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**Categorical Limits and Colimits**

Many types of universal constructions that appear in a wide

variety of mathematical contexts can be realized

as categorical limits and colimits. To define a limit

we first need the notion of a cone of a diagram.

A diagram of type $J$ in a category $\math...

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**TF-IDF for Document Clustering**

In this post we discuss a standard way to encode a text document

as a vector using a term frequency-inverse document frequency (tf-idf) score for each word,

with an aim to cluster similar documents in a corpus. Suppose we have a corpus of text documents $\...

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**The Continuum Hypothesis and Weird Probabilities**

I recently heard of an interesting ``proof'' that $(0,1)$ does not

have cardinality $\aleph_1$. This would disprove the Continuum Hypothesis

(\textbf{CH}), which asserts that any subset of $(0,1)$ is either countable or

has the same cardinality as $(0,1)$...

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**Simple Divisibility Tests**

In this post we will justify some common divisibility tests that most

school children are familiar with. Recall that a number such as

$111$ is divisible by $3$ if and only if the sum of the digits is

divisible by $3$. Since $1+1+1=3$ is divisible by $3$,...

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**A Simple Proof that the Harnoic Series Diverges**

One usually encounters the harmonic series

\[

\sum_{k=1}^{\infty} \frac{1}{k}

\]

as an example of a series that diverges for non-obvious reasons.

With $H_n$ defined to be the partial sum $H_n:=\sum_{k=1}^n \frac{1}{k}$,

a typical way to prove divergence i...

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**Flickr Tags are Useless**

Recently I began to code up an image classifier, and my hope was to use Flickr images as a source of real data. The Flickr API allows you to easily obtain the most recent 100 images that have a given tag. However, I found that the correspondence between t...

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**Sample Covariance Matrix**

Suppose $X_1, \dots, X_p$ are random variables and $X:=(X_1, \dots, X_p)$ is the random vector

of said random variables. Given a sample $\mathbf x_1, \dots \mathbf x_N$ of $X$, how do we obtain an estimate

for the covariance matrix $\text{Cov}(X)$? To this...

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**US Life Expectancy Data**

I took a look at the World Health Organization's data on the at birth life expectancies for various countries in the world. The number given for each year assumes that levels of mortality will remain roughly the same throughout the individual's life, and i...

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