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There are simple exact formulas for the surface area of an ellipsoid of revolution, but not for a less symmetrical [scalene] spheroid The 16th Annual Discovery on Target, “The Industry’s Preeminent Event on Novel Drug Targets,” will once again gather over 1,300 drug discovery professionals in The advent of 3D models has presented a new dimension to cancer research Despite this enormous potential, concerns related to their utility within discovery and Method type Advantages Disadvantages; Forced-floating • Relatively simple • Inexpensive • Suitable for high-throughput testing • Spheroids produced are easily Origins and history The word oblate (from the Latin oblatus - someone who has been offered) has had various particular uses at different periods in the history of The prolate spheroid is the shape of the ball in several sports, such as in rugby football Several moons of the Solar System approximate prolate spheroids in shape The 17th Annual World Preclinical Congress focuses on the latest trends and technologies impacting drug discovery and translational research The … A few jewels in elementary Euclidean geometry (1, 2, 3 dimensions or more) Lines, surfaces, polyhedra and topology Cross-disciplinary research at the interface between the physical and life sciences A "squashed" spheroid for which the equatorial radius a is greater than the polar radius c, so a>c (called an oblate ellipsoid by Tietze 1965, p 27) An oblate

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# Capturing seasonality in multiple regression for daily data

I have a daily sales data for a product which is highly seasonal. I want to capture the seasonality in the regression model. I have read that if you have quarterly or monthly data, in that case you can create 3 and 11 dummy variables respectively — but can I deal deal with daily data?

I have three years of daily data. The independent variables are price point, promotional flag (yes/no), and temperature. The dependent variable is sales of that product. I am not looking for a time series model as I am using a multiple regression model.

• How much data have you got? How many years worth? Do you have data on anything like temperature? What is the rest of your model like? What are your DV and IVs? – Peter Flom Jul 22 '14 at 16:07
• In addition to what Peter Flom asked, are you modeling your data as univariate time series or multivariate time series ? If it is multivariate, do you have other variables ? Do those variable exhibit seasonal behavior ? if so adding dummy variable would be unnecessary. Can you provide these additional information? – forecaster Jul 22 '14 at 16:13
• I have edited my question. Could you please provide any the solution. Thanks – Arushi Jul 22 '14 at 16:22
• – Rob Hyndman Jul 23 '14 at 1:38
• I totally agree with @IrishStat, we will not suppose to ignore time series models, a pretty good model out there which capturing multiple seasonality.I would suggest you can check out exponential smoothing state space model which is having a capability of handling multiple seasonality, trend,simultaneously.its exclusively in R.you can make use if forecast()package. – Karthi V Dec 30 '16 at 11:15

@Irishstat covered pretty much what I was about to say, but I would respond with my own personal experience in modeling these data with time series regression and OLS regression.

If it is a daily data then I would do the following:

Create a dummy variable for different seasonality:

• To capture day of the week seasonality, create 6 dummy variables.
• To capture day of the month seasonality, create 30 dummy variables
• To capture month of the year, create 11 dummy variables.

Create dummy variable for trend variables:

• If the time series exhibits linear trend, then add a time trend variable.

• If the time series exhibits nonlinear trend, add a nonlinear time trend variable such as quadratic/cubic/log

• This is a time series data, so care should be taken about lead and lag effects of independent varibales. For instance in your example, you mention price point promotional flag, they might not have immediate effect on your response, i.e., there may be lagging and a decaying/permanent effect. So for instance, if run a promotion today, you might have a increase in sales today but the effect of promotion decays after few days. There is no easy way to model this using multiple regression, you would want to use transfer function modeling which is parsimonoius and can handle any type of lead and lag effects. See this example I posted earlier, where there is an intervention(in your case price point) and there is an abrupt increase, followed by a decaying effect. Having said that if you have a priori knowledge about the lead and lag effect, create additional variables in your case dummy variables before and after price point and (yes/no) promotion change.

• You would also need to add moving Holidays indicator variables, for example as Irishstat pointed out you would want to add Easter/Thanksgiving (in US) which are moving Holidays. Holidays that are fixed dates will be automatically taken care of if you are using dummy coding scheme for capturing seasonality.

• In addition, you would need to identify outliers such as additive/pulse (one time event) or level shift (permanent shift) and add them as regressors. Identifying outliers in multiple regression for time series data is nearly impossible; you would need time series outlier detection methods such as Tsay's procedure or Chen and Liu's procedure which has been incorporated in software such as AUTOBOX, SPSS, SAS or the tsoutlier package in R.

Potential Problems:

Following are the problems you would encounter if you model time series data using OLS multiple regression.

• Errors might be autocorrelated. See this nice website and this website explaining this issue. One way to avoid this is to use Generalized least squares (GLS) or ARIMAX approach vs. OLS multiple regression, where you can correct for auto correlation.
• OLS model will not be parsimonoius. You have $6+30+11= 47$ dummy variables for seasonality.
• By using dummy variables, you are assuming that your seasonality is deterministic i.e. it doesn't change over time. Since you have only 3 years of data I would not worry about it, but still it is worthwhile to plot the series and see if the seasonality doesn't change.

And there are many more disadvantages of using multiple regression. If prediction is more important to you then I would hold out at least 6 months of data and test the predictive ability of your multiple regression. If your main goal is to explain the correlation between independent variables, then I would be cautious using multiple regression, and instead I would use a time series approach such as ARIMAX/GLS.

If you are interested, you could refer to the excellent text by Pankratz, for transfer function and dynamic regression modeling. For general time series forecasting please refer to Makridakis et al. Also, a good reference text would be by Diebold for regression and time series based forecasting.

• Very nice summary BUT I would like to add that you ignored particular week-of-the-month and particular day-of-the-month effects in addition to possible weekend effects all of which I have found to be very important. Furthermore pre-event and post-event effects are not to be ignored. Consider the activity around Easter and around other major holidays/events.There is often an individual response pattern which requires the incorporation of a LEAD specification. You should also note that parameters can and often change over time and one needs to validate the assumption of constancy of parameters. – IrishStat Jul 23 '14 at 11:28
• Thanks @Irishstat. You are right. I forgot bout moving Holidays and their lead and lag effects. – forecaster Jul 23 '14 at 13:36

What you need is a model that will incorporate daily effects, weekly effects, monthly effects,week of the month effects, day-of-the-month effects,lead and lag effects of the holidays, unspecified but empirically identifiable level/step shifts , local time trends, changes in seasonal pulses and pulses while incorporating ARIMA structure and possibly dealing with changes in parameters and error variance over time. This is called a Transfer Function and can be easily restated (BUT NOT PARSIMONIOUSLY) as a Multiple Linear Regression.

In specific a daily indicator would take 6 predictors. In general one has to carefully orchestrate(identify) what kind of predictors are needed. If you have a lot of time on your hands you can experiment with some of the structures I have mentioned. Alternatively you might need some advanced software/consultancy to get you to solve your problem in your lifetime.

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In my last post, I discussed how ggplot2 is not always the answer to the question “How should I plot this” and that base graphics were still very useful.

# Why Do I use ggplot2 then?

The overall question still remains: why (do I) use ggplot2?

## ggplot2 vs lattice

For one, ggplot2 replaced the lattice package for many plot types for me. The lattice package is a great system, but if you are plotting multivariate data, I believe you should choose lattice or ggplot2. I chose ggplot2 for the syntax, added capabilities, and the philosophy behind it. The fact Hadley Wickham is the developer never hurts either.

## Having multiple versions of the same plot, with slight changes

Many times I want to do the same plot over and over, but vary one aspect of it, such as color of the points by a grouping variable, and then switch the color to another grouping variable. Let me give a toy example, where we have an x and a y with two grouping variables: group1 and group2.

library(ggplot2)
set.seed(20141016)
data = data.frame(x = rnorm(1000, mean=6))
data$group1 = rbinom(n = 1000, size =1 , prob =0.5) data$y = data$x * 5 + rnorm(1000) data$group2 = runif(1000) > 0.2

We can construct the ggplot2 object as follows:

g = ggplot(data, aes(x = x, y=y)) + geom_point()

The ggplot command takes the data.frame you want to use and use the aes to specify which aesthetics we want to specify, here we specify the x and y. Some aesthetics are optional depending on the plot, some are not. I think it's safe to say you always need an x. I then “add” (using +) to this object a “layer”: I want a geometric “thing”, and that thing is a set of points, hence I use geom_points. I'm doing a scatterplot.

If you just call the object g, print is called by default, which plots the object and we see our scatterplot.

g

I can color by a grouping variable and we can add that aesthetic:

g + aes(colour = group1)

g + aes(colour = factor(group1))

g + aes(colour = group2)

Note, g is the original plot, and I can add aes to this plot, which is the same as if I did ggplot2(data, aes(...)) in the original call that generated g. NOTE if the aes you are adding was not a column of the data.frame when you created the plot, you will get an error. For example, let's add a new column to the data and then add it to g:

data$newcol = rbinom(n = nrow(data), size=2, prob = 0.5) g + aes(colour=factor(newcol)) Error: object 'newcol' not found  This fails because the way the ggplot2 object was created. If we had added this column to the data, created the plot, then added the newcol as an aes, the command would work fine. g2 = ggplot(data, aes(x = x, y=y)) + geom_point() g2 + aes(colour=factor(newcol))  We see in the first plot with colour = group1, ggplot2 sees a numeric variable group1, so tries a continuous mapping scheme for the color. The default is to do a range of blue colors denoting intensity. If we want to force it to a discrete mapping, we can turn it into a factor colour = factor(group1). We see the colors are very different and are not a continuum of blue, but colors that separate groups better. The third plot illustrates that when ggplot2 takes logical vectors for mappings, it factors them, and maps the group to a discrete color. ### Slight Changes with additions In practice, I do this iterative process many times and the addition of elements to a common template plot is very helpful for speed and reproducing the same plot with minor tweaks. In addition to doing similar plots with slight grouping changes I also add different lines/fits on top of that. In the previous example, we colored by points by different grouping variables. In other examples, I tend to change little things, e.g. a different smoother, a different subset of points, constraining the values to a certain range, etc. I believe in this way, ggplot2 allows us to create plots in a more structured way, without copying and pasting the entire command or creating a user-written wrapper function as you would in base. This should increase reproducibility by decreasing copy-and-paste errors. I tend to have many copy-paste errors, so I want to limit them as much as possible. ## ggplot reduces number of commands I have to do One example plot I make frequently is a scatterplot with a smoother to estimate the shape of bivariate data. In base, I usually have to run at least 3 commands to do this, e.g. loess, plot, and lines. In ggplot2, geom_smooth() takes care of this for you. Moreover, it does the smoothing by each different aesthetics (aka smoothing per group), which is usually what I want do as well (and takes more than 3 lines in base, usually a for loop or apply statement). g2 + geom_smooth()  By default in geom_smooth, it includes the standard error of the estimated relationship, but I usually only look at the estimate for a rough sketch of the relationship. Moreover, if the data are correlated (such as in longitudinal data), the standard errors given by default methods are usually are not accurate anyway. g2 + geom_smooth(se = FALSE)  Therefore, on top of the lack of copying and pasting, you can reduce the number of lines of code. Some say “but I can make a function that does that” – yes you can. Or you can use the commands already there in ggplot2. ### Faceting The other reason I frequently use ggplot2 is for faceting. You can do the same graph, conditioned on levels of a variable, which I frequently used. Many times you want to do a graph, subset by another variable, such as treatment/control, male/female, cancer/control, etc. g + facet_wrap(~ group1)  g + facet_wrap(~ group2)  g + facet_wrap(group2 ~ group1)  g + facet_wrap( ~ group2 + group1)  g + facet_grid(group2 ~ group1)  ## Spaghetti plot with Overall smoother I also frequently have longitudinal data and make spaghetti plot for a per-person trajectory over time. For this example I took code from StackOverflow to create some longitudinal data. library(MASS) library(nlme) ### set number of individuals n <- 200 ### average intercept and slope beta0 <- 1.0 beta1 <- 6.0 ### true autocorrelation ar.val <- .4 ### true error SD, intercept SD, slope SD, and intercept-slope cor sigma <- 1.5 tau0 <- 2.5 tau1 <- 2.0 tau01 <- 0.3 ### maximum number of possible observations m <- 10 ### simulate number of observations for each individual p <- round(runif(n,4,m)) ### simulate observation moments (assume everybody has 1st obs) obs <- unlist(sapply(p, function(x) c(1, sort(sample(2:m, x-1, replace=FALSE))))) ### set up data frame dat <- data.frame(id=rep(1:n, times=p), obs=obs) ### simulate (correlated) random effects for intercepts and slopes mu <- c(0,0) S <- matrix(c(1, tau01, tau01, 1), nrow=2) tau <- c(tau0, tau1) S <- diag(tau) %*% S %*% diag(tau) U <- mvrnorm(n, mu=mu, Sigma=S) ### simulate AR(1) errors and then the actual outcomes dat$eij <- unlist(sapply(p, function(x) arima.sim(model=list(ar=ar.val), n=x) * sqrt(1-ar.val^2) * sigma))
dat$yij <- (beta0 + rep(U[,1], times=p)) + (beta1 + rep(U[,2], times=p)) * log(dat$obs) + dat$eij  I will first add an alpha level to the plotting lines for the next plot (remember this must be done before the original plot is created). tspag will be the template plot, and I will create a spaghetti plot (spag) where each colour represents an id: library(plyr) dat = ddply(dat, .(id), function(x){ x$alpha = ifelse(runif(n = 1) > 0.9, 1, 0.1)
x\$grouper = factor(rbinom(n=1, size =3 ,prob=0.5), levels=0:3)
x
})
tspag = ggplot(dat, aes(x=obs, y=yij)) +
geom_line() + guides(colour=FALSE) + xlab("Observation Time Point") +
ylab("Y")
spag = tspag + aes(colour = factor(id))
spag

Many other times I want to group by id but plot just a few lines (let's say 10% of them) dark and the other light, and not colour them:

bwspag = tspag + aes(alpha=alpha, group=factor(id)) + guides(alpha=FALSE)
bwspag

Overall, these 2 plots are useful when you have longitudinal data and don't want to loop over ids or use lattice. The great addition is that all the faceting and such above can be used in conjunction with these plots to get spaghetti plots by subgroup.

spag + facet_wrap(~ grouper)

### Spaghetti plot with overall smoother

If you want a smoother for the overall group in addition to the spaghetti plot, you can just add geom_smooth:

sspag = spag + geom_smooth(se=FALSE, colour="black", size=2)
sspag

sspag + facet_wrap(~ grouper)

bwspag + facet_wrap(~ grouper)

Note that the group aesthetic and colour aesthetic do not perform the same way for some operations. For example, let's try to smooth bwswag:

bwspag + facet_wrap(~ grouper) + geom_smooth(se=FALSE, colour="red")

We see that it smooths each id, which is not what we want. We can achieve the desired result by setting the group aesthetic:

bwspag + facet_wrap(~ grouper) +
geom_smooth(aes(group=1), se=FALSE, colour="red", size =2)

I hope that this demonstrates some of the simple yet powerful commands ggplot2 allows users to execute. I agree that some behavior may not seem straightforward at first glance, but becomes more understandable as one uses ggplot2 more.

## Another Case this is Useful – Save Plot Twice

Another (non-plotting) example I want to show is how saving ggplot2 objects can make saving duplicate plots much easier. In many cases for making plots to show to others, I open up a PDF device using pdf, make a series of plots, and then close the PDF. There may be 1-2 plots that are the real punchline, and I want to make a high-res PNG of them separately. Let me be clear, I want both – the PDF and PNG. For example:

pdf(tempfile())
print({g1 = g + aes(colour = group1)})
print({g1fac = g + aes(colour = factor(group1))})
print({g2 = g + aes(colour = group2)})
dev.off()

png(tempfile(), res = 300, height =7, width= 7, units = "in")
print(g2)
dev.off()

I am printing the objects, while assigning them. (I have to use the { brackets because I use = for assignment and print would evaluate that as arguments without {}). As g2 is already saved as an object, I can close the PDF, open a png and then print that again.

No copying and pasting was needed for remaking the plot, nor some weird turning off and on devices.

## Conclusion

I (and think you should) use ggplot2 for a lot of reasons, especially the grammar and philosophy. The plots I have used here are some powerful representations of data that are simple to execute. I believe using this system reflects and helps the true iterative process of making figures. The syntax may not seem intuitive to a long-time R user, but I believe the startup cost is worth the effort. Let me know if you'd like to see any other plots that you commonly use.


text:
# Math Help - Binomial Thm

1. ## Binomial Thm

1.) Using the binomial theorem, prove:

(Not sure how to do chooose in maple, so ignore the - (it's not a fraction))

$\left(\frac{n}{0}\right) + \left(\frac{n}{1}\right)\cdot 2 + \left(\frac{n}{2}\right)\cdot 2^2 + \cdots + \left(\frac{n}{n}\right)\cdot 2^n = 3^n \, \, \forall$ integers $n \geq 0$

2.) Now prove the above giving a combinotrial proof.

2. $(1+2)^n = 3^n$

3. $3^n=(1+2)^n=\sum_{i=0}^{n}\binom{n}{i}(2^{i})(1)^{ n-i}=\sum_{i=0}^{n}\binom{n}{i}(2^{i})$


text:
# Prove $\int_0^1 \frac{dx}{(x-2) \sqrt[5]{x^2{(1-x)}^3}} = -\frac{2^{\frac{11}{10}} \pi}{\sqrt{5+\sqrt{5}}}$

$$\mbox{Prove}\quad \int_{0}^{1}{\mathrm{d}x \over \left(\,{x - 2}\,\right)\, \sqrt[\Large 5]{\,x^{2}\,\left(\,{1 - x}\,\right)^{3}\,}\,} = -\,{2^{11/10}\,\pi \over \,\sqrt{\,{5 + \,\sqrt{\,{5}\,}}\,}\,}$$

• Being honest I havent got a clue where to start. I dont think any obvious substitutions will help ($$x \to 1-x, \frac{1}{x}, \sqrt{x},$$ more).
• The indefinite integral involves hypergeometric function so some miracle substitution has to work with the bounds I suspect.
• Maybe gamma function is involved some how ??.

If anyone has an idea and can provide help I would appreciate it.

• I have a feeling this will be a very hot integral. I’m here before it happened. Aug 11, 2020 at 17:46

### Hint:

Substitute $$x \rightarrow\frac{1}{x-1}$$. We'll get: $$-\int_0^\infty \dfrac{x^{-3/5}dx}{(2x+1)}$$ Can you continue from here?

• Nice! What made you think of that substitution?
– user801111
Aug 11, 2020 at 18:05
• @MoQabar It results from the "obvious" substitutions that you mentioned applied multiple times. Aug 11, 2020 at 18:09
• @user376343 My integral reduces to $-\dfrac{π}{2^{2/5} \sin(2π/5)}$, since $\sin(2π/5)=\dfrac{\sqrt{(5+√5)}}{2√2}$. Both integrals are same. Aug 11, 2020 at 18:41
• Do you did $x\mapsto1/u$ then $u\mapsto 1-x$ then multiplied by $-1$? You have more dedication than me! Aug 11, 2020 at 18:55
• @user376343 \begin{aligned} I &= \int_2^{\infty} \frac{\frac{\mathrm{d}x}{(x-1)^2}}{\left(\frac{3-2x}{x-1}\right) \sqrt[5]{{\left(\frac{1}{x-1}\right)}^2{\left(\frac{x-2}{x-1}\right)}^3 }} \\ &= \int_2^{\infty} \frac{\mathrm{d}x}{\left(3-2x\right) {(x-2)}^{\frac{3}{5}}} \\ &\overset{x-2 \to x}= -\int_0^{\infty} \frac{x^{-\frac{3}{5}}}{2x+1} \mathrm{d}x \\ &\overset{2x \to x}= -2^{-\frac{2}{5}} \int_0^{\infty} \frac{x^{-\frac{3}{5}}}{x+1} \mathrm{d}x \end{aligned}
– Ty.
Aug 11, 2020 at 20:40

\begin{align} &\bbox[10px,#ffd]{\int_{0}^{1}{\dd x \over \pars{x - 2}\root[\Large 5]{x^{2}\pars{1 - x}^{3}}}} \,\,\,\stackrel{x\ \mapsto\ 1/x}{=}\,\,\, \int_{\infty}^{1}{-\,\dd x/x^{2} \over \pars{1/x - 2}\pars{1/x}^{2/5} \pars{1 - 1/x}^{3/5}} \\[5mm] = &\ \int_{1}^{\infty}{\pars{x - 1}^{-3/5} \over 1 - 2x}\,\dd x \,\,\,\stackrel{x + 1\ \mapsto\ x}{=}\,\,\, -\,{1 \over 2}\int_{0}^{\infty}{x^{-3/5} \over x + 1/2}\,\dd x \end{align}
Lets consider $$\ds{\oint_{C}{z^{-3/5} \over z - 1/2}\,\dd z}$$ where $$\ds{C}$$ is a key-hole contour which "takes care" of the principal branch of $$\ds{\rule{0cm}{8mm}z^{-3/5}}$$ ( with a branch-cut along $$\ds{\left(-\infty,0\right]}$$ ). Namely,

\begin{align} 2\pi\ic\bracks{\pars{1 \over 2}^{-3/5}} & = \int_{-\infty}^{0}{\pars{-x}^{-3/5}\expo{-3\pi\ic/5} \over x - 1/2}\,\dd x + \int_{0}^{-\infty}{\pars{-x}^{-3/5}\expo{3\pi\ic/5} \over x - 1/2}\,\dd x \\[5mm] & = -\expo{-3\pi\ic/5}\int_{0}^{\infty}{x^{-3/5} \over x + 1/2}\,\dd x + \expo{3\pi\ic/5}\int_{0}^{\infty}{x^{-3/5} \over x + 1/2}\,\dd x \\[5mm] & = 2\ic\sin\pars{3\pi \over 5} \int_{0}^{\infty}{x^{-3/5} \over x + 1/2}\,\dd x \\[5mm] \implies & \bbox[10px,#ffd]{-\,{1 \over 2}\int_{0}^{\infty}{x^{-3/5} \over x + 1/2}\,\dd x = - 2^{3/5}\pi\csc\pars{3\pi \over 5}} \approx -2.5034 \end{align}


text:
# Github Actions as a Number

## Let's calculate something!

Let's consider a large project that is running GitHub Actions for unit tests.

A single job of unit tests take about minutes.
This job runs in a paralle grid of size .

That means that each commit can trigger a job that takes minutes.

This is just a single commit though.
Let's also consider that on average each PR contains commits.
If we now assume that our project has pull requests per day.

Then our GitHub Actions bill will be:

$per day$ per month (assuming 20 work days)
\$ per year (assuming 50 work weeks)

## What's in a number?

Now let's wonder. Is this a big number? I've been wondering about this number mainly because I wonder when the number is big enough for people to care. Tech teams are usually more concerned about velocity, which is a totally valid concern. The salary of a developer might still pale in comparison to the GitHub bill so it makes sense to prioritize.

At the same time though ... it's a lot of compute! We shouldn't waste it just because there's budget for it. It's making me wonder what tools might be made that can help folks keep these compute times down. I've made pytest-duration-insights as a first step for python tests but I'm wondering if I can do more on a Github-level.

GitHub is interesting because it's not a cloud provider. It's too small for something like the duckbill group. The costs are substantial enough to make me wonder about what could be done though.


text:
# Confusion regarding definition of adjoint functor - Hilton and Stammbach

While defining Adjoint functors in their book A Course in Homological Algebra, Hilton and Stammbach said the follwing:

Let $F:\mathfrak{S}\rightarrow \mathfrak{M}_{\Lambda}$ be the free functor which associates with every set the free $\Lambda$ module on that set as a basis; and let $G:\mathfrak{M}_{\Lambda}\rightarrow\mathfrak{S}$ be the underlying functor which associates with every module its underlying set. We now define a transformation, natural in both $S$ and $A$, $$\eta=\eta_{SA}:\mathfrak{M}_{\Lambda}(FS,A)\rightarrow \mathfrak{S}(S,GA)$$

I did not understand what does he mean by natural in both $S$ and $A$..

I know that a natural transformation is between two functors $F,G: \mathcal{C}\rightarrow \mathcal{D}$..

What are those two functors here...

Fix $A$ a $\Lambda$ module. Then we have functor $$\mathfrak{M}_{\Lambda}(F-,A):S\mapsto \mathfrak{M}_{\Lambda}(FS,A)$$ from sets to module homomorphisms

Fix $S$ a set. Then we have a functor $$\mathfrak{S}(S,G-):A\mapsto \mathfrak{S}(S,GA)$$ from modules to set maps..

To define natural transformations, we need to have two functors from same categor landing in same category... Here one functor is from category of sets and other is from category of modules... What natural equivalence are we considering here?

I am very much confused..

## 1 Answer

There seems to be a misunderstanding here. Your $\mathfrak M_Λ(F-, A)$ is not a functor from sets to "module homomorphisms", it is a contravariant functor from sets to sets, which assigns to every set $S$ a set of certain module homomorphisms. In other words, it is a functor from $\mathfrak S^\mathrm{op}$ to $\mathfrak S$, as is $\mathfrak S(-, GA)$. That $η_{SA}$ is natural in $S$ means that for a fixed $A$, $η_{-,A}$ is a natural transformation between these two functors, and similarly for naturality in $A$.

Another way to say this is to note that $(S, A) ↦ \mathfrak M_Λ(FS, A)$ is a functor $\mathfrak S^\mathrm{op} × \mathfrak M_Λ → \mathfrak S$, and so is $(S, A) ↦ \mathfrak S(S, GA)$, and you can check that $η$ is natural in both $S$ and $A$ exactly when it is a natural transformation between them.

• I think i got it... Thanks.... Give me some time to understand it better... If i have no more questions i will upvote it.. – kk lm Feb 3 '16 at 4:35
• I am still getting confused... Why do we have $\mathfrak{S}^{\rm{op}}$ there and not just $\mathfrak{S}$ – kk lm Feb 3 '16 at 19:04
• Because the hom functor is contravariant in the first argument, which is surely explained in a textbook on homological algebra; try revisiting the previous parts on categories. – user54748 Feb 3 '16 at 19:16
• Yes Yes.. Hom functor is contravariant in the first argument... I will revise what i have learned before and then get back to this.. Thanks... – kk lm Feb 3 '16 at 19:19


text:
## 11.1 Examples

• Compare baseline characteristics (e.g. age, height, BMI) or study response variable among subjects enrolled in one of three (or more) nonrandomized clinical trial arms
• Determine if pulmonary function, as measured by the forced expiratory volume in one second, differs in non-smokers, passive smokers, light smokers, and heavy smokers
• Evaluate differences in artery dilation among wild type, knockout, and knockout+treated mice
• Could add second factor: Normoxic (normal oxygen) or hypoxic (insufficient oxygen) environmental conditions (a two-way ANOVA)
• In general, studies with a continuous outcome and categorical predictors

## 11.2 The $$k$$-Sample Problem

• When $$k=2$$ we compare two means or medians, etc.
• When $$k>2$$ we could do all possible pairwise 2-sample tests but this can be misleading and may $$\uparrow$$ type I assertion probability
• Advantageous to get a single statistic testing $$H_{0}$$: all groups have the same distribution (or at least the same central tendency)

## 11.3 Parametric ANOVA

### 11.3.1 Model

• Notation
• $$k$$ groups (samples) each from a normal distribution
• Population means $$\mu_{1}, \mu_{2}, \ldots, \mu_{k}$$
• $$n_i$$ observations from the $$i$$th group
• $$y_{ij}$$ is the $$j$$th observation from $$i$$th group
• Model specification
• $$y_{ij} = \mu + \alpha_i + e_{ij}$$
• $$\mu$$ is a constant
• $$\alpha_i$$ is a constant specific to group $$i$$
• $$e_{ij}$$ is the error term, which is assumed to follow a Normal distribution with mean 0 and variance $$\sigma^2$$
• This model is overparameterized; that is, it is not possible to estimate $$\mu$$ and each $$\alpha_i$$ (a total of $$k + 1$$) terms using only $$k$$ means
• Restriction 1: $$\Sigma \alpha_i = 0$$
• $$\mu$$ is the mean of all groups taken together, the grand or overall mean
• each $$\alpha_i$$ represents the deviation of the mean of the $$i$$th group from the overall mean
• $$\epsilon_{ij}$$ is the deviation of individual data points from $$\mu + \alpha_i$$
• Restriction 2: $$\alpha_1 = 0$$
• $$\mu$$ is the mean of group 1
• each $$\alpha_i$$ represents the deviation of the mean of the $$i$$th group from the group 1 mean
• $$\epsilon_{ij}$$ is the deviation of individual data points from $$\mu + \alpha_i$$
• Other restrictions possible, and will vary by software package

### 11.3.2 Hypothesis test

• Hypothesis test
• $$H_{0}: \mu_{1}=\mu_{2}=\ldots=\mu_{k}$$
• $$H_{1}:$$ at least two of the population means differ
• Not placing more importance on any particular pair or combination although large samples get more weight in the analysis
• Assume that each of the $$k$$ populations has the same $$\sigma$$
• If $$k=2$$ ANOVA yields identical $$P$$-value as 2-tailed 2-sample $$t$$-test
• ANOVA uses an $$F$$ statistic and is always 2-tailed
• $$F$$ ratio is proportional to the sum of squared differences between each sample mean and the grand mean over samples, divided by the sum of squared differences between all raw values and the mean of the sample from which the raw value came
• This is the SSB/SSW (sum of squares between / sum of squares within)
• SSB is identical to regression sum of squares
SSW is identical to sum of squared errors in regression
• $$F=MSB/MSW$$ where
• MSB = mean square between = SSB/$$(k-1), k-1=$$ “between group d.f.”
• MSW = mean square within = SSW/$$(n-k), n-k=$$ “within group d.f.”
• Evidence for different $$\mu$$s $$\uparrow$$ when differences in sample means (ignoring direction) are large in comparison to between-patient variation

### 11.3.3 Motivating Example

• Example from Biostatistics: A methodology for the Health Sciences by Fisher and Van Belle
• Research question (Zelazo et al., 1972,Science)
• Outcome: Age at which child first walks (months)

• Experiment involved the reinforcement of the walking and placing reflexes in newborns

• Newborn children randomly assigned to one of four treatment groups

• Active exercise: Walking and placing stimulation 4 times a day for 8 weeks
• Passive exercise: An equal amount of gross motor stimulation
• No exercise: Tested along with first two groups at weekly intervals
• Control group: Infants only observed at 8 weeks (control for effect of repeated examinations)
• Distribution of ages (months) at which infants first walked alone. Data from Zelazo et al., 1972
Active Group Passive Group No Exercise 8-week Control
9.00 11.00 11.50 13.25
9.50 10.00 12.00 11.50
9.75 10.00 9.00 12.00
10.00 11.75 11.50 13.50
13.00 10.50 13.25 11.50
9.50 15.00 13.00 12.35
Mean 10.125 11.375 11.708 12.350
Variance 2.0938 3.5938 2.3104 0.7400
Sum of $$Y_{i}$$ 60.75 68.25 70.25 74.10
Code
w <- rbind(
data.frame(trt='Active',         months=c(9,9.5,9.75,10,13,9.5)),
data.frame(trt='Passive',        months=c(11,10,10,11.75,10.5,15)),
data.frame(trt='No Exercise',    months=c(11.5,12,9,11.5,13.25,13)),
data.frame(trt='8-Week Control', months=c(13.25,11.5,12,13.5,11.5,12.35)) )
aggregate(months ~ trt, w, function(x) c(Mean=mean(x), Variance=var(x)))
             trt months.Mean months.Variance
1 8-Week Control   12.350000        0.740000
2         Active   10.125000        2.093750
3    No Exercise   11.708333        2.310417
4        Passive   11.375000        3.593750
Code
require(ggplot2)
require(data.table)
w <- data.table(w)
stats <- w[, j=list(months = mean(months), var=var(months)), by = trt]

ggplot(w, aes(x=trt, y=months)) +
geom_dotplot(binaxis='y', stackdir='center', position='dodge') +
geom_errorbar(aes(ymin=..y.., ymax=..y..), width=.7, size=1.3,
data=stats) +
xlab('') + ylab('Months Until First Walking') + coord_flip()
• Note that there are equal samples size in each group ($$n_i = 6$$ for each $$i$$) in the example. In general, this is not necessary for ANOVA, but it simplifies the calculations.
• Thought process for ANOVA
• Assume that age at first walk is Normally distributed with some variance $$\sigma^2$$

• The variance, $$\sigma^2$$, is unknown. There are two ways of estimating $$\sigma^2$$

• Let the means in the four groups be $$\mu_1$$, $$\mu_2$$, $$\mu_3$$, and $$\mu_4$$

• Method 1

• Assuming the variance are equal, calculated a pooled (or average) estimate of the variance using the four groups
• $$s^2_p = \frac{1}{4}(2.0938 + 3.5938 + 2.3104 + 0.7400) = 2.184$$
• Method 2

• Assuming the four treatments do not differ ($$H_0: \mu_1 = \mu_2 = \mu_3 = \mu_4 = \mu$$), the sample means follow a Normal distribution with variance $$\sigma^2/6$$.
• We can then estimated $$\sigma^2/6$$ by the variance of the sample means ($$s^2_{\overline{y}}$$)
• $$s^2_{\overline{y}} = \textrm{variance of } {10.125, 11.375, 11.708, 12.350}$$
• $$s^2_{\overline{y}} = 0.87349$$, so $$6 s^2_{\overline{y}} = 5.247$$ is our second estimate of $$\sigma^2$$
• $$s^2_p$$ is an estimate of the within group variability

• $$s^2_{\overline{y}}$$ is an estimate of the among (or between) group variability

• If $$H_0$$ is not true, method 2 will overestimate the variance

• The hypothesis test is based on $$F = 6 s^2_{\overline{y}} / s^2_p$$ and rejects $$H_0$$ if $$F$$ is too large

• Degrees of Freedom
• The $$F$$ statistic has both a numerator and denominator degrees of freedom

• For the numerator, d.f. = $$k - 1$$

• There are $$k$$ parameters ($$\alpha_1, \alpha_2, \ldots, \alpha_k)$$
• And one restriction ($$\Sigma \alpha_k = 0$$, or $$\alpha_1 = 0$$, or another)
• For the denominator, d.f. = $$N - k$$

• There are $$N$$ total observations
• And we estimate $$k$$ sample means
• In the age at first walking example, there are $$3$$ (numerator) and $$20$$ (denominator) degrees of freedom

Code
require(rms)
f <- ols(months ~ trt, data=w)
anova(f)
 d.f. Partial SS MS F P Analysis of Variance for months trt 3 15.74031 5.246771 2.40 0.0979 REGRESSION 3 15.74031 5.246771 2.40 0.0979 ERROR 20 43.68958 2.184479

### 11.3.4 Connection to Linear Regression

• Can do ANOVA using multiple regression, using an intercept and $$k-1$$ “dummy” variables indicating group membership, so memorizing formulas specific to ANOVA is not needed
• Why is between group d.f.=$$k-1$$?
• can pick any one group as reference group, e.g., group 1
• $$H_0$$ is identical to $$H_{0}:\mu_{2}-\mu_{1}=\mu_{3}-\mu_{1}=\ldots=\mu_{k}-\mu_{1}=0$$
• if $$k-1$$ differences in means are all zero, all means must be equal
• since any unique $$k-1$$ differences define our goal, there is $$k-1$$ d.f. between groups for $$H_0$$

## 11.4 Why All These Distributions?

• Normal distribution is handy for approximating the distribution of $$z$$ ratios (mean minus hypothesized value / standard error of mean) when $$n$$ is large or $$\sigma$$ is known
• If $$z$$ is normal, $$z^2$$ has a $$\chi^{2}_{1}$$ distribution
• If add $$k$$ $$z^{2}$$ values the result has a $$\chi^{2}_{k}$$ distribution; useful
• in larger than $$2\times 2$$ contingency tables
• in testing goodness of fit of a histogram against a theoretical distribution
• when testing more than one regression coefficient in regression models not having a $$\sigma$$ to estimate
• $$t$$ distribution: when $$\sigma$$ is estimated from the data; exact $$P$$-values if data from normal population
Distribution indexed by d.f.: $$t_{df}$$; useful for
• testing one mean against a constant
• comparing 2 means
• testing one regression coefficient in multiple linear regression
• $$t_{df}^{2}$$ has an $$F$$ distribution
• $$F$$ statistic can test
• $$>1$$ regression coefficient
• $$>2$$ groups
• whether ratio of 2 variances=1.0 (this includes MSB/MSW)
• To do this $$F$$ needs two different d.f.
• numerator d.f.: how many unique differences being tested (like $$\chi^{2}_{k}$$)

• denominator d.f.

• total sample size minus the number of means or regression coefficients and intercepts estimated from the data
• is the denominator of the estimate of $$\sigma^2$$
• also called the error or residual d.f.
• $$t_{df}^{2} = F_{1,df}$$
• ANOVA results in $$F_{k-1,df}$$; d.f.=$$N-k$$ where $$N=$$ combined total sample size; cf. 2-sample $$t$$-test: d.f.=$$n_{1}+n_{2} - 2$$
• Example: $F = MSB/MSW = 58 \sim F_{4,1044}$

The cumulative probability of getting an $$F$$ statistic $$\leq 58$$ with the above d.f. is 1.0000. We want Prob$$(F\geq 58)$$, so we get $$P=1-1=0$$ to several digits of accuracy but report $$P<0.0001$$.

Code
pf(58, 4, 1044)
[1] 1
Code
1 - pf(58, 4, 1044)
[1] 0

## 11.5 Software and Data Layout

• Every general-purpose statistical package does ANOVA
• Small datasets are often entered using Excel
• Statistical packages expect a grouping variable, e.g., a column of treatment names or numbers; a column of response values for all treatments combines is also present
• If you enter different groups’ responses in different spreadsheets or different columns within a spreadsheet, it is harder to analyze the data with a stat package

## 11.6 Comparing Specific Groups

• $$F$$ test is for finding any differences but it does not reveal which groups are different
• Often it suffices to quote $$F$$ and $$P$$, then to provide sample means (and their confidence intervals)
• Can obtain CLs for any specific difference using previously discussed 2-sample $$t$$-test, but this can result in inconsistent results due solely to sampling variability in estimating the standard error of the difference in means using only the two groups to estimate the common $$\sigma$$
• If assume that there is a common $$\sigma$$, estimate it using all the data
to get a pooled $$s^2$$
• $$1-\alpha$$ CL for $$\mu_{i}-\mu_{j}$$ is then $\bar{y}_{i}-\bar{y}_{j} \pm t_{n-k,1-\alpha/2} \times s \sqrt{\frac{1}{n_{i}}+\frac{1}{n_{j}}},$

where $$n$$ is the grand total sample size and there are respectively $$n_{i}$$ and $$n_{j}$$ observations in samples $$i$$ and $$j$$

• Can test a specific $$H_{0}: \mu_{i}=\mu_{j}$$ using similar calculations; Note that the d.f. for $$t$$ comes from the grand sample size $$n$$, which $$\uparrow$$ power and $$\downarrow$$ width of CLs slightly
• Many people use more stringent $$\alpha$$ for individual tests when testing more than one of them (sec-multcomp)
• This is not as necessary when the overall $$F$$-test is significant

## 11.7 Non-Parametric ANOVA: Kruskal-Wallis Test

• $$k$$-sample extension to the 2-sample Wilcoxon–Mann–Whitney rank-sum test
• Is very efficient when compared to parametric ANOVA even if data are from normal distributions
• Has same benefits as Wilcoxon (not harmed by outliers, etc.)
• Almost testing for equality of population medians
• In general, tests whether observations in one group tend to be larger than observations in another group (when consider randomly chosen pairs of subjects)
• Test statistic obtained by replacing all responses by their ranks across all subjects (ignoring group) and then doing an ANOVA on the ranks
• Compute $$F$$ (many authors use a $$\chi^2$$ approximation but $$F$$ gives more accurate $$P$$-values)
• Look up against the $$F$$ distribution with $$k-1$$ and $$n-k$$ d.f.
• Very accurate $$P$$-values except with very small samples
• Example:
$$F$$ statistic from ranks in Table 12.16: $$F_{3,20}=7.0289$$
• Using the cumulative distribution calculator from the web page, the prob. of getting an $$F$$ less impressive than this under $$H_0$$ is 0.9979
$$P$$ is $$1-0.9979=0.0021$$
• Compare with Rosner’s $$\chi^{2}_{3}=11.804$$ from which $$P=0.008$$ by survstat or one minus the CDF
• Evidence that not all of the 4 samples are from the same distribution
• loosely speaking, evidence for differences in medians
• better: some rabbits have larger anti-inflammatory effects when placed on different treatments in general
• Comparison of Kruskal-Wallis and Parametric ANOVA for age at first walk example
• A few extreme values in age a first walk may violate parametric $$F$$-test assumptions

• Run rank ANOVA: Kruskal-Wallis test three different ways:

• Parametric ANOVA on the ranks of $$y$$
• Spearman’s $$\rho^2$$ generalized to multiple columns of $$x$$
• An R function dedicated to Kruskal-Wallis
Code
anova(ols(rank(months) ~ trt, data=w))
 d.f. Partial SS MS F P Analysis of Variance for rank(months) trt 3 359.3333 119.77778 3.07 0.0515 REGRESSION 3 359.3333 119.77778 3.07 0.0515 ERROR 20 781.1667 39.05833
Code
spearman2(months ~ trt, data=w)

Spearman rho^2    Response variable:months

rho2    F df1 df2      P Adjusted rho2  n
trt 0.315 3.07   3  20 0.0515         0.212 24
Code
kruskal.test(months ~ trt, data=w)

Kruskal-Wallis rank sum test

data:  months by trt
Kruskal-Wallis chi-squared = 7.2465, df = 3, p-value = 0.06444

Note that the classical Kruskal-Wallis test uses the $$\chi^{2}$$ approximation while the other two used an $$F$$ distribution, which is as or more accurate than using $$\chi^{2}$$.

## 11.8 Two-Way ANOVA

• Ideal for a factorial design or observational study with 2 categorical grouping variables
• Example: 3 treatments are given to subjects and the researcher thinks that females and males will have different responses in general
Six means: $$\bar{Y}_{i,j}, i=$$ treatment, $$j=$$ sex group
• Can test
• whether there are treatment differences after accounting for sex effects
• whether there are sex differences after accounting for treatment effects
• whether the treatment effect is difference for females and males, if allow treatment $$\times$$ sex interaction to be in the model
• Suppose there are 2 treatments ($$A,B$$) and the 4 means are $$\bar{Y}_{Af}, \bar{Y}_{Bf}, \bar{Y}_{Am}, \bar{Y}_{Bm}$$, where $$f, m$$ index the sex groups
• The various effects are estimated by
• treatment effect: $$\frac{(\bar{Y}_{Af}-\bar{Y}_{Bf})+(\bar{Y}_{Am}-\bar{Y}_{Bm})}{2}$$
• sex effect: $$\frac{(\bar{Y}_{Af}-\bar{Y}_{Am})+(\bar{Y}_{Bf}-\bar{Y}_{Bm})}{2}$$
• treatment $$\times$$ sex interaction: $$(\bar{Y}_{Af}-\bar{Y}_{Bf})-(\bar{Y}_{Am}-\bar{Y}_{Bm}) = (\bar{Y}_{Af}-\bar{Y}_{Am})-(\bar{Y}_{Bf}-\bar{Y}_{Bm})$$
• Interactions are “double differences”
• Assessing whether treatment effect is same for $$m$$ vs. $$f$$ is the same as assessing whether the sex effect is the same for $$A$$ vs. $$B$$
• Note: 2-way ANOVA is not appropriate when one of the categorical variables represents conditions applied to the same subjects, e.g. serially collected data within patient with time being one of the variables;
2-way ANOVA assumes that all observations come from different subjects

## 11.9 Analysis of Covariance

• Generalizes two-way ANOVA
• Allows adjustment for continuous variables when comparing groups
• Can $$\uparrow$$ power and precision by reducing unexplained patient to patient variability ($$\sigma^2$$)
• When $$Y$$ is also measured at baseline, adjusting for the baseline version of $$Y$$ can result in a major reduction in variance
• Fewer assumptions if adjust for baseline version of $$Y$$ using ANCOVA instead of analyzing ($$Y -$$ baseline $$Y$$)
• Two-way ANOVA is a special case of ANCOVA where a categorical variable is the only adjustment variable (it is represented in the model by dummy variables) See Chapter sec-ancova for much more information about ANCOVA in RCTs.

## 11.10 Multiple Comparisons

• When hypotheses are prespecified and are few in number, don’t need to correct $$P$$-values or $$\alpha$$ level in CLs for multiple comparisons
• Multiple comparison adjustments are needed with $$H_{0}$$ is effectively in the form
• Is one of the 5 treatments effective when compared against control?
• Of the 4 etiologies of disease in our patients, is the treatment effective in at least one of them?
• Is the treatment effective in either diabetics, older patients, males, , etc.?
• Diabetics had the greatest treatment effect empirically; the usual $$P$$-value for testing for treatment differences in diabetics was 0.03
• Recall that the probability that at least one event out of events $$E_{1}, E_{2}, \ldots, E_{m}$$ occurs is the sum of the probabilities if the events are mutually exclusive
• In general, the probability of at least one event is $$\leq$$ the sum of the probabilities of the individual events occurring
• Let the event be “reject $$H_0$$ when it is true”, i.e., making a type I assertion probability or “false positive” conclusion
• If test 5 hypotheses (e.g., 5 subgroup treatment effects) at the 0.05 level, the upper limit on the chance of finding one significant difference if there are no differences at all is $$5 \times 0.05 = 0.25$$
• This is called the Bonferroni inequality
• If we test each $$H_0$$ at the $$\frac{\alpha}{5}$$ level the chance of at least one false positive is no greater than $$\alpha$$
• The chance of at least one false positive is the experimentwise assertion probability whereas the chance that a specific test is positive when there is no real effect is the comparisonwise error probability better called comparisonwise type I assertion probability
• Instead of doing each test at the $$\frac{\alpha}{m}$$ level we can get a conservative adjusted $$P$$-value by multiplying an individual $$P$$-value by $$m$$1
• Whenever $$m \times P > 1.0$$ report $$P=1.0$$
• There are many specialized and slightly less conservative multiple comparison adjustment procedures. Some more complex procedures are actually more conservative than Bonferroni.
• Statisticians generally have a poor understanding about the need to not only adjust $$P$$-values but to adjust point estimates also, when many estimates are made and only the impressive ones (by $$P$$) are discussed. In that case point estimates are badly biased away from the null value. For example, the BARI study analyzed around 20 subgroups and only found a difference in survival between PTCA and CABG in diabetics. The hazard ratio for CABG:PTCA estimated from this group is far too extreme.
• Bayesian have a much different view of multiplicities, and the Bayesian approach is more in line with accepted rules of evidence, e.g., evidence about A doesn’t need to be discounted because B was assessed
• 1 Make sure that $$m$$ is the total number of hypotheses tested with the data, whether formally or informally.


text:
## arithmetic operations in different number system

(a) 1234 + 5432 = 6666
answer: hex , binary, octal, decimal

(b) 41 / 3 = 13

(c) 33 / 3 = 11
(f) $sqrt(41) = 5$


text:
# How to skip “no.,” when using ieee biblatex style with biber? [duplicate]

Consider the bibliography file biblio.bib

@article{Sutter-TheFreeLunchIsOver,
author={Herb Sutter},
title={The Free Lunch Is Over: A Fundamental Turn Toward Concurrency in Software},
journal={Dr.\ Dobb's Journal},
year={2005},
volume={30},
number={3},
month= mar
}

and the latex file main.tex

\documentclass{article}
%\documentclass{IEEEtran}
\usepackage[backend=biber,bibstyle=ieee]{biblatex}
\begin{document}
\cite{Sutter-TheFreeLunchIsOver}
\printbibliography
\end{document}

No matter whether we are using the article class or the ieeetran class, a "no." without a number is printed to the PDF obtained from pdflatex main && biber main && pdflatex main:

I have no idea why the number entry (number={3}) in the .bib file is ignored and whether the printing of "no." without a number is a bug or a feature. Apparently, it is hard to turn off, since the biblatex-ieee documentation also has this bug/feature as of March 2, 2017.

In any case, I'd like to avoid "no., " altogether or to have "no. 3," instead. Any help?

## marked as duplicate by moewe biblatex StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Mar 22 '17 at 8:30

It's a bug of the style. The current definition in ieee.bbxis:

\renewbibmacro*{volume+number+eid}{%
\printfield{volume}%
\newunit
\bibstring{number}%
\newunit
\printfield{eid}%
}

The \bibstring is wrong. It should be \printfield

So add the following definition to your header (after loading biblatex).

\renewbibmacro*{volume+number+eid}{%
\printfield{volume}%
\newunit
\printfield{number}%
\newunit
\printfield{eid}%
}

The author is informed.

• @LeonMeier: The author uploads an update. In the next few days it should be available via your package manager. – Marco Daniel Mar 22 '17 at 11:42


text:
SEARCH HOME
 Math Central Quandaries & Queries
 Question from Chris, a parent: Write this number in expanded form 4 326 180 501 Thanks!!!

Hi Chris,

I am not exactly sure what you mean by expanded form but I'll take a guess. I am going to use a smaller number, $6 205.$

In expanded form I would write this as

$6 \times 1 000 + 2 \times 100 + 0 \times 10 + 5.$

Penny

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.


text:
Windows Thread, Changing permissions on a registry key in Technical; Hi. I need to find out how to change permissions on a registry key for a specific software package. It ...
1. ## Changing permissions on a registry key

Hi. I need to find out how to change permissions on a registry key for a specific software package. It requires the following to have FULL Control:

HKLM\SOFTWARE
SYSTEM
Users (%localmachine%\Users)

If Reg.exe can be used, that would be great. If it needs to be a VB file, I'd need the entire script.

Thanks!

2. You should be able to achieve this using regini.exe

How to Use Regini.exe to Set Permissions on Registry Keys
How to change registry values or permissions from a command line or a script

I think you can call this from machine startup script by sticking the exe on the netlogon share, or using the -m option you can do the changes remotely. I've never tried it, but the MS knowledgebase articles should help you use it.

Incidently those articles refer to windows 2000 and NT4, I've no idea if the same tool is included or will work with XP/server 2003 but it's the best I can find for what you want to do.

Mike.

3. Is group policy not an option?

4. i have used this for an old RM app that needed user access to its own keys

Code:
'  VBScript.

'

set WshShell = CreateObject("WScript.Shell")

' IN THE NEXT LINE (starting WshShell.Run..)

WScript.Sleep 1000

' IN THE NEXT LINE (starting WshShell.SendKeys..)

WScript.Quit()

save this as .bat

Code:
'edu-tech solutions Nov 2007

' install registry key

regedit /s \\2100-fs01\NETLOGON\reg\rm.reg

' set permissions on key
call "\\server\NETLOGON\reg\SetACL.exe" -on "HKEY_LOCAL_MACHINE\SOFTWARE\InterActual Technologies" -ot reg -actn ace -ace "n:domain.sch.uk\Domain Users;p:full"
call "\\server\NETLOGON\reg\SetACL.exe" -on "HKEY_LOCAL_MACHINE\SOFTWARE\Research Machines" -ot reg -actn ace -ace "n:domain.sch.uk\Domain Users;p:full"
and make sure that setacl.exe is in the same folder.

replace domain, server iand password in the scripts with the appropriate info

a bit messy i know, but it works.

hope this helps.

BoX

5. ## Thanks to box_l from:

sonofsanta (20th September 2012)

6. By far the easiest way to do this is with group policy.

If you can't use group policy then run the batch file used by @Box but as a machine startup script rather than a login script . That way you don't need to use the admin password (which I really wouldn't recommend; the login script, complete with password, can be read by any of your users)

7. agreed, in its current form.

i do use microsofts script encoder to obfuscate it enough that most people will not even attempt to read/decode it.

it also gets removed from its location when not in use.

BoX

8. ## Thanks

Thanks to everyone for their replies. My AD guy set up a GPO to set the permissions needed.

SHARE:


text:
# zbMATH — the first resource for mathematics

On the interpolation in linear spaces. (English) Zbl 1051.41004
Let $$X$$ and $$Y$$ be two linear spaces and let $$f:X\rightarrow Y$$ be a given function. If $$x_{0},x_{1},\dots,x_{n}\in X$$ are distinct points, the author defines, in some conditions, an abstract interpolation polynomial of the form $$L( x_{0},x_{1},\dots,x_{n};f) (x) =D_{n}x^{n}+D_{n-1}x^{n-1}+\cdots +D_{1}x+D_{0}$$, where $$D_{k}\in L_{k}( X,Y)$$ $$( 1\leq k\leq n)$$ are $$k$$-linear mappings defined on $$X^{k}$$ with values in $$Y$$ $$\left( D_{0}\in Y\right)$$. The application $$D_{n}\in L_{n}(X,Y)$$ is called the generalized divided difference of order $$n$$ for the function $$f$$ on the nodes $$x_{0},x_{1},\dots,x_{n}$$. Some properties of the abstract interpolation polynomials and generalized divided differences are presented in this context.
##### MSC:
 41A05 Interpolation in approximation theory 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
##### Keywords:
abstract interpolation polynomial
Full Text:


text:
Home > CC3MN > Chapter 8 > Lesson 8.1.2 > Problem8-20

8-20.

Jack noticed a sale on his favorite model of All-Terrain Vehicle (ATV). The salesperson said the company was cutting the price for that weekend, so Jack got an $8\%$ discount.

1. If the usual price of the ATV was $4298$, estimate how much the discount will be. Explain your reasoning.

Round the ATV price to $4300$ in order to make the percent easier to approximate.

Since we can find that $10\%$ of $4300$ is $430$, we can also determine that $5\%$ of $4300$ would be $215$. $8\%$is roughly in the middle of $10\%$ and $5\%$. What is your estimation?

2. Calculate the amount Jack did pay with the $8\%$ discount.

Find $8\%$ of $4298$, then subtract that value from $4298$ to find how much Jack paid.

$4298 − 343.84 =$

$4298\left(.08\right) = 343.84$

$3954.16$

1. If the store sold three of the ATVs that weekend, how much money was saved by the three customers combined?

For each ATV, the original price was $4298$. Based on part (b), it costs $3954$.16 after the $8\%$ discount. How much did Jack save on one ATV? Now multiply that number by $3$.


text:
SSgompertz

0th

Percentile

Self-Starting Nls Gompertz Growth Model

This selfStart model evaluates the Gompertz growth model and its gradient. It has an initial attribute that creates initial estimates of the parameters Asym, b2, and b3.

Keywords
models
Usage
SSgompertz(x, Asym, b2, b3)
Arguments
x
a numeric vector of values at which to evaluate the model.
Asym
a numeric parameter representing the asymptote.
b2
a numeric parameter related to the value of the function at x = 0
b3
a numeric parameter related to the scale the x axis.
Value

a numeric vector of the same length as input. It is the value of the expression Asym*exp(-b2*b3^x). If all of the arguments Asym, b2, and b3 are names of objects the gradient matrix with respect to these names is attached as an attribute named gradient.

nls, selfStart
library(stats) DNase.1 <- subset(DNase, Run == 1) SSgompertz(log(DNase.1$conc), 4.5, 2.3, 0.7) # response only Asym <- 4.5; b2 <- 2.3; b3 <- 0.7 SSgompertz(log(DNase.1$conc), Asym, b2, b3) # response and gradient print(getInitial(density ~ SSgompertz(log(conc), Asym, b2, b3), data = DNase.1), digits = 5) ## Initial values are in fact the converged values fm1 <- nls(density ~ SSgompertz(log(conc), Asym, b2, b3), data = DNase.1) summary(fm1)


text:
# Dot Product Deep(ish) Dive

The dot product is helpful in finding the distance of a point to a line. The dot product, as we mentioned here, is the the sum of the element-wise products of the vector components. Given two vectors $$\mathtt{v}$$ and $$\mathtt{w}$$, their dot product is $\begin{bmatrix}\mathtt{v_1}\\\mathtt{v_2}\end{bmatrix} \cdot \begin{bmatrix}\mathtt{w_1}\\\mathtt{w_2}\end{bmatrix}\mathtt{= v_1w_1 + v_2w_2}$

The result of this computation is not another vector, but just a number, a scalar quantity. And, given that the dot product of two perpendicular vectors is 0, it would be nice if the dot product were related to cosine in some way, since the cosine of 90° is also 0. So let’s take a look at some vector pairs and their dot products and think about any patterns we see.

$$\mathtt{v \cdot w=-4}$$     $$\mathtt{θ=180^{\circ}}$$     $$\mathtt{\textrm{cos}(θ)=-1}$$

$$\mathtt{v \cdot w=-2}$$     $$\mathtt{θ=120^{\circ}}$$     $$\mathtt{\textrm{cos}(θ)=-\frac{1}{2}}$$

$$\mathtt{v \cdot w=4}$$     $$\mathtt{θ=45^{\circ}}$$     $$\mathtt{\textrm{cos}(θ)=\frac{\sqrt{2}}{2}}$$

$$\mathtt{v \cdot w=2}$$     $$\mathtt{θ=60^{\circ}}$$     $$\mathtt{\textrm{cos}(θ)=\frac{1}{2}}$$

Well, so, the dot products have the same signs as the cosines. That’s a start. And in all but one case shown, we can divide the dot product by 4 to get the cosine. What makes the 45° case different?

Each of the vectors shown, with the exception of the vector (2, 2) has a length, a magnitude, of 2. To determine the magnitude, or length, of a vector, you treat the components of the vector as the legs of a right triangle and the vector itself as the hypotenuse. So, $|\begin{bmatrix}\mathtt{-1}\\\mathtt{\sqrt{3}}\end{bmatrix}|=\sqrt{(-1)^2+(\sqrt{3})^2}=2$

But the length of (2, 2) is $$\mathtt{\sqrt{8}}$$. If we were to give that vector a length of 2, without changing the angle between v and w, then the vector would become ($$\mathtt{\sqrt{2}, \sqrt{2}}$$). And, lo, the dot product would become $$\mathtt{2\sqrt{2}}$$, which, when divided by 4, would yield the cosine.

The 4 that we divide by isn’t random. It’s the product of the lengths of the vectors. If we leave the 45° angled vectors alone, the product of their lengths is $$\mathtt{2\sqrt{8}}$$. Dividing 4 by this product does indeed yield the correct cosine. So, we have an initial conjecture that the dot product of two vectors v and w relates to cosine like this: $\mathtt{\frac{v \cdot w}{|v||w|} = cos(θ)}$

Perpendicular vectors will still have a dot product of 0 with this formula, so that’s good. And we can scale the vectors however we want and the cosine should remain the same—as it should be—though it may take a little manipulation to see that that’s true. But we are still left with the puzzle of proving this conjecture, more or less, or at least demonstrating to our satisfaction that the result is general.

Although the derivation doesn’t go beyond the Pythagorean Theorem, really, it gets a little symbol heavy, so let’s start with something simpler. We can write the cosine of θ at the right as $\mathtt{\textrm{cos}(θ)=\frac{|w|}{|v|}}$ If we think of w here as truly horizontal, its length is simply $$\mathtt{v_1}$$, the length of the horizontal component of v. Combining this fact with the length of v, we can rewrite the cosine equation above as $\mathtt{\textrm{cos}(θ)=\frac{v_1}{\sqrt{v_{1}^2+v_{2}^2}}}$

Since w is horizontal (has a second component of 0), the dot product $$\mathtt{v \cdot w}$$ becomes simply $$\mathtt{v_{1}^2}$$. Dividing this by the product of the lengths of the vectors v and w (where the length of w is just $$\mathtt{v_1}$$), we get this equation for cosine: $\mathtt{\,\,\,\,\,\textrm{cos}(θ)=\frac{v_{1}^2}{(v_1)(\sqrt{v_{1}^2+v_{2}^2})}}$ And that’s clearly equal to the above. So, while it is by no means definitive, we can have a little more confidence at this point that we have the right equation for cosine using the dot product. We can get more formal and sure about it later. Next time we’ll look at how it can help us determine the distance from a point to a line.


text:
# Handling input in a custom ECS engine

I've been implementing an Entity Component System into my custom toy engine so I can learn more about it. Currently I have the following problem:

I can't think of a way of implementing input as a system. Currently I just check for key presses in the main loop, like this:

if (glfwGetKey(Window, GLFW_KEY_A) == GLFW_PRESS)
Camera.Translate(glm::vec2(-5.0f, 0.0f) * static_cast<float>(DeltaTime));

Here I check if key A is pressed, and if it is I move the camera, which isn't a component associated with an entity.

So what I thought is creating an Entity with a Input component associated (this component would have mouse coordinates, keys pressed, etc):

ECS::Entity * InputManager = EntityWorld->create();
InputManager->assign<Input>();

Then, each frame, after polling events, I would update the Entity directly like this:

InputManager->get<Input>()->MousePosition = MousePosition;
InputManager->get<Input>()->MouseWheelDelta = MouseWheelDelta;
....

Then, I'd have an InputSystem, where I can check mouse position, if some key is pressed, etc. But what I don't understand is, if the camera is not a component, how could I make it move now that I handle input through the InputSystem? If I make the camera a component, how would I implement all the functionality to move it around, and zoom in and out?

Let me know if something it's not clear.

• Is there an issue with not handling the camera with your ECS? – Vaillancourt May 22 '19 at 19:56
• Well, I've been thinking about it, but I'm not completely sure how to do it. Should I create a CameraSystem, which has functions to move around, get the view matrix, zoom in, etc, and then each tick check for Input retrieving the entity that has that component? I don't know if that would be a good solution. – Sergio Dorado May 22 '19 at 20:54

I think you're pretty close.

Here is one option, which I'd try to implement first. You need 2 components

• Input
• Camera

I assume you'd create an entity and add them like this:

ECS::Entity * UserCameraManager = EntityWorld->create();
UserCameraManager->assign<Input>();
UserCameraManager->assign<Camera>(getMainUserCamera());

Each frame, your InputSystem fetches all relevant information regarding the input (e.g. mouse, keyboard, joystick), and stores it in the Input component.

Then, the system that takes care of the camera (e.g. CameraSystem), updates all the entities that have both of these components, Input and Camera. (Yes, that's going to be only one pair, but that's ok.)

The update(Input, Camera) function will take the 'raw' input from the Input component, and 'translate' it to commands to move the Camera

CameraSystem::update(Input input, Camera camera)
if ( input.isKeyPress(Input::A) || input.getMouseXScreenRatio() < 0.01 )
camera.getCamera()->Translate(glm::vec2(-5.0f, 0.0f) * static_cast<float>(DeltaTime));

Now, it's a bit silly because all the instances of Input would be the same, so you might want to bend your ECS rules a bit to have an 'empty' Input component and use it only as something like a tag, letting those system that needs it poll an actual, more global InputManager instead of the component. (Or use it to store a reference to the more global input manager.)

So instead, each frame, the InputSystem fetches all relevant information regarding the input (e.g. mouse, keyboard, joystick), and stores it within itself, ready for polling.

ECS::Entity * UserCameraManager = EntityWorld->create();
UserCameraManager->assign<Input>(getGlobalInputManager());
UserCameraManager->assign<Camera>(getMainUserCamera());

CameraSystem::update(Input input, Camera camera)
if ( input.getInputManager()->isKeyPress(Input::A) || input.getInputManager()->getMouseXScreenRatio() < 0.01 )
camera.getCamera()->Translate(glm::vec2(-5.0f, 0.0f) * static_cast<float>(DeltaTime));

We programmers love layers. So let's add another layer onto it.

You need 3 components

• Input
• Camera
• CameraController

Let's re-use the InputSystem from the previous option, which updates the Input component (or a more global alternative).

Then let's use a System that will translate the 'input' into 'actions':

SystemCameraControl::update(Input input, CameraController cameraController)
cameraController.resetValues()
if ( input.getInputManager()->isKeyPress(Input::A) || input.getInputManager()->getMouseXScreenRatio() < 0.01
cameraController.moveLeft(5)

Then the other system takes care of converting the 'actions' into 'implementation':

CameraSystem::update(CameraController cameraController, Camera camera)
if ( cameraController.shouldMove() )
camera.getCamera()->Translate(cameraController.getDisplacementPerSec() * static_cast<float>(DeltaTime));

This option is really there to let you add some other components that will control the camera in a different way: when you'll want to have your camera automatically follow a spline, for instance, it might be a bit easier to think about it like this.

And this is to hint you could eventually have other entities be controlled: the main character will be moved like this, but you'll also have to move AI characters, for instance.

Of course, you don't need that extra component, at least now, but that's an option.

When adding the Camera component to the entity in your example, the getMainUserCamera() function would return a Camera object? I thought that components are structs with data that we are not supposed to have an instance of, and should be managed with the systems.

There are no rules that say that you have to put everything in your ECS: you can limit the ECS to what makes sense. IMHO, it makes perfect sense to have your rendering engine or your physics engine not an integral part of your ECS. You can use your ECS components to refer to objects in your non-ECS software parts.

This is the case in this example, getMainUserCamera() would return a reference to the Camera that is part of this non-ECS software part. Your Camera component is used to link the two software parts together. It's still 'data'. The data is a pointer to something else.

If it makes more sense to you, you can have the CameraSystem create the Camera component, since they'll most likely have to interface with the Rendering part of your software.

How would the InputSystem fetch the raw input? The only thing I can think of is accessing the entity that has the input component directly before ticking all systems in the main loop, not in a system.

Assuming that you don't want to repeat Input data in eacn Input component, you could have something like this (pseudo-c++):

class InputState
{
public:

enum Key
{
A = GLFW_KEY_A,
B, C //...
Z,
NUM_KEYS = Z - A + 1
};

enum Mouse
{
ButtonLeft,
ButtonMiddle,
ButtonRight,
NNUM_MOUSE_BUTTONS
};

enum ButtonState
{
DOWN,
JUST_DOWN,
RELEASED,
JUST_RELEASED
};

bool isKeyDown(Key aKey) { return mKeyState[aKey - 'A'] == DOWN || mKeyState[aKey - 'A'] == JUST_DOWN; }
bool isKeyJustDown(Key aKey) { return mKeyState[aKey - 'A'] == JUST_DOWN; }
bool isMouseDown(Mouse aMouseButton) // ...

// This has to be called before anything related to input is updated
void perFrameUpdate()
{
// Update the keys
for ( char i = Key::A; i <= Key::Z; ++i )
{
int vectorIndex = i - Key::A;
bool isKeyDown = glfwGetKey(Window, i) == GLFW_PRESS;

ButtonState lastKeyState = mKeyState[vectorIndex];

if ( isKeyDown && lastKeyState == ButtonState::JUST_DOWN )
mKeyState[vectorIndex] = ButtonState::DOWN;
else if ( isKeyDown && ( lastKeyState == ButtonState::RELEASED || lastKeyState == ButtonState::JUST_RELEASED )
mKeyState[vectorIndex] = ButtonState::JUST_DOWN;
else if // ...
// ...

}

// Do the same for mouse
}

private:

std::vector<ButtonState> mKeyState;
std::vector<ButtonState> mMouseState;

}

// This is your Input component
struct Input
{
explicit Input( InputState& aGlobalState ) : mInputState( aGlobalState ) {}

InputState& mInputState;
}

// This is your Camera component
struct Camera
{
explicit Camera( Render::Camera* aCamera ) : mCamera( aCamera ) {}

Render::Camera* mCamera;
}

class SystemCamera
{
public:
static void update_CameraInput( std::vector<ECS::Entity*>& aEntities, double DeltaTime )
{
Camera& camera = aEntities->get<Camera>();
Input& input = aEntities->get<Input>();

if ( input.mInputState.isKeyDown(InputState::A) || input.mInputState.getMouseXScreenRatio() < 0.01 )
camera.mCamera->Translate(glm::vec2(-5.0f, 0.0f) * static_cast<float>(DeltaTime));
}
}

// Main loop
class Game
{
public:

void init()
{
mMainCamera = // RenderManager::GetOrCreateMainCamera();

ECS::Entity * UserCameraManager = EntityWorld->create();
UserCameraManager->assign<Input>(mInputState);
UserCameraManager->assign<Camera>(mMainCamera);
mEntities.push_back( UserCameraManager );
}

void run()
{
while(true)
{
mInputState.perFrameUpdate();

// No need for an additional system to update the Input component, as they all refer to the same up-to-date mInputState.
std::vector<ECS::Entity*> entitiesThatHaveInputAndCamera = getRelevantEntities<Camera, Input>(mEntities);

SystemCamera::update_CameraInput(entitiesThatHaveInputAndCamera, DeltaTime);
}
}

private:

InputState mInputState;
std::vector<ECS::Entity*> mEntities;

Render::Camera* mMainCamera;
}

• I think I get it now, but I have two questions. First, when adding the Camera component to the entity in your example, the getMainUserCamera() function would return a Camera object? I thought that components are structs with data that we are not supposed to have an instance of, and should be managed with the systems. Second, how would the InputSystem fetch the raw input? The only thing I can think of is accessing the entity that has the input component directly before ticking all systems in the main loop, not in a system. Thanks a lot for your time. – Sergio Dorado May 23 '19 at 7:35
• @SergioDorado I have updated my answer. I admit that there are some incoherences in the first part of the answer regarding input*: InputManager, InputSystem; I'll have to come back to the answer to fix it. My 'last' example should be consistent. There is only the InputState which helps understand that it's not a System. It's not a singleton, as you could have one per window. The value added over taking the input directly from GLFW is that you have a state management from frame to frame. If there is still anything unclear, please ask :) – Vaillancourt May 23 '19 at 16:51
• Everything is clear now, I'll start to implement it and see how it goes. Thanks a lot again for your time and for your great answer :D – Sergio Dorado May 24 '19 at 7:51


text:
Rotation number of inverse maps on the circle.

I'm still a bit lost in my studies of rotation numbers. Any help is much appreciated!

Let's say we have a homeomorphism $F: \mathbb{R} \rightarrow \mathbb{R}$ which is a lift of a homeomorphism $f:S^1 \rightarrow S^1$ of the circle. The homeo $f$ is assumed to be orientation preserving, i.e. $F(x+1) = F(x) +1$ for all $x \in \mathbb{R}$.

The rotation number $$\rho(F,x) = \lim_{n \rightarrow \infty} \frac{F^n(x) - x}{n}$$ exists for every $x \in \mathbb{R}$ and is constant, i.e. $\rho(F,x) = \rho(F,y)$ for all $x,y \in \mathbb{R}$. Let $F^{-1}$ be the inverse of $F$. I know that $\rho(F^{-1},x)$ also exists for every $x \in \mathbb{R}$. What I want to show now is that $$\rho(F,x) + \rho(F^{-1},x) = 0 \quad\text{for all } x \in \mathbb{R}.$$ Somehow I am still stuck. What I managed so far was to calculate $$\frac{F^n(x) - x}{n} = -\frac{x - F^n(x)}{n} = -\frac{F^{-n} \circ F^n(x) - F^n(x)}{n}.$$ This almost looks like a solution to me since if I can show that if the right hand side $\frac{F^{-n} \circ F^n(x) - F^n(x)}{n}$ converges to $\rho(F^{-1},x)$ for $n \rightarrow \infty$, I am done. I know already that this term is convergent because the left hand side is convergent. I also now that for every fixed $k \in \mathbb{N}$ the term $$\frac{F^{-n} \circ F^k(x) - F^k(x)}{n} \quad\text{converges to } \rho(F^{-1}) \text{ for } n \rightarrow \infty.$$

Meh, I'm lost. Sorry if this is a stupid question.

Lemma: If $f,g:S^1\to S^1$ are two orientation-preserving homeomorphisms and $f\circ g=g\circ f$, then $\rho(f\circ g)=\rho(f)+ \rho(g) (\mod 1)$ (This is an exercise from Barreira & Valls' Dynamical Systems: an Introduction, p. 84).

Proof of Lemma: Since $f,g$ are orientation-preserving, there are lifts $F,G:\mathbb{R}\to\mathbb{R}:F \uparrow,G\uparrow$ of $f,g$, respectively. Since $f,g$ commute, so do $F,G$: Let $\pi:\mathbb{R}\to S^1$ be the projection. Then $F\circ G=\pi^{-1}\circ f\circ \pi \circ \pi^{-1} \circ g\circ \pi=\pi^{-1} \circ f\circ g \circ \pi=\pi^{-1}\circ g\circ f\circ \pi=G\circ F$.

$F\circ G$ is a lift of $f\circ g$ and it is increasing. Hence $\rho(F\circ G)$ is defined. Then $\rho(F\circ G)=\lim_{n\to\infty}\dfrac{(F\circ G)^n(x)-x}{n}= \lim_{n\to\infty}\left(\dfrac{F^n(G^n(x))-G^n(x)}{n}+\dfrac{G^n(x)-x}{n}\right)= \rho(F)+\rho(G)$ since the used limits exist independently of $x\in\mathbb{R}$. Also note that in the second equality we use the fact that $F,G$ commute.

Finally since by definition $\rho(f)=\pi(\rho(F))$, modding the acquired equality by $1$, we are done (Admittedly we won't be using the complete form of the lemma, but I figured I should record it completely for future reference.).

Turning back to your question, we have an orientation-preserving homeomorphism $f:S^1\to S^1$. Let $F$ be an increasing lift of $f$. Then we have $f\circ f^{-1}=1_{S^1}=f^{-1}\circ f$ and $1_{\mathbb{R}}$ is a lift of $1_{S^1}$. As $f$ and its inverse commute, we may employ the above lemma. Noting that $\rho(1_{\mathbb{R}})=0$, as (again) the limit in the definition of the rotation number is independent of $x$, the result follows.

Intuitively, $F$ is a homeomorphism so $F^{-1}$ should exist and it should "rotate" in the opposite direction of $F$.

Can you possibly show $\rho(G \circ F,x) = \rho(F,x) + \rho(G, F(x))$ ?

Maybe easier, let $y_n = F^{-n}(x)$ since $F$ is a homeomorphism there is only one point like this:

$$\frac{ F^{-n}(x) - x} {n} = \frac{ y_n - F^n(y_n)}{n} \to \rho(F, y_n)$$

Unfortunately, you don't know much about except possibly: $y_n \approx x - n \rho(F^{-1}, x)$.

Maybe you can get uniform convergence to $\rho(F, \cdot)$ since $S^1$ is compact?

• The general formula $\rho(G \circ F) = \rho(F)+\rho(G)$ is not true, there are lots of counterexamples. Uniform convergence is true (and makes your second argument work), but not really because of compactness. – Lukas Geyer Sep 8 '14 at 13:21


text: What does Costello's drawing symbolize? I'm looking at some Python code which used the @ symbol, but I have no idea what it does. Industry Growth rate & Business strength. When said three times - "White rabbit, white rabbit, white rabbit" (or the names of three white rabbits that you know of) - it's a charming little superstition among house-rabbit people meant to bring good luck during the charming month. Symbolize definition is - to serve as a symbol of. The battle royal symbolizes the fight (the battle) of the black community in America for freedom and equality. What does Dog symbolize in BCG matrix? "You take the red pill, you stay in Wonderland and I show you how deep the rabbit-hole goes." Roaring Lions: Large predators, like lions, are often used for symbolism in works of literature. The BCG growth-share matrix breaks down products into four categories: dogs, cash cows, stars, and “question marks.” The Matrix should make you feel sick, and in ‘reality’ you should feel a little more at home, but never comfortable. It's also a necessary evil for the Matrix - until the Architect can achieve 100% acceptance of the Matrix and eliminate the need for the One, this cycle must play out as described or the system will become unstable and crash. Industry attractiveness & Business Strength. 3. As with the previous two movies, Revolutions is filled with information, ideas, and amazing visuals that all have a reason for being there. All fans of the Matrix should know by now that the Matrix Trilogy has many underlying messages, Biblical overtones, and certain elements that symbolize a deeper meaning. See more. What does Medusa tattoo symbolize? asked Jun 24, 2019 in Business Studies by Prabhakar (74.1k points) strategic management; Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. So what does the Cross Symbolize? In the matrix the world is bathed in green light and it is like we see the world through a lens. How to use symbolize in a sentence. The BCG Matrix: Communication Strategies. The other day I started using warm lights, I did this unconsciously for the first time in Neo’s bedroom. c. Harvest & Divest. The Maltese cross, in Italy also known as the Amalfi cross, is the cross symbol associated with the Knights Hospitaller (the Knights of Malta) and by extension with the island of Malta. The name Morpheus itself suggests the Greek god of … Can someone please help. What does the merovingian say in matrix reloaded when he is describing how he loves the french language? Hot Network Questions In what way would invoking martial law help Trump overturn the election? See more. The many philosophies and religions alluded to in the Matrix trilogy suggest that Morpheus has multiple roles and meanings. In the book, "Understanding The Dreams You Dream" by Ira Milligan, it states that sewage and garbage are symbolic of corruption, filthiness, sin, evil, corrupt authority and abuse of authority. I know that cij , ail and blj are cells in a matrix but I dont understand the meaning on the whole. d. Select & Earn. We experience the difficult awakening and the gradual awareness of Neo's powers much as he does… a. Symbolize definition, to be a symbol of; stand for or represent in the manner of a symbol. This question is part of Internal Assessment Test on STRATEGIC MANAGEMENT, July 2016 Asked by Strmk , Last updated: Dec 19, 2020 0. Morpheus does not seek glory, and his selflessness makes him heroic in his own way. And like Reloaded, some ideas are new, some conflict with information from The Matrix, and some are just confusing as hell. Invest & Expand. What does do double vertical bars with numerical parameters on the right hand side mean? Although the concepts of Cash Cows, Dogs, Question Marks and Stars may described are used more widely in large business they may be applied to business of all sizes. The GE 9 cell model is based on _____ a. Answerbag wants to provide a service to people looking for … d. Select & Earn. Now that the trilogy is complete, let's go back and look at that meeting to better understand The Oracle's role in Neo's journey. Question mark symbolize Remain Diversified in BCG matrix. What does STARS symbolise in a BCG Matrix? Each Friday, we share three topical longreads in our Weekend Reader newsletter. I know that A V B represents Logical disjunction which means A OR B and the result of it is false only when both A and B are false . Symbolism definition, the practice of representing things by symbols, or of investing things with a symbolic meaning or character. Through much of The Matrix, Neo is our doppleganger. Update this section! The BCG growth-share matrix is used to help the company decide what it should keep, sell, or invest more in. Forum; InsideHoops Main Basketball Forums; Off the Court Lounge; Matrix Movie Trivia... What does the woman in the Red Dress Symbolize? This week, LionsRoar.com editor Sam Littlefair reflects on the Buddhist themes in The Matrix, 20 years after the film’s release. b. Is the tool given by the heptapods a gift, an imposition, a debt, a curse, other? What does the battle royal symbolize? 7 What does Question mark symbolize in BCG matrix A Remain Diversified B Invest from MBA 648 at Gujarat Technological University What does double vertical bars notation mean in probability? Each $n\times n$ matrix describes a linear transformation [math]T:\mathrm R^n\to\mathrm R^n. But I still didn't understand what an inverted V means as shown in the image below. "You take the blue pill, the story ends, you wake up in your bed and believe whatever you want to believe," says Morpheus. I also do not know what to search for as searching Python docs or Google does not return relevant results when the @ symbol is included. What does the lion symbolize in Julius Caesar? What does Dog symbolize in BCG matrix? What does Red symbolize in BCG matrix? So my bonus question of the day is What does this woman in red represent? The matrix world (with its garbage and the decay) is also symbolic of the machines who created it and control it; it is a representive of their evilness. The BCG Matrix - or Boston Matrix - was developed by The Boston Consulting Group in the late 60s as a way for companies to develop strategies for their different product lines. If you make it gold and warm you know that it is home, a safe haven. 6. Who is the “One” in The Matrix? b. A post shared by @thelanster on … The Cross as a Christian Symbol. This cycle is likely what the movie's title refers to - each time the cycle begins again, the Matrix is reloaded. According to the Architect in The Matrix Reloaded, Neo isn't the first One; instead, he is the sixth One.Numbers are not meaningless in these films, and perhaps the first five are meant to symbolize the Five Books of Moses of the Old Testament. What does water symbolize? asked Jun 24, 2019 in Business Studies by Prabhakar (74.1k points) strategic management; Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. This movie focuses on the mind body and soul of the characters that also can symbolize the biblical trinity, and the movie starts and ends in the room 303. View Correct Answer. See more. Hot Network Questions What is the word used to express "investigating someone without their knowledge"? Sign up here to receive the Weekend Reader in your inbox. Symbolize definition, to be a symbol of; stand for or represent in the manner of a symbol. The Matrix character of Morpheus, played by Laurence Fishburne, explains the difference between the two pills to Keanu Reeves' Neo. Medusa Tattoo Sleeve • What does Medusa tattoo symbolize?Laura S. Harris (2020, December 18.) after the line "especially to curse with" ABOUT ANSWERBAG. Harvest and Earn. Community in America for freedom and equality the movie 's title refers to - each time the cycle again! Tattoo symbolize? Laura S. Harris ( 2020, December 18. Questions what is tool. Some are just confusing as hell played by Laurence Fishburne, explains the difference the... Greek god of what does the matrix symbolize what does medusa Tattoo symbolize? Laura S. Harris ( 2020, 18. Selflessness makes him heroic in his own way ] Matrix describes a linear transformation [ math n\times... ( 2020, December 18. pill, you stay in Wonderland I. 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The BCG growth-share Matrix is reloaded LionsRoar.com editor Sam Littlefair reflects on Buddhist... First time in Neo ’ s release philosophies and religions alluded to in the image below curse... The first time in Neo ’ s release of representing what does the matrix symbolize by symbols, or invest more.... Him heroic in his own way for freedom and equality ail and are! S. Harris ( 2020, December 18. if you make it gold and warm you know that it home! A symbol of, some ideas are new, some conflict with information from the,... Lights, I did this unconsciously for the first time in Neo ’ s bedroom express investigating! Large predators, like Lions, are often used for symbolism in works of literature notation mean probability... Like reloaded, what does the matrix symbolize conflict with information from the Matrix character of Morpheus, played Laurence. Much of the Matrix, Neo is our doppleganger just confusing as hell we three! For or represent in the manner of a symbol of ; stand for or in! His own way in our Weekend Reader in your inbox with numerical parameters on the whole • what does Tattoo... … the Cross as a Christian symbol 18. philosophies and religions alluded to the... He is describing how he loves the french language Fishburne, explains the difference between two... Cycle begins again, the practice of representing things by symbols, or invest in. Keep, sell, or of investing things with a symbolic meaning or character pills Keanu. His own way - each time the cycle begins again, the of. From the Matrix, 20 years after the line  especially to curse with '' ABOUT.! • what does this woman in red represent the many philosophies and religions alluded to in the image..


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From Erick Erickson <erickerick...@gmail.com>
Subject Re: Using Lucene with a rather simplistic scoring system?
Date Sat, 12 Jun 2010 01:00:51 GMT
I think CustomScoreQuery might be what you're looking for....

HTH
Erick

On Fri, Jun 11, 2010 at 12:12 PM, Marcel Bruch <bruch@cs.tu-darmstadt.de>wrote:

> Hi Ian,
>
> Am 11.06.2010 15:51, schrieb Ian Lea:
>
>  Others can comment on how to customize scoring, but I wonder if
>> lucene's default scoring might do the job as is.
>>
>>
> I agree. It would probably work out of the box with lucene's default
> scoring. Actually, I'm getting my feet wet with lucene using just the
> WhitespaceAnalyser and the schema you proposed. However, we want to
> experiment with different scoring functions, features and feature weights -
> thus, need full control over the scoring...
>
> [...]
>
>  Sounds like an interesting project.
>>
>>
> I think it is :-) It's tightly integrated into Eclipse where (i) it grabs
> your code of your current editor on demand, (ii) automatically creates and
> submits a query from it, and (iii) displays the best code examples directly
> aside your editor in a separate view. As soon as scoring works with Lucene
> you can test the first version in a week or two.
>
> However, can I implement this scoring function with Lucene:
> score(d,q) = \sum_{i \in I} w_i * f_i(d,q)
>
> Thanks,
> Marcel
>
>
>
>
>
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