Today I Learned

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

Monthly Archives: December 2016

80: Multilinear Forms

In abstract algebra, a multilinear form is a mapping

f: V^n \rightarrow F

where V is a vector space over the field F, such that each argument of f is linear over F with the other arguments held fixed. A special case of this is when n = 2 and f is a bilinear form.

79: Hom-Set

In category theory, the hom-set between 2 objects X, Y in a category C, often denoted as

\textrm{hom}_C (X, Y)

or simply

\textrm{hom} (X, Y)

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

78: 2 Simple Topological Spaces

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

  1. \tau = P(X), the power set of X
  2. \tau = \{\O , X\}, the empty set and X itself

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

77: Mapping Continuity (Topology)

In topology, a mapping f: X \rightarrow Y between topological spaces X, Y is continuous if the pre-image of every open subset of Y is itself an open subset of X.

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

76: Probabilistic Classifiers

As used in machine learning, the term probabilistic classifier refers to a function f: X \rightarrow \pi _Y, where X is the set of objects to be classified, Y is the set of classes, and \pi _Y is the set of probability distributions over Y. 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 g: X \rightarrow Y that assigns a single class given an object. Often, the choice is simply that category with the highest probability.

75: Dyadic Rationals

The diadic rationals are the rational numbers which are of the form

\frac{a}{2^b}

where a is an integer and b 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)

2 sets X_1, X_2 of points in n-dimensional Euclidean space E^n are linearly separable if and only if there exists a non-zero vector \mathbf{w} \in E^n and a number k such that

\mathbf{w} \cdot \mathbf{x} < k

holds for every \mathbf{x} \in X_1, and does not hold for every \mathbf{x'} \in X_2. Intuitively, this means that two sets of points in an n-dimensional Euclidean space are linearly separable if there is an (n-1)-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 E^n, such as having a partial order, closure, etc., but E^n gives the simplest example.)

73: Eisenstein’s Criterion

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

P(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0

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

  1. p divides a_i where i \neq n
  2. p doesn’t divide a_n
  3. p^2 doesn’t divide a_0

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

72: Bilinear Maps

In linear algebra, a bilinear map is a function f: X \times Y \rightarrow Z, where X, Y, Z are vector spaces over a common field, which is linear in each of its 2 components when the other is held fixed. When f(x, y) = f(y, x) for all (x, y) \in X \times Y, 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)

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.