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All entries for Thursday 13 September 2012

## 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.