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The current version is TeX Live 2010. For beginners, I would recommend a dedicated TeX editor as it makes learning to write in TeX a lot easier! I usually type these commands at the same time, with four lines in between them. Envato Tuts+ tutorials are translated into other languages by our community members—you can be involved too! The editor does not offer out of the box capabilities that one is unlikely to use. Lead discussions. Once again, please feel free to download and play around with the sample document for this tutorial (all the examples listed here are in it) as it's a good way to learn how to use TeX. If you have perl/TK installed, open your terminal and type: sudo … A protip by itseranga about osx, vim, latex, and unix. And the nice thing is: you probably have it already installed on your computer, because it is part of MacTeX … After the installation is complete, you can delete the downloaded MacTeX.pkg file. Subscribe below and weâll send you a weekly email summary of all new Computer Skills tutorials. Introduction). Suppose you saved your document and named the file "mydocument.tex". Lists are relatively easy in TeX and there are three basic types. The commands for these are \subsection{Subsection Title here} and subsubsection{Subsubsection Title here}. Tip: I have uploaded the tutorial document so you can download and play around with it. Read and accept the conditions, then follow the on-screen instructions to install the program. In order to be able to write TeX documents on your Mac, you'll first need to download and install a distribution, which renders your documents. Design like a professional without Photoshop. TeXShop (v 4.58) requires Mac OS X 10.12 or later … Here are the steps to install TeX Live first and then SW without the TeX Live install: It may be necessary to disable any antivirus software. His aim with TeX was to create a system so that documents would look the same across all computers and that the look of the documents wouldn't diminish with time. Adobe Photoshop, Illustrator and InDesign. Once you've installed it (it'll pop up in a dedicated TeX folder in your applications) then run the TeX Live Utility, which will update your TeX distribution to the latest version. If you have any difficulty while installing the software, bring your laptop to the Help Desk in Clapp Library or email helpdesk@wellesley.edu with a description of the problem. There are quite a few TeX distributions out there that work with OS X, but my personal recommendation would be MacTeX, which is specifically written for the Mac and supports pretty much all OS X versions (including Mountain Lion). So, short answer: if all you're doing is add files to ~/texmf you simply do not need to run it. If you only use one set of curly brackets (like with other TeX commands), the entire sentence will be in that particular font size. If you want to write a little abstract for your document (if you are writing a report or academic paper, for example), then type it in after the \maketitle command: Sections are defined by the command \section, with the section title again going in between two curly brackets, for example: \section{Introduction}. Instead, you, like me and like all other weenies, will be working with LaTeX, whoseauthor Leslie Lamport has tamed the beast for the rest of us. Â© 2021 Envato Pty Ltd. So: TeX also includes three default fonts, which can be seen in the example below: Use the command \textsf{ } for text in the Sans Serif font and \texttt{ } for text in the Typewriter font. Are you starting to see that TeX really isn't that difficult? You don't have to, but I find it helps immensely. MacTeX installs the front end TeXShop in /Applications/TeX and most people will access TeX using that front end or a similar one. Prepare to set aside at least an hour of your time to install LaTeX. Double click on this downloaded … Its feature set has been selected to offer fast navigation and responsive editing experience in large text files. Everything you need for your next creative project. This … MacTeX and XEmTeX, is available for all three operating systems (Linux, Mac OS X, Windows). In addition, there are five apps include LaTeXit, BibDesk, Excalibur, Tex Live Utility, and TeXShop in MacTeX, when I installed & tested how to remove MacTeX, that means you need to remove … You can also write TeX in commercial code editors such as Sublime Text with the relevant syntax, but you will need to open any documents up in a TeX editor so that they can be rendered. With TeX, you have to learn the various commands before you can start creating documents. LaTeX documents are plain documents with a .tex extension (see the Creating a document in LaTeXarticle for examples), this plain text file has some markup commands that are meant to format the document but, how do you actually generate the final output?. I think it would be prudent to first give you a brief introduction of TeX and why it's so popular within certain circles. Latexmk is a Perl script which you just have to run once and it does everything else for you … completely automagically. … With MacTeX, you get a couple of free solutions (TeXworks, which I will use for this tutorial, and TeXShop). However, I would recommend a specific TeX editor as you often get a handy list of tools with them (such as auto-complete and error reporting, which flags up any errors in your TeX coding before the document is rendered). Basically, it is a typesetting language that was designed and developed (mostly) by Donald Knuth, a professor at Stanford University, back in the 1970s. TeX Users Group (TUG) and MacTeX TeXnical working group TeXShop is distributed under the GPL public license, GPLv2, and thus free. Don't update … The portable edition allows you to run MiKTeX from a portable storage device: you carry MiKTeX on a portable storage device for use on any Windows computer configuration settings are … If you're using MacTeX 2008, pop open your terminal and type: sudo tlmgr install package-name the package will be installed. Installed Latex distribution Programs Like 1. If you're following this example, then type: All your TeX coding should lie in between these two commands. If you run into an issue and want to debug that, it will be a complex task to track which Livy session belongs to which user. You will find the installed programs under. MacTeX-2020 requires macOS 10.13, 10.14, or 10.15 High Sierra, Mojave, or … A TEX file might instead be a Texture file. 1.2 Install LaTeX (TinyTeX) for PDF reports If you would like to create PDF documents from R Markdown, you will need to have a LaTeX distribution installed Actually, we … … Hopefully, this tutorial has provided you with enough information to start creating simple documents in TeX that look good. This wikiHow teaches you how to install a TeX distribution package with all the LaTeX components, using Windows, Mac or Linux. Yes, it takes a bit of extra time to learn but the main reason to use TeX is that you get standardized and professional-looking documents which can be used in a wide variety of scenarios. TeX supports text in bold and in italics through the \textbf and \textit commands. TLDR: This tutorial teaches you how to install LaTeX, R and R Markdown on Windows 10. The coding for the image above is as follows: You can mix and match fonts within the same document without any problems â just make sure you watch out for the correct placement of the curly brackets (some TeX editors will do this for you automatically). Once you've installed everything, then it's time to create your first TeX document! TeX will not automatically generate the numbering and each label must lie within the [ ] as in the example below: When rendered, this is how the three lists turn out: And that's it for your first TeX document! What scares people away from TeX is its learning curve. In our particular document class, article, sections are numbered (so the Introduction section would automatically be rendered as 1. MacTeX comes in two flavors from which you will need to choose: Full MacTeX : This version contains a complete LaTeX installation, but at the cost of being rather large … Although it can be used to edit any plain text file it shines when working with source code. Onlythe most seasoned computer veterans with a death wish would descend to that level. Miktex for Windows 2. You won't be interacting with TeX directly although everything boils down to it. A simple guide to LaTeX - Step by Step Learn about LaTeX in short lessons with full code examples. Can I avoid installing as root Sure you can. 8. You can also create subsections (1.1, 1.2 and so on) and subsubsections (1.1.1, 1.1.2 and so on). Never miss out on learning about the next big thing. You should run regularly the update manager in admin + user mode to check for new packages. This sets the layout and formatting of your document. This may sound a bit scary, but TeX is a lot simpler to learn than other programming languages and once you've used it for a while you'll soon pick it up really easy. Given the sheer extent of TeX, we can't cover everything in this tutorial, but in later articles we'll have a look at more advanced features, such as mathematical formulas and working with images (which I purposely haven't touched upon in this tutorial as it is quite complicated). You should also be on campus or using a high-speed internet connection, since you will have to download a large file. It depends on the type of document you want to generate. Share ideas. Looking for something to help kick start your next project? Note that you need a reasonably recent version of TeX … What scares people away from TeX is its learning curve. Remember to encase the text you would like in either bold or italics between the two curly brackets. There are loads of different classes to choose from, but for a standard document the article class is a good one to choose. Design templates, stock videos, photos & audio, and much more. If you are using cmd , type chcp … MacTeX installs a file called READ-ME-FIRST in … This guide provides an overview of how to get started with LaTeX, as well as … … Mactex for OSX Please note that there is a known issue with MacOSX El Capitan (10.11.x):LaTeX tools are now located in /Library/TeX/texbin instead of /usr/texbin.https://tug.org/mactex/elcapitan.html The MacTeX-2020 Distribution The current distribution is MacTeX-2020 This distribution requires Mac OS 10.13, High Sierra, or higher and runs on Intel and Arm processors. Host meetups. These fonts allow the author greater … This issimila… You can download MacTeX directly here (direct download link â the file is approximately 2.1 GB in size) or via the Torrent network. Trademarks and brands are the property of their respective owners. If you’re new to TeX and LaTeX or just want an easy installation, geta full TeX distribution. Type: So far, your coding should look like the following: Tip: Syntax colouring can make it a lot easier to spot individual TeX commands and functions. Here, we type: Of course, you have to tell TeX where your document ends and this is done by the command \end{document}. If anyone has had to write a long formula in, say, the Equation Editor of Microsoft Word, then they will know how fiddly it is. If you use the Windows 8 start screen instead, simply type cmd or powershell, and then run either the “Command Prompt” or “Windows Powershell” application. Now that we've defined the document class, it's time to start the document. It also guides you through creating your first R Markdown file and shows how to … Before clicking on the "update" button, use your brain: updates do change the system and so can break it. TeX is used heavily in the academic and scientific world, mostly due to the standardized layout and look of documents (though this can be changed), as well as the fact that TeX renders mathematical formulas extremely well (one of its fortes). For this tutorial, I'll create a couple of sample sections (the files above both contain some place filler text as well within each section) so you can see how they work: If you're not using sections, then you can simply start writing your main body text after the \maketitle command. When you render this (by clicking on the little play button), then it should look like this: If the document fails to render, then check the coding against my example. Before we get started with our first TeX document, there are a couple of things you need to install on your Mac first. Visual Studio Codeis a general purpose text editor. Most TeX editors have it on as default â if you're using TexWorks, then click on Format > Syntax Coloring > LaTeX to activate it! The TeX Users Group (TUG) has a list of notable distributionsthat are entirely, or least primarily, free software. Click here to download the raw document (in .tex format) or here for a PDF example. Just like HTML, you can write TeX is virtually any text editor or word processing package. Collaborate. All text for the individual sections goes straight underneath the section commands and unlike the abstract, you don't have to start every single one with the \begin and \end commands. LaTeX is a typesetting program useful for mathematical and scientific writings or publications. Installing mikitex and winedt and using it for writing mathematics and other thesis. These are images that some video games use to store the texture of objects so that they appear differently than other 2D or 3D objects. This may sound a bit scary, but TeX is a lot simpler to learn than other programming languages and once you've used it for a while you'll soon pick it up really easy. Unlike other programming languages, TeX commands usually are in plain English, so often it's clear what the command is meant to accomplish. This tutorial will show you the basics of TeX â how it works and how you can create a nice, simple document with very little effort. If you are behind a firewall, this may … It's not as simple as just opening up a word processor such as Pages or Microsoft Word and tapping away. Every TeX document starts by defining the document class. These tutorials, … TeX also supports different font sizes through a variety of commands. There are also other options out there, such as TeXstudio and the commercial Texpad offering, which I reviewed last year on Mac AppStorm. What is LaTeX LaTex is a document preparation system for high-quality typesetting It can also introduce as a document … Now that we've defined the document class and where the document starts and ends, we need to give our document a title and author. An itemized list uses simply bullet points for your items: An enumerated list uses the standard sequential numbering for your individual list items: A descriptive list uses specific labels for each item of your list (for example, a, b and so on). They often belong to one of categories: integration with third-party tools, linters and … Next steps Overview: Apache Spark on Azure … TeX is a wonderful and very versatile typesetting system that can be used to write almost anything â from reports to letters and even more complicated documents such as books and posters. Make sure you insert an empty line between paragraphs so that TeX knows when to start a new paragraph. LaTeX is a plain-text document … How to get MacTeX faster: Easily using BasicTeX Neel Somani - April 15, 2016 If you type equations regularly, you've probably heard of LaTeX. The MacTeX Working Group from the Tex User Group (TUG) constructed an install package which installs TeX Live and everything else needed to run TeX on Mac OS X in one step. pdflatex … Therefore, start by typing: All TeX commands are preceded by a backslash (\) and defined within {curly brackets}. A comprehensive guide to basic and advanced features. Features for a targeted audience are available through extensions. If you haven't included the command \maketitle, for example, then the document won't render as it will simply be empty (the command \maketitle actually places the title in the document, whereas the command \title simply defines it for the document). There is option of "Restrict where this project can be run", You can provide jenkins slave name on which this job should always run. With TeX, you have to learn the various commands before you can start creating documents. Users who run into trouble often write us without noticing that the solution is on this very page. Once the file has downloaded, double-click on MacTeX.pkg to begin the installation. To install LaTeX applications on your Mac: When you are ready to begin using LaTeX, you may find this website helpful:www.reed.edu/cis/help/latex/intro.html, Copyright © Trustees of Wellesley College   |   Wellesley College 106 Central Street â Wellesley, MA 02481 (781) 283-1000, Once the file has downloaded, double-click on. Install useful R packages in RStudio Download the file available at startup_packages.R.This is a text (script) file containing R commands that you will run. Design, code, video editing, business, and much more. Get access to over one million creative assets on Envato Elements. To create the document, I used the following: Note the use of a curly bracket before the \ command and the two curly brackets at the end. Just choose a TEXROOT you have write permission to. Read and accept the conditions, then follow the on-screen instructions to install the program. It's not as simple as just opening up a word processor such as Pages or Microsoft Word and tapping away. But the OSX distribution (MacTeX… There is standard … How to Install LaTeX Fonts on Mac OS X. LaTeX Fonts are high quality members of a family of fonts referred to as "Computer Modern." To create a .PDF file just run this command in the system terminal. Answer to your last doubt. LaTeX is a package that allows the use of markup in the document to structure itlogically in a way that users would never have to think about formatting. MiKTeX for Mac is distributed as a disk image (.dmg) file.If you double click the disk image file, you will see a Finder window: To install MiKTeX, simply drag the MiKTeX icon onto the … The \textbf and \textit commands example, then it 's not as simple just! Instructions to install LaTeX, R and R Markdown on Windows 10 system... Least primarily, free software file  mydocument.tex '' do n't have to but! 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# Competing risks

#### 2020-02-07

As of version 1.2.0, joineR fits the joint model proposed by Williamson et al. (2008) for joint models of longitudinal data and competing risks. Here, the longitudinal data submodel remains as per that analyzed in Henderson et al. (2000), namely

$Y_i(t) = X_i(t)^\top \beta + Z_i(t)^\top U_i + \epsilon_i(t),$

where $$Y_i(t)$$ is a repeated measurement on subject $$i$$ at time $$t$$, $$U_i$$ is a latent vector that follows a zero-mean multivariate normal distribution, $$X_i(t)$$ and $$Z_i(t)$$ are vectors of explanatory variables that may be time-constant or time-varying, and the $$\epsilon_i(t)$$ are mutually independent errors, $$\epsilon_i(t) \sim {\rm N}(0, \tau^2)$$.

For the competing risks data, a cause-specific hazards submodel is proposed, namely

$\lambda_{ig}(t) = \lambda_{0g}(t) \exp(W_i^\top \alpha_g + \gamma_g V_{ig}),$

where $$W_i$$ is a vectors of time-constant explanatory variables, $$V_i$$ is another latent vector that follows a zero-mean multivariate normal distribution, and $$\lambda_{0g}(t)$$ is the baseline hazard function for cause $$g = 1, 2$$. In joineR, when competing risks data are analysed, $$Z_i(t)^\top = (1, t)$$, We also note that currently only 2 failure types are be modelled, which must be coded as 1 and 2 (with 0 representing censoring).

# Example

We will perform the analysis from Williamson et al. (2008) to the SANAD (Standard And New Antiepileptic Drugs) study data – the largest trial of AEDs to date (Marson et al., 2007).

## Background

The time to withdrawal of a randomized drug or addition of another, has been recommended by the International League Against Epilepsy to be one of the primary endpoints for clinical trials of anti-epileptic drugs (AEDs) (Commission on Antiepileptic Drugs, 1998). Patients may decide to switch to an alternative AED because of inadequate seizure control (ISC) or to withdraw from a treatment because of unacceptable adverse effects (UAE). Overall analysis of treatment failures may miss differential effects of AEDs on the reasons for withdrawal, which may differ in terms of their relative importance for patients (Williamson et al., 2007).

The primary SANAD analysis of patients with partial epilepsy concluded that the newer drug lamotrigine (LTG) was preferable in terms of treatment failure to carbamazepine (CBZ), which had been the standard for many years (Williamson et al., 2006). LTG was significantly better in terms of withdrawal for UAE and not significantly worse in terms of withdrawal for poor seizure control. Subsequent correspondence after the trial results were published indicated that some readers were concerned that differential titration rates may have been to the disadvantage of CBZ (Cross et al., 2007). An AED that is titrated more quickly may bring benefits in terms of seizure control but be more likely to cause adverse effects. This criticism has also been leveled at previous AED trials (O’Donoghue & Sander, 1995).

## Data

SANAD was an unblinded randomized trial recruiting patients with epilepsy for whom CBZ was considered to be standard treatment and they were randomized to CBZ, gabapentin, LTG, oxcarbazepine or topiramate (Marson et al., 2007). Time to treatment failure was a primary outcome with the analysis of competing risks, i.e. treatment failure due to either ISC or UAE, an important secondary objective. Although LTG appeared to be superior to CBZ in the main analysis, concerns were raised that the results were biased in favor of the newer drug. CBZ was considered to have been titrated more quickly bringing benefits in terms of seizure control but being more likely to cause adverse effects. The data set analysed here, comparing CBZ and LTG, includes 605 patients. During the trial, 94 patients withdrew from the randomized drug due to UAE while 120 withdrew due to ISC. In order to compare the two AEDs after adjusting for titration rate, dose calibration was first undertaken by standardizing the dose of both drugs relative to the midpoint of the maintenance dose range for each particular drug. The maintenance dose recommended in the SANAD trial was independently deemed to be reasonable (Faught, 2007) and the approach to calibration considered sensible. These calibrated doses are taken to be the longitudinal measurements in the competing-risks joint model.

The data is available in the joineR package by running

library(joineR)
## Loading required package: survival
data(epileptic)
head(epileptic)
##   id dose time with.time with.status with.status2 with.status.uae
## 1  1 2.00   86      2400           0            0               0
## 2  1 2.00  119      2400           0            0               0
## 3  1 2.00  268      2400           0            0               0
## 4  1 2.67  451      2400           0            0               0
## 5  1 2.67  535      2400           0            0               0
## 6  1 2.67  770      2400           0            0               0
##   with.status.isc treat   age gender learn.dis
## 1               0   CBZ 75.67      M        No
## 2               0   CBZ 75.67      M        No
## 3               0   CBZ 75.67      M        No
## 4               0   CBZ 75.67      M        No
## 5               0   CBZ 75.67      M        No
## 6               0   CBZ 75.67      M        No

In the analysis, Williamson et al. (2008) considered an interaction term between time and treatment in the longitudinal data submodel. Therefore, we will manually code this term using

epileptic$interaction <- with(epileptic, time * (treat == "LTG")) A jointdata object is first constructed using the jointdata() function. Note that there are several status columns: with.status is a composite event status (treatment failure or censored), with.status.uae is a binary event status for UAE (with failure due to ISC treated as censoring), with.status.isc is a binary event status for ISC (with failure due to UAE treated as censoring), and with.status2 is a vector taking values 0, 1, and 2, with 0 denoting censoring and 1 and 2 denoting dropout due to ISC and UAE, respectively. longitudinal <- epileptic[, c(1:3, 13)] survival <- UniqueVariables(epileptic, c(4, 6), "id") baseline <- UniqueVariables(epileptic, "treat", "id") data <- jointdata(longitudinal = longitudinal, survival = survival, baseline = baseline, id.col = "id", time.col = "time") summary(data) ##$subjects
## [1] "Number of subjects: 605"
##
## $longitudinal ## class ## dose numeric ## time integer ## interaction integer ## ##$survival
##
## Number of subjects that fail due to cause 1: 120
## Number of subjects that fail due to cause 2:  94
## Number of subjects censored:                 391
##
## $baseline ## class ## treat factor ## ##$times
## [1] "Unbalanced longitudinal study or more than twenty observation times"
jointplot(data, Y.col = "dose", Cens.col = "with.status2")
## Warning in jointplot(data, Y.col = "dose", Cens.col = "with.status2"): jointplot
## does not display profiles for different failure events

## Model

To fit the joint model, we use the usual joint() function as

fit2 <- joint(data = data, long.formula = dose ~ time + treat + interaction,
surv.formula = Surv(with.time, with.status2) ~ treat,
longsep = FALSE, survsep = FALSE, gpt = 3)
## Competing risks detected
summary(fit2)
##
## Call:
## joint(data = data, long.formula = dose ~ time + treat + interaction,
##     surv.formula = Surv(with.time, with.status2) ~ treat, longsep = FALSE,
##     survsep = FALSE, gpt = 3)
##
## Random effects joint model
##  Data: data
##  Log-likelihood: -4056.936
##
## Longitudinal sub-model fixed effects: dose ~ time + treat + interaction
## (Intercept)  1.9730179636
## time         0.0003545439
## treatLTG    -0.1380959688
## interaction  0.0006185222
##
## Competing risks sub-model effects: Surv(with.time, with.status2) ~ treat
## Failure cause 1:
## treatLTG 0.02721022
## Failure cause 2:
## treatLTG -0.6601433
##
## Latent association:
## gamma_1  0.5894143
## gamma_2 -0.9258597
##
## Variance components:
##          U_0          U_1     Residual
## 7.321131e-01 1.211086e-06 1.964523e-01
##
## Convergence at iteration: 19
##
## Number of observations: 2797
## Number of groups: 605

To estimate standard errors, we use the jointSE() function as follows. However, we do not run the code in this vignette due to computational time required.

fit2.se <- jointSE(fit2, n.boot = 100)
fit2.se

## Conclusion

Significant $$\gamma$$ estimates in the model suggests that calibrated dose is associated with time to treatment failure for both particular causes. It is observed that these estimates have opposite signs for the two failure reasons. This was a situation anticipated previously in this application, since side effects are more likely at higher doses, whereas seizure control is poorer at low doses (Williamson et al., 2006). Thus, patients on higher doses may be more likely to be withdrawn from treatment due to poor seizure control since the reason they are on higher doses is that they are having continued seizures. Patients on higher doses may also be less likely to be withdrawn from treatment due to UAE since the reason a required dose increase is possible is because no such events have occurred

The results from all models fitted suggest that if LTG is titrated at the same rate as CBZ, the beneficial effect of LTG on UAE would still be evident and the two drugs still appear to provide similar seizure control. Thus, the conclusion from the original simpler analysis of the cause-specific hazards is found to be robust to the variation in titration between the drugs in the trial.

# Acknowledgements

The joineR package was funded by the UK Medical Research Council (MRC) under a grant with Principal Investigator (PI) Prof. Paula Williamson; Co-Investigators (Co-Is) Prof. Peter J. Diggle and Prof. Robin Henderson; and Research Associates Dr Ruwanthi Kolamunnage-Dona, Dr Peter Philipson, and Dr Ines Sousa (Grant numbers G0400615). Updates to it were made under a separate MRC grant with PI Dr Ruwanthi Kolamunnage-Dona, Co-Is Dr Peter Philipson and Dr Andrea Jorgensen, and Research Associate Dr Graeme L. Hickey (Grant number MR/M013227/1).

# References

1. Williamson PR, Kolamunnage-Dona R, Philipson P, Marson AG. Joint modelling of longitudinal and competing risks data. Statistics in Medicine, 2008; 27: 6426-6438.

2. Henderson R, Diggle PJ, Dobson A. Joint modelling of longitudinal measurements and event time data. Biostatistics, 2000; 1: 465-480

3. Marson AG, Al-Kharusi AM, Alwaidh M, Appleton R, Baker GA, Chadwick DW, Cramp C, Cockerell OC, Cooper PN, Doughty J, Eaton B, Gamble C, Goulding PJ, Howell SJL, Hughes A, Jackson M, Jacoby A, Kellett M, Lawson GR, Leach JP, Nicolaides P, Roberts R, Shackley P, Shen J, Smith DF, Smith PEM, Tudur Smith C, Vanoli A, Williamson PR on behalf of the SANAD Study group. Carbamazepine, gabapentin, lamotrigine, oxcarbazepine or topiramate for partial epilepsy: results from the SANAD trial. The Lancet, 2007; 369: 1000–1015.

4. Commission on Antiepileptic Drugs. Considerations on designing clinical trials to evaluate the place of new antiepileptic drugs in the treatment of newly diagnosed and chronic patients with epilepsy. Epilepsia, 1998; 39: 799–803.

5. Williamson PR, Smith CT, Josemir WS, Marson AG. Importance of competing risks in the analysis of anti-epileptic drug failure. Trials, 2007; 8: 12.

6. Williamson PR, Kolamnunnage-Dona R, Tudur Smith C. The influence of competing risks setting on the choice of hypothesis test for treatment effect. Biostatistics, 2006; 8: 689–694.

7. Cross H, Ferrie C, Lascelles K, Livingston J, Mewasingh L. Old versus new antiepileptic drugs: the SANAD study. The Lancet, 2007; 370: 314.

8. O’Donoghue MF, Sander JWAS. Lamotrigine versus carbamazepine in epilepsy. The Lancet, 1995; 345: 1300.

9. Faught E. Epilepsy drugs: getting it right the first time. Lancet Neurology, 2007; 6: 476–478.

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# math

When simplifying a rational fraction, why do you need to
factor the numerator and the denominator? Include two examples!

to get like terms

Can you explain a little more?

When you factor then you can cancel like terms

Example:

12/3

(3)(4)
------- = 4/1 =4
3

i need to ask you a dumb question
Is this an example of a ratonal Fraction?
because the question need to answer is
When simplifying a rational fraction, why do you need to factor the numerator and the denominator? Include two examples

Rational fractions are written as one number divided by another, such as 3/4.

1. 👍 0
2. 👎 0
3. 👁 73
1. When simplifying a rational fraction, why do you need to factor the numerator and the denominator? Include two examples

1. 👍 0
2. 👎 0
posted by shirley
2. this is 4 ques of jiskha

1. 👍 0
2. 👎 0
posted by Frank

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2. ### math

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More Similar Questions


text:
A body of density ${\mathrm{d}}_{1}$ is counterpoised by Mg of weights of density ${\mathrm{d}}_{2}$ in air of density d. Then the true mass of the body is

(a) M                                    (b) $\mathrm{M}\left(1-\frac{\mathrm{d}}{{\mathrm{d}}_{2}}\right)$

(c) $\mathrm{M}\left(1-\frac{\mathrm{d}}{{\mathrm{d}}_{1}}\right)$                       (d) $\frac{\mathrm{M}\left(1-\mathrm{d}/{\mathrm{d}}_{2}\right)}{\left(1-\mathrm{d}/{\mathrm{d}}_{1}\right)}$

Explanation is a part of a Paid Course. To view Explanation Please buy the course.

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# Find expected value of time to reach a state in Markov chain, by simulation

Consider a time homogeneous Markov chain $(X_n)_{n=0}$ with state space $E$, initial distribution $p(0)$ and transition probability matrix $P$ given by $E = \{0, 1, 2\}, p(0) = [1\;\; 0\;\; 0]$ and

P= $\begin{bmatrix}1/2 & 1/3 & 1/6\\0 & 2/3 & 1/3 \\0 & 0 & 1 \end{bmatrix}$

respectively. Find by a computer simulations an as good as is possible approximation of the expected value $\Bbb E(T)$ of the time $T = \min\{ n ∈ N : X_n = 2 \}$ it takes the chain to reach the state 2. Someone who can help me in doing some kind of pseudocode for this question, or some matlab code?

• Do you know how to sample from a discrete distribution? For instance, the second value of the process is given by sampling from the discrete distribution $(1/2,1/3,1/6)$. Then once you have the second value you sample from another distribution, etc.
– Ian
Dec 14, 2015 at 22:05
• No sorry. I have not taken any programming course yet. This is the main problem for me at this particulat moment. That is to solve some of these questions and get some kind of intuitive feeling in doing so for it. And taking a programming course after this christmas. I appreciate your'e answer but maybe you could if you have time demonstrate by some matlab code?
– user297559
Dec 14, 2015 at 22:14
• Do you have to use matlab? Is maple or C/C+/C++ also acceptable? Dec 14, 2015 at 22:24
• I prefer matlab in order to understand it myself. And the problem is for myself, to get a starter or intuition of maybe solve similar problems.
– user297559
Dec 14, 2015 at 22:29
• Then I will leave it to others :). Dec 14, 2015 at 22:31

In Mathematica:

myP = {{1/2, 1/3, 1/6}, {0, 2/3, 1/3}, {0, 0, 1}};

ListPlot[Transpose[NestList[#.myP &, {1, 0, 0}, 10]],
AxesLabel -> {"step", "Probability"},
Joined -> True,
PlotLegends -> {"P[0]", "P[1]", "P[2]"} ]

The successive differences in probabilities for being in state $2$ are:

difProbs = Differences@NestList[#.myP &, {1, 0, 0}, 100][[All, 3]];

The expected value is:

N@Range[100].difProbs

(* 4 *)

Probably easier for you to understand R than mathematica if you know matlab.

Ask if you dont understand what im doing.

Here is the code in R:

N = 10^4

T<-vector(length=N)

ptm <- proc.time()

for (k in 1:N) {
i<-0
X<-0

while (X==0) {
u=runif(1)

if (1/6<u & u<1/2) {X<-1}
else if (u<1/6) {X<-2}
else {X<-0}

i <- i+1
}

while (X==1) {
u=runif(1)

if (u<1/3) {X<-2}

i <- i+1
}

# Done Save

T[k]<-i

}

print(proc.time()-ptm)

mean(T)

• While X is in state 0 or 1, respectively, i generate uniform random numbers , u, and for the X=1 case i check if this u is less than 1/3. There is a 1/3 probability to go from 1 to 2 so if u fulfills this condition i set X=2 and we have reached our destination. If the condition is not fulfilled it stays in X=1 until i get a u that fulfills the condition. I add one to i for each time i check this. I do the same thing for X=0 and ofc always start in X=0. This procedure i repeat 10^4 times in the for-loop and for each time i save the i's and then i take the mean to get the simulated average time. Dec 15, 2015 at 9:20
• I added this comment before you removed yours. I leave it there if someone else need an explanation. Dec 15, 2015 at 9:22

Let $\psi(i)$ be the expected time to reach state 2 from state i, where i ={0,1,2}.

Then $\psi(2) =0$

$\psi(1) = 1+ \frac{2}{3}\psi(1)+\frac{1}{2}\psi(2)$

$\psi(0) = 1+\frac{1}{2}\psi(0)+\frac{1}{3}\psi(1)+\frac{1}{6}\psi(2)$

Solve the system to find $\psi(0)$

The initial state is 0 and the expected time to reach state 2 from 0 is $\boxed{4}$

• They asked to do it by simulation, though.
– Ian
Dec 14, 2015 at 22:19
• @DavidG.Stork This calculates the expected time to achieve the first hitting of state 2 (possibly doing so by passing through state 1).
– Ian
Dec 14, 2015 at 23:05
• From the transition matrix, I get to understand that state 2 is absorbing with 1 to itself and 0 elsewhere. If it reached 2, it ain't going anywhere Dec 14, 2015 at 23:06


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## FANDOM

236 Pages

Ultima is a recurring spell in the Final Fantasy series, usually classified as a Black Magic spell with the exception of Final Fantasy II where it is classified as a White Magic spell. It is often the most powerful spell in the game, hitting all enemies for non-elemental damage, which can ignore defense, Reflect and any other protection. Ultima usually manifests as an explosive sphere of blue or green energy consuming opponents.

## AppearancesEdit

### Final Fantasy Blackmoon ProphecyEdit

Ultima can be obtained through a vague sidequest or by reaching level 80 with Oalston. Only Oalston can learn this spell, and it can be the most damaging attack in the entire game depending on the enemy formation. Ultima is the only attack in Blackmoon Prophecy that is guaranteed to hit for 9999 non-elemental damage every time it is used. It costs 150 MP to cast, though this can be cut down to 75 MP if Oalston equips the Gem Box accessory.

### Final Fantasy DiscoveryEdit

Ultima is the most damaging Black Magic spell in the game. It costs 80 MP to cast, cannot be reflected, can pierce through Shell, and it ignores target's Magic Defense. It is one of only two Black Magic spells that Wedge Farron does not learn naturally by gaining levels. With the use of the Key Item, Enterprise, Ultima is capable of breaking the damage limit.

Its damage formula is as follows:

$Damage = 150 * 4 + (User's Level * User's Magic * 150 / 32) * (255 - 0) / 256 + 1$


text:
Title

# Using human partitionings of image sets to learn a similarity-based distance measure for the organization of image databases

Author
Published in SPIE Symposium on Voice, Video and Data Communications. Multimedia Storage and Archiving Systems III (VV02). Boston (USA). 1998, p. 80-88
Collection SPIE Proceedings; 3527
Abstract In this paper our goal is to employ human judgments of image similarity to improve the organization of an image database for content-based retrieval. We first derive a statistic, $\kappa_B$ for measuring the agreement between two partitionings of an image set into unlabeled subsets. This measure can be used both to measure the degree of agreement between pairs of human subjects, and also between human and machine partitionings of an image set. It also allows a direct comparison of database organizations, as opposed to the indirect measure available via precision and recall measurements. This provides a rigorous means of selecting between competing image database organization systems, and assessing how close the performance of such systems is to that which might be expected from a database organization done by hand. We then use the results of experiments in which human subjects are asked to partition a set of images into unlabeled subsets to define a similarity measure for pairs of images based on the frequency with which they were judged to be similar. We show that, when this measure is used to partition an image set using a clustering technique, the resultant clustering agrees better with those produced by human subjects than any of the feature space-based techniques investigated. Finally, we investigate the use of machine learning techniques to discover a mapping from a numerical feature space to this perceptual similarity space. Such a mapping would allow the ground truth knowledge abstracted from the human judgments to be generalized to unseen images. We show that a learning technique based on an extension of a Kohonen network allows a similarity space to be learnt which results in partitionings in excellent agreement with those produced by human subjects.
Keywords Image databasesDatabase organizationSimilarityLearningAgreement
Full text
This document has no fulltext available yet, but you can contact its author by using the form below.
Structures
Research groups Viper group
Computer Vision and Multimedia Laboratory
Citation
(ISO format)
SQUIRE, David. Using human partitionings of image sets to learn a similarity-based distance measure for the organization of image databases. In: SPIE Symposium on Voice, Video and Data Communications. Multimedia Storage and Archiving Systems III (VV02). Boston (USA). [s.l.] : [s.n.], 1998. p. 80-88. (SPIE Proceedings; 3527) https://archive-ouverte.unige.ch/unige:47928

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text:
Article Contents
Article Contents

# Solvability of nonlinear evolution equations generated by subdifferentials and perturbations

• The main objective of this paper is to discuss solvability of the Cauchy problem of an evolution equation with subdifferentials of convex functions which is generated by unknown functions and perturbations of the form:
$u'(t) + ∂ \varphi^t(u;u(t)) + G(u(t)) \ni f(t)$   0 < t < T,      in     H.
where H is a Hilbert space, $u'=\frac{du}{dt}$, and $∂ \varphi^t(u;\cdot )$ is a subdifferential operator of convex function $\varphi^t(u;\cdot )$. The evolution equation corresponds to parabolic quasi-variational inequalities.
Mathematics Subject Classification: Primary: 47J35; Secondary: 47J20.

 Citation:

•  [1] A. Attouch, "Variational Convergence for Functions and Operators," Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. [2] A. Azevedo and L. Santos, Convergence of convex sets with gradient constraint, J. Convex Anal., 11 (2004), 285-301. [3] M. Biroli, Sur les inéquations paraboliques avec convexe dépendant du temps: Solution forte et solution faible, Riv. Mat., Univ. Parma, 3 (1974), 33-72. [4] H. Brézis, "Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert," (French) North-Holland Mathematics Studies, No. 5, Notas de Matema'tica (50), North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. [5] H. Brézis, Problémes unilatéraux, (French) J. Math. Pures Appl., 51 (1972), 1-168. [6] K. H. Hoffmann, M. Kubo and N. Yamazaki, Optimal control problems for elliptic-parabolic variational inequalities with time-dependent constraints, Numer. Funct. Anal. Optim., 27 (2006), 329-356.doi: 10.1080/01630560600686116. [7] A. Kadoya, N. Kenmochi and Y. Murase, A class of nonlinear parabolic systems with environmental constraints, Adv. Math. Sci. Appl., 20 (2010), 281-313. [8] R. Kano, Applications of abstract parabolic quasi-variational inequalities to obstacle problems, in "Nonlocal and Abstract Parabolic Equations and their Applications," Banach Center Publ., 86, Polish Acad. Sci. Inst. Math., Warsaw, (2009), 163-174.doi: 10.4064/bc86-0-10. [9] R. Kano and A. Ito, The existence of time global solutions for tumor invasion models with constraints, Discrete Contin. Dyn. Syst., 2011, Dynamical Systems, Differential Equations and Applications, $8^{th}$ AIMS Conference, Suppl. Vol. II, 774-783. [10] R. Kano, A. Ito, K. Yamamoto and H. Nakayama, Quasi-variational inequality approach to tumor invasion models with constraints, GAKUTO Internat. Ser. Math. Sci. Appl., 32 (2010), 365-388. [11] R. Kano, N. Kenmochi and Y. Murase, Existence theorems for elliptic quasi-variational inequalities in Banach spaces, in "Recent Advance in Nonlinear Analysis," World Sci. Publ., Hackensack, NJ, (2008), 149-169.doi: 10.1142/9789812709257_0010. [12] R. Kano, N. Kenmochi and Y. Murase, Nonlinear evolution equations generated by subdifferentials with nonlocal constraints, in "Nonlocal and Abstract Parabolic Equations and their Applications," Banach Center Publ., 86, Polish Acad. Sci. Inst. Math., Warsaw, (2009), 175-194.doi: 10.4064/bc86-0-11. [13] R. Kano, N. Kenmochi and Y. Murase, Parabolic quasi-variational inequalities with non-local constraints, Adv. Math. Sci. Appl., 19 (2009), 565-583. [14] R. Kano, N. Kenmochi and Y. Murase, Elliptic quasi-variational inequalities and applications, Discrete Contin. Dyn. Syst., 2009, Dynamical Systems, Differential Equations and Applications, $7^{th}$ AIMS Conference, suppl., 583-591. doi: 10.4064/bc86-0-15. [15] N. Kenmochi, Some nonlinear parabolic variational inequalities, Israel J. Math., 22 (1975), 304-331.doi: 10.1007/BF02761596. [16] N. Kenmochi, Solvability of nonlinear evolution equations with time-dependent constraints and applications, Bull. Fac. Education, Chiba Univ., 30 (1981), 1-87. [17] N. Kenmochi, "Monotonicity and compactness methods for nonlinear variational inequalities," in "Handbook of Differential Equations: Stationary Partial Differential Equations," Vol. IV, Elsevier/North Holland, Amsterdam, (2007), 203-298.doi: 10.1016/S1874-5733(07)80007-6. [18] M. Kunze and J. F. Rodrigues, An elliptic quasi-variational inequality with gradient constraints and some of its applications, Math. Methods Appl. Sci., 23 (2000), 897-908.doi: 10.1002/1099-1476(20000710)23:10<897::AID-MMA141>3.0.CO;2-H. [19] F. Mignot and J.-P. Puel, Inéquations d'évolution paraboliques avec convexes d'épe- ndant du temps. Applications aux inéquations quasi variationnelles d'évolution, Arch. Rat. Mech. Anal., 64 (1977), 59-91.doi: 10.1007/BF00280179. [20] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Advances in Math., 3 (1969), 510-585.doi: 10.1016/0001-8708(69)90009-7. [21] Y. Murase, Abstract quasi-variational inequalities of elliptic type and applications, in "Nonlocal and Abstract Parabolic Equations and their Applications," Banach Center Publ., 86, Polish Acad. Sci. Inst. Math., Warsaw, (2009), 235-246.doi: 10.4064/bc86-0-15. [22] J. F. Rodrigues, On a quasi-variational inequality arising in semiconductor theory, Revista Mat., Univ. Complutense Madrid, 5 (1992), 137-151. [23] J. F. Rodrigues and L. Santos, A parabolic quasi-variational inequality arising in a superconductivity model, Ann. Scuola Norm. Sup. Pisa, 29 (2000), 153-169. [24] U. Stefanelli, Nonlocal quasivariational evolution problems, J. Differential Equations, 229 (2006), 204-228.doi: 10.1016/j.jde.2006.05.004. [25] Y. Yamada, On nonlinear evolution equations generated by the subdifferentials, J. Fac. Sci., Univ. Tokyo, Sect. IA, 23 (1976), 491-515.


text:
# Category:Definitions/Hyperbolic Cosine Function

This category contains definitions related to Hyperbolic Cosine Function.
Related results can be found in Category:Hyperbolic Cosine Function.

The hyperbolic cosine function is defined on the complex numbers as:

$\cosh: \C \to \C$:
$\forall z \in \C: \cosh z := \dfrac {e^z + e^{-z} } 2$

## Pages in category "Definitions/Hyperbolic Cosine Function"

The following 3 pages are in this category, out of 3 total.


text:
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Okihiro Sawada (Waseda Univ.) An improvement of the a priori estimate on the 2-D Navier-Stokes flow with nondecaying initial velocity
Paolo Secchi (Brescia Univ.) Nonlinear compressible vortex sheets in two space dimensions
Adelia Sequeira (Instituto Superior Técnico, Portugal) A numerical study of shear-thinning effects of generalized Newtonian flows in curved pipes
Gregory Seregin (Steklov Institute) Navier-Stokes equations: almost $L_{3,\infty}$-case
Senjo Shimizu (Shizuoka Univ.) Lp-Lq maximal regularity and viscous incompressible flows with free surface
Ana Leonor Silvestre (Instituto Superior Técnico, Portugal) On the motion of a self-propelled body in a Navier-Stokes liquid
Zdenek Skalák Modes in Solutions of the Homogeneous Navier-Stokes Equations
Hermann Sohr (Univ. Paderborn) Recent regularity results on the Navier-Stokes equations for general smooth domains
Troy L. Story (Morehouse College) Navier-Stokes dynamics on a differential one-form
Yasushi Taniuchi (Shinshu Univ.) On heat convection equations in a half space with non-decaying data
Tomomasa Tatsumi (Kyoto Univ.) Inertial similarity of homogeneous isotropic Navier-Stokes turbulence
Gudrun Thaeter (Univ. Dortmund) Natural Convection between two horizontal coaxial cylinders
Edriss S. Titi (Weizmann Institute of Science & Univ. of California, Irvine) On the Global Regularity of the 3D Navier-Stokes and Other Relevant Equations
Xiao-Ping Wang (Hong Kong Univ. Sci. Tech.) Continuum model for moving contact line and analysis of sharp interface limit
Zhouping Xin (Chinese Univ. Hong Kong) Boundary Layers for the NS with various boundary conditions

Abstracts of talks [pdf]

 Organizer: Hisashi OKAMOTO (RIMS, Kyoto Univ.) Research Institute for Mathematical Sciences


text:
Fermat's Little Theorem/Corollary 4

Corollary to Fermat's Little Theorem

Let $p^k$ be a prime power for some prime number $p$ and $k \in \Z_{\gt 0}$.

Let $n \in \Z_{\gt 0}$ with $p \nmid n$.

Then:

$n^{p^k - 1} \equiv 1 \pmod p$

Proof

$n^{p^k} \equiv n \pmod p$

That is:

$p \divides \paren {n^{p^k} - n} = n \paren {n^{p^k - 1} - 1}$

Since $p \nmid n$, by Corollary to Divisors of Product of Coprime Integers:

$p \divides \paren {n^{p^k - 1} - 1}$

That is:

$n^{p^k-1} \equiv 1 \pmod p$

$\blacksquare$


text:
# Skill networks and measures of complex human capital

1. aTepper School of Business, Carnegie Mellon University, Pittsburgh, PA 15213
1. Edited by Matthew O. Jackson, Stanford University, Stanford, CA, and approved October 24, 2017 (received for review April 20, 2017)

## Significance

The relationship between worker human capital and wages is a question of considerable economic interest. Skills are usually characterized using a one-dimensional measure, such as years of training. However, in knowledge-based production, the interaction between a worker’s skills is also important. Here, we propose a network-based method for characterizing worker skill sets. We construct a human capital network, wherein nodes are skills and two skills are connected if a worker has both or both are required for the same job. We then illustrate the method by analyzing an online freelance labor market, showing that workers with diverse skills earn higher wages and that those who use their diverse skills in combination earn the highest wages of all.

## Abstract

We propose a network-based method for measuring worker skills. We illustrate the method using data from an online freelance website. Using the tools of network analysis, we divide skills into endogenous categories based on their relationship with other skills in the market. Workers who specialize in these different areas earn dramatically different wages. We then show that, in this market, network-based measures of human capital provide additional insight into wages beyond traditional measures. In particular, we show that workers with diverse skills earn higher wages than those with more specialized skills. Moreover, we can distinguish between two different types of workers benefiting from skill diversity: jacks-of-all-trades, whose skills can be applied independently on a wide range of jobs, and synergistic workers, whose skills are useful in combination and fill a hole in the labor market. On average, workers whose skills are synergistic earn more than jacks-of-all-trades.

The relationship between worker skills and wages is a problem of tremendous economic interest, making it critical to have effective measures of the skills, knowledge, and experience that a worker brings to production: a bundle of worker characteristics that economists refer to as human capital. Traditionally, human capital measures either divide workers into broad categories (e.g., laborers and management) or count years of experience, training, or education (1). However, treating skills as interchangeable removes some of the richness of human capital: workers’ skills are clearly heterogeneous and multidimensional as are the skills required for jobs. A considerable body of literature shows the importance of skill diversity, specialization, and recombination in problem-solving and knowledge generation (2?????8). This plus continued growth in knowledge-based industry (9) have generated interest in more nuanced measures of human capital (10????????19).

Determining the relationship between wages and factors like skill diversity requires us to not only look at a worker’s individual skills but also, her skill combinations. However, considering skills in combination makes measuring human capital much more difficult (20). Some skills (e.g., programing and user interface design) are synergistic, meaning that the combination is more valuable than the sum of its parts: each skill enhances the effectiveness of the other. Other skills (e.g., programing and Russian translation) are no more valuable together than they are individually. On the supply side, some skills (e.g., programing and management skills) are quite common individually but extremely rare in combination. Taken together, these factors mean that the value of an additional skill will depend on the skills that the worker already has (16, 21).

Here, we propose a network-based framework for the characterization of human capital that complements existing notions of human capital and production. Given a pool of workers with multidimensional skill baskets, we construct a network in which skills are nodes and two skills are connected by a link if a worker has both. Links are weighted according to how often the two skills co-occur. We construct a similar network using the skill sets required to perform different jobs, wherein two skills are connected if they are required by an employer in combination. Together, these two networks provide a more complete picture of the supply and demand for human capital in a particular job market. Most importantly, they suggest a number of measures of human capital that account for both the relationships between skills and the context dependency of human capital.

We then use data from an online freelance labor market as an illustration of the method. Using information drawn from worker profiles and employer job advertisements, we construct a human capital network and several network-based measures of worker skills. We use an algorithmic method to split the human capital networks into clusters of closely related skills, providing an entirely endogenous categorization of skills. There is considerable variation in wages between workers specializing in these different skill categories. Workers with more diverse skills tend to earn higher wages than specialists. We then compare the skill measures on the supply and demand sides of the market to show that workers with diverse skills fall into two categories: those who exploit gaps in the market tend to earn higher wages than “jacks-of-all-trades,” who use their skills for multiple different jobs. Finally, we illustrate the value of this approach by showing that our network-based human capital measures explain variation in worker wages, even after controlling for individual skills.

## Methods: Constructing a Human Capital Network

Network science provides a means of making sense of the relationships between skills in the labor market. Here, we construct two different networks: one representing the skills that workers have and the other representing the skills that employers require. Nodes in these networks are skills present in the labor market. On the workers’ side, two skills are connected by a link if the same worker has both. On the employers’ side, two skills are connected if they are required for the same job. We will call these networks the worker (supply) side and the employer (demand) side human capital networks.

More formally, let <mml:math><mml:mrow><mml:mi>I</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mrow><mml:mo>…</mml:mo><mml:mi>N</mml:mi></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math>I={1,2,…N} be a pool of workers, each endowed with a skill set <mml:math><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mo>…</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math>Ai={s1,s2,…sk}. Let <mml:math><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>W</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mo largeop="true" stretchy="false" symmetric="true">?</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math>SW=?i∈I{Ai} denote the set of all skills possessed by workers, with <mml:math><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mi>W</mml:mi></mml:msub><mml:mo>|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>W</mml:mi></mml:msub></mml:mrow></mml:math>|SW|=MW. Let <mml:math><mml:mrow><mml:mi>A</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mo>…</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mi>N</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math>A={A1,A2,…AN} denote the set of all worker skill sets. Let <mml:math><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math>ni be the number of skill sets containing skill <mml:math><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math>si, and let <mml:math><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub></mml:math>nij be the number of skill sets containing both <mml:math><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math>si and <mml:math><mml:msub><mml:mi>s</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math>sj. (Note that we have assumed that skills are binary. Our data do not allow us to consider the ability level of an individual in a particular skill, and therefore, we do not consider it explicitly. However, the intensive margin could easily be incorporated into either the link weights or the measures derived from the network.) In a worker human capital network, <mml:math><mml:mrow><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>A</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>g(A), the nodes are the skills in the set <mml:math><mml:msub><mml:mi>S</mml:mi><mml:mi>W</mml:mi></mml:msub></mml:math>SW, and two skills, <mml:math><mml:msub><mml:mi>s</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math>sj and <mml:math><mml:msub><mml:mi>s</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math>sk, are connected if <mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mo>∈</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math>sj,sk∈Ai for some <mml:math><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math>Ai∈A. This network can be represented as an <mml:math><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi>W</mml:mi></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>W</mml:mi></mml:msub></mml:mrow></mml:math>MW×MW matrix, where <mml:math><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>gjk=wjk if <mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mo>∈</mml:mo><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math>sj,sk∈Ai for some <mml:math><mml:mrow><mml:msub><mml:mi>A</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>A</mml:mi></mml:mrow></mml:math>Ai∈A and is zero otherwise. The value <mml:math><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:math>wjk is the weight on the link between skills <mml:math><mml:mi>i</mml:mi></mml:math>i and <mml:math><mml:mi>j</mml:mi></mml:math>j.

Let <mml:math><mml:mrow><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mrow><mml:mo>…</mml:mo><mml:mi>F</mml:mi></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math>K={1,2,…F} be the set of vacancies in the labor market. Let <mml:math><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mi>f</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mo>…</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math>Bf={s1,s2,…sk} be the set of skills required for job <mml:math><mml:mi>f</mml:mi></mml:math>f, and let <mml:math><mml:mrow><mml:mi>B</mml:mi><mml:mo>=</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mrow><mml:mo>…</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>F</mml:mi></mml:msub></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math>B={B1,B2,…BF}. Let <mml:math><mml:mrow><mml:msub><mml:mi>S</mml:mi><mml:mi>J</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:msub><mml:mo largeop="true" stretchy="false" symmetric="true">?</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:math>SJ=?i∈I{Bi} denote the set of all skills requested by all employers in the market with <mml:math><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:msub><mml:mi>S</mml:mi><mml:mi>J</mml:mi></mml:msub><mml:mo>|</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>J</mml:mi></mml:msub></mml:mrow></mml:math>|SJ|=MJ. Then, one can construct a network similar to that above, <mml:math><mml:mrow><mml:mi>g</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>B</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>g(B), an <mml:math><mml:mrow><mml:msub><mml:mi>M</mml:mi><mml:mi>J</mml:mi></mml:msub><mml:mo>×</mml:mo><mml:msub><mml:mi>M</mml:mi><mml:mi>J</mml:mi></mml:msub></mml:mrow></mml:math>MJ×MJ matrix where <mml:math><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>w</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:math>gjk=wjk if <mml:math><mml:mrow><mml:mrow><mml:msub><mml:mi>s</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:mrow><mml:mo>∈</mml:mo><mml:msub><mml:mi>B</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math>sj,sk∈Bi for some <mml:math><mml:mrow><mml:msub><mml:mi>B</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>∈</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:math>Bi∈B and is <mml:math><mml:mrow><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>j</mml:mi><mml:mi>k</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>gjk=0 otherwise. Note that this network might be much different from the human capital possessed on the supply side.

We will weight each link in the network to reflect how closely related the two skills are in the labor market. Here, our weights will be a modification of conditional probability, which we will call skill similarity weights (alternative weighting schemes are discussed in SI Appendix): <mml:math><mml:mrow><mml:msubsup><mml:mi>w</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi><mml:mi>i</mml:mi><mml:mi>m</mml:mi></mml:mrow></mml:msubsup><mml:mo>=</mml:mo><mml:mi>P</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo stretchy="false">|</mml:mo><mml:msub><mml:mi>s</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>/</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:mrow></mml:math>wijsim=P(si|sj)=nij/nj, where <mml:math><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:math>ni and <mml:math><mml:msub><mml:mi>n</mml:mi><mml:mi>j</mml:mi></mml:msub></mml:math>nj are the numbers of workers who have skills <mml:math><mml:mi>i</mml:mi></mml:math>i and <mml:math><mml:mi>j</mml:mi></mml:math>j, respectively, and <mml:math><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo><</mml:mo><mml:msub><mml:mi>n</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math>nj<ni. Skill similarity weights have three properties that are desirable in this context.

• i) If skill <mml:math><mml:mi>A</mml:mi></mml:math>A never co-occurs with skill <mml:math><mml:mi>B</mml:mi></mml:math>B (<mml:math><mml:mrow><mml:mrow><mml:mi>A</mml:mi><mml:mo>∩</mml:mo><mml:mi>B</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mi mathvariant="normal">?</mml:mi></mml:mrow></mml:math>A∩B=?), then the link weight is zero.

• ii) If skill <mml:math><mml:mi>A</mml:mi></mml:math>A always occurs with skill <mml:math><mml:mi>B</mml:mi></mml:math>B (<mml:math><mml:mrow><mml:mi>A</mml:mi><mml:mo>?</mml:mo><mml:mi>B</mml:mi></mml:mrow></mml:math>A?B), then the link weight is one.

• iii) The weight between two skills is strictly increasing as they co-occur more frequently.

## Illustration of Network-Based Measures of Human Capital

The worker and job networks effectively summarize the human capital in a particular job market. A major advantage of this method is that it is context-dependent—a human capital network is constructed using data from a particular labor pool, meaning that a worker’s human capital measures will also be context-dependent, which reflects differences in how her skills are valued in different markets (17). We will now show this method in a particular context: an online freelance labor market called UpWork. (These data both are publically available and do not require interaction with any individual. Thus, it does not qualify as human subjects research according to the Institutional Review Board at Carnegie Mellon University.)

### Data.

The world of online freelancing is a natural data source for this method. Online freelance labor has been a growing part of the economy fueled by a combination of better technology and evolving attitudes toward career change, and that growth is expected to continue (22). As a result, UpWork and associated markets (e.g., Elance and ODesk) have been the subject of much recent study (23???27).

In the UpWork market, workers apply for jobs, and employers hire and pay workers through the site. The information on a worker’s profile includes a list of her skills and a list of previous jobs with associated hourly wages. Job postings contain information about the job, including a list of required skills. The job opportunities range from small tasks, such as data entry and software testing, to large-scale projects, like application development and website design.

We use a sample of 26,046 worker profiles and 365,561 job listings collected over a period of 3 mo between November 2013 and January 2014. We use this full population to construct our skill networks, because all of the workers are visible to employers and thus, part of the labor market; 18,283 workers have a wage history on the site. A worker’s average hourly wage is calculated from her wage history, which both is publicly visible and cannot be altered (26). While both hourly and flat rate jobs are listed, we only consider hourly jobs in calculating the wage rate, because we do not observe the hours worked on fixed price jobs. The average worker on the site makes $16.74/h, has worked 765 h on the site, and lists six to seven skills. The distribution of workers’ hourly wages shows typical inequality (SI Appendix has summary statistics). The worker profiles include 2,197 different skills, and the job postings include 2,447. Workers and employers must choose their listed skills from the site’s database of allowable skills. Adding a skill to this database requires a petition by a worker/employer and is only granted if the skill is not redundant. This eliminates any ambiguity in skills caused by spelling errors or synonymous entries and makes the data ideal for this application (23). We drop skills that occur only once in our sample. (The results that follow are no different than they would be with those skills included, because the skill similarity weights for these links are, by definition, one.) We are left with 1,933 worker skills and 2,293 job skills. We then use these data to construct supply side and demand side human capital networks as detailed above. ### An Endogenous Skill Taxonomy. Placing the skills on a network allows us to use the deep toolbox of network analysis to reveal underlying structure in the job market, such as subpools of labor that are evaluated similarly by employers and categories of jobs for which there is no dedicated labor pool. Here, we partition the human capital networks into groups of related skills using the Louvain method: a standard community-finding algorithm (28). We show that this division is significant using modularity. (SI Appendix has additional analysis concerning the significance of this categorization.) The modularity of a partition is proportional to the number of links within a group, relative to what would be expected in a random network. The modularity of the worker network is 0.47, and the modularity of the job network is 0.5, indicating that this division is very strong, and represents real community structure in the network. [It is widely held that any network with modularity above 0.3 has significant community structure (29).] We will call these skill groups “categories.” The skill categories are easily identified using the list of skills in each (SI Appendix has the most common skills in each category). The categories are represented by the colors in Fig. 1. For clarity, we have limited the visualization to skills listed by at least 0.5% of worker profiles and at least 0.2% of job listings (the full networks containing all communities are pictured in SI Appendix). We have attached names to each of the categories based on that identification. On the worker side, the skill categories are (i) administration, writing, and marketing; (ii) art and design; (iii) software testing; (iv) statistics and mobile development; (v) information technology (IT) administration; and (vi) general programing. Jobs divide into far more categories, presumably because jobs are more specific than workers. The categories here are (i) administrative, (ii) writing, (iii) translation, (iv) marketing, (v) art and design, (vi) music and audio, (vii) software testing, (viii) engineering and physical design, (ix) data handling and statistics, (x) mobile and game development, (xi) IT administration, and (xii) general programing. Note that jobs divide into more categories than the workers, suggesting that there are well-defined jobs that lack a well-defined labor pool (e.g., while there are significant numbers of jobs that could be categorized under “translation,” few workers could be definitively identified as “translators”). This categorization of skills is useful for several reasons. It allows us to quantify the diversity of a worker’s skills. Those whose skills fall into a single category are specialists, while those who bridge categories are generalists. The specialists can be further divided by their area of specialty. These denote different subpools of labor, which are likely identifiable to employers observing the labor market. Employers use this kind of “low-bandwidth” information as a substitute for more costly search mechanisms (26). In the case of UpWork, there is substantial overlap between the endogenous categories derived from the network and their exogenously generated job categories. (They do not provide information about worker categories.) However, there are some notable differences between the two. The network suggests that data analysis and software testing require very different sets of skills: the workers in those areas are in distinct labor subpools and are qualified for different jobs. Given that worker search and employer search are guided by these categories, employer/employee matches might be improved by allowing jobs to be categorized by the labor market itself. (SI Appendix has more on the differences between the endogenous and exogenous skill taxonomies.) ### Human Capital and Wages. We will now illustrate how human capital networks can shed light on the relationship between human capital and wages. We provide three examples where skill interactions might be relevant: worker type, skill diversity, and synergies between skills. By definition, these are aspects of human capital that cannot be captured with independent skills. Note that what follows is not a complete treatment of any of these questions. Our simple illustration faces many of the typical problems that arise in identifying wage determinants, and we will not attempt to make any causal statements. Our intent is to simply illustrate how more nuanced measures of human capital can provide greater insight into the correlations between worker skills and wages. We leave the identification of causal relationships to future work. ### Skill Categories. We consider how different types of workers are valued using the skill categories defined above. In the worker network, workers whose skills are in a single area are specialists, representing a worker “type.” Fig. 2A shows wages for workers who specialize in each skill category: technical workers earn, on average,$3.40/h more than artists and designers, who in turn, make an average of $3.00/h more than writers and administrative workers. View larger version: Fig. 2. Average wages for freelance workers with different kinds of human capital. Wages are calculated for all workers who average more than$1/h. Error bars are a 95% confidence interval on the point estimate. (A) Average wages for workers in different areas. (B) Average wages for workers qualified for jobs in different areas (we have omitted two categories with fewer than 20 workers). (C) Average wages for workers with different levels of skill diversity. (D) Average wages for workers with synergistic skills and those whose skills must be applied independently as a jack-of-all-trades.

In the job network, workers whose skills are in a single category are qualified for one type of job. Fig. 2B shows that there is a similar pattern in wages: workers qualified for technical jobs make more than those whose employment opportunities are in creative fields, who in turn make more than those who qualify for administrative tasks (a full breakdown of average wages in each worker and job category is in SI Appendix). These results are similar to those found in previous studies using UpWork’s exogenous categorization of worker skills (22) and the literature using occupation-level data from O*net and similar sources (17).

### The Value of Considering Skill Interactions.

Finally, we address whether these network-based measures provide insight into wages beyond that provided by individual skills. Our baseline will be the most flexible specification using only individual skills. Specifically, we relate log wages to a vector of skill dummies, where <mml:math><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>di=1 if the worker has that skill and <mml:math><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>di=0 otherwise. We include terms for skills that appear in at least 2% of worker profiles: a total of 62 dummies. (When too many dummies are included, the terms become colinear.) As expected, some dummies (e.g., project management) are associated with higher wages, and others (e.g., data entry) are associated with lower wages (Model 1 in SI Appendix, Table S10).

We then consider a model that includes both the dummies for individual skills and the network-based worker skill categories: mobile development, testing, programing, IT administration, and art and design (we omit administration as a comparison). The coefficients on these additional terms are significant with signs in the expected directions (Model 2 in SI Appendix, Table S10). The effect sizes are quite large: for example, workers in programing fields earn 51% (<mml:math><mml:mo>∽</mml:mo></mml:math>$5.40) more than administrative workers with the same skills, while workers in software testing earn 30% (<mml:math><mml:mo>∽</mml:mo></mml:math>$4.00) less. The adjusted <mml:math><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math>R2 of this model is higher than that of the baseline model, indicating that the network-based measures explain variance in worker wages, even when controlling for the skills individually.

The story is similar for the other network-based measures. In models containing both the skill dummies and the number of worker categories/job categories crossed, the network-based measures have significant coefficients (Models 3 and 4 in SI Appendix, Table S10). The effect sizes on these terms are smaller but still notable. Workers with skills in two different worker categories earn an average of $0.63 more than their more specialized peers. Workers who qualify for two different types of job earn an average of$0.62 more than those with more constrained options. Again, the adjusted <mml:math><mml:msup><mml:mi>R</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:math>R2 of both models is higher than that of the baseline.

Ideally, we would also examine a model with a more traditional one-dimensional measure of human capital, such as years of education or experience. Unfortunately, because of the nature of this dataset, we do not observe either. The closest that we have to such a measure is the number of skills listed on a worker’s profile, which is obviously problematic in this context. However, as this type of data becomes more common, we expect that future studies will perform that comparison.

## Conclusion

The complex nature of human capital in knowledge-based industry has made the measurement of worker skills increasingly difficult. Placing skills relevant for employment decisions onto a network provides both a way to operationalize the interrelationships between skills and a deep toolbox with which to measure them. As online marketplaces play a greater role in matching workers to employers, human capital networks have an increasing number of practical applications. The detailed skill data in these markets lend themselves to aggregation and algorithmic search (30?32). Even small improvements in these algorithms would reduce search frictions and improve employer–employee matches. The networks can also reveal which skills are complementary to an existing skill set, which would help workers decide on which new skills to acquire and how to best appeal to potential employers.

## Acknowledgments

We thank Brian Kovak, Rebecca Lassem, Sarah Feldt-Muldoon, Seth Richards-Shubik, Ben Tengelsen, Ross O’Connell, and the attendees of the 2016 Networks in Economics Conference.

## Footnotes

• ?1Email: andersok{at}andrew.cmu.edu.

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text:
# Wiener criteria for existence of large solutions of nonlinear parabolic equations with absorption in a non-cylindrical domain

created by nguyen on 22 Jul 2018

[BibTeX]

Published Paper

Inserted: 22 jul 2018
Last Updated: 22 jul 2018

Journal: J. Differential Equations
Pages: 40
Year: 2016
Doi: https://www.sciencedirect.com/science/article/pii/S002203961500649X

Abstract:

We obtain a necessary condition and a sufficient condition, both expressed in terms of Wiener type tests involving the parabolic $W_{q'}^{2,1}$- capacity, where $q'=\frac{q}{q-1}$, for the existence of large solutions to equation $\partial_tu-\Delta u+u^q=0$ in non-cylindrical domain, where $q>1$. Also, we provide a sufficient condition associated with equation $\partial_t u-\Delta u+e^u-1=0$. Besides, we apply our results to equation: $\partial_tu-\Delta u+a \nabla u ^p+bu^{q}=0$ for $a,b>0$, $1<p<2$ and $q>1$.

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