## Archive of posts with tag “preference relation”

### The distribution when indistinguishable balls are put into boxes

Suppose there are $n$ distinguishable boxes and $k$ indistinguishable balls. Now, we randomly put the balls into the boxes. For each of the boxes, what is the probability that it contains $m$ balls? This is a simple combanitorics problem that can be solved by the stars and bars method. It turns out that in the limit $n,k\to\infty$ with $k/n$ fixed, the distribution tends to be a geometric distribution.

- Categories: math
- Tags: probability, combinatorics, ruby

### Labeled

`break`

,`next`

, and`redo`

in RubyMany languages support breaking out of nested loops, such as Perl, Java, JavaScript, C#, etc. Languages that have

`goto`

can also do this easily. However, in most other languages, it is not easy to break out of nested loops. I want to introduce a way to do this in Ruby.- Categories: programming
- Tags: meta programming, ruby

### Free trade (single good case)

I set up a simple model to determine the production and consumption in free trade between nations.

- Categories: economics
- Tags: global economy, from zhihu

### A measure-theoretic formulation of statistical ensembles (part 2)

For sake of rigor and generalizability, I feel it necessary to try to have a mathematical formulation for statistical ensembles. I chose measure spaces as the underlying mathematical structure of thermal systems and tried to justify the method of statistical ensembles by deducing them from some axioms.

- Categories: physics
- Tags: mathematical physics, statistical mechanics, functional analysis, measure theory, probability, long paper

### Summarizing my methods for studying

I summarized my methods for studying. This can be served as a guide for those who have started learning math and science.

### A measure-theoretic formulation of statistical ensembles (part 1)

For sake of rigor and generalizability, I feel it necessary to try to have a mathematical formulation for statistical ensembles. I chose measure spaces as the underlying mathematical structure of thermal systems and tried to justify the method of statistical ensembles by deducing them from some axioms.

- Categories: physics
- Tags: mathematical physics, statistical mechanics, functional analysis, measure theory, probability, long paper

### The core of a voting system is the intersection of Pareto sets

There is a very neat relation between the core (the set of proposals that defeats every proposal) of a voting system and the Pareto sets of the voters. Suppose there is a voting system of quota $q$, then the core is the intersection of all such sets: the Pareto set of $q$ of the voters.

- Categories: economics
- Tags: voting system, preference relation, pareto efficiency

### Relationship between the Gini coefficient and the variance

Both the Gini coefficient and the variance are measures of statistical dispersion. We are then motivated to find the relationship between them. It turns out that there is a neat mathematical relationship between them.

- Categories: economics
- Tags: from zhihu, calculus, probability

### Distinguishing all the letters in handwritten math/physics notes

Personally, I have the demand of handwriting math/physics notes, but an annoying fact about this is that I usually cannot distinguish every letter that may be possibly used well enough. In this article, I will try to settle this problem.

- Categories: misc
- Tags: from zhihu, font, tex, handwriting, long paper

### 手纸本

何为

*手纸本*?- Categories: literature
- Tags: from zhihu, literary composition, short story, chinese