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In the algebraic geometry of Grothendieck, a stack refers to a sheaf of categories. In particular, a stack is a presheaf of categories in which the following descent ...

There are at least two results known as "the area principle." The geometric area principle states that (|A_1P|)/(|A_2P|)=(|A_1BC|)/(|A_2BC|). (1) This can also be written in ...

An Argand diagram is a plot of complex numbers as points z=x+iy in the complex plane using the x-axis as the real axis and y-axis as the imaginary axis. In the plot above, ...

A theorem which states that the analytic and topological "indices" are equal for any elliptic differential operator on an n-dimensional compact smooth C^infty boundaryless ...

The rank of a vector bundle is the dimension of its fiber. Equivalently, it is the maximum number of linearly independent local bundle sections in a trivialization. ...

A catastrophe which can occur for four control factors and one behavior axis. The butterfly catastrophe is the universal unfolding of the singularity f(x)=x^6 of codimension ...

A fractal which can be constructed using string rewriting beginning with a cell [1] and iterating the rules {0->[0 1 0; 1 1 1; 0 1 0],1->[1 1 1; 1 1 1; 1 1 1]}. (1) The size ...

The term "Cartesian" is used to refer to anything that derives from René Descartes' conception of geometry (1637), which is based on the representation of points in the plane ...

For any system that seeks to minimize a function, only seven different local forms of catastrophe "typically" occur for four or fewer variables: 1. Fold catastrophe, 2. Cusp ...

The Cesàro fractal is a fractal also known as the torn square fractal. The base curves and motifs for the two fractals illustrated above are shown below. Starting with a unit ...