*Urss*

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## September 25, 2012

### My Researcher Profile: Finishing

#### What have you enjoyed about your URSS project?

#### What have you found difficult?

#### How do you feel that completing a project has/can help you?

#### How did you find out about the URSS scheme?

I found it first through the poster on the wall in the computer workroom but I did not take it serious. Later I found its notice on the department website, then I explored more on its homepage and queried my personal tutor about it.

#### What are your 3 top tips for URSS students about to start work on their research projects?

1. Try more and harder when stuck.

2. Explaining your problem to others may help to clarify your problem.

3. Store your relevant files and documents in an organised way.

#### What's next for you?

I will finish my degree next year. Currently I am interested in further study in statistics and computer science.

#### How would you sum up your URSS experience in one sentence?

It is enjoyable to learn something useful and overcome chanlleges in a relatively relaxing situation.

#### Recommendations for the URSS

The URSS is indeed a great chance for students to know more about research work and experience a different way of learning. It motivates students to learn and think about their projects. It also carefully supports students in various ways, such as workshops and mentors.

## September 24, 2012

### Final Reflective Blog

#### What were your key objectives for completing the URSS and have they been met?

My key objectives included my two researcher goals and experience about research through the whole project. My two researcher goals were understanding the connections between random walks and electric network and deriving explicit formula for cover times of random conductance model on a polygon. I accomplished these two goals and learned a lot about how research was done in various aspects.

#### What skills have you gained by completing the URSS project that you will utilise in your studies or other aspects of your working life?

4. Using Latex and Latex poster template. This is a great tool for typesetting mathematical articles and posters, which will be applied on my fourth year dissertation.

#### What will you do differently as a result of your URSS experience?

Through my project, I found that research work indeed interests me though there may be unforeseen difficulties along the way. Thus I prefer to do research related work in the future and I feel more confident to do research work.

#### Have there been any unexpected outcomes of taking part in the URSS?

Not many. I did not expect to write such a long report since the template report contained only several pages. But I thought this was indeed a very good practice. My report contained all essential details to understand the content, including propositions, lemmas and theorems. This was what my supervisor required, i.e. self-contain.

#### What would you consider as your highlights of the URSS process?

My supervisor denied one of my findings in the middle of the project. My findings came from my calculations, which were not based on proof. But my finding suggested that two irrelevant terms should be equal. Although I knew that in theory they represented two different concepts, I guessed that they might be equal due to some unknown reasons. So I tried to prove it. However, my supervisor thought that those two terms could not be equal and suggested another method to solve the problem. Instead of learning and applying my supervisor's method directly, I struggled to prove it for a few days. Finally, I found my calculations were correct only in the simple symmetric case and those two terms were only equal in such case. For the general model, which was my target, it could not be generalised. Hence I still needed to use my supervisor's method.

I had another finding which was also based on calculations, similar to the case above. However, this time the two terms were highly related to each other. Even I could see the relationship, but I could not describe them precisely. When I told my supervisor this finding, he responded rapidly and gave a logical deduction.

#### Did you encounter any challenges, issues or difficulties whilst partaking in the URSS? How did you overcome them?

#### How do you feel about the URSS now that you have completed your project?

The URSS is indeed a great chance for student to know more about research work and experience a different way of learning. I learned a lot through the project and my supervisor helped and encouraged me a lot. The workshops let us know other URSS researchers and their projects. My mentor's comments on my blogs gave me beneficial feedbacks.

I think the promotion of URSS was not very intense, at least in my department. There were only a few notices on the noticeboard, in the workrooms and on the department website. And the difficulty of the application part was to find a supervisor. I felt nervous when I asked an professor to help me to be my supervisor, especially for a professor who never taught me before. I might think whether they would care such a undergraduate project when I looked through their homepage, which were full of their publications. I do feel lucky to find such a nice professor to be my supervisor, who provided me a suitable topic just before the deadline.

Thank you, Dr David Croydon and other people who help me through the URSS.

## September 14, 2012

### Researcher Development Goal 2

#### What was the objective you set yourself?

To derive explicit formula for cover times of random conductance model on a polygon.

#### What actions did you take to achieve it?

First I read and understood the formula derived for cover times of simple random walks on a polygon. Since the random conductance model was more general, some methods could not be applied to it from the simple model. This required more careful analysis. However, the ideas of decomposing the chains and conditioning on the last visited vertex were similar. With the helps of my supervisor, the complicated parts were finally solved by a simple and clever intuition and Doob-h-transform. Though the final explicit formula was extreme complicated, it was indeed an explicit formula which could be calculated for given weights.

#### How do you know you have achieved your goal?

The terms constituting the cover time agree with the formula for the simple model, if we reduce our model to the simple model. Hence, the formula for our model is verified through the simple model.

The simulations showed that the cover times for some uniform random variables grows along with the formula for the simple model. And for some types of random variables, the cover times grows also in O(m^2), which agrees with the formula for the simple model.

#### What new or existing skills have you developed as a result of achieving this objective?

I learned to use Matlab to do symbolic calculations, analyse Markov chains in details and apply Doob-h-transform to find certain probabilities.

#### How will these support your research project, studies or career?

Complicated symbolic calculations can be annoying. Matlab helps to release me from the tedious work. This will save me lots of time if there are symbolic calculations in the future.

Markov chains is an important topic through my study. Hence a detailed analysis deepens my understanding.

Doob-h-transform is a method which can simplify the original problem significantly, which can be helpful in various situations. And there is also a continous version of it, which was applied in one of the talks of P@W workshop.

#### If you were to set yourself the same objective again, what would you do differently?

Since the method involving the Green's function failed, I either should examine it more carefully or study the Doob-h-transform in advance. However, these problems happened along the progress of project, which could be hardly foreseen.

## September 13, 2012

### Researcher Development Goal 1

#### What was the objective you set yourself?

Understanding the connections between random walks and electric networks.

#### What actions did you take to achieve it?

I read textbooks and papers which are related to this topic and tried to solve the similar questions in our extended model. I explained to my supervisor what I thought. My naive thinking about random walk was to break the chain directly to force the walk to go into certain directions. However, this was a too strong condition, since we were looking at walks that could go in both directions but happen to go certain one of them. This understanding was obtained through my discussion with my supervisor.

#### How do you know you have achieved your goal?

I successfully applied two key results about the connections between random walks and electric network in my project. One of them is about the probability of hitting certain state before another state, starting at a state between those two states. The other is about the expected times of visiting certain state before returning to the starting state.

These two results helped me to simplify the original problem, which were very useful.

#### What new or existing skills have you developed as a result of achieving this objective?

One of the skills is through reading papers to understand what the problem is and what the author tries to do. Papers are similar to lecture notes, but they are more compact and maybe refer to other sources. Hence they are more difficult and require more patience to read through them.

There are important connections between maths and physics. They may help each other to solve the problem. This project is a live example that renews my understanding about the interaction between physics and maths.

#### How will these support your research project, studies or career?

The ability to read papers is definitely an important skill in all research areas. It helps to know the new development of the subject and avoid the repeat of works. Also it gives me an example about how to present ideas clearly and precisely.

The understanding of connections between maths and physics may help me to solve difficult problems in the future, especially for providing ideas.

#### If you were to set yourself the same objective again, what would you do differently?

I think I probably should look at more complicated network and do some exercises. This is because the network in my project become complicated in the later stage. So a deep understanding would be more helpful.

### Second reflection on Urss project

After the draft of my Urss project was finished, I explained it to several friends of mine. Most of them could at least get a basic understanding of what is going on and some of them could see how the mathematics work in my project. However, some of them were not interested in it or confused by the aim of my project. This may be because there was not enough time to give a detailed explantion. In addition, I designed my poster through latex poster template.

### Change of previous plan

Our previous plan included a further investigation of a more general model. However, due to a unforeseen complexity of our first model and the time limitation, we decided to concentrate on our first model so that we can at least obtain some useful results.

### Learning from mistakes and bugs

###### 1. Matlab functions

I used some Matlab functions on symbols. It saved me some time. Then under the pressure of a rise in the amount of calculations, I explored the functions further and found more useful functions which saved me more time. However, there was some limitation in the input of the functions, which I did not notice at the first time. Later it gave out some error since I input some too complicated formula.

Therefore, to test functions before using and get to know the limitation of the functions are important.

###### 2. Bugs.

When I did the simulation, there were annoying errors reported. The problem was that the error reported was not the actual error. It was caused by another error. This mislead me and cost much time to understand the problem.

###### 3. Replacing files wrongly.

I worked both on computers on campus and on my own laptop. I replaced some files wrongly, which cost me one night to recover what I had done. So remember to be careful when replacing files.

### Learning from my supervisor

Analysis on how the cover time changes asymptotically and plots against some parameters give me a more clear picture of what we are investigating and what the results are. My supervisor also explained to me how the underlying theory supports our results. This makes our results more rigorous.

### Learning from P@W workshop

First the lectures were attractive, compact and highly related to my project. Although I can only get a general idea without knowing some details in calculations and proofs, I did feel that I learned a lot from the lectures.

The talks from the young researchers gave me a pretty straight forward picture about what PhD students were doing. And they showed me how to present your own work to the audience. Some of them gave really good presentations. Even I could understand some of the technical details.

My supervisor helped me to solve the problem set and he encouraged to try to think about the problems, though the problems may be a bit hard for my level.

I am really glad to have such an opportunity to attend this workshop. It was extremely beneficial. I hope I can attend such workshops again next time.

### Progress of the project

Since I had a disagreement with my supervisor, I thought the problem deeply. Finally I found that my answer was only correct for the symmetric case. It indeed could not be interpreted meaningfully. On the other hand, my supervisor's method which involved a so-called Doob-h-transformation worked well on the calculation. The only drawback of this method was that it required a massive amount of calculations for large chains. Hence, the simulation could only be done for chains of size about 100. But this is the best we could do.

The other term which was difficult to calculated was simplified and solved by me through several attempts. Then my supervisor gave a very precise deduction which verified my result.

The methods above worked well, but there was a failure in our first method which involved Green's function. It was almost correct except that some terms appeared twice in the calculations. And we could not find the problem.

There were fluctuations in some of the simulations of the cover times, which was counterintuitive. This is because of my method of generating random variables. I generate new random variables each time. So if the random variables vary dramatically, the cover time of large chain can be smaller than that of small chain. I decide to generate random variables gradually. This will make sure the cover time is strictly increasing as the size increase.

### Improvement in typesetting(Latex)

Besides the powerful referencing functions of Latex, I learned how to draw simple graphs and create different types of texts, such as proposition, proof, remark.

### Learning poster design through examples

I learned poster design through examples using Latex poster template. It looked odd at first glimpse. The guide was short so I worked on an example to understand the settings of the package. Then I found it actually very easy to use and lots of options could be set to adjust the poster.

My problem about my poster was that it was hard to fit everything into a A1 paper, though I already gave very brief summaries. I was worried about that the pictures and texts would be blurry if they were set to be too small.

### Reflections on other things

More time should be left to finish the project at the end, since there are lots of trivial and time-consuming stuffs need to be done. Basically, the time management should be improved.

## July 10, 2012

### First reflection on Urss project

My Urss project has gone more than half way through. I think it is the time for me to have a little pause and review what I have done. Moreover, I shall record what I gained and where I need to improve.

**Background reading **

I spent about one week on reading background material, which included three papers and three chapters in a textbook. The difficulty of the reading was about all right, but it did require lots of efforts to concentrate to understand. For a few hard proofs, I can only scratch the main idea. Compared with attending lectures, this self-studying process is more independent with a bit more stress.

**Communication with my supervisor**

I feel that I can understand most of my supervisor's explanations and provide responses related to the topic to show my understanding. I am not afraid to ask questions even if it is sometimes basic. My supervisor explains to me with details, so everything looks very clear.

**Learning from my supervisor**

After obtaining some results, he will check it with a general estimate. If they are different, then he will look up the previous deduction for mistakes. It helps when the final result is a combination of several basic results. It narrows down the range of the mistakes if the final result is incorrect.

Due to lack of understanding of random walks, I often decompose the chain wrongly by removing some edges. This condition sometimes is so strong that the walks are pushed into one direction. Hence the later calculation go wrong. A more careful investigation of the Markov chain can help to solve ths problem.

**Progress of the project**

After learning the background knowledge about the relationship between random walks and electri networks, the analysis of random conductance model on probabilities went smooth. But the analysis on expectations did offer us some challenges. The formula became complicated and messy. We analysed the chain and calculated the formula for several times. Finally we get most of it done except one term in the formula. The simulations for the probabilities run well.

However, I had a disgreement with my supervisor. Based on my calculation, the term we want to find coincides with another term I find. So I guess they are the same, but we need more explanations. but my supervisor thinks they cannot be equal theoretically. Hence he provides me another method to find the term. I will try to investigate this term further to see the result and organise all the drafts.

**Improvement in typesetting(learning latex and bibtex)**

Previously I know TeX, which a basic typesetting program for mathematics. Due to a problem of inserting graphs, I learn latex, which is a more powerful one. It enables me to do something I cannot do before, e.g. the cross referencing within a paper.

I learned how to make a proper reference to a mathematical article. This involves saving the references in a seperate file through bibtex.

**Reflections on other things**

Most of the time, I can meet the goal set by our meeting. But when there is a long break between our meetings, I tends to be inefficient, especially when I am stuck in some mistake. Perhaps I should set a daily goal to improve self-motivation and leave extra time for the goals.

## June 22, 2012

### URSS Action Plan

### Summary

### Actions

### Researcher Development Goal 1

#### What is the challenge or skill that I would like to develop?

Understanding the connections between random walks and electric networks.

#### What am I going to do to achieve this?

To read textbooks and papers, try to solve specific questions on this topic and ask my supervisor questions when I cannot understand after my own thinking.

#### Why have I chosen this particular objective?

#### Is this achievable within the timeframe and resources available?

#### When will I achieve this by?

#### How am I going to know that I have been successful?

### Researcher Development Goal 2

#### What is the challenge or skill that I would like to develop?

#### What am I going to do to achieve this?

#### Why have I chosen this particular objective?

#### Is this achievable within the timeframe and resources available?

#### When will I achieve this by?

#### How am I going to know that I have been successful?

## April 18, 2012

### My Researcher Profile

**Name:** Jianjun Zhao

**Department:** Statistics

**Project Department:** Statistics

**Supervisor:** Dr David Croydon

#### Why does your URSS project interest you?

I am interested in the Markov process, random walk and analysing some graph properties with the help of probability. So I would like to investigate them more, such as cover time on graphs. Also I wish to study the subject in a different way from attending lectures. Doing such a project can stimulate me some problem-oriented thinking and motivate me to learn everything I need to do the project.

#### What are you hoping to achieve through your URSS project?

I hope I can gain a better understanding of the nature of the subjects and an experience about how to gather different sources of knowledge to solve the problem.

#### What new skills are you hoping to develop?

I would like to see how research work is carried out. The project will develop my problem-solving and communication skills. Since in the course of conducting the project, there will appear various difficulties, I will try to solve them by myself and usually need to ask for my supervisor¡¯s help. Therefore, my communication skill is practised as well.

#### What contribution to knowledge could your URSS project make?

I would like to investigate how large cover time is on some graph and how will it change for different graphs.