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Two fractions are said to be adjacent if their difference has a unit numerator. For example, 1/3 and 1/4 are adjacent since 1/3-1/4=1/12, but 1/2 and 1/5 are not since ...

A fundamental result of de Rham cohomology is that the kth de Rham cohomology vector space of a manifold M is canonically isomorphic to the Alexander-Spanier cohomology ...

An analytic function f(z) satisfying the irreducible algebraic equation A_0(z)f^k+A_1(z)f^(k-1)+...+A_k(z)=0 with single-valued meromorphic functions A_j(z) in a complex ...

An impossible hexnut modeled after the impossible torus. The figure above shows three impossible figures: the ambihelical hexnut in the lower left-hand corner, tribox in the ...

Analytic number theory is the branch of number theory which uses real and complex analysis to investigate various properties of integers and prime numbers. Examples of topics ...

The point of concurrence of the four maltitudes of a cyclic quadrilateral. Let M_(AC) and M_(BD) be the midpoints of the diagonals of a cyclic quadrilateral ABCD, and let P ...

The area element for a surface with first fundamental form ds^2=Edu^2+2Fdudv+Gdv^2 is dA=sqrt(EG-F^2)du ^ dv, where du ^ dv is the wedge product.

The orthogonal polynomials on the interval [-1,1] associated with the weighting functions w(x) = (1-x^2)^(-1/2) (1) w(x) = (1-x^2)^(1/2) (2) w(x) = sqrt((1-x)/(1+x)), (3) ...

If P(x) is an irreducible cubic polynomial all of whose roots are real, then to obtain them by radicals, you must take roots of nonreal numbers at some point.

A transformation of the plane which transforms collinear points into collinear points. A projective collineation transforms every one-dimensional form projectively, and a ...