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Let A=a_(ij) be a matrix with positive coefficients so that a_(ij)>0 for all i,j=1, 2, ..., n, then A has a positive eigenvalue lambda_0, and all its eigenvalues lie on the ...

The hacoversine, also known as the hacoversed sine and cohaversine, is a little-used trigonometric function defined by hacoversin(z) = coversinz (1) = 1/2(1-sinz), (2) where ...

Given three circles, each intersecting the other two in two points, the line segments connecting their points of intersection satisfy (ace)/(bdf)=1 (Honsberger 1995).

The following three pieces of information completely determine the homeomorphic type of a surface (Massey 1996): 1. Orientability, 2. Number of boundary components, 3. Euler ...

The identric mean is defined by I(a,b)=1/e((b^b)/(a^a))^(1/(b-a)) for a>0, b>0, and a!=b. The identric mean has been investigated intensively and many remarkable inequalities ...

An imaginary quadratic field is a quadratic field Q(sqrt(D)) with D<0. Special cases are summarized in the following table. D field members -1 Gaussian integer -3 Eisenstein ...

An integer kappa equal to 0 or 1 which vanishes iff the product manifold M^4×R can be given a smooth structure. Here, M^n is a compact connected topological four-manifold.

An algebraically soluble equation of odd prime degree which is irreducible in the natural field possesses either 1. Only a single real root, or 2. All real roots.

It is conjectured that every tree with e edges whose nodes are all trivalent or monovalent can be given a "magic" labeling such that the integers 1, 2, ..., e can be assigned ...

Given a convex plane region with area A and perimeter p, A-1/2p<N<=A+1/2p+1, where N is the number of enclosed lattice points (Nosarzewska 1948). This improves on Jarnick's ...