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A square matrix U is a unitary matrix if U^(H)=U^(-1), (1) where U^(H) denotes the conjugate transpose and U^(-1) is the matrix inverse. For example, A=[2^(-1/2) 2^(-1/2) 0; ...

It is possible to describe a set of positive integers that cannot be listed in a book containing a set of counting numbers on each consecutively numbered page. Another form ...

A two-dimensional binary (k=2) totalistic cellular automaton with a von Neumann neighborhood of range r=1. It has a birth rule that at least 2 of its 4 neighbors are alive, ...

For algebraic alpha |alpha-p/q|<1/(q^(2+epsilon)), with epsilon>0, has finitely many solutions. Klaus Roth received a Fields medal for this result.

By way of analogy with the usual tangent tanz=(sinz)/(cosz), (1) the hyperbolic tangent is defined as tanhz = (sinhz)/(coshz) (2) = (e^z-e^(-z))/(e^z+e^(-z)) (3) = ...

Let kappa_1 and kappa_2 be the principal curvatures, then their mean H=1/2(kappa_1+kappa_2) (1) is called the mean curvature. Let R_1 and R_2 be the radii corresponding to ...

A formal mathematical theory which introduces "components at infinity" by defining a new type of divisor class group of integers of a number field. The divisor class group is ...

Let t be a nonnegative integer and let x_1, ..., x_t be nonzero elements of Z_p which are not necessarily distinct. Then the number of elements of Z_p that can be written as ...

A group having continuous group operations. A continuous group is necessarily infinite, since an infinite group just has to contain an infinite number of elements. But some ...

A branch point whose neighborhood of values wrap around an infinite number of times as their complex arguments are varied. The point z=0 under the function lnz is therefore a ...