text:
# Adele and Hello - Math and a Carbon Footprint Analysis

I mentioned I wasn't going to post about the environment as much anymore, but I can't stop. I honestly started this just to show how fun math can be in looking at the number of views on Adele's Hello on youtube. It ended up with me being curious what the environmental impact was.

So today I looked at the youtube video of this song. For your convenience, here it is so you can listen to it while reading this post. Apparently it's blowing up, with an insane number of views per day.

It has 309,000,000 views and has been up for all of 20 days. Let's use some basic math to figure out how popular this song is!

Let's first see how many seconds there are in a day.

$24 \frac{hr}{day} \cdot 60 \frac{min}{hr} \cdot 60 \frac{sec}{min} = 86,400 \frac{sec}{day}$

And over 20 days:

$86,400 \frac{sec}{day} \cdot 20 days = 1,728,000 seconds$

With 309,000,000 views in that time, we end up with 178 views initiated per second, on average, for 20 days, nonstop.

But, hold up, it takes 6 minutes for this song to complete. So that means, on average, there are:

$6 min \cdot 60 \frac{sec}{min} \cdot 178 \frac{views}{sec} = 64,000$

Let's round that up to 65,000. On average, at any given time, there are 65,000 people watching the Adele Hello video on youtube. There are 15 million views per day. That's just one video on one source. In eight whole day, as many people listen to the song, on just youtube, as watch the superbowl.

If this clip keeps up (which is unlikely), she will surpass Gangam Style, with over 2 billion views, in less than 150 days. And medical stocks will go up, cause I worry that listening to this song too frequently may drive people into depression.

"But Jay," you say, "what about the energy and environment component?!?"

Let's figure out Adele's carbon footprint from youtube! Let's assume that Google's hosting emissions are negligible. Let's just focus on the energy used for a laptop. Let's say that it is 50W (this is conservative).

From a prior post, we know that a 100W lightbulb uses ~1kg of coal per day. So a laptop will use about 0.5kg per day. How much CO2 is this?

This is basic chem! Let's get really basic. Coal is mostly made up of carbon. So when it is burnt, every carbon molecule breaks off and bonds with O2. So we assume that all 0.5kg of carbon becomes CO2. You need to add that weight of the O2. C weighs 12 AMU (atomic mass units, pretty much the mass of a single molecule) and O2 weighs 32 AMU.

So we need to multiply the weight of coal, which is pure carbon, by the proportional increase of mass of CO2, cause each carbon atom gains O2 weight (12+32=44)

$0.5 kg-CO_2 \cdot \frac{44}{12} = 1.85 \frac{kg-CO_2}{day}$ for a 50 watt computer running all the time.

Also this song is 6 minutes, or 1/10th an hour, and an hour is 1/24th a day, so this song takes 1/240th of a day.

$1.85 \frac{kg-CO_2}{day} * \frac{1}{240} = .00771 \frac{kg-CO_2}{view}$

Don't forget, though, that we have 15,000,000 views per day!

$0.00771 \frac{kg-CO_2}{view} \cdot 15,000,000 \frac{views}{day} = 115625 \frac{kg-CO_2}{day}$

Or, you know, ~125 tons of CO2 per day. Just from Youtube and Adele. This is the generation rate of about 2000 Americans.

And why this analysis overestimates Youtube's and Adele's contribution to CO2 emissions

Most people are multi-tasking while listening to Adele (ie those 50 watts they are using are also going towards whatever else they do while listening to Adele, such as reaching for tissues to blot their tears), so you have to take a fraction of this number  🙂

Second, much of the power use in the US is now coming from natural gas, which is more efficient than coal, so you can cut down this number.

Third, tissues take a whole lot of energy to make. I'm betting that the raw number of tissues used while listening to Adele will increase the CO2 footprint.

How many times did you restart the song while reading this blog, BTW?

- Jason Munster

# Why Giant Houses Always Use More Energy

Big houses use more energy to heat and cool, for reasons you might not suspect. Houses lose heat to the outside. Nearly all houses are drafty in some form or another, and they need to be somewhat drafty, as we will soon find out.

When energy prices skyrocketed in the 70s due to price gouging and market manipulation of oil (thanks, OPEC), there was a big movement to make it so houses didn't leak air (and leak their heat energy in the process). The idea is that for every bit of air you heat and then let out into the environment, you have just wasted energy. So the process of sealing houses began.

OPEC oil embargoes of '73 and '79. The prices of energy spiked worldwide.

Some groups bragged that they could build houses that only exchanged 1% of their air per hour with the outside. In other words, it would take 4 full days to lose all the heat or AC energy of a house to the outdoors. Excellent, right?

It was excellent in terms of energy savings. But anyone with a flatulent spouse/significant other can tell you that being stuck in a place that is producing unhealthy fumes is dangerous if you don't vent it. It turns out that a lot of basic human activity, like cooking and heating, produce things that are bad for humans and need to be vented.

Much more importantly for advanced cultures*, cooking (it boils water, yo) and breathing and sweating make the air inside a house humid. Humidity in a house causes mold that can make you ill or, in extreme cases, kill you. One of the most effective ways to remove all this humidity is to let the air exchange with the outside.

So here we have a problem. We need to seal our houses well in order to save energy on heating and cooling, yet we also need to allow loss of all this heated and cooled air so we don't sweat ourselves out and cause bad mold to grow.

And we arrive to the crux of the matter. A good exchange rate is .6, or that 60% of a houses air is exchanges per hour. Sounds like a lot? It kind of is. But it's what is healthy for normal technology (we aren't all going to install CO and CO2 scrubbers and dehumidifiers in our houses). So in 24 hours, we have

$24$ hours $\cdot .6 \frac{exchanges}{hour} = 14.4$ exchanges per day. Of your entire house volume.

So. You have to exchange air in your house. About 15 times per day. Otherwise you might start falling ill. If you have a gigantic house that is 2x larger than you need, then you will use 2x as much energy to keep the place heated and cooled as you need to. So, in short, living in a giant house is a bad thing for energy conservation (take notice, Al Gore**)

Next week we will suspend our assumption that all houses have decent exchange rates, and discuss why this is a huuuuge policy gap.

You don't really need to live in a place like this, do you?

- Jason Munster

*Developing countries still use coal. By 2020 there will be up to an estimated 400,000 deaths per year in China from indoor air pollution associated with burning coal for heat and cooking in poor rural homes (160,000 median estimate). Obviously this is more pressing than mold.

**I was going to rip Al Gore a new one for having had a huge electricity bill just after making An Inconvenient Truth, but it turns out that in 2007, before it was cheaper or easier, he elected to power his home, in TN, with solar and wind power almost exclusively, jacking up the price to a level higher than most Americans pay. So yeah, he did have a much higher electricity bill than the average American, but he only used about 4x the electricity, apparently. Which is still a lot. Except that he and Tipper both also work out of their houses. And now they have solar panels all over it. So it's not that bad. Though it is still huge.

It was  below 0C in August in my first two days here.

The North Slope of Alaska is flat and has lots of lakes and running water

Deadhorse, Prudhoe Bay, AK, is different from most other places. Deadhorse is practically devoid of women, and has zero children. I asked a friend who works here on the oil rigs what dating was like: "The same as working everywhere in Alaska. Get in line and wait your turn."

Deadhorse is a dry town. You can't buy alcohol here. If someone working here is found with alcohol, they are fired or kicked out. The workers here will occasionally buy cool-aid, throw some champagne yeast in, and brew their own awful alcohol that way.

There are no hotels here. Instead there are Man-Camps. There are no restaurants in Deadhorse, and no place to buy real food to cook. You eat at a mancamp. There was an outbreak of the flu here a few weeks back, and an entire camp was quarantined. From then on, every camp requires that you use hand sanitizer upon entrance, and you use disposable plastic gloves on top of that when you get your food. All the hangar facilities here also have hand sanitizer everywhere.

This is not a town, it's a construction site.

There is no pavement. Dirt roads lead everywhere. You get dust and mud on your boots (you don't wear shoes). Upon entering a camp, you rub your shoes on some aggressive mud-and-dust removing blocks. After that you are required to put fabric booties over your boots. You are not allowed to walk around in camps in sandals or barefoot. Everyone has big manly boots with fabric booties over them.

Bears roam freely through town. I wanted to take a picture of one, but I have been assured that a grizzly does not look majestic while rummaging through a dumpster, emerging with an ice cream wrapper stuck to its head.

The only passenger vehicle used here is a 4WD king cab diesel pickup. All trucks have heating blocks in them, because it is so cold in the winter that the oil in the engine would turn to a gel if left unheated. In every single parking lot there are dozens upon dozens of plugs for these trucks to keep warm during the forever night of an Arctic winter.

Every truck needs to be plugged in during the winter, or it won't start again.

Because of the bears, we are required to keep our keys in the trucks and the trucks unlocked. If someone is walking along, they need the safety of a heavy truck to dive into immediately, and be able to drive off.

No one is worried about theft here. We don't get keys to our hotel rooms. They are left unlocked when we leave. There is no black market here, so what would someone do with something they stole? We freely leave most of our possessions out, knowing that no one will touch them.

8/7-Research

Our plane has arrived. We had to make repairs to the cooling of the Laser Pressure Vessel (it is an airplane, so as it flies higher, the ambient air pressure decreases. A pressure vessel holds the instrument at one pressure to keep everything precise) on the methane detection instrument.

I forgot my boots at lab. My advisor is flying out on 8/8. He is putting them in his luggage. That's right, my advisor thinks nothing of carrying my hiking boots across the country for field work. He's pretty cool that way. Would your PhD advisor do that?

8-10

We flew!

8-11

We had some heat control issues with the laser system, and two of our three instrument overheated.

That's all for now. Thanks for reading!

- Jason Munster


text:
putting tikzpicture inside beamer

I am trying to add tikzpicture inside Beamer. However, I cannot use such functions as \pause, enumerate etc with TikZ.

Here is what I got thus far. I would like to change the enumerate environment and \pause after each point.

\documentclass{beamer}
%\usecolortheme{beaver}
\usepackage{textpos}
\usepackage{comment}
\usepackage{tikz}
\usepackage{graphicx}
%\usepackage[a4paper,left=2.0cm, right=2.0cm, top=4.5cm, bottom=2.5cm]{geometry}
\usetikzlibrary{snakes}

\begin{document}

\begin{frame}{Time Line of the Model}

\begin{tikzpicture}
%draw horizontal line
\draw (-2,0) -- (10,0);

%draw vertical lines
\foreach \x in {-1.5,3,8}
\draw (\x cm,3pt) -- (\x cm,-3pt);

%draw nodes
\draw (-1.5,0) node[above=3pt] {$0$};
\draw (3,0) node[above=3pt] {$1$};
\draw (8,0) node[above=3pt] {$2$};

\small
\node[align=left, below] at (0,-.5)%
{
1) The board of directors \\ offers the manager a \\ contract ($\omega_0, \omega_p, \omega_v$).\\\\
2) The short-term $v_S$, \\and the long-term $v_L$  \\ projects are initiated \\ by the manager.\\\\
3) The manager is \\ (partially) compensated \\ a fixed wage of $\omega_0$.\\\\
4) The IPO price of the \\stock  $p_0$ is determined \\by the investors
};

\node[align=left, below] at (4.5,-.5)%
{
1) Speculators observe\\ costly noisy signals $\theta_S$ \\ and $\theta_L)$ of the value of \\ short- and long-term \\ project. \\\\
2) The market for the firm's \\ shares opens for trade;\\ and the resulting market \\ price is $p$.\\\\
3) The manager is \\ (partially) compensated \\ an amount of $\omega_p p$.
};

\node[align=left, below] at (8.5,-.5)%
{
1) The terminal value \\of the firm $v$ is realized.\\\\
2) The manager is \\  (partially) compensated \\an amount of $\omega_v v$.
};
\end{tikzpicture}
\end{frame}
\end{document}

-
Welcome to TeX.SX. A tip: If you indent lines by 4 spaces, then they're marked as a code sample. You can also highlight the code and click the "code" button ({}). –  Claudio Fiandrino Sep 10 '12 at 14:58

I would use the columns environment to split your page; the tikzpicture works well for the line across the top, but there's no need to use it for all of the other parts.

Note that the columns environment takes an optional argument to specify the vertical alignment; I've used [t] for top. The environment allows you to specify each column using

\begin{column}{<width>}
content
\end{column}

which I have done, and changed all of your manual lists to enumerate. You can now add \pause at the end of each \item and get the display you want :) See section 12.7 of the documentation for more details.

\documentclass{beamer}
\usepackage{tikz}

\begin{document}

\begin{frame}{Time Line of the Model}

\begin{tikzpicture}
%draw horizontal line
\draw (-2,0) -- (10,0);

%draw vertical lines
\foreach \x in {-1.5,3,8}
\draw (\x cm,3pt) -- (\x cm,-3pt);

%draw nodes
\draw (-1.5,0) node[above=3pt] {$0$};
\draw (3,0) node[above=3pt] {$1$};
\draw (8,0) node[above=3pt] {$2$};

\end{tikzpicture}
\small
\begin{columns}[t]
\begin{column}{.4\textwidth}
\begin{enumerate}
\item The board of directors offers the manager a  contract ($\omega_0, \omega_p, \omega_v$).
\item The short-term $v_S$, and the long-term $v_L$   projects are initiated  by the manager.
\item The manager is  (partially) compensated  a fixed wage of $\omega_0$.
\item The IPO price of the stock  $p_0$ is determined by the investors
\end{enumerate}
\end{column}%
\begin{column}{.35\textwidth}
\begin{enumerate}
\item  Speculators observe costly noisy signals $\theta_S$  and $\theta_L)$ of the value of  short- and long-term  project.
\item The market for the firm's  shares opens for trade; and the resulting market  price is $p$.
\item The manager is  (partially) compensated  an amount of $\omega_p p$.
\end{enumerate}
\end{column}%
\begin{column}{.25\textwidth}
\begin{enumerate}
\item The terminal value of the firm $v$ is realized.
\item  The manager is   (partially) compensated an amount of $\omega_v v$.
\end{enumerate}
\end{column}
\end{columns}
\end{frame}
\end{document}

-
@ClaudioFiandrino I understand your point, but I think the question is a little ambiguous I would like to change the enumerate environment and \pause after each point. Let's wait for clarification from the OP –  cmhughes Sep 10 '12 at 15:39
I actually deleted my previous comment because, thinking a bit over your answer, it's clear it also provides the basics to add later on overlays in whatever manner the OP wants. –  Claudio Fiandrino Sep 10 '12 at 15:43
Great!! It is awesome!!!! Thanks for the help. –  Elias Sep 12 '12 at 3:05
@Elias you're welcome :) If my answer helped, don't forget to upvote it using the up arrow; if it solved the problem, please consider accepting it using the green check mark :) –  cmhughes Sep 12 '12 at 14:40

format: plain


text:
## Background

Colorectal cancer is one of the most common cancers of developed societies.1 Obesity is one of the risk factors for colorectal cancer.2 Ready access to high-calorie foodstuffs combined with a sedentary lifestyle means that obesity has become a major public health problem in developed countries, further contributing to increased colorectal cancer incidence.

Evidence from observational epidemiological studies is consistent with the premise that increased physical activity (PA) reduces colorectal cancer risk. Furthermore, the magnitude of the effect is sufficient to be meaningful for the individual.3 An umbrella review of 22 anatomical cancer sites concluded that there is strong evidence for a protective association between self-reported recreational PA and colorectal cancer.4 In addition, a recent meta-analysis of 17 cohorts and 21 case–control studies found that occupational activity, recreational activity, transport-related PA and reduced occupational sedentary behaviour were each associated with lower colorectal cancer risk.5 The estimated effects of increased recreational PA and occupational sedentary behaviour for colon cancer risk were 0.80 (95% confidence interval (CI): 0.71, 0.89) and 1.44 (95% CI: 1.28, 1.62), respectively, and for rectal cancer risk were 0.87 (95% CI: 0.75, 1.01) and 1.02 (95% CI: 0.82, 1.28) respectively.5 Despite the strength of such correlative evidence on the effect of PA on colorectal cancer risk, causality cannot be ascribed by observational studies, since the observed association could be due to confounding factors or residual confounding. Furthermore, PA is routinely measured in observational studies as a self-reported activity, and this may be systematically overestimated.6

Given the lack of randomised clinical trials (RCTs) to formally test the effect of a PA intervention on colorectal cancer risk, one approach is to apply Mendelian randomisation (MR) approaches to test whether the association is causal. MR explores the effect of the exposure (PA) on colorectal cancer risk through a genetic instrumental variable.7 Since the instrumental variable is randomly assorted at conception, it can overcome the aforementioned shortcomings such as confounding effects. Common genetic variants shown in genome-wide association studies (GWASs) to be associated with PA can be used as instrumental variables for various measures of PA. One recent MR study supports a causal association between higher PA measured by accelerometer and lower colorectal cancer risk.8 However, potential confounding or mediating effect of body fatness was not taken into account.8

As the most commonly used measure of body fatness, body mass index (BMI) is often considered as a proxy of overall body fat. The body fat percentage measured by bioimpedance is another proxy of overall body fat to compare with the results from BMI. However, for people with the same overall body fat (i.e. BMI and body fat percentage), body fat distribution changes with factors such as sex, age, ethnicity, nutritional status and fitness training level.9 Evidence shows that people with normal BMI but excess trunk fat are at higher risk of metabolic diseases,10 while those with a normal BMI but with excess leg fat are at lower myocardial infarction risk.11 The effect of excess body fat (BMI, body fat percentage, waist circumference and body fat distribution) on colorectal cancer risk has been well described in observational2 and MR studies.12,13

Here, using MR approaches, we have investigated the observed association between PA and colorectal cancer to establish, or refute, causality. Further, we have tested whether the effects are confounded by, or mediated through, measures of body fat. In order to comprehensively assess the influence of body fat, we employed measures of body fat including BMI, body fat percentage, waist circumference and three types of body fat distribution measured by bioimpedance.

## Methods

### Genotype data resources for colorectal cancer case–control genome-wide association analysis

We used genome-wide summary-level genotyping data imputed to a merged reference panel comprising the 1000 Genome Project and UK10K from a meta-analysis of 15 GWAS datasets14 from populations of European ancestry (Supplementary Method and Table S1). Briefly, the colorectal cancer GWAS meta-analysis included the following GWASs: NSCCG, the SCOT study, SOCCS/GS, SOCCS/LBC and UK Biobank GWAS, as well as ten previously published GWASs: UK1, Scotland1, VQ58, CCFR1, CCFR2, COIN, Finnish GWAS, CORSA, DACHS and Croatia. Standard quality-control measures were applied to each GWAS and summary statistic data from 31,197 cases and 61,770 controls were included in the analyses.14 GWAS data from the UK Biobank was excluded from the sensitivity analysis to avoid bias caused by sample overlap between exposure and outcome datasets. All studies were approved by respective ethics/institutional review committees, in accordance with the Declaration of Helsinki. Participants of all the included studies have signed the relevant consent forms.

### Generation of genetic instruments

We implemented genetic instrumental variables for three validated measures of continuous PA: self-reported moderate-to-vigorous PA (MVPA), overall acceleration vector magnitude PA (AMPA) and sedentary time. Instrumental variables for PA were extracted from two GWASs that established associations between common genetic variants (minor allele frequency ≥ 5%) and PA (Table S2): (i) a meta-analysis of GWASs of 337,234 UK Biobank participants on habitual PA;15 (ii) a GWAS of average acceleration vector magnitude on 91,105 UK Biobank participants.16 All summary-level statistics from the GWASs used in the current work were restricted to CEU (northern and western European) populations. Therefore, population stratification is not a potential bias in our study. Data for the MVPA measure were derived from self-reported questionnaires completed as part of the UK Biobank dataset collected between 2006 and 2010.17 AMPA and sedentary time were collected from a subset of 91,105 UK Biobank participants wearing an accelerometer 7 days between 2013 and 2015.18 Details of MVPA, AMPA and sedentary time were described in the Supplementary Methods. Klimentidis et al.15 detected eight single-nucleotide polymorphism (SNPs) for MVPA at P < 5 × 10−9 (Table 1). Doherty et al.16 detected three SNPs at P < 5 × 10−9 and five SNPs at P < 5 × 10−8 for AMPA and six SNPs for the sedentary time at P < 5 × 10−8 (Table 1). The SNP-based heritability estimate was 5% for MVPA, 21% for AMPA and 12.9% for sedentary time.15,16

Several genetic variants associated with MVPA, AMPA and sedentary time were previously found to be also associated with the weight, BMI and arm/body/leg/trunk fat percentage (Table S3). We included BMI, body fat percentage, waist circumference and body fat distribution as indicators of obesity in our analysis. The body fat distribution consists of arm fat ratio (AFR), trunk fat ratio (TFR) and leg fat ratio (LFR) measured by segmental bio-electrical impedance (sBIA). The information about  instrumental variables for measures of body fatness can be found in Supplementary Method, Table 1 and Fig. 1. The threshold of linkage disequilibrium (LD) was set as R2 > 0.2. Due to the limited variance explained by the MVPA SNPs, and the fact that the inverse variance-weighted, MR-Egger and MR-Robust methods can incorporate correlation between variants, we included all the SNPs to generate instrumental variable and added correlation matrix into these analyses (available in R package ‘MendelianRandomization’). We excluded rs149943 from the median-based method because of LD (r2 = 0.35) with rs3094622.

### Two-sample MR

Having determined the effect estimates of SNPs on PA and each measure of body fat from GWASs (Table 1), we tested the effects of these SNPs on colorectal cancer risk using genome-wide data from our previous colorectal cancer GWAS meta-analysis.14 The causal effects and the corresponding standard errors of exposures on colorectal cancer were calculated by using the random-effect inverse variance-weighted method.19 We then evaluated the heterogeneity among the causal effects of each variant (Cochran’s Q statistic). P values < 0.10 were considered indicative of significant heterogeneity.20

### Multivariable MR

We applied multivariable MR21 to elucidate the causal relationship between PA and colorectal cancer while keeping measures of body fatness constant. First, we analysed the beta–beta correlations of SNP effects for each exposure through Pearson’s correlation coefficient analysis, including significant SNPs of all measures of PA and body fatness. Next, we ran multivariable MR for measures of PA that significantly associated with colorectal cancer risk in the two-sample MR by adding measures of body fatness that significantly associated with colorectal cancer risk in the two-sample MR as covariates. In total, eight models were fitted (Fig. 1). All SNPs associated with each trait were included to generate instrumental variables for each model (Fig. 1). The pairwise LD threshold between all of these SNPs was set at R2 > 0.2.

The rationale of the study design is shown in Fig. 1.

### Sensitivity analysis

We applied a variety of sensitivity analyses testing different MR assumptions.22 Specifically, we performed MR-Pleiotropy Residual Sum and Outlier (MR-PRESSO),23 MR-Robust,24 MR-Egger,25 leave-one-out method,26 mode-based estimate27 and the median-based method.28 MR-Robust applies MM-estimation (modified maximum-likelihood estimation) with Tukey’s bisquare function, which efficiently limits the contribution of outliers.24 MR-Egger was applied to explore any potential bias introduced by pleiotropy. In particular, when the intercept of MR-Egger differs from zero (at p < 0.05), then either directional pleiotropy is indicated or the InSIDE assumption is violated.25 We also applied the mode-based estimate, which works well when most estimates of identical individual-instrument causal effects are derived from valid instrumental variables and the weighted median-based method, which allows for 50% of invalid weights.27,28

MR-PRESSO was applied to identify horizontal pleiotropic outliers.23 When both MR-PRESSO and leave-one-out method indicated an outlier, we took the analysis after removing the outlier as our main analysis. Two-sample MR estimates can be biased when samples between exposure and outcome GWASs overlapped.29 To minimise the risk of this bias, we performed sensitivity analysis after excluding UK Biobank cohort participants from colorectal cancer GWAS meta-analysis. In addition, we checked the GWAS Catalogue30,31 (https://www.ebi.ac.uk/gwas/home, accessed on 2 February 2020) and PhenoScanner32,33 (http://www.phenoscanner.medschl.cam.ac.uk/, accessed on 2 February 2020) to determine whether the PA instrumental variables were associated with other traits, consistent with pleiotropic effects. For all MR analyses, the P value threshold was set at 0.05. All statistical analyses were performed on R v3.6.1 with packages ‘MendelianRandomization’ and ‘TwoSampleMR’.26,34

### Power calculation

The non-centrality parameter-based approach was applied to estimate the power of this study.35 The R2 (the variance explained by each genetic instrument) was estimated by the following formula: $$2 \times {\mathrm{EAF}} \times \left( {1 - {\mathrm{EAF}}} \right) \times \beta ^2$$ and F-statistic $$F = R^2 \times (N - 2)/(1 - R^2)$$ was applied to estimate the strength of genetic instrument,36,37 where EAF is effect allele frequency, beta is the effect size of instrumental variables per standard deviation (SD) change of PA and N is the sample size of PA GWAS. The MVPA-, AMPA- and sedentary time-related variants explained ~0.07%, 0.2% and 0.1% of the phenotypic variance, respectively. We fixed the type I error at α < 0.05 and listed the effect estimates that could be detected for each SD increase of the PA time. We required 80% power to detect any effects. Effect sizes that can be detected with the power of 0.8, as well as F-statistics for instrumental variables are presented in Table S4. The F-statistic for all the analyses did not indicate weak instruments (F > 10).

## Results

### Two-sample MR

The MR result indicated a decreased colorectal cancer risk through the effect of MVPA. The odds ratio (OR) for inverse variance-weighted MR was 0.56 for colorectal cancer risk per 1 SD increase of MVPA (95% CI: 0.31, 1.01) (Figs. 2 and S1) and each sensitivity analysis method generated similar effect estimates (Table S5). The intercept of MR-Egger regression test did not identify any horizontal pleiotropy and/or violation of the InSIDE assumption (P = 0.75) and the Q-statistic did not indicate heterogeneity (P = 0.56). After removing the UK Biobank data case–control study (4800 cases and 20,289 controls) from the outcome population, the CI of effect sizes was wider (Table S5).

Implementing the MR-PRESSO and leave-one-out methods highlight rs429358 as an outlier (Fig. S2). This SNP maps to the APOE gene and showed the strongest association with PA. It is also associated with multiple traits, including total cholesterol, low-density lipoprotein cholesterol, triglyceride and Alzheimer’s disease.38,39 This finding was supported by searching from GWAS Catalogue and PhenoScanner (Table S3). Therefore, we generated the instrumental variable by using seven SNPs for MVPA after removing rs429358. The effect sizes of the seven MVPA SNPs with MVPA and with colorectal cancer were presented in Table S6.

Evidence for a causal association was detected between AMPA and colorectal cancer risk by using both five SNPs at P < 5 × 10−8 and three SNPs at P < 5 × 10−9 as genetic instruments (Fig. 2). In particular, the ORs of inverse variance-weighted MR were 0.60 (95% CI: 0.41, 0.88) and 0.54 (95% CI: 0.35, 0.86) for each genetic instrument (Figs. 2 and S1). All sensitivity analyses showed a similar effect size of the association between AMPA and colorectal cancer risk (Table S5). The Q-statistic suggested no heterogeneity (P = 0.34 and 0.35, respectively) and the intercept of MR-Egger suggested no pleiotropy. Removing UK Biobank colorectal cancer GWAS from the outcome populations narrowed the CIs of the effect sizes (Table S5). No association was detected between sedentary time and colorectal cancer risk and no pleiotropy or heterogeneity was indicated (Fig. 2 and Table S5). The effect estimates for each instrumental variable on AMPA and colorectal cancer or sedentary time and colorectal cancer were listed in Table S6.

Two-sample MR on the associations between measures of body fatness and colorectal cancer risk, as well as the effect estimates of each SNP on exposures were listed in Table S710. BMI, body fat percentage, waist circumference and AFR were significantly associated with increased colorectal cancer risk, with OR (95% CI) 1.24 (1.11, 1.39), 1.14 (1.03, 1.25), 1.19 (1.04, 1.36) and 1.50 (1.17, 1.93), respectively (Fig. 2). We did not observe any significant associations between TFR or LFR and colorectal cancer risk.

### Multivariable MR

Instrumental variables for BMI, waist circumference and AFR were highly correlated as their correlation coefficients (r) range from 0.67 to 0.79 (Fig. 3). Body fat percentage was positively correlated with BMI (r = 0.13), waist circumference (r = 0.41), AFR (r = 0.33) and TFR (r = 0.16). LFR and TFR were highly inversely correlated (r = −0.90). MVPA and AMPA were positively correlated with each other (r = 0.16) and negatively correlated with sedentary time (r = −0.23 and −0.42 respectively). The three measures of PA were not correlated with colorectal cancer risk.

We estimated the direct effect of MVPA and AMPA and four different measures of obesity and overweight on colorectal cancer risk by applying the multivariable MR method (Table 2). The direct effect sizes (OR) of MVPA or AMPA on colorectal cancer when BMI was kept constant were 0.56 (95% CI: 0.36, 0.89) and 0.70 (95% CI: 0.52, 0.96), respectively. The direct effect sizes (OR) of BMI on colorectal cancer when MVPA or AMPA was kept constant were 1.24 (95% CI: 1.12, 1.37) and 1.22 (95% CI: 1.09, 1.36), respectively. There was no statistically significant heterogeneity and MR-Egger did not indicate a significant pleiotropy effect. Compared to the effect estimates from two-sample MR, the estimated CIs of the two types of PA from multivariable MR became much wider, while the BMI effects were not affected (Table 2 and Fig. 2). Results from two-step network MR also indicated a partial mediation effect of BMI on the PA-colorectal cancer association (Table S11).

The direct effect sizes (OR [95% CI]) of MVPA or AMPA on colorectal cancer when AFR was kept constant were 0.58 (0.33, 1.02) and 0.64 (0.45, 0.92), respectively. The direct effect sizes of AMPA on colorectal cancer when waist circumference was kept constant were 0.61 (0.39, 0.97), while no association observed for MVPA after adjusting for waist circumference. The associations between MVPA or AMPA and colorectal cancer risk disappeared when keeping body fat percentage constant.

Based on the total effect estimates from two-sample MR and direct effect estimates from multivariable MR, we evaluated the proportion of effects of PA on colorectal cancer risk mediated through measures of body fat. The attenuated direct effects indicated that part of the effects of the two measures of PA on colorectal cancer was mediated through BMI or AFR. AMPA also can affect colorectal cancer risk through waist circumference. For the effects of MVPA and AMPA on colorectal cancer risk, the proportion mediated through BMI was 2% (95% CI: 0, 14) and 32% (95% CI: 12, 46), respectively, while the proportion mediated through AFR was 8% (95% CI: 0, 16) and 14% (95% CI: 0, 35), respectively (Fig. 4). The proportion of effects mediated through waist circumference was 5% (95% CI: 0, 22) for AMPA (Fig. 4).

## Discussion

Colorectal cancer is a common cancer with appreciable morbidity and mortality. Prospective cohort, case–control, and cross-sectional observational studies support an inverse association between PA and colorectal cancer risk.40,41,42 The association between AMPA and colorectal cancer risk has been reported in a previous MR study.8 However, the mechanism(s) through which PA influences colorectal cancer risk is not clear yet, especially the relative role of body weight and distribution in this association. Applying two-sample and multivariable MR approaches, we used genetic variants from large GWAS as instruments to explore whether the apparent beneficial effects of three measures of PA (MVPA, AMPA and sedentary time) on colorectal cancer risk are mediated through body fatness. Our results show that both lower MVPA and AMPA increase colorectal cancer risk, both independently and through body fatness. In line with our findings, a meta-analysis has indicated that PA is associated with colorectal cancer risk in both high and low BMI groups.43

It is important to note that we found that increasing PA causes decreased colorectal cancer risk independent of measures of obesity and body fat distribution. One SD of MVPA is ~4.96 metabolic equivalent task (MET)-h/day. MET is an objectively measured ratio of energy expenditure relative to the mass of a person when performing PA compared to sitting quietly. Although there is no standard method to transform milli-gravities (accelerometer measurement units) to energy expenditure, each SD of AMPA was 8.14 milli-gravities (or 0.08 m/s2), which approximates to 3 MET-h/day.18 For each SD increase in MVPA or AMPA, colorectal cancer risk decreased by 44% and 40%, respectively. These estimates imply that if individuals replace daily sedentary behaviour with 20–90 min of MVPA or with 13–60 min accumulated MVPA, their risk of colorectal cancer will decrease by 40% (Table S12).44 In our study, MVPA estimated a longer time spending on PA compared to the estimation by AMPA to achieve a similar benefit of colorectal cancer risk, which is consistent with the existing evidence that MVPA tends to overestimate time engaged in PA in the general population.45 Part of the decreased effect on colorectal cancer risk (2% [95% CI: 0, 14] and 8% [95% CI: 0, 16] for MVPA and 32% [95% CI: 12, 46] and 14% [95% CI: 0, 35] for AMPA) was mediated through the effects of BMI and AFR, respectively. Since the genetic instrument of MVPA is weaker compare to AMPA, the BMI and AFR mediation effect for MVPA may be underestimated.46

Diverse biological mechanisms have been proposed to explain the observed inverse association between PA and colorectal cancer. These include beneficial effects on bowel transit time,47 immune system reactions,48 metabolisms of bile acid, better insulin sensitivity49 and the reduction of prostaglandin E2 levels in colonic mucosa.50 Evidence from RCTs supports that PA can reduce the bowel transit time and therefore reduce the time of contact between carcinogens and colonic mucosa.47 The decrease of prostaglandin E2 synthesis may also promote intestinal peristalsis and hence reduce transit time.51 Besides, prostaglandin E2 can promote tumour generation directly or through its multifaceted effects on inflammation.52 PA also results in a lower concentration of bile acid, which is an essential mediator of the cholesterol mechanism and the lower bile acid concentration is associated with lower blood triglycerides.53 The effect between PA and colorectal cancer risk could be through these pathways, although none of the genetic variants included as an instrumental variable for PA was located within genes involved in indicators of above-mentioned metabolism pathways. In addition, regular moderate PA may have a benefit on natural cytotoxicity and T-lymphocyte proliferation, on reducing the production of pro-inflammatory cytokines and on increasing the count of T cells, B cells and immunoglobulins.48

### Strengths and limitations

One of the strengths of this study was that we explored both subjective and objective measures of PA (MVPA, AMPA and sedentary time). Previous studies showed that there are discrepancies between MVPA and AMPA.54 Compared to MVPA where recall and reporting bias are problematic, AMPA explains 44–47% variance of energy expenditure.55 MVPA tends to overestimate time engaged in PA in the general population.45 Nevertheless, MVPA is commonly used in epidemiological and observational studies, because it is data that is readily collected and inexpensive. Our results for MVPA and AMPA were consistent, with both supporting a causal effect of PA in reducing cancer risk. The confidence intervals for the effect estimates observed for AMPA were appreciably narrower than for MVPA, suggesting the possibility of recall bias, but the health-promoting effect of actually wearing an accelerometer might also influence our results.56 Nonetheless, our use of a variety of instrumental variable methods provides new insight into the effect of PA on cancer risk. Another main advantage of our study is that we have clarified the association pathways among three measures of PA, different measures of body fatness and colorectal cancer for the first time, including the total effect and direct effect, as well as the proportion of the indirect effect. Several measures of body fatness have been considered in our study, BMI and body fat percentage represent two different measures of total body fat while waist circumference, AFR, TFR and LFR represent body fat in different areas. In the two-sample MR study, BMI, body fat percentage, waist circumference and AFR are associated with colorectal cancer risk, which is consistent with the previous evidence.12,57 Based on the results of multivariable MR, AMPA can affect colorectal cancer risk both through and independent of BMI, waist circumference and AFR, while MVPA can affect colorectal cancer risk both through and independent of BMI and AFR. After adjusting for body fat percentage, associations between AMPA or MVPA and colorectal cancer risk disappeared, which may be due to the high heterogeneity introduced by using hundreds of SNPs as the instrumental variable. Papdimitriou et al. performed an MR on PA and CRC risk and they also considered the effects of BMI in the sensitivity analysis. In the current study, we have expanded previous work by additionally including genetic instruments for measures of body fatness along with PA in multivariable MR.

We acknowledge that the study has several limitations. First, although we derived instrumental variables from the largest available GWAS for PA, the SNPs for self-reported PA explain only 0.07% of the variance in MVPA. However, the calculated F-statistic (F-statistic = 273) reached a widely-accepted threshold level.37 As a result, our analysis of MVPA was underpowered (<0.8). Second, with a 21% SNP heritability for AMPA and 5% for MVPA, the low variance of genetic instruments (AMPA: 0.2%; MVPA: 0.07%) may imply that the current discovered SNPs cannot be considered as powerful proxies for PA. Furthermore, although we applied the most up-to-date MR methods, we cannot completely rule out any potential horizontal pleiotropy until we know all biological functions for each SNP. Third, in two-sample MR analysis, weak instrument bias is in the direction of the null while the partial overlapping data between exposure and outcome from UK Biobank may bias against the null.29 However, the sensitivity analysis by removing UK Biobank participants from the colorectal cancer GWAS broadens the CI for association with MVPA while narrowing the CI for AMPA slightly. The overall results did not change. Fourth, sBIA is not a perfect method to measure body fat distribution. However, the correlation between body fat measured by sBIA and MRI is ~0.858 and there is no available GWAS with enough power on MRI or DEXA measured body fat distribution or body fat percentage available. Fifth, because we do not have access to individual-level data we did not perform stratified analysis by gender or tumour site, even though evidence from a European multinational cohort study showed that PA was associated with proximal colon cancer and distal colon cancer risk but not with rectal cancer risk.59 Hence, we may have underestimated the effect of PA on colonic cancer risk in our dataset by including rectal cancer. Finally, all analyses were performed using instrumental variables and summary level data derived from the GWAS on individuals of European ancestry, which may impact the generalisability of our findings to non-European populations.

## Conclusions

The results of this study establish a causal role of both subjectively and objectively measured PA in colorectal cancer risk, independent of the obesity and body fat distribution. Our results suggest that promoting and facilitating exercise could result in a decrease in colorectal cancer incidence, regardless of individuals’ weight or body fat distribution.


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# A circle and a pentagon

Geometry Level 3

Pictured here is a $${\color{Blue} \text{regular pentagon}}$$ which has one vertex on a $${\color{Purple} \text{unit circle}}$$, and the $${\color{Green} \text{green line segments}}$$ connect vertices of the pentagon:

What is the length of the $${\color{Orange} \text{orange line segment}}$$?

Note: The circle isn't necessarily tangent to any of the line segments drawn.

×


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# Lnx _ lnx

Browse Model # LNX-S01-A006-10, LNX Series Remote Condensers in the Dehumidifier Corporation of America catalog including Model #,Item Name

The Free Dictionary. https://acronyms. thefreedictionary.com/LNX. Printer Friendly · Dictionary, Encyclopedia and Thesaurus  Tags: LNX-800AG, LNX-800AG-T, LNX800AG, LNX800AGT. Related Products. Antaira LNX-1002G-SFP (-T) 10-Port Unmanaged Gigabit Ethernet. Add to Wish   All Product Data Sheets · About · Blog · Contact · Newsletter Sign-up · Careers · Brands · 800.255.4629 · Home / Hoists / LNX Hoist / LNX 1250 Hoist.

Plus, the LNX is now available in 5 capacities to meet a broader range of work site needs. LNX Hoist Features. Cast Aluminum outer shell provides strength and durability; Sealed aluminum shell keeps moisture and debris out of electrical and mechanical housings Jan 06, 2009 · Ln(x) = 7 means that e^7=x where e is Euler's number (about 2.718). Normally, you have to use a calculator to solve it. Most scientific calculators have an e^x button. Antaira Technologies LNX-500AG 5-Port Gigabit Industrial Unmanaged Ethernet Switch supports 10/100/1000 Mbps Auto-Negotiation and has a wide range DC power input design (12 – 48 VDC).

## Since x is a variable , your question is incorrect because $ln(ln(x))$ represents a curve not a particular value . $ln(ln(x))$ is basically the natural log of the natural log of x . Example: if $x=100$, then [math]

Feb 06, 2016 · #color(brown)("Total rewrite as changed my mind about pressentation.")# #color(blue)("Preamble:")# Consider the generic case of #" "log_10(a)=b#. Another way of Solve your math problems using our free math solver with step-by-step solutions.

### Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history

Websitesi. Alarm bildirimi olarak; Bu özellikten  Jan 31, 2020 After their Grande Finale at the last LnX on sunday jan 26th, Herrhausen & Treindl were so kind as to let us pass this special edit on to you! Linuxcare was founded in San Francisco in 1998 by Dave Sifry, Arthur Tyde and Dave LaDuke. The company's initial goal was to be "the 800 number for Linux"  sans utiliser ce théorème je dirais que (lnx)² = lnx.lnx donc en dérivant on trouve \ frac{2lnx}{x} sinon en l'utilisant tu as fait une erreur puisque je  We would love to receive additional details or corrections to our description of LNX (a type of LNX file). Your Input: Contact Email (optional):. Submit.

Micropayments. Micropayments with comparatively low price.

It has a powerful set of tools for creating music, all of which can be networked. Co-location gigs or real-time collaborations don't have to be in the same room. This calculus video tutorial explains how to find the integral of lnx/x using the u-substitution integration technique. Subscribe:https://www.youtube.com/ch Sep 23, 2017 LNX Platform.

Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history The notations ln x and log e x both refer unambiguously to the natural logarithm of x, and log x without an explicit base may also refer to the natural logarithm. This usage is common in mathematics, along with some scientific contexts as well as in many programming languages. $\ln^2 x$ ought to mean $\ln\ln x$ but some people write $\ln^2 x$ when they mean $(\ln x)^2$.

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Animation | Climb. Rthro Fall. Animation | … Solve your math problems using our free math solver with step-by-step solutions. Our math solver supports basic math, pre-algebra, algebra, trigonometry, calculus and more. About LNX; Empty Legs; Destination; Blog; Contact; Charter a private jet now. We fly you back home. Private jet charter at your service Fly private for leisure or business.

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## Invariant forms of a binary quintic

Does anyone have a reference for the invariant binary forms of a quintic? That is, what are the $SL_2(C)$ invariant polynomial functions on the space of binary quintics.

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Complementing Noam's answer: for $d\leq 6$ the algebras of $SL_2$-invariants of degree $d$ forms in 2 variables can be found in 10.2 of Dolgachev's Lectures on invariant theory, as well as a reference in English for $d=5$. – algori Oct 27 2011 at 10:41

According to this page there are independent invariants of degree 4, 8, and 12, plus a degree-18 invariant whose square is a polynomial in the first three. This is attributed to Gordan (1868):

P. Gordan, Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Funktion mit numerischen Coeffizienten einer endlichen Anzahl solcher Formen ist, Journ. f. Math. 69 (1868), 323–354.

That page also describes the invariants, and even the covariants, for several other degrees, and also variations such as multiple forms.

[All I did was Google it; the present MO question already turned up in the first page of search results only a few hours after it was posted.]

-
 The complete list of invariants was discovered by Hermite in "Sur la theorie des fonctions homogenes a deux indeterminees" Cambridge and Dublin Mathematical Journal, vol 9, pp. 172-217, 1854. Gordan discovered the larger list of covariants. Perhaps the quickest justification of the list of invariants is using the Sylvester canonical form ax^5+by^5-c^(x+y)^5 and writing the invariants in terms of a, b, c. It is done for instance in Salmon's higher algebra book. – Abdelmalek Abdesselam Feb 2 2012 at 14:31


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# Combinatorics Question

• Nov 16th 2009, 05:24 PM
fholik
Combinatorics Question
In how many ways can I package 6 colors of posterboard in packages of 5 posterboards each ?
• Nov 16th 2009, 07:32 PM
Soroban
Hello, fholik!

Quote:

In how many ways can I package 6 colors of posterboard in packages of 5 posterboards each?

Assuming that the order of the colors in a package does not matter,
. . we are choosing 5 colors from the available 6 colors.

There are: . $_6C_5 \:=\:{6\choose6} \:=\:6$ ways.

• Nov 24th 2009, 12:34 PM
fholik
Combinatorics Question
I misstated the original problem. If you have an endless supply of posterboard in 6 different colors, in how many ways can you package 5 posterboards ?
• Nov 24th 2009, 12:48 PM
Plato
Quote:

Originally Posted by fholik
I misstated the original problem. If you have an endless supply of posterboard in 6 different colors, in how many ways can you package 5 posterboards ?

If the first answer is not what you meant, then I assume that a package could contain say three yellow, one green and one red.
If that is the case this is a multi-selection problem.
We can select $K$ items from $N$ different kinds in $\binom{K+N-1}{K}$ ways.
• Nov 24th 2009, 01:46 PM
fholik
Combinatorics Question
That is the solution I was looking for. Thank you. fholik


text:
SAM Algorithm

The data is , i = 1, 2, ... p genes, j = 1, 2, ... n samples, and response data , j = 1,2, ... n ( may be a vector).

Generic SAM procedure :

• Compute a statistic : ; i = 1, 2, ... p
• is a score.
• is a standard deviation.
• is a fudge factor.

Details of and for Two Class, Unpaired Data response type :

• .
• .
• ; k = 1,2. = 1 or 2.
• is the number of observations in .

Computation of :

1. Compute the 100 quantiles of the values, denoted by .
2. For (0, .01, .02, ... 1.0)
1. Let , where be the percentile of the values.
2. Compute , j = 1, 2, ... 100, where _mad_ is the median absolute deviation from the median, multiplied with 1.4826.
3. Compute cv() = .
3. Choose = argmin[cv()].
4. Finally compute = . is henceforth fixed at the value .
Topic revision: r12 - 06 Feb 2012, WillGray

• Biostatistics Webs

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## Files in this item

FilesDescriptionFormat

application/pdf

5425.pdf (17kB)
(no description provided)PDF

## Description

 Title: Cold Ion Spectroscopy Of B12 Derivatives: Stretching The Limits Of Theoretical Interpretation Author(s): Chen, Peter Contributor(s): Miloglyadova, Larisa; Tsybizova, Alexandra Subject(s): Non-covalent interactions Abstract: Aryl-corrins is a class of B$_{12}$ derivatives that embodies “antivitamin” properties and represents a biologically important class of compounds that hence demands investigation. In our previous study, we experimentally determined bond strength in a series of aryl-corrins by the threshold CID experiments and compared these BDE’s with those calculated with DFT.[1] We found that the computed BDEs were very much dependent on the chosen structure. Given that aryl-cobinamides have many side chains that increase their conformational space, and that the question of a specific structure, measured in the gas phase was important for further evaluation of our T-CID experiment, we proceeded to analyze structural properties of arylcobinamides using CIVP spectroscopy. We explored BOMD simulations with the semi-empirical GFN-xTB methods[2-3] for interpretation of experimental spectra in these conformationally flexible systems. \newline References: \newline [1] A. Tsybizova, C. Brenig, C. Kieninger, B. Kräutler, and P. Chen, Chem. Eur. J., in press. (DOI: 10.1002/chem.202004589) \newline [2] S. Grimme, C. Bannwarth, and P. Shushkov, J. Chem. Theory Comput. 13, 1989 (2017). \newline [3] C. Bannwarth, S. Ehlert, and S. Grimme, J. Chem. Theory Comput. 15, 1652 (2019). Issue Date: 2021-06-24 Publisher: International Symposium on Molecular Spectroscopy Genre: Conference Paper / Presentation Type: Text Language: English URI: http://hdl.handle.net/2142/111236 Date Available in IDEALS: 2021-09-24


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# filter¶

The function filter(function, list) offers a convenient way to filter out all the elements of an iterable, for which the function returns True.

The function filter(function(),l) needs a function as its first argument. The function needs to return a Boolean value (either True or False). This function will be applied to every element of the iterable. Only if the function returns True will the element of the iterable be included in the result.

Lets see some examples:

In [4]:
#First let's make a function
def even_check(num):
if num%2 ==0:
return True

Now let's filter a list of numbers. Note: putting the function into filter without any parenthesis might feel strange, but keep in mind that functions are objects as well.

In [3]:
lst =range(20)

filter(even_check,lst)

Out[3]:
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]

filter() is more commonly used with lambda functions, this because we usually use filter for a quick job where we don't want to write an entire function. Lets repeat the example above using a lambda expression:

In [5]:
filter(lambda x: x%2==0,lst)

Out[5]:
[0, 2, 4, 6, 8, 10, 12, 14, 16, 18]

Great! You should now have a solid understanding of filter() and how to apply it to your code!


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# List numbers within given ranges [duplicate]

## Goal:

Your goal is to print out numbers within a given range (Also including given values).

## Input Format:

The input will look like: (int-int, int-int, ...)

... meaning the user can add as many int-int as they'd like.

## Other Info:

• You will only have to handle non-negative integer numbers.
• The first integer in int-int will always be smaller, or the same, then the next integer in int-int
• The input will always have at least two int-int
• Each int-int will followed by a comma and a space, except for the last one.
• This is meaning the shortest answer wins!

## Examples / Test Cases:

• Example 1:
• Input: (10-13, 11-15, 0-3)
• Output: 10 11 12 13 11 12 13 14 15 0 1 2 3

• Example 2:

• Input: (1-5, 1-15)
• Output: 1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

• Example 3:

• Input: (1-2, 1-3, 1-3, 1-2)
• Output: 1 2 1 2 3 1 2 3 1 2

• Example 4:

• Input: (4-6, 0-1, 9-9)
• Output: 4 5 6 0 1 9
• Will the input have all the ( and the ,  or can we take an array of n-n for the input? – Downgoat Nov 25 '15 at 6:33
• @Downgoat It will have all the other (, , , ) too. – Alien G Nov 25 '15 at 6:34
• Will the first integer always be less than the second or can it be reversed and still be valid? e.g. 11-10 would give 11 10 I'd recommend adding a test case for that and for when the integers are the same (e.g. 10-10 -- though I'd imagine this would give 10) – cole Nov 25 '15 at 6:43
• @Cole Great question(s), yes the first integer will always be smaller and in the case of the numbers being the same it should just print that number (8-8 => 8). – Alien G Nov 25 '15 at 6:49
• Unfortunate that this was closed. It isn't a complete duplicate, as the other one requires comma-separated output. I have a cute answer (valid?) in ngn/apl: Instead of a program/function to do the work, I redefine minus to give the desired output: -←{(⊃~⍨/⍳¨⍺,1+↑⍵),1↓⍵}. Now (10-13, 11-15, 0-3) evaluates to 10 11 12 13 11 12 13 14 15 0 1 2 3! – Adám Nov 25 '15 at 17:34

# CJam, 19

qS/{'.,Ser~),>~}%S*

Martin Büttner killed 2 bytes again, thanks :)
Try it online

Explanation:

q       read input
S/      split by space
{…}%    transform each string (containing a pair of numbers and some other chars)
'.,   enumerate all characters smaller than '.'
Ser   replace them all with a space in the string
~     evaluate the resulting string, pushing the 2 numbers on the stack
let's call them A and B
),    create an array [0 1 … B]
>     slice it from A: [A … B]
~     dump it onto the stack
S*      join the resulting array with spaces

## Minkolang 0.13, 12 bytes

nd1+?.n~Lr$N  Try it here. ### Explanation n Take number from input (-1 if input is devoid of numbers) d1+ Duplicate and add 1 ?. If this is 0, stop. n~ Get number from input and negate it (because of the '-') L Pops b,a and pushes a,a+1,a+2,...,b-1,b r Reverses list$N     Outputs whole stack as numbers

# Julia, 75 bytes

s->print((j=join)(map(i->j(i," "),eval(parse(replace(s,"-",":"))))," "))

This is an anonymous function that accepts a string and prints to STDOUT. To call it, give it a name, e.g. f=s->....

Ungolfed:

function f(s::AbstractString)
# Replace dashes with colons in the input
c = replace(s, "-", ":")

# Parse as a tuple of UnitRange objects
r = eval(parse(c))

# Join each range with spaces
m = map(i -> join(i, " "), r)

# Join into one string and print
print(join(m, " "))
end

## Javascript (ES 2015), 86 83 bytes

f=r=>{z="";r.replace(/(?:(\d+).(\d+))/g,(m,n,x)=>{do{z+=n+" "}while(n++<x)});alert(z)}

Edit 1: saved 3 bytes changing the body of the inner function into for(;n<=x;)z+=n+++" " (and I feel good when the horizontal scrollbar of my snippet at last disappears :) )

f=r=>{z="";r.replace(/(?:(\d+).(\d+))/g,(m,n,x)=>{for(;n<=x;)z+=n+++" "});alert(z)}

Ungolfed

f=r=>{                                       // define a function with 'r' arg

z="";                                      // initialize a variable to collect
// the output

r.replace(/(?:(\d+).(\d+))/g,(m,n,x)=>{    // for each range
// where (m=range, n=min, x=max)
for(; n<=x; ) {                          // until min <= max
z += n++ + " "                         // add to z the value and a space
}
});

}

f("(10-13, 11-15, 0-3, 9-9)");  // Output: 10 11 12 13 11 12 13 14 15 0 1 2 3 9

## Python 2, 125 bytes

import re
print' '.join(map(lambda x:' '.join([str(y)for y in x]),eval(re.sub('(\d+)-(\d+)',r'range(\1,\2+1)',raw_input()))))

Try it online

I'm certain this could be golfed more...

## CJam, 41 bytes (too long!)

Definitely can be golfed, but its a start. Test it online here.

l'(/e_')/e_',/{['-' er~]2,.+:,W%:-}%e_' *

l                                           e# Take string input
'(/e_                                      e# Remove '( characters
')/e_                                 e# Remove ') characters
',/                              e# Split on commas
{                   }%        e# Map over array...
['-' er~]                       e# Convert A-B to [A B]
2,.+                   e# Convert [A B] to [A B+1]
:,W%               e# Convert [A B+1] to [[0,1,2, ... B][0, 1, 2, ... A]]
:-             e# Remove element 1 of the previous array from element 2
e_' *   e# Format nicely
e# CJam prints stack after execution

• A few tips: 1. To remove '(', do '(-; even better in this case: 1>; similarly, ')- or W< 2. To convert A-B to [A B], do '-/:~ 3. To convert [A B] to [A … B], do ~),> 4. Instead of e_ after the loop, you can do ~ at the end of the loop 5. Use S instead of '  – aditsu quit because SE is EVIL Nov 25 '15 at 7:36

# TeaScript, 18 + 1 = 19 bytes 20

+1 byte for "Inputs are arrays / lists?" checkbox

Such abuse...

xßer(${ls-}¬))  I've never abused JavaScript's casting more in my life. Try it online ## Ungolfed && Explanation My favorite part is JavaScript arrays when cast to a string have commas auto-inserted. We abuse this so the array elements are inserted as numbers and as separate arguments. So 1-2, would become ['1','2'] but when string casted 1,2 and this can be put directly in a function. xm(#er(${ls-}+1))

// The checkbox makes the input like ['1-2','3-4']
xm(#                )   // Loop through input
e                 // Eval string...
r(        +1)     // range function...
${ls-} // Convert range -> args // See above for more details  ## Python 3, 76 bytes l=eval(input().replace(*'-,')) while l:a,b,*l=l;print(*range(a,b+1),end=' ')  The input is evaluated as a tuple by replacing each - with , (10-13, 11-15, 0-3) --> (10,13, 11,15, 0,3)  Then, the first two elements of the list are repeatedly lopped off and the numbers between them are printed. Saved 2 bytes thanks to xsot. • '-',',' can be shortened to *'-,'. – xsot Nov 25 '15 at 8:19 • @xsot Thanks. I wonder if there's something to be done with replace('-',',1+') to get the ranges right... – xnor Nov 25 '15 at 8:24 # Mathematica, 80 bytes Join@@BlockMap[Range@@FromDigits/@#&,StringCases[#,DigitCharacter..],2]~Row~" "&  # O, 75 74 bytes I'(-')-", "/]{[[n'-/]{n#}d[$,.$.0=0={.@=0={;.@=0=}w;}{;;}?r]]{n{noTo}d}d}d  This makes logical sense to anyone who built the language (i.e. me). The logic behind this is intense and I'm just happy that I got it working. Explanation: I'(-')-", "/] Format inputs into array { Iterate over each input range [[n'-/]{n#}d Split ranges into sets of numbers [$,              Get the second number and get a range from 0 to it
.\$.0=0=          Is the first number 0?
{.@=0={;.@=0=}w;} True: Remove any unneeded numbers
{;;}?             False: Pop some random stuff that shouldn't have been on there to begin with
r]]              Reverse the range and close some array stuff or whatever
{n{noTo}d}d}     Format output
}d

• No idea what any of it does but good job. – Alien G Nov 25 '15 at 7:36
• Maybe would be better to specify the version too. As I see, your answers work only with the new C interpreter, but until you document it I will keep posting answers which work only with the old Java interpreter (like the 99 bottles of beer one). – manatwork Nov 25 '15 at 9:05
• @manatwork That answer actually works on the IDE. The C interpreter has most of the same features as the old one, and most old programs will work unless they do crazy stuff. – phase Nov 25 '15 at 9:08
• Hmm… Indeed. Maybe I tested only an older version of the code. But on thing is sure, was not the only time we didn't understood each other with the online interpreter. – manatwork Nov 25 '15 at 9:54


text:
Question
(a) A narrow beam of light containing yellow (580 nm) and green (550 nm) wavelengths goes from polystyrene to air, striking the surface at a $30.0^\circ$ incident angle. What is the angle between the colors when they emerge? (b) How far would they have to travel to be separated by 1.00 mm?
1. $0.043^\circ$
2. $2.00 \textrm{ m}$
Solution Video


text:
Category:Vector Analysis

This category contains results about Vector Analysis.
Definitions specific to this category can be found in Definitions/Vector Analysis.

Vector analysis is the branch of linear algebra concerned with differentiation and integration of vector spaces, primarily in Euclidean space of $3$-dimensional space.

Subcategories

This category has the following 8 subcategories, out of 8 total.


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# Fungrim entry: 9fbe4f

$\left\{{n + 1 \atop k}\right\} = k \left\{{n \atop k}\right\} + \left\{{n \atop k - 1}\right\}$
Assumptions:$n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 1}$
TeX:
\left\{{n + 1 \atop k}\right\} = k \left\{{n \atop k}\right\} + \left\{{n \atop k - 1}\right\}

n \in \mathbb{Z}_{\ge 0} \;\mathbin{\operatorname{and}}\; k \in \mathbb{Z}_{\ge 1}
Definitions:
Fungrim symbol Notation Short description
StirlingS2$\left\{{n \atop k}\right\}$ Stirling number of the second kind
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
Source code for this entry:
Entry(ID("9fbe4f"),
Assumptions(And(Element(n, ZZGreaterEqual(0)), Element(k, ZZGreaterEqual(1)))))


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dp06-15.pdf (157.7 kB)

# The utility function under Prospect Theory

report
posted on 02.02.2010, 12:17 by Ali al-Nowaihi, Ian Bradley, Sanjit Dhami
Prospect theory is the main behavioral alternative to expected utility. Tversky and Kahnemann (1992) motivate the utility function for gains and losses under prospect theory by using the axiom of preference homogeneity. However, they do not provide the formal proof. We provide the relevant proof. Furthermore, we show that the utility function under preference homogeneity obeys an additional and important restriction that is not noted by Tversky and Kahnemann (1992). This simplifies the use of prospect theory by reducing the number of free parameters by one.

## Publisher

Dept. of Economics, University of Leicester

02/02/2010

## Publisher version

http://www.le.ac.uk/economics/research/discussion/papers2006.html

## Book series

Papers in Economics;06/15

en


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I have forgotten

• https://me.yahoo.com

Trigonometrical Ratios

Describes the derivation of the basic ratios and their relationship one with another
View other versions (3)

Trigonometric Ratios For Any Angle.

A knowledge of the Pythagoras theorem and the properties of Similar triangles, is assumed.

Definition Of The Basic Ratios.

Consider the two right angled triangles shown. As they are equiangular the following relationships exist between the lengths of their sides.

Diagram

Re-arranging:-

These two ratios are clearly independent of the size of the triangle and depend solely upon the size of the angles of a right angled triangle. By definition the value of equalities shown in equation (2) is called .

Using the same analysis two other ratios can be identified. These are

In Conclusion, considering the following right angled triangle:-

Other Identities

For convenience the following identities exist:-

The Relationships Between Sin; Cos And Tan.

By Inspection of equations (3:4:5)it can be seen that

In addition if we put a = 1 . The following diagram can be drawn. The two perpendicular Coordinates are Ox and Oy. The radius OP is of unit length and the angle xOP is measured clockwise from xOP and the coordinates of P are defined as whatever the position of P.

The signs are defined as in ordinary Algebraic graphs.

From the above Graph it can be seen that :-

Dividing through by

Or dividing by gives:-

Special Angles

From the definitions and from an inspection of the graph it is possible to rite down the Ratios for the following angles.

It can also be seen that Sin 45 = Cos 45 and that as the sum of their squares = 1

From the above diagrams it is possible to see that :-

Similarly

Complementary Angles

If angles xOP and xOQ are complementary then Angles xOP and QOy are equal and so the projection of OP onto the x axis is equal to the projection of OQ onto the y-axis.

and similarly

NOTE The prefix co- in the ratios stands for "complementary" and means that the ratio of any angle is equal to the co-ratio of the complementary angle. e.g.

Angles Larger Than 90 Degrees.

From the definitions and from the graph it can be seen that:-

• In the first quadrant, sin is +ve; cos is +ve; tan is +ve.
• In the second quadrant, sin is +ve; cos is -ve; tan is -ve.
• In the third quadrant, sin is -ve; cos is -ve; tan is +ve.
• In the forth quadrant, sin is -ve; cos is +ve; tan is -ve.

There are various methods of remembering the above table. One way is by using the CAST diagram.

Each letter stands for the positive ratios. e.g. In the first quadrant all ratios are positive whilst in the third quadrant only tan is positive.

Since the magnitude of the projections of the unit radius on the axies depend solely on the acute angle which the radius makes with the x-axis, any ratio of any angle is equal numerically to the same ratio of the acute angle which the radius makes with the x-axis. The sign must be found from the Cast circle.

Example

Find the value of sin 210 degrees.

As 210 is in the third quadrant the sin is negative and the acute angle which the radius makes with the x-axis is 30 degrees.

To simplify expressions such as follow the same logic but for simplicity imagine that to be acute. The formula will apply for all values of but the argument is simplified by supposing that it is acute.

Examples

It should be noted that cosec; sec; and cot follow the same rules as sin;cos; and tan respectively since they are their reciprocals.

Example 1

Prove the following identities:-

a)

The Graphs Of The Trigonometrical Ratios

In defining the ratios the following graph was used.

It can be seen that the value of is given by the y-ordinate. Thus by drawing a circle of unit radius, the value of of the sine of any angle can be found. In this way the following graph was drawn. On the left hand circle is measured from Ox in an anticlockwise direction whilst on the right hand graph is measured along the x axis in the normal way.

the values taken by the cosine as the angle increases from 0 to 90 degrees will be the same as those taken by the sine as the angle decreases from 90 degrees to 0. The two graphs are identical in shape and magnitude but displaced by 90 degrees.

To construct the graph of is slightly more complicated. By inspection it can be seen that as:-

The tan of the angle will be infinite whenever the cosine is zero.

To construct the graph, CX is drawn at unit length. The points are markedoff on the line XY and correspond to the various angles chosen for CP. If a point Q is plotted such that its abscissa on ON is equal to the number of degrees in the chosen angle XCP. This process is repeated for each value of the angle XCP.


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# How can I simplify this circuit?

My attempt: I first converted the delta network into a wye network. The 4 ohm and 2.69 ohm resistors are in series, that series equivalent is series with the 0.38 ohm and 0.54 ohm resistors. The 3 ohm and 2 ohm resistors are also in series. Finally the 8 ohm and 6 ohm resistor are parallel. Is there an easier way in which I could reduce the circuit keeping the 7 Ohm resistor? If I were to leave the delta network without performing any simplification on it, how can I go about solving this problem?


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## Jacobson density theorem

Posted: December 19, 2009 in Noncommutative Ring Theory Notes, Primitive Rings
Tags: , , , ,

We showed in the previous section that every dense subring of the ring of linear transformations of a vector space over a division ring is left primitive. Now, we’d like to prove the converse: every left primitive ring is a dense subring of the ring of linear transformations of some vector space over some division ring. We will assume that $R$ is a ring, $M$ is a simple left $R$ module and $D=\text{End}_R(M).$ As usual, $M$ is considered as a right vector space over $D.$

Remark. If $\varphi \in \text{Hom}_R (M^k, M),$ then there exist $d_i \in D$ such that for all $x_i \in M$ we have $\varphi (x_1, \cdots , x_k) = \sum_{i=1}^k x_i d_i.$

Proof. For any $1 \leq i \leq k$ let $\mu_i : M \longrightarrow M^k$ be the $i$-th injection map, i.e. $\mu_i(x)=(0, \cdots , x, \cdots , 0),$ where $x$ is the $i$-th coordinate. Clearly $d_i=\varphi \mu_i \in D.$ Now

$\varphi (x_1, \cdots , x_k)=\sum_{i=1}^k \varphi \mu_i(x_i)=\sum_{i=1}^k x_i d_i.$

Density Theorem. (Chevalley – Jacobson) Let $R$ be a left primitive ring, $M$ a faithful simple left $R$ module and $D=\text{End}_R(M).$ Then $R$ is a dense subring of $S=\text{End}_D(M).$

Proof. We have already showed that $R$ is a subring of $S.$ So we need to prove that for any $D$-linearly independent set $\{x_1, \cdots , x_n \} \subset M$ and any set $\{y_1, \cdots , y_n \} \subset M,$ there exists $f \in R$ such that $f(x_j)=y_j,$ for all $j.$ The proof is by induction over $n$: if $n = 1,$ then, since $M$ is simple and $x_1 \neq 0,$ we have $Rx_1=M$ and thus there exists $f \in R$ such that $fx_1=f(x_1)=y_1.$ Assuming that the result is true for $n-1,$ we will have $R(x_1, \cdots , x_{n-1})=M^{n-1}$  (density condition!). We now prove a claim:

Claim.  There exists $f \in R$ such that $f(x_n) \neq 0$ and $f(x_1) = \cdots = f(x_{n-1})=0.$

Proof of the claim. Suppose to the contrary that the claim is not true. Then $f(x_1)= \cdots f(x_{n-1})=0$ will imply that $f(x_n)=0,$ for any $f \in R.$ Define $\varphi : M^{n-1} \longrightarrow M$ by $\varphi (f(x_1), \cdots , f(x_{n-1}))=f(x_n), \ f \in R.$ See that $\varphi$ is well-defined, that is $\varphi \in \text{Hom}_R (M^{n-1},M).$ Therefore, if we put $f=1_R,$ then by the above remark there must exist $d_i \in D$ such that $x_n = \varphi (x_1, \cdots , x_{n-1}) = \sum_{i=1}^{n-1} x_i d_i,$ which contradicts $D$-linear independence of $x_1, \cdots , x_n. \ \Box$

So, using the above claim,  for any $1 \leq i \leq n,$ we can choose $f _i \in R$ such that $f_i(x_i) \neq 0$ and $f_i (x_j)=0,$ for all $j \neq i.$ Thus, since $M$ is simple, $Rf_i(x_i)=M,$ for all $1 \leq i \leq n.$ Hence there exist $g_i \in R$ such that $g_if_i(x_i)=y_i.$ Let $f = \sum_{i=1}^n g_i f_i.$ Then for any $1 \leq j \leq n$

$f(x_j)=\sum_{i=1}^n g_if_i(x_j)=g_jf_j(x_j)=y_j. \ \Box$

The Structure Theorem For Primitive Rings. Let $R$ be a left primitive ring, $M$ a faithful simple left $R$ module and $D=\text{End}_R(M).$
1) If $\dim_D M = n < \infty,$ then $R \cong M_n(D).$
2) If $\dim_D M = \infty,$ then for any positive integer $n$, there exists a subring $R_n$ of $R$ and an onto ring homomorphism $\varphi : R_n \longrightarrow M_n(D).$

Proof. It follows from the density theorem and remarks 3 and 4 in the previous post. $\Box$


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# Step-by-step Solution

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## Step-by-step explanation

Problem to solve:

$\int\left(xln\left(x^3\right)\right)dx$

Learn how to solve trigonometric integrals problems step by step online.

$\int3x\ln\left(x\right)dx$

Learn how to solve trigonometric integrals problems step by step online. Solve the trigonometric integral int(x*ln(x^3))dx. Using the power rule of logarithms: \log_a(x^n)=n\cdot\log_a(x). The integral of a constant by a function is equal to the constant multiplied by the integral of the function. Use the integration by parts theorem to calculate the integral \int x\ln\left(x\right)dx, using the following formula. First, identify u and calculate du.

$\frac{1}{4}\left(-\frac{3}{2}x^2+3x^2\ln\left(x\right)\right)+C_0$

### Problem Analysis

$\int\left(xln\left(x^3\right)\right)dx$

### Main topic:

Trigonometric integrals

~ 0.35 seconds