text:
# Cognitive Models

This lesson demonstrates how to control the “golem” (McElreath, 2016), the canonical linear ballistic accumulation (LBA) model (Brown & Heathcote, 2008). Please refer to the LBA paper for more details. Here we focus only on how to use this model in the context of Bayesian MCMC.

The LBA model posits a latent matching (M) factor and a response factor (R) on top of regular experimental factors. For most people who are not familiar with the LBA model, the two factors are unfortunately confusing.

Also for the modelling technicalities, the LBA model must fix one of the parameters in the mean_v or sd_v in at least one design cell. This is to serve as scaling purpose, similar to the moment-to-moment variability in the decision diffusion model. For example, the following code fixes sd_v = 1.

require(ggdmc)
model <- BuildModel(
p.map     = list(A = "1", B = "1", t0 = "1", mean_v = "M", sd_v = "1",
st0 = "1"),
match.map = list(M = list(s1 = 1, s2 = 2)),
factors   = list(S = c("s1", "s2")),
constants = c(st0 = 0, sd_v = 1),
responses = c("r1", "r2"),
type      = "norm")

In the above model, I define only one experimental factor, S, for stimulus, which has two levels, s1 and s2. The accuracy, reflected by the M factor, is mapped by s1 = 1 and s2 = 2, meaning that a correct response for s1 (or s2) stimulus is response r1 (or r2) and an error response for s1 (or s2) stimulus is r2 (or r1). Below I use simulate to generate an example data set.

p.vector <- c(A = .75, B = 1.25, t0 = .15, mean_v.true = 2.5, mean_v.false = 1.5)
dat <- simulate(model, nsim = 30, ps = p.vector)
dmi <- BuildDMI(dat, model)  ## DMI stands for data model instance.

dplyr::tbl_df(dmi)
# A tibble: 60 x 3
##    S     R        RT
##    <fct> <fct> <dbl>
##  1 s1    r1    0.608
##  2 s1    r1    0.972
##  3 s1    r2    0.817
##  4 s1    r1    0.718
##  5 s1    r1    0.618
##  6 s1    r1    1.17
##  7 s1    r1    0.730
##  8 s1    r2    0.727
##  9 s1    r1    0.711
## 10 s1    r1    0.688

I use an imaginary experiment with a design of one binary stimulus factor (S), such as left vs. right motion random dots.

match.map = list(M = list(left = “LEFT”, right = “RIGHT”)), responses = c(“LEFT”, “RIGHT”),

In another tutorial, I will fit the model to an empirical data set (Cox & Criss, 2017) to demonstrate fitting HLBA model.

The above match.map code shows the usage of strings, instead of numbers. The “left” and “LFET” could mean the random dots moving left and a left response. From an experimenter’s perspective, this imaginary experiment only has one stimulus (S) factor, which has two levels, random dots moving towards right and moving towards left as defined below.

factors = list(S = c(“left”, “right”)),

Below is the complete model definition.

model <- BuildModel(p.map = list(A = "1", B = "1", t0 = "1", mean_v = "M",
sd_v = "M", st0 = "1"),
constants = c(st0 = 0, sd_v.false = 1, mean_v.false = 0),
match.map = list(M = list(left = "LEFT", right = "RIGHT")),
factors   = list(S = c("left", "right")),
responses = c("LEFT", "RIGHT"),
type      = "norm")

The first option in the BuildModel function, p.map indicates the experimental design. In this example, I assumed the S factor does not affect any LBA latent variables operations. Therefore, I entered p.map as:

p.map = list(A = “1”, B = “1”, t0 = “1”, mean_v = “M”, sd_v = “M”, st0 = “1”),

The notations of the parameters in the LBA model refer to:

1. A, the variability of the starting point,
2. B, the travelling distance of accumulators,
3. b, (not shown in the p.map) the decision threshold,
4. t0, the non-decision time
5. mean_v, the means of the drift rates,
6. sd_v, the standard deviations of the drift rates,
7. st0, the variability of the non-decision time component.

The A = “1”, for instance, indicates that the variability of the starting point is fit by the intercept, 1. The M factor, because it is defined by the LBA model as a latent factor, you still see it in the p.map. It indicates there are two drift rate means in mean_v, one for each accumulator: the accumulator for the correct / matched responses and the accumulator for the error / mismatched responses. Similarly, this is also applied to the standard deviation of the drift rates, sd_v.

The only effect in the model defined in the p.map is that the drift rate for a correct response is larger than that for an error response. This is an assumption based on, in general, psychological literature. This is artificially set at

constants = c(st0 = 0, sd_v.false = 1, mean_v.false = 0),

which enforces mean_v.false = 0. This is to presume (also frequent observed phenomenon) that manifested accuracy rate should usually be greater than chance (50%).

mean_v.false stands for the mean of the drift rate of the error (false) accumulator. Because it is always zero, the correct drift rate, mean_v.true, if drawn from a truncated normal distribution bounded by 0 and Inf, will always be larger than the error drift rate.

### Demo 1

1. Fast and error prone performance This demonstration shows how I control the LBA golem to simulate fast and error prone RT distributions. I defined a true parameter vector, defining sd_v.true = (0.66), which is smaller than sd_v.false = 1. This seems often seen in empirical data.
pvec1 <- c(A = 1, B = 0, t0 = .2, mean_v.true = 1, sd_v.true = 0.66)
dat1  <- simulate(model, ps = pvec1, nsim = 1e4)
dmi1  <- BuildDMI(dat1, model)

In the following, I used functions in dplyr to print out the mean response times and accuracy for each stimulus types. The results showed:

1. Error and correct responses have similar average RTs.
2. Stimulus type 1 and stimulus type 2 have similar rates of correctness.
library(dplyr)

## dplyr
library(dplyr)
dat1$C <- dat1$S == tolower(dat1\$R)
d <- dplyr::tbl_df(dat1)

## Print average RTs and accuracy rates for each condition
group_by(d, S, C) %>% summarize(m = mean(RT))
## A tibble: 4 x 3
## Groups:   S [?]
##   S     C         m
##   <fct> <lgl> <dbl>
## 1 left  FALSE 0.624
## 2 left  TRUE  0.645
## 3 right FALSE 0.639
## 4 right TRUE  0.634

group_by(d, S, C) %>% summarize(m = length(RT) / 1e4)
## A tibble: 4 x 3
## Groups:   S [?]
##   S     C         m
##   <fct> <lgl> <dbl>
## 1 left  FALSE 0.391
## 2 left  TRUE  0.609
## 3 right FALSE 0.392
## 4 right TRUE  0.608

## data.table
library(data.table)
DT <- data.table(dat1)

## Print average RTs and accuracy for each condition
DT[, .(MRT = round(mean(RT), 3)), .(S, C)]
##        S     C   MRT
## 1:  left  TRUE 0.645
## 2:  left FALSE 0.624
## 3: right  TRUE 0.634
## 4: right FALSE 0.639

prop <- DT[, .N, .(S, C)]
prop[, NN := sum(N), .(S)]
prop[, acc := round(N/NN, 2)]
## Print accuracy rates for each condition
prop

##        S     C    N    NN  acc
## 1:  left  TRUE 6092 10000 0.61
## 2:  left FALSE 3908 10000 0.39
## 3: right  TRUE 6079 10000 0.61
## 4: right FALSE 3921 10000 0.39


text:
# zbMATH — the first resource for mathematics

On the growth of entire and meromorphic functions of infinite order. (English) Zbl 0722.30016
Let $$\phi$$ (x) be a positive increasing function satisfying $$\int^{\infty}dx/\phi (x)<\infty$$. It is shown that if f is a meromorphic function, then there exists a set E of logarithmic density 1 such that $\lim_{r\to \infty,r\in E}\frac{\log M(r)}{T(r)\phi (\log T(r))\log (\phi (\log T(r)))}=0.$ Here M(r) and T(r) denote the maximum modulus and Nevanlinna characteristic of f. If f is entire, then $\lim_{r\to \infty,r\in E}\frac{\log M(r)}{T(r)\phi (\log T(r))}=0.$ The proof depends on the Poisson-Jensen formula, a growth lemma for real functions, and - in the case that f is meromorphic - Cartan’s lemma. Next it is shown that the condition that $$\int^{\infty}dx/\phi (x)<\infty$$ cannot be weakened. To construct examples which show this, the authors first use conformal mappings of strips to construct a function which is harmonic in the plane except on certain curves where it is subharmonic or superharmonic. Then they approximate this function by log$$| f|$$ where f is meromorphic and has zeros (respectively poles) on these curves. The estimation of the asymptotic behavior of the conformal mapping of the strip seems to require some regularity hypothesis on $$\phi$$ (x) which is not stated in the paper, but will be given in a forthcoming correction (same journal, to appear).

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory 30D15 Special classes of entire functions of one complex variable and growth estimates 30D20 Entire functions of one complex variable, general theory 30D30 Meromorphic functions of one complex variable, general theory
##### Keywords:
infinite order; maximum modulus; Nevanlinna characteristic
Full Text:
##### References:
 [1] W. Al-Katifi,On the asymptotic values and paths of certain integral and meromorphic functions, Proc. London Math. Soc. (3)16 (1966), 599–634. · Zbl 0145.30801 · doi:10.1112/plms/s3-16.1.599 [2] V. F. Azarin,The asymptotic behavior of subharmonic and entire functions, Dokl. Akad. Nauk SSSR229 (1976), 1289–1291. · Zbl 0352.31001 [3] W. Bergweiler,Maximum modulus, characteristic and area on the sphere, preprint (1990). · Zbl 0703.30025 [4] C. T. Chuang,Sur la croissance des fonctions, Kexue Tongbao26 (1981), 677–684. · Zbl 0496.30019 [5] W. H. J. Fuchs,Topics in Nevanlinna theory, Proc. of the NRL Conference on Classical Function Theory, 1970. · Zbl 0294.30021 [6] N. V. Govorov,On Paley’s problem, Funkts. Anal. Prilozh.3 (1969), 35–40. [7] W. K. Hayman,Meromorphic Functions, Oxford, 1964. [8] W. K. Hayman,Subharmonic Functions, Vol. 2, Academic Press, London, 1989. · Zbl 0699.31001 [9] T. Kövari,A gap theorem for entire functions of infinite order, Michigan Math. J.12 (1965), 133–140. · Zbl 0152.06604 · doi:10.1307/mmj/1028999302 [10] I. I. Marchenko and A. I. Shcherba,Growth of entire functions, Siberian Math. J. (Engl. Transl.)25 (1984), 598–606. · Zbl 0581.30025 · doi:10.1007/BF00968899 [11] R. Nevanlinna,Analytic Functions, Springer, New York, 1974. · Zbl 0014.16304 [12] V. P. Petrenko,The growth of meromorphic functions of finite lower order, Izv. Akad. Nauk SSSR33 (1969), 414–454(in Russian). · Zbl 0194.11001 [13] G. Valiron,Sur un theorème de M. Wiman, in: Opuscula Math. A. Wiman dedicata, 1930, pp. 1–12. [14] A. Wahlund,über einen Zusammenhang Zwischen dem Maximalbetrage der ganzen Funktion und seiner unteren grenze nach dem Jensensche Theoreme, Arkiv Math.21A (1929), 1–34. · JFM 55.0775.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.

format: plain


text:
# XML Schema File that validates my XML File

I just want to know if my schema file is correct:

XML File:

  <?xml version="1.0" encoding="UTF-8"?>
<xs:books xmlns="http://www.w3.com/1999/XML/prjXMLValidation"
xmlns:xs="http://www.w3.org/2001/XMLSchema/prjXMLValidation"
xs:noNamespaceSchemaLocation="collection.xsd">
<xs:book title="A Lesson Before Dying" author="Ernest J. Gaines" version="1">
<xs:stars>
5
</xs:stars>
<xs:description>
This is a story about a young man that was wrongly
accused of killing a white man in the south at a time
when blacks were not given the same rights as whites.
This is the story of how Jefferson becomes a man.
</xs:description>
<xs:thoughts>
Great book!
</xs:thoughts>
</xs:book>
</xs:books>

Here is my XSD File:

<?xml version="1.0" encoding="utf-8"?>
<xs:schema elementFormDefault="qualified" xmlns:xs="http://www.w3.org/2001/XMLSchema"
targetNamespace="http://www.w3.org/2001/XMLSchema/prjXMLSchema">
<xs:element name="books">
<xs:complexType>
<xs:element name="book">
<xs:complexType>
<xs:sequence minOccurs="1" maxOccurs="unbounded">
<xs:element name="stars">
<xs:complexType>
<xs:choice minOccurs="0" maxOccurs="1">
<xs:element ref="1" />
<xs:element ref="2" />
<xs:element ref="3" />
<xs:element ref="4" />
<xs:element ref="5" />
</xs:choice>
</xs:complexType>
</xs:element>
<xs:element name="description" type="xs:string" />
<xs:element name="thoughts" type="xs:string" />
</xs:sequence>
<xs:attribute name="title" type="xs:string" use="required" />
<xs:attribute name="author" type="xs:string" use="required" />
<xs:attribute name="version" type="xs:string" />
</xs:complexType>
</xs:element>
</xs:complexType>
</xs:element>
</xs:schema>

• Well, you may want to include the XSD file collection.xsd ... ;-) – rolfl Feb 11 '14 at 5:01

Overall your structures look OK. Your actual XML document could be indented more neatly, but it's OK.

I would recommend that you change your stars element to have an attribute. Using the PCDATA space is not a great place to put a numeric value.... Actually, I would recommend adding a stars attribute to the thoughts element (seems like a logical place to put it)....

<xs:thoughts stars="5" >
Great book!
</xs:thoughts>

This brings up the validation of the stars value too.... you have broken references in your XSD:

          <xs:complexType>
<xs:choice minOccurs="0" maxOccurs="1">
<xs:element ref="1" />
<xs:element ref="2" />
<xs:element ref="3" />
<xs:element ref="4" />
<xs:element ref="5" />
</xs:choice>
</xs:complexType>

The references 1 .... 5 are not declared. This is a problem. But, you should be using references (just not those ones....). You should be separating out your XSD types and then merging them up in complex-types with ref references. You should also be declaring the types outside of their usage... which makes things easier to manage.

Consider the following (which changes the stars validation, and location).

<xs:simpleType name="startype">
<xs:restriction base="xs:string">
<xs:pattern value="[1-5]"/>
</xs:restriction>
</xs:simpleType>

.....
<xs:element name="thoughts">
<xs:complexType>
<xs:simpleContent>
<xs:extension base="xs:string">
<xs:attribute name="stars" type="startype" />
</xs:extension>
</xs:simpleContent>
</xs:complexType>
</xs:element>

• Sorry about the formatting of my XML file. It was not like that originally, but when I put my code into the site it refused to format correctly. At least my XSD turned out the way I made it. And thank you for your advice! – tylerjholmgren Feb 11 '14 at 11:46


text:
# The locale of real numbers

## Summary

This page gives an elementary description of the locale of real numbers, that is the localic real line. The development is manifestly constructive and even predicative over the natural numbers (although we are somewhat careless with the language and do not always point out when a set may predicatively be a proper class). Ideally, we will show that our construction satisfies the seven ‘headline properties’ of the real line described by Bauer & Taylor (although so far we cover only the Heine–Borel theorem).

The exposition here is pretty much my own work, although of course the basic ideas are well known to many. In particular, the technical zigzag lemma is mine (but it is not a very deep result!).

## Idea

To describe the locale of the real numbers, we need first of all to describe what an open in the real line is. The key insight is that an open set is determined by the open intervals of rational numbers that it contains. This is analogous to the key insight of Richard Dedekind's definition of real number: that a point in the real line is determined by which open intervals of rational numbers that it belongs to. (Once you're talking about rational numbers, things are manageable.)

## Opens

An open in the real line is a binary relation $\sim$ on the rational numbers that satisfies the four properties listed below. Intuitively, we have $a\sim b$ iff the open interval $\left(a,b\right)$ is contained in the corresponding open set.

1. If $a\ge b$, then $a\sim b$.
2. If $a\ge b\sim c\ge d$, then $a\sim d$.
3. If $a\sim b>c\sim d$, then $a\sim d$.
4. If $b\sim c$ whenever $a and $c, then $a\sim d$.

Property (1) is motivated because $\left(a,b\right)$ is empty whenever $a\ge b$. Property (2) is motivated because $\left(a,d\right)\subseteq \left(b,c\right)$ whenever $a\ge b$ and $c\ge d$. Property (4) is somewhat technical; it keeps $\left\{a,b\phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}a from being an open, while $\left\{a,b\phantom{\rule{thickmathspace}{0ex}}\mid \phantom{\rule{thickmathspace}{0ex}}a is allowed.

The really interesting property is property (3). It in fact generalises as follows:

• If

(1)${a}_{1}\sim {b}_{1}>{a}_{2}\sim {b}_{2}>\cdots >{a}_{n}\sim {b}_{n},$a_1 \sim b_1 \gt a_2 \sim b_2 \gt \cdots \gt a_n \sim b_n ,

then ${a}_{1}\sim {b}_{n}$.

We call the combined hypothesis of this property a zigzag; each hypothesis ${a}_{i}\sim {b}_{i}$ is a zig, and each hypothesis ${b}_{i}>{a}_{{i}^{+}}$ is a zag. To indicate the length of a zigzag, we will count the zigs; the zigzag (1) has $n$ zigs (and $n-1$ zags). A typical nondegenerate zigzag with $3$ zigs is shown below; it consists of $3$ overlapping open intervals, each of which belongs to a given open set; we are motivated to conclude that the entire interval from ${a}_{1}$ to ${b}_{3}$ belongs to that open set.

Thus the zigzag property for $n=2$ is property (3), the zigzag property for $n=1$ is trivial (if $a\sim b$, then $a\sim b$), and the zigzag property for $n>2$ may be proved by induction. (There is also a sense in which the zigzag property for $n=0$ is property (1), but I haven't quite got my head around that. It would be awesome if the zigzag property for $n=\infty$ could be interepreted as property (4), but that seems unlikely so far.)

If $G$ is an open in the real line, then we write $\left(a,b\right)\subseteq G$ to mean that $a$ is related to $b$ through $G$ and say that $G$ contains the interval from $a$ to $b$.

## The zigzag lemma

We will often have occasion to consider a zigzag as in (1), but where the various zigs are taken relative to different opens. Nevertheless, we will usually want to say something about the relationship of ${a}_{1}$ to ${b}_{n}$, and it will often be convenient to assume, for all $i$, that ${a}_{1}\le {a}_{i}$ and ${b}_{i}\le {b}_{n}$. We may do so without loss of generality as follows: consider the smallest value of $i$ such that ${b}_{i}\ge {b}_{n}$, truncate the zigzag so that it now has only $i$ zigs, and change the new endpoint ${b}_{i}$ to the old endpoint ${b}_{n}$. The final zig $\left({a}_{i},{b}_{i}\right)\subseteq {G}_{i}$ becomes $\left({a}_{i},{b}_{n}\right)\subseteq {G}_{i}$, which will hold by property (2) since ${b}_{i}\ge {b}_{n}$. Now consider the largest value of $j\le i$ such that ${a}_{1}\ge {a}_{j}$, and truncate similarly on the other side. We now have a zigzag with $i-j+1$ zigs that never falls below ${a}_{1}$ on a zag nor rises above ${b}_{n}$ on a zig.

To be precise:

###### Zigzag lemma

Given a collection $𝒞$ of opens and a zigzag in which each zig holds relative to some open in $𝒞$, there exists a zigzag (1) in which each zig holds relative to some open in $𝒞$, ${a}_{1}\le {a}_{i}$ for each $i$, and ${b}_{i}\le {b}_{n}$ for each $i$.

(Since each ${b}_{i}>{a}_{{i}^{+}}$, it follows further that ${a}_{i}<{b}_{n}$ for each $i>1$ and ${a}_{1}<{b}_{i}$ for each $i, although this corollary doesn't seem to be particularly useful.)

At the moment, this lemma is used only in the proof of the infinite distributivity law.

## The frame of opens

The opens form a sub-poset of the power set $𝒫\left(ℚ×ℚ\right)$. This poset is in fact a frame, as we will now show. (It is not a subframe of the power set, since the joins are different. It is a sub-inflattice of the power set, although this seems to be a red herring at least for infinitary meets, since those are not part of the frame structure that we need.)

The top open, denoted $ℝ$, is the binary relation which is always true. Given two opens $G$ and $H$, their meet in the poset of opens, denoted $G\cap H$, is simply their conjunction, that is their intersection as subsets of $ℚ×ℚ$. (In fact, given any collection of opens, their meet is their conjunction.) It is straightforward to check that $ℝ$ and $G\cap H$ are opens and to prove that these are the desired meets. Intuitively, this all works because an open interval will be contained in the intersection of a family of open sets if and only if it is contained in each individual open set.

The bottom open, denoted $\varnothing$, is the binary relation $\ge$. That is, $\left(a,b\right)\subseteq \varnothing$ iff $a\ge b$. It is easy to check that this is an open; it precedes every open by property (1). Intuitively, this corresponds to the empty subset of the real line; $\left(a,b\right)$ is empty if and only if $a\ge b$. However, note that $\varnothing$ is not the empty subset of $ℚ×ℚ$; the notation follows our topological intuition rather than the algebra of relations.

Given two opens $G$ and $H$, their join in the poset of opens, denoted $G\cup H$, is defined as follows: $\left(a,b\right)\subseteq G\cup H$ if and only if there exists a zigzag (1) with $a={a}_{1}$ and $b={b}_{n}$ in which each zig is of the form $\left({a}_{i},{b}_{i}\right)\subseteq G$ or $\left({a}_{i},{b}_{i}\right)\subseteq H$. It is immediate that this is an open in which $G$ and $H$ are both contained. Conversely, any open in which $G$ and $H$ are contained must contain this open $G\cup H$, by property (3).

More generally, given any family $\left({G}_{k}{\right)}_{k}$ of opens, their join in the poset of opens, denoted ${\bigcup }_{k}{G}_{k}$, is defined as follows: $\left(a,b\right)\subseteq {\bigcup }_{k}{G}_{k}$ if and only if $a\ge b$ or there exists a zigzag (1) with $a={a}_{1}$ and $b={b}_{n}$ in which each zig is of the form $\left({a}_{i},{b}_{i}\right)\subseteq {G}_{k}$ for some $k$. (We can leave out the $a\ge b$ clause if the family is inhabited.) The same argument applies as before. Note that each individual zigzag has finitely many zigs, and therefore involves finitely many of the opens ${G}_{k}$, even when taking the join of an infinite family.

Finally, we must check the distributive law $G\cap {\bigcup }_{k}{H}_{k}\subseteq {\bigcup }_{k}\left(G\cap {H}_{k}\right)$. That is, if $a\sim b$ directly through $G$ and $a\sim b$ through a zigzag of $H$s (or $a\ge b$), then we need that $a\sim b$ through a zigzag in which each zig is related both through $G$ and through some $H$ (or $a\ge b$). To prove this (ignoring the trivial case where $a\ge b$), start with the zigzag of $H$s, and apply the zigzag lemma to get a zigzag of $H$s in which each zig involves values bounded by $a$ and $b$. Then these zigs hold for $G$ as well, by property (2). Therefore, we may interpret each zig using $G\cap {H}_{k}$ for some $k$, proving the desired result.

This frame of opens, interpreted as a locale, is the locale of real numbers. As usual, we denote this locale with the same symbol as the top element of its frame, in this case $ℝ$. (Of course, the true etymology of the symbols runs in the other order.)

## Open intervals as opens

Given rational numbers $a$ and $b$, the open interval $\left(a,b\right)$ may itself be interpreted as an open in the real line, also denoted $\left(a,b\right)$, as follows: let $\left(c,d\right)\subseteq \left(a,b\right)$ hold if every rational number strictly between $c$ and $d$ (in that order) is also strictly between $a$ and $b$ (in that order). In other words, we interpret ‘$\subseteq$’ literally as comparing subsets of $ℚ$. It is straightforward to check that this condition does indeed define an open. There is now a third way to interpret ‘$\left(c,d\right)\subseteq \left(a,b\right)$’; interpreting both intervals as opens in the real line, this states that the first is contained in the second. But again, it is easy to check that this is equivalent; $\left(c,d\right)\subseteq \left(a,b\right)$ (in the set-theoretic sense) if and only if $\left(e,f\right)\subseteq \left(a,b\right)$ whenever $\left(e,f\right)\subseteq \left(c,d\right)$. Notice that $\left(a,b\right)=\varnothing$ whenever $a\ge b$.

We can actually generalise this somewhat. Given any set $L$ of rational numbers and any set $U$ of rational numbers, we may define the open $\left(\mathrm{inf}U,\mathrm{sup}L\right)$ as follows: let $\left(a,b\right)\subseteq \left(\mathrm{inf}U,\mathrm{sup}L\right)$ hold if every rational number strictly between $a$ and $b$ (in that order) is greater than some element of $U$ and less than some element of $L$. If the infimum of $U$ and the supremum of $L$ exist in the usual sense as rational numbers, then this agrees with the previous paragraph. If instead $U$ or $L$ is the set of all rational numbers, then we write $-\infty$ for $\mathrm{inf}U$ or $\infty$ for $\mathrm{sup}L$. In general, we may interpret $\mathrm{inf}U$ as an extended upper real and $\mathrm{sup}L$ as an extended lower real. Classically, every extended upper or lower real is either a real number, $-\infty$, or $\infty$; only the converse holds constructively. Notice that $\left(-\infty ,\infty \right)=ℝ$.

Since we will refer to them below, we will state for the record the complete definitions of $\left(-\infty ,a\right)$ and $\left(a,\infty \right)$ for a rational number $a$. We have $\left(b,c\right)\subseteq \left(-\infty ,a\right)$ iff every rational number strictly between $b$ and $c$ is less than $a$, that is iff $b\ge c$ or $c\le a$. Similarly, we have $\left(b,c\right)\subseteq \left(a,\infty \right)$ iff every rational number strictly between $b$ and $c$ is greater than $a$, that is iff $b\ge c$ or $b\ge a$. A fortiori, $\left(b,c\right)\subseteq \left(-\infty ,0\right)$ if $c=0$, and $\left(b,c\right)\subseteq \left(1,\infty \right)$ if $b=1$.

## Open sets, closed sets, and points

We think of each open as defining an open subset of the real line, but we can equally well think of it as defining a closed subset. The difference between these perspectives is reflected in complementary criteria for when a real number belongs to the set. So to make sense of this, we must identify the points of the real line.

Recall that a real number may be defined as a pair $\left(L,U\right)$ of inhabited subsets of $ℚ$ satisfying the following properties:

1. If $a\in L$, then $a for some $b\in L$.
2. If $b\in U$, then $a for some $a\in U$.
3. If $a\in L$ and $b\in U$, then $a.
4. If $a, then $a\in L$ or $b\in U$.

We define a point of the real line to be a real number in this sense. Given such a point $x=\left(L,U\right)$ and a rational number $a$, we write $a to mean that $a\in L$ and $a>x$ to mean that $a\in U$. If we wish to refer to $L$ and $U$ directly, we may call $L$ the lower set of $x$ and $U$ the upper set.

Given a point $x$ and an open $G$, we say that $x$ belongs to $G$, written $x\in G$, if $\left(a,b\right)\subseteq G$ for some $a and $b>x$; that is, $G$ contains an interval from some element of the lower set to some element of the upper set of $x$. We have $x\in ℝ$ since its lower and upper sets are inhabited. If $x\in G$ and $x\in H$, with $\left(a,b\right)\subseteq G$ and $\left(c,d\right)\subseteq H$, then $\left(\mathrm{max}\left(a,c\right),\mathrm{min}\left(b,d\right)\right)\subseteq G\cap H$, so $x\in G\cap H$; note that this argument fails for infinitary intersections. (The converse, that $x\in G$ and $x\in H$ if $x\in G\cap H$, is immediate.) Dually, suppose that $x\in {\bigcup }_{k}{G}_{k}$, as shown by some zigzag (since $a\ge b$ is impossible when $a and $b>x$, by 3). Applying condition (4) of the definition of real number to each zag, we have ${a}_{{i}^{+}} or ${b}_{i}>x$, for each $i. Checking all ${2}^{n-1}$ possibilities, and knowing that ${a}_{1} and ${b}_{n}>x$ in any case, we must have ${a}_{i} and ${b}_{i}>x$ for some $i$. Then we have $x\in {G}_{k}$ for some $k$, whichever corresponds to the $i$th zig. (The converse, that $x\in {\bigcup }_{k}{G}_{k}$ if $x\in {G}_{k}$ for some $k$, is immediate.)

Therefore, each point defines a completely prime filter on the frame of all opens, which is the definition of a point in general locale theory. Conversely, given such a completely prime filter $P$, let $L$ be the set of all rational numbers $a$ such that the open interval $\left(a,\infty \right)$ (when interpreted as an open in the real line as defined above) is in $P$, and symmetrically let $U$ be the set of all $a$ such that $\left(-\infty ,a\right)\in P$.

Given a point $x$ and an open $G$, we say that $x$ co-belongs to $G$, written $x\notin G$, if we never have $a, $b>x$, and $\left(a,b\right)\subseteq G$, which is precisely the negation of the property that $x\in G$. We think of this condition as defining a closed set to which $x$ does belong. Notice that $x\notin {\bigcup }_{k}{G}_{k}$ if and only if $x\notin {G}_{k}$ for each $k$, giving the desired behaviour for an arbitrary intersection of closed sets (which corresponds to union of open sets under de Morgan duality). We also have that $x\notin ℝ$ always fails, and $x\notin G\cap H$ if $x\notin G$ or $x\notin H$. To prove that $x\notin G$ or $x\notin H$ whenever $x\notin G\cap H$, however, we must use excluded middle; constructively, closed sets don't behave well under union.

A related question is whether we can reconstruct $G$ from the set of points which belong to it. This should be equivalent to the fan theorem, which is classically true and also accepted by Brouwer's school of intuitionism, but refuted in the Russian school in which all real numbers are assumed to be computable?. (I should check this.) Arguably, the real lesson of these logical technicalities is that we should remember that opens, not points, are the fundamental concept in a locale.

## The Heine–Borel theorem

The classical Heine–Borel theorem, as a statement about sets of real numbers, may fail constructively; this is related to the comments above about the fan theorem. But the beauty of the localic approach is that Heine–Borel necessarily holds when interpreted as a statement about opens in the locale of real numbers. To state the theorem, we must define what it means for a collection of opens to cover the unit interval. We will give an ad-hoc definition, but this may also be derived from the general theory of closed sublocales which allows us to interpret the unit interval as a compact locale in its own right.

###### Definition

An open cover of the unit interval is a collection $𝒞$ of opens in the real line such that $ℝ$ is the join of $\left(-\infty ,0\right)$, $\left(1,\infty \right)$, and the elements of $𝒞$.

###### Theorem

(Heine–Borel)

Every open cover of the unit interval has a finite subcover.

###### Proof

The proof is almost embarrassingly simple. The key point is that the construction of joins in terms of zigzags involves only finite zigzags, even for an infinitary join.

Let $J$ be the join of $\left(-\infty ,0\right)$, $\left(1,\infty \right)$, and the elements of $𝒞$. If this equals $ℝ$, then in particular $\left(-1,2\right)\subseteq J$, a fact which is given by some zigzag $\zeta$. This zigzag involves only finitely many of the elements of $𝒞$; let $𝒟$ be the collection of these, and let $K$ be the join of $\left(-\infty ,0\right)$, $\left(1,\infty \right)$, and $𝒟$. Now if $\left(a,b\right)$ is any pair of rational numbers, we construct a zigzag showing that $\left(a,b\right)\subseteq K$ as follows: the zig $\left(a,0\right)\subseteq \left(-\infty ,0\right)$, the zag $0>-1$, the zigzag $\zeta$ from $-1$ to $2$, the zag $2>1$, and the zig $\left(1,b\right)\subseteq \left(1,\infty \right)$. This is always a valid zigzag, so $K=ℝ$. Therefore, the finite collection $𝒟$ covers the unit interval.

This proof generalises immediately to any closed interval $\left[a,b\right]$, for $a$ any upper real and $b$ any lower real. But note that we do not say ‘extended’ here; we need to find some rational number (analogous to $-1$ in the proof above) smaller than $a$ and some rational number (analogous to $2$ above) larger than $b$. So the Heine–Borel theorem applies only to bounded closed intervals.

Revised on August 12, 2011 21:20:01 by Toby Bartels (64.89.53.88)


text:
# Cubic identity 1

Objective
 To verify $(a+b)^3 = a^3+3a^2b+3ab^2+b^3$ using unit cubes

27 Unit cubes

## What is a unit cube?

A unit cube is nothing but a cube all of whose sides are 1 unit long as shown below.
This is a unit cube

1. We say its dimension is 1 X 1 X 1.
2. The volume of a 3-dimensional unit cube is 1 cubic unit.
3. By joining unit cubes we can form cubes and cuboids of varied dimension.


text:
location:  Publications → journals → CJM
Abstract view

Author's Draft

# Relative discrete series representations for two quotients of $p$-adic $\mathbf{GL}_n$

We provide an explicit construction of representations in the discrete spectrum of two $p$-adic symmetric spaces. We consider $\mathbf{GL}_n(F) \times \mathbf{GL}_n(F) \backslash \mathbf{GL}_{2n}(F)$ and $\mathbf{GL}_n(F) \backslash \mathbf{GL}_n(E)$, where $E$ is a quadratic Galois extension of a nonarchimedean local field $F$ of characteristic zero and odd residual characteristic. The proof of the main result involves an application of a symmetric space version of Casselman's Criterion for square integrability due to Kato and Takano.
 Keywords: $p$-adic symmetric space, relative discrete series, Casselman’s Criterion
 MSC Classifications: 22E50 - Representations of Lie and linear algebraic groups over local fields [See also 20G05] 22E35 - Analysis on $p$-adic Lie groups


text:
# Measuring coaxial cable loss by reflection with a VNA

At Measuring coaxial cable loss with a voltmeter I discussed measuring terminated coax cable loss with an RF voltmeter, and it had some real practical limitations.

This article explores using a nanovna to measure line loss in a similar scenario. We will use 6m of Belden 8359 (RG58A/U) @ 3.6MHz.

The same technique could be used with a quality antenna analyser.

## Expectation

A short digression, what is the specification Matched Line Loss (MLL) at 3.6MHz? Using TLLC we get 0.171dB, that is our expectation.

## Return Loss of SC section

A common method proposed is to measure Return Loss (RL) of a section with load end RL=0dB and halve it. Many experts advise that the section should be terminated in a short circuit (S) because short circuits are more reliable than open circuits. So let’s get cracking.

Above is measured |s11| using a nanovna with recent OSL calibration from 1-30MHz. |s11| @ 3.6MHz is by eye -0.651dB, RL=-|S11|, so RL/2=0.651/2=0.325dB.

Hmmm, that is nearly double the expected 0.171dB, time to chuck it?

No not yet, let’s do an open circuit sweep.

## Return Loss of OC section

Above is measured |s11| using a nanovna with recent OSL calibration from 1-30MHz. |s11| @ 3.6MHz is by eye -0.138dB, RL=-|S11|, so RL/2=0.138/2=0.069dB.

Hmmm, that is less than half the expected 0.171dB, a quarter of the SC measurement, what is going on?

## The problem

The problem is that the nanovna was calibrated using a 50+j0Ω load, and its reference impedance is now 50+j0Ω meaning that s11 and therefore RL is wrt 50+j0Ω, and since Zo is more like 51.42-j1.33Ω there is error in the two results that should theoretically agree with each other.

## An approximate solution

A good approximation when the departure of actual Zo from the reference impedance is small is that $$MLL\approx\frac{RL_{sc}+RL_{oc}}4dB$$.

So $$MLL\approx\frac{0.651+0.138}4=0.197dB$$, not a lot worse than spec at 0.171dB, and given measurement uncertainties, we could not say if fails spec.

This is discussed in more detail at On Witt’s calculation of Matched Line Loss from Return Loss .


text:
# iterative method to solve nonlinear equations

I'd like to know whether there are any methods like the Gauss-Seidel method to solve nonlinear equations.

For example, I'd like to solve $f(\textbf x) = 0$, where $f(\textbf x)$ is a nonlinear function and can be split into two parts $f(\textbf x)=f_1(\textbf x)+f_2(\textbf x)$. Solving $f_1(\textbf x) = c$ is quite simple, where $c$ is an arbitrary constant.

As a result, can I solve $f(\textbf x) = 0$ iteratively by solving

$f_1(\textbf x^{(i+1)}) = - f_2(\textbf x^{(i)})$

in each iteration $i$?

When does this method converge?

-
How are $f_1$ and $f_2$ related to the original $f$? What do you mean by "split"? –  bubba Aug 14 '13 at 11:01
@bubba $f = f_1+f_2$ –  Hugo Aug 14 '13 at 11:50
I don't know if your algorithm would work. I suspect not. But there are many available methods for iteratively solving systems of non-linear equations. An internet search should give you plenty to choose from. For example, both the Newton-Raphson algorithm and the secant method can be generalized to the multi-dimensional case. ocw.usu.edu/Civil_and_Environmental_Engineering/… –  bubba Aug 14 '13 at 12:38
@Amzoti No, not Newton-Raphson algorithm. I want to find a method for nonlinear systems corresponding to the Gauss-Seidel method for linear systems. –  Hugo Aug 14 '13 at 13:49
@Hugo: You mean like GS in this? math.fullerton.edu/mathews/n2003/newtonsystem/… –  Amzoti Aug 16 '13 at 5:40


text:
# Urgent HW help

• Mar 13th 2008, 09:12 PM
vballnyy2
Urgent HW help
Provide a relation on the integers that
a) is reflexive, is symmetric, but is not transitive

b) is not reflexive, not symmetric, and is transitive

Help?!
• Mar 14th 2008, 03:16 AM
Plato
Try this relation for part a.
$nRm\, \Leftrightarrow \,\left| {n - m} \right| \le 1$

For part b, think about 'less than'.
• Mar 14th 2008, 05:52 AM
vballnyy2
Thank you so much! I've got it now


text:
# Setting up Hammer

## A Tutorial for Left 4 Dead

SEX! Ok, we got your attention. We are one of the oldest mod sites on the Interwebs. Sign up and maximize your browsing experience.

• Subscribe: Get notified of new submissions like this.
• Say Thanks: Show your appreciation by sending Fearlezz points.
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• Flag: Alert moderators and warn members of a problem with this Tutorial.
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Join the community!

Start mapping for L4D today!

This tutorial will guide you how to set up hammer for Left 4 Dead so you can create your own awesome l4d maps >:D. Written by Fearlezz
Step 1.0 Creating a new mod
1.1
Open up Source SDK. Choose The Orange Box as the engine, and press Create a Mod.
1.2
Choose Half-Life 2 Multiplayer. However, if you can not choose this option, choose anything esle, it does not matter.Now fill in the directory. Doesn't matter where you install it. I choose to install it C:\Program\Left4Fake\. Name; Left4Fake. The name is totally random and you can put in whatever you want. However, I recomend that you do not put Left 4 Dead, or Left4dead. Since it might screw up your SDK when the official SDK is released for L4D
1.3
When its done copying the new files, click continue and then next. Now close down Source SDK and follow the step below.
Written by Occlusion
Step 2.0 Extracting Materials and Models
2.1 2.2
Okay, now you need to add the materials and models from l4d to your new mod. Download the VPK extractor and run it (Green Icon). Now press extract and find the file called "pak01_dir.vpk" which is in this directory; drive:\Program Files\Steam\SteamApps\Common\left4dead\left4dead
2.3
Then another browse box will come up. Now find the mod you just created in your SourceMods Folder (SteamApps\SourceMods\Left4Fake) and press ok. The process of extraction can take around 10 minutes, so don't worry if it takes a while. Now if you were to open your mod with hammer you would see the new textures, but no new models or entities. That's what we are going to fix now!
Written by Fearlezz
Step 3.0 Fixing the models
3.1
First off, you need to download a file that I've created. But Corion on L4Dmaps.net has written this code, I haven't. Download it here;http://www.savefile.com/files/1908708
3.2
Extract this .Bat file to your mod model folder, where you have extracted the L4D models (SteamApps\SourceMods\Left4Fake\models\). If there isn't any models folder, you've done something wrong :S.
3.3
Now Run this file, double click it and a new batch window should pop up. This process will take a while, it copies and deletes different files to make the models work for you =).
3.4
When it is done, you should now have working textures and models in hammer. Although, something is missing, doesn't it? Oh right, the entities, well, lets get started on that then :)
Written by Fearlezz with help from NovaSilisko on L4Dmaps.net
4.1
4.2
Open up hammer, with your custom mod selected. Go to Tools>Options A new window should pop up
4.3
It should look something like this:
I have my .fgd on the desktop, not in the same directory as the other .fgd's.
Written by Fearlezz
Step 5.0 Making Hammer automatically launch L4D after Compile
5.1
Welcome to the final step, to make Hammer launch Left 4 Dead after you have compiled your map, this will also save your maps in the maps folder. Open up the options menu again.
5.2
The other options should be following;
$SteamDir\steamapps\SourceMods\Left4Fake Game Directory:$SteamDir\steamapps\SourceMods\Left4Fake
Hammer VMF Directory:
\$SteamUserDir\sourcesdk_content\l4f
You'll need to create a new folder.
Written by Fearlezz
Step 6.0 Restart and Enjoy!
Guess what? Your done, enjoy the new hammer and dont forget to say thanks! Thanks to: Fearlezz Occlusion NovaSilisko (at l4dmaps.net) Corion (at l4dmaps.net) www.l4dmaps.net
You know how pissed I would be if they would release the official SDK tomorrow? >.>
THE END

### Posts

• 8y
Have any other place that I can download this file (. bat)?
Walker bait
• 8y
Step 2.3....0 files in package. Whats going on here, the main game plays fine, set my friend up using this tut, worked perfect. Any ideas.
Bananite
• 8y
This is the problem, I have no money to buy the game!
Walker bait
• 8y
Posted by estevoo

Hey man, can I extract this files with gfc or whatever???

The official SDK has been released. If you bought L4D then you should have no problems.
• 8y
Hey man, can I extract this files with gfc or whatever???
Walker bait
• 8y
Posted by reyrey723

HELP!!!!! ermmmmmmm i do the starting but the file when i making it,it will always corrupt i type C:\Program\Left4Fake on install and Left4Fake on name but always to the last part of the downloading it corrupt HELP!!!!!!

The official SDK has been released. If you bought L4D then you should have no problems.
• 8y
HELP!!!!! ermmmmmmm i do the starting but the file when i making it,it will always corrupt i type C:\Program\Left4Fake on install and Left4Fake on name but always to the last part of the downloading it corrupt HELP!!!!!!
Bananite
• 8y
Posted by dr.Cola

Why can't i download L4Dfgd.zip? All i get is a 404 error. :/

google it :) i did it too.

But why cant i choose Left4Fake under the orange box. i can even run my "created mod" and stuff -.-
• 8y
Why can't i download L4Dfgd.zip? All i get is a 404 error. :/
• 8y
Ok i did everything it said. i have the folders and everything! models entities.

BUT when i start sdk it wont let me select anything as "current game" :(

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# Is there a unique line bundle in the Kummer surface which pulls back to a totally symmetric line bundle?

Let $X=Jac(C)$ be an abelian surface over $\mathbb{C}$, the Jacobian of a genus 2 curve. Let $L$ be a symmetric line bundle. Let $Y$ be the Kummer surface, quotient of $X$ by the action of involution. Then $L^2$ is totally symmetric, hence there is a line bundle $L'$ on $Y$ which pulls back to $L^2$. Further since $L^2$ embeds $Y$ in $\mathbb{P}^3$, $L'$ is very ample.

1) Is $L'$ the unique line bundle which pulls back to $L^2$?

2) Consider a curve $C\in |L^2|$, which is smooth, preserved under involution and avoiding the 16 double points of $X$. Then the image of $C'$ is smooth and avoids the 16 singular points of $Y$. Can we say that $C'\in |L'|$?

Thanks!

1) Yes. If there is another one, it differs from $L'$ by a line bundle $M$ with $M^{2}\cong \mathcal{O}_Y$. Consider the resolution $\pi :\hat{Y}\rightarrow Y$ obtained by blowing up the double points $p_1,\ldots ,p_{16}$. Since $\hat{Y}$ is simply connected, we have $\pi^* M\cong \mathcal{O}_{\hat{Y}}\$.
Thus $M_{|Y\smallsetminus \{p_i\}}$ is trivial, and this implies that $M$ is trivial.
2) Yes by 1), since the pull-back of $\mathcal{O}_Y(C')$ to $X$ is $\mathcal{O}_X(C)=L^2$.


text:
# User:345Kai

## Sine and Cosine

Fig. 1a - Sine and cosine of the angle θ in the unit circle of a cartesian coordinate system.

In a Cartesian coordinate system, consider the unit circle, which is of radius 1 and centered at the origin (see Figure 1a). The ray (blue) forming angle θ with the positive x-axis intersects the unit circle at a point whose x-coordinate (red) is the cosine and whose y-coordinate (green) is the sine of θ. This defines $\sin\theta$ and $\cos\theta$ for all angles between 0 and 360°. Sine and cosine of θ are real numbers between -1 and +1.

Fig. 1b - Angle θ in the second quadrant. The sine is positive, the cosine negative.
Fig. 1c - Angle θ in the third quadrant. Both sine and cosine are negative.
Fig. 1d - Angle θ in the fourth quadrant. The sine is negative, the cosine positive.

## Cartesian Coordinates

Fig 1 - Cartesian coordinate system with the points (5,12) marked in green, (-3,1) in red, (-1.5,-2.5) in blue and (0,0), the origin, in violet.

## other stuff

$\begin{pmatrix}1&1\\-1&1\end{pmatrix}$

$\begin{pmatrix}2&4\\-1&2\end{pmatrix}$

$\begin{pmatrix}1&0&1\\0&1&0\end{pmatrix}$

$\begin{pmatrix}0&0\\2&2\end{pmatrix}$

$\begin{pmatrix}0\\1\\1\end{pmatrix}$

$\vec x ' (t) = A \, \vec x(t)$

$\vec x (t) = c_1 e^{\lambda_1 t}\vec v_1+\ldots+c_ne^{\lambda_n t} \vec v_n$

$\vec x (k+1) = A \, \vec x(k)$

$\vec x (k) = c_1 \lambda_1^n \vec v_1+\ldots+c_n \lambda_n k \vec v_n$

$\langle T\vec v,\vec w\rangle = \langle \vec v, T\vec w\rangle$ Superscript text $3x+4x+20=$ $z^{m/n}=|z|^{m/n}e^{i\arg(z)m/n}$


text:
# Essential self-adjointness of powers of first-order differential operators on non-compact manifolds with low-regularity metrics

#### Autoren: Lashi Bandara, Hemanth Saratchandran (2017)

We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator possesses higher regularity coefficients, we show that higher powers are essentially self-adjoint if and only if this condition is satisfied. In the case that the low-regularity Riemannian metric induces a complete length space, we demonstrate essential self-adjointness of the operator and its higher powers up to the regularity of its coefficients. We also present applications to Dirac operators on Dirac bundles when the metric is non-smooth.

Zeitschrift:
Journal of Functional Analysis
Reihe:
273
Seiten:
3719-3758

zur Übersicht der Publikationen


text:
# Question #d1389

Jun 1, 2017

For ideal gas on a pV diagram, an adiabatic curve is always steeper than an isotherm if they intersect.

#### Explanation:

The ideal gas equation: $p V = n R T$

• $p$ is pressure
• $V$ is volume
• $n$ is number of moles
• $R$ is ideal gas constant
• $T$ is the absolute temperature

For an isothermal process, the right hand side of the ideal gas equation does not change. Which means

$p V = \text{constant}$

The slope at any point is given by

$\frac{\mathrm{dp}}{\mathrm{dV}} = - \frac{p}{V}$

using implicit differentiation or any other methods.

For adiabatic processes, the path would follow a curve of the following form:

$p {V}^{\gamma} = \text{constant}$

$\gamma$ in the above equation is treated as a constant and is meant to be the ratio of heat capacities at constant pressure, ${c}_{p}$, to heat capacity at constant volume, ${c}_{V}$. This can be derived using the first Law of Thermodynamics.

Differetiating to find the slope

$\frac{\mathrm{dp}}{\mathrm{dV}} = - \gamma \frac{p}{V}$

This looks very similar to the expression for the slope of isotherm, except that gamma is always > 1 and therefore the slope would be steeper.


text:
Hey! I'm David, a security engineer at the Blockchain team of Facebook, previously a security consultant for the Cryptography Services of NCC Group. I'm also the author of the Real World Cryptography book. This is my blog about cryptography and security and other related topics that I find interesting.

# Writing a book is hard posted October 2019

I am now half-way in the writing of my book (I wrote 8 chapters out of 16) and I am already exhausted. It doesn't help that I started writing right before accepting a new position for a very challenging (and interesting) project. But here I am, half-way there, and I think I'm onto something. I can't wait to get there and look at the finished project as a real paper book :)

To give you some insight into this process, let me share some thoughts.

Writing is hard. I have realized that I need at least a full day to write something. It does take time to get into the zone, and writing in the morning before work just doesn't work for me (and writing after work is even worse). As JP Aumasson put it (about his process of writing Serious Cryptography):

I quickly realized that I didn’t know everything about crypto. The book isn’t just a dump of my own knowledge, but rather the fruit of hours of research—sometimes a single page would take me hours of reading before writing a single word.

So when I don't have a full day ahead of me, I use my limited time to read articles and do research in topics that I don't fully understand. This is useful, and I make more progress during the week end once I have time to write.

Revising is hard. If writing a chapter takes some effort X, revising a chapter takes effort X^3 . After each chapter, several people at Manning, and in my circle, provide feedback. At the same time, I realize that there's much more I want to write about subject Y and I start pilling up articles and papers that I want to read before I revise the chapter. I end up spending a TON of effort revising and re-visiting chapters.

Getting feedback is hard. I am lucky, I know a lot of people with different levels of knowledge in cryptography. This is very useful when I want to test how different audiences read different chapters. Unfortunately people are good at providing good feedback, and bad at providing bad feedback. And only the bad feedback ends up being useful feedback. If you want to help, [the first chapters are free to read](https://www.manning.com/books/real-world-cryptography?a_aid=Realworldcrypto&a_bid=ad500e09 ) and I'm ready to buy you a beer for some constructive negative feedback.

Laying out a chapter is hard. Writing a blog is relatively easy. It's short, self-contained, and often something I've been thinking about for weeks, months, before I put it into writing. Writing a chapter for a book is more like writing a paper: you want it to be perfect. Knowing a lot about the subject makes this even more difficult: you know you can make something great and not achieving that would be disappointing. One strategy that I wish I would have more time to spend on is the following one:

• create a presentation about the subject of a chapter
• give the presentation and observe what diagrams need revisiting and what parts are hard for an audience to understand
• after many iterations put the slides into writing

I'm convinced this is the right approach, but I am not sure how I could optimize for this. If you're in SF and wants me to give you a presentation on one of the chapter of the book, leave a comment here :)

Well done! You've reached the end of my post. Now you can leave me a comment or read something else.

## Comments

#### Dan

David -- I can totally relate to what you're saying. One other thing that I find hard is to figure out how to "dial in" the right level of detail (basically, this comes down to choosing your intended audience, and "sticking to it").

When you understand something really well (as with you and cryptography), it can be difficult to figure out where to "draw the line". Sometimes you regret using "hand waving" ("magic happens here" or "trust me, it really just works") but sometimes, going into too much detail, you lose the audience's interest.

Good luck! I'm eager to read your book.

#### Raja Nagendra Kumar

Which is your LinkedIn profile

#### Daniel Li

As an author of a 20-chapter 800+ technical book, I can totally relate to your experience. I used to write one or two blog posts every week, and I thought writing a book would be a similar experience - just put all your blog posts together and add some materials to fill the gap.

I quit my job to write the book, which I'd anticipated would take 3 months. After 6 months I still didn't manage to finish it and had to take on a new job to support myself. I ended up spending a total of 16 months, with many 3-4am nights to get it done. Many times there seems to be no light at the end of the tunnel, but there was.

I am glad to see you've made some progress and now we are on the 11th chapter! Looking forward to reading the whole book!

(Would also be happy to be the audience if you'd like to do a presentation on a chapter virtually, I am quite the crypto n00b though, I must warn you)


text:
"Never doubt that a small group of thoughtful, committed citizens can change the world. Indeed, it is the only thing that ever has."

Original article
peer-reviewed

## Evaluation of a Modified Objective Structured Assessment of Technical Skills Tool for the Assessment of Pediatric Laceration Repair Performance

### Abstract

#### Introduction

The Accreditation Council for Graduate Medical Education (ACGME) has developed milestones including procedural skills under the core competency of patient care. Progress in training is expected to be monitored by residency programs. To our knowledge, there exists no tool to evaluate pediatric resident laceration repair performance.

#### Methods

The Objective Structured Assessment of Technical Skills was adapted to evaluate resident laceration repair performance using two components: a global rating scale (GRS) and a checklist. Pediatric and family medicine residents at a tertiary care children's hospital were filmed performing a simulated laceration repair. Videos were evaluated by at least five physicians trained in laceration repair. Concordance correlation coefficients (CCC) were calculated for the GRS and checklist scores. Scores for each resident were compared across levels of training and procedural experience. Spearman's rank order correlations were calculated to compare the checklist and GRS.

#### Results

Thirty residents were filmed performing laceration repair procedures. The CCC showed fair concordance across reviewers for the checklist (0.55, 95% CI: 0.38–0.69) and the GRS (0.53, 95% CI: 0.36–0.67). There was no significant difference in scores by self-reported experience or training level. There was correlation between the median GRS and checklist scores (Spearman ρ = 0.730, p < .001).

#### Conclusions

A novel tool to evaluate resident laceration repair performance in a pediatric emergency department showed fair agreement across reviewers. The study tool is not precise enough for summative evaluation; however, it can be used to distinguish between trainees who have and have not attained competence in laceration repair for formative feedback.

### Introduction

The Accreditation Council for Graduate Medical Education (ACGME) has developed milestones for trainees, which include procedural or technical skills under the core competency of patient care [1]. The ACGME Pediatric Residency Review Committee states laceration repair is a procedure pediatric residents should receive training on, and progress in competency should be monitored by pediatric residency programs [2]. Clinical competence has previously been established by procedural experience in the form of procedure logs. However, it has been demonstrated that resident clinical experience is not an accurate proxy for skill [3]. Also, to our knowledge there are no validated tools for evaluation of laceration repair skills in non-surgical residents.

The Objective Structured Assessment of Technical Skills (OSATS) is a tool developed for evaluation of surgical technical skills, including suturing. It consists of checklist and global rating scale (GRS) components in the evaluation of surgical trainees [4-8]. Similar tools have been developed for the evaluation of emergency medicine residents' management of pediatric airways and pediatric residents' performance of lumbar punctures [9-11]. While suturing technique is a central component of both surgery and pediatric emergency department (PED) laceration repair, there are significant differences between the two areas of practice. PED physicians suture less frequently in less controlled environments than surgeons. It is therefore unclear if tools used to evaluate surgical technique are applicable to PED trainees.

To address the above-described need, we developed and evaluated the validity of a modified OSATS tool for the evaluation of residents' laceration repair skills in the PED. Validity, as described by Messick, refers to evidence that an assessment is an accurate and complete representation of a specific construct [12]. A tool is not “valid” or “invalid”; instead, evidence is obtained that either supports or refutes the concept of validity [13]. Sources of validity evidence are content, response process, internal structure, relationship to other variables, and consequences [13]. In this paper, we provide content evidence for our tool by discussing its development. We examine the internal structure of the tool by examining the reproducibility of assessment scores across raters, and compare the performances of components and individual items in the tool. We assess the relationship of tool scores to variables presumed to have a relationship to procedural competence, level of training and procedural experience. Finally, we use this data to establish the appropriate level of consequence for the results obtained by using this assessment tool.

### Materials & Methods

This was a study to prospectively evaluate the validity evidence of a novel tool for the evaluation of pediatric resident laceration repair performance. Procedures were performed on suturing task trainers and evaluated via blinded video review. Procedures were recorded at a primary single center, while video reviewers were from multiple centers. The study was reviewed and approved by the Institutional Review Board at the primary study institution, while the study was exempted from review by the Institutional Review Boards at the other study institutions.

#### Tool development

The modified OSATS tool developed for this study was adapted from OSATS tool used to measure technical skills of surgical residents [5]. Like the OSATS, the modified OSATS tool contains both checklist and GRS components (Figures 1, 2), the content of which was adapted for PED practitioners from published resources on laceration repair [14-16]. Changes to the checklist included items on completion of pre- and post-suturing tasks, as well as inclusion of laceration repair specific tasks and removal of surgery specific tasks. Changes to the GRS also included evaluation of pre- and post-suturing procedures, specific evaluation of needle insertion and bite size and knot tying techniques, and omission of knowledge of tools and use of assistants. The checklist component allows for measurement of knowledge and completion of tasks associated with the laceration repair procedure, while the GRS tool is designed to capture proceduralists' proficiency and efficiency in performing procedural tasks. Both tools were designed to evaluate the entire construct of laceration repair, including pre-procedure preparation, wound irrigation and anesthesia, suturing, and post-procedure clean-up. While at the test site all proceduralists were scored on the proper application of both topical and injected anesthetics, the authors recognize that some centers have a standard of wound care where topical anesthetics alone are employed. Thus, a choice of N/A was included under Wound Anesthesia to support the generalization of the tool.

The modified tool was primarily written by NU, and was reviewed by AT and JR for content validity. The GRS tool was piloted by several pediatric emergency medicine (PEM) physicians evaluating laceration repairs performed by resident physicians during clinical encounters, while the checklist tool was piloted as an evaluative tool for resident just-in-time training by several PEM physicians. Feedback was solicited from PEM physicians who had used the tools, and modifications were made to the tools based on this feedback.

#### Study recruitment, simulation, and video recording

Study subjects were recruited from the pediatric residents at the study institution and family medicine residents rotating through the study institution's pediatric emergency department. Participants were enrolled in the study between January and November of 2016. Participants were volunteers who were recruited through in-person appeals at resident conferences and targeted e-mails to residents rotating in the pediatric emergency medicine department. Data was collected on residents' area of specialty, training year, and self-estimated number of lacerations previously repaired in an acute care setting. Subjects received a \$5 Starbucks gift card as compensation for participating in the study.

Subjects were given a case vignette of a child who required three simple, interrupted sutures to repair a laceration. They were presented with standardized equipment to perform the laceration repair, including two suture materials: fast absorbing plain gut and Vicryl (polyglactin 910) (Figure 3). Laceration repairs were performed on Tissue Suture Pads (SKU: TSP-10, Simulab Corp, Seattle WA) in a standardized location (Figure 4). Subjects were instructed to articulate individual steps of the procedure they were performing as they completed the laceration repair.

Videos of the laceration repair procedure were simultaneously recorded on two different cameras: one filming at a wide angle, the second filming with a zoomed-in view. The two videos were edited into one video comprised of elements of both recordings using video editing software (Movavi Video Editor 11, Movavi, St. Louis, MO, USA). The videos were filmed by authors NU and AT, and were edited by the study principal investigator (PI) (NU) to optimize views of the procedure. No components of the procedure were edited out of the videos.

#### Video scoring and data analysis

Eight physicians – five PEM attendings, two PEM fellows, and one emergency medicine attending – were recruited to evaluate and score study videos. Reviewers were familiarized with the rating tool, but did not receive training in its use. Five physicians (AT, RB, MJ, RK, JR) were from the study institution, while three (IG, RW, JU) were from outside centers. The study PI (NU) did not participate in video reviews. Video order was randomized, and then videos were randomly distributed to the reviewers. Reviewers then evaluated each video using the GRS tool. Each video was reviewed by at least five reviewers. The order of the videos was then randomized again, and each reviewer was sent videos to review using the checklist tool, with each video being once again reviewed by at least five reviewers.

For each study subject, overall GRS and checklist scores were averaged across reviewers. Average scores were compared by study subject level of training and previous self-reported procedural experience using linear regression. To assess agreement in scores across reviewers, concordance correlation coefficients (CCCs) were calculated for both the overall GRS and the overall checklist scores using variance components estimated from fitting linear mixed effects models [17,18]. CCCs were also calculated for each item for the GRS to evaluate agreement across reviewers for individual GRS items. Intraclass correlation coefficients (ICCs) were calculated using the variance components of generalized mixed effects models with a binomial distribution and logistic link function to determine agreement on individual checklist items [19-21]. Spearman's correlation coefficients were calculated to measure the association between the mean GRS and mean checklist scores for each proceduralist. For a sensitivity analysis, a linear mixed effects model was fit on 124 cases where the rater used both the GRS and the checklist for an individual study subject.

For the GRS, scores were tiered based on the mean performance per task, presuming a score on the last item, "Post-procedure" of five, as this item can only be scored a one or five. This resulted in following tiers: high performance (55–60, average score >4.5 on each item), good performance (49–54, average score >4 on each item), fair (39–48, average score >3 on each item), and remedial (12–38). Study subjects with individual reviewer scores in the lowest tier had their lowest and highest individual reviewer scores excluded and had their remaining mean scores compared to the mean GRS scores of all subjects with their lowest and highest scores excluded using a two-sided t-test. Statistical analyses were performed using SPSS Version 19 (IBM Co, Armonk, NY) and R Version 3.4.0 (R Foundation for Statistical Computing, Vienna, Austria). Study sample size was generated to detect a 10-point difference between first year residents vs second and third year residents in mean GRS scores with $$\alpha$$ = 0/05 and 80% power, presuming a mean score standard deviation of 9. This required the enrollment of 14 subjects in each group. Residents were split into two groups based on natural divisions in experience and to maximize study power. We a priori chose five video raters and 30 study subjects such that with the above standard deviation and 80% power we could detect an intraclass correlation coefficient of up to 0.89 between raters with 95% confidence, although CCC was ultimately used to assess agreement in scores across reviewers.

### Results

A total of 30 subjects participated in the study. Study subjects included 14 first year residents (interns), along with four second year residents and 12 third year residents (seniors). Twenty-eight participants self-reported previous laceration repair experience, while this data was not collected for two participants. Median intern experience was four procedures (Interquartile range (IQR) 4–12), median second year resident experience was 17.5 procedures (IQR 13.2–21.2), and median third year resident experience was 15 procedures (IQR 11.5–17.5).

There was no statistically significant difference in either GRS or checklist scores by years of training or procedural experience (Tables 1, 2). Figure 5 and Figure 6 show box plots of mean GRS and checklist scores by years of training and reported procedural experience. However, there were greater ranges in scores for the GRS and checklist tool for the more junior and less experienced trainees than in the more senior and more experienced trainees. Calculated CCCs showed fair concordance across reviewers for both the checklist (0.55, 95% CI: 0.38–0.69) and the GRS (0.53, 95% CI: 0.36–0.67).

 First Year Residents (n = 15), mean (SD) Second and Third Year Residents (n = 15), mean (SD) p-value Global Rating Scale Score (max = 60) 45.3 (7.2) 48.5 (4.0) 0.15 Checklist Scale Score (max = 18) 12.8 (2.3) 14.2 (1.9) 0.09
 Estimate 95% CI p-value Global Rating Scale (max score = 60) 3.0 (-0.1, 6.1) 0.06 Checklist Scale (max score = 18) 0.8 (-0.4, 1.9) 0.20

For each item on the GRS, mean scores for interns and seniors and the CCC for the item were calculated (Table 3). For each item on the checklist tool, mean percentages of interns and seniors completing each task and intraclass correlation coefficients were calculated (Table 4).

 Mean Score, First Year Residents Mean Score, Second and Third Year Residents CCC (95% CI) Pre-procedure Preparation 3.84 4.19 0.35 (0.17-0.50) Wound Anesthesia 3.76 3.76 0.60 (0.42-0.73) Wound Irrigation 3.71 4.32 0.60 (0.42-0.73) Use of Instruments 3.44 3.90 0.36 (0.18-0.51) Time and Motion 3.46 3.59 0.30 (0.13-0.46) Needle Insertion and Bite Size 3.44 3.85 0.32 (0.16-0.46) Knot Tying 3.71 3.93 0.30 (0.13-0.45) Self-correction 3.85 4.11 0.24 (0.08-0.38) Overall Knowledge of Procedure 3.84 4.20 0.37 (0.20-0.52) Independence 3.95 4.26 0.38 (0.21-0.52) Overall Performance 3.61 3.98 0.40 (0.22-0.56) Post-procedure* 4.78 4.45 0.12 (0.03-0.22)
 Item Number Achieving Task, First Year Residents (% checked) Achieving Task, Second and Third Year Residents (% checked) ICC Prepares for Laceration - 1 64% 71% 0.98 Positioning - 2 67% 65% 0.37 Anesthetizes - 3 56% 58% 0.20 Cleanses - 4 59% 83% 0.55 Maintains Sterility - 5 56% 68% 0.54 Use of Suturing Equipment - 6a 48% 56% 0.28 6b 85% 86% 0.53 6c 81% 82% 0.16 Needle Insertion - 7a 77% 77% 0.37 7b 72% 73% 0.23 7c 59% 66% 0.55 7d 55% 53% 0.06 Knot Tying - 8a 77% 86% 0.42 8b 82% 85% 0.93 8c 59% 56% 0.19 8d 60% 56% 0.28 Makes Corrections - 9 71% 80% 0.05 Discards Sharps - 10 83% 77% 0.13

There was a significant positive correlation in the mean GRS and mean checklist scores for each subject (Spearman ρ = 0.730, p < .01). In the linear mixed effects model for the sensitivity analysis, the association between the GRS and checklist scores was significant (p < 0.001).

For the GRS, scores were placed into descriptive tiers based on average scores per item (Table 5). Ten subjects had a score from at least one reviewer in the lowest tier (<39). Excluding their lowest and highest scores, these subjects had mean scores that were significantly lower than the overall mean score of all subjects excluding their lowest and highest scores (40.4 vs 47.3, p = .006).

 Cutoff Range Tier Mean Question Score at Cutoff Comments 55 55+ High Performance 4.58 High performance on most items 49 49-54 Good performance 4.08 High performance on some items 39 39-48 Fair 3.25 Understands the basics, but needs oversight on most items 38 or less 12-38 Remedial Needs oversight/instruction on most items

### Discussion

In this pilot study of a modified OSATS adapted to evaluate residents' suturing skills in a pediatric ED, we found the tool only had a fair concordance in scores across reviewers. The scores across the two components of the OSATS, the GRS and checklist, on average had a significant positive correlation, but there was inconsistency among reviewers. While scores increased by experience and training level, these increases were not significant. Based on this limited evidence of validity in internal structure and relationship to other variables the modified OSATS is not appropriate for summative testing in PED trainees.

The results of this study differ from a number of studies providing validity evidence for the use of the OSATS tool and its derivatives. The OSATS was initially developed for simulated assessment of surgical residents' technical skills. It showed good interrater reliability on both bench and live animal models for both the GRS and checklist scores (ranging from 0.64 to 0.72), providing evidence of its validity in these settings [4]. The OSATS was then adapted for use in multiple surgical and gynecologic settings. A review of surgical assessments of technical skills in 2010 found 26 studies that utilized some form of the OSATS tool [22]. Many of these studies showed construct validity, high internal consistency, and high inter-rater reliability. However, as validity of these constructs had not yet been translated to the operating room, the review's authors recommended using the tools only for formative feedback. Another review focused solely on the OSATS identified 29 studies utilizing this tool. They again found sufficient evidence to endorse use of the OSATS for formative feedback. However, an absence of evidence linking OSATS scores to performance in real clinical settings or regarding implications and decisions based on OSATS scores made the OSATS inappropriate for higher stakes decisions [23].

In pediatrics, the procedure for which the greatest development of procedural assessment tools has occurred is lumbar puncture (LP). Gerard et al. developed a tool consisting of a four-point GRS and a 15-item checklist to evaluate infant LP procedural performance with fair to good correlation in scores between raters [10]. Iyer et al. developed a seven-point GRS tool with similar interrater reliabilities [11].

The specific task of laceration repair had been previously evaluated by Acton et al. in medical students on a surgical rotation [7]. This study also used an OSATS-based tool with GRS and checklist components. This tool was validated by having 11 raters evaluate a demonstration examination, and checklist items with less than 75% agreement were revised. The authors found greater than 80% agreement for each of their seven GRS items. However, raters were considered in agreement if they were within ± one point of each other on a five-point GRS scale. Additionally the focus of the study was not validating the OSATS, but instead comparing trainee performance before and after the implementation of a new surgical curriculum.

In developing and attempting to validate our modified OSATS tool, we attempted to adhere to the model of construct validity as developed by Messick and applied to medical assessment by Downing [12,13]. The content of the evaluation tool was based on recognized resources for laceration repair education [12-14], and was reviewed by experts in PEM who offered their feedback on the tool's validity. It was designed to encompass the totality of the laceration repair process, and not just specific aspects of it. We analyzed the reliability of scoring by employing multiple reviewers and comparing their scores. We examined the internal consistency of scoring by comparing the independent scores of the GRS and checklists tools. We compared trainees' scores to training and experience, presumed proxies of technical proficiency. Our inability to establish evidence of validity in many of these examined domains makes our tool inappropriate for use in summative evaluation.

There are many potential reasons why the results we obtained examining our modified OSATS tool differed from the findings of other researchers. As the goal of our study was to develop a tool to discriminate between resident trainees in a PED, we limited our study enrollees to this population. Other studies have used enrollees with a broader range of experience levels [10]. While we did not find significant differences among groups based on level of training or reported clinical experience, these may be poor proxies for procedural skill [3].

Notably, while there was no overall difference in performance between interns and more senior trainees, the range of average scores on both the GRS and checklist tools was much greater for interns than more senior residents. It may be that individuals entering residency have a wide range of procedural competency which narrows as residents engage in specialty training. No proceduralists reported performing less than two laceration repair procedures. It may be in our population this may be sufficient experience for many learners to gain competence. Alternatively, the infrequency of laceration repair procedures in pediatric training, as opposed to in surgical practice, may lead to a reduced differential in skill in pediatric trainees, as opposed to surgical trainees. This lack of difference between study enrollees may also have resulted in lower CCCs, as the lack of true difference in the skill level of study enrollees resulted in more arbitrary scoring by raters.

We developed our modified OSATS tool with the hope it would be widely applicable. We therefore attempted to make the tool as standalone as possible. Reviewers did not receive additional training in the use of the tool. This potentially resulted in some of the observed variation in scores, as interpretation of the scoring tool was left to each reviewer. This may have been exacerbated by the subjective nature of many of the checklist GRS items, as well as variation in displayed technical skills by trainees within a simulated procedure when placing different sutures. Additionally, reviewers, trained in PEM and emergency medicine, may have less uniform practices and standards than surgical specialists. This may have led to more variation in scoring than seen in surgical OSATS tools. Orientation of reviewers to the standards for laceration repair and calibrating scoring standards may reduce the observed variability in scoring.

There also existed additional limitations specific to this study. While we attempted to create as high fidelity simulation environment as possible, differences between the task trainer and an actual patient care scenario may have limited the fidelity of our construct and therefore our ability to measure actual proficiency in laceration repair. Specific aspects of laceration repair, such as patient movement, ambient noise, lighting issue, and the stress associated with performing a procedure on an actual patient, could not be reproduced in our scenario. Additionally, reviewers scored filmed procedures, allowing multiple reviewers to review the same procedure. Review accuracy, however, may have been limited by video quality. We attempted to mitigate this concern by filming procedures from multiple angles and having trainees place three sutures. Nevertheless, video quality may have led to ambiguity in scoring. We attempted to blind video reviewers to the identity of filmed subjects. We did not film subjects' faces and attempted to remove identifiers from subjects. However, blinding may not have been one-hundred percent complete as reviewers may have recognized participants' voices. We utilized video reviewers external to the study institution to mitigate this concern, and there were no apparent differences in scoring between internal and external video reviewers. Nevertheless, undetected biases in scoring may have been present.

Finally, not every reviewer reviewed each video using each tool. This was done to both decrease the video reviewing burden for each reviewer and to minimize video repetition to allow for independent scoring. There may have been biases in scoring based on the particular reviewer assigned to each video. We attempted to mitigate this by randomizing video distribution and developing distinct randomization schemes for the GRS and checklist review procedures. This also somewhat limited our ability to perform exact comparisons between GRS and checklist scores. Despite this limitation, we were able to establish correlation between the scores obtained using these two tools.

While scores from the modified OSATS were not precise enough for summative evaluations, the tool may have utility for formative evaluation. While only fair, there was correlation between reviewers' scores. Averages scores between the GRS and the checklist tools also correlated. While not significant, there was a trend to higher scores in more experienced trainees, and specifically an absence of extremely low scores in more advanced trainees. Finally, proceduralists with individual GRS scores in the lowest scoring tier, determined a priori, had scores that were significantly lower than the GRS scores of all participants, excluding the highest and lowest scores received by each proceduralist in both groups. This means that receipt of GRS score in the lowest tier (<38) from an individual reviewer could potentially be used to identify proceduralists in need of additional attention to their training.

### Conclusions

We developed a modified OSATS tool for the evaluation of resident laceration repair performance in the PED and evaluated its validity. We based the tool on established resources for laceration repair and had experts in PEM review the tool to establish content validity. The tool showed fair concordance in scores across reviewers, but there was a significant correlation of scores between the GRS and checklist portion of the tool, providing some evidence of validity in internal structure. There were no significant differences in scores by self-reported laceration repair experience or level of training, limiting our ability to establish validity in relationship to other variables.

In this study the modified OSATS’ performance was inferior to its reported performance in the evaluation of surgical trainees, and it is therefore not appropriate for use in summative evaluation in the PED. The tool may have utility for formative feedback in this setting. Further study is needed to determine if question refinement and improved reviewer training would lead to greater score concordance.

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Original article
peer-reviewed

### Author Information

###### Ethics Statement and Conflict of Interest Disclosures

Human subjects: Consent was obtained by all participants in this study. Seattle Children's Hospital Institutional Review Board issued approval STUDY00000008. The IRB at Seattle Children's Hospital determined the study was exempt from needing IRB approval on 12/22/2015. Animal subjects: All authors have confirmed that this study did not involve animal subjects or tissue. Conflicts of interest: In compliance with the ICMJE uniform disclosure form, all authors declare the following: Payment/services info: All authors have declared that no financial support was received from any organization for the submitted work. Financial relationships: All authors have declared that they have no financial relationships at present or within the previous three years with any organizations that might have an interest in the submitted work. Other relationships: All authors have declared that there are no other relationships or activities that could appear to have influenced the submitted work.

###### Acknowledgements

The authors would like to acknowledge the University of Washington Center for Leadership in Medical Education for supporting this work.

Original article
peer-reviewed


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# zbMATH — the first resource for mathematics

An almost linear-time algorithm for graph realization. (English) Zbl 0654.05023
Summary: Given a $$\{0,1\}$$-matrix $$M$$, the graph-realization problem for $$M$$ is to find a tree $$T$$ such that the columns of $$M$$ are incidence vectors of paths in $$T$$, or to show that no such $$T$$ exists. An algorithm is presented for this problem the time complexity of which is very nearly linear in the number of ones in $$M$$.

##### MSC:
 05C05 Trees 90C08 Special problems of linear programming (transportation, multi-index, data envelopment analysis, etc.)
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# Is the mole fraction used in Raoult's law calculated initially or at equilibrium?

Raoult’s law states that:

The partial pressure of any volatile constituent of a solution at a constant temperature is equal to the vapour pressure of pure constituent multiplied by the mole fraction of that constituent in the solution

My question:

Is the mole fraction of the constituent of the solution to be considered at the time when equilibrium between vapours dissolving into the solution and rate of evaporation has been achieved or at the beginning when the solution is just formed and evaporation has just begun change the mole fraction of the constituent which we have just calculated?

Also, experimentally, is there much difference in the mole fractions at equilibrium and at the beginning when there is apparently no vapour pressure (for zero vapour pressure in the beginning, I think we can just close the lid of the container)?

• Jun 16 '20 at 14:52

Application of Raoult's law assumes an equilibrium has been established between gas and liquid phases (otherwise the standard equations of thermodynamics, of which it forms a part, would not apply). For a mixture of components one can write a set of equations $$\gamma_i p = \chi_i p_i^\circ$$ that can be solved for a self-consistent composition. For instance, for two components one can write two equations in two unknowns:
\begin{align} \gamma_1 p &= \chi_1 p_1^\circ \\(1-\gamma_1) p &= (1-\chi_1) p_2^\circ \end{align}
\begin{align} \chi_1 &= \frac{p-p_2^\circ}{p_1^\circ-p_2^\circ} \\ \gamma_1 &= \left(\frac{p-p_2^\circ}{p_1^\circ-p_2^\circ}\right)\left(\frac{p_1^\circ}{p}\right) \end{align}