Some of the things I've learned every day since Oct 10, 2016
Monthly Archives: December 2016
December 31, 2016Posted 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.
December 30, 2016Posted by on
In category theory, the hom-set between 2 objects in a category , often denoted as
is the collection of arrows (morphisms) in from to . Note that despite the name, the hom-set is not a set in general.
December 29, 2016Posted 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 .
December 28, 2016Posted 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 .)
December 27, 2016Posted 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.
December 26, 2016Posted 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).
December 25, 2016Posted 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.)
December 24, 2016Posted 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.
December 23, 2016Posted 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.
December 22, 2016Posted 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.