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A module having dual properties with respect to a free module, as enumerated below. 1. Every free module is projective; every cofree module is injective. 2. For every module ...

After a half rotation of the coin on the left around the central coin (of the same radius), the coin undergoes a complete rotation. In other words, a coin makes two complete ...

A function H that maps an arbitrary length message M to a fixed length message digest MD is a collision-free hash function if 1. It is a one-way hash function. 2. It is hard ...

The colon is the symbol ":". It is used in a number of different ways in mathematics. 1. To denote ratio or odds, as in 2:1 (voiced "two to one"). 2. To mean such that in ...

The vector space generated by the columns of a matrix viewed as vectors. The column space of an n×m matrix A with real entries is a subspace generated by m elements of R^n, ...

The subset C of the Euclidean plane formed by the union of the x-axis, the line segment with interval [0,1] of the y-axis, and the sequence of segments with endpoints (1/n,0) ...

Consider a combination lock consisting of n buttons that can be pressed in any combination (including multiple buttons at once), but in such a way that each number is pressed ...

Given a complex Hilbert space H with associated space L(H) of continuous linear operators from H to itself, the commutant M^' of an arbitrary subset M subset= L(H) is the ...

If V and W are Banach spaces and T:V->W is a bounded linear operator, the T is said to be a compact operator if it maps the unit ball of V into a relatively compact subset of ...

A subset S of a topological space X is compact if for every open cover of S there exists a finite subcover of S.