{-x+4}{2} \\y = -\frac{1}{2}x+2$ The equation above has: $\text{slope}=m=-\frac{1}{2}; \\\text{y-intercept}=b=2$ To graph the line, perform the following steps: (1) Plot the y-intercept point $(0, 2)$. (2) Use the slope $-\frac{1}{2}$ to find another point on the line. From $(0, 2)$, move down 1 unit (the rise) and move 2 units to the right (the run) to arrive at the point $(2, 1)$. (3) Complete the graph by connecting the two points using a straight line. (refer to the attached image in the answer part above for the graph)
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2018-04-25 12:55:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7267943620681763, "perplexity": 477.9225315362848}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125947803.66/warc/CC-MAIN-20180425115743-20180425135743-00572.warc.gz"} |
https://ask.sagemath.org/question/8470/evaluating-an-expression/ | evaluating an expression
This post is a wiki. Anyone with karma >750 is welcome to improve it.
Hello,
started to use open source mathematical software. And i have some questions since i can not find in google what i am searching for (maybe i am a bad searcher :S).
The thing is if i write in maple (with predefined F, M, q and Q):
Sum('F[ix]', 'i' = 1 .. n) = 0; '-2*F*cos(45*deg)-3*Q+4*RB*cos(45*deg)-RB*sin(45*deg)-M = 0'; evalf[4](solve(%, RB))
I get
RB := 19.51
And in Sage:
solve(-2*F*cos(45*deg)-3*Q+4*RB*cos(45*deg)-RB*sin(45*deg)-M,RB)
i get:
[23/3*sqrt(2) + 26/3 == 23/3*sqrt(2) + 26/3]
Now i do not find any equivalent to evalf in sage. Could someone help me, please?
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Hi! I think you must have assigned one too many of your variables (like assigning a value to RB) to come up with your last expression. Also, I don't see what predefined, lowercase q you're referring to.
In any case, here's roughly what I'd do. First, define the variables and the equation, and then solve it:
sage: var("F, Q, M, RB")
(F, Q, M, RB)
sage: deg = pi/180;
sage:
sage: eq = -2*F*cos(45*deg)-3*Q+4*RB*cos(45*deg)-RB*sin(45*deg)-M == 0
sage: eq
-sqrt(2)*F + 3/2*sqrt(2)*RB - M - 3*Q == 0
sage:
sage: solve(eq,RB)
[RB == 1/3*sqrt(2)*M + sqrt(2)*Q + 2/3*F]
sage: for sol in solve(eq,RB):
....: print sol.rhs()
....:
1/3*sqrt(2)*M + sqrt(2)*Q + 2/3*F
Then solve it. By default, solve returns a list of equations, so you can loop over them like I did or index into them using [0], [1], [2], etc. I prefer the solution_dict style:
sage: solve(eq,RB, solution_dict=True)
[{RB: 1/3*sqrt(2)*M + sqrt(2)*Q + 2/3*F}]
sage: sols = solve(eq,RB, solution_dict=True)
sage: sols[0][RB]
1/3*sqrt(2)*M + sqrt(2)*Q + 2/3*F
in which you get a list of dictionaries, with each dictionary corresponding to a possible solution, and then you simply use the variable name as the index. We can then extract a solution:
sage: rb = sols[0][RB]
sage: rbval = rb.subs(M=23, Q=6, F=1/4)
sage: rbval
41/3*sqrt(2) + 1/6
and I've substituted some basically random M, Q, and F values into it. This can then be evaluated numerically (the evalf equivalent) by using the numerical_approx method:
sage: rbval.n()
19.4942520190990
sage: rbval.numerical_approx()
19.4942520190990
sage: rbval.n(100)
19.494252019098965666956412564
There are lots of ways to turn a symbolic expression into a "number", though. See my answer to this question on a similar subject.
Does that make sense?
more | 2017-11-20 19:15:36 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3063272535800934, "perplexity": 4308.509710558263}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934806124.52/warc/CC-MAIN-20171120183849-20171120203849-00221.warc.gz"} |
https://www.gradesaver.com/textbooks/math/algebra/elementary-and-intermediate-algebra-concepts-and-applications-6th-edition/chapter-10-exponents-and-radicals-10-6-solving-radical-equations-10-6-exercise-set-page-669/60 | ## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)
$$m=-191800$$
Using the equation for slope to find the rate of change, we obtain: $$\mathrm{Slope}=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{419000-1378000}{5-0}\\ m=-191800$$ | 2019-12-10 07:51:44 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7959735989570618, "perplexity": 1268.824382717865}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-51/segments/1575540527010.70/warc/CC-MAIN-20191210070602-20191210094602-00104.warc.gz"} |
https://stats.stackexchange.com/questions/522949/are-the-beta-and-bernoulli-processes-normalized-random-measures-with-independent | # Are the Beta and Bernoulli Processes Normalized Random Measures with Independent Increments?
James et al. 2009 introduced the notion of "Normalized Random Measures with Independent Increments." Do the Beta process and Bernoulli process belong to this family? | 2021-08-05 03:38:25 | {"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8824450373649597, "perplexity": 1822.4654138047847}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046155322.12/warc/CC-MAIN-20210805032134-20210805062134-00100.warc.gz"} |
https://stats.stackexchange.com/questions/54766/relationship-between-poisson-and-gamma | # Relationship between poisson and gamma
Supposing that I have an interval $$n$$ units long with $$m$$ arrivals, and I model arrival using a poisson distribution with $$\lambda = m/n$$, it's pretty easy to show that the distance between successive arrivals has an exponential distribution $$\lambda e^{-\lambda x}$$.
However, I'm having a hard time generalizing this to a variation of the gamma distribution when I want to look more than $$2$$ arrivals (say I have $$3$$ successive or four successive arrivals). Can someone explain the logic there?
I believe the density should be $$\frac{\lambda^{k-1}} {(k-1)!} x^{k-2}e^{-\lambda x}$$ but I am not sure how!
The sum of independent Gamma random variables with the same scale factor (equivalently, same rate factor) is a Gamma random variable with the same scale factor, and the order is the sum of the orders. See, for example, this question. So, apply this to your problem using the fact that exponential random variables are Gamma random variables of order $1$. Your density is not quite right, by the way: it should be $$\frac{\lambda(\lambda x)^{k-1}}{\Gamma(k)}\exp(-\lambda x) = \frac{\lambda^{k}x^{k-1}}{(k-1)!}\exp(-\lambda x)~~ \text{for}~ x>0.$$ | 2020-10-22 15:35:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 7, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8526459336280823, "perplexity": 139.06726471870286}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107879673.14/warc/CC-MAIN-20201022141106-20201022171106-00124.warc.gz"} |
http://tex.stackexchange.com/questions/128606/includegraphics-inside-a-tikz-node-with-xelatex-is-incorrectly-positioned-when-w | # includegraphics inside a TikZ node with XeLaTeX is incorrectly positioned when width or height is given [closed]
Consider the following code:
\documentclass{article}
\usepackage{tikz}
\begin{document}
a\includegraphics[height=100pt]{example-image-10x16}b
\begin{tikzpicture}
\node[anchor=south west] at (0,0) {a\includegraphics[height=100pt]{example-image-10x16}b};
\end{tikzpicture}
\end{document}
Run with pdflatex and it compiles just fine. Run with xelatex and the image in the node is displaced. It appears to be where the included graphic is located, since the a and b are always in the correct place. It also has something to do with the explicit height=100pt since removing that also fixes the problem (width=<something> also causes the problem to occur).
1. What's happening?
2. How can I make it stop?
Note that the following are related:
Nor does the answer to Centering full-page Tikz image on page without margins with xelatex? work.
Edit: Following on from David and Harish's comments, I updated TL2013 and PGF. With the PGF that comes with TL2013 then it compiles correctly. With the CVS version of PGF then it does not. So the problem appears to lie with the PGF code for xelatex. I also discovered that the page changed from A4 paper (TL version of PGF) to letter paper (CVS version). As a Brit, my TL is set to produce A4 by default. Manually putting in a4paper changes the paper size back to A4, but doesn't correct the image displacement. So my guess is that the error is in PGF's driver file for XeTeX.
-
## closed as off-topic by Loop Space, Peter Jansson, Martin Schröder, Benedikt Bauer, barbara beetonNov 10 '13 at 21:17
• This question does not fall within the scope of TeX, LaTeX or related typesetting systems as defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.
works for me with a tL2013 updated today – David Carlisle Aug 16 '13 at 22:38
Strange. I get it right with miktex 2.9. – Harish Kumar Aug 16 '13 at 22:38
Curiouser and curiouser ... I'm updating TL2013 and PGF as I type, will report back. – Loop Space Aug 16 '13 at 22:50
Edited in the results of a bit of research ... – Loop Space Aug 16 '13 at 23:36
This has now been fixed in the CVS version of PGF. – Loop Space Nov 10 '13 at 20:36 | 2015-09-02 02:30:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.923708438873291, "perplexity": 3260.2104075559614}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-35/segments/1440645241661.64/warc/CC-MAIN-20150827031401-00045-ip-10-171-96-226.ec2.internal.warc.gz"} |
https://socratic.org/questions/is-it-possible-for-a-molecule-to-be-nonpolar-even-though-it-contains-polar-coval | Is it possible for a molecule to be nonpolar even though it contains polar covalent bonds?
Molecular polarity is conceived to result from the VECTOR sum of the individual bond dipoles. For a molecule such as $C {X}_{4}$ $\left(\text{X = halide}\right)$, the individual bond dipoles are polar, due to the difference in electronegativity between carbon and halogen. However, because the vector sum of the $C \rightarrow X$ dipoles is ZERO, the molecule is non-polar.
And thus chloroform, ${\text{CHCl}}_{3}$ is polar, while ${\text{CCl}}_{4}$ is non-polar. | 2020-11-24 18:38:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 5, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6539695262908936, "perplexity": 1077.1296460158208}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141176922.14/warc/CC-MAIN-20201124170142-20201124200142-00356.warc.gz"} |
http://mathhelpforum.com/algebra/54389-partial-fraction-decomposition.html | 1. ## partial fraction decomposition
I need help with these two problems!!
Write the partial fraction decomposition for the rational expression. Check your result algebraically and graphically using a graphing utility.
1. (x^2-1) / (x(x^2 + 1))
2. (x^2) / (x^4 - 2x^2 - 8)
2. Originally Posted by juldancer
I need help with these two problems!!
Write the partial fraction decomposition for the rational expression. Check your result algebraically and graphically using a graphing utility.
1. (x^2-1) / (x(x^2 + 1))
2. (x^2) / (x^4 - 2x^2 - 8)
1.$\displaystyle \frac{x^2}{x^4 - 2x^2 - 8}$
$\displaystyle \frac{x^2}{(x^2+2)(x+2)(x-2)} = \frac{A}{x-2} + \frac{B}{x+2} + \frac{Cx +D}{x^2+2}$
2. $\displaystyle \frac{x^2-1}{x(x^2+1)}$
$\displaystyle \frac{x^2-1}{x(x^2+1)} = \frac{A}{x} + \frac{Bx +c}{x^2+1}$
Can you finish up? | 2018-04-25 05:07:01 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.377402126789093, "perplexity": 2665.116913958103}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125947693.49/warc/CC-MAIN-20180425041916-20180425061916-00443.warc.gz"} |
http://mscroggs.co.uk/puzzles/LXIV | mscroggs.co.uk
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# Sunday Afternoon Maths LXIV
Posted on 2018-05-06
## Equal lengths
The picture below shows two copies of the same rectangle with red and blue lines. The blue line visits the midpoint of the opposite side. The lengths shown in red and blue are of equal length.
What is the ratio of the sides of the rectangle?
## Digitless factor
Ted thinks of a three-digit number. He removes one of its digits to make a two-digit number.
Ted notices that his three-digit number is exactly 37 times his two-digit number. What was Ted's three-digit number?
## Backwards fours
If A, B, C, D and E are all unique digits, what values would work with the following equation?
$$ABCCDE\times 4 = EDCCBA$$
If you enjoyed these puzzles, check out Sunday Afternoon Maths LXVI,
puzzles about pascal's triangle, or a random puzzle. | 2018-10-21 08:29:25 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5094708800315857, "perplexity": 1285.5638073521968}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-43/segments/1539583513804.32/warc/CC-MAIN-20181021073314-20181021094814-00110.warc.gz"} |
http://www.reference.com/browse/having+say | Definitions
# Error threshold (evolution)
The error threshold is a concept in the study of evolutionary biology and population genetics and is concerned with the origins of life, in particular of very early life, before the advent of DNA. The first self-replicating molecules were probably small ribozyme-like RNA molecules. These molecules consist of strings of base pairs or "digits", and their order is a code that directs how the molecule interacts with its environment. All replication is subject to mutation error. During the replication process, each digit has a certain probability of being replaced by some other digit, which changes the way the molecule interacts with its environment, and may increase or decrease its fitness, or ability to reproduce, in that environment.
It was noted by Manfred Eigen in his 1971 paper (Eigen 1971) that this mutation process places a limit on the number of digits a molecule may have. If a molecule exceeds this critical size, the effect of the mutations become overwhelming and a runaway mutation process will destroy the information in subsequent generations of the molecule. The error threshold is also controlled by the fitness landscape for the molecules. Molecules which differ only by a few mutations may be thought of as "close" to each other, while those which differ by many mutations are distant from each other. Molecules which are very fit, and likely to reproduce, have a "high" fitness, those less fit have "low" fitness. These ideas of proximity and height form the intuitive concept of the "fitness landscape". If a particular sequence and its neighbors have a high fitness, they will form a quasispecies and will be able to support longer sequence lengths than a fit sequence with few fit neighbors, or a less fit neighborhood of sequences. Also, it was noted by Wilke (Wilke 2005) that the error threshold concept does not apply in portions of the landscape where there are lethal mutations, in which the induced mutation yields zero fitness and prohibits the molecule from reproducing.
The concept of this critical mutation rate, or "error threshold" is crucial to understanding "Eigen's paradox" which is discussed in the next section.
Eigen's paradox is one of the most intractable puzzles in the study of the origins of life. It is thought that the error threshold concept described above limits the size of self replicating molecules to perhaps a few hundred digits, yet almost all life on earth requires much longer molecules to encode their genetic information. This problem is handled in living cells by the presence of enzymes which repair mutations, allowing the encoding molecules to reach sizes on the order of millions of base pairs. These large molecules must, of course, encode the very enzymes that repair them, and herein lies Eigen's paradox, first put forth by Manfred Eigen in his 1971 paper (Eigen 1971). Simply stated, Eigen's paradox amounts to the following:
• Without error correction enzymes, the maximum size of a replicating molecule is about 100 base pairs.
• In order for a replicating molecule to encode error correction enzymes, it must be substantially larger than 100 bases.
This is a chicken-or-egg kind of a paradox, with an even more difficult solution. Which came first, the large genome or the error correction enzymes? A number of solutions to this paradox have been proposed:
• Stochastic corrector model (Szathmáry & Smith, 1995). In this proposed solution, a number of primitive molecules of say, two different types, are associated with each other in some way, perhaps by a capsule or "cell wall". If their reproductive success is enhanced by having, say, equal numbers in each cell, and reproduction occurs by division in which each of various types of molecules are randomly distributed among the "children", the process of selection will promote such equal representation in the cells, even though one of the molecules may have a selective advantage over the other.
• Relaxed error threshold (Kun et al., 2005) - Studies of actual ribozymes indicate that the mutation rate can be substantially less than first expected - on the order of 0.001 per base pair per replication. This may allow sequence lengths of the order of 7-8 thousand base pairs, sufficient to incorporate rudimentary error correction enzymes.
## A simple mathematical model illustrating the error threshold
Consider a 3-digit molecule [A,B,C] where A, B, and C can take on the values 0 and 1. There are eight such sequences ([000], [001], [010], [011], [100], [101], [110], and [111]). Let's say that the [000] molecule is the most fit; upon each replication it produces an average of $a$ copies, where $a>1$. This molecule is called the "master sequence". The other seven sequences are less fit; they each produce only 1 copy per replication. The replication of each of the three digits is done with a mutation rate of μ. In other words, at every replication of a digit of a sequence, there is a probability $mu$ that it will be erroneous; 0 will be replaced by 1 or vice versa. Let's ignore double mutations, and divide the eight molecules into three classes depending on their Hamming distance from the master sequence:
Hamming
distance
Sequence(s)
0 [000]
1 [001]
[010]
[100]
2 [110]
[101]
[011]
3 [111]
Note that the number of sequences for distance d is just the binomial coefficient $C\left(L,d\right)$ for L=3, and that each sequence can be visualized as the vertex of an L=3 dimensional cube, with each edge of the cube specifying a mutation path in which the change Hamming distance is either zero or ±1. It can be seen that, for example, one third of the mutations of the [001] molecules will produce [000] molecules, while the other two thirds will produce the class 2 molecules [011] and [101]. We can now write the expression for the child populations $n\text{'}_i$ of class i in terms of the parent populations $n_j$.
$n\text{'}_i=sum_\left\{j=0\right\}^3 w_\left\{ij\right\}n_j$
where the fitness matrix w is given by:
$mathbf\left\{w\right\}=$
begin{bmatrix} (Q+a-1)&frac{1}{3}m&0&0 m&Q&frac{2}{3}m&0 0&frac{2}{3}m&Q&m 0&0&frac{1}{3}m&Q end{bmatrix}
where $Q=\left(1-mu\right)^L$ is the probability that an entire molecule will be replicated successfully. The eigenvectors of the w matrix will yield the equilibrium population numbers for each class. For example, if the mutation rate μ is zero, we will have Q=1, and the equilibrium concentrations will be $\left[n_0,n_1,n_2,n_3\right]=\left[1,0,0,0\right]$. The master sequence, being the fittest will be the only one to survive. If we have a replication fidelity of Q=0.95, then the equilibrium concentrations will be roughly $\left[0.35,0.37,0.22,0.055\right]$. It can be seen that the master sequence is not as dominant. If we have a replication fidelity of Q=0, then the equilibrium concentrations will be roughly $\left[0.131,0.376,0.371,0.123\right]$. This is almost an equal population of all sequences. (If we had perfectly equal population of all sequences, we would have populations of [1,3,3,1]/8.)
If we now go to the case where the number of base pairs is large, say L=100, we obtain behavior that resembles a phase transition. The plot below on the left shows a series of equilibrium concentrations divided by the binomial coefficient $C\left(100,k\right)$. (This multiplication will show the population for an individual sequence at that distance, and will yield a flat line for an equal distribution.) The selective advantage of the master sequence is set at a=1.05. The horizontal axis is the Hamming distance d . The various curves are for various total mutation rates $\left(1-Q\right)$. It is seen that for low values of the total mutation rate, the population consists of a quasispecies gathered in the neighborhood of the master sequence. Above a total mutation rate of about 1-Q=0.05, the distribution quickly spreads out to populate all sequences equally. The plot below on the right shows the fractional population of the master sequence as a function of the total mutation rate. Again it is seen that below a critical mutation rate of about 1-Q=0.05, the master sequence contains most of the population, while above this rate, it contains only about $2^\left\{-L\right\}approx 10^\left\{-30\right\}$ of the total population.
It can be seen that there is a sharp transition at a value of 1-Q just a bit larger than 0.05. For mutation rates above this value, the population of the master sequence drops to practically zero. Above this value, it dominates.
In the limit as L approaches infinity, the system does in fact have a phase transition at a critical value of Q: $Q_c=1/a.$. One could think of the overall mutation rate (1-Q) as a sort of "temperature", which "melts" the fidelity of the molecular sequences above the critical "temperature" of $1-Q_c$. In order for faithful replication to occur, the information must be "frozen" into the genome. | 2013-05-25 15:26:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 17, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6537555456161499, "perplexity": 701.5140823165684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368705957380/warc/CC-MAIN-20130516120557-00013-ip-10-60-113-184.ec2.internal.warc.gz"} |
https://math.stackexchange.com/questions/3160498/existence-of-3-natural-numbers-that-divide-each-other-when-squared-and-have-1-ta | # Existence of 3 natural numbers that divide each other when squared and have 1 taken away from them
Does there exist natural numbers, $$a,b,c > 1$$, such that;
$$a^2 - 1$$ is divisible by $$b$$ and $$c$$, $$b^2 - 1$$ is divisible by $$a$$ and $$c$$ and $$c^2 - 1$$ is divisible by $$a$$ and $$b$$.
• Hello and welcome to math.stackexchange. This is a nice question. What are your thoughts? What have you tried? Any results from searching for examples? – Hans Engler Mar 24 at 12:59
• Unless $b,c$ are prime it is not true that $b,c$ must divide one of $a+1,a-1$. – lulu Mar 24 at 13:03
• If you think that the first part of the problem statement implies that $b$ and $c$ must each divide $a-1$ or $a+1$, and similarly for the other two parts, I think you will quickly find that it can't work that way. – David K Mar 24 at 13:06
• The only triples with five out of six; and all three below 1000, are: $(3,4,5),(3,7,8),(8,21,55),(24,115,551),(15,56,209)$ – Empy2 Mar 24 at 13:29
• Those triples include $(n^2-1,n^3-2n,n^4-3n^2+1)$ – Empy2 Mar 24 at 13:59
First, observe that $$a,b,c$$ are pairwise coprime: For example, since $$a$$ divides $$b^2-1$$, any prime $$p$$ that divides $$a$$ also divides $$b^2 - 1$$; hence $$p$$ does not divide $$b^2$$ and also does not divide $$b$$. Thus $$a$$ and $$b$$ are coprime; and likewise for the other pairs.
Since $$b,c$$ are coprime and each divide $$a^2 - 1$$, so does their product $$bc$$. Since all the quantities are positive, that implies $$bc \le a^2 - 1 \lt a^2$$. So we have three strict inequalities: \begin{align} bc &\lt a^2\\ ac &\lt b^2\\ ab &\lt c^2 \end{align} Multiplying these inequalities together yields $$a^2 b^2 c^2 \lt a^2 b^2 c^2$$ which is impossible. | 2019-05-21 16:38:33 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 27, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8973905444145203, "perplexity": 287.7597439273434}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256494.24/warc/CC-MAIN-20190521162634-20190521184634-00347.warc.gz"} |
https://in.mathworks.com/help/econ/estimate-multiplicative-arima-model-using-econometric-modeler.html | ## Estimate Multiplicative ARIMA Model Using Econometric Modeler App
This example shows how to estimate a multiplicative seasonal ARIMA model by using the Econometric Modeler app. The data set `Data_Airline.mat` contains monthly counts of airline passengers.
### Import Data into Econometric Modeler
At the command line, load the `Data_Airline.mat` data set.
`load Data_Airline`
At the command line, open the Econometric Modeler app.
`econometricModeler`
Alternatively, open the app from the apps gallery (see Econometric Modeler).
Import `DataTable` into the app:
1. On the Econometric Modeler tab, in the Import section, click .
2. In the Import Data dialog box, in the Import? column, select the check box for the `DataTable` variable.
3. Click .
The variable `PSSG` appears in the Time Series pane, its value appears in the Preview pane, and its time series plot appears in the Time Series Plot(PSSG) figure window.
The series exhibits a seasonal trend, serial correlation, and possible exponential growth. For an interactive analysis of serial correlation, see Detect Serial Correlation Using Econometric Modeler App.
### Stabilize Series
Address the exponential trend by applying the log transform to `PSSG`.
1. In the Time Series pane, select `PSSG`.
2. On the Econometric Modeler tab, in the Transforms section, click .
The transformed variable `PSSGLog` appears in the Time Series pane, its value appears in the Preview pane, and its time series plot appears in the Time Series Plot(PSSGLog) figure window.
The exponential growth appears to be removed from the series.
Address the seasonal trend by applying the 12th order seasonal difference. With `PSSGLog` selected in the Time Series pane, on the Econometric Modeler tab, in the Transforms section, set Seasonal to `12`. Then, click .
The transformed variable `PSSGLogSeasonalDiff` appears in the Time Series pane, and its time series plot appears in the Time Series Plot(PSSGLogSeasonalDiff) figure window.
The transformed series appears to have a unit root.
Test the null hypothesis that `PSSGLogSeasonalDiff` has a unit root by using the Augmented Dickey-Fuller test. Specify that the alternative is an AR(0) model, then test again specifying an AR(1) model. Adjust the significance level to 0.025 to maintain a total significance level of 0.05.
1. With `PSSGLogSeasonalDiff` selected in the Time Series pane, on the Econometric Modeler tab, in the Tests section, click > Augmented Dickey-Fuller Test.
2. On the ADF tab, in the Parameters section, set Significance Level to `0.025`.
3. In the Tests section, click .
4. In the Parameters section, set Number of Lags to `1`.
5. In the Tests section, click .
The test results appear in the Results table of the ADF(PSSGLogSeasonalDiff) document.
Both tests fail to reject the null hypothesis that the series is a unit root process.
Address the unit root by applying the first difference to `PSSGLogSeasonalDiff`. With `PSSGLogSeasonalDiff` selected in the Time Series pane, click the Econometric Modeler tab. Then, in the Transforms section, click .
The transformed variable `PSSGLogSeasonalDiffDiff` appears in the Time Series pane, and its time series plot appears in the Time Series Plot(PSSGLogSeasonalDiffDiff) figure window.
In the Time Series pane, rename the `PSSGLogSeasonalDiffDiff` variable by clicking it twice to select its name and entering `PSSGStable`.
The app updates the names of all documents associated with the transformed series.
### Identify Model for Series
Determine the lag structure for a conditional mean model of the data by plotting the sample autocorrelation function (ACF) and partial autocorrelation function (PACF).
1. With `PSSGStable` selected in the Time Series pane, click the Plots tab, then click .
2. Show the first 50 lags of the ACF. On the ACF tab, set Number of Lags to `50`.
3. Click the Plots tab, then click .
4. Show the first 50 lags of the PACF. On the PACF tab, set Number of Lags to `50`.
5. Drag the ACF(PSSGStable) figure window above the PACF(PSSGStable) figure window.
According to [1], the autocorrelations in the ACF and PACF suggest that the following SARIMA(0,1,1)×(0,1,1)12 model is appropriate for PSSGLog.
`$\left(1-L\right)\left(1-{L}^{12}\right){y}_{t}=\left(1+{\theta }_{1}L\right)\left(1+{\Theta }_{12}{L}^{12}\right){\epsilon }_{t}.$`
Close all figure windows.
### Specify and Estimate SARIMA Model
Specify the SARIMA(0,1,1)×(0,1,1)12 model.
1. In the Time Series pane, select the `PSSGLog` time series.
2. On the Econometric Modeler tab, in the Models section, click the arrow > SARIMA.
3. In the SARIMA Model Parameters dialog box, on the Lag Order tab:
• Nonseasonal section
1. Set Degrees of Integration to `1`.
2. Set Moving Average Order to `1`.
3. Clear the Include Constant Term check box.
• Seasonal section
1. Set Period to `12` to indicate monthly data.
2. Set Moving Average Order to `1`.
3. Select the Include Seasonal Difference check box.
4. Click .
The model variable `SARIMA_PSSGLog` appears in the Models pane, its value appears in the Preview pane, and its estimation summary appears in the Model Summary(SARIMA_PSSGLog) document.
The results include:
• Model Fit — A time series plot of `PSSGLog` and the fitted values from `SARIMA_PSSGLog`.
• Residual Plot — A time series plot of the residuals of `SARIMA_PSSGLog`.
• Parameters — A table of estimated parameters of `SARIMA_PSSGLog`. Because the constant term was held fixed to 0 during estimation, its value and standard error are 0.
• Goodness of Fit — The AIC and BIC fit statistics of `SARIMA_PSSGLog`.
## References
[1] Box, George E. P., Gwilym M. Jenkins, and Gregory C. Reinsel. Time Series Analysis: Forecasting and Control. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994. | 2022-01-25 22:17:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 1, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7492677569389343, "perplexity": 4382.964787453159}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320304876.16/warc/CC-MAIN-20220125220353-20220126010353-00621.warc.gz"} |
https://tongfamily.com/2005/03/30/microwave-oven-reviews/ | "GE Spacemaker II 1 cu ft 800 watts stainless steel JEM31SF":http://www.pricegrabber.com/search_getprod.php/masterid=928352/search=spacemaker%2520ii. It figures that 18 months after we buy a microwave that it dies.
We are stuck on GE mainly because they haven't change the user interface and the thought of learning more than one is terrifying. Lists for $249 (stainless), but the regular colors (white, black and bisque) are$199 list.
The cool features on this one are that if you press the 1 button, it just turns on for one minute, etc. There is also an add 30 seconds button. Most hilarious is that there is a child lock that only "Alex":https://tongfamily.com/family/alex knows how to use 🙂
Main gotcha is that it is very easy to make a mistake and reset the clock (you press clock, then clear then a number and then clock and you've reset it).
This one is only 1 cubic foot and 800 watts, but that is all we really need, don't do roasts in ours. Having it spin inside is nice too, but we don't have that. It is very small and compact.
I'm sure there are cheaper ones, but Pricegrabber shows that "Sears":http://www.sears.com/sr/javasr/product.do?BV_UseBVCookie=Yes&vertical=SEARS&pid=02073129000 has it for $199 and shipping is very expensive for these. This is actually the cheapest price that pricegrabber shows. Another review mentions that theirs died too, I guess too many parts made by my cousins in China 🙂 Now if you don't want the same interface, there are certainly cheaper ones. For instanct the Panasonic "NN-T654SF":http://www.amazon.com/exec/obidos/tg/detail/-/B000277X8S/qid=1112244191/br=1-4/ref=br_lf_k_4//104-4605664-8295141?v=glance&s=kitchen&n=289935 is silver, costs just$100. It is alos 1300 watts and still pretty small at 16"x21x12". The main issue is that one review on Amazon notes the thing dies in a couple of months. Not surprising. That is really cheap.
These things are pretty much commodities where reliability matters. It is amazing to me that epinions et al just have nothing on them. Are there no general reviews of things like this on the web? If so, google can't find them. The only one I saw was from "about.com":http://busycooks.about.com/cs/microwavemagic/tp/microwave.htm where busycooks picks out a few models. Most interesting piece was about Panasonics inverter technology. Where instead of running 50 percent power by turning off half the time (that's what all those fancy power setting really do, the things runs full bore but shuts off part of the time), apparently, they actually turn the microwave power down. She says that the cooking works better (makes sense to me) so there are fewer cold spots and burnt edges.
The Panasonic "NNS564SF":http://www2.panasonic.com/webapp/wcs/stores/servlet/vModelDetail?displayTab=O&storeId=15001&catalogId=13401&itemId=70990&catGroupId=25069&modelNo=NN-T654SF&surfModel=NN-T654SF has this is is about \$120 direct in stainless steel. 21"x12"X16" and is 1300 watts. Costco, Target and Sears all apparently stock Panasonic, but I'll bet they don't have every model :-). The direct price doesn't look too bad either. The main problem is the GE is 17x9x13 so it is much narrower and will fit into our narrow counter. | 2022-08-17 16:04:38 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3025796413421631, "perplexity": 1743.2211657190771}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573029.81/warc/CC-MAIN-20220817153027-20220817183027-00259.warc.gz"} |
https://ijnaa.semnan.ac.ir/?_action=article&kw=18820&_kw=Banach+contraction | International Journal of Nonlinear Analysis and Applications
Dislocated quasi rectangular $b$-metric spaces and fixed point theorems of cyclic and weakly cyclic contractions
Volume 12, Issue 2, July 2021, Pages 173-191
Fixed points for Banach and Kannan contractions in $G$-metric spaces endowed with a graph
Volume 12, Issue 2, July 2021, Pages 297-304 | 2022-12-04 01:22:03 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.17376995086669922, "perplexity": 1002.6657292391586}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710953.78/warc/CC-MAIN-20221204004054-20221204034054-00108.warc.gz"} |
https://mathoverflow.net/questions/256830/component-dimensions-of-a-tensor-product-of-irreducible-representations | # component dimensions of a tensor product of irreducible representations
Is there a criterion (or, more generally, an algorithm) to decide whether given non-negative integers $n_1, n_2$ and $m_1,...,m_k$ there is a group $G$ with irreducible reps $V_1, V_2$ over $\mathbb C$ of dimensions $n_1$ and $n_2$ whose tensor product $V_1\otimes V_2$ decomposes into $k$ irreducible components of dimensions $m_1,....,m_k$? (I am also interested in versions of this problem when $G$ is required to be finite or Lie.)
I imagine someone studied that in the 100+ years of history of representation theory.
• There is an obvious necessary condition coming from the equality of the dimensions. Perhaps, the theory of symmetric monoidal categories can be used to show that it is also sufficient in the case of finite groups. For compact connected Lie groups (or equivalently, reductive Lie algebras), I am doubtful that anything significantly better than exhaustive search using Weyl character formula is available in general. If you impose further restrictions, such as multiplicity-freeness (e.g., all $m_i$s are distinct), the situation changes dramatically: full classification is available. – Victor Protsak Dec 9 '16 at 23:29
• In the case of simple Lie groups (or rather their Lie algebras), given a sequence of dimensions you'd probably need to rely on the Weyl dimension formula coupled with the known methods of decomposing tensor products. Presumably there is an algorithm lurking here, if one starts with the well-known classifications of simple Lie algebras and their irreducible finite dimensional representations over $\mathbb{C}$. Like Victor I wouldn't expect anything easy if the $n_i$ and $m_j$ can be arbitrary. – Jim Humphreys Dec 10 '16 at 16:03
• Related: the more specific mathoverflow.net/questions/254537/… – YCor Dec 10 '16 at 17:45
• As I explained as a comment in the link, assuming that $G$ is Lie with finitely many components, or that $G$ is compact Lie, is no restriction. Indeed, once a $(k+2)$-tuple $(n_1,n_2,m_1,\dots,m_k)$ is achieved, you can pass to the Zariski closure of its image in $V_1\oplus V_2$ (which is reductive), and then to a maximal compact subgroup (which is Zariski-dense), yielding the same decomposition on the tensor product. – YCor Dec 10 '16 at 17:49
• @VictorProtsak: Dear Victor, could you provide a reference to your multiplicity-free case answer to my question? – Adam Dec 10 '16 at 18:36 | 2019-03-22 09:18:23 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8249058127403259, "perplexity": 257.1677273903849}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-13/segments/1552912202640.37/warc/CC-MAIN-20190322074800-20190322100800-00294.warc.gz"} |
https://discourse.mc-stan.org/t/loo-cross-validation-with-correlated-data/22845 | # Loo/cross-validation with correlated data
I have N data points which are most easily modeled as a multivariate normal distribution, with the mean/covariance dependent on the parameters of the model \theta i.e.
y \sim \mathcal{N}(\mu (x;\theta),\Sigma(\theta))
My question is, is there a way to cross-validate this model effectively? I see that the Vehtari, Gelman, and Gabry paper on loo makes the assumption that the data are independent conditional on the parameters. Are there alternatives that would work for this case?
1 Like
This will work with LOO out of the box, as long as all the y vectors are independent (conditional on the parameters). Correlations between the components of the y vectors won’t cause errors in the results. As an example, if y is a vector consisting of [wealth, income] where wealth and income are correlated, this will be ok. However, if you have several vectors of [wealth, income] coming from the same family, such that the vectors are correlated with each other by common confounders (e.g. rich parents tend to have rich children), LOO will not be an accurate representation of the out-of-sample error.
Apologies that I didn’t clarify this in the initial post, but I’ve got only got the one data vector, with about N=1000 data points in it. The other thing I guess I should specify is that my basic googling has mostly turned up suggestions that I try to select fitting and hold-out sets for cross-validation such that correlations are small between them, but my goal is specifically to evaluate the correlation structure of the data; in particular, whether various mitigation strategies (based on external datasets) are successful in removing the correlations between our measurements (which are a systematic uncertainty in our final analysis), and where the mitigations really only affect the mostly strongly correlated part of the data vector. Thus it seems to me that if I select hold-outs that are (effectively) uncorrelated with the fitted data points, I won’t have a useful answer to the question I’m asking (i.e. are the mitigations really working?). Is there a sensible way to go about answering these questions?
3 Likes
Sorry, but I had a follow up question about this paper. This worked perfectly when I was using a normal distribution, but I’m now considering whether using a multidimensional t-distribution would be better suited to the data set. However, I’m encountering a difficulty implementing the procedure given, specifically Eq. 18. It seems like \Sigma is an N\times N matrix, while the vector y_{-i}-\mu_{-i} is a vector of size N-1, so the multiplication described here doesn’t work. Is there something I’m missing, or is there a typo in the equation?
ping @paul.buerkner
Yes, I think it should have been \Sigma^{-1}_{-i} (instead of \Sigma^{-1}) in the sense that we first compute the full inverse \Sigma^{-1} and then remove the $i$th row and column. Sorry for that mistake.
This is then also consistent with the proof of Proposition 3.
1 Like | 2022-07-03 13:58:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8575759530067444, "perplexity": 523.8186311603205}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656104244535.68/warc/CC-MAIN-20220703134535-20220703164535-00695.warc.gz"} |
http://accessphysiotherapy.mhmedical.com/content.aspx?bookid=333§ionid=40013258 | Chapter 61
###### Figure 61–1.
Structure of the skin.
### Epidermis
The epidermis is composed of stratified squamous epithelium in which several cell types can be identified.
#### Keratinocytes
Keratinocytes are subdivided according to degree of differentiation into the following four types: (1) basal cells, the germinative, epidermal stem cells, which are fixed to the basement membrane by attachment plaques (modified desmosomes) and anchoring filaments (stratum basale); (2) prickle or spinous cells (stratum spinosum), characterized by the presence of keratin and desmosomes and named for the desmosomal intercellular attachments, which appear as spines or prickles connecting cells; (3) granular cells (stratum granulosum), containing numerous large basophilic keratohyalin granules in the cytoplasm; and (4) cornified cells (stratum corneum)—flat, anucleate cells that form the most superficial layer of the epidermis. The stratum corneum is continuously shed at the surface and varies greatly in thickness—it is greatest in the soles and palms.
The normal rate of maturation from a basal cell to a surface cornified cell is 2 weeks. The cell remains in the stratum corneum another 2 weeks before it is shed.
#### Melanocytes
Melanocytes are epidermal cells derived from the embryologic neural crest. They produce melanin and are situated singly in the basal layer, appearing as larger clear cells. The dopa reaction, which is positive in the presence of enzymes of the tyrosine–melanin pathway is a histochemical means of identifying melanocytes. Melanocytes have dendritic processes that ramify in the epidermis and transfer melanin to keratinocytes.
Melanocytes can be identified through electron microscopy by the presence of melanosomes, which are membrane-bound ellipsoidal structures containing concentric internal lamellae. Positive staining for S100 protein and melanosomal antigen (HMB45) are useful immunohistochemical markers for melanocytes.
The number of melanocytes in the skin is relatively constant. Skin pigmentation is dependent on the rate of synthesis of melanin, which is governed by (1) racial factors, being greater in dark-skinned races; (2) ultraviolet radiation, which increases melanin synthesis; and (3) hormones—melanocyte-stimulating hormone and adrenocorticotropin both increase melanin pigmentation.
#### Langerhans Cells
Langerhans cells are clear dendritic cells situated among the cells of the stratum spinosum. They are believed to be antigen-processing cells. They are S100 protein-positive on immunohistochemical studies. On electron microscopy, they lack melanosomes but contain a characteristic organelle known as a Birbeck granule.
#### Merkel Cells
Merkel cells are neuroendocrine cells that are present in the basal layer of the epidermis. They cannot be recognized in routine histologic sections but can be identified in electron micrographs by the presence of cytoplasmic neurosecretory granules.
### The Dermis
The dermis is a layer of connective tissue 1–4 mm thick subjacent to the epidermis. ...
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Connect to the full suite of AccessPhysiotherapy content and resources including interactive NPTE review, more than 500 videos, Anatomy & Physiology Revealed, 20+ leading textbooks, and more. | 2017-03-28 04:14:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.19143714010715485, "perplexity": 9004.833161676703}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-13/segments/1490218189667.42/warc/CC-MAIN-20170322212949-00174-ip-10-233-31-227.ec2.internal.warc.gz"} |
https://www.math.ucdavis.edu/research/seminars?talk_id=402 | # Mathematics Colloquia and Seminars
We consider the model problem: given an affine matrix family $A(x)$ parameterized by a vector $x$, minimize the spectral abscissa (maximum real part of the eigenvalues) of $A(x)$, subject to bounds on $\|x\|$. Local minimizers of this class of problems typically have the following property: the spectral abscissa is not differentiable, in fact not even Lipschitz, at the minimizer, because of the presence of multiple eigenvalues. We give some explanation for this phenomenon; among our tools are optimality conditions derived using nonsmooth analysis. We also present a gradient bundle algorithm for approximating local minimizers, and we compare the results to those obtained using a Newton barrier method to minimize a related "robust" spectral abscissa. This work is joint with James Burke (U. Washington) and Adrian Lewis (U. Waterloo). | 2023-03-24 15:59:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8269070386886597, "perplexity": 747.3066705307817}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296945287.43/warc/CC-MAIN-20230324144746-20230324174746-00301.warc.gz"} |
https://murray.cds.caltech.edu/index.php?title=Hw5_ex1_-_norm_minimization&oldid=5732 | # Hw5 ex1 - norm minimization
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Q How do we solve the minimization problem for $$\sqrt{ || \dot{x} ||}$$?
A The minimization of the square root of the norm is difficult; though one will have the same result by minimizing just $$|| \dot{x} ||$$, where with this notation we mean $$\dot{x}^T \dot{x}$$ | 2021-03-01 18:35:32 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9824109673500061, "perplexity": 1566.853186115197}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178362899.14/warc/CC-MAIN-20210301182445-20210301212445-00322.warc.gz"} |
https://web2.0calc.com/questions/macaroons | +0
# Macaroons
0
343
3
You are baking macaroons for your class. There are 9 total students in your class and you have baked 17 macaroons. Write and solve an equation to find the additional number x of macaroons you need to bake in order to have 4 macaroons for each student. Write your equation so that the units on each side of the equation are macaroons per student.
Equation: ___=4
Number of macaroons: x=___
Apr 13, 2020
#1
+80
0
The number of macaroons you need total is 36. This is found by multiplying 9 and 4, which is 36. 36-17=19 (You need to make 19 more macaroons).
$${x +17 \over 9}=4$$
$$x+17=36$$ Subtract 17 from both sides
$$x = 19$$
Apr 13, 2020
#2
0
Thank you so much, I have been stuck on this question all day
Guest Apr 13, 2020
#3
+80
0
No Problem! I find it easier to do these problems when you know how many you are supposed to end with first. :)
LovelyRock Apr 13, 2020 | 2021-01-17 01:01:21 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8634620308876038, "perplexity": 767.687098374959}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703507971.27/warc/CC-MAIN-20210116225820-20210117015820-00396.warc.gz"} |
https://www.rdocumentation.org/packages/Rssa/versions/1.0.2/topics/plot | # plot
0th
Percentile
##### Plot SSA object
This function plots various sorts of figures related to the SSA method.
##### Usage
# S3 method for ssa
plot(x,
type = c("values", "vectors", "paired", "series", "wcor"),
…,
vectors = c("eigen", "factor"),
plot.contrib = TRUE,
numvalues = nsigma(x),
numvectors = min(nsigma(x), 10),
idx = 1:numvectors,
idy,
groups)
##### Arguments
x
SSA object holding the decomposition
type
Arguments to be passed to methods, such as graphical parameters
vectors
For type = 'vectors', choose the vectors to plot
plot.contrib
logical. If 'TRUE' (the default), the contribution of the component to the total variance is plotted. For ossa' class, Frobenius orthogonality checking of elementary matrices is performed. If not all matrices are orthogonal, corresponding warning is risen
numvalues
Number of eigenvalues to plot (for type = 'values')
numvectors
Total number of eigenvectors to plot (for type = 'vectors')
idx
Indices of eigenvectors to plot (for type = 'vectors')
idy
Second set of indices of eigenvectors to plot (for type = 'paired')
groups
Grouping used for the decomposition (see reconstruct)
##### Details
This function is the single entry to various plots of SSA objects. Right now this includes:
values
plot the graph of the component norms.
vectors
plot the eigenvectors.
paired
plot the pairs of eigenvectors (useful for the detection of periodic components).
series
plot the reconstructed series.
wcor
plot the W-correlation matrix for the reconstructed objects.
Additional (non-standard) graphical parameters which can be transfered via …:
plot.type
lattice plot type. This argument will be transfered as type argument to function panel.xyplot.
ref
logical. Whether to plot zero-level lines in series-plot, eigenvectors-plot and paired-plot. Zero-level isolines will be plotted for 2d-eigenvectors-plot.
symmetric
logical. Whether to use symmetric scales in series-plot, eigenvectors-plot and paired-plot.
useRaster
logical. For 2d-eigenvector-plot and wcor-plot, indicating whether raster representations should be used. 'TRUE' by default.
col
color vector for colorscale (for 2d- and wcor-plots), given by two or more colors, the first color corresponds to the minimal value, while the last one corresponds to the maximal value (will be interpolated by colorRamp)
zlim
for 2d-plot, range of displayed values
at
for 2d-eigenvectors-plot, a numeric vector giving breakpoints along the range of z, a list of such vectors or a character string. If a list is given, corresponding list element (with recycling) will be used for each plot panel. For character strings, values 'free' and 'same' are allowed: 'free' means special breakpoints' vectors (will be evaluated automatically, see description of cuts argument in 'Details') for each component. 'same' means one breakpoints' vector for all component (will be evaluated automatically too)
cuts
for 2d-reconstruction-plot, the number of levels the range of z would be divided into.
fill.color
color or 'NULL'. Defines background color for shaped 2d-eigenvectors plot. If 'NULL', standard white background will be used.
ssa-object, ssa plot.reconstruction,
• plot.ssa
##### Examples
# NOT RUN {
# Decompose 'co2' series with default parameters
s <- ssa(co2)
# Plot the eigenvalues
plot(s, type = "values")
# Plot W-cor matrix for first 10 reconstructed components
plot(s, type = "wcor", groups = 1:10)
# Plot the paired plot for first 6 eigenvectors
plot(s, type = "paired", idx = 1:6)
# Plot eigenvectors for first 6 components
plot(s, type = "vectors", idx = 1:6)
# Plot the first 4 reconstructed components
plot(s, type = "series", groups = list(1:4))
# Plot the eigenvalues by points only
plot(s, type = "values", plot.type = "p")
# Artificial image for 2dSSA
mx <- outer(1:50, 1:50,
function(i, j) sin(2*pi * i/17) * cos(2*pi * j/7) + exp(i/25 - j/20)) +
rnorm(50^2, sd = 0.1)
# Decompose 'mx' with default parameters
s <- ssa(mx, kind = "2d-ssa")
# Plot the eigenvalues
plot(s, type = "values")
# Plot eigenvectors for first 6 components
plot(s, type = "vectors", idx = 1:6,
ref = TRUE, at = "same", cuts = 50,
plot.contrib = TRUE, symmetric = TRUE)
# Plot factor vectors for first 6 components
plot(s, type = "vectors", vectors = "factor", idx = 1:6,
ref = TRUE, at = "same", cuts = 50,
plot.contrib = TRUE, symmetric = TRUE)
# Plot wcor for first 12 components
plot(s, type = "wcor", groups = 1:12, grid = c(2, 6))
# 3D-SSA example (2D-MSSA)
data(Barbara)
ss <- ssa(Barbara, L = c(50, 50, 1))
plot(ss, type = "values")
plot(ss, type = "vectors", idx = 1:12, slice = list(k = 1),
cuts = 50, plot.contrib = TRUE)
plot(ss, type = "vectors", idx = 1:12, slice = list(k = 1, i = 1))
plot(ss, type = "vectors", vectors = "factor", idx = 1:12, slice = list(k = 3),
cuts = 50, plot.contrib = FALSE)
plot(ss, type = "series", groups = 1:12, slice = list(k = 1))
plot(ss, type = "series", groups = 1:12, slice = list(k = 1, i = 1))
plot(ss, plot.method = "xyplot", type = "series", groups = 1:12, slice = list(k = 1, i = 1))
# }
` | 2020-10-24 23:49:14 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.44721338152885437, "perplexity": 14515.693683003496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107885059.50/warc/CC-MAIN-20201024223210-20201025013210-00456.warc.gz"} |
http://www.koreascience.or.kr/article/ArticleFullRecord.jsp?cn=CPTSCQ_2016_v21n3_1 | Three Color Algorithm for Two-Layer Printed Circuit Boards Layout with Minimum Via
Title & Authors
Three Color Algorithm for Two-Layer Printed Circuit Boards Layout with Minimum Via
Lee, Sang-Un;
Abstract
The printed circuit board (PCB) can be used only 2 layers of front and back. Therefore, the wiring line segments are located in 2 layers without crossing each other. In this case, the line segment can be appear in both layers and this line segment is to resolve the crossing problem go through the via. The via minimization problem (VMP) has minimum number of via in layout design problem. The VMP is classified by NP-complete because of the polynomial time algorithm to solve the optimal solution has been unknown yet. This paper suggests polynomial time algorithm that can be solve the optimal solution of VMP. This algorithm transforms n-line segments into vertices, and p-crossing into edges of a graph. Then this graph is partitioned into 3-coloring sets of each vertex in each set independent each other. For 3-coloring sets $\small{C_i}$, (i
Keywords
Via;Crossing;Coloring;Independent set;Two layer;
Language
Korean
Cited by
References
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X. Munoz, W. Unger, and I. Vrt'o, "One Sided Crossing Minimization is NP-hard for Sparse Graphs," In P. Mutzel and M. Junger, eds., Graph Drawing GD'01, LNCS 2265, pp. 115-123, Springer, 2001.
23.
M. Newton, O. Sykora, and I. Vrt'o, "Two New Heuristics for Two-Sides Bipartite Graph Drawing," Lecture Notes in Computer Science, Vol. 2528, pp. 465-485, Springer Berlin/Heidelberg, 2002. | 2017-08-16 15:37:54 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 1, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.2557089924812317, "perplexity": 4867.527989961627}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886102307.32/warc/CC-MAIN-20170816144701-20170816164701-00357.warc.gz"} |
https://math.stackexchange.com/questions/3011031/solving-cauchys-problem-with-a-discontinuous-function | Solving Cauchy's problem with a discontinuous function
I have the Cauchy problem: $$\begin{cases} f'=g(t)+2(f-5) \\ f(0)=2\end{cases}$$
Now $$g(t)$$ is a periodic function: $$g(t)=\begin{cases} 0,t\in(24k,24k+8)\\ 2,t\notin(24k,24k+8) \end{cases}$$
for $$k=1,2,3...$$ How can I solve this Cauchy Problem? Since $$g$$ is not continuous I don't know what to do.
I am using sagemath to solve it.
• Use Laplace transform. – Jacky Chong Nov 24 '18 at 0:47
Rewrite as
$$f' - 2f = g - 10$$
A Fourier method is appropriate here. Since $$g$$ has a period of $$24$$, we can assume a series solution of similar periodicity
$$f(t) = f_0 + \sum_{n=1}^\infty \left[f_n\cos\left(\frac{n\pi}{12}t\right) + f_n^*\sin\left(\frac{n\pi}{12}t\right) \right]$$
Next, decompose $$g$$ to its Fourier series
$$g(t) = g_0 + \sum_{n=1}^\infty \left[g_n\cos\left(\frac{n\pi}{12}t\right) + g_n^*\sin\left(\frac{n\pi}{12}t\right) \right]$$
From the usual definitions:
$$g_0 = \frac{1}{24}\int_0^{24} g(t)\ dt = \int_8^{24} 2\ dt = \frac{4}{3}$$ $$g_n = \frac{1}{12}\int_ 8^{24} 2\cos\left(\frac{n\pi}{12}t\right) dt = -\frac{2}{n\pi}\sin \frac{2n\pi}{3}$$ $$g_n^* = \frac{1}{12}\int_8^{24} 2\sin\left(\frac{n\pi}{12}t\right)dt = -\frac{2}{n\pi}\left[1-\cos \frac{2n\pi}{3} \right]$$
Plug in the 2 series and compare coefficients, we get:
$$-2f_0 = g_0 - 10$$ $$-2f_n + \frac{n\pi}{12}f_n^* = g_n$$ $$-2f_n^* - \frac{n\pi}{12}f_n = g_n^*$$
Solve the above system of the equations to find $$f_n$$ and $$f_n^*$$ | 2019-05-22 04:38:50 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 19, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9618250131607056, "perplexity": 287.6397255847299}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256763.42/warc/CC-MAIN-20190522043027-20190522065027-00446.warc.gz"} |
https://beta.aofm.gov.au/publications/annual-reports/part-2-performance-and-outcomes | Skip to main content
# Introduction
This part of the annual report is presented in three sections: Section 1 provides an Annual Performance Statement as required by the PGPA Act; Section 2 details the AOFM’s issuance operations, debt and cash portfolio management task and its engagement with the market (namely investors); and Section 3 presents the main considerations for how the AOFM approaches strategy development to underpin its operational objectives.
## Section 1: Annual Performance Statement
As the accountable authority of the Australian Office of Financial Management, I present the 2018-19 Annual Performance Statement of the Australian Office of Financial Management, as required under paragraph 39(1)(a) of the Public Governance, Performance and Accountability Act 2013 (PGPA Act).
In my opinion this annual performance statement accurately reflects the performance of the Australian Office of Financial Management, is based on properly maintained records and complies with subsection 39(2) of the PGPA Act.
Rob Nicholl
Chief Executive Officer
27 September 2019
## Purpose
The AOFM’s purpose is to ensure the government’s debt financing needs are met each year while managing the cash, debt and other portfolios over the medium-long term at low cost subject to acceptable risk. The AOFM takes into account the potential for its operations to impact domestic financial markets.
The AOFM has adopted three key objectives to achieve its purpose:
1. meet the budget financing task in a cost-effective manner subject to acceptable risk;
2. facilitate the government’s cash outlay requirements as and when they fall due; and
3. be a credible custodian of the AGS market, and meet other portfolio responsibilities as directed by government.
The AOFM balances cost and risk considerations but its overriding aim is to ensure that the financing requirements of government are able to be met in full and on time. The AOFM has minimal appetite for failure in any function associated with issuance, settlement and cash management. The design and conduct of its core business processes (including its business continuity arrangements) reflect this risk appetite.
The AOFM monitors its performance against the performance indicators presented in Table 1, sourced from the AOFM’s Corporate Plan 2018-19 and Portfolio Budget Statements 2018-19. Sections 2 and 3 of this part of the Annual Report provide detail on a range of outcomes important to achievement by the AOFM of its annual and longer-term aims. This detail is provided separately to the Performance Statement because it is aimed at financial market participants as the relevant audience.
### Measure (b)
Objective 1: Meet the budget financing task in a cost effective manner subject to acceptable risk
1. Term issuance
Shortfall in volume ($) between actual Treasury Bond issuance and planned issuance announced at the Budget and subsequent releases. 2.1 Financing cost (portfolio) 2.2 Financing cost (issuance) The cost of the long-term debt portfolio compared to the 10 year average of the 10-year bond rate. The cost of Treasury Bond issuance over the past 12 months compared to the average 10-year bond rate over the same period. 3. New issuance yields Weighted average issue yield at Treasury Bond and Treasury Indexed Bond tenders less prevailing mid-market secondary yields. Objective 2: Facilitate the government’s cash outlay requirements as and when they fall due 4. Use of the overdraft facility Number of instances the RBA overdraft facility was utilised to the extent that it required Ministerial approval during the assessment period. Objective 3: AOFM is a credible custodian of the AGS market and other portfolio responsibilities 5. A liquid and efficient secondary market Annual turnover in the secondary market for Treasury Bonds and Treasury Indexed Bonds. 6. Market commitments Number of times the AOFM failed to undertake actions consistent with public announcements. (a) Source: AOFM Corporate Plan 2018-19; Portfolio Budget Statements 2018-19 Budget Related Paper No. 1.16 — Treasury Portfolio, p. 108 (b) Source: AOFM measures performance against indicators using data captured from its market transactions; its financial systems recording portfolio composition; official notices to the market; and secondary financial market turnover data requested from intermediaries. ## Performance Results 2018–19 ### Objective 1: Meet the budget financing task in a cost effective manner subject to acceptable risk Indicator 1 Term Issuance: Shortfall in volume ($) between actual Treasury Bond issuance and planned issuance announced at the Budget and subsequent releases
Target
Zero
Result
Target met
At the time of the 2018–19 Budget, Treasury Bond and Treasury Indexed Bond issuance for the year was expected to total around $77 billion in face value terms. This volume was revised at the time of MYEFO in accordance with a revised improvement in the government’s fiscal position compared to Budget forecasts. #### Treasury Bond issuance Gross Treasury Bond issuance for the year totalled$55.0 billion. This was a significant reduction from the $75.5 billion of Treasury Bonds issued in 2017–18. The bulk of this issuance was into existing bond lines in order to enhance market liquidity. In addition, two new Treasury Bond lines were launched in 2018–19: • a new bond line maturing in June 2031 was issued to support the operation of the 10-year Treasury Bond futures contract and to reduce growth in the amount outstanding in surrounding bond lines, which will make it easier to manage maturity of those bonds lines; and • a new bond line maturing in May 2041 was issued to support the operation of the 20-year Treasury Bond futures contract. In selecting the bond lines to issue each week, the AOFM continued to follow its standard practice of taking into account the debt issuance strategy, prevailing market conditions, information from financial market contacts about investor demand, relative value considerations, the liquidity of outstanding bond lines, and managing the maturity structure to limit funding risk. One or two tenders were held during most weeks. Large transaction volumes were achieved at issues of new Treasury Bonds. The May 2041 ($3.6 billion) was issued by syndication, while a tender was held for the initial issue of the June 2031 ($3.0 billion) bond line. At the end of the year, there were 25 Treasury Bond lines, with 13 of these lines having over$20 billion on issue and 21 having over $10 billion on issue. Chart 2 shows Treasury Bonds outstanding as at 30 June 2019 and the allocation of issuance across bond lines during 2018–19. ### Chart 2: Treasury Bonds outstanding as at 30 June 2019 and issuance in 2018–19 #### Treasury Bond buybacks A total of$23.1 billion of Treasury Bonds were repurchased ahead of maturity in 2018-19, of which $16.7 billion were bonds maturing after 30 June 2019: • 25 Treasury Bond buyback tenders were conducted, at which$14.7 billion of bonds were repurchased;
• the AOFM repurchased $2.05 billion of bonds in conjunction with the syndicated issue of the new 21 May 2041 Treasury Bond; •$6.3 billion of bonds were repurchased from the RBA; and
• a small amount of bonds were repurchased from retail investors who sold their holdings via the Australian Government Securities Buyback Facility.
Buyback tenders are effectively a reverse of normal competitive issuance tenders. The AOFM sets the total volume of bonds it is prepared to buy back and offers from intermediaries are accepted from the highest yield (lowest price) in descending order until the total volume is reached. All Treasury Bond buybacks other than those from retail investors were of lines shorter than the three-year futures basket.
The volume outstanding in short-dated Treasury Bonds was reduced as illustrated in Chart 3.
### Chart 3: Volume outstanding in short-dated Treasury Bonds as at 30 June 2018 and 30 June 2019
#### Treasury Indexed Bond issuance
The AOFM maintains less than 10 per cent of the long-term debt portfolio in the form of Treasury Indexed Bonds, the capital values of which are adjusted with changes in the CPI. The issuance of these bonds typically attracts a different (and predominantly domestic) class of investor compared to Treasury Bonds, providing a source of diversification in the funding base. While the indexed bond portfolio has declined marginally as a share of the long term funding, the total stock of indexed bonds has continued to grow steadily (as shown in Chart 4).
### Chart 4: Treasury Indexed Bonds — average term to maturity and share of the long-term funding base
Treasury Indexed Bond issuance for the year totalled $5.9 billion, of which$2.2 billion was conducted via tender. Two tenders for the issue of Treasury Indexed Bonds were conducted in most months. The volume of each line outstanding, relative yields and other prevailing market conditions were considered in the selection of which line to offer. A syndicated offer for $3.8 billion of a new February 2050 Treasury Indexed Bond line was conducted in September 2018. In conjunction with the syndication, around$2.1 billion of the August 2020 Treasury Indexed Bond line was repurchased.
Chart 5 shows the amount outstanding in each of the eight Treasury Indexed Bond lines as at 30 June 2019, and the allocation of issuance during the 2018–19 year.
### Chart 5: Treasury Indexed Bonds outstanding as at 30 June 2019 and issuance in 2018–19
#### Efficiency of issuance
Table 2 summarises the results of Treasury Bond tenders conducted during the year. The results are shown as averages for each half-year and grouped by the maturity dates of the bonds offered.
### Table 2: Summary of Treasury Bond tender results
Period
Maturity
Face value amount allocated ($m) Weighted average issue yield (%) Average spread to secondary market yield (basis points) Average times covered July - December 2018 Up to 2026 4,900 2.1670 -0.51 5.44 2027 - 2031 20,900 2.6644 -0.30 3.39 2032 - 2047 900 3.0703 -0.15 2.38 January - June 2019 Up to 2026 3,200 1.5485 -0.39 4.45 2027 - 2031 19,600 1.8815 -0.10 3.25 2032 - 2047 1,900 2.3736 0.08 2.64 The average coverage ratio for all Treasury Bond tenders in 2018–19 was 3.57, a decrease from 4.44 in 2017–18. The average tender size of$829 million was higher than in 2017-18, reflecting a move to less frequent tenders. Shorter-dated bond tenders generally received a greater volume of bids (higher than average coverage ratios), which reflected both core investor base interest and a lower supply of short-dated bonds.
The strength of bidding at tenders was also reflected in competitive issue yield spreads to secondary market yields. At most Treasury Bond tenders the weighted average issue yields were below prevailing secondary market yields.
The average coverage ratio was 4.40 for Treasury Indexed Bond tenders, an increase from 3.53 in 2017–18. At most tenders, the weighted average issue yields were below prevailing secondary market yields.
Full tender details are available in Part 5 of this annual report.
#### Market liquidity and efficiency
The Treasury Bond market operated smoothly during 2018-19 with liquidity and efficient price discovery being maintained throughout the year. Repo rates during the first half of the year remained high compared to long term averages with continued pressure around quarter ends, raising the costs to some investors and market makers of holding AGS. The elevated rates experienced during 2018 eased somewhat during the second half of the year.
AOFM monitoring of the market indicates that liquidity in Treasury Indexed Bonds has continued to prove noticeably more challenging than for Treasury Bonds. This is consistent with the relative liquidity of nominal and inflation-linked securities in other sovereign debt markets. Treasury Indexed Bond turnover in 2018-19 was around $52 billion, an increase of 3 per cent from 2017-18. This was driven by a decrease of Interbank turnover Australian investors and in Asia (ex Japan). Intermediaries are responsible for the bulk of trades. ### Chart 7: Annual Treasury Indexed Bond Turnover There was tightness in several indexed bond lines at times during the year, requiring some market participants to borrow these from the securities lending facility. Turnover in the Treasury Bond futures market is significantly higher than in the underlying Treasury Bonds. The three and 10-year Treasury Bond futures contracts are highly liquid: over 60 million three-year contracts (representing$6.0 trillion face value of bonds) and over 52 million 10-year contracts ($5.2 trillion face value of bonds) were traded in 2018–19. Turnover in the 20-year contract is considerably lower: 259,000 contracts ($15.8 billion face value of bonds) were traded. All contract close-outs in 2018–19 occurred smoothly.
The AOFM’s securities lending facility allows market participants to borrow Treasury Bonds and Treasury Indexed Bonds for short periods when they are not otherwise available in the secondary market. This enhances the efficiency of the market by improving the capacity of intermediaries to continuously make two-way prices, reduces the risk of settlement failures, and supports market liquidity. The facility was used 21 times for overnight borrowing in 2018–19 compared with 52 times during 2017-18. The volumes borrowed were lower than in 2017–18, with the total face value amount lent in 2018–19 being $393 million, a decrease from$1,667 million in the previous year.
## Debt portfolio management
### Aims
This section details the outcomes of the AOFM’s portfolio management activities. In managing the debt portfolio and meeting the government’s financing requirements, the AOFM aims for low and stable debt servicing costs over the medium-long term. It also seeks to maintain liquid bond lines to facilitate cost-effective issuance of debt through time and to effectively manage future funding and refinancing risks.
### Approach to achieving the aims
To meet these aims the AOFM endeavours to execute a debt issuance strategy that appropriately accounts for the trade-offs between cost and risk while providing consistent and transparent stewardship of the AGS market in order to underpin confidence and promote market liquidity. Through its operations the AOFM contributes to an efficient and resilient market while seeking to maintain continuity of access to financial markets for the Australian Government.
The AOFM uses cost and risk measures that reflect the considerations faced by sovereign debt managers generally. The primary cost measure used is historic accrual debt service cost. This includes interest payments made on AGS, realised market value gains and losses on repurchases, capital indexation of indexed debt, and the amortisation of any issuance premiums and discounts. Total accrual debt service cost can be expressed as a percentage of the stock of debt outstanding to provide the effective yield of the portfolio. The use of an historic accrual debt service cost measure excludes unrealised market value gains and losses.
An alternative measure of cost is ‘fair value’, which takes account of unrealised gains and losses resulting from movements in the market value of physical debt and assets. Debt service cost outcomes are presented in the AOFM’s financial statements on this basis. A comprehensive income format is used that allows revenues and expenses on an historic basis to be distinguished from the effects of unrealised market value fluctuations. Fair value facilitates an assessment of financial risk exposures and changes in those exposures from year to year, the value of transactions managed and the economic consequences of alternative strategies. It is most useful in the context of trading for profit making purposes.
The AOFM calculates and compares several metrics to assess risk. In general, an acceptable level of risk can be characterised as an acceptable level of variation in interest cost outcomes over time. Debt issuance decisions made today impact the variability of future interest cost outcomes because of their influence on the maturity profile of the portfolio and hence the amount of debt that needs to be refinanced (and ‘re-priced’) through time.
### Outcomes
#### Portfolio cost
The debt servicing cost[1] of the net AGS portfolio managed by the AOFM in 2018–19 was $17.42 billion on an average book volume of$527.01 billion, representing a net cost of funds of 3.31 per cent for the financial year. The largest component of net AGS debt is the Long Term Debt Portfolio (LTDP), comprised primarily of Treasury Bonds and Treasury Indexed Bonds, which incurred debt servicing costs of $17.92 billion on an average book volume of$550.16 billion, implying a cost of funds of 3.26 per cent. The difference between net AGS debt and the LTDP is attributable to the short term assets and liabilities the AOFM uses for liquidity management purposes (term deposits and Treasury Notes) and other residual assets (such as state housing advances).
Table 3 provides further details of the cost outcomes for the portfolio of debt and assets administered by the AOFM broken down by instrument and portfolio for 2018-19 as well as 2017–18.
### Table 3: Commonwealth debt and assets administered by the AOFM
Debt servicing cost
$million Book volume$ million
Effective yield
per cent per annum
2017-18
2018-19
2017-18
2018-19
2017-18
2018-19
Contribution by instrument
Treasury Bonds
(15,854)
(16,136)
(485,291)
(505,825)
3.27
3.19
Treasury Indexed Bonds
(1,598)
(1,785)
(43,207)
(44,337)
3.70
4.03
Treasury Notes
(67)
(63)
(3,950)
(3,398)
1.70
1.85
Gross physical AGS debt
(17,519)
(17,984)
(532,448)
(553,560)
3.29
3.25
Term deposits with the RBA
650
459
37,095
24,748
1.75
1.85
RMBS investments
37
915
4.04
0.00
State Housing Advances
110
106
1,880
1,801
5.86
5.89
Gross assets
797
565
39,890
26,548
2.00
2.13
Net AGS debt
(16,722)
(17,419)
(492,558)
(527,011)
3.39
3.31
Contribution by portfolio
Long Term Debt Portfolio
(17,452)
(17,921)
(528,498)
(550,162)
3.30
3.26
Cash Management Portfolio
583
396
33,145
21,350
1.76
1.85
RMBS Portfolio
37
915
4.04
0.00
State Housing Portfolio
110
106
1,880
1,801
5.86
5.89
Total debt and assets
(16,722)
(17,419)
(492,558)
(527,011)
3.39
3.31
Re-measurements (a)
581
(43,550)
Total after re-measurements
(16,141)
(60,969)
(492,558)
(527,011)
Note: Sub totals and totals are actual sum results, rounded to the nearest million dollars. Effective yields are based on actual results before rounding, rounded to two decimal places. Book volume is a through the year average.
(a) Interest expense and effective yield on foreign loans incorporates foreign exchange revaluation effects.
(b) Re-measurements refer to unrealised gains and losses from changes in the market valuation of financial assets and liabilities.
The cost of gross debt increased in dollar terms by $0.47 billion compared to the previous year. This was primarily due to an increase in the average volume of debt on issue by$21.11 billion to $553.65 billion. However, in percentage terms the funding cost of gross debt declined by 4 basis points to 3.25 per cent. This improvement was driven by the issuance of new bonds at yields that were below the average of pre-existing (and maturing) debt. The return on gross assets in dollar terms for the period was$565 million, a decrease of $232 million compared to 2017–18. This was driven by a$191 million decrease in income from term deposits (resulting from smaller holdings) as well as a $37 million reduction in income following the completion of the RMBS divestment process in February 2018. However, in percentage terms the return of gross assets increased by 13 basis points to 2.13 per cent. The net servicing cost of the combined portfolio of debt and assets was$17.42 billion. This was higher in dollar terms compared to 2017–18, primarily due to the higher volume of debt on issue. In percentage terms, net debt servicing costs fell from 3.39 per cent to 3.31 per cent, slightly larger than the fall in gross debt servicing costs.
Movements in market interest rates had an unfavourable impact on the market value of the portfolio in 2018–19. Unrealised losses from re-measurements amounted to $43.55 billion. This compares to an unrealised gain of$0.58 billion in the previous year. Most of the re-measurement losses are attributable to changes in the market value of Treasury Bonds. Re-measurement items are highly volatile from one year to the next and have no bearing on the AOFM’s debt issuance strategy. Indeed, were the AOFM to adopt a strategy designed to minimise the ‘noise’ from re-measurements, issuance would be limited to only very short-term debt securities, for example Treasury Notes and near maturity bonds. However, this would create a portfolio structure that would maximise expected variability in debt servicing costs when measured in cash, accrual and public debt interest terms, while also maximising exposure to refinancing and funding risk. In practice the AOFM has been seeking to reduce these risks through allocating a greater proportion of issuance to long dated bond lines.
#### Portfolio risk management
Chart 8 shows the funding cost profile of the net AGS debt portfolio and the LTDP back to 2007-08. These profiles are contrasted with the cash rate and the 10-year moving average of the 10-year bond yield. With interest rates trending down, funding costs on net debt and the LTDP have declined by 188 and 200 basis points respectively since 2010-11. This compares to declines of 350 basis points in the cash rate and 205 basis points in the 10 year average of the 10-year bond rate over the same period. Given the largely fixed cost structure of net debt and the LTDP, changes in funding cost will always lag changes in the overnight cash rate (changing only when existing debt securities or assets mature or new securities are issued/investments placed).
### Chart 8: Net AGS debt and LTDP cost of funds analysis (per cent)
The reduced risk levels of the portfolio in terms of funding, refinancing and interest rate risk are demonstrated in Chart 9 below. The chart shows a steady decline in the short to medium term Treasury Bond refinancing task, measured as the proportion of the stock of Treasury Bonds on issue through time[2]. At 30 June 2010 the structure of the portfolio was such that 43 per cent and 65 per cent of bonds required refinancing over the next three and five year periods respectively; these have continued to decline and have now fallen to 21 per cent and 39 per cent.
## Cash management
### Aims
The AOFM manages the daily cash balances of the Australian Government in the OPA.[3] This is undertaken in a manner that ensures the government is able to meet its financial obligations as and when they fall due. Other objectives are to minimise the cost of funding and the carrying cost of holding cash balances (which centres on holding only balances assessed as prudent to cover forecast needs and contingencies, while investing excess balances at low or minimal risk). In minimising cost, the AOFM seeks to avoid use of the overdraft facility provided by the RBA.[4]
### Approach to achieving the aims
Achieving the cash management objective involves formulating forecasts of government cash flows, and developing and implementing appropriate strategies for short-term investments and debt issuance.
A precautionary asset balance is maintained to manage the forecasting risk associated with potentially large unexpected cash requirements (or shortfalls in revenue collections) and the funding risk associated with market constraints.
Cash balances not required immediately were invested in term deposits at the RBA, with the magnitudes and tenors of the term deposits determined by the AOFM. Maturity dates of term deposits were selected to most efficiently finance net outflows. Interest rates for term deposits at the RBA reflect the rates earned by the RBA in its open market operations.
Treasury Notes are issued to assist with management of the within-year funding requirement. The volume of Treasury Notes on issue ranged from $2.5 billion to$5.0 billion during 2018–19.
The size and volatility of the within-year funding requirement are reflected in changes in the short-term financial asset holdings managed by the AOFM, after deducting Treasury Notes on issue. Chart 10 shows movement in the funding requirement over the year.
### Outcomes
The task of meeting the government’s financial obligations as and when they fall due was fully met. The overdraft facility was not utilised in 2018–19.
During 2018–19, the AOFM placed 390 term deposits with the RBA. The stock of term deposits fluctuated according to a range of factors influencing the AOFM’s cash portfolio management needs. The balance of term deposits ranged from a maximum of $46.0 billion in July 2018 to a minimum of$11.3 billion in January 2019.
The average yield obtained on term deposits during 2018–19 was 1.85 per cent, compared with 1.75 per cent in 2017–18. The increase in average yield reflects the higher average level of short-dated interest rates that prevailed during 2018–19.
## Market engagement
### Aims
Consistent and regular market engagement assists the AOFM to maintain a comprehensive understanding of market related issues including major announcements and events, impacts on the global flow of capital, changing investor preferences, and the performance of banks that play the role of intermediaries — particularly in the AGS market. While this latter aim can in part be served by assessing announced regulatory changes, there remains the need for ongoing direct engagement with market participants. The AOFM also understands the importance of regular engagement with investors to ensure that they understand the key considerations underpinning the AOFM’s issuance and market maintenance/development strategies and appreciable changes in operation should these arise from time to time.
Market engagement by the AOFM continues to place a heavy emphasis on maintaining lines of communication with investors and bank intermediaries. This is done directly with both, and indirectly, with investors through feedback from the banks. Ongoing engagement assists greatly in understanding how investors view financial markets generally and in turn their view on the outlook for AGS and how intermediaries interact with and service the end investor.
### Approach to achieving the Aims
Market engagement is based on an investor relations program underpinned by an investor relations strategy that is reviewed annually. This review takes account of changes in market conditions, investor activity, known changes in the key investor base, and the AOFM’s planned issuance strategy.
The Investor Relations strategy has three themes:
• collecting and analysing market intelligence from investors;
• managing and maintaining updates to key investors about the AOFM’s program of activities and its intended operations; and
• a deepening of the investor base through engagement with new or potentially new investors.
Diversification of the AGS investor base is expected to change with appreciable shifts in financial market conditions. Given the complexity of influences on the attractiveness of AGS relative to alternative investment options it is difficult to predict with a high degree of confidence full detail of the AGS investor composition at any point in time. However, the AOFM is highly active in looking to understand changes as they occur. It does this through high frequency and comprehensive communication with market participants. In this regard the major focus of investor engagement continues to be, engaging with our key investors and collecting and analysing the information on their portfolio activities as we receive it from them.
### Outcomes
For the last two years the AOFM’s market engagement has been heavily guided by the shift in its operations away from active market development through yield curve extensions and the introduction of a high number of new Treasury Bond and Treasury Indexed Bond maturities, to more of a market maintenance role. There were effectively two key drivers for this change; one was achieving the market and portfolio objectives of establishing 30-year benchmark yield curves for both the nominal and indexed yield curves (with no strong incentive to extend beyond 30-years), and the ongoing reduction in the budget deficit with consequent smaller funding tasks. Together these factors have made the AOFM’s discussion with market participants appreciably more straightforward and this has allowed for an easier task of updating investors via teleconference and videoconferencing as substitutes for the regular face-to-face engagement that had previously been more appropriate. Due to the April Budget followed by the federal election, direct engagement over the months of March through to June was not conducted.
For the year overall, discussions were held with 95 investors directly (compared with 128 in 2017–18). This comprised of 65 face to face meetings and 30 video/teleconferencing calls. Although direct investor meetings were down from the year before, the AOFM was still able to cover a larger number of geographically diverse key AGS investors from differing sectors of the market. Key investors engaged via teleconference and videoconference were from 19 cities outside of Australia.
The AOFM intends to continue these methods where appropriate as a supplement to direct face-to-face meetings. However for some areas or regions the AOFM did not feel that conference calls were an appropriate method to use due to the lack of familiarity with certain investors and other practical considerations.
Face-to-face meetings with investors were held with the domestic investor base; around 30 meetings with large fund managers, bank balance sheets and superannuation funds. Domestic investors continue to hold around 40 per cent of total AGS outstanding. There was one overseas series of meetings conducted in London, Paris and Madrid.
To also supplement communication with investors, the AOFM launched a quarterly investor note (Investor Insights) on the 1st of June this year via the AOFM website. ‘Investor Insights’ will provide AOFM views and background thinking on a range of AGS related matters.
The AOFM continued its past practice of using appropriate opportunities offered through conferences and speaking events. These events offer AOFM the opportunity to engage briefly but directly with investors and to reiterate key messages and themes, regarding AGS issuance and its market impacts. Each year the Australian Business Economists hosts a post-Budget speech by the CEO. It remains an important platform to provide information to the market for the upcoming year by giving the market some detail around the AOFM’s intentions for forthcoming issuance and operations; (June) 2019 was the ninth consecutive year for this event.
The conference and investor missions hosted by the financial intermediaries in which the AOFM participated included a CBA Fixed Income Conference, ANZ Hunter Valley Debt Conference, the Deutsche Bank Investor Mission and an ANZ Investor Tour. The AOFM also spoke at the annual KangaNews roundtable and again participated in its annual year book publication. While these events do not substitute for the benefits derived from face-to-face meetings, they remain useful in their own right and typically offer opportunity for short face-to-face meetings with selected attendees.
The current approach to maintaining investor engagement is considered appropriate during lower and stable issuance programs and with the current operational approach to achieving these programs having been comprehensively explained to and understood by AGS investors. In the event that the AOFM foreshadowed a significant change in issuance or the nature of its approach to the market, a return to more intensive and widespread face-to-face investor engagement would be considered as appropriate.
### Table 4: Summary of investor relations activities in 2018-19
Activity Details Conferences, speaking engagements and investor roadshows 9 events Presentations: large engagements/ roundtables 5 presentations Approximate total audience size: large presentations 220 attendees Individual investor meetings 65 investor meetings Individual investor Tele/ Video Calls 30 Individual cities visited 7 cities AOFM staff participating in investor relation activities CEO, Head of Investor Relations, Head Portfolio Strategy & Research, Head Funding & Liquidity. Head of Market Intelligence, Senior Analyst, Investor Relations, Analyst Funding and Liquidity Hosting banks: Investor roadshows, conferences, roundtable discussions ANZ, Citi Commonwealth Bank of Australia, Deutsche Bank UBS, Westpac
## Section 3: Portfolio and Operational Strategy
### Debt Portfolio Management
#### Aims
The AOFM is a price-taker in global capital markets but influences the cost and risk profile of the AGS portfolio through the maturity structure of the securities it issues (and to a lesser extent, the mix between nominal and inflation-linked securities). Issuing longer-term securities will typically involve paying higher debt service costs (in the presence of a positive term premium)[5] although this is compensated by reduced variability in future interest cost outcomes and lower exposure to refinancing risk.[6] Issuing shorter term debt securities by contrast will typically incur less interest cost (avoiding a term premium), but result in higher variability in cost outcomes through time and a greater debt refinancing task. Striking the right balance between these cost and risk considerations is the debt manager’s ongoing challenge.
Developing a medium to long-term view on appropriate portfolio management and then translating that into annual decision making on a strategy to implement that portfolio management objective is informed by an ongoing research program. This program is focussed on exploring the cost and risk characteristics of alternative portfolio structures and issuance strategies under a wide range of scenarios. This is done in light of prevailing fiscal and economic conditions, as well as an assessment of broader market trends. Drawing on this research, the AOFM formulated a strategy for the structure and composition of issuance for 2018-19 that was approved by the Treasurer at the time of the Budget. Separately, a range of complementary limits, thresholds, guidelines and targets governing the AOFM’s operations were submitted to the Secretary to the Treasury for approval through an Annual Remit. These governance arrangements provide appropriate oversight for the impact of AOFM’s gross issuance decisions each year on overall debt policy.
Implementing the annual issuance strategy involves weekly decisions such as determining how much and which lines to issue, or when a new maturity should be established. These operational decisions are influenced by several factors including prevailing market conditions, relative value considerations and feedback from intermediaries and investors. The ongoing suitability of the annual debt issuance and portfolio strategies is under constant review, but the strategies would only be changed during the year in the face of substantive changes to market conditions or the fiscal outlook.
#### Debt Issuance Strategy
The AOFM’s strategy for 2018-19 was formulated amid a strengthening global economic environment, and was in large part influenced by a continuation of low outright bond yields (from a historical perspective), and a low term premium (which increases the cost-effectiveness of longer term issuance).
In the first half of 2018-19 the synchronised strengthening of global economic data continued and expectations of higher policy rates tended to result in higher government bond yields. However, there was an abrupt change in the outlook in mid-2018-19 as expectations for global growth and monetary policy were revised to reflect a weakening outlook, leading to a significant rally in government bond yields (Chart 12).
### Chart 12: Evolution of Treasury Bond benchmark yields
The yield curve flattened through the first half of 2018-19 before re-steepening in the second half of the year. Most of this change in longer bond yields reflected changes in other markets more than appreciable changes in the demand for AGS. Short-term AGS yields decreased in 2018-19 reflecting market expectations for a lower RBA monetary policy rate. The cash rate was subsequently lowered in June 2019 to 1.25 per cent to a new historic low. The 10-year and 30-year benchmark yield ended the financial year at 1.32 and 1.94 per cent respectively (also near historic lows).
In light of prevailing market conditions and funding requirements, the AOFM’s strategy in 2018–19 followed a broadly similar theme to recent years, with a bias toward longer term issuance and further lengthening of the average term to maturity of the debt portfolio. Low outright rates and a very low (or zero) term premium reinforced this strategy from a cost effectiveness perspective. At its core, the 2018–19 portfolio strategy was designed to preserve the AOFM’s operational flexibility under a wide variety of circumstances, while continuing to benefit from the relatively low interest rates on offer. The strategy was complemented by a regular program of bond buyback tenders. The strategy also aimed to support diversity in the AGS investor base.
#### Debt Portfolio and Issuance Metrics
Chart 13 demonstrates the lengthening bias implicit in the AOFM’s issuance strategy with the average Treasury Bond issued in 2018–19 having a term to maturity of 11.27 years[7]. The issuance program continued to benefit from low interest rates, with an average yield on new issuance of 2.29 per cent. [8]
### Chart 13: Treasury Bond issuance — average yield, term to maturity and 10-year bond yield
Chart 14 shows that the average term to maturity of the Treasury Bond portfolio as a whole lengthened by 0.06 years to 7.44 years over 2018-2019. Duration was also higher by 0.43 years finishing the year at 6.60 years. The effective cost of funds (or yield) on the Treasury Bond portfolio fell from 3.12 to 2.99 over the same period.[9]
### Chart 14: Treasury Bond portfolio — modified duration, average term to maturity and cost of funds
The structure and effective yield on the Treasury Bond portfolio as at 30 June 2019 is a product of issuance undertaken since the 2007-08 fiscal year. Around two thirds of the current portfolio for instance was issued in the last four financial years at average yields below the portfolio average of 2.99 percent as depicted in Chart 15.
### Chart 15: Treasury Bond portfolio — composition and average yield by issuance year, as at 30 June 2019
Chart 16 shows that more than half of current Treasury Bonds were issued with an original term to maturity of between 9 and 12 years. When issuance beyond 12 years is included, around three quarters of the portfolio has been issued with an original term to maturity of 9 years or longer. The predominance of longer term bonds in the portfolio is reflective of the AOFM lengthening bias since the start of the decade. This has contributed considerably to aim of reducing funding risk and the potential for high volatility in future interest rate outcomes.
### Chart 16: Treasury Bond portfolio — composition and average yield by original term to maturity, as at 30 June 2019
The AOFM’s strategy for the indexed bond proportion of the portfolio has been to provide sufficient supply to meet investor requirements while supporting liquidity and the continuing development of the market over time. Over 2018-19, Treasury Indexed Bonds comprised on average around 8 per cent of total term debt (nominal and indexed bonds) on issue. This share has been stable for several years now although the overall size of the market has continued to grow in dollar terms. Issuance of Treasury Indexed Bonds in 2018-19 was $5.9 billion gross and$0.5 billion in net terms after buybacks and maturities. The real yield curve was extended to 30 years through the successful launch of the 2050 line in September 2018.
In 2018–19, the AOFM continued to favour a relatively defensive liquidity strategy by maintaining an asset buffer (in the form of term deposits with the RBA) to act as a precaution against a possible deterioration in funding conditions. The AOFM anticipated that it would have sufficient cash and liquid assets available, each business day of the fiscal year, to fund the next four or more weeks of projected net government outlays and AGS maturities.
[1] Debt servicing cost includes net interest expense (measured on an accruals basis and includes realised gains and losses on the disposal of assets or liabilities) plus foreign exchange revaluation gains and losses (now minimal). Unrealised changes in the market valuation of domestic debt and assets are not part of this measure.
[2] In absolute dollar terms, the quantum of three and five year maturities in the portfolio has still grown although this has occurred at a considerably slower pace compared to growth in the overall stock of Treasury Bonds.
[3] The OPA is the collective term for the core bank accounts maintained at the RBA for Australian Government cash balance management.
[4] The overdraft facility is more costly than equivalent short-term borrowing (for example, issuance of Treasury Notes). The terms of the facility provide that it is to cover only temporary shortfalls of cash and is to be used infrequently and, in general, only to cover unexpected events.
[5] The term premium is the additional yield demanded by investors in order to hold a long-term bond instead of a series of shorter-term bonds.
[6] Refinancing risk, also referred to as rollover risk or re-pricing risk, is the risk that renewed borrowings to replace maturing debt occurs on unfavourable terms (or perhaps not at all).
[7] Calculation is based on the term to maturity of each bond issued during the year, weighted by book value.
[8] Calculation is based on issue yields during the year weighted by book value.
[9] These are point in time measures as at 30 June each year, in contrast to the debt servicing cost incurred throughout the year captured in Table 3. Figures are calculated by weighting Treasury Bond issuance yields by book volume. | 2020-05-29 23:32:19 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.20057500898838043, "perplexity": 5305.1889280201785}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-24/segments/1590347406785.66/warc/CC-MAIN-20200529214634-20200530004634-00539.warc.gz"} |
http://ptreview.sublinear.info/?cat=2&paged=2 | # News for April 2017
April was a busy month for property testing. A mixture of distribution testing, junta testing, Fourier transforms, matrix problems, and graph testing. Let’s go!
(Update: thanks to Omri Ben Eliezer for correcting some errors in our post. Also, he pointed out that we missed posting about by a property testing paper by Gishboliner-Shapira that appeared in Nov 2016.)
Fourier-Based Testing for Families of Distributions, by Clement Canonne, Ilias Diakonikolas, and Alistair Stewart (ECCC). Readers of PTReview will be familiar with great progress made over the past few years in distribution testing. We wish to determine some property $$\mathcal{P}$$ of a discrete distribution $$D$$ of support size $$n$$, using iid samples from the distribution. The challenge is that the number of samples should be $$o(n)$$. This paper gives a general framework for any property $$\mathcal{P}$$, such that all members (distributions) have sparse Fourier transforms. One of the main tools is a tester for determining if $$D$$ has small Fourier support (and find the restriction to this support). This is a really neat unifying methods that subsumes much of the past work. Moreover, this method leads to new testers for a variety of classes of distributions, including that of multinomial distributions.
Testing from One Sample: Is the casino really using a riffle shuffle, by Constantinos Daskalakis, Nishanth Dikkala, and Nick Gravin (arXiv). This paper introduces a novel twist on distribution testing: suppose $$\mathcal{D}$$ was generated through a Markov Chain, and we only get to see a sequence of samples generated through a random walk. Observe that we do not have the classic iid assumption any more. What can be said now? There is a challenge in defining “closeness” of distributions, which is done by looking at the variation distance between sequences generated through random walks of a carefully chosen length. The focus is on identity testing in two different regimes. First, with no assumptions on the Markov chain, there are general results that take $$O(n)$$ queries (note that the Markov chain has size $$O(n^2)$$). Secondly, for special “sparse” Markov Chains such as various models of card shuffling, one can get the number of samples to be logarithmic in size of the state space. Thus, one effectively gets to see a single shuffle process (which is like a walk in a Markov Chain), and can still decide if it is generating a uniform distribution.
Settling the query complexity of non-adaptive junta testing, by Xi Chen, Rocco A. Servedio, Li-Yang Tan, Erik Waingarten, and Jinyu Xie (arXiv). Ah, progress on a classic open problem! A function $$f:\{0,1\}^n \to \{0,1\}$$ is a $$k$$-junta if it only depends on $$k$$ variables. After a long line of work, Blais provided a non-adaptive $$\widetilde{O}(k^{3/2})$$ tester and in a separate result of Blais gave an adaptive (ignoring $$\epsilon$$ factors) $$\widetilde{O}(k)$$ tester. Previous to this paper, the best non-adaptive lower bound was essentially $$\Omega(k)$$, ignoring polylog factors. This paper finally provides a matching non-adaptive lower bound of $$\Omega(k^{3/2})$$, up to polylog factors! This basically settles the knowledge (adaptive and non-adaptive) of this problem, up to polylogs.
An Adaptive Sublinear-Time Block Sparse Fourier Transform, by Volkan Cevher, Michael Kapralov, Jonathan Scarlett, and Amir Zandieh (arXiv).
(We should have caught this when it was first published in Feb 2017, but at least we caught the update!) Consider a vector $$X$$ whose Fourier transform we wish to compute. Suppose there are only $$k$$ non-zero coefficients (typically, this is what we care about). We can actually compute these coefficients using $$O(k\log n)$$ queries, which is strongly sublinear when $$k \ll n$$. An important question is when the sample complexity can go below $$k\log n$$, which is optimal in general. Often sparsity Fourier coefficients have structure that can be exploited. This paper studies block-sparsity, where all non-zero coefficients occur in $$k_0$$ contiguous blocks (in Fourier space) of size $$k_1$$. Thus, the total sparsity is $$k = k_0k_1$$. This paper designs an algorithm with adaptive query complexity $$O(k_0k_1 + k_0\log n)$$, thereby circumventing the $$k\log n$$ barrier for $$k_0 \ll k$$. Interestingly, the paper gives lower bounds showing that adaptivity is necessary for removing the $$\log n$$ factor.
Sublinear Time Low-Rank Approximation of Positive Semidefinite Matrices, by Cameron Musco and David P. Woodruff (arXiv). Few problems occur as often as computing a low-rank approximation of a matrix $$A$$. Formally, one wishes to compute a rank $$k$$-matrix $$B$$ such that $$\|A-B\|_F$$ is minimized. SVD is the obvious method to do this, but is computationally expensive. There are a number of sampling methods, but all of them require reading all of $$A$$. This paper gives a truly sublinear-time algorithm (for small $$k$$) for PSD matrices. Formally, the algorithm presented here requires $$O(n \cdot \mathrm{poly}(k))$$ running time, where $$n$$ is the dimension size of $$A$$. It is well-known from previous work that there exist a subset of $$k$$ columns that form a good low-rank approximation. Yet finding them requires reading the whole matrix to compute various sampling strategies. The intuition here is that large entries cannot “hide” in arbitrary locations in a PSD matrix.
Testing hereditary properties of ordered graphs and matrices, by Noga Alon, Omri Ben-Eliezer, and Eldar Fischer (arXiv). Classic results by Alon-Shapira and Alon-Fischer-Newman-Shapira have shown that hereditary graph properties are testable in constant time. There is a deep connection between testing such properties and the existence of subgraph removal lemmas. The latter assert that, given some family $$\mathcal{F}$$ of forbidden subgraphs, if an $$n$$-vertex graph is $$\epsilon$$-far from being $$\mathcal{F}$$-free, then it must contain $$\delta n^q$$ copies of some $$F \in \mathcal{F}$$ (where $$F$$ has $$q$$ vertices, $$\delta$$ is a function only of $$\epsilon$$). This paper generalizes these theorems to general matrices over finite domains and ordered edge-colored graphs, where vertices in the graphs have a fixed ordering. Thus, one does not require the symmetry (over isomorphism) of graph properties for testability. These theorems lead to testability results for large classes of properties defined through forbidden substructures.
Removal Lemmas with Polynomial Bounds, by Lior Gishboliner and Asaf Shapira (arXiv). (Paper originally appeared in Nov 2016, but we missed it.) As mentioned above, there is a close connection between property testing of dense graphs and graph removal lemmas. Yet the property testers that result from these lemmas, like the celebrated triangle removal lemma, have terrible tower-like dependences on the parameter $$\epsilon$$. When can we get polynomial dependences on $$\epsilon$$ (which is called “easy testability”)? This fundamental question is answered in this paper. Consider the case when $$\mathcal{F}$$ is finite. If $$\mathcal{F}$$ contains a bipartite graph, the complement of a bipartite graph, and a split graph, then there are polynomially bounded graph removal lemmas (and hence easy property testers) for this family. Furthermore, one cannot drop any of these requirements! (A split graph is one where vertices can be partitioned into a set spanning a clique, and a set spanning an independent set.) There are generalizations to infinite families of graphs. An important corollary of this result is a proof of easy testability of semi-algebraic properties. This captures properties of graphs where vertices are points in Euclidean space and edges are present when the points satisfy some polynomial constraint (in the coordinates).
# News for March 2017
March was a relatively good month for property testing, with 3 papers — including a foray in differential privacy.
Optimal Unateness Testers for Real-Valued Functions: Adaptivity Helps, by Roksana Baleshzar, Deeparnab Chakrabarty, Ramesh Krishnan S. Pallavoor, Sofya Raskhodnikova, and C. Seshadhri (arXiv). Testing unateness (that is, whether a function is monotone after a rotation of the hypercube) has recently received significant attention. Understanding the power of adaptivity in property testing, on the other hand, has been a long-standing question, which has sparked much interest over the past decade. In this paper, the authors obtain optimal results for testing unateness of real-valued functions $$f\colon \{0,1\}^d\to \mathbb{R}$$ and $$f\colon [n]^d\to \mathbb{R}$$ (up to the dependence on the proximty parameter $$\varepsilon$$), and show that — unlike what happens for monotonicity testing of real-valued functions — adaptivity helps. Namely, adaptive testing of unatess of functions $$f\colon \{0,1\}^d\to \mathbb{R}$$ has query complexity $$O(d/\varepsilon)$$, but non-adaptive testers require $$\Omega(d\log d)$$ queries.
This work is a merged, and significantly improved version of two papers we covered a few months back, by the same authors.
Switching from function to distribution testing, last month saw two independent samples appear online:
Near-Optimal Closeness Testing of Discrete Histogram Distributions, by Ilias Diakonikolas, Daniel M. Kane, and Vladimir Nikishkin (arXiv). Closeness testing of discrete distributions is the question of testing, given sample access to two unknown distributions $$p,q$$ over a known discrete domain $$[n] \stackrel{\rm def}{=} \{1,\dots,n\}$$, whether $$p=q$$ or $$d_{\rm TV}(p,q)> \varepsilon$$ (are the two distributions the same, or differ significantly in total variation distance?). This quite fundamental question is by now fully understood in the general case, where $$p,q$$ are allowed to be arbitrary distributions over ; but what if one has some additional knowledge about them? Specifically, what if we were guaranteed that both distributions have some “nice structure” — e.g., that they are piecewise constant with $$k$$ pieces ($$k$$-histograms)? Leveraging machinery the authors developed in a series of previous work (for the continuous case), the authors show that in this case, the sample complexity of “testing closeness under structural assumptions” is quite different than in the general case; and that the resulting (near tight) bound is a rather intricate tradeoff between the three parameters $$n,k,\varepsilon$$.
Priv’IT: Private and Sample Efficient Identity Testing, by Bryan Cai, Constantinos Daskalakis, and Gautam Kamath (arXiv). Distribution testing is a vibrant field; differential privacy (DP) is an incredibly active and topical area. What about differentially private distribution testing — a.k.a. testing the data without learning too much about any single sample? In this paper, the authors address the task of performing identity testing of distributions (“given samples from an unknown arbitrary distribution, decide whether it matches a fixed known probability distribution/model”) in the DP setting, focusing on getting the best of both worlds: how to guarantee privacy without sacrificing efficiency. Not only do they obtain asymptotically near-optimal bounds in both the testing and differential privacy parameters — they also run some experiments to validate their approach empirically. (Plus, the title is rather cute.)
Usual disclaimer: if we forgot or misrepresented your paper, please signal it in the comments — we’ll address it as quickly as we can.
# News for February 2017
This February, we had four exciting testing papers, covering domains from Boolean functions to distributions to graphs. Read on to catch up on the latest results!
Property Testing of Joint Distributions using Conditional Samples, by Rishiraj Bhattacharyya and Sourav Chakraborty (arXiv). This paper focuses on the problems of identity testing and independence testing for multivariate distributions in the conditional sampling model. Multivariate distribution testing has not previously been considered in the conditional sampling model. While previous algorithms for identity testing would generally carry over to this setting, the authors restrict themselves to algorithms where the queries are on subcubes of the domain. Interestingly, the authors show that the complexity of these problems is polynomial in the dimension (and that a polynomial dependence is necessary). That is, despite the “simple” structure of the queries, they are still powerful enough to avoid the curse of dimensionality which is present in vanilla multivariate distribution testing.
An Improved Dictatorship Test with Perfect Completeness, by Amey Bhangale, Subhash Khot, and Devanathan Thiruvenkatachari (arXiv, ECCC). A function is a dictator if it depends on exactly one variable. This paper provides an improved tester with perfect completeness for this property. In particular, they give a (non-adaptive) $$k$$-query test which never rejects a dictator function, and accepts a far-from-dictator function with probability at most $$\frac{2k +1}{2^k}$$. This improves upon the previous best result, which accepts a far-from-dictator function with probability at most $$\frac{2k+3}{2^k}$$.
An Adaptivity Hierarchy Theorem for Property Testing, by Clément L. Canonne and Tom Gur (arXiv, ECCC). This paper investigates the power of adaptivity in property testing. We most frequently think of algorithms as either adaptive or non-adaptive. In the former, an algorithm may specify each query after viewing the results of all previous queries, while in the latter, all the queries must be specified in advance. The authors focus on the natural interpolation, in which the algorithm is allowed to make queries in $$k$$ rounds, where the queries in a round may depend on the result of queries in all previous rounds. They show that the power of an algorithm can shift dramatically with just one more round of adaptivity — there exist properties which can be tested with $$\tilde O(k)$$ queries with $$k$$ rounds of adaptivity, but require $$\Omega(n)$$ queries with $$k-1$$ rounds of adaptivity.
Beyond Talagrand Functions: New Lower Bounds for Testing Monotonicity and Unateness, by Xi Chen, Erik Waingarten, and Jinyu Xie (arXiv). The main result in this paper is an improved lower bound for the well-studied problem of testing monotonicity of Boolean functions. The authors show that any two-sided and adaptive algorithm requires $$\tilde \Omega(n^{1/3})$$ samples to test whether a function is monotone or far from monotone. This improves upon the previous result of $$\tilde \Omega(n^{1/4})$$ by Belovs and Blais, which was the first to show that adaptive monotonicity testing requires polynomially-many queries. The construction of Belovs and Blais uses Talagrand’s random DNFs, for which they also showed $$\tilde O(n^{1/4})$$ queries are sufficient. This paper analyzes an extension of this family, denoted as two-level Talagrand functions. The result leaves open the tantalizing question: does adaptivity help with monotonicity testing? Or does the complexity remain $$\tilde \Theta(\sqrt{n})$$? The authors also prove lower bounds for the problem of testing unateness, a problem of recent interest and a natural generalization of monotonicity.
# News for December 2016
December was indeed a merry month for property testing, with seven new papers appearing online.* Let’s hope the trend continues!
Cube vs. Cube Low Degree Test, by Amey Bhangale, Irit Dinur, and Inbal Livni Navon (ECCC). This work provides a new and improved analysis of the “low-degree test” of Raz and Safra, tightening the dependence on the alphabet size of the soundness parameter. Specifically, the goal is as follows: given query access to the encoding of a function $$f\colon \mathbb{F}^m\to \mathbb{F}$$ as a “cube table,” decide whether $$f$$ is a low-degree polynomial (i.e., of degree at most $$d$$). With direct applications to PCP theorems, this question has a long history; here, the authors focus on a very simple and natural test, introduced by Raz–Safra and Arora–Sudan. In particular, they improve the soundness guarantee, which previously required the error to be at least $$\textrm{poly}(d)/\mathbb{F}^{1/8}$$, to obtain a dependence on the field size which is only $$\mathbb{F}^{-1/2}$$.
Robust Multiplication-based Tests for Reed-Muller Codes, by Prahladh Harsha and Srikanth Srinivasan (arXiv). Given a function $$f\colon \mathbb{F}_q^n\to\mathbb{F}_q$$ purported to be a degree-$$d$$ polynomial, deciding whether this is indeed the case is a question with applications to hardness of approximation and — as the title strongly hints— to testing of Reed—Muller codes. Here, the authors generalize and improve on a test originally due to Dinur and Guruswami, improving on the soundness: that is, they show a robust version of the soundness guarantee of this multiplication test, answering a question left open by Dinur and Guruswami.
A Note on Testing Intersection of Convex Sets in Sublinear Time, by Israela Solomon (arXiv). In this note, the author addresses the following testing question: given $$n$$ convex sets in $$\mathbb{R}^d$$, distinguish between the case where (i) the intersection is non-empty, and (ii) even after removing any $$\varepsilon n$$ sets, the intersection is still empty. Through the use of a generalization of Helly’s Theorem due Katchalski and Liu (1979), the author then provides and analyzes an algorithm for this question, with query complexity $$O(\log 1/(d\varepsilon))$$.
A Characterization of Constant-Sample Testable Properties, by Eric Blais and Yuichi Yoshida (arXiv). A lot of work in property testing has been to understand which properties can be locally tested, i.e. admit testers with constant query complexity. Here, the authors tackle — and answer — the variant of this question in the sample-based testing model, that is restricting the ability of the algorithm to only obtain the value of the function on independently and uniformly distributed locations. Namely, they provide a full characterization of the properties of Boolean-valued function which are testable with constant sample complexity, resolving the natural question of whether most properties are easily tested from samples only. As it turns out, only those properties which are (essentially) $$O(1)$$-part-symmetric have this nice testability — where a property $$\mathcal{P}$$ is $$k$$-part-symmetric if one can divide the input variables in $$k$$ blocks, such that membership to $$\mathcal{P}$$ is invariant by permutations inside each block.
The last three are all works on distribution testing, and all concerned with the high-dimensional case. Indeed, most of the distribution testing literature so far has been focusing in probability distributions over arbitrary discrete sets or the one-dimension ordered line; however, when trying to use these results in the (arbitrary) high-dimensional case on is subjected to the curse of dimensionality. Can one leverage (presumed) structure to reduce the cost, and obtain computationally and sample-efficient testers?
Testing Ising Models, by Costis Daskalakis, Nishanth Dikkala, and Gautam Kamath (arXiv). In this paper, the authors take on the above question in the context of Markov Random Fields, and more specifically in the Ising model. Modeling the (unknown) high-dimensional distributions as Ising models with some promise on the parameters, they tackle the two questions of identity and independence testing; respectively, “is the unknown Ising model equal to a fixed, known reference distribution?” and “is the high-dimensional, a priori complex distribution a product distribution?”. They obtain both testing algorithms and lower bounds for these two problems, where the soundness guarantee sought is in terms of the symmetrized Kullback—Leibler divergence (a notion of distance more stringent that the statistical distance).
Square Hellinger Subadditivity for Bayesian Networks and its Applications to Identity Testing, by Costis Daskalakis and Qinxuan Pan (arXiv). Another natural model to study structured high-dimensional distribution is that of Bayesian networks, that is directed graphs providing a succinct description of the dependencies between coordinates. This paper studies the closeness testing problem (testing if two unknown distributions are equal or far, with regard to the usual statistical distance) for Bayesian networks, parameterized by the dimension, alphabet size, and (promise on the) maximum in-degree of the unknown Bayes net. At its core is a new inequality relating the (squared) Hellinger distance of two Bayesian networks to the sum of the (squared) Hellinger distances between their marginals.
Testing Bayesian Networks, by Clément Canonne, Ilias Diakonikolas, Daniel Kane, and Alistair Stewart (arXiv). Tackling the testing of Bayesian networks as well, this second paper considers two of the most standard testing questions — identity (one-unknown) and closeness (two-unknown) testing — under various assumptions, in order to pinpoint the exact sample complexity in each case. Specifically the goal is to see when (and under which natural restrictions) does testing become easier than learning for Bayesian networks, focusing on the dimension and maximum in-degree as parameters.
* As usual, if we forgot one or you find imprecisions in our review of a paper, please let us now in the comments below.
# News for November 2016
November was quite eventful for property testing, with six exciting new results for you to peruse.
Alice and Bob Show Distribution Testing Lower Bounds (They don’t talk to each other anymore.), by Eric Blais, Clément L. Canonne, and Tom Gur (ECCC). The authors examine distribution testing lower bounds through the lens of communication complexity, a-la Blais, Brody, and Matulef, who previously showed such a connection for property testing lower bounds in the Boolean function setting. In this work, the authors’ main result involves testing identity to a specific distribution $$p$$. While Valiant and Valiant showed tight bounds involving the $$\ell_{2/3}$$-quasinorm of $$p$$, this paper gives tight bounds using a different quantity, namely Peetre’s $$K$$-functional. Their techniques also give lower bounds for several other properties (some old and some new), including monotonicity, sparse symmetric support, and $$k$$-juntas in the PAIRCOND model.
Fast property testing and metrics for permutations, by Jacob Fox and Fan Wei (arXiv). This paper proves a general testing result for permutations. In particular, it shows that any hereditary property of permutations is two-sided testable with respect to the rectangular distance with a constant number of queries. While in many such testing results on combinatorial objects (such as graphs), a “constant number of queries” may be exorbitantly large (due to complexities arising from an application of the strong regularity lemma), surprisingly, the complexity obtained in this paper is polynomial in $$1/\varepsilon$$.
A Unified Maximum Likelihood Approach for Optimal Distribution Property Estimation, by Jayadev Acharya, Hirakendu Das, Alon Orlitsky, and Ananda Theertha Suresh (arXiv, ECCC). There has been a considerable deal of work recently on estimating several symmetric distribution properties, namely support size, support coverage, entropy, and distance to uniformity. One drawback of these results is that, despite the similarities between these properties, seemingly different techniques are required to obtain optimal rates for each. This paper shows that one concept, pattern maximum likelihood (PML), unifies them all. A PML distribution of a multiset of samples is any distribution which maximizes the likelihood of observing the multiplicities of the multiset, after discarding the labels on the elements. This can behave quite differently from the sequence maximum likelihood (SML), or empirical distribution. In particular, if a multiset of samples on support $$\{x, y\}$$ is $$\{x, y, x\}$$, then the SML is $$(2/3, 1/3)$$, while the PML is $$(1/2, 1/2)$$. The main result of this paper is, if one can approximate PML, then applying the plug-in estimator gives the optimal sample complexity for all of the aforementioned properties. The one catch is that efficient approximation of the PML is currently open. Consider the gauntlet thrown to all our readers!
Statistical Query Lower Bounds for Robust Estimation of High-dimensional Gaussians and Gaussian Mixtures, by Ilias Diakonikolas, Daniel M. Kane, and Alistair Stewart (arXiv, ECCC). While the main focus of this work is on lower bounds for distribution estimation in the statistical query (SQ) model, this paper also has some interesting lower bounds for multivariate testing problems. Namely, they show that it is impossible to achieve a sample complexity which is significantly sublinear in the dimension for either of the following two problems:
• Given samples from an $$n$$-dimensional distribution $$D$$, distinguish whether $$D = \mathcal{N}(0,I)$$ or $$D$$ is $$\varepsilon/100$$-close to any $$\mathcal{N}(\mu, I)$$ where $$\|\mu\|_2 \geq \varepsilon$$.
• Given samples from an $$n$$-dimensional distribution $$D$$, distinguish whether $$D = \mathcal{N}(0,I)$$ or $$D$$ is a mixture of $$k$$ Gaussians with almost non-overlapping components.
Collision-based Testers are Optimal for Uniformity and Closeness, by Ilias Diakonikolas, Themis Gouleakis, John Peebles, and Eric Price (arXiv, ECCC). In the TCS community, the seminal results in the field of distribution testing include the papers of Goldreich and Ron and Batu et al., which study uniformity testing and $$\ell_2$$-closeness testing (respectively) using collision based testers. While these testers appeared to be lossy, subsequent results have attained tight upper and lower bounds for these problems. As suggested by the title, this paper shows that collision-based testers actually achieve the optimal sample complexities for uniformity and $$\ell_2$$-closeness testing.
Testing submodularity and other properties of valuation functions, by Eric Blais, and Abhinav Bommireddi (arXiv). This paper studies the query complexity of several properties which have been studied in the context of valuation functions in algorithmic game theory. These properties are real-valued functions over the Boolean hypercube, and include submodularity, additivity, unit-demand, and much more. The authors show that, for constant $$\varepsilon$$ and any $$p \geq 1$$, these properties are constant-sample testable. Their results are obtained via an extension of the testing by implicit learning method of Diakonikolas et al.
# News for October 2016
Alas, a rather dry month for property testing. We did find one quantum computing result based on the classic linearity testing theorem.
Robust Self-Testing of Many-Qubit States, by Anand Natarajan and Thomas Vidick (arXiv). (Frankly, my understanding of quantum computation is poor, and this summary may reflect that. Then again, some searching on Google and Wikipedia have definitely broadened my horizons.) One of the key concepts in quantum computation is the notion of entanglement. This allows for correlations between (qu)bits of information beyond what can be classically achieved. Given some device with supposed quantum properties (such as sets of entangled bits), is there a way of verification by measuring various outcomes of the device? This is referred to as self-testing of quantum states. This paper proves such a result for a set of $$n$$ EPR pairs, which one can think of $$n$$ pairs of “entangled” qubits. The interest to us property testers is the application of a quantum version of the seminar Blum, Luby, Rubinfeld linearity test.
# News for September 2016
September has just concluded, and with it the rise of Fall in property testing: no less than 5 papers* this month.
Removal Lemmas for Matrices, by Noga Alon and Omri Ben-Eliezer (arXiv). The graph removal lemma, and its many variants and generalizations, is a cornerstone in graph property testing (among others), and is a fundamental ingredient in many graph testing results. In its simplest form, it states that for any fixed $$h$$-vertex pattern $$H$$ and $$\varepsilon > 0$$, there is some constant $$\delta=\delta(\varepsilon)$$ such that any graph $$G$$ on $$n$$ vertices containing at most $$\delta n^h$$ copies of $$H$$ can be “fixed” (made $$H$$-free) by removing at most $$\varepsilon n^2$$ edges.
The authors introduce and establish several analogues of this result, in the context of matrices. These matrix removal lemmas have direct implications in property testing (discussed in Section 1.2); for instance, they imply constant-query testing of $$F$$-freeness of (binary) matrices, where $$F$$ is any family of binary matrices closed under row permutations.
Improving and extending the testing of distributions for shape-restricted properties, by Eldar Fischer, Oded Lachish, and Yadu Vasudev (arXiv). Testing membership to a class (family) is a very natural question in distribution testing, and one which has received significant attention lately. In this paper, the authors build on, improve, and generalize the techniques of Canonne, Diakonikolas, Gouleakis, and Rubinfeld (2016) to obtain a generic, one-size-fits-all algorithm for this question. They also consider the same question in the “conditional oracle” setting, for which their ideas yield significantly more sample-efficient algorithms. At the core of their approach is a better and simpler learning procedure for families of distributions that enjoy some decomposability property.
The Bradley―Terry condition is $$L_1$$–testable, by Agelos Georgakopoulos and Konstantinos Tyros (arXiv). An incursion in probability models: say that the directed weighted graph $$((V,E),w)$$ on $$n$$ vertices, with $$w\colon E_n\to[0,1]$$ such that $$w(x,y)=w(y,x)$$ for all $$(x,y)\in V^2$$, satisfies the Bradley—Terry condition if there exists an assignment of probabilities $$(a_x)_{x\in V}$$ such that
$$\qquad\displaystyle \frac{w(x,y)}{w(y,x)} = \frac{a_x}{a_y}$$
for all $$(x,y)\in V^2$$. This condition captures whether such a graph corresponds to a Bradley—Terry tournament.
This paper considers this characterization from a property testing point of view: that is, it addresses the task of testing, in the $$L_1$$-sense of Berman, Raskhodnikova, and Yaroslavtsev (2012), if a $$((V,E),w)$$ satisfies the Bradley—Terry condition (or is far from any such graph).
On SZK and PP, by Adam Bouland, Lijie Chen, Dhiraj Holden, Justin Thaler, and Prashant Nalini Vasudevan (arXiv, ECCC). Another (unexpected) connection! This work proves query and communication complexity separations between two complexity classes, $$\textsf{NISZK}$$ (non-interactive statistical zero knowledge) and $$\textsf{UPP}$$ (a generalization of $$\textsf{PP}$$ with unbounded coin flips, and thus arbitrary gap in the error probability). While I probably cannot even begin to list all that goes over my head in this paper, the authors show in Section 2.3.2 some implications of their results for property testing, for instance in distribution testing. In more detail, their results yield some lower bounds on the sample complexity of (tolerant) property testers with success probability arbitrarily close to $$1/2$$ (as opposed to the “usual” setting of $$1/2+\Omega(1)$$).
Quantum conditional query complexity, by Imdad S. B. Sardharwalla, Sergii Strelchuk, and Richard Jozsa (arXiv). Finally, our last paper of the list deals with both quantum algorithms and distribution testing, by introducing a new distribution testing model in a quantum setting, the quantum conditional oracle. This model, the quantum analogue of the conditional query oracle of Chakraborty et al. and Canonne et al., allows the (quantum) testing algorithm to get samples from the probability distribution to test, conditioning on chosen subsets of the domain.
In this setting, they obtain speedups over both (i) quantum “standard” oracle and (ii) “classical” conditional query testing for many distribution testing problems; and also consider applications to quantum testing of Boolean functions, namely balancedness.
* As usual, in case we missed or misrepresented any, please signal it in the comments.
# News for August 2016
This summer is a very exciting one for property testing, with seven (!) papers uploaded during the month of August. In particular, there have been a number of results on testing unateness.
An $$\tilde O(n)$$ Queries Adaptive Tester for Unateness, by Subhash Khot and Igor Shinkar (arXivECCC). A Boolean function is unate if, for each $$i \in [n]$$, it is monotone non-increasing or monotone non-decreasing in coordinate $$i$$. This is a generalization of monotonicity, one of the most studied problems in the property testing literature. This paper gives an adaptive algorithm for testing if a Boolean function is unate. The algorithm requires $$\tilde O(n)$$ queries, improving on the $$O(n^{1.5})$$ query tester of Goldreich et al. from 2000.
A $$\tilde O(n)$$ Non-Adaptive Tester for Unateness, by Deeparnab Chakrabarty and C. Seshadhri (arXivECCC). The unateness testing continues! This paper also gives a $$\tilde O(n)$$ algorithm for this problem, but improves upon the previous work by making non-adaptive queries. The algorithm and analysis are very clean and elegant — with a theorem by Chakrabarty et al. in place, they take up just a single page!
Testing Unateness of Real-Valued Functions, by Roksana Baleshzar, Meiram Murzabulatov, Ramesh Krishnan S. Pallavoor, and Sofya Raskhodnikova (arXiv). The final paper this month on testing unateness, this work studies real-valued functions on the hypergrid $$[n]^d$$ (as opposed to Boolean functions on the hypercube). They give an adaptive algorithm requiring $$O\left(\frac{d \log (\max (d,n))}{\varepsilon}\right)$$ queries, generalizing the above result by Khot and Shinkar, which works only for Boolean functions on the hypercube and requires $$O\left(\frac{d \log d}{\varepsilon}\right)$$ queries. Additionally, they prove lower bounds for this setting (of $$\Omega(\min(d, |R|^2))$$, where $$R$$ is the range of the function) and the Boolean hypercube setting (of $$\Omega(\sqrt{d}/\varepsilon)$$). The latter lower bound leaves open a tantalizing question: does a $$O(\sqrt{d})$$ query algorithm exist? In other words, is testing unateness no harder than testing monotonicity?
Testing k-Monotonicity, by Clément L. Canonne, Elena Grigorescu, Siyao Guo, Akash Kumar, Karl Wimmer (arXivECCC). And now, a different generalization of monotonicity! This paper studies testing $$k$$-monotonicity, where a Boolean function over a poset $$\mathcal{D}$$ is $$k$$-monotone if it alternates between the values $$0$$ and $$1$$ at most $$k$$ times on any ascending chain in $$\mathcal{D}$$. Classical monotonicity is then $$1$$-monotone in this definition. On the hypercube domain, the authors prove a separation between testing monotonicity and testing $$k$$-monotonicity and a separation between testing and learning. They also present a tolerant test for $$k$$-monotonicity over the hypergrid $$[n]^d$$ with complexity independent of $$n$$.
The Dictionary Testing Problem, by Siddharth Barman, Arnab Bhattacharyya, and Suprovat Ghoshal (arXiv). The dictionary learning problem has enjoyed extensive study in computer science. In this problem, given a matrix $$Y$$, one wishes to construct a decomposition $$Y = AX$$ such that each column of $$X$$ is $$k$$-sparse. In the settings typically studied, such a decomposition is guaranteed to exist. This paper initiates the dictionary testing problem, in which one wishes to test whether $$Y$$ admits such a decomposition, or is far from it. The authors provide a very elegant characterization — it admits a decomposition iff the columns of $$Y$$ have a sufficiently narrow Gaussian width.
The Sparse Awakens: Streaming Algorithms for Matching Size Estimation in Sparse Graphs, by Graham Cormode, Hossein Jowhari, Morteza Monemizadeh, and S. Muthukrishnan (arXiv). This paper provides streaming algorithms for estimating the size of the maximum matching in graphs with some underlying sparsity. They give a one-pass algorithm for insert-only and dynamic streams with bounded arboricity, and a two-pass algorithm for random order streams with bounded degree. Interestingly, for the latter result, they use local exploration techniques, which have seen previous use in the property testing literature.
Testing Assignments to Constraint Satisfaction Problems, by Hubie Chen, Matt Valeriote, and Yuichi Yoshida (arXiv). Given a CSP from a finite relational structure $$A$$ and query access to an assignment, is the CSP satisfied? This paper shows a dichotomy theorem which characterizes which $$A$$ admit a constant-query test. They go on to show a trichotomy theorem for when instances may include existentially quantified variables, classifying which $$A$$ admit a constant-query test, which admit only a sublinear-query test, and which require a linear number of queries.
Minimizing Quadratic Functions in Constant Time, by Kohei Hayashi and Yuichi Yoshida (arXiv). This paper investigates the problem of (approximately) minimizing quadratic functions in high-dimensions. Prior approaches to this problem scale poorly with the dimension — at least linearly, if not worse. This work uses a sampling-based method to avoid paying this cost: the resulting time complexity is completely independent of the dimension!
# News for July 2016
After a few slow months, property testing is back with a bang! This month we have a wide range on results ranging from classic problems like junta testing to new models of property testing.
Tolerant Junta Testing and the Connection to Submodular Optimization and Function Isomorphism, by Eric Blais, Clément L. Canonne, Talya Eden, Amit Levi, and Dana Ron (arXiv). The problem of junta testing requires little introduction. Given a boolean function $$f:\{-1,+1\}^n \mapsto \{-1,+1\}$$, a $$k$$-junta only depends on $$k$$ of the input variables. A classic problem has been that of property testing $$k$$-juntas, and a rich line of work (almost) resolves the complexity to be $$\Theta(k/\epsilon)$$. But what of tolerant testing, where we want to tester to accept functions close to being a junta? Previous work has shown that existing testers are tolerant, but with extremely weak parameters. This paper proves: there is a $$poly(k/\epsilon)$$-query tester that accepts every function $$\epsilon/16$$-close to a $$k$$-junta and rejects every function $$\epsilon$$-far from being a $$4k$$-junta. Note that the “gap” between the junta classes is a factor of 4, and this is the first result to achieve a constant gap. The paper also gives testers when we wish to reject functions far from being a $$k$$-junta (exactly matching the definition of tolerant testng), but the tester has an exponential dependence on $$k$$. The results have intriguing connections with isomorphism testing, and there is a neat use of constrained submodular minimization.
Testing Pattern-Freeness, by Simon Korman and Daniel Reichman (arXiv). Consider a string $$I$$ of length $$n$$ and a “pattern” $$J$$ of length $$k$$, both over some alphabet $$\Sigma$$. Our aim is to property test if $$I$$ is $$J$$-free, meaning that $$I$$ does not contains $$J$$ as a substring. This problem can also be generalized to the 2-dimensional setting, where $$I$$ and $$J$$ are matrices with entries in $$\Sigma$$. This formulation is relevant in image testing, where we need to search for a template pattern in a large image. The main results show that these testing problems can be solved in time $$O(1/\epsilon)$$, with an intriguing caveat. If the pattern $$J$$ has at least $$3$$ distinct symbols, the result holds. If $$J$$ is truly binary, then $$J$$ is not allowed to be in a specified set of forbidden patterns. The main tool is a modification lemma that shows how to “kill” a specified occurrence of $$J$$ without introducing a new one. This lemma is not true for the forbidden patterns, resulting in the dichotomy of the results.
Erasure-Resilient Property Testing, by Kashyap Dixit, Sofya Raskhodnikova, Abhradeep Thakurta, and Nithin Varma (arXiv). Property testing begins with a query model for $$f: \mathcal{D} \mapsto \mathcal{R}$$, so we can access any $$f(x)$$. But what if an adversary corrupted some fraction of the input? Consider monotonicity testing on the line, so $$f: [n] \mapsto \mathbb{R}$$. We wish to test if $$f$$ is $$\epsilon$$-far from being monotone, but the adversary corrupts/hides an $$\alpha$$-fraction of the values. When we query such a position, we receive a null value. We wish to accept if there exists some “filling” of the corrupted values that makes $$f$$ monotone, and reject if all “fillings” keep $$f$$ far from monotone. It turns out the standard set of property testers are not erasure resilient, and fail on this problem. This papers gives erasure resilient testers for properties like monotonicity, Lipschitz continuity, and convexity. The heart of the techniques involves a randomized binary tree tester for the line that can avoid the corrupted points.
Partial Sublinear Time Approximation and Inapproximation for Maximum Coverage, by Bin Fu (arXiv). Consider the classic maximum coverage problem. We have $$m$$ sets $$A_1, A_2, \ldots, A_m$$ and wish to pick the $$k$$ of them with the largest union size. The query model allows for membership queries, cardinality queries, and generation of random elements from a set. Note that the size of the input can be thought of as $$\sum_i |A_i|$$. Can one approximate this problem without reading the full input? This paper gives a $$(1-1/e)$$-factor approximation that runs in time $$poly(k)m\log m$$, and is thus sublinear in the input size. The algorithm essentially implements a greedy approximation algorithm in sublinear time. It is shown the the linear dependence on $$m$$ is necessary: there is no constant factor approximation that runs in $$q(n) m^{1-\delta}$$ (where $$n$$ denotes the maximum cardinality, $$q(\cdot)$$ is an arbitrary function, and $$\delta > 0$$).
Local Testing for Membership in Lattices, by Karthekeyan Chandrasekaran, Mahdi Cheraghchi, Venkata Gandikota, and Elena Grigorescu (arXiv). Inspired by the theory of locally testable codes, this paper introduces local testing in lattices. Given a set of basis vectors $$b_1, b_2, \ldots$$ in $$\mathbb{Z}^n$$, the lattice $$L$$ is the set of all integer linear combinations of the basic vectors. This is the natural analogue of linear error correcting codes (where everything is done over finite fields). Given some input $$t \in \mathbb{Z}^n$$, we wish to determine if $$t \in L$$, or is far (defined through some norm) from all vectors in $$L$$, by querying a constant number of coordinates in $$t$$. We assume that $$L$$ is fixed, so it can be preprocessed arbitrarily. This opens up a rich source of questions, and this work might be seen as only the first step in this direction. The papers shows a number of results. Firstly, there is a family of “code formula” lattices for which testers exists (with almost matching lower bounds). Furthermore, with high probability over random lattices, testers do not exist. Analogous to testing codes, if a constant query lattice tester exists, then there exists a canonical constant query tester.
# News for June 2016
This month isn’t the most exciting for property testing. Just one paper to report (thanks to Clément for catching!), of which only a minor part is actually related to property testing.
Approximating the Spectral Sums of Large-scale Matrices using Chebyshev Approximation, by Insu Han, Dmitry Malioutov, Haim Avron, and Jinwoo Shin (arXiv). The main results of the paper deal with approximating the trace of matrix functions. Consider symmetric matrix $$M \in \mathbb{R}^{d\times d}$$, with eigenvalues $$\lambda_1, \lambda_2, \ldots, \lambda_d$$. The paper gives new algorithms to approximate $$\sum_i f(\lambda_i)$$. Each “operation” of the algorithm is a matrix-vector product, and the aim is to minimize the number of such operations. The property testing application is as follows. Suppose we wish to test if $$M$$ is positive-definite (all $$\lambda_i > 0$$). We consider a matrix $$\epsilon$$-far from being positive-definite if the smallest eigenvalue is less than $$-\epsilon \|M\|_F = -\epsilon \sum_i \lambda^2_i$$. The main theorems in this paper yield a testing algorithm (under this definition of distance) that makes $$o(d)$$ matrix-vector products. While this is not exactly sublinear under a standard query access model, it is relevant when $$M$$ is not explicitly represented and the only access is through such matrix-vector product queries. | 2018-03-19 05:07:48 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7566843032836914, "perplexity": 834.2478396016055}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-13/segments/1521257646375.29/warc/CC-MAIN-20180319042634-20180319062634-00007.warc.gz"} |
https://www.albert.io/ie/topology/cylinder | Free Version
Easy
# Cylinder
TOPO-KVYYWV
What is the fundamental group of a hollow cylinder?
A
$0$
B
$\mathbb Z$
C
$\mathbb Z/2\mathbb Z$
D
$\mathbb Z\times\mathbb Z/2\mathbb Z$
E
$\mathbb Z^2$ | 2017-02-22 08:35:13 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5189033150672913, "perplexity": 6131.304996278216}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501170925.44/warc/CC-MAIN-20170219104610-00369-ip-10-171-10-108.ec2.internal.warc.gz"} |
http://doc.cgal.org/latest/Box_intersection_d/index.html | CGAL 4.4 - Intersecting Sequences of dD Iso-oriented Boxes
User Manual
# Introduction
Simple questions on geometric primitives, such as intersection and distance computations, can themselves become quite expensive if the primitives are not so simple anymore, for example, three-dimensional triangles and facets of polyhedral surfaces. Thus algorithms operating on these primitives tend to be slow in practice. A common (heuristic) optimization approximates the geometric primitives with their axis-aligned bounding boxes, runs a suitable modification of the algorithm on the boxes, and whenever a pair of boxes has an interesting interaction Boxes represent volumes or point-sets. So, intersection means intersection of the point-set enclosed by the box and not only intersection of the boundary, of course., only then the exact answer is computed on the complicated geometric primitives contained in the boxes.
We provide an efficient algorithm [2] for finding all intersecting pairs for large numbers of iso-oriented boxes, i.e., typically these will be such bounding boxes of more complicated geometries. One immediate application of this algorithm is the detection of all intersections (and self-intersections) for polyhedral surfaces, i.e., applying the algorithm on a large set of triangles in space, we give an example program later in this chapter. Not so obvious applications are proximity queries and distance computations among such surfaces, see Section Example for Point Proximity Search with a Custom Traits Class for an example and [2] for more details.
# Definition
A $$d$$-dimensional iso-oriented box is defined as the Cartesian product of $$d$$ intervals. We call the box half-open if the $$d$$ intervals $$\{ [lo_i,hi_i) \,|\, 0 \leq i < d\}$$ are half-open intervals, and we call the box closed if the $$d$$ intervals $$\{ [lo_i,hi_i] \,|\, 0 \leq i < d\}$$ are closed intervals. Note that closed boxes support zero-width boxes and they can intersect at their boundaries, while non-empty half-open boxes always have a positive volume and they only intersect iff their interiors overlap. The distinction between closed and half-open boxes does not require a different representation of boxes, just a different interpretation when comparing boxes, which is selected with the two possible values for the topology parameter:
• Box_intersection_d::HALF_OPEN and
• Box_intersection_d::CLOSED.
The number type of the interval boundaries must be one of the built-in types int, unsigned int, double or float.
In addition, a box has an unique id-number. It is used to order boxes consistently in each dimension even if boxes have identical coordinates. In consequence, the algorithm guarantees that a pair of intersecting boxes is reported only once. Note that boxes with equal id-number are not reported since they obviously intersect trivially.
The box intersection algorithm comes in two flavors: One algorithm works on a single sequence of boxes and computes all pairwise intersections, which is called the complete case, and used, for example, in the self-intersection test. The other algorithm works on two sequences of boxes and computes the pairwise intersections between boxes from the first sequence with boxes from the second sequence, which is called the bipartite case. For each pairwise intersection found a callback function is called with two arguments; the first argument is a box from the first sequence and the second argument a box from the second sequence. In the complete case, the second argument is a box from an internal copy of the first sequence.
# Software Design
The box intersection algorithm is implemented as a family of generic functions; the functions for the complete case accept one iterator range, and the functions for the bipartite case accept two iterator ranges. The callback function for reporting the intersecting pairs is provided as a template parameter of the BinaryFunction concept. The two principle function calls utilizing all default arguments look as follows:
#include <CGAL/box_intersection_d.h>
template< class RandomAccessIterator, class Callback >
Callback callback);
template< class RandomAccessIterator1,
class RandomAccessIterator2,
class Callback >
void box_intersection_d(RandomAccessIterator1 begin1, RandomAccessIterator1 end1,
RandomAccessIterator2 begin2, RandomAccessIterator2 end2,
Callback callback);
Additional parameters to the functions calls are a cutoff value to adjust performance trade-offs, and a topology parameter selecting between topologically closed boxes (the default) and topologically half-open boxes.
The algorithm reorders the boxes in the course of the algorithm. Now, depending on the size of a box it can be faster to copy the boxes, or to work with pointers to boxes and copy only pointers. We offer automatic support for both options. To simplify the description, let us call the value_type of the iterator ranges box handle. The box handle can either be our box type itself or a pointer (or const pointer) to the box type; these choices represent both options from above.
In general, the algorithms treat the box type as opaque type and just assume that they are models of the Assignable concept, so that the algorithms can modify the input sequences and reorder the boxes. The access to the box dimension and box coordinates is mediated with a traits class of the BoxIntersectionTraits_d concept. A default traits class is provided that assumes that the box type is a model of the BoxIntersectionBox_d concept and that the box handle, i.e., the iterators value type, is identical to the box type or a pointer to the box type (see the previous paragraph for the value versus pointer nature of the box handle).
Two implementations of iso-oriented boxes are provided; Box_intersection_d::Box_d as a plain box, and Box_intersection_d::Box_with_handle_d as a box plus a handle that can be used to point to the full geometry that is approximated by the box. Both implementations have template parameters for the number type used for the interval bounds, for the fixed dimension of the box, and for a policy class [1] selecting among several solutions for providing the id-number.
The function signatures for the bipartite case look as follows. The signatures for the complete case with the box_self_intersection_d() function look the same except for the single iterator range.
#include CGAL/box_intersection_d.h
template< class RandomAccessIterator1,
class RandomAccessIterator2,
class Callback >
void box_intersection_d(RandomAccessIterator1 begin1, RandomAccessIterator1 end1,
RandomAccessIterator2 begin2, RandomAccessIterator2 end2,
Callback callback, std::ptrdiff_t cutoff = 10,
template< class RandomAccessIterator1,
class RandomAccessIterator2,
class Callback, class BoxTraits >
void box_intersection_d(RandomAccessIterator1 begin1, RandomAccessIterator1 end1,
RandomAccessIterator2 begin2, RandomAccessIterator2 end2,
Callback callback, BoxTraits box_traits,
std::ptrdiff cutoff = 10,
# Minimal Example for Intersecting Boxes
The box implementation provided with Box_intersection_d::Box_d<double,2> has a dedicated constructor for the CGAL bounding box type Bbox_2 (similar for dimension 3). We use this in our minimal example to create easily nine two-dimensional boxes in a grid layout of $$3 \times 3$$ boxes. Additionally we pick the center box and the box in the upper-right corner as our second box sequence query.
The default policy of the box type implements the id-number with an explicit counter in the boxes, which is the default choice since it always works, but it costs space that could potentially be avoided, see the example in the next section. We use the id-number in our callback function to report the result of the intersection algorithm. The result will be that the first query box intersects all nine boxes and the second query box intersects the four boxes in the upper-right quadrant. See Section Example Using the Topology and the Cutoff Parameters for the change of the topology parameter and its effect.
#include <CGAL/box_intersection_d.h>
#include <CGAL/Bbox_2.h>
#include <iostream>
typedef CGAL::Bbox_2 Bbox;
// 9 boxes of a grid
Box boxes[9] = { Bbox( 0,0,1,1), Bbox( 1,0,2,1), Bbox( 2,0,3,1), // low
Bbox( 0,1,1,2), Bbox( 1,1,2,2), Bbox( 2,1,3,2), // middle
Bbox( 0,2,1,3), Bbox( 1,2,2,3), Bbox( 2,2,3,3)};// upper
// 2 selected boxes as query; center and upper right
Box query[2] = { Bbox( 1,1,2,2), Bbox( 2,2,3,3)};
void callback( const Box& a, const Box& b ) {
std::cout << "box " << a.id() << " intersects box " << b.id() << std::endl;
}
int main() {
CGAL::box_intersection_d( boxes, boxes+9, query, query+2, callback);
return 0;
}
# Example for Finding Intersecting 3D Triangles
The conventional application of the axis-aligned box intersection algorithm will start from complex geometry, here 3D triangles, approximate them with their bounding box, compute the intersecting pairs of boxes, and check only for those if the original triangles intersect as well.
We start in the main function and create ten triangles with endpoints chosen randomly in a cube $$[-1,+1)^3$$. We store the triangles in a vector called triangles.
Next we create a vector for the bounding boxes of the triangles called boxes. For the boxes we choose the type Box_with_handle_d<double,3,Iterator> that works nicely together with the CGAL bounding boxes of type Bbox_3. In addition, each box stores the iterator to the corresponding triangle.
The default policy of this box type uses for the id-number the address of the value of the iterator, i.e., the address of the triangle. This is a good choice that works correctly iff the boxes have unique iterators, i.e., there is a one-to-one mapping between boxes and approximated geometry, which is the case here. It saves us the extra space that was needed for the explicit id-number in the previous example.
We run the self intersection algorithm with the report_inters function as callback. This callback reports the intersecting boxes. It uses the handle and the global triangles vector to calculate the triangle numbers. Then it checks the triangles themselves for intersection and reports if not only the boxes but also the triangles intersect. We take some precautions before the intersection test in order to avoid problems, although unlikely, with degenerate triangles that we might have created with the random process.
This example can be easily extended to test polyhedral surfaces of the Polyhedron_3 class for (self-) intersections. The main difference are the numerous cases of incidences between triangles in the polyhedral surface that should not be reported as intersections, see the examples/Polyhedron/polyhedron_self_intersection.cpp example program in the CGAL distribution.
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/intersections.h>
#include <CGAL/point_generators_3.h>
#include <CGAL/Bbox_3.h>
#include <CGAL/box_intersection_d.h>
#include <CGAL/function_objects.h>
#include <CGAL/Join_input_iterator.h>
#include <CGAL/algorithm.h>
#include <vector>
typedef Kernel::Point_3 Point_3;
typedef Kernel::Triangle_3 Triangle_3;
typedef std::vector<Triangle_3> Triangles;
typedef Triangles::iterator Iterator;
Triangles triangles; // global vector of all triangles
// callback function that reports all truly intersecting triangles
void report_inters( const Box& a, const Box& b) {
std::cout << "Box " << (a.handle() - triangles.begin()) << " and "
<< (b.handle() - triangles.begin()) << " intersect";
if ( ! a.handle()->is_degenerate() && ! b.handle()->is_degenerate()
&& CGAL::do_intersect( *(a.handle()), *(b.handle()))) {
std::cout << ", and the triangles intersect also";
}
std::cout << '.' << std::endl;
}
int main() {
// Create 10 random triangles
typedef CGAL::Random_points_in_cube_3<Point_3> Pts;
Pts points( 1); // in centered cube [-1,1)^3
Triangle_gen triangle_gen( points, points, points);
CGAL::cpp11::copy_n( triangle_gen, 10, std::back_inserter(triangles));
// Create the corresponding vector of bounding boxes
std::vector<Box> boxes;
for ( Iterator i = triangles.begin(); i != triangles.end(); ++i)
boxes.push_back( Box( i->bbox(), i));
// Run the self intersection algorithm with all defaults
CGAL::box_self_intersection_d( boxes.begin(), boxes.end(), report_inters);
return 0;
}
# Example for Using Pointers to Boxes
We modify the previous example, finding intersecting 3D triangles, and add an additional vector ptr that stores pointers to the bounding boxes, so that the intersection algorithm will work on a sequence of pointers and not on a sequence of boxes. The change just affects the preparation of the additional vector and the call of the box intersection function. The box intersection function (actually its default traits class) detects automatically that the value type of the iterators is a pointer type and not a class type.
// Create the corresponding vector of pointers to bounding boxes
std::vector<Box *> ptr;
for ( std::vector<Box>::iterator i = boxes.begin(); i != boxes.end(); ++i)
ptr.push_back( &*i);
// Run the self intersection algorithm with all defaults on the
// indirect pointers to bounding boxes. Avoids copying the boxes.
CGAL::box_self_intersection_d( ptr.begin(), ptr.end(), report_inters);
In addition, the callback function report_inters() needs to be changed to work with pointers to boxes. The full example program looks as follows:
#include <CGAL/Exact_predicates_inexact_constructions_kernel.h>
#include <CGAL/intersections.h>
#include <CGAL/point_generators_3.h>
#include <CGAL/Bbox_3.h>
#include <CGAL/box_intersection_d.h>
#include <CGAL/function_objects.h>
#include <CGAL/Join_input_iterator.h>
#include <CGAL/algorithm.h>
#include <vector>
typedef Kernel::Point_3 Point_3;
typedef Kernel::Triangle_3 Triangle_3;
typedef std::vector<Triangle_3> Triangles;
typedef Triangles::iterator Iterator;
Triangles triangles; // global vector of all triangles
// callback function that reports all truly intersecting triangles
void report_inters( const Box* a, const Box* b) {
std::cout << "Box " << (a->handle() - triangles.begin()) << " and "
<< (b->handle() - triangles.begin()) << " intersect";
if ( ! a->handle()->is_degenerate() && ! b->handle()->is_degenerate()
&& CGAL::do_intersect( *(a->handle()), *(b->handle()))) {
std::cout << ", and the triangles intersect also";
}
std::cout << '.' << std::endl;
}
int main() {
// Create 10 random triangles
typedef CGAL::Random_points_in_cube_3<Point_3> Pts;
Pts points( 1); // in centered cube [-1,1)^3
Triangle_gen triangle_gen( points, points, points);
CGAL::cpp11::copy_n( triangle_gen, 10, std::back_inserter(triangles));
// Create the corresponding vector of bounding boxes
std::vector<Box> boxes;
for ( Iterator i = triangles.begin(); i != triangles.end(); ++i)
boxes.push_back( Box( i->bbox(), i));
// Create the corresponding vector of pointers to bounding boxes
std::vector<Box *> ptr;
for ( std::vector<Box>::iterator i = boxes.begin(); i != boxes.end(); ++i)
ptr.push_back( &*i);
// Run the self intersection algorithm with all defaults on the
// indirect pointers to bounding boxes. Avoids copying the boxes.
CGAL::box_self_intersection_d( ptr.begin(), ptr.end(), report_inters);
return 0;
}
A note on performance: The algorithm sorts and partitions the input sequences. It is clearly costly to copy a large box compared to a simple pointer. However, the algorithm benefits from memory locality in the later stages when it copies the boxes, while the pointers would refer to boxes that become wildly scattered in memory. These two effects, copying costs and memory locality, counteract each other. For small box sizes, i.e., small dimension, memory locality wins and one should work with boxes, while for larger box sizes one should work with pointers. The exact threshold depends on the memory hierarchy (caching) of the hardware platform and the size of the boxes, most notably the type used to represent the box coordinates. A concrete example; on a laptop with an Intel Mobile Pentium4 running at 1.80GHz with 512KB cache and 254MB main memory under Linux this version with pointers was 20% faster than the version above that copies the boxes for 10000 boxes, but the picture reversed for 100000 boxes, where the version above that copies the boxes becomes 300% faster.
Note that switching to the built-in type float is supported by the box intersection algorithm, but the interfacing with the CGAL bounding box Bbox_3 would not be that easy. In particular, just converting from the double to the float representation incurs rounding that needs to be controlled properly, otherwise the box might shrink and one might miss intersections.
# Example Using the Topology and the Cutoff Parameters
Boxes can be interpreted by the box intersection algorithm as closed or as half-open boxes, see also Section Definition. Closed boxes support zero-width boxes and they can intersect at their boundaries, while half-open boxes always have a positive volume and they only intersect iff their interiors overlap. The choice between closed or half-open boxes is selected with the topology parameter and its two values:
• Box_intersection_d::HALF_OPEN and
• Box_intersection_d::CLOSED.
The example program uses a two-dimensional box with int coordinates and id-numbers that are by default explicitly stored. We create the same boxes as in the minimal example in Section Minimal Example for Intersecting Boxes. We create a $$3 \times 3$$ grid of boxes, and two boxes for the query sequence, namely the box at the center and the box from the upper-right corner of the grid.
We write a more involved callback function object Report that stores an output iterator and writes the id-number of the box in the first argument to the output iterator. We also provide a small helper function report() that simplifies the use of the function object.
We call the box intersection algorithm twice; once for the default topology, which is the closed box topology, and once for the half-open box topology. We sort the resulting output for better readability and verify its correctness with the check1 and check2 data. For the closed box topology, the center box in query intersects all boxes, and the upper-right box in query intersects the four boxes of the upper-right quadrant in boxes. Almost all intersections are with the box boundaries, thus, for the half-open topology only one intersection remains per query box, namely its corresponding box in boxes. So, the output of the algorithm will be:
0 1 2 3 4 4 5 5 6 7 7 8 8
4 8
For the second box intersection function call we have to specify the cutoff parameter explicitly. See the Section Runtime Performance below for a detailed discussion.
#include <CGAL/box_intersection_d.h>
#include <vector>
#include <algorithm>
#include <iterator>
#include <cassert>
// coordinates for 9 boxes of a grid
int p[9*4] = { 0,0,1,1, 1,0,2,1, 2,0,3,1, // lower
0,1,1,2, 1,1,2,2, 2,1,3,2, // middle
0,2,1,3, 1,2,2,3, 2,2,3,3};// upper
// 9 boxes
Box boxes[9] = { Box( p, p+ 2), Box( p+ 4, p+ 6), Box( p+ 8, p+10),
Box( p+12, p+14), Box( p+16, p+18), Box( p+20, p+22),
Box( p+24, p+26), Box( p+28, p+30), Box( p+32, p+34)};
// 2 selected boxes as query; center and upper right
Box query[2] = { Box( p+16, p+18), Box( p+32, p+34)};
// callback function object writing results to an output iterator
template <class OutputIterator>
struct Report {
Report( OutputIterator i) : it(i) {} // store iterator in object
// We write the id-number of box a to the output iterator assuming
// that box b (the query box) is not interesting in the result.
void operator()( const Box& a, const Box&) { *it++ = a.id(); }
};
template <class Iter> // helper function to create the function object
Report<Iter> report( Iter it) { return Report<Iter>(it); }
int main() {
// run the intersection algorithm and store results in a vector
std::vector<std::size_t> result;
CGAL::box_intersection_d( boxes, boxes+9, query, query+2,
report( std::back_inserter( result)));
// sort, check, and show result
std::sort( result.begin(), result.end());
std::size_t check1[13] = {0,1,2,3,4,4,5,5,6,7,7,8,8};
assert(result.size() == 13 && std::equal(check1,check1+13,result.begin()));
std::copy( result.begin(), result.end(),
std::ostream_iterator<std::size_t>( std::cout, " "));
std::cout << std::endl;
// run it again but for different cutoff value and half-open boxes
result.clear();
CGAL::box_intersection_d( boxes, boxes+9, query, query+2,
report( std::back_inserter( result)),
std::ptrdiff_t(1),
// sort, check, and show result
std::sort( result.begin(), result.end());
std::size_t check2[2] = {4,8};
assert(result.size() == 2 && std::equal(check2, check2+2, result.begin()));
std::copy( result.begin(), result.end(),
std::ostream_iterator<std::size_t>( std::cout, " "));
std::cout << std::endl;
return 0;
}
# Runtime Performance
The implemented algorithm is described in [2] as version two. Its performance depends on a cutoff parameter. When the size of both iterator ranges drops below the cutoff parameter the function switches from the streamed segment-tree algorithm to the two-way-scan algorithm, see [2] for the details.
The streamed segment-tree algorithm needs $$O(n \log^d (n) + k)$$ worst-case running time and $$O(n)$$ space, where $$n$$ is the number of boxes in both input sequences, $$d$$ the (constant) dimension of the boxes, and $$k$$ the output complexity, i.e., the number of pairwise intersections of the boxes. The two-way-scan algorithm needs $$O(n \log (n) + l)$$ worst-case running time and $$O(n)$$ space, where $$l$$ is the number of pairwise overlapping intervals in one dimensions (the dimension where the algorithm is used instead of the segment tree). Note that $$l$$ is not necessarily related to $$k$$ and using the two-way-scan algorithm is a heuristic.
Unfortunately, we have no general method to automatically determine an optimal cutoff parameter, since it depends on the used hardware, the runtime ratio between callback runtime and segment-tree runtime, and of course the number of boxes to be checked and their distribution. In cases where the callback runtime is dominant, it may be best to make the threshold parameter small. Otherwise a cutoff $$=\sqrt{n}$$ can lead to acceptable results. For well distributed boxes the original paper [2] gives optimal cutoffs in the thousands. Anyway, for optimal runtime some experiments to compare different cutoff parameters are recommended.
To demonstrate that box intersection can be done quite fast, different box sequences are intersected in the range between 4 and 800000 boxes total. We use three-dimensional default boxes of closed topology with float coordinates and without additional data fields. The algorithm works directly on the boxes, not on pointer to boxes. Each box intersection is reported to an empty dummy callback.
For each box set, a near-optimal cutoff parameter is determined using an adaptive approximation. The runtime required for streaming is compared against usual scanning. Results on a Xeon 2.4GHz with 4GB main memory can be seen in Figure 62.1. For a small number of boxes, pure scanning is still faster than streaming with optimal cutoff, which would just delegate the box sets to the scanning algorithm. As there are more and more boxes, the overhead becomes less important.
Figure 62.1 Runtime comparison between the scanning and the streaming algorithm.
# Example Using a Custom Box Implementation
The example in the previous Section Example Using the Topology and the Cutoff Parameters uses an array to provide the coordinates and then creates another array for the boxes. In the following example we write our own box class Box that we can initialize directly with the four coordinates and create the array of boxes directly. We also omit the explicitly stored id-number and use the address of the box itself as id-number. This works only if the boxes do not change their position, i.e., we work with pointers to the boxes in the intersection algorithm.
We follow with our own box class Box the BoxIntersectionBox_d concept, which allows us to reuse the default traits implementation, i.e., we can use the same default function call to compute all intersections. See the example in the next section for a self-written traits class. So, in principle, the remainder of the example stays the same and we omit the part from the previous example for brevity that illustrates the half-open box topology.
The requirements for the box implementation are best studied on the reference manual page of BoxIntersectionBox_d. In a nutshell, we have to define the type NT for the box coordinates and the type ID for the id-number. Member functions give access to the coordinates and the id-number. A static member function returns the dimension.
#include <CGAL/box_intersection_d.h>
#include <vector>
#include <algorithm>
#include <iterator>
#include <cassert>
struct Box {
typedef int NT;
typedef std::ptrdiff_t ID;
int lo[2], hi[2];
Box( int lo0, int lo1, int hi0, int hi1) { lo[0]=lo0; lo[1]=lo1; hi[0]=hi0; hi[1]=hi1;}
static int dimension() { return 2; }
int min_coord(int dim) const { return lo[dim]; }
int max_coord(int dim) const { return hi[dim]; }
// id-function using address of current box,
// requires to work with pointers to boxes later
std::ptrdiff_t id() const { return (std::ptrdiff_t)(this); }
};
// 9 boxes of a grid
Box boxes[9] = { Box( 0,0,1,1), Box( 1,0,2,1), Box( 2,0,3,1), // low
Box( 0,1,1,2), Box( 1,1,2,2), Box( 2,1,3,2), // middle
Box( 0,2,1,3), Box( 1,2,2,3), Box( 2,2,3,3)};// upper
// 2 selected boxes as query; center and upper right
Box query[2] = { Box( 1,1,2,2), Box( 2,2,3,3)};
// With the special id-function we need to work on box pointers
Box* b_ptr[9] = { boxes, boxes+1, boxes+2, boxes+3, boxes+4, boxes+5,
boxes+6, boxes+7, boxes+8};
Box* q_ptr[2] = { query, query+1};
// callback function object writing results to an output iterator
template <class OutputIterator>
struct Report {
Report( OutputIterator i) : it(i) {} // store iterator in object
// We write the position with respect to 'boxes' to the output iterator
// assuming that box b (the query box) is not interesting in the result.
void operator()( const Box* a, const Box*) {
*it++ = ( reinterpret_cast<Box*>(a->id()) - boxes);
}
};
template <class Iter> // helper function to create the function object
Report<Iter> report( Iter it) { return Report<Iter>(it); }
int main() {
// run the intersection algorithm and store results in a vector
std::vector<std::size_t> result;
CGAL::box_intersection_d( b_ptr, b_ptr+9, q_ptr, q_ptr+2,
report( std::back_inserter( result)),
std::ptrdiff_t(0));
// sort and check result
std::sort( result.begin(), result.end());
std::size_t chk[13] = {0,1,2,3,4,4,5,5,6,7,7,8,8};
assert( result.size()==13 && std::equal(chk,chk+13,result.begin()));
return 0;
}
# Example for Point Proximity Search with a Custom Traits Class
Given a set of 3D points, we want to find all pairs of points that are less than a certain distance apart. We use the box intersection algorithm to find good candidates, namely those that are less than this specified distance apart in the $$L_\infty$$ norm, which is a good approximation of the Euclidean norm.
We use an unusual representation for the box, namely pointers to the 3D points themselves. We implement a special box traits class that interprets the point as a box of the dimensions $$[-$$eps $$,+$$eps $$]^3$$ centered at this point. The value for eps is half the specified distance from above, i.e., points are reported if their distance is smaller than 2*eps.
The requirements for the box traits class are best studied on the reference manual page of BoxIntersectionTraits_d. In a nutshell, we have to define the type NT for the box coordinates, the type ID for the id-number, and the type Box_parameter similar to the box handle, here Point_3* since we work with the pointers. All member functions in the traits class are static. Two functions give access to the max and min coordinates that we compute from the point coordinates plus or minus the eps value, respectively. For the id-number function the address of the point itself is sufficient, since the points stay stable. Another function returns the dimension.
The report callback function computes than the Euclidean distance and prints a message for points that are close enough.
Note that we need to reserve sufficient space in the points vector to avoid reallocations while we create the points vector and the boxes vector in parallel, since otherwise the points vector might reallocate and invalidate all pointers stored in the boxes so far.
#include <CGAL/Simple_cartesian.h>
#include <CGAL/box_intersection_d.h>
#include <CGAL/point_generators_3.h>
#include <CGAL/algorithm.h>
#include <vector>
#include <algorithm>
#include <iterator>
#include <cmath>
typedef Kernel::Point_3 Point_3;
typedef CGAL::Random_points_on_sphere_3<Point_3> Points_on_sphere;
std::vector<Point_3> points;
std::vector<Point_3*> boxes; // boxes are just pointers to points
const float eps = 0.1f; // finds point pairs of distance < 2*eps
// Boxes are just pointers to 3d points. The traits class adds the
// +- eps size to each interval around the point, effectively building
// on the fly a box of size 2*eps centered at the point.
struct Traits {
typedef float NT;
typedef Point_3* Box_parameter;
typedef std::ptrdiff_t ID;
static int dimension() { return 3; }
static float coord( Box_parameter b, int d) {
return (d == 0) ? b->x() : ((d == 1) ? b->y() : b->z());
}
static float min_coord( Box_parameter b, int d) { return coord(b,d)-eps;}
static float max_coord( Box_parameter b, int d) { return coord(b,d)+eps;}
// id-function using address of current box,
// requires to work with pointers to boxes later
static std::ptrdiff_t id(Box_parameter b) { return (std::ptrdiff_t)(b); }
};
// callback function reports pairs in close proximity
void report( const Point_3* a, const Point_3* b) {
float dist = std::sqrt( CGAL::squared_distance( *a, *b));
if ( dist < 2*eps) {
std::cout << "Point " << (a - &(points.front())) << " and Point "
<< (b - &(points.front())) << " have distance " << dist
<< "." << std::endl;
}
}
int main() {
// create some random points on the sphere of radius 1.0
Points_on_sphere generator( 1.0);
points.reserve( 50);
for ( int i = 0; i != 50; ++i) {
points.push_back( *generator++);
boxes.push_back( & points.back());
}
// run the intersection algorithm and report proximity pairs
CGAL::box_self_intersection_d( boxes.begin(), boxes.end(),
report, Traits());
return 0;
}
# Design and Implementation History
Lutz Kettner and Andreas Meyer implemented the algorithms starting from the publication [2]. We had access to the original C implementation of Afra Zomorodian, which helped clarifying some questions, and we are grateful to the help of Afra Zomorodian in answering our questions during his visit. We thank Steve Robbins for an excellent review for this package. Steve Robbins provided an independent and earlier implementation of this algorithm, however, we learned too late about this implementation. | 2014-09-02 06:45:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.33263131976127625, "perplexity": 3733.32068869519}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1409535921869.7/warc/CC-MAIN-20140901014521-00025-ip-10-180-136-8.ec2.internal.warc.gz"} |
https://www.gradesaver.com/textbooks/math/trigonometry/trigonometry-10th-edition/appendix-d-graphing-techniques-exercises-page-447/68 | ## Trigonometry (10th Edition)
Published by Pearson
# Appendix D - Graphing Techniques - Exercises: 68
#### Answer
The graph $f(x) = |x|$ has been transformed horizontally to the right by 2 units, vertically down by 1 unit and with a vertical shrink by $\frac{1}{2}$. The equation of the graph will be $g(x) = \frac{1}{2} |x - 2| -1$.
#### Work Step by Step
As seen, the graph $f(x) = |x|$ has been transformed horizontally to the right by 2 units, vertically down by 1 unit and with a vertical shrink by $\frac{1}{2}$. The equation of the graph will be $g(x) = \frac{1}{2} |x - 2| -1$.
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback. | 2018-07-22 20:56:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8110941052436829, "perplexity": 819.2695392902605}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676593586.54/warc/CC-MAIN-20180722194125-20180722214125-00631.warc.gz"} |
https://pure.mpg.de/pubman/faces/ViewItemOverviewPage.jsp?itemId=item_2226624_2 | English
# Item
ITEM ACTIONSEXPORT
Released
Thesis
#### Discrete quantum geometries and their effective dimension
##### MPS-Authors
/persons/resource/persons145939
Thürigen, Johannes
Quantum Gravity & Unified Theories, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;
##### Locator
There are no locators available
1510.08706.pdf
(Preprint), 3MB
##### Supplementary Material (public)
There is no public supplementary material available
##### Citation
Thürigen, J. (2015). Discrete quantum geometries and their effective dimension. PhD Thesis.
Cite as: http://hdl.handle.net/11858/00-001M-0000-0028-FE72-8
##### Abstract
In several approaches towards a quantum theory of gravity, such as group field theory and loop quantum gravity, quantum states and histories of the geometric degrees of freedom turn out to be based on discrete spacetime. The most pressing issue is then how the smooth geometries of general relativity, expressed in terms of suitable geometric observables, arise from such discrete quantum geometries in some semiclassical and continuum limit. In this thesis I tackle the question of suitable observables focusing on the effective dimension of discrete quantum geometries. For this purpose I give a purely combinatorial description of the discrete structures which these geometries have support on. As a side topic, this allows to present an extension of group field theory to cover the combinatorially larger kinematical state space of loop quantum gravity. Moreover, I introduce a discrete calculus for fields on such fundamentally discrete geometries with a particular focus on the Laplacian. This permits to define the effective-dimension observables for quantum geometries. Analysing various classes of quantum geometries, I find as a general result that the spectral dimension is more sensitive to the underlying combinatorial structure than to the details of the additional geometric data thereon. Semiclassical states in loop quantum gravity approximate the classical geometries they are peaking on rather well and there are no indications for stronger quantum effects. On the other hand, in the context of a more general model of states which are superposition over a large number of complexes, based on analytic solutions, there is a flow of the spectral dimension from the topological dimension $d$ on low energy scales to a real number $0<\alpha<d$ on high energy scales. In the particular case of $\alpha=1$ these results allow to understand the quantum geometry as effectively fractal. | 2019-08-20 03:31:34 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.5454499125480652, "perplexity": 681.22425103173}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027315222.14/warc/CC-MAIN-20190820024110-20190820050110-00033.warc.gz"} |
http://vmu.phys.msu.ru/en/toc/2008/4 | Faculty of Physics
M.V.Lomonosov Moscow State University
Theoretical and mathematical physics
## Algebra of individual inaccuracies of intervals
### V.I. Grigor’ev
Moscow University Physics Bulletin 2008. 63. N 4. P. 229
A version of the geometry of non-arithmetizable physical space-time based on the hypothesis of individual inaccuracies of the spatial and temporal intervals is discussed.
Show Abstract
## Calculating lateral distribution functions of the Cherenkov light from extensive atmospheric showers in terms of a multilevel scheme
### L.G. Dedenko$^2$, T.M. Roganova$^2$, G.F. Fedorova$^2$, D.A. Podgrudkov$^2$, G.P. Shoziyoev$^2$
Moscow University Physics Bulletin 2008. 63. N 4. P. 232
A new technique for calculating the lateral distribution function of the Cherenkov light in very-high-energy extensive atmospheric showers is proposed. It is shown that the calculation results are in good agreement with the experimental data.
Show Abstract
## Internal energy of ordered multicomponent systems in a correlation approximation
### P.N. Nikolaev
Moscow University Physics Bulletin 2008. 63. N 4. P. 238
An expression for the internal energy of a multicomponent ordered many-body system has been obtained by means of the correlation expansion technique. This expression has proved to be useful for strongly anharmonic crystals in wide temperature and pressure ranges. The relation between the fundamental characteristics of a solid that are used in the correlation expansion and in the phonon approximation has been found. This is, in particular, the relation between the Debye and Einstein temperatures and the average speed of sound.
Show Abstract
## A study of the applicability of a method for successive determination of the kinetic coefficients of sorption on the basis of mathematical modeling
### M.G. Tokmachev$^1$, M. Riegel$^2$, W. Hoell$^2$
Moscow University Physics Bulletin 2008. 63. N 4. P. 242
This paper presents the results of mathematical modeling and experimental studies of ion-exchange process between a weakly concentrated uranium solution and sorbent grains. On the basis of mathematical modeling, we analyze the method of successive determination of the kinetic coefficients of sorption, determining its practicality.
Show Abstract
## Energy spectrum and optical properties of $D^{-}$− centers in structures with quantum disks
### V.Ch. Zhukovsky$^1$, V.D. Krevchik$^2$, M.B. Semenov$^2$, V.A. Proshkin$^2$
Moscow University Physics Bulletin 2008. 63. N 4. P. 245
The impurity absorption of light in a quasi-zero-dimensional structure with disk-like quantum dots is theoretically investigated using the zero-range potential model. It is shown that the singularity of the geometric and potential confinement of a quantum disk manifests itself in spatial anisotropy of the coupling energy of the $D^{-}$ − state and in a significant dependence of the impurity absorption edge on the characteristic dimensions of disk-like quantum dots.
Show Abstract
## Diffraction of monochromatic waves on a metallic cylinder with longitudinal slots
### A.V. Lychev
Moscow University Physics Bulletin 2008. 63. N 4. P. 251
The projection matching method is applied to the calculation of a circular cylindrical slot line with one and two slots. The diffraction problem is studied at near-resonance frequencies.
Show Abstract
Physics of nuclei and elementary particles
## Compact linear electron accelerator for radiation technologies
### A.S. Alimov, B.S. Ishkhanov, V.I. Shvedunov
Moscow University Physics Bulletin 2008. 63. N 4. P. 256
A prototype of a commercial compact continuous-beam linear electron accelerator has been designed and constructed. With its energy of 600 keV and a beam with 30 kW maximum power, the accelerator can be used in various radiation technologies. Its compact size, easy operation, reliability, wide adjustment range for the main parameters, and possible local shielding allow it to be integrated into the majority of radiation technology processes.
Show Abstract
## Mixing of $K^0$ and $B^0$ neutral mesons within the minimum supersymmetry model
### M.N. Dubinin, A.I. Sukachev
Moscow University Physics Bulletin 2008. 63. N 4. P. 259
Mixing of $K^0$ and $B^0$ mesons is studied in the scope of the minimum supersymmetry model (MSSM) with a type II Yukawa sector and explicit violation of $CP$ invariance in the Higgs potential. The mixing parameters $\Delta m_{LS}$ and $\varepsilon$ are calculated in the limit of the low-energy four-fermion approximation with a charged Higgs boson exchange. It is shown that supersymmetric effects are very small for $K^0$ mesons and may be quite significant for $B_s^0$ and $B_d^0$ mesons, which imposes constraints on the MSSM parameter space.
Show Abstract
## Experimental determination of sensitivity of piezoquartz microweighing using an electrochemical method
### Yu.K. Aleshin, A.P. Sukhorukov
Moscow University Physics Bulletin 2008. 63. N 4. P. 264
The sensitivity of a piezoquartz microweighing device was determined experimentally using an electrochemical method working in a real time mode. The results obtained confirmed the linearity of the dependence of the piezoquartz generator’s basic frequency drift on the mass of an applied metal for a wide measurement range.
Show Abstract
Optics and spectroscopy. Laser physics
## Peculiar properties of hot plasma formation under the influence of intense femtosecond pulses on the surface of molten metal
### D.S. Uryupina$^{1,2}$, M.V. Kurilova$^2$, N. Morshedian$^2$, R.V. Volkov$^{1,2}$, A.B. Savel’ev$^{1,2}$
Moscow University Physics Bulletin 2008. 63. N 4. P. 267
The peculiar properties of plasma formation on the surface of different liquid metals by femtosecond laser radiation have been studied. It is shown that plasma formation and generation of hard X-radiation on the surface of molten metal depends substantially on the contrast and weakly on the polarization of laser radiation, which clearly distinguishes the plasma produced in our experiments from the plasma generated on the surface of a solid target.
Show Abstract
Condensed matter physics
## Frequency dependence of low-temperature phononless conductivity in disordered systems
### I.P. Zvyagin, M.A. Ormont
Moscow University Physics Bulletin 2008. 63. N 4. P. 272
The frequency dependence of resonant phononless hopping conductivity in disordered systems with pointlike centers of localization in the low-temperature limit is considered within the framework of the pair approximation. It is shown that the existing theory that predicts the power-mode frequency dependence of the low-frequency phononless hopping conductivity $\sigma(\omega)$ and a transition from linear to quadratic dependence (crossover) when frequency increases may become invalid, and the quadratic frequency dependence may never manifest itself at all.
Show Abstract
Astronomy, astrophysics, and cosmology
## Primordial black holes and the asteroid hazard
### A.A. Shatskii
Moscow University Physics Bulletin 2008. 63. N 4. P. 276
The probability of a primordial black hole (PBH) entering into the Kuijper asteroid belt in the Solar System is calculated. It is shown that primordial black holes of certain masses can significantly modify asteroid trajectories and orbits. Such events can result in local solar-system catastrophes and global terrestrial catastrophes (e.g., the Tunguska event). The rate of such events is estimated.
Show Abstract
Theoretical and mathematical physics
## Generation of second and third harmonics by a rotating pulsar in parametrized post-Maxwellian electrodynamics
### V.A. Sokolov
Moscow University Physics Bulletin 2008. 63. N 4. P. 279
We consider the generation of multiple harmonics by a rotating pulsar in parametrized post-Maxwellian electrodynamics, and evaluate angular distributions of the emission intensity and conversion ratios for emission at a doubled or tripled pulsar rotation frequency.
Show Abstract
## Conformal behavior of a single chain $АВ$ block copolymer with mobile $В$-blocks
### O.S. Pevnaya, E.Yu. Kramarenko, A.R. Khokhlov
Moscow University Physics Bulletin 2008. 63. N 4. P. 281
Monte-Carlo computer modeling was used to investigate the conformal behavior of a single chain $AB$ block copolymer with mobile hydrophobic $B$-blocks and hydrophilic $A$-links. The formation of a “tailed globule” was observed with an increase of the energy of attraction between hydrophobic links. A comparison between the collapsing of a chain with mobile blocks and the behavior of regular and random block copolymers of the same structure was conducted.
Show Abstract
Physics of nuclei and elementary particles
## Macroscopic calculation of air shower radio emission
### N.N. Kalmykov$^1$, A.A. Konstantinov$^1$, R. Engel$^2$
Moscow University Physics Bulletin 2008. 63. N 4. P. 284
Two patterns of calculation of radio emission from extensive air showers (EAS), the so-called microapproach and macroapproach, are considered. The predictions of these approaches are compared by calculating the spatial distribution of the 40 MHz radio emission at the EAS energy $10^{16}$ eV.
Show Abstract
Optics and spectroscopy. Laser physics
## Local transverse diffusion of light due to laguerre laser beam diffraction
### Yu.V. Vasil’ev, A.V. Kozar’, D.A. Kuvshinov, A.E. Luk’yanov, A.V. Seliverstov
Moscow University Physics Bulletin 2008. 63. N 4. P. 287
When shading half of a Laguerre laser light beam with a semi-infinite metal screen with an imperfect edge the effect of fine structure formation in a transverse diffusion of “half-beam” light and its further evolution in the process of half-beam space propagation were experimentally revealed.
Show Abstract
Condensed matter physics
## Redistribution of chromium atoms between the components of an intermetallic/oxide nanocomposite in the course of its production
### T.Yu. Kiseleva, A.A. Novakova, A.N. Falkova, T.L. Talako, T.F. Grigor’eva
Moscow University Physics Bulletin 2008. 63. N 4. P. 290
Structural studies of nanocomposites produced by the method combining mechanical preactivation of the mixture comprising 8.1% Сr$_2$O$_3, 65.9% Fe, and 25% Al by mass and self-propagating high-temperature synthesis (SPHTS) have been carried out by Mössbauer spectroscopy methods. It was found that a FeAlCr$_2$O$_3$composite with a small Fe$_2$Al$_5$intermetallic impurity is produced at the mechanical activation stage. At the SPHTS stage, interaction between the activated components of the mixture results in formation of the Fe$_{0.70-x}$Сr$_{x}$Al$_{0.3}$($x=$0-0.2)/Al$_2$O$_3$composite. Show Abstract Chemical physics, physical kinetics, and plasma physics ## Combustion of a supersonic flowing propane-air mixture within a DC longitudinal-transverse discharge ## Combustion of a supersonic flowing propane-air mixture within a DC longitudinal-transverse discharge ### A.F. Aleksandrov, A.Yu. Baurov, A.P. Ershov, A.A. Logunov, V.A. Chernikov Moscow University Physics Bulletin 2008. 63. N 4. P. 293 The findings of experimental investigations on the combustion of a supersonic flowing propane-air mixture with initiation by a dc discharge are given. It was found that the pattern of combustion depended on a number of initial conditions associated with the generation of plasma and with the external parameters of the supersonic flow. Show Abstract Biophysics and medical physics ## Dynamics of collagen molecules in aqueous solutions containing metal ions with different ionic radii ## Dynamics of collagen molecules in aqueous solutions containing metal ions with different ionic radii ### G.P. Petrova$^1$, Y.M. Petrusevich$^2$, I.A. Perfil’eva$^1$, M.S. Ivanova$^1$, Chzhan Syaolei$^2\$
Moscow University Physics Bulletin 2008. 63. N 4. P. 296
The dynamic parameters of collagen molecules in aqueous solutions were assessed by the photon-correlation spectroscopy method. The dependences of translational diffusion coefficients as a function of pH in pure aqueous collagen solutions and in solutions containing salts of different metals such as Na, Ca, K, Pb were established. It has been reported that the size of a metal’s ionic radius affects intermolecular interactions and the mobility of collagen molecules in solution.
Show Abstract | 2021-11-27 12:23:40 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.4964406192302704, "perplexity": 4107.854872758238}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964358180.42/warc/CC-MAIN-20211127103444-20211127133444-00486.warc.gz"} |
https://www.physicsforums.com/threads/intro-to-the-ionization-energy-of-atomic-hydrogen.980823/ | Homework Helper
Introduction
In previous articles relating to various transition energies in Hydrogen, Helium and Deuterium we have employed the following formula for electron energy given a particular primary quantum number n:
$$E_{n}=\mu c^2\sqrt{1-\frac{Z^2\alpha^2}{n^2}}$$
where ## \alpha ## is the fine structure constant and ## \mu ## the reduced electron mass for a single electron bound to whichever nucleus. Z=1 for Hydrogen and Deuterium, Z=2 for Helium. Reduced mass is calculated from electron mass and nuclear mass as follows: $$\mu = \frac{m_e\times m_{n}}{m_e+m_{n}}$$.
Calculation of Ionization Energy: Atomic Hydrogen
Perhaps one of the simplest applications of this formula is the determination of the ionization energy of...
Last edited:
mfb
Mentor
as many significant figures as Wolfram Alpha is prepared to calculate!
Unless you click on "more digits".
The WolframAlpha image has a low resolution in the article and you could have added a link to WA with the calculation filled in.
Homework Helper
Many thanks - the article wasn't actually finished and I accidentally pressed submit. Anyway I'll chat to Greg and either temporarily 'withdraw' it or just do some online editing. It's very 'junky' as it stands - not even any references.
Re "more digits": I'm not sure if this option is available when you are using scientific constants as in this calculation. WA seems to put some kind of limit on the number of significant figures it will display.
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Dragrath | 2021-08-04 05:36:37 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6579647064208984, "perplexity": 1545.2088777606182}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154796.71/warc/CC-MAIN-20210804045226-20210804075226-00278.warc.gz"} |
https://math.stackexchange.com/questions/1052905/ab-square-leq-1-implies-ba-triangle-leq-1 | # $\| AB\|_\square \leq 1$ implies $\| BA\|_\triangle \leq 1$
In this post norm denotes a matrix norm, i.e. it is sub-multiplicative. All matrices are real. $A$ is of size $n \times k$ with independent columns ($k \leq n$). $B$ is of size $k \times n$.
Let $\| \cdot \|_\square$ be some arbitrary matrix norm. Assume $\| AB\|_\square \leq 1$. Then we know $\rho(AB) \leq 1$. Then we know from this post that $\rho(BA) \leq 1$.
Edit: thanks to @loupblanc I realized this was incorrect. From this we know that there must be a matrix norm $\| \cdot \|_\triangle$ such that $\| BA\|_\triangle \leq 1$. . In fact we only know that there is a norm which is arbitrarily close to the spectral radius.
I am interested in whether there is an easy way of defining $\| \cdot \|_\triangle$ from $\| \cdot \|_\square$. For example, is it true that $\| \cdot \|_\triangle = \|F (\cdot) Q\|_\square$ for some choice of $F$ and $Q$?
Edit: when I say the norm $\| \cdot \|_\square$ is sub-multiplicative I mean it is sub-multiplicative in the space of $n \times n$ matrices. I do not mean $\| A B \|_\square \leq \|A\|_\square \| B \|_\square$, which indeed doesn't make sense. What I mean that if, say, one decomposed $AB$ into a product of square matrices, i.e. $AB = C_1C_2$, where $C_1,C_2$ are square, then we would have $\| A B\|_\square = \| C_1 C_2 \|_\square \leq \| C_1 \|_\square \| C_2 \|_\square$.
Submultiplicative norms for non-square matrices obviously do not make sense, but with square matrices, I don't think what you originally wrote was wrong. More specifically, we have the following
Proposition. Suppose $A,B\in M_n(\mathbb C)$. If $\rho(AB)=\|AB\|_\square\le1$ for some submultiplicative norm $\|\cdot\|_\square$, then there exists some submultiplicative matrix norm $\|\cdot\|_\triangle$ such that $\|BA\|_\triangle\le1$.
If $\rho(AB)<1$, the proposition is obvious because $\rho(AB)=\rho(BA)$ and the spectral radius is the infinum of all submultiplicative matrix norms. So, it suffices to consider only the case $\rho(AB)=\rho(BA)=\|AB\|_\square=1$, and we need to find some norm such that $\|BA\|_\triangle=1$.
As $\rho(AB)=1$, we have $1 = \left(\rho(AB)\right)^k = \rho((AB)^k) \le \|(AB)^k\|_\square \le \|AB\|_\square^k = 1$ and in turn $\|(AB)^k\|_\square=1$ for all natural number $k$. Hence $AB$ is power bounded. As $(BA)^{k+1}=B(AB)^kA$, the product $BA$ and in turn its Jordan form are power bounded too. Since all matrix norms (submultiplicative or not) on $M_n(\mathbb C)$ are equivalent, the Frobenius norms of the powers of the Jordan form of $BA$ are uniformly bounded. It follows that for each unit eigenvalue of $BA$, its algebraic and geometric multiplicities coincide. That is, in the Jordan decomposition $BA=PJP^{-1}$, we may assume that $$J=\pmatrix{T&0\\ 0&D},$$ where $T$ is a direct sum of (upper triangular) Jordan blocks for eigenvalues whose magnitudes are strictly smaller than $1$, and $D$ is a diagonal matrix whose diagonal entries have unit moduli.
Now suppose the size of $T$ is $m\times m$. Let $\epsilon>0$ and $\Lambda=\operatorname{diag}(1,\epsilon,\epsilon^2,\ldots,\epsilon^{m-1})\oplus I_{n-m}$. When $\epsilon$ is sufficiently small, the superdiagonal of $\Lambda^{-1}J\Lambda$ can be made arbitrarily small and the diagonal entries remain invariant. Therefore the spectral norm $\|\Lambda^{-1}J\Lambda\|_2=\sigma(\Lambda^{-1}J\Lambda)$ is equal to $1$. Consequently, if we define $$\|X\|_\triangle = \|\Lambda^{-1}P^{-1}XP\Lambda\|_2,$$ we get $\|BA\|_\triangle=1$.
• @ user1551 , pretty proof. In fact you show that: there is a sub-mult. norm $||.||$ s.t. $\rho(U)=||U||$ iff the eigenvalues of $U$ of maximal modulus are semi-simple. – user91684 Dec 7 '14 at 23:26
• @loupblanc Thanks. Initially I thought this is a known result, but I couldn't find it anywhere in the books offhand. Google showed nothing either. The closest thing I can find is the characterisation of radial matrices. Do you know if this is a known result? – user1551 Dec 7 '14 at 23:46
• @ user1551 , I did not know this equivalence. Householder speaks about this result in one of his paper; you can read for free the first 2 pages as follows: with Google, do "S. Householder, Minimal matrix norms" and click on the Springer link. Note that it is a reference cited by Goldberg-Zwaz. – user91684 Dec 8 '14 at 11:04
• @loupblanc Thanks a lot. Minimal norms (for fixed matrices!) ... of course ... that makes much sense. I have looked for useful results about minimal norms before answering the question, but quickly abandoned the search because such norms (the usual ones, not Householder's) are minimal for all matrices. – user1551 Dec 8 '14 at 11:37
• @user1551: The book Numerical Analysis by Ortega has a proof of the same. In it such matrices are said to be " of class M". – me10240 Mar 3 '16 at 4:39
@ ziutek , a matrix norm $||.||$ is defined on a space $M_{n,m}$. If $||.||$ is sub-multiplicative, then necessarily $m=n$ and $A$ cannot be rectangular. Moreover there is a big mistake in your line 5. Indeed if $\rho(BA)\leq 1$, then, we have only the following: for every $\epsilon>0$, there is an induced norm $N()$ s.t. $N(BA)\leq \rho(BA)+\epsilon$.
Assume for example that $B$ is invertible and let $N(U)=||B^{-1}UB||$. Then $N()$ is sub-multiplicative norm (easy). Moreover $N(BA)=||AB||=1$. It remains the case when $A,B$ are non-invertible.
• Thanks for the useful answer. I clarified the question a little bit and stated explicitly what I mean by sub-multiplicative. – ziutek Dec 7 '14 at 10:44 | 2021-03-05 01:50:08 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9750245809555054, "perplexity": 146.92784597768693}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178369553.75/warc/CC-MAIN-20210304235759-20210305025759-00292.warc.gz"} |
http://www.tutorcircle.com/what-are-rational-numbers-sgUi.html | # What are Rational numbers?
Math is all about Numbers, and whenever we deal with mathematics we have to deal with different Types of Numbers. In the different categories of numbers, Rational numbers play an important and vital role. Now the query is What is a Rational Number?
The word 'Rational' is derived from the most popular Math term Ratio. Rational number can be defined as the number that is formed by dividing one Integer by the other. When a number is written as a simple fraction like $\frac{p}{q}$ where p and q are integers, then it is a Rational number. It is a Set of positive Fractions, negative fractions and zero. It is dented by I. where I is:
I = { 0, $\pm$ 1, $\pm$ 2, $\pm$ 3..................}.
Note:
Rational numbers can be both positive and negative. As we all know, there are two type real numbers exist in mathematics – rational number and irrational number. So, except irrational number all numbers like natural numbers(1,2,3…..), whole numbers(0,1,2,3..) and integers(….-2,-1,0,1,2….) are define in rational numbers and we all know that integers include all kind of numbers whether number is positive or negative. Now we discuss when rational numbers are positive and when rational numbers are negative: If a and b both are positive, then this kind of rational numbers (a/b) are called as a positive rational number. Like we have an integer ‘a’, which is equal to +5 and integer ‘b’, which is equal to +7 in rational number form (a/b), then this produces (5/7) which is a positive rational number. If one of integer between a and b are negative, then this kind of rational numbers (a/b) are called as a negative rational number. Like we have an integer a is equal to -7 and integer b is equal to +9 in rational number form (a/b), then this produces (-7/9), which is a negative rational number. If both integer a and b are negative, then this kind of rational numbers (a/b) are called as a positive rational number. For instance, we have an integer a, which is equal to -3 and integer b is equal to -4 in rational number form (a/b), then this produces (3/4) which is a positive rational number. Now, we can say that rational number can be positive and negative and it can be represented on number line. | 2013-05-25 18:20:58 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.6380268931388855, "perplexity": 286.4816671451374}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-20/segments/1368706082529/warc/CC-MAIN-20130516120802-00071-ip-10-60-113-184.ec2.internal.warc.gz"} |
http://abe-taha.blogspot.com/2014/12/ | Saturday, December 27, 2014
Randomized algorithms: Verifying matrix multiplication
Another great example of randomized algorithms in the introductory chapters of the book "Probability and Computing" is verifying matrix multiplication. Suppose we have 3 matrices of compatible dimensions $A, B$, and $C$, and we want to verify that
$A\cdot B = C$
For simplicity, let's assume that all the matrices are square, and of dimension $n \times n$. The straightforward way to do the verification is to explicitly multiple the matrices $A$ and $B$ together, an operation that is of the order of magnitude of $O(n^3)$.
As an aside, we can do better than $O(n^3)$ for matrix multiplication for larger matrices. Wikipedia has an excellent writeup on Strassen's algorithm which accomplishes matrix multiplication in $O(n^{2.8074})$ through divide and conquer. First the matrices $A$, $B$, and $C$ are padded with zero columns and rows so that their dimensions are of the form $2^m \times 2^m$. The matrices are then divided into equally sized block matrices of the form:
$A = \left[ \begin{array}{cc} A_{1,1} & A_{1,2} \\ A_{2,1} & A_{2,2} \end{array} \right]$
$B = \left[ \begin{array}{cc} B_{1,1} & B_{1,2} \\ B_{2,1} & B_{2,2} \end{array} \right]$
$C = \left[ \begin{array}{cc} C_{1,1} & C_{1,2} \\ C_{2,1} & C_{2,2} \end{array} \right]$
Then
$C_{i,j} = \Sigma_{k=1}^2 A_{i,k} B_{k,j}$
The trick of the Strassen's algorithm is decreasing the number of required multiplications (8) by defining the auxiliary matrices $M_i$ such that:
$\begin{eqnarray} M_1 & = & (A_{1,1}+A_{2,2})(B_{1,1}+B_{2,2}) \\ M_2 & = & (A_{2,1}+A_{2,2}) B_{1,1} \\ M_3 & = & A_{1,1}(B_{1,2}-B_{2,2}) \\ M_4 & = & A_{2,2}(B_{2,1}-B_{1,1}) \\ M_5 & = & (A_{1,1}+A_{1,2})B_{2,2}\\ M_6 & = & (A_{2,1} - A_{1,1})(B_{1,1}+B_{1,2}) \\ M_7 & = & (A_{1,2} - A_{2,2})(B_{2,1}+B_{2,2}) \end{eqnarray}$
which have $7$ multiplications instead, and expressing the result in terms of the $M$ matrices as:
$\begin{eqnarray} C_{1,1} & = & M_1 + M_4 - M_5 + M_7\\ C_{1,2} & = & M_3 + M_5 \\ C_{2,1} & = & M_2 + M_4 \\ C_{2,2} & = & M_1 - M_2 + M_3 + M_6 \end{eqnarray}$
The algorithm continues to divide matrices $A$, $B$, and $C$ into smaller blocks in a similar manner until there is only one element, and the multiplication becomes straightforward.
There are faster algorithms for matrix multiplication, such as the Coppersmith–Winograd algorithm with complexity of $O(n^{2.375477})$ and yet faster ones with $O(n^{2.3728639})$, but what if these are not fast enough?
Randomized algorithms come to the rescue. The idea is to choose a vector $r$ of dimension $n$, whose elements $r_i \in \{0,1\}$ and compute the quantities $ABr$ and $Cr$. If $ABr \ne Cr$ then $AB \ne C$ and the verification of matrix multiplication is done.
On the other hand if $ABr = Cr$ then either $AB=C$ or we might have a false positive. The false positive occurs when $(AB-C)r=0$, with the first term in the equation $AB-C$ not zero. The book computes the probability of this happening as less than $\frac{1}{2}$. The argument goes as follows.
Define the first term as a matrix $D=AB-C$. Since $D \ne 0$ in the false positive case, it must contain a non-zero element somewhere in the matrix. Assume this element is $d_{11}$. We can then write $Dr=0$ as the set of $n$ simultaneous equations:
$\Sigma_{j=1}^{n} d_{ij}r_j=0$
Or for $r_1$ in the false positive case
$r_1 = -\frac{\Sigma_{j=2}^{n} d_{1j}r_j}{d_{11}}$
Using the principle of deferred decisions we can figure out the probability of the false positive. We start by selecting $r_i$ for $i \in \{2, n\}$ randomly and independently from the set $\{0,1\}$, and we pause before selecting $r_1$ from the same set. Since $r_1$ can have one of two values $\{0,1\}$, the probability of selecting $r_i$ that satisfies the prior equation to give us the false positive case is $\frac{1}{2}$
Since a false positive probability of $\frac{1}{2}$ is not good enough, we can repeat the algorithm with a new random vector $r$. After $k$ repetitions of the algorithm the false positive probability drops to $\frac{1}{2^k}$, which we can make small to our liking with many repetitions. The cost of the verification algorithm is on the order of $O(kn^2)$ multiplications, which is better than the fastest direct algorithm for explicit matrix multiplication.
Saturday, December 20, 2014
Classic Algorithms: Page Rank and the importance of web pages
Think of a search query, and go to your favorite search engine and conduct the search. Of the millions of matching documents returned, it is most likely that the first page or two contain what you are looking for. Have you ever wondered how the search engines know how to present the most relevant results for us in the first couple of pages, when all the results returned match the query we were looking for?
There are a lot of signals search engines use to determine the importance of the matched results: some are intrinsic to the page such as where in the page the match occurs, whether it is highlighted or not, and the importance of the web page; and some are extrinsic such as how many users clicked on the result for a similar query. The signals, their weights, and the formulas to rank the results are usually guarded secrets by search engines as they are the secret sauce for giving users the best results to their queries.
One such signal is the importance of a web page. How would you compute that? In 1996, Larry Page and Sergey Brin came up with a way called PageRank. The idea they proposed is that in a connected graph structure such as the world wide web, the importance of a web page is related to the importance of other web pages that point to it. If we denote the importance of a web page, or its PageRank by $r_i$, Page and Brin proposed that $r_i = \Sigma_{j \in N} \frac{r_j}{d_j}$ where $N$ is the set of pages that link to page $i$, $r_j$ is the PageRank for each of the pages that point to $r_i$, and $d_j$ is the number of outgoing links for each $r_j$ page.
If we write these equations for every node on the web, we come up with a system of simultaneous linear equations that we can solve, and get the PageRank for each web page. We can also write the simultaneous equations in vector and matrix notations as:
$\mathbf{r} = \mathbf{M r}$
where $\mathbf{r}$ is the vector of PageRanks for all the web pages, and $\mathbf{M}$ is the matrix whose column entries are either $0$ or $\frac{1}{d_j}$ for pages that link to page $i$.
It turns out that by rewriting the PageRank problem in this form, we discover that it resembles the eigenvalue/eigenvector problem in linear algebra, with the PageRank $\mathbf{r}$ as the eigenvector corresponding to an eigenvalue of $1$. There are many ways to solve the eigenvalue problem, and the power method is one of them: where we start with an initial value of the PageRank $\mathbf{r}^{(0)}$, and compute subsequent values through:
$\mathbf{r}^{(t+1)} = \mathbf{M} \mathbf{r}^{(t)}$
until convergence occurs, i.e. the difference between the PageRanks in two subsequent iterations becomes very small:
$\| \mathbf{r}^{(t+1)} - \mathbf{r}^{(t)} \|_n < \epsilon$
where $\|\cdots\|_n$ is the $L_n$ norm, and $\epsilon > 0$. Typically the Euclidean or $L_2$ norm is used.
There is a wrinkle in the simultaneous equations or eigenvalue/eigenvector formulations of the PageRank though: not all simultaneous equations or eigenvalue/eigenvector problems are well posed to have a solution. This happens for example when you have linear dependence between the equations or rows/columns of the matrix. An illustrative example would be a sink page, which does not have any outbound links to any other pages. Its effect on the PageRanks of other pages would be zero, and all the respective columns in the matrix $\mathbf{M}$ associated with the page would be $0$. Another example would be a page that links to itself, where during the power iteration, all the other page PageRanks are leaked to $0$. How would you tackle such problems? Through a bit of creativity.
It turns out that if we model browsing the Internet as a random walk over its connected graph structure, where a random surfer moves from page $i$ to another page, by following one of the outbound links from page $i$, we arrive at a similar formulation
$\mathbf{r}=\mathbf{Mr}$
as before.
Of course the random surfer would not stop when there are no outbound links in the page they are at, and would whimsically go to any other page on the Internet. The model is then augmented to accommodate such behavior:
$\mathbf{r} = \beta \mathbf{Mr} + \frac{(1-\beta)}{N}\mathbf{r}$
where $0<\beta<1$, and $N$ is the total number of pages in the web graph. The first term in the equation models the random surfer picking one of the outbound links at random to visit a subsequent page, and the second term models them jumping to any other page on the Internet at random.
We can rewrite the prior formulation as:
$\mathbf{r} = \mathbf{M'r}$
which is well formed, and has a solution since $\mathbf{M'}$ is a stochastic matrix. Using the power method on the last equation will yield the PageRanks $\mathbf{r}$ of all the web pages in the Internet.
Of course that computation requires a lot of engineering magic, since there are billions of web pages on the Internet, and each have many outbound connections.
Thursday, December 11, 2014
Randomized Algorithms: Polynomial equivalence
We are all accustomed to deterministic algorithms; we work with them every day, and feel comfortable in knowing that the results of running them are predictable, barring a coding error of course. The idea of randomized algorithms feels remote and uncomfortable, despite their usefulness and elegance. There are a couple of great examples in the introductory chapters of the book "Probability and Computing" that are an eye opener.
One is verifying polynomial identities: how can you tell that two different representations of polynomials are the same? For example, if we have two polynomials $P(x)$ and $Q(x)$, both of degree $d$ described by the following formulas:
$P(x) = \Sigma_{i=0}^{i=d} a_i x^i \\ Q(x) = \Pi_{i=1}^{i=d} (x-b_i)$
how can we determine that they are the same polynomial?
Intuitively we first check that the degrees are the same, then we could try to transform one form into the other, either by multiplying out the terms for $Q(x)$, collecting like terms and reducing it to $P(x)$, or finding the $d$ roots $r$ of $P(x)$, and expressing it as a product of $d$ terms of the form $(x-r_i)$ and comparing them to $Q(x)$. The first approach is easier, but can we do better?
The book presents a randomized algorithm to do the same. What if we pick a random number $r$ from the range $[0, Nd]$, where $N$ is a natural number greater than zero. For example if $N$ is $100$, we pick a random number $r$ from the range $[0,100d]$, and evaluate $P(r)$ and $Q(r)$, which could be done in $O(d)$ time. What would the result tell us?
If $P(r) \ne Q(r)$ then the polynomials are not the same. If $P(r) = Q(r)$, then there is a chance that the polynomials are equivalent, but there is also a chance that they are not, and that $r$ in this case is a root of the equation $P(x)-Q(x)=0$. The chance that we picked an $r$ that satisfies the last equation is no more than $1/N$---($1/100$ in the concrete example).
How do we minimize that chance? By repeating the evaluation by drawing another random value $r$ from the interval $[0,Nd]$. The book describes the probability of producing a false result as the evaluations are repeated with and without replacement, and they are less than $(1/N)^k$, where $k$ is the number of evaluations.
Isn't it quite elegant to find out that two polynomials are the same or not simply through repeated evaluations, and not through algebraic manipulations to transform either or both to a canonical form?
Monday, December 8, 2014
Down memory lane: Approximate counting
At work we do a lot of counting. For some counts we need an accurate result, while for others we can get by with an approximate count. For these approximate counts, as long as the result is within the same order of magnitude of the true count, we're ok.
There is a lot of literature on approximate counting techniques, and the blog post "Probabilistic Data Structures for Web Analytics and Data Mining" does a great job at explaining some of them in detail, with references to the original papers.
One of the original approximate counting papers was the 1978 paper: Counting large numbers of events in small registers by Robert Morris. In the paper Morris explains the context for the problem, which might seem foreign today. Back in 1978 he was faced with the problem of counting a large number of different events, and because of memory restrictions he could only use 8-bit registers for each counter. Since he was not interested in exact counts, but with accurate enough counts, he came with the technique of probabilistic counting.
The first solution he explored in the paper was to count every other event based on a flip of a coin. If the probability of the coin flip is $p$, for a true value $n$, the expected value of the counter is $n/2$, with a standard deviation of $\sqrt{n/4}$. The error for such a solution is high for small values of $n$.
Morris then presents his second solution, by introducing a counter that stores the logarithm of the number of events that have occurred $v$, and not the raw count of the events. He increments the counter value $v$ probabilistically based on a flip of a coin. The stored value in the counter is $\log{1+n}$ and the probability of increasing the counter is $1/\Delta = \exp{(v+1)}-\exp{(v)}$.
The paper is an enjoyable short read, and it is amazing that the techniques introduced in the 70's are still applicable to technology problems today. | 2017-12-13 07:04:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8059048056602478, "perplexity": 222.76317982585243}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948522205.7/warc/CC-MAIN-20171213065419-20171213085419-00042.warc.gz"} |
https://www.gamedev.net/forums/topic/614299-mouse-to-ray-sphere-collision-detection-help/ | # mouse to ray - sphere collision detection help
This topic is 2568 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.
## Recommended Posts
Hi all
I have been trying to get a working algorithm that detects intersection between a ray (representing the bullets from a gun) and a sphere around the enemy.... I tried a few found on the net but none seems to work properly, maybe I am doing something wrong...
This is the one I am currently using:
//// Ray-sphere intersection. // p=(ray origin position - sphere position), // d=ray direction, // r=sphere radius, // Output: // i1=first intersection distance, // i2=second intersection distance // i1<=i2 // i1>=0 // returns true if intersection found,false otherwise.// bool Player::RaySphereIntersect(const Vector3 &p, const Vector3 &d, double r, double &i1, double &i2){ double det,b; b = -Vector3::dot(p,d); det = b*b - Vector3::dot(p,p) + r*r; if (det<0){ return false; } det= sqrt(det); i1= b - det; i2= b + det; // intersecting with ray? if(i2<0) return false; if(i1<0) i1=0; return true; }
Where I use the position of the enemy as sphere position, roughly the position of the player's gun as ray origin and the projected mouse coordinates for ray direction... This is the OpenGL code I am using to project the mouse coords to the the far plane:
Vector3 projectedMouse(float mx, float my){ GLdouble model_view[16]; GLint viewport[4]; GLdouble projection[16]; GLfloat winX, winY, winZ; GLdouble dx, dy, dz, bx, by, bz; glGetDoublev(GL_MODELVIEW_MATRIX, model_view); glGetDoublev(GL_PROJECTION_MATRIX, projection); glGetIntegerv(GL_VIEWPORT, viewport); winX = (float)mx; winY = (float)viewport[3] - (float)my; glReadPixels ((int)mx, (int)winY, 1, 1, GL_DEPTH_COMPONENT, GL_FLOAT, &winZ); gluUnProject(winX, winY, 1, model_view, projection, viewport, &dx, &dy, &dz); projectedAim = Vector3(dx, dy, dz); return projectedAim; }
Which seems right cos I am drawing a GL Line with it and it looks fine... So maybe it's the intersection code, but nothing seems to work.... I tried this other one that should return the intersection point distance, but for any given enemy position, it still gives me very random results:
double intersectRaySphere(Vector3 rO, Vector3 rV, Vector3 sO, double sR) Vector3 Q = sO-rO; double c = Q.magnitude(); double v = Vector3::dot(Q,rV); double d = sR*sR - (c*c - v*v); // If there was no intersection, return -1 if (d < 0.0) return (-1.0f); // Return the distance to the [first] intersecting point return (v - sqrt(d));
they have both been slightly modified to match the Math function in the library that I am using.... can anyone spot something wrong with them, or suggest another one? this is driving me crazy....
Thank you!
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It looks like the algorithms you have assume that the line direction is a normal vector. Are you normalizing the vector before you pass it in?
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from what I gather:
c is magnitude of ray origin to sphere center.
v is c converted along vector rv since rv is a unit vector the dot product wth lengthed vector Q would return length along rv direction
There is a line segment perpendicular to rV passing through sphere center at a length v along rV direction. The length of this segment is (c*c) - (v*v).
the ray intersecting with the sphere will have the first (closest) intersection point at distance sR (radius of shphere) from sphere center sO.
ths is also alon vector rV. so now we have two points along rV. we know the length to one is sRr and the other forms a segment perpendicular to rV. The collision point and v distance point form a right triangle wth sR as the length of the hypotenuse.
so the distance ALONG rV to the collision point is v- sqrt( (sr*sr) - ((c*c)-(v*v)) ). they represented d as the big portion in the brackets.
This function appears to work fine.
Are you applying your final distance along rV to indicate collision point? if rV is not normalized you may end up with strange results.
your collision point should be rO + (rV*finaldist).
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It looks like the algorithms you have assume that the line direction is a normal vector. Are you normalizing the vector before you pass it in?
Thank you for your response! You know I suspected that, but I was kind of confused by it.... I assumed that the ray direction vector would simply be the ray from the gun to the projected mouse coordinates on the far plane, and have therefore used the difference of the two for the vector that I pass...
So what would I normalize then? simply the unprojected mouse coordinates? Or the difference between ray origin and unprojected cursor?
Thank you very much
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This function appears to work fine.
Are you applying your final distance along rV to indicate collision point? if rV is not normalized you may end up with strange results.
your collision point should be rO + (rV*finaldist).
Thanks a lot for your explanation! Well, still no luck tho. So here is what I am passing into the function:
double d = intersectRaySphere(entity.getPosition(), shootingRay, Vector3(0, 100, -300), 30);
where the first paramenter is the position of the character, the second is the unprojected mouse coordinates on the far plane according to the function above, the third is the position of the stationary enemy and the fourth is a radius... I usually get similar d values in the order of 0.1 to 0.3 when I hover the mouse over the center of the object when positioned in different spots...still it's not really precise enough for shooting....
I tried normalizing the unprojected mouse coords but I then get it never returns any collision.... The Normalization code i have is:
inline void Vector3::normalize() { float invMag = 1.0f / magnitude(); x *= invMag, y *= invMag, z *= invMag; }
any ideas?
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[color="#000088"]double d [color="#666600"]= intersectRaySphere[color="#666600"](entity[color="#666600"].getPosition[color="#666600"](), shootingRay[color="#666600"], [color="#660066"]Vector3[color="#666600"]([color="#006666"]0[color="#666600"], [color="#006666"]100[color="#666600"], [color="#666600"]-[color="#006666"]300[color="#666600"]), [color="#006666"]30[color="#666600"]);you have the unprojected coordinates on the far plane and the source of the gun (character). what you are passing in is not the proper ray.
to get the ray do
farCoords = projectedMouse[color="#666600"]( mx[color="#666600"], [color="#000088"]my[color="#666600"]);
then shootingRay = farCoords - entity.getPosition(); now i know this isn't the gun muzzle position but you can easily sub that in for the getposition(),, im just using stuff from your code that I can name.
then shootingRay.normalize();
then do
[color="#000088"]double d [color="#666600"]= intersectRaySphere[color="#666600"](entity[color="#666600"].getPosition[color="#666600"](), shootingRay[color="#666600"], [color="#660066"]Vector3[color="#666600"]([color="#006666"]0[color="#666600"], [color="#006666"]100[color="#666600"], [color="#666600"]-[color="#006666"]300[color="#666600"]), [color="#006666"]30[color="#666600"]);
this will fire a ray from your entity at a sphere of radius 30 around the enemy character's coordinates of (0,100,-300)
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[color="#000088"]double d [color="#666600"]= intersectRaySphere[color="#666600"](entity[color="#666600"].getPosition[color="#666600"](), shootingRay[color="#666600"], [color="#660066"]Vector3[color="#666600"]([color="#006666"]0[color="#666600"], [color="#006666"]100[color="#666600"], [color="#666600"]-[color="#006666"]300[color="#666600"]), [color="#006666"]30[color="#666600"]);you have the unprojected coordinates on the far plane and the source of the gun (character). what you are passing in is not the proper ray.
to get the ray do
farCoords = projectedMouse[color="#666600"]( mx[color="#666600"], [color="#000088"]my[color="#666600"]);
then shootingRay = farCoords - entity.getPosition(); now i know this isn't the gun muzzle position but you can easily sub that in for the getposition(),, im just using stuff from your code that I can name.
then shootingRay.normalize();
then do
[color="#000088"]double d [color="#666600"]= intersectRaySphere[color="#666600"](entity[color="#666600"].getPosition[color="#666600"](), shootingRay[color="#666600"], [color="#660066"]Vector3[color="#666600"]([color="#006666"]0[color="#666600"], [color="#006666"]100[color="#666600"], [color="#666600"]-[color="#006666"]300[color="#666600"]), [color="#006666"]30[color="#666600"]);
this will fire a ray from your entity at a sphere of radius 30 around the enemy character's coordinates of (0,100,-300)
That worked. It's the normalization bit that I was missing....gosh that took a whole day of my life trying to figure this out... you're a star ;)
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× | 2018-11-14 14:08:24 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.3360469937324524, "perplexity": 5378.5891864628165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039742020.26/warc/CC-MAIN-20181114125234-20181114151234-00221.warc.gz"} |
https://x-engineer.org/undergraduate-engineering/mathematics/arithmetics/binary-to-decimal-conversion/ | # Binary to Decimal Conversion
In this tutorial we are going to learn about converting a binary number into a decimal number with:
• Method 1: Multiply bits with powers of two
• Method 2: Using Doubling
• Scilab programming: using for loop
• Scilab programming: using the build-in function bin2dec
• C programming: using for loop
Before going through this article, it is recommended to have a basic understanding about:
Numbers Representation Systems – Decimal, Binary, Octal and Hexadecimal
A binary number is a series of ones (1) and zeros (1). The ones (1) and zeros (0) are called bits. Let’s take as example the binary number 111001. The extreme right bit is bit number 0, the extreme left bit is bit number 5.
#### Method 1: Multiply bits with powers of two
Before converting to decimal let’s write down the powers of two. We will use only 8 bits for this example:
$2^7$ $2^6$ $2^5$ $2^4$ $2^3$ $2^2$ $2^1$ $2^0$ 128 64 32 16 8 4 2 1
Under each power of two result we’ll write the corresponding bit value:
128 64 32 16 8 4 2 1 0 0 1 1 1 0 0 1
Now we’ll multiply each bit value with the corresponding power of two and add the products together:
$0 \cdot 128 + 0 \cdot 64 + 1 \cdot 32 + 1 \cdot 16 + 1 \cdot 8 + 0 \cdot 4 + 0 \cdot 2 + 1 \cdot 1$
The result of the sum is the decimal number:
$32 + 16 + 8 + 1 = 57$
The binary number converted to decimal is:
$111001_{2} = 57_{10}$
#### Method 2: Using Doubling
This method doesn’t use the power of two. For this reason it should be simpler to convert lager binary numbers into decimal.
As an example we’ll use the same binary number as in first method: 111001
This method uses a concept named previous total. For the first step the previous total is 0.
We start by taking the previous total, multiply it by 2 and add the extreme left bit (bit number 5).
$0 \cdot 2 + 1 = 1$
The result of the above operations is the previous total for the next step. In our case the previous total becomes 1.
Next, take the previous total, multiply it by 2 and add the following bit (bit number 4). We get:
$1 \cdot 2 + 1 = 3$
We do the same operations until we run out of bits:
$\begin{equation*} \begin{split} 3 \cdot 2 + 1 = 7\\ 7 \cdot 2 + 0 = 14\\ 14 \cdot 2 + 0 = 28\\ 28 \cdot 2 + 1 = 57 \end{split} \end{equation*}$
After we run out of bits, the latest previous total is our converted decimal number: 57.
As expected, the binary number converted to decimal is:
$111001_{2} = 57_{10}$
#### Scilab: using for loop
Scilab implementation of Method 1: Multiply bits with powers of two
// Binary number to be converted
binNo = '111001';
// Initialization of the decimal number
decNo = 0;
// Loop all bits in the binary number
for i=1:length(binNo)
if part(binNo,i) == '1'
decNo = decNo + 1 * 2^(length(binNo)-i);
else
decNo = decNo + 0 * 2^(length(binNo)-i);
end
end
// Display binary and decimal numbers
mprintf("Binary number %s \nDecimal number: %d", binNo, decNo);
Scilab implementation of Method 2: Using Doubling
// Binary number to be converted
binNo = '111001';
// Initialization of the decimal number
prevTot = 0;
// Loop all bits in the binary number
for i=1:length(binNo)
if part(binNo,i) == '1'
prevTot = prevTot * 2 + 1;
else
prevTot = prevTot * 2 + 0;
end
end
// Display binary and decimal numbers
mprintf("Binary number %s \nDecimal number: %d", binNo, prevTot);
#### Scilab programming: using the build-in function bin2dec
In order to verify if the above conversion algorithm are properly designed, we can use the build-in Scilab function bin2dec to convert from binary to decimal numbers:
-->bin2dec('111001')
ans =
57.
-->
#### C programming: using for loops
C implementation of Method 1: Multiply bits with powers of two
#include <stdio.h>
#include <string.h>
#include <math.h>
int main(void)
{
char binNo[] = "111001";
int decNo = 0;
int i;
int stringLen;
stringLen = strlen(binNo); // Length of string
for (i=0; i<stringLen; i++){
if (binNo[i]=='1'){
decNo = decNo + 1 * pow(2,(stringLen-1-i));
}
else{
decNo = decNo + 0 * pow(2,(stringLen-1-i));
}
}
printf("Binary: %s", binNo);
printf("\nDecimal: %d", decNo);
return 0;
}
C implementation of Method 2: Using Doubling
#include <stdio.h>
#include <string.h>
int main(void)
{
char binNo[] = "111001";
int prevTot = 0;
int i;
int stringLen;
stringLen = strlen(binNo); // Length of string
for (i=0; i<stringLen; i++){
if (binNo[i]=='1'){
prevTot = prevTot * 2 + 1;
}
else{
prevTot = prevTot * 2 + 0;
}
}
printf("Binary: %s", binNo);
printf("\nDecimal: %d", prevTot);
return 0;
}
As reference save the image below which contains a summary of both methods for binary to decimal conversion.
Image: Binary to Decimal Conversion Poster
For any questions, observations and queries regarding this article, use the comment form below.
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https://forum.bebac.at/forum_entry.php?id=22610 | ## Changing the regulations: Hope dies last. [NCA / SHAM]
Hi PharmCat,
❝ I don't think that our regulators will be able to understand this, …
Well, the majority of regulators is not stupid.
That’s basic PK and $$\small{CL=\textrm{const}}$$ is a pretty strong assumption, which might be outright false. Then the entire concept is built on sand: Studies are substantially larger than necessary, exposing innocent subjects to nasty drugs.
❝ … or there will be a desire and dare to change the foundations.
Correct. However, the FDA1 gives in its regulatory definitions of BE ‘rate and extent of absorption’ without mentioning any PK metrics, the EMA2 ‘BE with the reference medicinal product […] demonstrated by appropriate bioavailability studies’. Only the WHO3 gives specifically AUC for the extent of absorption. Too lazy to check other jurisdictions.
Somehow it reminds me on the comparison of $$\small{AUC_{t_\textrm{last}}}$$, which is known to be biased if $$\small{t_\textrm{last,T}\neq t_\textrm{last,R}}$$. Nobody gives a shit. Remember the endless story of the inflated Type I Error in SABE? For HVDs (not HVDPs) probably we could counteract the high variability, use ABE with fixed limits, and all is good.
Of course, it would mean changing the guidelines.
Abdallah (FDA/CDER):
It is recommended that area correction be attempted in bioequivalence studies of drugs where high intrasubject variability in clearance is known or suspected. […] The value of this approach in regulatory decision making remains to be determined.
Lucas et al.:
Performance of the AUC·k ratio test […] indicate that the regulators should consider the method for its potential utility in assessing HVDs and lessening unnecessary drug exposure in BE trials.
❝ Also, I think NLME modeling can be applied.
Hhm, can you elaborate?
1. CFR 21–320.23(b)(1). 2021.
2. Directive 2001/83/EC, Article 10(2)(b). 2001.
3. TRS 992, Annex 6. 2017.
P.S.: $$\small{C_\textrm{max}}$$ is a lousy metric for comparing rates of absorption, since it is composite of $$\small{k_\textrm{a}}$$ and $$\small{AUC_{0-\infty}}$$. Lots of publications… What about: $$\small{\frac{F_\textrm{T}}{F_\textrm{R}}\sim \frac{C_\textrm{max,T}\big{/}\left(AUC_{0-\infty,\textrm{T}}\cdot k_\textrm{T} \right)}{C_\textrm{max,R}\big{/}\left(AUC_{0-\infty,\textrm{R}}\cdot k_\textrm{R} \right)}}$$ Deserves a paper.
Dif-tor heh smusma 🖖🏼 Довге життя Україна!
Helmut Schütz
The quality of responses received is directly proportional to the quality of the question asked. 🚮
Science Quotes | 2023-03-28 15:09:59 | {"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 2, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7599042057991028, "perplexity": 4900.6115711571365}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296948867.32/warc/CC-MAIN-20230328135732-20230328165732-00038.warc.gz"} |