text:
# 54

Followed By: Payor's Lemma in Natural Language

TL;DR: This post introduces a novel logical approach to achieving group-scale cooperation, based on modal fixpoint theory.  This approach is both easier to understand and roughly 3x more efficient than previous approaches that factored through Löb's Theorem, measured in terms of the length / complexity of the proofs involved.

The following theorem was inspired by Scott Garrabrant, and uses a lemma of James Payor in place of Löb's Theorem to prove cooperation between a group of agents.  I'll state the theorem for three agents because that's most illustrative of what's going on:

Theorem: Suppose  and  are agents that return "true" to signify cooperation and "false" to signify defection.   Let , so  is the statement that "everyone cooperates".  Let , and  denote proof systems that extend Peano Arithmetic, let  stand for , and suppose the agents behave in a manner satisfying the following conditions:

1. Then it follows that , i.e., everyone cooperates (provably!).

(Intuitively, the strategy of the agents in this theorem is to check that the group is trustworthy in a certain way before joining (cooperating with) the group.  The theorem shows that the collective check on trustworthiness nests inside itself in a way that self-validates and yields cooperation.)

Proof:
2. , by combining 1, 2, and 3 with .
3. , from 4 by the definition of E and .
4. , by the lemma below with .

[end proof]

The key to the theorem is the following lemma, due to James Payor:

Lemma: If  then .

Historical note: Originally this lemma assumed  , but then I realized only the forward implication is needed, so I rewrote it.

Proof: The proof uses the same modal rules of inference for  as Löb's theorem, namely, necessitation and distributivity:

1. , by tautology ().
2. , from 1 by  necessitation and distributivity.
3. , by assumption.
4. , from 2 and 3 by modus ponens.
5. , from 4 by  necessitation.
6. , from 5 and 3 by modus ponens.

[end proof]

Sweet!  In comparison to Löb's Theorem, two things are beautiful about Payor's lemma:

• It sidesteps the use of an auxiliary fixpoint , by examining a proposition of interest () that itself has the fixpoint structure needed to self-validate; and
• It also allows the construction of unexploitable modal agents without Löb's Theorem, as in the theorem above.

## Discussion

In the proof of the Theorem, you might be wondering if it really makes sense to be thinking of  as a logical system of its own.  It doesn't need to be, but the answer is yes if , and  are all finite extensions of PA.  Then the axioms of  are just [the conjunction of axioms of ][the conjunction of axioms of  [the conjunction of axioms of ].

You also might wonder if an alternative approach to group cooperation might be to instead use the following strategies:

Then you'd be right!  Here it also follows that .  However, the proof involves a lot more nesting, with A thinking about what B's thinking about what C's thinking about (etc.), and it's not as easy or short as the proof of the Theorem above.

## Writing implications backwards

If you think of  in Payor's Lemma as "defining"  as a program, it's fun to write all the implication arrows from right to left, so they look a little like variable assignments.  (And, under the Curry-Howard correspondence, they do actually correspond to maps.)  Here's the lemma and proof again, written that way:

Lemma: If  then .

Proof: The same as above, but with the arrows written leftward!

1. , by tautology ().
2. , from 1 by  necessitation and distributivity.
3. , by assumption.
4. , from 2 and 3 by modus ponens.
5. , from 4 by  necessitation.
6. , from 5 and 3 by modus ponens.

[end proof]

I find the above somewhat more elegant than the version with rightward arrows, albeit a less standard way of writing.

## Conclusion

In my opinion, what's great about the lemma and theorem above is that they're both relatively short and simple (relative to proving and using Löb's Theorem), and they allow a proof of unexploitable group cooperation that's roughly three times shorter than than one that starts by proving Löb's Theorem (only ~6 lines of logic, vs ~18).

PS James says his next idea will be even better ;)

# 54

New Comment

Here's a simple Kripke frame proof of Payor's lemma.

Let  be a Kripke frame over our language, i.e.  is a set of possible worlds,  is an accessibility relation, and  judges that a sentence holds in a world. Now, suppose for contradiction that  but that , i.e.  does not hold in some world .

A bit of De Morganing tells us that the hypothesis on  is equivalent to , so . So, there is some world  with  such that . But again looking at our equivalent form for , we see that , so , a contradiction.

Both this proof and the proof in the post are very simple, but at least for me I feel like this proof tells me a bit more about what's going on, or at least tells me something about what's going on that the other doesn't. Though in a broader sense there's a lot I don't know about what's going on in modal fixed points.

Kripke frame-style semantics are helpful for thinking about lots of modal logic things. In particular, there are cool inductiony interpretations of the Gödel/Löb theorems. These are more complicated, but I'd be happy to talk about them sometime.

Thanks! I also wrote up my proof in another comment, which should help triangulate intuition.

Nice, I like this proof also. Maybe there's a clearer way to say thing, but your "unrolling one step" corresponds to my going from  to . We somehow need to "look two possible worlds deep".

Also, here's a proof that a bot is never exploited. It only cooperates when its partner provably cooperates.

First, note that , i.e. if cooperates it provably cooperates. (Proof sketch: .)

Now we show that (i.e. if chooses to cooperate, its partner is provably cooperating):

1. We get by distributing.
2. We get by applying internal necessitation to .
3. By (1) and (2), .

(PS: we can strengthen this to , by noticing that .)

A point of confusion: is it enough to prove that ? What about ? I'm not sure I can say this well, but here goes:

We might not be able to prove in the theory, or even that (which would mean "there are no proofs of inconsistency"). But if we believe our theory is sound, we believe that it can't show that a copy of itself proves something false.

So tells us that if is true, the theory would show that a copy of itself proves is true. And this is enough to convince us that we can't simultaneously have true and false.

This is cool (and fwiw to other readers) correct. I must reflect on what it means for real world cooperation... I especially like the A <-> []X -> [][]X <-> []A trick.

In case this helps folks' intuition, my generating idea was something like: "look at what my opponent is thinking, but assume that whenever they check if I cooperate, they think the answer is yes". This is how we break the loop.

This results in a proof of the lemma like thus:

1. (given)
2. (unrolling one step)
3. is straightforward intuitively, and we can get there by applying box-distributivity to

(EDIT: This is basically the same proof as in the post, but less simple. Maybe the interesting part is the "unroll once then break the loop" intuition.)

Relatedly, with two agents and , we can have:

1. (given)
2. (given)
3. (substituting into )
4. is straightforward as before

(In fact, I think "assume they will think I cooperate" turns out to be too strong, and leads to unnecessary defection. I'm still working through the details.)

So, ideally you would like to assume only

1. □A→B
2. □B→A

and conclude A and B ?

If I follow what you mean, we can derive:

So there's a Löbian proof, in which the provability is self-fulfilling. But this isn't sufficient to avoid this kind of proof.

(Aside on why I don't like the Löbian method: I moreso want the agents to be doing "correct" counterfactual reasoning about how their actions affect their opponent, and to cooperate because they see that mutual cooperation is possible and then choose it. The Löbian proof style isn't a good model of that, imo.)

Very cool! How does this affect your quest for bounded analogues of Löbian reasoning?

I'm working on it :) At this point what I think is true is the following:

If ShortProof(x \leftrightarrow LongProof(ShortProof(x) \to x)), then MediumProof(x).

Apologies that I haven't written out calculations very precisely yet, but since you asked, that's roughly where I'm at :)

It looks like you're investigating an angle that I can't follow, but here's my two cents re bounded agents:

My main idea to port this to the bounded setting is to have a bot that searches for increasingly long proofs, knowing that if it takes longer to find a proof then it is itself a bit harder to reason about.

We can instantiate this like:

The idea is that if there is a short way to prove that the opponent would cooperate back, then it takes just a constant steps more to prove that we cooperate. So it doesn't open us up to exploitation to assume that our own cooperation is provable in steps.

The way in which this works at all is by cutting the loop at the point where the opponent is thinking about our own behaviour. This bot cuts it rather aggressively: it assumes that no matter the context, when thinks about whether cooperates, it's provable that does cooperate. (I think this isn't great and can be improved to a weaker assumption that would lead to more potential cooperation.)

If you construct similarly, I claim that and mutually cooperate if and are large enough, and mutually defect otherwise.

Similarly, I claim can mutually cooperate with other bots like .


text:
# This problem is FRUSTRATING me SO Much!

by BuBbLeS01
Tags: frustrating
 P: 602 1. The problem statement, all variables and given/known data Find the indefinite integral... X^2/ X-1 In my book they set it up as 2 separate integrals... X+1 + 1/X-1 I figured they got that from long division but I am not getting that answer. So can someone please tell me how they got those 2 separate integrals
 P: 1,754 $$\int\frac{x^2}{x-1}dx$$ $$\int\frac{x^2-1+1}{x-1}dx$$ $$\int\left[\frac{(x+1)(x-1)}{x-1}+\frac{1}{x-1}\right]dx$$ $$\int\left(x+1+\frac{1}{x-1}\right)dx$$ Does this make a little more sense?
 P: 602 Yes it makes more sense but I just want to make sure I understand what you did.. In part B you completed the square...(But I thought it would be (X^2 + 1) - 1)
 P: 1,754 This problem is FRUSTRATING me SO Much! I did not complete the square, and plus, you do not have a linear factor - so you shouldn't even consider it. $$x^2-1=(x+1)(x-1)=\mbox{Difference of Squares}$$
 P: 602 Oh....I don't think we've used that before...I will have to look it up...thank you!
P: 1,754
 Quote by BuBbLeS01 Oh....I don't think we've used that before...I will have to look it up...thank you!
BuBbLeS, you have used it plenty of times (you learn it in junior high or HS). You just forgot! No worries, go look it up and you'll be like oh yeah ... I remember that :p
 P: 602 I hope so lol! I have another question on another problem... I am trying to complete the square... I have 6x - x^2 Then I did (6/2)^2 = 9 So I end up with (-x^2 + 6x + 9) - 9 I need to end up with 9 - (x - 3)^2 Can I make (-x^2 + 6x + 9) to (x^2 + 6x + 9) because the x is squared so it will be positive or is that negative mean like -(x^2)?? And (x - 3)^2 gets me x^2 - 6x + 9 so I am a little confused...
 P: 1,754 Before you complete the square, make sure that the coefficient of your leading term is a positive one. $$6x-x^2$$ $$-(x^2-6x+9-9)$$ $$-(x^2-6x+9)+9$$ $$9-(x-3)^2$$
 P: 602 Ohhhh....thank you so much!!!
 P: 1,754 So you solved it, yes?

 Related Discussions Introductory Physics Homework 2 General Discussion 6 Calculus & Beyond Homework 2 General Discussion 3 Linear & Abstract Algebra 4

format: plain


text:
how would i find the zeros of this equation

y=2xsquared - 27x

2. Originally Posted by svaningen
how would i find the zeros of this equation

y=2xsquared - 27x
factor ...

$2x^2 - 27x = 0$

$x(2x - 27) = 0$

set each factor equal to zero and find the two zeros.


text:
# How do you condense 3 ln 6 + 3 lnx?

Apr 4, 2016

$3 \ln \left(6\right) + 3 \ln \left(x\right) = \ln \left(216 {x}^{3}\right)$

#### Explanation:

We will use the following properties:

• $\ln \left(a\right) + \ln \left(b\right) = \ln \left(a b\right)$
• $x \ln \left(a\right) = \ln \left({x}^{a}\right)$

With those, we have

$3 \ln \left(6\right) + 3 \ln \left(x\right) = 3 \left(\ln \left(6\right) + \ln \left(x\right)\right)$

$= 3 \ln \left(6 x\right)$

$= \ln \left({\left(6 x\right)}^{3}\right)$

$= \ln \left(216 {x}^{3}\right)$


text:
# 3.9: Using Logical Operators

Most students have only seen logical operators used as branch conditions in if, while, and for statements, and so they cannot see why these operators are useful. This section will show some uses of these operators.

## 3.9.1 Storing immediate values in registers

In MARS, the li instruction is translated into an "addui Rd, $zero, Immediate" instruction. This is perfectly valid, however when doing addition there always has to be propagation of a carry-bit, and so addition can take a relatively long time (O(logn)..O(n)). However the OR operation is a bit-wise operation and does not have any carry bits, and can be executed in O(1) time. In MARS the OR is not faster than addition, but in some architectures the OR is in fact faster than addition. If the OR is faster than addition, the li instruction can be implemented as "ori Rd,$zero, Immediate" with some advantage. Be aware that an ori is not the same as an addui. This difference is shown in problem 14 at the end of the chapter.

## 3.9.2 Converting a character from upper case to lower case

Chapter 1 pointed out that in ASCII the difference between upper case and lower case letters is the 0x20 bit. So an upper case letter can be converted to lower case by setting this bit to 1, and an upper case letter can be set to a lower case letter by turning this bit off. One of the exercises in Chapter 1 asked for an implementation of this program in a HLL. Many novice programmers see this problem and realize that convert uppercase letters to lowercase, they can add 0x20. The problem is that if the letter is already uppercase, the result is no longer a letter.

To correctly handle converting from uppercase to lowercase, the letters should be OR'ed with 0x20 (converting from lowercase to uppercase is left as an exercise at the end of the chapter). If this is done, letters which are already lowercase are not changed. This is illustrated in the following program.

Program 3-5: Letter case conversion in assembly

# File:       Program3-5.asm
# Author:     Charles Kann
# Purpose:    This program shows that adding 0x20 to a character can result in an error
#             when converting case, but or'ing 0x20 always works.
.text
.globl main
main:
# Show adding 0x20 only works if the character is upper case.
ori $v0,$zero, 4
la $a0, output1 syscall ori$t0, $zero, 0x41# Load the character "Z" addi$a0, $t0, 0x20 # Convert to "z" by adding ori$v0, $zero, 11 # Print the character syscall ori$v0, $zero, 4 la$a0, output2
syscall
ori $t0,$zero,0x61 # Load the character "z"
addi $a0,$t0, 0x20 # Attempt to convert to lower case
ori $v0,$zero, 11 # Print the character, does not work
syscall

# Show or'ing 0x20 works if the character is upper or lower case.
ori $v0,$zero, 4
la $a0, output1 syscall ori$t0, $zero, 0x41# Load the character "Z" ori$a0, $t0, 0x20 # Convert to "z" by adding ori$v0, $zero, 11 # Print the character syscall ori$v0, $zero, 4 la$a0, output1
syscall
ori $t0,$zero,0x61 # Load the character "z"
ori $a0,$t0, 0x20 # Attempt to convert to lower case
ori $v0,$zero, 11 # Print the character, does not work
syscall

ori $v0,$zero, 10  # Exit program
syscall
.data
output1: .asciiz "\nValid conversion: "
output2: .asciiz "\nInvalid conversion, nothing is printed: "

## 3.9.3 Reversible operations with XOR

Often it is nice to be able to invert the bits in a value in a way that they are easily translated back to the original value. Inverting the bits and restoring them is shown in the program below. One use of this is to swap two values without using a temporary storage space, is given at the end of the chapter.

# File:        Program3-6.asm
# Author:      Charles Kann
#Purpose:      To show the XOR operation is reversible
.text
.globl main
main:
ori $s0,$zero, 0x01234567 # the hex numbers

# Write out the XOR'ed value
la $a0, output1 li$v0, 4
syscall
xori $s0,$s0, 0xffffffff  # the results in $t1 will be fedcba98 move$a0, $s0 li$v0, 34
syscall

# Show the original value has been restored.
la $a0, output2 li$v0, 4
syscall
xori $s0,$s0, 0xffffffff  # the results in $t1 will be fedcba98 move$a0, $s0 li$v0, 34
syscall

ori $v0,$zero, 10         # Exit program
syscall
.data
output1: .asciiz "\nAfter first xor: "
output2: .asciiz "\nAfter second xor: "

This page titled 3.9: Using Logical Operators is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Charles W. Kann III.


text:
# How-to guide on R Shiny app for Bayesian Diagnostic Test Meta-Analysis

DTA meta-analysis
app
rshiny
Bayesian
Authors

Ian Schiller

Nandini Dendukuri

Published

November 17, 2021

## OVERVIEW

Many researchers who wish to carry out diagnostic accuracy meta-analysis are not familiar with rjags. To address this we developed a Shiny app that is available on the net here. The programs behind this are the ones described in our previous blogs articles illustrating Rjags programs for diagnostic meta-analysis model (Bivariate model and Bivariate latent class model).

This blog article is intended to serve as a guide for the usage of the R Shiny app that implement the models discussed in the blog articles mentioned above. The main goal will be to describe the various options in the app via applied examples. The interested reader can learn more about the statistical theory in Dendukuri et al[-@Dendukuri2012].

The R Shiny app provides options to perform both Bayesian Bivariate meta-analysis and latent class meta-analysis.

## MOTIVATING EXAMPLE AND DATA

For our motivating example we will use the data from a systematic review of studies evaluating the accuracy of GeneXpertTM (Xpert) test for tuberculosis (TB) meningitis (index test) and culture (reference test) for 29 studies (Kohli et al[-@Kohli2018]). Each study provides a two-by-two table of the form

reference_test_positive Reference_test_negative
Index test positive tp fp
Index test negative fn tn

where:

• tp is the number of individuals who tested positive on both Xpert and culture tests
• fp is the number of individuals who tested positive on Xpert and negative on culture test
• fn is the number of individuals who tested negative on Xpert and positive on culture test
• tn is the number of individuals who tested negative on both Xpert and culture tests

We will explain later how to correctly supply the data into the app. Let’s first look at the 4 tab menus of the Shiny app.

Through this blog article, we will walk you through the 4 tab menus;

• Home
• Manual
• Bayesian Bivariate Meta-Analysis
• Latent-Class Meta-Analysis

and we will explore features and options of each of them, starting with the Home tab menu.

## HOME

The Home tab menu gives a brief presentation of the app’s functionalities. It lists the restrictions of the app regarding the prior distributions used in the models or the initial values, as well as references to papers describing, among others, the theoretical models this app is based on.

It also details how the app expects the user to organize the data to be imported in the app at a later step. Basically, the data must be stored into a .txt file with 4 columns. The 4 column’s headers must be spelled exactly as tp , fp , fn and tn who were defined earlier in the OVERVIEW section, but the order in which the columns are presented in the file does not matter. Missing values or NA values are not supported. For example, the file Xpert_Data below would be a well defined .txt file to load into the app (we will see in a later section how to load the data in the app).

tp <- c(7, 6, 2, 5, 0, 1, 4, 3, 3, 1, 7, 35, 103, 1, 2, 22, 6, 25, 27, 15, 5, 2, 2, 11, 2, 0, 8, 3, 1)
fp <- c(5, 4, 0, 0, 0, 0, 6, 0, 3, 0, 0, 5, 6, 1, 0, 1, 1, 3, 11, 3, 9, 3, 0, 2, 4, 0, 2, 0, 2)
fn <- c(5, 4, 0, 1, 2, 0, 8, 1, 1, 0, 0, 1, 18, 2, 0, 5, 0, 20, 25, 7, 10, 1, 1, 2, 2, 3, 5, 0, 0)
tn <- c(63, 115, 2, 44, 132, 4, 83, 250, 67, 14, 150, 119, 252, 107, 8, 31, 148, 687, 204, 205, 115, 53, 4, 118, 22, 16, 186, 28, 43)
cell <- cbind(tp, fp, fn, tn)
n <- length(tp)  #  Number of studies
write("tp fp fn tn","Xpert_Data.txt")
for (j in 1:n) write(cell[j,],"Xpert_Data.txt",append=T)

The user must note that when loading the data in the app, if the headers are not correctly spelled, if the number of columns differs from 4, if there are missing values or NA values in the file, then the Shiny App will fail and a Disconnected from the server error message like the one below will be displayed instead.

## MANUAL

The MANUAL tab menu gives some basic guidelines on how to use the app features. It splits itself into 3 sub tab menus

• Example Analysis of Dataset: Which gives a brief overview on how to prepare the data file to be loaded into the app.
• Bayesian Bivariate Meta-Analysis: how-to guide on how to use the app to model Bivariate meta-analysis.
• Latent-Class Meta-Analysis: how-to guide on how to use the app to model latent class meta-analysis.

## BAYESIAN BIVARIATE MODEL

The Bayesian Bivariate Meta-Analysis is the tab menu that must be selected in order to perform a Bayesian bivariate meta-analysis. In this section, we will guide the user step by step to analyse the Xpert data presented in the MOTIVATING EXAMPLE AND DATA section in order to estimate Xpert’s accuracy under the assumption that culture is a perfect reference test.

In the red rectangle of the screen shot above, uploading the text file that contains the data is the only feature that is mandatory before pressing the submit button. This is done by browsing your folders in order to load the data. Once done, an uploading bar will fill up and let you know that your data has been successfully loaded. Note that if there is a problem with the way you provided your data (missing values for instance), it is not until you press the submit button that the app will return the Disconnected from the server error message.

### RANDOM SEED

The Set Seed feature is used to set the random number generator (RNG) state for random number generation in R. The default setting is to let the app randomly generates the RNG state. By checking the box next to Set Seed, the user can set up the RNG state for reproducibility of the results by entering a postive integer value. A value of 123 is suggested, but can be changed by the user.

### MONITORING PARAMETERS

Below the seed option are check-box options to select the various parameters of the model. Although it is specified to select parameters you want to monitor, there is actually no need to do that as the model will monitor every parameters of the model by default. All parameters monitored by default by the app are regrouped under 3 categories, the parameters shared across studies, the parameters varying across studies and the shared parameters on the logit scale. Checking any of those boxes enable more check box options as follow:

Under the Parameters shared across studies check box,

• Mean sensitivity of index test (later referred as Mean_Se in the results)
• Mean specificity of index test (later referred as Mean_Sp in the results)
• Predicted sensitivity of index test (later referred as Predicted_Se in the results)
• Predicted specificity of index test (later referred as Predicted_Sp in the results)

Under the Parameters varying across studies check box,

• Sensitivity of index test (later referred as se in the results)
• Specificity of index test (later referred as sp in the results)

Under the Shared parameters on the logit scale check box,

• Mean logit (Sensitivity) of index test (later referred as mu[1] in the results)
• Mean logit (specificity) of index test (later referred as mu[2] in the results)
• Between-study precision in logit (Sensitivity) of index test (later referred as prec[1] in the results)
• Between-study precision in logit (specificity) of index test (later referred as prec[2] in the results)
• Correlation between logit sensitivity and logit specificity (later referred as rho in the results)

Actually, the only parameters that are not being monitored by default are the Positive predicted value of the index test and Negative predicted value of the index test parameters under the Parameters shared across studies check box. In order to monitor them, the user must check the Parameters shared across studies box which will enable more check box options, including the Positive predicted value of the index test and Negative predicted value of the index test options. To tell the app you want to monitor them, you will check those boxes which will enable the option to specify a guess value for the disease Prevalence. By entering a value between 0 and 1, this will enable the app to monitor the positive and negative predicted values.

### OPTIONAL SETTINGS

The bottom check box option, named Optional settings provides the user with more control over the MCMC process with options like

• Setting the Number of burn-in iterations. It consists of a number of iterations that are discarded during the early MCMC sampling process. The default setting is to discard the first 5,000 iterations.
• Seeting the Number of iterations to monitor parameters. This controls how many iterations will be perform to create the posterior sample of each parameter. The default number of iterations is 10,000.
• Setting the Number of chains. In a MCMC run, the current iteration value of a given parameter, depends on the previously drawn value and so on, such that all those values form what we call a chain. Therefore, this provides the user with the option to control how many chains to monitor. Selecting to have more than 1 chain will generally help assess convergence. The default is to monitor 3 chains.
• Setting the Number of decimal places. This option allows the user to determine how precise they want the results to be. The default is to display 3 digits.

### SUBMIT AND REFRESH BUTTONS

At the bottom of the page, the Submit and Refresh buttons will launch the MCMC iterative sampling process and clear the app in order to start from scratch, respectively. On the right side beneath the 4 main tab menus, there are 5 sub-tab menus. Those are the tab menus reserved for the result outputs and will only be filled once after the Submit button has been pressed.

When the app is running the MCMC iterative sampling process (after having pressed the Submit button), there are 2 visible signs that the app is running in the background. First, at the top of the app, you will see a progress bar moving back and fourth, ensuring that the app isn’t stuck. The other sign appears in the lower right corner. The app will display a small window with the words Iterating Please be patient. The time the app takes to run will largely vary with the number of studies, the number of iterations to monitor parameters and the model used (Bayesian Bivariate Meta-Analysis or Latent-Class Meta-Analysis tab menu).

Once the app has finished running in the background, the Data tab menu will get filled with the complete data file that was uploaded earlier in the LOADING THE DATA sub-section.

The second tab menu History Plot, will display a plot of the posterior sample values of a given parameter (Y axis) against the number of iterations (X axis). If no boxes where checked (See Monitoring Parameters subsection above), then there will be as many plots as there are parameters monitored displayed like in the figure below. Below the plots there are check box options for selecting chain(s). If no boxes are selected (the default), all chains are depicted in the parameters history plots. Although the chain selection boxes are only available in this tab menu, any boxes checked (or not) here will carry over to the other sub tab menus (Density Plot, Summary, SROC plot).

It is possible to display plots for certain parameters by selecting those specific parameters in the check box options discussed in the Monitoring Parameters sub-section. As seen in the picture below, by checking the Shared parameters on the logit scale box and then the Mean (logit) sensitivity of index test (mu[1] parameter) and Between-study precision in logit sensitivity of index test (prec[1] parameter) boxes, this will remove every other history plots from the History plot sub tab menu, except for mu[1] and prec[1].

Similar to the History Plot tab menu, the Density Plot tab menu will display a plot of the posterior distribution of a given parameter based on its posterior sample. When no boxes where checked (See Monitoring Parameters subsection above), then density plots for every parameters monitored will be displayed (see figure below). When more than one chains were run, each chain will have its own density curve and they will be represented in the density plot by overlapping density curves of different colors. Recall that the check box options for selecting chain(s) can be found in the History Plot.

Zoom-in of certain parameter density plots can be achieved by selecting those specific parameters in the check box options discussed in the Monitoring Parameters sub-section. For instance, by checking the Parameters shared across studies box and then the Mean sensitivity of index test (Mean_Se parameter) and Predicted sensitivity of index test (Predicted_Se parameter) boxes, only the density plots for Mean_Se and Predicted_Se will be displayed in the Density plot sub tab menu (see image below).

The posterior summary statistics can be viewed in the Summary tab menu. For each parameter monitored, the following statistics are being returned:

• Mean: This is the posterior mean.
• sd: The posterior standard deviation.
• 2.5%: The 2.5% posterior quantile.
• 50%: The 50% posterior quantile, also referred to the posterior median.
• 97.5%: The 97.5% posterior quantile.
• Rhat: The Gelman-Rubin statistic. It is enabled when 2 or more chains are generated. It evaluates MCMC convergence by comparing within- and between-chain variability for each model parameter. Rhat tends to 1 as convergence is approached (Gelman and Rubin[-@Gelman1992], Brooks and Gelman[-@Brooks1998]).
• n.eff: The effective sample size. Because the MCMC process causes the posterior draws to be correlated, the effective sample size is an estimate of the sample size required to achieve the same level of precision if that sample was a simple random sample. When draws are correlated, the effective sample size will generally be lower than the actual numbers of draws resulting in poor posterior estimates (Gelman et al[-@Gelman2013]).

The 2.5% and 97.5% posterior quantiles are commonly reported as the 95% credible interval.

The final sub tab menu is the SROC PLOT tab menu. By default, it will plot each individual studies (blue shaded squares) as well as a summary point (black circle with coordinates (Mean_Sp, Mean_Se)) in a ROC space. The blue squares are proportional to the sample size of their respective study, with the largest squares corresponding to the largest studies.

Further options below the plot allows the user to customize the figure by adding an SROC curve, credible and/or prediction regions.

In order to create the SROC curve, the conversion from bivariate parameters to HSROC parameters is done according to Harbord([-@Harbord2007]). The calculation of the credible and prediction regions are also derived from that paper.

## LATENT CLASS META-ANALYSIS

Like in the Bayesian bivariate model, we are still interested in estimating the index test’s propertises, but the main difference is that we are now acknowledging that the reference test is imperfect. This model is referred to as latent class model and it extends the bivariate model by allowing the reference test to be imperfect (Xie et al[-@Xie2017], Dendukuri et al[-@Dendukuri2012]). This model is accessible under the Latent-Class Meta-Analysis tab menu.

The Latent-Class Meta-Analysis tab menu shares a certain number of options with the Bayesian Bivariate Meta-Analysis tab menu. In this section, we will therefore highlights the new options that are related to the latent class model. Anything not covered here works exactly as in the Bayesian Bivariate Meta-Analysis tab menu.

The first visual differences with the Bayesian Bivariate Meta-Analysis tab menu are the presence of new check boxes underneath the Set Seed option. The new options are as follow:

• Informative prior distribution check box.
• Set initial values to help convergence check box.
• Model conditional dependence check box.

We will go over those 3 new options in the next sub sections

### Informative prior distribution check box

Through our many years of experience working with these models, we found that the latent class models can be sensitive to prior information. The app allows certain freedom for that by giving the option to tweak prior information for the index and reference tests. So when checking the box (as seen in the image below), you will be given with specifying the lower and upper limit of a 95% credible interval for sensitivity and/or specificity. Nothing is being estimated at this step as this is based on prior knowledge one might have on the test’ sensitivity and specificity. So for instance, if it is believed that the sensitivity ranges between 0.6 and 0.8 with probability 0.95, then Lower should bet set to 0.6 and Upper sould be equal to 0.8. So bascially, if any prior knowledge exists, it can pay off to include it in the model at this step. Otherwise, the default values Lower=0.025 and Upper=0.975 were selected such that the prior is approximately uniform over the range [0,1], which corresponds to a vague prior. This vague prior is also used by default if the Informative prior distribution box is left unchecked.

For the more advanced user who wants to connect this with the theory behind the model, this uniform prior on sensitivity/specificity corresponds to a normal prior with mean=0 and variance=4 placed on the logit-transformed sensitivity/specificity.

### Set initial values to help with convergence check box

Latent class models can also be affected by the starting values. To mitigate that, the app allows to setup initial values for the following parameters:

• Mean logit sensitivity of the index test, noted by mu[1]
• Mean logit specificity of the index test, noted by mu[2]
• Mean logit sensitivity of the reference test, noted by mu2[1]
• Mean logit specificity of the reference test, noted by mu2[2]

There are no “one size fit all” set of initial values here. The app provides some by default, but the user should really not rely on them and instead provide the ones that makes more sense from their own model perspective.

Contrary to the prior check box in the previous sub-section, here the initial values must be supplied on the logit scale which might not be intuitive at first. As a guide line, keep in mind that a value of 0 on the logit scale back-transforms into 0.50 on the probability scale. This means that if you want to select an initial value of 0.50 for the sensitivity of the reference test, then you would have to enter the value 0 under the mu2[1] box. Keeping that logic in mind, if we increase values on the logit scale above 0 we get closer to a probability of 1. Similarly, decreasing the logit scale value below 0 (negative values), we get closer to a probability of 0.

Here are some more examples:

• A value of 0 on the logit scale corresponds to a probability of 0.5
• A value of 1 on the logit scale corresponds to a probability of 0.7310586
• A value of 1.25 on the logit scale corresponds to a probability of 0.7772999
• A value of 2 on the logit scale corresponds to a probability of 0.8807971
• A value of 3 on the logit scale corresponds to a probability of 0.9525741
• A value of -1 on the logit scale corresponds to a probability of 0.2689414
• A value of -1.25 on the logit scale corresponds to a probability of 0.2227001
• A value of -2 on the logit scale corresponds to a probability of 0.1192029
• A value of -3 on the logit scale corresponds to a probability of 0.0474259

For the more advanced user, the equation transforming a probability onto a logit value is given by the following expression

$l = log\left( \frac{p}{1-p} \right),$ where $$l$$ and $$p$$ are the logit and probability values respectively.

### Model conditional dependence check box

When the reference test is no longer assumed to be perfect, it means that it is likely to make some errors, i.e. being positive when the subject receiving the test is actually free of the disease (commonly referred to as a false positive result) or come out negative when the subject does have the disease (a false negative result). When both the reference test and the index test are making the same error, this is referred to as conditional dependence.

It is possible to allow for conditional dependence by checking the Model conditional dependence check box. By doing so, it will prompt 2 new check boxes which basically allow the user to select what type of error both the index and reference tests are believed to make more often simultaneaously.

• Among disease positive check box means that the conditional dependence is occurring due to false negative results
• Among disease negative check box means that the conditional dependence is occurring due to false positive results

### Extra parameters of the model

Modelling the reference test to be imperfect means that our latent class model has more parameters than the simpler Bayesian bivariate model. In this section we simply highlight the new parameters that are monitored by the model with their labels for easy reference in the Summary tab menu.

The following parameters shared across studies are being added by the latent class model.

• Mean sensitivity of reference test denoted by mu2[1]
• Mean specificity of reference test denoted by mu2[2]
• Predicted sensitivity of reference test denoted by Predicted_Se2
• Predicted specificity of reference test denoted by Predicted_Sp2
• Positive predicted value of the reference test denoted by Summary_PPV2
• Negative predicted value of the reference test denoted by Summary_NPV2

As we have seen in the Monitoring parameters subsection of the Bayesian bivariate model section above, Summary_PPV2 and Summary_NPV2 will not be monitored by default unless their respective boxes were checked before running the app.

The model also monitors disease prevalence parameters for every study. This means that the parameters varying across studies will add a bunch of disease prevalence parameters that will be denoted by prev indexed by the study number, i.e. prev[1] for the estimate of the disease prevalence of the first study, prev[2] for the estimate of the disease prevalence of the second study and so on.

If conditional dependence was modeled as seen in the Model conditional dependence subsection, then 2 extra check boxes will become available under the Parameters varying across studies check box option. By default, the summary statistics for both covariance terms will not appear in the various output tabs unless the 2 boxed are properly checked. Similar to the disease prevalence, there will be as many covariance terms as there are studies.

• Covariance in disease positive will be denoted by covp indexed by the study number, i.e. covp[1] for the covariance in disease positive for study 1, covp[2] for the covariance in disease positive for study 2, so on.
• Covariance in disease negative will be denoted by covn indexed by the study number, i.e. covn[1] for the covariance in disease negative for study 1, covn[2] for the covariance in disease negative for study 2, so on.

Finally, the following shared parameters on the logit scale will also be added by default to the list of parameters monitored as part of the latent class model:

• Mean logit (Sensitivity) of reference test denoted by mu2[1]
• Mean logit (specificity) of reference test denoted by mu2[2]
• Between-study precision in logit sensitivity of reference test denoted by prec2[1]
• Between-study precision in logit specificity of reference test denoted by prec2[2]

## Citation

BibTeX citation:
@online{schiller2021,
author = {Ian Schiller and Nandini Dendukuri},
title = {How-to Guide on {R} {Shiny} App for {Bayesian} {Diagnostic}
{Test} {Meta-Analysis}},
date = {2021-11-17},
url = {https://www.nandinidendukuri.com/blogposts/2021-11-17-bayesdta-shiny-app/},
langid = {en}
}


text:
## WC '18 Contest 1 S2 - Essay Generator

View as PDF

Points: 10 (partial)
Time limit: 1.0s
Memory limit: 16M

Author:
Problem type
##### Woburn Challenge 2018-19 Round 1 - Senior Division

Alice has a -word essay due tomorrow , but she's too busy programming to bother with that! However, Alice happens to know that H.S. High School's English teacher is sick of reading and grading long essays, so she figures that if she just submits a "reasonable" essay which fulfills the requirements but is as short as possible, she may get some pity marks!

As such, Alice wants to write a program to generate a sequence of words to pass off as her essay, where each word is any string consisting of or more lowercase letters (a...z) (not necessarily a real English word). The essay will have no punctuation or formatting, as those seem unnecessary to Alice. In an attempt to disguise the essay's generated nature, Alice will insist that all words are distinct. Finally, for her plan to come together, she'll make the sum of the words' lengths as small as possible.

Help Alice generate any essay which meets the above requirements.

In test cases worth 5/20 of the points, .
In test cases worth another 5/20 of the points, .

#### Input Specification

The first and only line of input consists of a single integer, .

#### Output Specification

Output a single line containing Alice's essay: a sequence of distinct space-separated words, with the sum of their lengths minimized.

#### Sample Input

2

#### Sample Output

i a

#### Sample Explanation

The sum of the lengths of the two words i a is , which is the minimum possible total length of a two-word essay. Various other essays (such as x y) would also be accepted. However, the essay i i would not be accepted due to its words not being distinct, and the essay i am would not be accepted due to the total length of its words not being as small as possible.


text:
### Home > PC > Chapter 1 > Lesson 1.4.1 > Problem1-132

1-132.

A wheel is spinning at $500$ revolutions per minute. Remember $1$ revolution equals $2\pi$ radians. How many radians per second is that? Give the exact value.

First, calculate the number of Radians per minute.

Second, divide by $60$ to calculate the number of Radians per second.

$\frac{50\pi}{3}$


text: AbstractOn 6 October 2020 Sir Roger Penrose, Reinhard Genzel and Andrea Ghez were jointly awarded the Nobel Prize in Physics for their work on black holes. Among Professor Penrose’s many contributions to mathematical physics is his work on a form of aperiodic tiling which bears his name: Penrose tiling. Using the LaTeX package penrose Overleaf created this LaTeX template project as our tribute to Professor Penrose on his award of a Nobel Prize and for anyone wishing to explore the fascinating and beautiful patterns created by Penrose tilings.


text:
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Hoses A and B spout water at different constant rates, and hose A can

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Hoses A and B spout water at different constant rates, and hose A can  [#permalink]

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07 Jan 2015, 05:37
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Hoses A and B spout water at different constant rates, and hose A can fill a certain pool in 6 hours. Hose A filled the pool alone for the first 2 hours and the two hoses, working together, then finished filling the pool in another 3 hours. How many hours would it have taken hose B, working alone, to fill the entire pool?

A 18
B 15
C 12
D 6
E 3

Thanks!

My approach is the follwing and I don't understand what my mistake is:
A filled 1/3 of the pool, so 2/3 are left to be filled in 3 hours by both hoses.

$$\frac{2}{3} / 3 = \frac{1}{6} + \frac{1}{B}$$
Math Expert
Joined: 02 Sep 2009
Posts: 49965
Re: Hoses A and B spout water at different constant rates, and hose A can  [#permalink]

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07 Jan 2015, 06:57
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noshtafoyza wrote:
Hoses A and B spout water at different constant rates, and hose A can fill a certain pool in 6 hours. Hose A filled the pool alone for the first 2 hours and the two hoses, working together, then finished filling the pool in another 3 hours. How many hours would it have taken hose B, working alone, to fill the entire pool?

A 18
B 15
C 12
D 6
E 3

Thanks!

My approach is the follwing and I don't understand what my mistake is:
A filled 1/3 of the pool, so 2/3 are left to be filled in 3 hours by both hoses.

$$\frac{2}{3} / 3 = \frac{1}{6} + \frac{1}{B}$$

Since hose A can fill the pool in 6 hours, then in 2 + 3 = 5 hours it will fill 5/6th of the pool. Thus the remaining 1/6th is filled by hose B in 3 hours. This means that hose B,working alone, to fill the entire pool will need 3*6 = 18 hours.

Hope it's clear.
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Re: Hoses A and B spout water at different constant rates, and hose A can  [#permalink]

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07 Jan 2015, 20:39
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Rate of hose A $$= \frac{1}{6}$$

Work done by hose A in 2 hrs $$= \frac{1}{6} * 2 = \frac{1}{3}$$

Pending work $$= 1 - \frac{1}{3} = \frac{2}{3}$$

Let rate of hose B $$= \frac{1}{b}$$

$$(\frac{1}{b} + \frac{1}{6}) * 3 = \frac{2}{3}$$

b = 18

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Re: Hoses A and B spout water at different constant rates, and hose A can  [#permalink]

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07 Jan 2015, 07:57

Could you please comment on my approach which I posted in the spoiler? I don't get the flaw in my logic there
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Re: Hoses A and B spout water at different constant rates, and hose A can  [#permalink]

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07 Jan 2015, 08:01
noshtafoyza wrote:

Could you please comment on my approach which I posted in the spoiler? I don't get the flaw in my logic there

My approach is the follwing and I don't understand what my mistake is:
A filled 1/3 of the pool, so 2/3 are left to be filled in 3 hours by both hoses.

$$\frac{2}{3} / 3 = \frac{1}{6} + \frac{1}{B}$$[/spoiler]

(2/3)/3=1/6 + 1/x
2/3 = 1/2 + 3/x
1/6 = 3/x
x = 18.

The same answer as in my solution.
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Re: Hoses A and B spout water at different constant rates, and hose A can  [#permalink]

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07 Jan 2015, 08:09
1
Thanks for verifying bunuel! Turns out I just didnt get my math straight solving the equation
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Re: Hoses A and B spout water at different constant rates, and hose A can  [#permalink]

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07 Jan 2015, 08:39
2
noshtafoyza wrote:
Hoses A and B spout water at different constant rates, and hose A can fill a certain pool in 6 hours. Hose A filled the pool alone for the first 2 hours and the two hoses, working together, then finished filling the pool in another 3 hours. How many hours would it have taken hose B, working alone, to fill the entire pool?

A 18
B 15
C 12
D 6
E 3

Thanks!

My approach is the follwing and I don't understand what my mistake is:
A filled 1/3 of the pool, so 2/3 are left to be filled in 3 hours by both hoses.

$$\frac{2}{3} / 3 = \frac{1}{6} + \frac{1}{B}$$

Another approach.

let work = 18 units
therefore, amount of work done by A in 1 hour = 18/6 = 3 units

now, since A work alone for two hours. therefore amount of work completed by A in two hours = 2*3 =6 units. thus we have 18-6 = 12 units remaining.

the remaining 12 units are completed by both A and B in 3 hours.

Amount of work completed by A in 3 hours = 3*3 = 9units . Thus remaining 3 units must have been contributed by B.

i.e. B completed 1unit in 1 hour. Therefore total time taken by B to complete 18 units of work alone = 18/1 = 18 hrs.
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Hoses A and B spout water at different constant rates, and hose A can  [#permalink]

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03 Nov 2016, 10:12
Hoses A and B spout water at different constant rates, and hose A can fill a certain pool in 6 hours. Hose A filled the pool alone for the first 2 hours and the two hoses, working together, then finished filling the pool in another 3 hours. How many hours would it have taken hose B, working alone, to fill the entire pool?

A) 18
B) 15
C) 12
D) 6
E) 3
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Posts: 49965
Re: Hoses A and B spout water at different constant rates, and hose A can  [#permalink]

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03 Nov 2016, 11:00
SW4 wrote:
Hoses A and B spout water at different constant rates, and hose A can fill a certain pool in 6 hours. Hose A filled the pool alone for the first 2 hours and the two hoses, working together, then finished filling the pool in another 3 hours. How many hours would it have taken hose B, working alone, to fill the entire pool?

A) 18
B) 15
C) 12
D) 6
E) 3

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Re: Hoses A and B spout water at different constant rates, and hose A can  [#permalink]

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12 Nov 2016, 00:30
noshtafoyza wrote:
Hoses A and B spout water at different constant rates, and hose A can fill a certain pool in 6 hours. Hose A filled the pool alone for the first 2 hours and the two hoses, working together, then finished filling the pool in another 3 hours. How many hours would it have taken hose B, working alone, to fill the entire pool?

A 18
B 15
C 12
D 6
E 3

Thanks!

My approach is the follwing and I don't understand what my mistake is:
A filled 1/3 of the pool, so 2/3 are left to be filled in 3 hours by both hoses.

$$\frac{2}{3} / 3 = \frac{1}{6} + \frac{1}{B}$$

Please check the solution in attachment
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File comment: www.GMATinsight.com

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Re: Hoses A and B spout water at different constant rates, and hose A can  [#permalink]

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31 May 2018, 16:19
noshtafoyza wrote:
Hoses A and B spout water at different constant rates, and hose A can fill a certain pool in 6 hours. Hose A filled the pool alone for the first 2 hours and the two hoses, working together, then finished filling the pool in another 3 hours. How many hours would it have taken hose B, working alone, to fill the entire pool?

A 18
B 15
C 12
D 6
E 3

We are given that hose A can fill the pool in 6 hours alone. Thus, the rate of hose A is 1/6.

Since we are not given the rate of hose B, we can let B = the number of hours it takes hose B to fill the pool alone. Thus, the rate of hose B is 1/B.

We are given that hose A worked alone for the first 2 hours and then the two hoses worked together for another 3 hours to complete the job. Thus, the time worked by hose A is 5 hours and the time worked by hose B is 3 hours. Since work = rate x time, we can calculate the work done by hose A and hose B.

Work done by hose A = (1/6) x 5 = 5/6

Work done by hose B = (1/B) x 3 = 3/B

Finally, since the completed job = 1, we can sum the work values of hoses A and B and set the sum to 1.

5/6 + 3/B = 1

Multiplying the entire equation by 6B gives us:

5B + 18 = 6B

18 = B

Thus, hose B, when working alone, can complete the job in 18 hours.

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Hoses A and B spout water at different constant rates, and hose A can  [#permalink]

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15 Sep 2018, 03:34
Quote:
Hoses A and B spout water at different constant rates, and hose A can fill a certain pool in 6 hours. Hose A filled the pool alone for the first 2 hours and the two hoses, working together, then finished filling the pool in another 3 hours. How many hours would it have taken hose B, working alone, to fill the entire pool?

A 18
B 15
C 12
D 6
E 3

A worked for a total of 5 hrs (2 hrs alone and remaining 3 hrs with B).
A alone can complete the task in 6 hrs.
amount of work 'A' can do in 1 hr = $$\frac{1}{6}$$
amount of work done by 'A' in 5 hrs = $$\frac{5}{6}$$
this means that the remaining $$\frac{1}{6}$$ of the work was done by 'B' in 3 days.
this means that 'B' can complete $$\frac{1}{6}$$ of the work in 3 days.
this means that 'B' can complete $$\frac{1}{18}$$ of the work in 1 day.
Hence 'B' can complete the entire work alone in 18 days.
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Re: Hoses A and B spout water at different constant rates, and hose A can  [#permalink]

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25 Sep 2018, 17:16
Where am i going wrong: my approach is long but i wd like to know where am i going wrong in my thinking:

1. after 2 hours, the remaining task is: 4/6--> 2/3
2. in the next 3 hours, A fills (3/6)(2/3): 1/3
3. which means 2/3rd of the remaining 2/3 (from 1) is filled by B
4. thus 4/9 is filled by B in 3 hours
5. thus the rate of B is (4/9)/3 --> 4/27
6. which means the # of hours is: 27/4--> approx 8
Re: Hoses A and B spout water at different constant rates, and hose A can &nbs [#permalink] 25 Sep 2018, 17:16
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text:
# Latex thesis bibliography at each chapter

I am helping a colleague with his phd thesis and we need to present the bibliography at the end of each chapter the question is: does anyone have a minimal working example for this case using la. Tips on writing a thesis in latex this example contains all modifications discussed in bibliography, biblatex customization in my thesis a typical chapter includes about 20 pages, and i assumed citereset=chapter to be quite acceptable. Example thesis latex files including unbthesistex, dedicationtex, abstracttex tables and bibliography, all of which may be single-spaced) start each chapter with numbering by using: \chapter{chapter title. Tips on writing a thesis in latex this example contains all modifications discussed in bibliography, biblatex contains the only settings for entry types i used in my thesis appearance of each entry in the final document together with the corresponding entry fields from bib file.

Latex thesis front page -combining instruction on driving questions have to spend one and a google form collected by 204 students isolating each 201 engaging students with the educational research and in this chapter, we discuss some of the father of the the secondary. Writing-phd-thesis-using-latex - download as pdf file (pdf), text file (txt) some people prefer to place each chapter of a large document in a separate file each bibliography item in the thebibliography environment must be separated by a blank line. Natbib is a package for latex that allows you to format citations and bibliographies latex makes it possible to have different files for each chapter 6 thoughts on writing a phd thesis in latex jan says. Any latex version required by your department thesis or by chapter that is 11, 12, 21, 22 return to top references or bibliography each of these parts may be subdivided into sections and subsections. Formatting guide thesis formatting guide in this section tables and/or figures may be grouped together at the end of each chapter a thesis must include a bibliography or reference section listing all works which are referred to in the text.

Writing your thesis with r markdown (2) - text, citations and equations and equations in your thesis chapter first up, writing plain text and headings: select 'escape latex special characters'. Latex for thesis, unveil the step by step guide on latex thesis writing open modal it would be a good idea for you to split each chapter of you large document into separate files you can do that by using the command: \include{file name. I am posting on behalf of a friend, whose thesis advisor wants her to add a bibliographic section at the end of each chapter of her thesis she. A comprehensive latex guide with easy to understand examples and how-tos bibliography management with bibtex each bibliography record holds relevant information for a single entry @article {einstein.

## Latex thesis bibliography at each chapter

Latex your thesis general thesis tips about the thesis template common queries you can add a reed thesis citation option in fact, your bibliography style (bst) you can have separate files for each chapter and then include them in the thesis template.

• I want to generate the bibliography for each section using latex, how can i have a list of references at the end of each section [closed] ask question something that makes sense if there is a bibliography in each chapter.
• I am trying to compile a thesis in latex and i need to put in references after each chapter can anyone give a detailed example of how to use chapterbib properly to obtain a bibliography at the end of each chapter.
• Tex - latex stack exchange is a question and answer site for users of tex putting bibliographies at the end of each chapter up vote 10 down vote favorite 4 problem in creating bibliography for each chapter 1.
• Writing your thesis with r markdown (1) - getting started in the next blog post, i will explain how to do each of these things in a thesis chapter (update: i was thinking of latex for my phd thesis.

An online latex editor that's easy to use no installation register log in sharelatex blog how to write a thesis in latex pt 4 - bibliographies with biblatex to see the corresponding video for this blog post click here in this post we are going to look at adding a bibliography to. I'm using latex + bibtex to write my dissertation i have one separate tex file for each chapter and i'll generate one bibliography for each chapter but all the references are from the same file. Latex thesis bibliography at each chapter just imagine if you can create your own resume like a professional resume writer and save on cost now you can. So i like to break it up by chapters with each chapter having a separate folder than contains its own tex \addcontentsline{toc}{chapter}{bibliography} well i hope you found these helpful if one thought on latex tips and tricks for writing a thesis or dissertation.

Latex thesis bibliography at each chapter
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# Tag Info

50

Colloquially, there's a lot of conceptual overlap between all of these terms, but "sameness" is not a well-defined mathematical property. Congruent shapes need not be "the same" or "equal" in all respects - they can be rotated differently, or be in different positions, or be different colors, or have different names, or differ ...

42

Because the aim of geometry is to study properties related to shape, size and length. Therefore, in the context of geometry, we cannot deform our objects because deformation changes these properties. On the other hand, the properties in which graph theory is interested are independent of size, shape and length. Therefore, we do not care about them. (However, ...

37

I'm confused. Are you really going to try to make this sort of distinction when teaching geometric figures to students "around 9-13 years old"? Students that age (and engineers my age -- much, much older) think that a triangle is a triangle. It's a polygon formed by three non-colinear points. A triangle has many ways you can think about it. ...

28

The smart-aleck answer is that most congruent triangles, or congruent figures more generally, aren't actually "the same" or "equal". Usually when we say two things are "the same", we mean that they are not just indistinguishable, but that they are literally the same exact thing. "Equality" means two numbers are the ...

24

One encounters exactly the same issue teaching multivariable calculus when one treats integrals over three-dimensional regions and integrals over the surfaces that are their boundaries. In particular the word sphere is particularly confusing in this context. (Mathematicians use sphere to mean the the two-dimensional surface; colloquial speech and some ...

22

Generally, this math falls under the scope of what is commonly called Taxicab Geometry. I would use taxicab path as a noun to describe the specific paths illustrated in the original question; whereas taxicab geometry would be a term I'd use for the subset of mathematics covering these types of scenarios.

22

As an answer so that I can paste in a picture: I think that the problem with the one on the left is that it is possibly "good enough" that someone might think that it is to scale and that apparent regularities are real. (Well, maybe not one of those "right angles".) In particular, someone might become under the impression that the length ...

21

I would use this to help students understand three "meta" ideas: (1) Math is not about memorizing lots of random trivia. In the real world, if you go up to a mathematician and ask them which definition of a trapezoid is right, they will just smile indulgently. They don't know or care. (2) There is not always a consensus about definitions. Get over ...

19

There is no value in drawing the figure exactly to scale, but the left-hand figure is inaccurate to the point where it is positively misleading. Since the angle marked 30 degrees is actually drawn greater than 45 degrees, it gives the impression that a is less than 3 (or maybe equal to it, if the student takes the "30 degree" angle as being shown ...

19

Common knowledge The formula $a^2+b^2 = c^2$ is common knowledge and the words for hypotenuse and leg (is "cathetus" not used in English?) are basic mathematical vocabulary. Including these seems a good idea. Connections to other mathematics The notation with AB, CA and BC might be something the students have used or will use in less analytical ...

17

I think the distinction you are raising is not natural to students at this age. I teach undergraduates and graduate students, not elementary schoolers, but I find that it is not natural for undergraduates who have not had a theoretical math course. In my experience, students do not naturally think of geometric figures as sets of points. If $P = (-1,-1)$, $Q =... 12 I really like the idea of a "discovery fiction" -- it gives a name to something I often try to use when teaching. Here is one suggestion. I will try to come back and write a more elaborated version of this answer later, with diagrams and proper notation, but briefly: (a) Don't let on that you are going to prove the Pythagorean Theorem -- don't ... 11 I would split the difference by creating an accurate diagram in Desmos and then giving the students a hand drawing based on that diagram. That way, students could estimate their answer before solving and check their answer against that original estimate. But they couldn't conclude that$b=6$by measuring the length of the known side. Pedagogically, the two ... 10 In the field of micro/nano-lithography, such geometry would be called Manhattan geometry; containing only two directions of edges orthogonal to each other. If diagonals were included, it would be classified as having skew edges. Finally, if it were truly freeform, it would be called curvilinear. Example: https://doi.org/10.1117/12.2243030 10 In Olympiad geometry,$a$,$b$,$c$is the so-called standard notation for the sides of a triangle, so it makes sense to use it consistently when referring to a triangle (in isolation). However, in any case the general principle is introducing all your notation. Writing Pythagoras' theorem states that$a^2+b^2=c^2$. or Pythagoras' theorem states that$...

8

These are pretty much universally called lattice paths, which Will Orrick comments on the question. A quick search on the arXiv reveals a large number of papers from combinatorics and computer science using the term in this way. (The main competition is from papers about path integrals on lattices, which gets unfortunately concatenated to "lattice path ...

8

Unfortunately, we don't have a set of universally agreed upon definitions in mathematics. It might seem like we do (or should), especially in Geometry with its long history and so much agreement, but the truth is that we use different definitions frequently. One proof of this is the differing definitions in your textbooks! That is just the nature of a ...

8

I recommend either Excursions in Geometry by Ogilvy or Geometry Revisited by Coxeter and Greitzer. Both are cheap too.

8

Two alternatives I have seen used (am not necessarily recommending them, but will list some pros and cons) which don't seem to have been mentioned yet. Don't denote it algebraically at all! Draw a picture instead For the lower end of the 12-16 age range, I've seen this work really well. You literally draw the squares sticking out from the triangle. Write the ...

8

I confess, when you start talking about 1-D triangles, my own first thought is "how can you have non-colinear points in 1-D?". So, I imagine most students that age will have a far more difficult time with that. Keep in mind age appropriateness. For 9-12 year old children, you are generally looking at a level of psychological development ...

7

A mathematics teacher once told me, "at a ratio of 4:5, a rectangle becomes a square." While this is obviously not true in a strict sense, it holds the deeper truth that a rectangle at 4:5 or worse is not a good example to represent a generic rectangle to young students, unless the goal is to teach or test that squares are also rectangles. That ...

7

Note: as in the originally referenced article, I'll take $a=1$ to keep it less complicated. I think it's important to first establish two key issues about the graphical setting: What does it mean to "be a square" in a graph. What information are we assuming we can "read" from the graph. For (1), in the geometric setting this notion is ...

7

Many of the geometric figures are so elementary that they are deeply rooted in daily language, and there seems to be no great solution. I agree with you here, and I think this is the key point. To me they are clearly well-defined: "Triangle", "square", and polygons in general, are bounded regions on the Euclidean plane, i.e., 2D figures. ...

7

Educator here who has worked with many students in the aforementioned age range (9-13) on triangles and squares. In my experience, it has never come up that a confusion between the boundary and the interior of a plane region was relevant to problem solving at that grade level. For these types of elementary shapes, the boundary and the interior completely ...

6

This paper calls a horizontal/vertical path a rook path, for the movement of a rook on a chess board. If the paths connect the lower-left corner to the upper-right corner, this would be a North-East Lattice Path. Edit: If the path didn't need to go through points with integer coordinates, I would submit the name "Cardinal Paths" in reference to the ...

6

This is a great question. Love it love it love it!! The following is just what occurs to me off the top of my head. Show a graph paper grid with a dot at the origin and a dot at (3,4). Say we want to find the distance between the dots. We could guess that it would be 3+4=7. Well, that would be right if these were city blocks, but it's not the right answer as ...

6

Since your job is helping the teacher with technology and the teacher should be making the pedagogical decisions, ask her her preference and do it that way. Maybe she meant for the pictures not to be to scale, or maybe they came out that way due to her poor technology skills. Just ask.

6

Finland is a small country with a centralized political system and a university system that is essentially public and free, and highly centralized on a couple of big schools. The US is a big country with a federal system of government and a higher education system that is a mix of public and private, spread across an entire continent. In the US, the federal ...

6

The only one of these that looks objectionable to me is the one that calls the hypotenuse $h$, since in a triangle the letter $h$ usually refers to the triangle's height (which could be either one of the legs but could not be the hypotenuse).

6

Historically (and by historically I mean "in Euclid's Elements") the word "equal", when applied to geometric figures, meant "equal in magnitude". So for example: Euclid refers to two segments as equal if they have the same length Two triangles are equal if they have the same area Two solids are equal if they have the same ...

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# Maximum value of a function with iterated logarithms

Consider this function $f(x)=-x\log\log(bx)+x\log\log\log(bx)+ax$, where $a$ and $b$ are positive constants, and the logarithms are 2-based.

Is it possible to find the maximum value (or even with approximation) of $f(x)$, in terms of $a$ and $b$, for $x>\frac{16}{b}$?

Note, it can be shown that $f(x)$ is concave for $x>\frac{16}{b}$.

-
Why would you want to do that? – Will Jagy Feb 6 '12 at 8:37

If you call that maximum value $M(a,b)$, you can first of all get rid of $b$ since $M(a,b)= M (a,1) / b$. And $M(a,1) = g ^ \star (a)$, the Legendre transform of the convex function $g(x):= x\log \log x -x\log\log\log x$ for $x \ge 16$, and $g=\infty$ for $x < 16$. Then, you may at least estimate $g ^ \star$ integrating $(g^\star)'=(g')^{-1}$. Note that you may also consider the somehow more natural definition of $g$ with finiteness domain $]2,\infty[$, which seems still convex.
Thanks for you suggestion. However, I think to find the inverse mapping $(g^')^{-1}$ is even harder. – Allen Feb 6 '12 at 17:50


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Tag Info

8

Monocular vision is a difficult and very interesting, particularly in its application to the general navigation problem. I will make an attempt at answering your questions, but if you find anything lacking, you can read through Szeliski's book Computer Vision: Algorithms and Applications. What is the core principle of a monocular visual odometry algorithm? ...

5

The most important point is the scale. If you do monocular SLAM, your map will only be accurate up to scale so that you e.g. cannot compute the length of the travelled path in meters. The scale between your map and the world is not even constant over time so that if you come back to your starting point, it's going to be difficult to match the beginning and ...

2

From the webpage for the dataset: All sequences contain mostly exploring camera motion, starting and ending at the same position: this allows to evaluate tracking accuracy via the accumulated drift from start to end, without requiring ground-truth for the full sequence. It appears as though they didn't intend the ground truth for these datasets to be ...

2

NaN values are sometimes used to indicate unavailable data. Is it possible that these are simply portions of the dataset for which ground truth was not available, but you can still evaluate on the remainder of the sequence (and that this is in fact the standard evaluation domain)? It looks like their evaluation code checks for trajectory entries with any ...

1

The problem of the pure strong rotation is that the image will become easily blurred unless you use 1000FPS camera. This kind of super-strong rotation often occurs in hand-held camera motion. Simply turn on your mobile camera in a slightly dark place and try to make a translational and rotational motion. You will see rotation make huge blurriness in the ...

1

This is not the answer of your question but ORB is a feature detection algorithm. Feature detection and tracking are different. Once you found a feature it is better to track them by a feature tracking algorithm such as KLT. You can also track the features by feature matching algorithms but it is not very efficient. Since features come with their own ...

1

I will assume, similar to OpenCV, that each camera is a pinhole camera, so you already corrected for things like lens distortion. In this case each visible point in 3D space $(x,y,z)$ gets projected into camera coordinates using  \begin{bmatrix} x'_i \\ y'_i \\ z'_i \end{bmatrix} = R_i \, \begin{bmatrix} x \\ y \\ z \end{bmatrix} + \vec{t}_i, \\ u_i = \...

1

DSO initializes the scene and camera poses with a specific scale factor such that the average inverse depth of the pointHessians is one. After the initialization the first two frameHessians are led into the backend to do a bundle adjustment like optimization in which, however, the previous determined scale can change (because the absolute scale is not ...

1

Researchers from Davide Scaramuzza's group authored a paper in 2014 about a probabilistic technique called REMODE that estimates depth from a single moving camera. The algorithm performs per pixel depth estimation from a camera stream and regularization based on uncertainty estimation. It should be suitable for your application involving a moving camera and ...

1

It turns out this can easily be done with OpenCV - just find image features (FAST etc.) in first image, track them to the second image (get a set of corresponding features between two images) and then use triangulatePoints function to get the 3D scene. triangulatePoints accepts two projection matrices - one for each image. Each projection matrix defines ...

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# organise references alphabetically

I have organised my bibliography as follows:

\begin{thebibliography}{99}

\bibitem[Author (date)]{code}

Author Name (date).

\textit{Title}.

Publisher.

etc

\end{thebibliography}

Is there any way to organise the references alphabetically?

• Welcome to Tex.se!
– M S
Aug 30, 2016 at 4:10
• Well, if you do it this way you have to sort it manually. Why don't you use biblatex and an appropiate bibliographystyle?
– user31729
Aug 30, 2016 at 4:21

You may want to look into creating bibliographies via tools such as BibTeX, which has been around for decades, and biblatex, which is (much!) more modern than BibTeX and offers more formatting and sorting options than BibTeX does. If you choose the BibTeX route, do also look into using a citation management package (e.g., natbib) for options for generating citation callouts.


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# Prove that $\lim\limits_{x \to 2} \frac{x-1}{x^2}$= $\frac{1}{4}$, using the $\varepsilon$-$\delta$ definition

So I have to prove $\lim\limits_{x \to 2} \frac{x-1}{x^2}$= $\frac{1}{4}$.

So I need to find $0<|x-2|<\delta$, such that $\left\vert\frac{x-1}{x^2}-\frac{1}{4}\right\vert<\epsilon$.

So $\left\vert\frac{x-1}{x^2}-\frac{1}{4}\right\vert=\frac{(x-2)^2}{4x^2}$.

Now I try to set $\delta=1$, so $0<|x-2|<1$ and $4<4x^2<36$.

And so $\frac{(x-2)^2}{4x^2}<\frac{(x-2)^2}{4}$.

And so $\frac{(x-2)^2}{4}<\epsilon$, when $\delta=\sqrt{4\epsilon}$.

• How did you arrive at the lower bound of $0$ in $\color{red}{0}<4x^2<36$? If $|x-2| < 1$, then $1 < x < 3$ so $4 < 4x^2 < 36$. – StackTD Jun 10 '16 at 11:54
• Edited. I was one step ahead of myself, thinking 4*0=0. I forgot to add across both sides first, – stackdsewew Jun 10 '16 at 11:57
• Wait so I can just set delta to be square root of 4*epsilon, right? – stackdsewew Jun 10 '16 at 11:59
• You use it in the step where you replace the denominator $4x^2$ by the (lower bound, to get an upper bound on the fraction) $4$. – StackTD Jun 10 '16 at 12:04
• @anna_xox You can just set $\delta$ to be $2 \sqrt{\epsilon}$ since inequality is carried on from the previous steps. Since you also want $\delta = 1$, we take the minimum. That is, $\delta = \text{min} \{ 1, 2 \sqrt{\epsilon}\}$. Do you understand why we take the minimum? – Jester Tran Jun 10 '16 at 12:09

Looks good, because you can use $\color{blue}{4<4x^2}<36$ to say: $$|f(x)-L|=\frac{(x-2)^2}{4x^2} \le \frac{(x-2)^2}{4}$$ And since $|x-2| < \delta$, this means: $$\frac{(x-2)^2}{4} < \frac{\delta^2}{4}$$ You want this under $\varepsilon$, so: $$\frac{\delta^2}{4} < \varepsilon \iff \delta < 2\sqrt{\varepsilon}$$ Note that you know require $\delta \le \min\left\{ 1,2\sqrt{\varepsilon} \right\}$.

Coming back to the '<' vs '=' (see comments as well): if you set $\delta = 1$ along the way and you end up with requiring $\delta = 2\sqrt{\varepsilon}$, note that you need the strongest (i.e. smallest) bound on $\delta$.

If you have a good $\delta$, any $\delta' < \delta$ will work as well; which is why you'll often see $\delta$ taken to be smaller than (or equal to) all the upper bounds you set / need in your proof.

May be you set $\delta$ to 1,and $0<|x-2|<1$, so $4 x^2$ will between 4 and 36, that is $1/4 x^2<1/4$. Then we can get $(x-2)^2/(4 x^2)< \frac{1}{4} (x-2)^2$. Set $\frac{1}{4} (x-2)^2 < \epsilon$, then $|x-2|<2 \sqrt{ \epsilon}$. Let $\delta = Min \{ 2\sqrt \epsilon ,1 \}$

• I think your reasoning is circular. We are trying to find a delta in terms of an epsilon. You have found a delta in terms of a delta. – stackdsewew Jun 10 '16 at 12:06
• Oh~It's my first edition.I can explain it. Set $1/16 \delta^2<\epsilon$,get the $\delta$,can solve it. – user346926 Jun 10 '16 at 12:12
• Your 1/16 should be 1/4. – stackdsewew Jun 10 '16 at 12:13
• I correct it as you say. I also a beginner learn Calculus and $\LaTeX$. – user346926 Jun 10 '16 at 12:24


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# Sample mean and covariance

sample meansample covariancesample covariance matrixempirical meanmeansSample mean and sample covariancemean of a random samplemeans of two samplessamplesample average
The sample mean or empirical mean and the sample covariance are statistics computed from a collection (the sample) of data on one or more random variables.wikipedia
105 Related Articles

### Statistic

sample statisticempiricalmeasure
The sample mean or empirical mean and the sample covariance are statistics computed from a collection (the sample) of data on one or more random variables.

### Bias of an estimator

unbiasedunbiased estimatorbias
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a row vector whose j th element (j = 1, ..., K) is one of the random variables.
If the sample mean and uncorrected sample variance are defined as

### Estimation of covariance matrices

Covariance estimationShrinkage estimationestimate of the covariance
The maximum likelihood estimate of the covariance
Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix.

### Normal distribution

normally distributedGaussian distributionnormal
for the Gaussian distribution case has N in the denominator as well.
The following table gives the quantile z_p such that X will lie in the range with a specified probability p. These values are useful to determine tolerance interval for sample averages and other statistical estimators with normal (or asymptotically normal) distributions:.

### Covariance matrix

variance-covariance matrixcovariance matricescovariance
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a row vector whose j th element (j = 1, ..., K) is one of the random variables.
The expected values needed in the covariance formula are estimated using the sample mean, e.g.

### Estimator

estimatorsestimateestimates
The sample mean and sample covariance are estimators of the population mean and population covariance, where the term population refers to the set from which the sample was taken.
The central limit theorem implies asymptotic normality of the sample mean \bar X as an estimator of the true mean.

### Unbiased estimation of standard deviation

sample standard deviationanti-biasingbias
where is the sample (formally, realizations from a random variable X) and is the sample mean.

### Scatter matrix

Given n samples of m-dimensional data, represented as the m-by-n matrix, the sample mean is

### Sample (statistics)

samplesamplesstatistical sample
The sample mean or empirical mean and the sample covariance are statistics computed from a collection (the sample) of data on one or more random variables.

### Mean

mean valueaveragepopulation mean
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a row vector whose j th element (j = 1, ..., K) is one of the random variables. The sample mean and sample covariance are estimators of the population mean and population covariance, where the term population refers to the set from which the sample was taken. The sample mean is a vector each of whose elements is the sample mean of one of the random variables – that is, each of whose elements is the arithmetic average of the observed values of one of the variables.

### Covariance

covariantcovariationcovary
The sample mean and sample covariance are estimators of the population mean and population covariance, where the term population refers to the set from which the sample was taken. The sample covariance matrix is a square matrix whose i, j element is the sample covariance (an estimate of the population covariance) between the sets of observed values of two of the variables and whose i, i element is the sample variance of the observed values of one of the variables.

### Vector (mathematics and physics)

vectorvectorsvectorial
The sample mean is a vector each of whose elements is the sample mean of one of the random variables – that is, each of whose elements is the arithmetic average of the observed values of one of the variables.

### Average

Rushing averageReceiving averagemean
The sample mean is a vector each of whose elements is the sample mean of one of the random variables – that is, each of whose elements is the arithmetic average of the observed values of one of the variables.

### Matrix (mathematics)

matrixmatricesmatrix theory
The sample covariance matrix is a square matrix whose i, j element is the sample covariance (an estimate of the population covariance) between the sets of observed values of two of the variables and whose i, i element is the sample variance of the observed values of one of the variables.

### Location parameter

locationlocation modellocation parameters
Due to their ease of calculation and other desirable characteristics, the sample mean and sample covariance are widely used in statistics and applications to numerically represent the location and dispersion, respectively, of a distribution.

### Statistical dispersion

Due to their ease of calculation and other desirable characteristics, the sample mean and sample covariance are widely used in statistics and applications to numerically represent the location and dispersion, respectively, of a distribution.

### Probability distribution

distributioncontinuous probability distributiondiscrete probability distribution
Due to their ease of calculation and other desirable characteristics, the sample mean and sample covariance are widely used in statistics and applications to numerically represent the location and dispersion, respectively, of a distribution.

### Definiteness of a matrix

positive definitepositive semidefinitepositive-definite
:Like covariance matrices for random vector, sample covariance matrices are positive semi-definite.

### Multivariate random variable

random vectorvectormultivariate
The sample mean and the sample covariance matrix are unbiased estimates of the mean and the covariance matrix of the random vector, a row vector whose j th element (j = 1, ..., K) is one of the random variables. :Like covariance matrices for random vector, sample covariance matrices are positive semi-definite.

### Bessel's correction

Bessel-correctedBessel corrected variance
The sample covariance matrix has in the denominator rather than due to a variant of Bessel's correction: In short, the sample covariance relies on the difference between each observation and the sample mean, but the sample mean is slightly correlated with each observation since it is defined in terms of all observations.

### Random variable

random variablesrandom variationrandom
The sample mean or empirical mean and the sample covariance are statistics computed from a collection (the sample) of data on one or more random variables. This is an example of why in probability and statistics it is essential to distinguish between random variables (upper case letters) and realizations of the random variables (lower case letters).

### Realization (probability)

realizationrealizationsobserved data
This is an example of why in probability and statistics it is essential to distinguish between random variables (upper case letters) and realizations of the random variables (lower case letters).

### Maximum likelihood estimation

maximum likelihoodmaximum likelihood estimatormaximum likelihood estimate
The maximum likelihood estimate of the covariance

### Statistical population

populationsubpopulationsubpopulations
Of course the estimator will likely not be the true value of the population mean since different samples drawn from the same distribution will give different sample means and hence different estimates of the true mean.

### Normalizing constant

normalizednormalization constantnormalization
Without loss of generality, assume that the weights are normalized: