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# Masters vs. Staying a 9th semester / 5th year

1. Mar 10, 2008

### JasonJo

All the admissions decisions are in, and I got rejected from most of my schools but I did get into one of my top choice's....Masters program.

So I called this school's math department today and I got the rundown of being in the Masters program. I made it clear that it is not my intention to get a Masters or even be a masters student, I ultimately want to be a fully funded PhD student. I got the following response:
1) There is a way to be favorably admitted to the PhD program via doing well in classes and getting a high mark on written exams.
2) Chances of getting funded are extremely tough and there have been examples of stellar PhD (not even Masters) students who decided to enter without funding for 1 or 2 years who still have not gotten funding. Usually this school funds only 10-12 students a year, and regularly have 400-500 applicants.

Here is my stance on this situation:
I can work my butt off for my Masters, which would add up to ~ $40k in loans. Even if I get into the PhD program, my chances of funding are not great. The person who I spoke to in the department told me I might have to wait until my Oral exam to be funded. This is one of my top choices for grad school, but the possibility to having to wait 2-3 years to get funding, amassing about$60-70k in loans (if I can even borrow that much) does not seem very appealing.

I'm considering turning down their Masters admissions and staying a 9th semester, or maybe a full 5th year.

What I would do in the 5th year:
1) Over this upcoming summer, one of the things I would do is study for my GRE Math subject exam score again. It has become clear to me that I need a great GRE score. Many of the schools said that once applications come in, unless the person has great letters of recommendations, the first way to cut them down is by GRE scores. I had an atrocious GRE Math score, but I'm confident if I study over the summer, I might be able to get a great GRE score.
2) Take a grad course and two independent studies: one in general relativity and one in Lie Algebras and Particle Physics. Both independent studies have been confirmed and the professors have agreed to it. The general relativity independent study will probably involve me writing an undergraduate thesis and hopefully display my knowledge of both geometry, tensor analysis and physics. The Lie Algebra and Particle Physics independent study might also be used to write an undergraduate thesis, i.e. rigorously solve one of the problems in the graduate textbook.

Over the summer I would also study a bit for these research projects, more in the way of background material.

Here is my timeline:
Summer
- Work 20-30 hours a week
- Study for the GRE Math Subject exam 10 hours a week
- Study for my independent studies 15-25 hours a week
That is at worst a 65 hour week, which isn't too bad at all. I'm usually bored out of my mind during the summer vacation anyway.

Fall
- Take a grad course (Topology I or Algebra I) and the two independent studies

If I do well in these independent studies and my grad course, and I mean really well, I see my graduate application as this:

1) High GRE Math Score (I'm aiming for an 85+% percentile score)
2) 3 letters of recommendation - one from a professor that I worked with for 3 semesters and this semester might publish a paper on an unsolved problem, one from my general relativity professor (in the math department) that I would've worked with for 2 semesters (including Fall 2008) and one from a prominent physics professor at my school who I did my Lie Algebra independent study with, who I only did a semester with. But if I do really well and get a paper published and 2 undergraduate projects done, I see this as 3 really good letters.

The recommenders that I might get a paper with is extremely well known in their field, my general relativity professor got their PhD a few years ago and is not as so well known and the professor I might get a recommendation from is very very well known in theoretical particle physics and string theory.
3) High GPA from a top 10 (according to rankings) geometry school, ~ 3.8
4) Done well in a grad course (though not comparable to some students who have taken 5-6 grad courses).

Overall, I view this is as a very strong graduate school application and A MUCH stronger application than what I had in Fall 2007.

PS: I know I made a similar thread about staying a 5th year, but now this is staying a 5th year vs. doing a Masters.

Last edited: Mar 10, 2008
2. Mar 10, 2008

### Gtay

I would say staying for a 5th year. I think you have a much better chance next year. ( also to get financial support)

Last edited: Mar 10, 2008
3. Mar 11, 2008

### cristo

Staff Emeritus
What's the difference, money wise, between staying another year, and going somewhere else for a year to get a master's degree? It seems to me that you'd have to pay tuition where ever you are, and so I'd go for the masters to get another degree to my name before applying grad school again, rather than staying for another year.

4. Mar 11, 2008

### JasonJo

Staying a 5th year would cost me about $5000, this includes everything, down to food. One year of a masters is about$27k, that's only tuition. I would be living at home, so I would only need to pay for books and food occasionally.

But I am really thinking I should at the very least get my Masters at this school and then apply to a PhD program somewhere else if they won't fund me.

Now another question, if I get my masters at a well known institution, will they waive the first year courses for me at another PhD institution? In total, the masters would cost me about \$40k in debt.

I would also need to retake the GRE Math anyway wouldn't I? So maybe I should reapply for schools without staying a fifth year???

Last edited: Mar 11, 2008
5. Mar 11, 2008

### tuna_wasabi

The way I see it, there are three options:

1. You go to grad school now and accrue debt. In two years you'll have a masters degree and have ~40k in debt.

2. You wait a year. You nail your GRE. You publish a paper. You get great letters of recommendation. Your grad school app is much stronger. You get into a school that offers you support. You'll have a masters in three years, and have paid ~5k.

3. You wait a year. For some reason or another, your situation does not improve. Worst of both worlds. You'll have a masters in three years and ~40k worth of debt.

To me, #2 is the best choice, and it sounds like you're willing to work hard enough to make it happen.

6. Mar 23, 2008

### JasonJo

I got some more specifics of the masters program. I'll be taking a bunch of non-PhD courses and at most take 3-4 courses with PhD students. My main goal was to do the Masters, but take the full differential geometry course sequence, do really well, get great letters and transfer to a really good school for mathematical physics (Northwestern, Columbia) for my PhD. But it does not seem possible to do this.

I don't know which way to go. My parents want me to do a masters because then I can stay by them and go to a "prestigious" school, but the school does not necessarily have what I want, and i'm still only a masters student there.

My goal is to study mathematical physics. I am wondering what the hell I have to do to get this done.

7. Mar 23, 2008

### ircdan

I think you should definitely stay where you are and

- study for the math gre, this should be your main focus
- take some classes for fun, maybe just one or two, don't overdo it and don't get your masters here
- reapply next year, but this time don't be so limited in your choices, think about all the factors when choosing schools
- read this book https://www.amazon.com/Mathematicia...=sr_1_1?ie=UTF8&s=books&qid=1206318758&sr=8-1

Last edited: Mar 23, 2008
8. Mar 23, 2008

### JasonJo

Hey thanks for the reply.

I am definitely trying to get my GRE Score WAY UP. I am aiming for a 90% percentile score. I am reviewing calculus, algebra and other topics from undergrad and I plan on taking it October 2008 (whether or not i'm in grad school).

If I stay a fifth year, I feel like I could get into a school that is strong in mathematical physics (Northwestern for example) and maybe even a better school if I work really really hard (Columbia hopefully). But if I do reapply, I would apply to some lower range schools like UCSB.

I have Krantz's book and I got a lot from it, however, I gained nearly nothing from his admissions part. Krantz took seven grad courses and was a hot shot at his school. I cannot relate to that and he has no section about my situation.

I think taking 2 grad courses and doing an independent study with a well known physics professor on Lie Algebras and particle physics will really boost my graduate school application. I would have all that, a possible publication, a way better GRE score and 2 really really good letters of recommendation and possibly a 3rd one.

I am thinking about staying, but it would kill my parents to turn down the Masters. But I want to study at a school that is good in mathematical physics. This is my dream. I know even after all this, I'm still a long shot for Columbia or MIT or UCLA, but I HAVE TO TRY. I only have a few chances at something like this. I view this as one of my last chances to get into a great math-physics program.

9. Mar 23, 2008

### Crosson

There was a time when I remember saying each of those things!

In my case it indicated that I was very stressed out about being accepted to grad school. What I chose to do was remain where I was as an undergrad for another (5th) year, and although it was surely the right decision, it was also a very difficult year. Drawing out what should be a 9 month process into an 18 month process is always bound to take it's toll. My advice to you is to take the 5th year at home, but make sure not to over-stress out about getting into MIT. Instead of telling yourself that you will study more intensely that year, tell yourself that you will study at your normal rate and take on a new activity unrelated to math-phys (again, the reason is to prevent over-stress).

P.S. It sounds like letters of recommendation was the weak-link in your applications.

Tell your parents that you are determined to get a PhD, and the master's route doesn't actually look viable.

10. Mar 23, 2008

### ircdan

I would really turn down the masters, I just thinking paying is a waste of money. The name of the school you go to isn't going to really matter too much when it comes time to finding a job, it will be your research, but I'm sure you already know this.

About the math gre, i think if you score 700+ or even 650+/600+ that's enough to get into a good school(ie, a group I school by AMS rankings) because you seem have good grades and great recommendations. Remember once you get to graduate school all this will be behind you, so try not to worry(I know it's hard not too). With a better gre score and your other qualifications you will definitely get into several schools imo if you apply to the right places. Goodluck.

11. Mar 30, 2008

### JasonJo

So here is where I am at now:

I have landed a possible independent study for Fall 2008 with a well known geometer at my school and a well known theoretical physicist at my school. Do you think 1 semester is enough to garner a great letter from these two professors? If I can get a good letter from either of these guys, I feel like I can get into a good PhD program with funding.

I would study for the GRE Math this summer (along with this I would be brushing up on my calculus, algebra, linear algebra, analysis, etc) and study geometry and physics hardcore to prepare for these independent studies. I'm hoping that I can prepare myself well enough to really just demolish these independent studies, really impress them every meeting and get great recommendations.

12. Mar 30, 2008

### tuna_wasabi

[slightly off topic]

I read the http://books.google.com/books?id=-U...alled+the"&source=gbs_quotes_s&cad=2#PPP1,M1" chapter on Google Books. It looks really good, and the majority of the material seems to translate pretty readily to advice on physics grad schools. Do you think this would also be a wise investment for a physics major? Or is there a similar book for physics majors? I did some brief searching and didn't turn up anything particularly compelling.

Last edited by a moderator: Apr 23, 2017
13. Mar 30, 2008

### JasonJo

I have the book and I have read through it various times. I don't know how relevant the whole book is. Parts of it definitely will be relevant as they relate to graduate life in general. But other pars focus on prelim exams, how to obtain tenure track positions (which I don't know if the procedure is the same for physics as it is in math).

But it can't hurt to buy the book. It is worth it just for the advice on being a graduate student. The tenure track stuff might differ and the last few chapters go over material a math grad student should know.

Last edited by a moderator: Apr 23, 2017
14. Mar 30, 2008

### tuna_wasabi

I just ordered it. Thanks!


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# zbMATH — the first resource for mathematics

Singular perturbations for parabolic equations with unbounded coefficients leading to ultraparabolic equations. (English) Zbl 1164.35312
Summary: Linear parabolic equations with coefficients of the lower-order terms unbounded and with a small parameter multiplying some of the second (highest) space derivatives are considered, in the limiting case when such a parameter goes to zero. This yields a degenerate parabolic (ultraparabolic) equation with one space-like variable, $$x$$, and two time-like variables, $$y$$ and $$t$$. No boundary-layer is found to be needed in the case of the boundary-value problem on the $$x$$-unbounded domain $$Q_T=\{(x,y,t)\in \mathbb R\times [0,1]\times [0,T]\}$$ with a periodic boundary condition in the variable $$y$$ and initial data at $$t=0$$.

##### MSC:
 35B25 Singular perturbations in context of PDEs 35F15 Boundary value problems for linear first-order PDEs 35F30 Boundary value problems for nonlinear first-order PDEs 35K20 Initial-boundary value problems for second-order parabolic equations

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# Minimize sin(x) – exp(cos(x^2)) – x on [0,1]

A quick note for those who have been regular readers here. I’ve been out of commission for the last month mostly because of getting sick (twice) and then having to do quadruple time for catching up on work, etc.

Now on to some writing …

I recently taught a genetic algorithms lesson as part of a larger lecture series I gave on Data Science and Python. This was a broad survey course that touched on statistical analysis, regression, neural nets, and genetic algorithms.

In the genetic algorithms course, I wanted to show how they can be used as generic solvers [with a little bit of work]. But I also wanted to show some off-the-shelf solvers that already exist. One example is with Python’s scipy‘s optimize module that contains a function called minimize. If we let $$h(x) = \sin(x) – e^{\cos(x^{2})} – x$$ and provide an initial guess of $$x = 0.5$$ with the constraint that we wish to minimize $$h(x)$$ on $$[0,1]$$:

>>> scipy.optimize.minimize(h,[0.5,],bounds=((0,1),))

minimize gives $$x = 0$$ and $$h(x) = e$$.

Now what’s fun about this is what minimize does with different initial guesses. With an initial guess of $$x_{0} = 0.48$$, we have $$h(x) = -2.71831419$$ at $$x = 0.09182462$$.

>>> scipy.optimize.minimize(h,[0.48,],bounds=((0,1),))

Test this out on Wolfram Alpha which gives $$x \approx .0919095$$.

And more hack work with the initial guesses: for an initial guess of $$x_{0}=0.01$$, scipy.optimize.minimize tells me the minimum is at $$x=0.01004454$$ with a value of $$−2.71828198$$. However, an initial guess of $$x_{0}=0.02$$ gives $$(0.09188086,−2.71831419)$$.

Interestingly, if we graph $$h(x)$$ on $$[0,1]$$ we see this

And if we zoom in to $$[0,0.1]$$ we see this

So a simple word to the wise, even when running (effectively) black box optimization algorithms, it may be helpful to provide multiple initial guesses.

As an aside, any of the results given by scipy.optimize.minimize were in agreement with each other to 4 places for minimizing $$h(x)$$. So, depending on the context, any of the solutions given by scipy.optimize.minimize are either good enough or not good enough. If we recall, $$\sin(x) \approx x$$ and $$\cos(x) \approx 1$$ for small $$x$$. Thus, $$h(x) \approx x – e – x = -e$$ for small $$x$$.

New here? Check out the "About The Blog" page and say "Hello" to @shahlock!

Do you enjoy this blog?
Consider supporting with a contribution!


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Hello.

I am a bit rusty on my natural log rules. How would I go about solving for x:

.25ln(11000 + x) + .75ln(10000 + x) = .25ln(11000) + .75ln(10000)

Any help would be greatly appreciated.

2. If you had $\displaystyle a\ln (b + x) + c\ln (d + x) = a\ln (b) + c\ln (d)$ then it follows that:
$\displaystyle \begin{gathered} \ln \left[ {(b + x)^a (d + x)^c } \right] = \ln \left[ {(b)^a (d)^c } \right] \hfill \\ (b + x)^a (d + x)^c = (b)^a (d)^c \hfill \\ \end{gathered}$.
That at least get it down to x’s.


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# Why isn't the $\lvert\phi(x)=0\rangle$ Fock eigenstate the vacuum state?

If, in a QFT of a scalar field $\phi$, a Fock space $n$-particle position eigenstate $\lvert x_1\cdots x_n\rangle$ is given by $$\lvert x_1\cdots x_n\rangle =\hat\phi^\dagger(x_1)\cdots\hat\phi^\dagger(x_n)\lvert 0\rangle \,,$$ where $\lvert 0\rangle$ is the vacuum state, then we have $$\langle x_1\cdots x_n\lvert\phi\rangle =\phi(x_1)\cdots\phi(x_n)\langle 0\lvert\phi\rangle \,,$$ with $\hat\phi(x)\lvert\phi\rangle =\phi(x)\lvert\phi\rangle$.

Now, the question is what is the value of $\langle 0\lvert\phi\rangle$?

If there's a vacuum configuration state $\lvert\phi_0\rangle = \lvert0\rangle$, then $\lvert 0\rangle$ is orthogonal to any other configuration in the basis, and we would have $$\langle x_1\cdots x_n\lvert\phi\rangle = 0$$ except for the case $\langle 0\lvert 0\rangle$ and thus a joined basis $\{\lvert x_1\cdots x_n\rangle ,\lvert\phi_0\rangle \}$ for a space of higher dimension than the Fock space. So, I conclude there's no vacuum configuration. But the zero configuration, $\phi(x)=0$, is anyway orthogonal to every Fock space state $\lvert x_1\cdots x_n\rangle$ different from $\lvert 0\rangle$, and thus excludes the presence of any particle. In other words, $$\lvert\phi(x)=0\rangle=\lvert0\rangle\langle0\lvert\phi(x)=0\rangle$$ So, the zero field configuration corresponds to the vacuum state, a contradiction; what am I missing? What's wrong in all this?

The state $|\phi\rangle$ is a coherent state, which has a non-zero overlap with the vacuum state $|0\rangle$.
The vacuum state is defined by $\hat \phi (x)|0\rangle=0$ for all $x$. The vacuum state is also given by the coherent state $|\phi(x)=0\rangle$ for all $x$. Furthermore, one should keep in mind that the coherent state basis is overcomplete, and therefore $\langle \phi|\phi'\rangle\neq0$ for any $\phi$ and $\phi'$. All this properties solve the OP's problem.
• The coherent state basis is overcomplete, and two different coherent state are never orthonormal ($\langle\phi|\phi'\rangle\neq 0$). The vacuum is just the coherent state $\phi(x=)0$ for all $x$. As I said, look at the case of a single harmonic oscillator. Then the (free) field theory is just the case of an infinite number of harmonic oscillator. – Adam Apr 21 '14 at 0:10


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# Binomial Tree American Put example

#### Eustice_Langham

##### Active Member
Hi David, I have a question concerning the example that appears in the Hull reading on page 287, Figure 13.10. I have attached a screenshot.

I am able to calculate the size of an up move factor, and down move factor as 1.35, .741 respectively, and the risk neutral probabilities of both up and down moves as .490 and .509 respectively.

I calculate the correct forward values:

U = $67.5, D =$37.04
UU = $91.13, UD =$50 and DD = $24.55 and terminal values of$0, $2 and$24.55 respectively

I however get different values for nodes B and C, as:

Node B: exp -.05 * sqrt (1) * (0 * .490) + (2 *.509) = $0.96 Node C: exp -.05 * sqrt (1) * (2 * .490) + (24.55 * .509) =$12.80

then at Node A my values will be different.

I have two questions:
1. are my calculations correct (its not just down to rounding differences) and
2. when considering American options, you need to consider two aspects/calculations..a. how much of the option is worth if exercised at the node AND b. how much is the option worth if not exercised. My question is then how do you make the comparison if the option should be exercised early or not? Thanks

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0
Views
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# Evaluation of Predictive Models for Early Identification of Dropout Students

JongHyuk Lee* , Mihye Kim** , Daehak Kim* and Joon-Min Gil**

## Abstract

Abstract: Educational data analysis is attracting increasing attention with the rise of the big data industry. The amounts and types of learning data available are increasing steadily, and the information technology required to analyze these data continues to develop. The early identification of potential dropout students is very important; education is important in terms of social movement and social achievement. Here, we analyze educational data and generate predictive models for student dropout using logistic regression, a decision tree, a naïve Bayes method, and a multilayer perceptron. The multilayer perceptron model using independent variables selected via the variance analysis showed better performance than the other models. In addition, we experimentally found that not only grades but also extracurricular activities were important in terms of preventing student dropout.

Keywords: Educational Data Analysis , Student Dropout , Predictive Model

## 1. Introduction

In recent years, the field of data analytics in education is attracting increasing attention with the rapid growth of educational data and the spread of big data technology. With the increasing use of online and software-based learning tools, the amounts and types of educational data have increased greatly, and new methods are required to analyze them. Thus, educational data mining [1] and learning analytics [2] are being expanded to explore large-scale data generated by universities and intelligent tutoring systems to better understand students; all of data mining, machine learning, and statistics are employed to this end.

The prediction and prevention of student dropout are very important in terms of continuing education. In other words, we seek not only to identify students at risk of leaving school but also to understand why dropout occurs, which will aid the design of future educational policies. Although many educators have statistically explored relationships between student lifestyle factors and dropout rates, they have focused on ensuring that students graduate, rather than an accurate prediction of dropout. As a result, there are many kinds of researches [3-12] on the features affecting the student dropout and the evaluation of predictive models. However, it is also required the study of the new features and the performance evaluation results when they are reflected in the predictive model. In this paper, we find the new features that prevent the students from dropping out and evaluate the models that predict the dropout candidates so that they can get appropriate measures.

Data mining extracts important patterns or knowledge from large amounts of data and is used in retailing, finance, telecommunications, education, fraud detection, stock market analysis, and text mining. Here, to predict and prevent student dropout, we developed and evaluated predictive models using various data-mining methods, including logistic regression (LR), a decision tree (DT), a naïve Bayes (NB) method, and a multilayer perceptron (MP). In particular, we focused on features that exert major influences on dropout; we used analysis of variance to this end and created an optimized model via feature selection. As a result, we found that engagement in extracurricular activities is an important feature to prevent students from dropping out. The results of MP among the four data-mining methods were the best in terms of performance metrics such as F-score and area under the curve.

The remainder of the paper is organized as follows: in Section 2, we present related work on dropout prevention; in Section 3, we introduce the architecture that we used for model creation and evaluation. Section 4 describes the four methods used to generate predictive models, and Section 5 describes the evaluation of the models. Finally, Section 6 contains our conclusions and future plans.

## 2. Related Work

In data mining, classification is a problem of determining which category a new observation belongs to. Classification is generally regarded as supervised learning using predefined classes. On the other hand, clustering corresponds to unsupervised learning, which groups objects without prior knowledge of classes. Classification and clustering are data-mining techniques by which data are grouped into several classes, and it has many applications. Classification and clustering models can be used to identify tumors [13], fingerprint large numbers of people [14], evaluate employee performance [15], explore student dropout [16-19], monitor anti-phishing [20], and DoS attack detection [21]. Several studies have focused on the causes and features of dropout in terms of prevention, construction of predictive dropout models, and development of dropout prevention systems.

Hoff et al. [3] considered various variables affecting dropout, discussed procedures and tools for the prevention of dropout, and showed examples of early warning systems termed EWIMS (Early Warning Intervention and Monitoring System) [4], DEWS (Dropout Early Warning System) [5], and NDPC-SD (National Dropout Prevention Center for Students with Disabilities) [6]. The variables used to identify dropouts and trigger preventative procedures were attendance, behavior, course performance, race, ethnicity, socioeconomic status, disability status, grade retention, school climate, engagement, and mobility. In this paper, we discovered new variables such extra-curricular activities as to identify dropouts. Yukselturk et al. [7] investigated the applicability of data-mining techniques (k-nearest neighbors, DT, NB, and neural networks) in predicting dropout among online students. Although the differences did not attain statistical significance, the k-nearest neighbors and DT classifiers were somewhat more sensitive than the other models. Manhaes et al. [8] developed an architecture using EDM techniques (an NB model, an MP, a support vector machine, and DT) to predict student dropout. The use of time-varying data aided the prediction of student achievement. The true-positive rate of the NB model was the highest among the four techniques. Guarin et al. [9] evaluated the data-mining models featuring an NB approach and a DT to predict dropout among students with low academic achievement. Dropout prediction performance was improved when academic data were included. Omoto et al. [10] reviewed institutional research (IR) and Fujitsu trends. IR involves the measurement of many activities via oncampus data collection and analysis, planning of appropriate measures, and implementation and verification of management improvements, student support, and higher-quality educational techniques [10]. Fujitsu developed several statistical methods for the analysis of trends in quality improvement and dropout prevention. Support vector machines were used in data mining. Kuznar and Gams [11] developed a Metis system for the prediction of student dropout and prevention of associated negative consequences. The Metis system uses machine learning algorithms to analyze data from school information systems, identifies students who are likely to drop out, and triggers appropriate action from educational experts. Costa et al. [12] compared the effectiveness of four data-mining techniques (an NB method, a DT, a neural network, and a support vector machine) in predicting the likelihood that students would leave the Brazilian Public University introductory curriculum on programming. The support vector machine performed better than the other techniques.

This study is similar to related work in that we evaluated predictive models using data-mining methods as shown in Table 1. We also chose methods generally used for classification in the related work. However, it differs from related work in that we used new features such as extra-curricular activities and generated a model after selecting the optimal features thereof via LR.

Data-mining methods used for identifying dropouts

## 3. System Architecture

Our dropout prevention system architecture is divided into five layers (collection, storage, processing, analysis, and visualization), as shown in Fig. 1.

Collection layer: In this layer, data are collected from various systems inside and outside the organization. Our collector uses various interfaces (e.g., REST, HTTPS, SFTP) to connect different databases and files within the school affairs, learning, and library systems; and logs and documents.

Storage layer: In this layer, the collected data are held permanently or temporarily in distributed storage. The HDFS and NoSQL approaches are used to permanently store large files and large amounts of messaging data, respectively. Personal data are anonymized.

Process layer: In this layer, the data are formalized and normalized to render them suitable for analysis. For example, the learning management system processes unstructured log files to generate structured data, such as the numbers and times of logins per student.

Analysis layer: In this layer, new patterns in large datasets are sought and interpreted to provide novel insights. For example, a predictive model is created to detect students who are likely to drop out.

Visualization layer: In this layer, the big data results are presented in an easy-to-understand manner. For example, new student information is entered into a predictive model to explore whether particular students dropped out.

Fig. 2 illustrates the model creation and deployment process. Here, we selected significant features with the aid of the chi-squared test and analysis of variance (ANOVA) and generated models using four data-mining methods. ANOVA [22] is a method used when comparing two or more groups in statistics. In this paper, we try to find features that affect the dropout group through the ANOVA. This process is necessary for improving the prediction performance and reducing the learning time. When model performance exceeded the desired threshold, the best model was deployed, and new student data were input to identify students who might drop out.

The system architecture.
Model creation and deployment.

## 4. Classification Methods

We predicted student dropout by analyzing existing data to create a predictive model and then entered new student data into this model to predict dropout. Such data analysis creates a classifier predicting whether dropout occurs (e.g., yes or no). We used LR, a DT, an NB method, and an MP to this end. Although examples for using student data in each method are briefly described in this section, experiments, where actual student data is used on a variety of cases, are shown in the next section.

##### 4.1 Logistic Regression

LR is a well-known classification method for the derivation of relationships between dependent and independent variables, like linear regression (which explains a dependent variable as a linear combination of independent variables), but differing in that LR uses categorical (discrete) dependent variables, rather than numerical (continuous) variables. For example, assuming that student dropout can be categorized into two states (yes or no) by grade point average (GPA), the independent variable is the GPA and the dependent variable is the dropout. When we analyzed the relationship between the GPA and the probability of dropout (Fig. 3), that probability decreased gradually as the GPA rose from zero to a certain point, then rapidly decreased, and later gradually decreased further. Thus, we performed LR analysis employing the logit (or log-odds ratio) to describe the curve mathematically. The following equation is a logistic function created by the logit:

##### (1)
[TeX:] $$\operatorname{logistic~function~}=\frac{e^{\beta_{i} x_{i}}}{1+e^{\beta_{i} x_{i}}}$$

where the values are regression coefficients [TeX:] $$\left(\beta_{0}, \ldots, \beta_{i}\right) \text { and } x$$ values are independent variables [TeX:] $$\left(x_{0}, \ldots\right.\left.x_{i}\right).$$

The odds ratios generated by LR can be used to determine the extent to which independent variables affect dependent variables. Generally, LR analysis is used when the dependent variables fall into two categories; multinomial LR is employed when two or more categories are in play, and ordinal LR is used when the dependent variable is sequential.

The relationship between the GPA and the probability of dropout.
##### 4.2 Decision Tree

A DT is a predictive model linking attributes (independent variables) to a class label (the dependent variable). In a DT, an internal node (that is not a leaf node) is used to test the value of an attribute, and this information is used to branch to another internal or leaf node, ultimately determining a class. A DT learns by induction, employing training data with class labels analyzed with the aid of algorithms such as ID3, C4.5, or CART. The algorithms differ in terms of the attribute selection methods (e.g., information gain, gain ratio, and Gini index) used to identify criteria that divide the training data well. A DT is constructed as follows.

First, an appropriate split criterion and a stopping rule are defined, depending on the purpose of the analysis and the data structure. Here, the split criterion was a classification tree emphasizing the purity of the child node at the expense of that of the parental node. Parameters sharing high-level purity are more likely to belong to the same category. The split criterion varies depending on the type of dependent variable. For example, when the dependent variable is discrete, the p-value is used for the chi-squared test, the Gini index, and the entropy index. When the dependent variable is continuous, the F statistic is used for analysis and reduction of variance. Here, we employed the Gini index because the dependent variable (dropout) was discrete. The Gini index is based on a binary split of all attributes. In terms of the discrete value attribute, the subset with the lowest Gini index is selected. For continuous attributes, all possible separable values are considered. The independent variable with the smallest Gini index forms a branch of the DT. The following equation is used to obtain the Gini index:

##### (2)
[TeX:] $$\operatorname{Gini}(T)=1-\sum_{i=1}^{k} p_{i}^{2},$$

where is the dataset, and is the probability that a tuple in belongs to the i-th class. For example, if the two categories are in a ratio of 0.8:0.2, the Gini index is [TeX:] $$1-\left(0.8^{2}+0.2^{2}\right)=0.32.$$ Fig. 4 shows an example of a DT considering the only GPA as an independent variable.

The tree is pruned because non-standard learning data (noise and outliers) may be present and overfitting is possible. The pruned tree is smaller, less complex, and simpler than the original tree.

The DT is evaluated against the test data using a cross-validation method.

An example of a DT.
##### 4.3 The NB Approach

The NB approach is based on Bayes’ theorem, as follows:

##### (3)
[TeX:] $$P(H \mid X)=\frac{P(X \mid H) P(H)}{P(X)}$$

where H and X are events,[TeX:] $$P(H \mid X)$$ is the posterior probability of H given that X is true, P(H) is the prior probability of H, [TeX:] $$P(X \mid H)$$ is the posterior probability of X given that H is true, and P(X) is the prior probability of X. The NB approach assumes that the value of any variable is independent of the values of other variables (i.e., lack of dependency among features). For example, independent variables (terms, or the GPA) that affect student dropout are independent; each independent variable is assumed to contribute uniquely to the probability that a student will drop out. An NB classifier is generated as follows:

If a tuple X with a class label (a dependent variable indicating whether or not dropout occurs) is composed of n attributes (independent variables), that tuple is expressed as a vector [TeX:] $$X=\left(x_{1}, x_{2}, \ldots\right.\left.x_{n}\right).$$ We need to find the class label with maximum [TeX:] $$P\left(X \mid C_{i}\right) P\left(C_{i}\right)(i=1,2)$$ for two labels: [TeX:] $$C_{1}$$ (enrolled students) and [TeX:] $$\mathrm{C}_{2}$$(expelled students).

When the elements of the vector X are continuous, the continuous value attribute is assumed to follow a standard Gaussian distribution with a mean [TeX:] $$\mu$$ and a standard deviation [TeX:] $$\sigma.$$ The probability of the standard normal distribution is as follows:

##### (4)
[TeX:] $$g x, \mu, \sigma=12 \pi \sigma 2 e-(x-\mu) 22 \sigma 2$$

Therefore, [TeX:] $$P\left(x_{k} \mid C_{i}\right)$$ can be written as:

##### (5)
[TeX:] $$P\left(x_{k} \mid C_{i}\right)=g\left(x_{k}, \mu_{C_{i}}, \sigma_{C_{i}}\right)$$

When the tuple X satisfies the following expression condition, it is considered to a class label [TeX:] $$C_{1}:$$

##### (6)
[TeX:] $$P\left(X \mid C_{1}\right) P\left(C_{1}\right)>P\left(X \mid C_{2}\right) P\left(C_{2}\right)$$

##### 4.4 The MP

An MP is used to model the neurons of the human brain involved in learning. Several hidden layers lie between an input and an output layer and identify data that are not linearly separable. Recently, artificial neural networks featuring MPs have been termed deep neural networks, and the algorithm used to study such networks is said to evaluate engagement in “deep learning.” The value of an independent variable is input in an input layer node. Usually, the numbers of independent variables and input nodes are identical. The outputs of the input layer are used as inputs in hidden layer nodes. To calculate outputs of the hidden layer, a weighted sum of inputs is calculated and delivered to an activation function that may be linear, exponential, or sigmoid. Here, we used a sigmoid logistic function. The MP is constructed as follows:

Initialization is performed by assigning connection weights to arbitrary values, calculating the inputs to each layer for a set of learning data (e.g., GPA, terms, etc.), and, finally, calculating outputs employing the activation function.

After comparing the outputs to the expected values, the connection weight is adjusted via backpropagation to ensure that outputs lie within the error limit.

This procedure is repeated using other learning data and terminated when the differences between the outputs and the target values are within the acceptable error range.

Here, we used an input layer in which the numbers of nodes and independent variables were equal; thus, 14 nodes with two hidden layers, and one output layer node (Fig. 5).

Although the MP usually shows better predictive performance than LR, the extent to which the input affects the output is difficult to determine. Therefore, we used LR to identify variables that should be preferentially considered. In addition, we selected, from the four methods, that method with the best real predictive performance.

Architecture of MP.

## 5. Experiments

##### 5.1 Experimental Environment

We implemented a collector based on Sqoop (https://sqoop.apache.org) for the collection layer in our system architecture. Using the collector, we collected data from our university system and stored it in HBase (https://hbase.apache.org) for the storage layer. The data size is several hundred MB, and the number of samples is about fourteen thousand. Two categories for enrolled students and expelled students are in a ratio of 8:2. Table 2 shows variables affecting dropout in our experiment. We selected thirteen independent variables out of a total of 155 variables, Next, we used Spark (https://spark.apache.org) for the process layer to clean the dependent and independent variables. The cleaned training and test data were randomly divided at a ratio of 7:3. Finally, using the Spark for the analysis layer, we created and evaluated LR, DT, NB, and MP models.

Variables and data cleaning

As shown in Table 2, we used the independent variables from five cases to explore how the evaluations changed according to the characteristics and numbers of independent variables employed for dropout prediction.

Case #1: GPA

Case #2: Case #1 + {age, semester, sex}

Case #3: Case #2 + {engagement in club activities}

Case #4: Case #3 + {nationality, parental address, extracurricular activity score, number of volunteer activities, number of surveys completed evaluating satisfaction with extracurricular activities, number of surveys completed evaluating satisfaction with the department, number of consultations, and engagement in freshman camp activities}

Case #5: Case #3 + {extracurricular activity score, number of surveys completed evaluating satisfaction with extracurricular activities, number of surveys completed evaluating satisfaction with the department, number of consultations, and engagement in freshman camp activities}

Case #5 was derived from Case #4 (i.e., full feature set) by excluding independent variables (i.e., nationality, parental address, and number of volunteer activities) that did not significantly affect dropout, as revealed by ANOVA (Table 3). ANOVA can be used to improve model performance when an independent variable is included, depending on the difference between the null and residual deviance. We found that engagement in extracurricular activities significantly reduced dropouts.

Analysis of variance
##### 5.2 Experimental Results

We used the accuracy, precision, recall, F-score, and area under ROC curve (AUC) parameters to evaluate the four models, as follows. In particular, we selected the F-score and AUC because the class ratio of the experimental data is imbalanced.

##### (7)
[TeX:] $$\text { Accuracy }=\frac{T P+T N+F P+F N}{T P+T N}$$

##### (8)
[TeX:] $$\text { Precision }=\frac{T P}{T P+F P}$$

##### (9)
[TeX:] $$\operatorname{Recall}=\frac{T P}{T P+F N}$$

##### (10)
[TeX:] $$F-\text { score }=2 * \frac{\text { Precision } * \text { Recall }}{\text { Precision }+\text { Recall }}$$

##### (11)
[TeX:] $$A U C=\int_{\infty}^{-\infty} T P R(T) F P R^{\prime}(T) d T$$

To facilitate comprehension of the above equations, Table 4 shows the confusion matrix. The true positive (TP), true negative (TN), false positive (FP), and false negative (FN) are defined as follows.

TP: The model predicted that students dropped out, and they did in fact drop out.

TN: The model predicted that students did not drop out, and this was in fact the case.

FP: The model predicted that students dropped out, but they did not drop out (type I error).

FN: The model predicted that students would not drop out, but they did drop out (type II error).

The confusion matrix

Both TP and TN (only) must be in play if a predictive model is to match all real truths, associated with an accuracy of unity. In this sense, precision is a measure of the extent of type I error, and recall is a measure of the extent of type II error.

Fig. 6 compares the four methods in terms of accuracy. The NB model was less accurate than the other models; the application of the MP method to Case #5 yielded the greatest accuracy (i.e., 0.95). The accuracies of the LR and MP methods increased as more independent variables were added, but the accuracies of the DT and NB methods did not.

Comparison of the four methods in terms of accuracy.

Fig. 7 compares the four methods in terms of precision. The NB method was less precise than the other methods; the application of the DT method to Case #5 yielded the greatest precision (i.e., 0.91). Thus, the use of the NB method resulted in more type I errors than did the use of other methods. The precisions of the LR, DT, and MP methods increased as more independent variables were added, whereas the precision of the NB method did not. The weakness of the NB method is that the dependence between independent variables has a relatively poor predictive performance. As a result of this experiment, the dependency between independent variables such as (age, semester) and (engagement in club activities, extracurricular activities score) seemed to have influenced the performance of the NB method.

Comparison of the four methods in terms of precision.

Fig. 8 compares the four methods in terms of recall. The degree of recall was lower for the NB method than for the other methods for Cases #1, #2, #3, and #4; however, the degree of recall for the NB method for Case #5 was the highest (i.e., 0.92). Thus, type II errors created using the NB method were significantly reduced by optimizing variable selection (i.e., from Case #4 to Case #5). The degrees of recall of the LR, NB, and MP methods increased as more independent variables were added, whereas the degree of recall of the DT method did not.

Comparison of the four methods in terms of recall.

Fig. 9 compares the four methods in terms of the F-score. The F-score was lower for the NB method than for the other methods. As shown in Fig. 9, the highest F-score was 0.87, obtained by using the MP method to analyze Case #5. Fig. 10 compares the four methods in terms of AUCs. As shown in Fig. 10, the highest AUC was 0.98, obtained when the MP method was used to analyze Case #5. Thus, the predictive model generated by analyzing the independent variables of Case #5 via the MP method showed the best performance. However, the results of the MP method do not differ much from those of the LR method. The LR method is similar to the one-layer neural network and divides the pattern space linearly into two regions. On the other hand, the MP method, which is a two-layer neural network used in this paper, divides the pattern space into convex regions, which is theoretically better than the LR method. We leave it to future studies that the MP method yields much better results than the LR method.

Comparison of the four methods in terms of the F-score.
Comparison of the four methods in terms of the AUC.

## 6. Conclusions and Future Work

Here, we used LR, a DT, an NB model, and an MP to create predictive models that might provide information for the prevention of student dropout. Predictive models using independent variables selected with the aid of variance analysis and the MP method showed the best performance (the F-score and AUC were 0.87 and 0.98, respectively).

We will improve the performance of the MP model and apply the optimized model to our school management system to better prevent dropout. We will counsel students who are at risk (as revealed by data analysis), and establish a data-driven campus management plan embracing student guidance, the living environment, and campus activities.

## Acknowledgement

This work was supported by research grants from Daegu Catholic University in 2017.

## Biography

##### JongHyuk Lee
https://orcid.org/0000-0002-8163-9388

He received his Ph.D. of Computer Science Education from Korea University where he did research in distributed systems. He was previously a research professor at Korea University, a research scientist at University of Houston, and a senior engineer at Samsung Electronics. He is currently working as an assistant professor in the Depart-ment of Artificial Intelligence and Big Data Engineering at Daegu Catholic University. In the past, he authored and co-authored over publications covering research problems in distributed systems, computer architecture systems, mobile computing, p2p computing, grid computing, cloud computing, computer security, and computer science education.

## Biography

##### Mihye Kim
https://orcid.org/0000-0001-5581-6179

She received her Ph.D. degree in Computer Science and Engineering from New South Wales University, Sydney, Australia in 2003. She is currently a professor in the School of Computer Software at Daegu Catholic University, South Korea. Her research interests include knowledge acquisition, machine learning, knowledge management and retrieval, computer science education, and Internet of Things.

## Biography

##### Daehak Kim
https://orcid.org/0000-0003-4406-3233

He received his Ph.D. degree in Statistics from Korea University, Korea in 1989 where he did research in kernel estimation of density function and regression function. He is currently working as a professor in the Department of Artificial Intelligence and Big Data Engineering at Daegu Catholic University since 1994. In the past, he authored and co-authored over publications covering research problems in various statistical computing and computational works based on statistical simulation, computer science and big data areas. His recent research interests include artificial intelligence, big data computing and data science.

## Biography

##### Joon-Min Gil
https://orcid.org/0000-0001-6774-8476

He received his Ph.D. degree in Computer Science and Engineering from Korea University, Korea in 2000. Before joining in School of Computer Software, Daegu Catholic University, he was a senior researcher in Supercomputing Center, Korea Insti-tute of Science and Technology Information (KISTI), Daejeon, Korea from October 2002 to February 2006. From June 2001 to May 2002, he was a visiting research associate in the Department of Computer Science at the University of Illinois at Chicago, USA. His recent research interests include artificial intelligence, big data computing, cloud computing, distributed and parallel computing, and wireless sensor networks.

## References

• 1 A. Dutt, M. A. Ismail, T. Herawan, "A systematic review on educational data mining," IEEE Access, vol. 5, pp. 15991-16005, 2017.doi:[[[10.1109/ACCESS.2017.2654247]]]
• 2 D. Gasevic, V. Kovanovic, S. Joksimovic, "Piecing the learning analytics puzzle: a consolidated model of a field of research and practice," Learning: Research and Practice, vol. 3, no. 1, pp. 63-78, 2017.custom:[[[-]]]
• 3 N. Hoff, A. Olson, R. L. Peterson, "Dropout screening and early warning," University of Nebraska-LincolnNE, USA, 2015.custom:[[[-]]]
• 4 American Institutes for Research, 2019 (Online). Available:, http://www.earlywarningsystems.org/
• 5 Wisconsin Department of Public Instruction, c2021 (Online). Available:, https://dpi.wi.gov/ews/dropout
• 6 National Dropout Prevention Center for Students with Disabilities,, https://dropoutprevention.org/
• 7 E. Y ukselturk, S. Ozekes, Y. K. Turel, "Predicting dropout student: an application of data mining methods in an online education program," European Journal of OpenDistance and e-learning, vol. 17, no. 1, pp. 118-133, 2014.custom:[[[-]]]
• 8 L. M. B Manhaes, S. M. S. Cruz, G. Zimbrão, "W A VE: an architecture for predicting dropout in undergraduate courses using EDM," in Proceedings of the 29th Annual ACM Symposium on Applied Computing, Gyeongju, South Korea, 2014;pp. 243-247. custom:[[[-]]]
• 9 C. E. L. Guarin, E. L. Guzman, F. A. Gonzalez, "A model to predict low academic performance at a specific enrollment using data mining," IEEE Revista Iberoamericana de tecnologias del Aprendizaje, vol. 10, no. 3, pp. 119-125, 2015.doi:[[[10.1109/RITA.2015.2452632]]]
• 10 A. Omoto, Y. Lwayama, T. Mohri, "On-campus data utilization: working on institutional research in universities," Fujitsu Science Technology, vol. 1, no. 51, pp. 42-49, 2015.custom:[[[-]]]
• 11 D. Kuznar, M. Gams, "Metis: system for early detection and prevention of student failure," in Proceedings of the 6th International Workshop on Combinations of Intelligent Methods and Applications (CIMA), Hague, Holland, 2016;custom:[[[-]]]
• 12 E. B. Costa, B. Fonseca, M. A. Santana, F. F. de Araujo, J. Rego, "Evaluating the effectiveness of educational data mining techniques for early prediction of students' academic failure in introductory programming courses," Computers in Human Behavior, vol. 73, pp. 247-256, 2017.doi:[[[10.1016/j.chb.2017.01.047]]]
• 13 R. Gurusamy, V. Subramaniam, "A machine learning approach for MRI brain tumor classification," ComputersMaterials and Continua, vol. 53, no. 2, pp. 91-109, 2017.custom:[[[-]]]
• 14 C. Y uan, X. Li, Q. J. Wu, J. Li, X. Sun, "Fingerprint liveness detection from different fingerprint materials using convolutional neural network and principal component analysis," ComputersMaterials Continua, vol. 53, no. 3, pp. 357-371, 2017.custom:[[[-]]]
• 15 J. Kaur, K. Kaur, "A fuzzy approach for an IoT-based automated employee performance appraisal," ComputersMaterials and Continua, vol. 53, no. 1, pp. 24-38, 2017.custom:[[[-]]]
• 16 N. Iam-On, T. Boongoen, "Generating descriptive model for student dropout: a review of clustering approach," Human-centric Computing and Information Sciences, vol. 7, no. 1, 2017.doi:[[[10.1186/s13673-016-0083-0]]]
• 17 C. A. Christle, K. Jolivette, C. M. Nelson, "School characteristics related to high school dropout rates," Remedial and Special Education, vol. 28, no. 6, pp. 325-339, 2007.custom:[[[-]]]
• 18 J. V asquez, J. Miranda, "Student desertion: What is and how can it be detected on time?," in Data Science and Digital Business. ChamSwitzerland: Springer, pp. 263-283, 2019.custom:[[[-]]]
• 19 D. Olaya, J. V asquez, S. Maldonado, J. Miranda, W. V erbeke, "Uplift Modeling for preventing student dropout in higher education," Decision Support Systems, vol. 134, no. 113320, 2020.doi:[[[10.1016/j.dss..113320]]]
• 20 D. Jampen, G. Gur, T. Sutter, B. Tellenbach, "Don’t click: towards an effective anti-phishing training: a comparative literature review," Human-centric Computing and Information Sciences, vol. 10, no. 33, 2020.doi:[[[10.1186/s13673-020-00237-7]]]
• 21 D. Tang, R. Dai, L. Tang, X. Li, "Low-rate DoS attack detection based on two-step cluster analysis and UTR analysis," Human-centric Computing and Information Sciences, vol. 10, no. 6, 2020.doi:[[[10.1186/s13673-020-0210-9]]]
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Table 1.

Data-mining methods used for identifying dropouts
k-nearest neighbors DT NB MP (neural network) Support vector machine LR
Yukselturk et al. [7] o o o o
Manhaes et al. [8] o o o o
Guarin et al. [9] o o
Omoto et al. [10] o
Kuznar and Gams [11] o o o o
Costa et al. [12] o o o o
Present study o o o o

Table 2.

Variables and data cleaning
Variable Data cleaning Cases used
Dependent variable
Registration Enrolled and expelled students. All cases
Independent variable
GPA Calculated grade point average for each student All cases
Age Cases #2–#5
Semester Cases #2–#5
Sex Cases #2–#5
Engagement in club activities Cases #3–#5
Nationality Koreans and foreigners Case #4
Parental address Nearby and distant Case #4
Number of consultations Case #4
Number of volunteer activities engaged in Case #4
Number of surveys completed evaluating satisfaction with extra-curricular activities Cases #4, #5
Number of surveys completed evaluating satisfaction with the department Cases #4, #5
Extracurricular activities score Cases #4, #5
Engaged in freshman camp activities Cases #4, #5

Table 3.

Analysis of variance
df Deviance Residual Pr(>Chi)
df Deviance
NULL 10,052 10,204.7
Age 1 1,367.89 10,051 8,836.8 [TeX:] $$2.2 \mathrm{e}-16$$
Semester 1 2,568.04 10,050 6,268.7 [TeX:] $$2.2 \mathrm{e}-16$$
Sex 1 25.08 10,049 6,243.7 5.503e-07
GPA 1 456.07 10,048 5,787.6 [TeX:] $$2.2 \mathrm{e}-16$$
Engagement in club activities 1 567.49 10,046 5,189.4 [TeX:] $$2.2 \mathrm{e}-16$$
Nationality 1 0.12 10,045 5,189.3 0.72881
Parental address 1 3.41 10,044 5,185.9 0.06478
Extracurricular activities score 1 643.46 10,043 4,542.4 [TeX:] $$2.2 \mathrm{e}-16$$
Number of volunteer activities engaged in 1 3.40 10,042 4,539.0 0.06518
Number of surveys completed evaluating satisfaction with extra-curricular activities 1 249.53 10,041 4,289.5 [TeX:] $$2.2 \mathrm{e}-16$$
Number of surveys completed evaluating satisfaction with the department 1 72.42 10,040 4,217.1 [TeX:] $$2.2 \mathrm{e}-16$$
2.2 \mathrm{e}-16 1 3.42 10,039 4,213.6 0.06447
Engaged in freshman camp activities 1 38.62 10,038 4,175.0 5.148e-10

Table 4.

The confusion matrix
Actual truth
True False
Prediction result True TP FP (type I error)
False FN (type II error) TN
The system architecture.
Model creation and deployment.
The relationship between the GPA and the probability of dropout.
An example of a DT.
Architecture of MP.
Comparison of the four methods in terms of accuracy.
Comparison of the four methods in terms of precision.
Comparison of the four methods in terms of recall.
Comparison of the four methods in terms of the F-score.
Comparison of the four methods in terms of the AUC.


text:
Which one to choose Type-I, Type-II, or Type-III ANOVA? [duplicate]

I don't understand what the difference is between TypeI and TypeIII? Since my background is not very mathematics, it's very difficult for me to understand this mathematical notation; Type I SS: SS(A) SS(B|A) SS (AB|A, B) Type II SS: SS(A|B) SS(B|A) SS (AB|A, B) Type III SS: SS (A|B, AB) SS (B|A, AB) SS (A*B|A, B)?

I want to check the interaction between my two factors (genotype*FuncBIN), but my output comes as Type II ANOVA with interaction. Is this normal?

structure( list(genotype = structure(c(3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 3L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L), .Label = c("WT", "Mut2", "Mut1" ), class = "factor"), FuncBIN = structure(c(1L, 1L, 1L, 2L, 2L, 2L, 3L, 3L, 3L, 1L, 1L, 1L, 2L, 2L, 2L, 3L, 3L, 3L, 1L, 1L, 1L, 2L, 2L, 2L, 3L, 3L, 3L), .Label = c("Mitochondria ", "Golgi", "ER"), class = "factor"), k = c(0.2, 0.3, 0.4, 0.5, 0.3, 0.2, 0.5, 0.6, 0.7, 0.4, 0.6, 0.9, 0.2, 0.3, 0.5, 0.4, 0.2, 0.3, 0.5, 0.6, 0.8, 0.5, 0.4, 0.6, 0.2, 0.3, 0.1)), .Names = c("genotype", "FuncBIN", "k"), class = "data.frame", row.names = c(NA, -27L) )

library(car)
library(lsmeans)
library(multcomp)
fit <- lm(k ~ genotype + FuncBIN + genotype*FuncBIN, df)
plot(fit) shapiro.test(residuals(fit))
Anova(fit)
lsmip(fit, FuncBIN ~ genotype)
lsmeans(fit, pairwise ~ genotype)[[2]]
fit.tukey <- lsmeans(fit, pairwise ~ genotype | FuncBIN)[[2]]
fit.tukey
cld(fit.tukey)

How can I run an ANOVA with interaction followed my Tukey PostHOC?

• Thank you, EdM, but I need a simple explanation to understand this + how to improve my R coding for this? – Kynda May 11 '18 at 14:53
• Read the "|" sign as "given. That is, for type II ANOVA, "SS(A|B)" means to test A given what is known about B; "SS(B|A)" means to test B given what is known about A; "SS (AB|A, B)" means to test the AB interaction given what is known about A and B individually. So type II (as recommended by @dbwilson) does provide a test for the interaction itself; in type II the interaction term, unlike in type III tests, is not included in the tests for main effects of A and B. – EdM May 11 '18 at 16:08
• I think you will find the information you need in the linked thread. Please read it. If it isn't what you want / you still have a question afterwards, come back here & edit your question to state what you learned & what you still need to know. Then we can provide the information you need without just duplicating material elsewhere that already didn't help you. – gung - Reinstate Monica May 11 '18 at 16:26

fit <- aov(k ~ genotype*FuncBIN, data=df)

• In practice, however, there will appear to be a difference when this is done with Anova() in the car package. Its Type II tests do not include an intercept term, while its Type III tests do. See this R-help thread. – EdM May 11 '18 at 18:02


text:
# If $Ax=b$, for $b\ne 0$, has more than one solution, then $Ax=0$ does as well. T or F. Prove this.

I get that this is true, because there's one free variable, so no matter what the augmented matrix is, there always will be an infinite amount of solutions. Right? But how to I explain this as a proof?

-

If $x_1$ and $x_2$ are both solutions of $Ax=b$ then what can you say about $A(x_2-x_1)$?
Which means $x_2-x_1$ is a solution and so is any multiple of it. So $k(x_2-x_1)$ is a solution for any $k$. – Maesumi Feb 1 '13 at 20:30
So final answer is Yes. If there are two distinct solutions $x_1,x_2$, then the solution set of $Ax=0$, the so-called null space, contains all multiples of $x_2-x_1$, so it has infinitely many solutions. – Maesumi Feb 1 '13 at 22:45
$$\rm\begin{eqnarray} ax = &\rm b&\rm = ax',\ \ x\ne x'\\ \rm \Rightarrow\ a(x\!-\!x') = &0&\rm = a\,0,\ \ \,x -x'\ne 0\end{eqnarray}$$


text:
# Tag Info

## New answers tagged latexmk

0

Hmm. After resetting my computer, everything was fixed. No additional steps were taken.

2

It seems to me that you have more than one version of perl installed. In your path settings the wrong one will be find first. Most likley the reason for your task. Check path settings on your machine...

1

My current solution without using gnuplottex is a modification of one I found a while ago (How to include an SVG image in LaTeX) for the inclusion of SVGs (with an automatic call to Inkscape if necessary). \newcommand{\executeiffilenewer}[3]{% \ifnum\pdfstrcmp{\pdffilemoddate{#1}}% {\pdffilemoddate{#2}}>0% {\immediate\write18{#3}}\fi% } \...

3

Load lastpage after pdfcomment. \documentclass{report} \usepackage{graphicx} \usepackage[a3paper,landscape]{geometry} \usepackage[author={Max Schlezpig}]{pdfcomment} \usepackage[rgb]{xcolor} \usepackage{lastpage}% \usepackage{hyperref,bookmark} \begin{document} \pdfmarkupcomment[markup=Highlight,color=yellow]{This highlighted Text Has Tool Tip With ...

6

You can not need \include functionality here as there is no way to use \includeonly to specify just one of the inclusions as they have the same name. So use \input not \include then you can input the file as often as you like.

Top 50 recent answers are included


text:
# ExtX file system on my usb key

if I format my usb key with an extX file system, copy some files on it and then give it to a friend for him to add files or modify existing one on this key, then he is rejected by its own system. Because its User ID (UID) nor GID are the same as mine on my machine.

How to get rid of this limitation? Is it possible to disable user rights on a ext2/ext3 partition?

Of course, I would really like not to rely on any other file system.

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just curios but why do you want ext2/ext3? sure i agree that fat fs is junk but it do work better for sharing? –  Joakim Elofsson Jul 24 '09 at 9:54
I'made the same reflexion as yours, a fat-like file system better suits on an usb key. But it is a propietary file system AFAIK (MS one?). And I guess that linux addict people might propose a better solution for that ? Just wondering because I would really like to do without it me too. –  yves Baumes Jul 24 '09 at 18:30
Another reason for using extx2/3 is support for symbolic links. Doesn't matter for the odd file but if you want an untar'd collection of files that use these it can be important. –  mas Aug 4 '09 at 7:14

## 5 Answers

you can chmod 777 all of the files, to remove all of the restrictions.

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cha ching! hit the jackpot! :D –  RCIX Jul 24 '09 at 11:36
glad to hear it helped –  Babu Jul 24 '09 at 15:44

Mount the drive with an option of umask=000. This tells the system to give every file newly created/copied on the system permissions of 777, unless told otherwise(such as via cp -p).

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This solution is much better than the chmod suggestion, lay yier votes hire dear folks! –  Deleted Aug 1 '09 at 19:13
oooh, I was entirely unaware of this. That's very useful! –  Babu Aug 1 '09 at 19:16
I can't find that option for ext2/ext3 fs .. sigh ... Why ? I don't have any linux box at hand. Does they forget that in man page? linux.die.net/man/8/mount –  yves Baumes Aug 6 '09 at 18:47
That's a vfat/ntfs/hfs/hpfs mount option, not an xfs/ext2/ext3/ext4/reiserfs option! Stop confusing people! –  user62017 Jan 8 '11 at 22:05

Funnily enough a kernel patch to do some sort of UID remapping on ext mounts "to make them actually useful on removable media" turned up very recently. Doesn't seem to be too controversial in the subsequent discussion so it should make it into your favourite distro at some point.

BTW, I've used ext2/3 on removable media myself for years but it's almost entirely for rsynced backups where the whole point is to preserve all the file metadata.

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very instersting link .. I wonder why anyone make that before . I will check it out when having some free time . thanx! –  yves Baumes Aug 3 '09 at 17:27

"Is it possible to disable user rights on a ext2/ext3 partition?" It is not but as more or less said before:

standing in the filesystem root (/media/<name>/) and doing chmod -R 777 . will set permission to allow anything on all filesystem.

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Why not just chmod -R 777 /media/<name>/? –  grawity Jul 24 '09 at 8:54
The chmod won't have any effect on newly created files though - permissions are not inherited in ext3. (Setting all directories to setgid may help a little.) –  grawity Jul 24 '09 at 8:55
nothing wrong with doing it directly on /media/<name>, works too ofc –  Joakim Elofsson Jul 24 '09 at 9:38
just though the answer more generic as '/media/<name>' is just an example mountpoint (most likely correct in this case but...) –  Joakim Elofsson Jul 24 '09 at 9:45

You would want to use

chmod -R 777 /dev/sdc1/

In order to give permission for someone else to create in the root directory, as well.

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this sets premission on the device entry, not the filesystem –  Joakim Elofsson Jul 23 '09 at 23:43
I think you actually want to set the mount point to 777 not the device entry. –  Jim OHalloran Jul 24 '09 at 0:54
Since a specific distro wasn't mentioned, I tried to do a generic, but, you're right, of course. –  Joshua Nurczyk Jul 24 '09 at 12:37


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# Another adjoint pair: Definable sets and set-builder formulas

I see adjointness between the two concepts of "being a definable set" and "being a set-builder formula":

A set $X$ is definable when there is a formula $\phi(x)$ such that $X = \lbrace x : \phi(x)\rbrace$.

A formula $\phi(x)$ is a set-builder when there is a set $X$ such that $X = \lbrace x : \phi(x)\rbrace$.

In which (category theoretical) contexts this adjointness is made explicit and valuable?

Given any pair of sets $X, Y$ (let me not be too specific about what "sets" means because what follows is robust with respect to changes in the definition), any relation $R : X \times Y \to 2$ whatsoever induces a contravariant adjunction between the poset of subsets of $X$ and the poset of subsets of $Y$ given by

$$2^X \ni S \mapsto F(S) = \{ y \in Y : xRy \forall x \in S \} \in 2^Y$$

$$2^Y \ni T \mapsto G(T) = \{ x \in X : xRy \forall y \in T \} \in 2^X.$$

A contravariant adjunction between posets is also known as an antitone Galois connection. Any such thing determines two closure operators, one on $2^X$ and one on $2^Y$, and it is often interesting to determine the corresponding closed subsets. Examples:

1. Let $L/K$ be a Galois extension, let $X = L$, and let $Y = \text{Gal}(L/K)$. The relation is "the element $y \in Y$ fixes the element $x \in X$." The adjunction sends a subset of $L$ to the subgroup of the Galois group fixing it and sends a subset of the Galois group to its fixed field. The fundamental theorem of Galois theory tells us that the closed subsets are the subgroups of the Galois group on the one hand and the subextensions of the Galois extension on the other.

2. Let $X = k[x_1, ..., x_n]$ be a polynomial ring over an algebraically closed field and let $Y = \mathbb{A}^n$ be the corresponding affine $n$-space. The relation is "the polynomial $x \in X$ vanishes on the point $y \in Y$." The adjunction sends a set of polynomials to the variety it cuts out and sends a set of points to the ideal of functions vanishing on it. The Nullstellensatz tells us that the closed subsets are the radical ideals on the one hand and the Zariski-closed subsets on the other (the second statement is tautologous; this is the definition of being Zariski closed. But the first one has content).

3. Let $X$ be the set of models of a first-order theory $T$ and let $Y$ be the set of possible sentences that could be added to $T$. The relation is "the sentence $y \in Y$ is true in the model $x \in X$." The adjunction sends a set of models to the set of sentences true in every model and sends a set of sentences to the set of models of those sentences (together with $T$). The completeness theorem tells us that the closed subsets are the deductively closed sets of sentences on one hand and whatever their sets of models are on the other.

In your example one of the "sets" is the set of all sets, the other is the set of all formulas, and the relation is "the set $x \in X$ satisfies the formula $y \in Y$." It is a version of the slogan "syntax is adjoint to semantics," an observation first made, as far as I know, by Lawvere.

• Further reading: logicmatters.net/resources/pdfs/Galois.pdf Oct 29, 2013 at 23:23
• By the way, IIRC I learned this from Todd Trimble, possibly on MathOverflow. Oct 30, 2013 at 2:29

Your question is not as precise as you portray it, and apart from the question of adjointness, a naive treatment of definability like this leads easily to contradictions. Specifically, I claim that your definitions do not actually succeed in defining the concepts of definability and of being a set-builder.

To convince you of this, let me prove that the property of a set-theoretic formula $\varphi$ being a set-builder is not expressible in the language of set theory.

Specifically, let me prove that no expressible property $P(\varphi)$ aligns with your concept of being a set-builder. Consider any property $P(\varphi)$ of formulas that is expressible in the language of set theory. By the Gödel fixed point lemma, there is a sentence $\sigma$ such that we can prove the equivalence $\sigma\iff P(\sigma)$; and we can prove this equivalence in a very weak system. Note that since $\sigma$ has no free variables, it follows that $\{ x \mid \sigma\}$ is either the empty set or everything, depending on whether $\sigma$ is false or true, respectively. In particular, $\sigma$ is a set builder in your sense just in case $\sigma$ is false. The point now is that this is equivalent to saying that $P(\sigma)$ is false. Thus, this particular formula $\sigma$ is a set-builder if and only if $P(\sigma)$ fails, and so $P$ gets the wrong answer in this instance. Since $P$ was arbitrary, there simply is no way to express the property of a formula that it is a set-builder.

(One could alternatively argue like this: suppose that we could express the property of being a set-builder. By the fixed-point lemma, there is a sentence $\sigma$ that is equivalent to "$\sigma$ is a set-builder". And so $\{x \mid \sigma\}$ is a set if and only if it isn't.)

Moving to a richer theory or to category theory does not solve the fundamental problem, since the Gödel fixed point lemma applies to any sufficiently rich system, including GBC, KM or ETCS. In these systems, there simply is no way to express the concept of being a set-builder in that system.

In particular, you haven't actually defined what it means for a formula to be a set-builder. And similar issues arise with the concept of being a definable set.

The basic obstacle here, of course, is Tarski's theorem on the non-definability of truth, which can be thought of as the claim that we have no definable way to express when a particular sentence is true. Both of your definitions in effect appeal to such a predicate, since you are defining a property of $\varphi$, but then making assertions about the truth of instances of $\varphi$. Although we may speak of specific formulas being set-builders or not, there simply is no general concept of a formula being a set-builder that is expressible in the same language.

Lastly, and perhaps this will be good enough for you, one can address part of the problem by working in a stronger system, but being satisfied with notions of definability and set-builders only for formulas in a weaker system. For example, in Kelley-Morse set theory KM, we can define a truth predicate for first-order truth in the language of set theory. In this case, working in KM we have robust concepts of definability and set-builders, but only for formulas in the first-order language of set theory, and not for formulas in the language of KM.

Another way to address the issues is to work with the notions of definability and set-builders over a specific set model. If $M$ is a set model of some theory, then we have notions of definable-in-$M$ and of a formula $\varphi$ being a set-builder-in-$M$. This model-theoretic treatment of definability is used throughout logic. But when using it, one has in effect an outside-the-universe account of the topic, since the notions are not expressible inside the model $M$ being considered. The truly problematic issues arise only when one wants to refer to definability in the whole universe.

These definability issues arose before on MathOverflow, in my answers to Anixx's question Is analysis in fact the analysis of definable numbers? and to Scott Aaronson's question Succinctly naming big numbers. In those answers, I show some examples showing that there are models of set theory with a range of paradoxical behavior with respect to definability. Most of those examples appear also in my paper, J. D. Hamkins, D. Linetsky, J. Reitz, Pointwise definable models of set theory, JSL 78(1):2013, which includes a relatively accessible account of some of these metamathematical issues with definability.

• Quick reply: I did not argue for these concepts to be expressible in (first-order) set theory. I am aware they are not. Oct 29, 2013 at 23:36
• Well, you presented a naive account, and my main point is that the naive account is flawed. We can't really take that kind of talk about definability at face value. If you had meant something sophisticated, then you should explain more fully what in fact you did mean. But notice that the argument of my post shows that there is no nontrivial system that is able to formalize these notions for that system itself. And so you will have a hard time of it, unless you adopt one of the methods for getting around the problem. In that case, you should say so explicitly what it is that you mean. Oct 29, 2013 at 23:43

This is really just an answer stub but it's worth mentioning anyway...

In some way, this is one of the many facets and extensions of Stone Duality. Henrik Forssell's 2008 thesis is one of the latest attempts that I am aware of to formalize this general idea to the fullest.

• I hope experts will chime in and I will gladly delete my stub in favor of their input. Oct 30, 2013 at 1:37


text:
# How to get an eigenvector of a $3\times 3$ matrix that has first column and a row of zeros

I have the following matrix $$\begin{bmatrix} 1& 0& 0\\ 0& 1& 1\\ 0& 1& 1 \end{bmatrix}$$

First I got the eigenvalues which are $0$, $1$, $2$.

I tried to get the eigenvectors associated with the above eigenvalues but I cannot in case of the eigenvalue $1$ as I got the following matrix $$\begin{bmatrix} 0& 0& 0\\ 0& 0& 1\\ 0& 1& 0 \end{bmatrix}$$ So, how can I get an eigenvector for this matrix?

• Just by looking at the matrix one can see that $\left(1, \begin{bmatrix}1 \\ 0 \\ 0 \end{bmatrix}\right)$ is an eigenpair of the first matrix and the other eigenvalues $0$ and $2$ follow easily. – Git Gud Apr 16 '14 at 22:03
• I tried online matrix calculator for checking eigen values and was correct. – Eng .. Abdalmonem Apr 16 '14 at 22:06
• @Eng..Abdalmonem $0,2$ and $4$ are not the eigenvalues, sorry. – Git Gud Apr 16 '14 at 22:07
• Yes i am sorry , you are correct. – Eng .. Abdalmonem Apr 16 '14 at 22:09
• It's a pair in which the first entry is an eigenvalue and the second is an eigenvector associated to the first entry. – Git Gud Apr 16 '14 at 22:12

The RREF of $[A-1I]v_1=0$ is:

$$\begin{bmatrix}0 & 1 & 0\\0 & 0 & 1 \\ 0 & 0 & 0\end{bmatrix}v_1 = 0$$

What if we chose:

$$v_1 = (1,0,0)$$

Update If we write:

$$0a + 1b + 0c = 0 \\ 0 a + 0b + 1 c = 0 \\ 0 a + 0 b + 0c = 0$$

What choices will actually make all three equations true and not be a zero eigenvector? What if we choose $(a , b , c) = (1, 0 , 0)$? Substitute those values back in and see if it satisfies the system.

Do you see how that satisfies the system? Recall that you cannot have a zero eigenvector. Would any other choice for $b = 1$, or $c=1$ or $b = c = 1$ work? No.

Also, see how this is the null space of the RREF?

• No I do not , can you explain a bit more ? – Eng .. Abdalmonem Apr 16 '14 at 22:07
• and how you got v1=(1,0,0) ? – Eng .. Abdalmonem Apr 16 '14 at 22:07
• Judging from your "last seen" (on this site) datum, you were up WAY TOO LATE last night! ;-) – Namaste Apr 17 '14 at 11:29
• I've done the same. I actually function best (both when I was a student) and onward, when I take a long "nap" (at the end of the work day), then awaken refreshed and working late. (Early rising, late bedtime - but a nice long nap in between!) – Namaste Apr 17 '14 at 12:35


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# equivalence relation topology

It turns out that this is true, and it's very easy to prove. Introduction to Algebraic Topology Page 1 of28 1Spaces and Equivalences In order to do topology, we will need two things. De nition 1.2.2. Contents 1 Introduction 5 2 The space of closed subgroups 7 3 Full groups 9 4 The space of subequivalence relations 13 4.1 The weak topology Remark 3.6.1. Do you have any reference to this equivalence relation or a similar one? a = a (reflexive property),; if a = b then b = a (symmetric property), and; if a = b and b = c then a = c (transitive property). 1. Similarly, the equivalence relation E 1 is the relation of eventual agreement on R ω. Firstly, we will need a notation of ‘space’ that will allow us to ask precise questions about objects like a sphere or a torus (the outside shell of a doughnut). 38 D. Fernández-Ternero et al. Let R be the equivalence relation … The relation bjaon f1;2;:::;10g. Equivalence relation and partitions An equivalence relation on a set Xis a relation which is reﬂexive, symmetric and transitive A partition of a set Xis a set Pof cells or blocks that are subsets of Xsuch that 1. Equivalence Relation Proof. That's in … Examples: an equivalence relation is a subset of A A with certain properties. Equivalence Relations and Functions October 15, 2013 Week 13-14 1 Equivalence Relation A relation on a set X is a subset of the Cartesian product X£X.Whenever (x;y) 2 R we write xRy, and say that x is related to y by R.For (x;y) 62R,we write x6Ry. See also partial equivalence relation. Theorem 1.2.5 If R is an equivalence relation on A, then each element of A is in one and only one equivalence class. Of course this can be generalized to any set of binary relations, but I want to understand it in the case of the plane. 2) is an equivalence relation. T contains the following “equivalence classes” (we don’t know yet that these are equivalence classes before we show that T is an equivalence relation, but within these subsets every element is related to every element, while no elements from different subsets are related): for , , and for and . Relations. equivalence relation can be defined in a more general context entail-ing functions from a compact Hausdorff space to a set, which need not have a topology, provided the functions satisfy a certain compati-bility condition. A relation R on a set X is said to be an equivalence relation if Given below are examples of an equivalence relation to proving the properties. But before we show that this is an equivalence relation, let us describe T less formally. Section 14 deals with ultraproducts of equivalence relations and in Section 15 we de ne and study various notions of factoring for equivalence relations. Define x 1 ≈ x 2 if π(x 1) = π(x 2); we easily verify that this makes ≈ an equivalence relation on X. If C∈ Pthen C6= ∅ 2. Let $X:=\mathbb R^2/\sim$ and $\tau_X$ its quotient topology. As an example, ¿can you describe the equivalence class of a disk? Establish the fact that a Homeomorphism is an equivalence relation over topological spaces. The equivalence class [a] of an element a A is defined by [a] = {b e A aRb}. In linear algebra, matrices being similar is an equivalence relation; when we diagonalize a matrix, we choose a better representative of the equivalence class. 5 A relation R on a set including elements a, b, c, which is reflexive (a R a), symmetric (a R b => b R a) and transitive (a R b R c => a R c). Various quotient objects in abstract algebra and topology require having equivalence relations first. Exercise 3.6.2. The set of all elements of X equivalent to xunder Ris called an equivalence class x¯. The equivalence relation E 0 is the relation of eventual agreement on {0, 1} ω, i.e., for x, y ∈ {0, 1} ω, x E 0 y ⇔ ∃ m ∀ n > m (x (n) = y (n)). Equivalence relations are an important concept in mathematics, but sometimes they are not given the emphasis they deserve in an undergraduate course. Let Xand Y be Polish spaces, with Borel equivalence relations Eand F de ned on each space respectively. U;E is just the equivalence relation of being in the same orbit for the subgroup generated by E. However, if Uis a proper subset of Xthen U;E equivalence classes will generally be smaller than the intersection of Uwith the orbits for the subgroup of generated by E. Here is our main de nition. Another class of equivalence relations come from classical Banach spaces. As a set, it is the set of equivalence classes under . It has a domain and range. R ∈ T. Then (R ,T ) is an AF-equivalence relation, where T is the relative topology. If A is an inﬁnite set and R is an equivalence relation on A, then A/R may be ﬁnite, as in the example above, or it may be inﬁnite. Math 3T03 - Topology Sang Woo Park April 5, 2018 Contents 1 Introduction to topology 2 ... An equivalence relation in a set determines a partition of A, namely the one with equivalence classes as subsets. (ii) Let R = (R,T) be an AF-equivalence relation on X, and let R ⊂ R be a subequivalence relation which is open, i.e. Suppose is a topological space and is an equivalence relation on .In other words, partitions into disjoint subsets, namely the equivalence classes under it. The quotient space of by , or the quotient topology of by , denoted , is defined as follows: . Here is an equivalence relation example to prove the properties. Of course, the topology which corresponds to an equivalence relation which is not just the identity relation is not To. Let us consider that R is a relation on the set of ordered pairs that are positive integers such that … Consider the family of distinct equivalence classes of X under R. It is easily veriﬁed that they are pairwise disjoint and that their union is X. Actually, every equivalence relation … (1.47) Given a space $$X$$ and an equivalence relation $$\sim$$ on $$X$$, the quotient set $$X/\sim$$ (the set of equivalence classes) inherits a topology called the quotient topology.Let $$q\colon X\to X/\sim$$ be the quotient map sending a point $$x$$ to its equivalence class $$[x]$$; the quotient topology is defined to be the most refined topology on $$X/\sim$$ (i.e. Prove that the open interval (,) is homeomorphic to . De nition 1.2. As the following exercise shows, the set of equivalences classes may be very large indeed. Homeomorphism is an equivalence relation; Exercises . The class of continuous functions from a compact Lemma 1.11 Equivalence Classes Let ‡ be any equivalence relation on S. Then (a) If s, t é S, then [s] = [t] iff s ‡ t. (b) Any two equivalence classes are either disjoint or equal (6) [Ex 3.5] (Equivalence relation generated by a relation) The intersection of any family of equivalence relations is an equivalence relation. relation is an equivalence relation that is a Borel subset of X Xwith the inherited product topology. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c: . In fact your conception of fractions is entwined with an intuitive notion of an equivalence relation. If C 1,C 2 ∈ Pand C 1 6= C 2 then C 1 … A relation Rbetween Aand Bis a subset RˆA B. Let now x∈ Xand Ran equivalence relation in X. Quotient space: ˘is an equivalence relation for elements (i.e., points) in X, then we have a quotient space X=˘de ned by the following properties: i) as a set, it’s the set of equivalence classes; ii) open sets in X=˘are those with open "pre-images" in X[as in Hillman notes, it is exactly the topology making sure the Let π be a function with domain X. A relation can be visualized as a directed graph with vertices A[Band with an edge from ato bexactly when (a;b) 2R. partial orders 'are' To topological spaces. C. The equivalence classes in ZZ of equivalence mod 2. The equivalence classes associated with the cone relation above. / Topology and its Applications 194 (2015) 37–50 such theory allows us to establish relations between simplicial complexes and ﬁnite topological spaces. Two Borel equivalence relations may be compared the following notion of reducibility. We have studied the nature of complementation in these lattices in [20] and The idea of an equivalence relation is fundamental. The equivalence classes are Aand fxgfor x2X A. random equivalence relations on a countable group. This self-contained volume offers a complete treatment of this active area of current research and develops a difficult general theory classifying a class of mathematical objects up to some relevant notion of isomorphism or equivalence. AF-equivalence relation on X. Equivalence relations are preorders and thus also topological spaces. Having a good grasp of equivalence relations is very important in the course MATHM205 (Topology and Groups) which I'm teaching this term, so I have written this blog post to remind you what you need to know about them. This is an equivalence relation. Going back to (R,T)from Example 4 it is easy to establish that it is not CEER. The largest equivalence relation is the universal relation, defined in 3.3.b; that is, x ≈ y for all x and y in X. b. Example7 (Example 4 revisited). One writes X=Afor the set of equivalence classes. In a very real sense you have dealt with equivalence relations for much of your life, without being aware of it. Munkres - Topology - Chapter 1 Solutions Munkres - Topology - Chapter 1 Solutions Section 3 Problem 32 Let Cbe a relation on a set A If A 0 A, de ne the restriction of Cto A 0 to be the relation C\(A 0 A 0) Show that the restriction of an equivalence relation is an equivalence relation Homework solutions, 3/2/14 - OU Math This indicates that equivalence relations are the only relations which partition sets in this manner. Conversely, a partition1 fQ j 2Jgof a set Adetermines an equivalence relation on Aby: x˘yif On the one hand, ﬁnite T0-spaces and ﬁnite partially ordered sets are equivalent categories (notice that any ﬁnite space is homotopically equivalent to a T0-space). Definition Quotient topology by an equivalence relation. The intersection of all equivalence relations containing a given relation The relation i