# Today I Learned

Some of the things I've learned every day since Oct 10, 2016

## Monthly Archives: December 2016

## 80: Multilinear Forms

December 31, 2016

Posted by on In abstract algebra, a **multilinear form** is a mapping

where is a vector space over the field , such that each argument of is linear over with the other arguments held fixed. A special case of this is when and is a *bilinear* form.

## 79: Hom-Set

December 30, 2016

Posted by on In category theory, the **hom-set** between 2 objects in a category , often denoted as

or simply

is the collection of arrows (morphisms) in from to . Note that despite the name, the hom-set is not a set in general.

## 78: 2 Simple Topological Spaces

December 29, 2016

Posted by on Using the open set definition of a topology as a pair , where is a set and is a collection of subsets of satisfying certain axioms, if is non-empty and finite then we immediately can define 2 simple and valid topologies on it:

- , the power set of
- , the empty set and itself

The former is called the **discrete topology** on , and the latter is called the **trivial topology** on .

## 77: Mapping Continuity (Topology)

December 28, 2016

Posted by on In topology, a mapping between topological spaces is **continuous** if the pre-image of every open subset of is itself an open subset of .

(An equivalent condition/definition is that the pre-image of every *closed* subset of is itself a closed subset of .)

## 76: Probabilistic Classifiers

December 27, 2016

Posted by on As used in machine learning, the term **probabilistic classifier **refers to a function , where is the set of objects to be classified, is the set of classes, and is the set of probability distributions over . That is, a probabilistic classifier takes an object to be classified and gives the probability of that object belonging to each of the possible classes.

This contrasts with a *non-probabilistic classifier*, which instead is simply a function that assigns a single class given an object. Often, the choice is simply that category with the highest probability.

## 75: Dyadic Rationals

December 26, 2016

Posted by on The **diadic rationals** are the rational numbers which are of the form

where is an integer and is a non-negative integer. With the standard operations of addition and multiplication, these numbers form a subring of the rationals (and an overring of the integers).

## 74: Linearly Separable Values (Euclidean)

December 25, 2016

Posted by on 2 sets of points in -dimensional Euclidean space are **linearly separable **if and only if there exists a *non-zero* vector and a number such that

holds for every , and does not hold for every . Intuitively, this means that two sets of points in an -dimensional Euclidean space are linearly separable if there is an -dimensional plane that when inserted into the same space separates the two sets.

(This concept could probably be extended to spaces which share certain properties with , such as having a partial order, closure, etc., but gives the simplest example.)

## 73: Eisenstein’s Criterion

December 24, 2016

Posted by on **Eisenstein’s criterion** provide a sufficient (but not necessary) set of criterion for a polynomial with integer coefficients

to be irreducible over the rationals. The criterion are that there exists a prime such that

- divides where
- doesn’t divide
- doesn’t divide

If all these conditions are true, then can’t be reduced over the rationals.

## 72: Bilinear Maps

December 23, 2016

Posted by on In linear algebra, a **bilinear map** is a function , where are vector spaces over a common field, which is linear in each of its 2 components when the other is held fixed. When for all , it is referred to as a *symmetric* bilinear map.

Examples of bilinear maps include matrix multiplication, the inner product, and the cross product.

## 71: Ropes (Data Structure)

December 22, 2016

Posted by on A **rope **is a data structure which stores long strings as a type of binary tree rather than a ‘monolithic’ list of characters. The nodes of the tree contain weights, and the leaves of the tree contain small substrings of the larger string.

A rope is advantageous over the latter structure with respect to speed of destructive concatenation, insertion, deletion, as well as extra memory required *during operations*, but is disadvantageous in speed of splitting, appending, and extra memory required while the structure is *not* being operated on.

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